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The normative side of this problem asks what moral values or +principles, if any, we should encode in AI. To this end, we present a framework +to consider the question at four levels: Individual, Organizational, National, and +Global. We aim to illustrate how AI alignment is made up of value alignment +problems at each of these levels, where values at each level affect the others and +effects can flow in either direction. We outline key questions and considerations +of each level and demonstrate an application of this framework to the topic of AI +content moderation. +1 +Introduction +AI is used on a global scale in a multitude of ways, from social media algorithms and cybersecurity +to smart home devices and increasingly-more-autonomous vehicles. This poses risks of both direct +and indirect negative effects on our political, economic, and social structures. With this new realm of +technology, we must thoroughly understand and work to address the risks in order to navigate the +space and use the technology wisely. This is the field of AI ethics, and here, specifically, AI safety. +1.1 +AI Alignment +The AI alignment problem considers how we can encode AI systems in a way that is compatible +with human moral values. The problem becomes complex when there are multiple values that we +want to prioritize in a system. For example, we might want both speed and accuracy out of a system +performing a morally relevant task, such as online content moderation. If these values are conflicting +to any extent, then it is impossible to maximize for both. AI alignment becomes even more important +when the systems operate at a scale where humans cannot feasibly evaluate every decision made to +check whether it was performed in a responsible and ethical manner. +The alignment problem has two parts [1]. The first is the technical aspect which focuses on how +to formally encode values and principles into AI so that it does what it ought to do in a reliable +manner. Cases of unintended negative side effects and reward hacking can result if this is not done +properly [2]. The second part of the alignment problem is normative, which asks what moral values +or principles, if any, we should encode in AI. To this end, we present a framework to consider the +question at four levels.1 While other notable recent papers have focused more on the content of the +1For another take on the problem, see the “multiscale alignment” section of Max Tegmark’s interview with +the 80,000 Hours Podcast [3]. Tegmark’s framework does not yet seem to be published, so we cannot know in +exactly what ways his and our frameworks are similar or different. +ML Safety Workshop, 36th Conference on Neural Information Processing Systems (NeurIPS 2022). +arXiv:2301.03740v1 [cs.CY] 10 Jan 2023 + +problem solution (e.g. Hendrycks et al., 2021, [4] and Hendrycks et al., 2022, [5]), this framework +focuses on the social context in which those content-focused solutions must exist, and how that +context has multiple layers across which solutions should be integrated and coherent. +2 +Breaking Down the Alignment Problem +AI alignment is made up of value alignment problems at multiple different levels, not just in the +technology itself, how it is built, and the design methods. In order for AI to truly be aligned +with human moral values, all levels must be aligned with each other as well. The following is an +approach to AI alignment in which the values at each level affect the others, with effects flowing both +downwards and upwards. At each level, there are key questions that need to be answered. +Figure 1: Four Levels of Alignment +Individual & Familial +On the individual level, the framework invites individuals and families +to ask questions about values and flourishing. In our everyday actions, we are shaping our own +definitions of individual flourishing—what makes life fulfilling and brings contentment. We must +consider what role models and lifestyles we seek to emulate, how we define success for ourselves, +what sacrifices we are willing to make, and what ethical values we prioritize. +Organizational +The organizational level refers to corporations, state and local governments, uni- +versities, churches, social movements, and various other groups in civil society. When considering +alignment at this level, we must determine what values the organization operates on, what values +are instilled in its products and services, and what role the organization plays within society. For +institutions, important considerations are what constitutes success, what metrics are used to evaluate +success, and how they are involved in the broader movements for AI alignment. +National +The next level is the national level. Each nation has either implicitly or explicitly defined +values that determine the country’s goals and objectives pertaining to AI. A country aiming to assert +itself as a global power may invest resources into building a domestic AI industry, as well as regulate +the usage of AI to moderate and nudge users’ behaviors towards particular views. On the other hand, +a country aiming to promote freedom may follow a decentralized approach to AI production, giving +firms freedom and privacy while allowing for competition amongst firms. Alternatively, countries +may try to build an AI initiative in a way that not only ensures that they are aligned with moral values, +but also encourages or requires other countries to do so. +Global +Globally, humankind must think about the kind of future we want to have. The recently +articulated United Nations Sustainable Development Goals (SDGs) offer a good starting point, but +these goals are merely the preconditions necessary for survival and flourishing, so they are not +enough [6]. A further step is needed to determine our common goals as a civilization, and more +philosophically, the purpose of human existence, and how AI will fit into it. Is it to survive, raise +2 + +Individual +: How do we define success and flourishing for ourselves? +What ethical values do we operate on? +Organizational +: What are the core values of the organization? +: What role does the organization play within society? +National + What are the country's values, priorities, and objectives? + How does the nation affect and rely on other nations? +Global +. What should our common goals be as a civilization? + What does global flourishing look like and entail?children, live in society, seek the truth, etc.? Related to this are the end goals of economic and +political structures, as well as what powerful nations and corporations need to give up in order to +attend to the needs of the poor and the earth. +2.1 +Putting the Levels Together +All of these levels interact with each other. Because AI typically originates from the organizational +level, often in profit driven corporations, the primary motivation is often simply to make money. +However, when put in the context of these other levels, further goals should become visible: 1) AI +development should be aligned to individual and familial needs, 2) AI development should align with +national interests, and 3) AI development should contribute to human survival and flourishing on the +global level. +But other layers in the framework also interact with each other, through inputs and outputs. For +example, looking at the same organizational layer from the inbound perspective, individuals can +choose whether or not to buy certain kinds of technologies, nations can pass laws and regulations +to control what technology companies can do, and at the global level, international pressure (for +example from the UN through ideas such as the Sustainable Development Goals) can also influence +technology company behavior. Of note, these levels can have intermediate levels too, such as the +European Union, which is above national but below global, and which has, through the General Data +Protection Regulation (GDPR) [7], had a major influence on the internet, data, and through those, AI. +Examining the individual level, we have already seen how it influences and is influenced by the +organizational level. The individual level can influence the national through elections, and the global +through organizations such as the UN, although these influences are quite underdeveloped. Similarly, +the global can influence individuals through international treaties, while nations obviously exert +significant control over their citizens through laws and other behavioral expectations. +Lastly, the national and global levels interact. Nations influence the global state of the Earth, for +example through war and other national policies with global effects (such as energy policies which +can drive or mitigate climate change.) The global level can exert power back, whether through the +UN or other international expectations of national behavior. +To get a more practical view of the framework, we look at the problem of social media content +moderation. +3 +Content Moderation as an Example +A global debate has emerged on the risks posed by certain types of content on the internet. User +generated content is not subject to the same editorial controls as traditional media, which enables +users to post content that could harm others, particularly children or vulnerable people. This includes +but is not limited to content promoting terrorism, child abuse material, hate speech, sexual content, +and violent or extremist content. Yet at the same time, attempts to restrict this content can seem to +some like it violates user freedom of expression and freedom to hear certain kinds of expression. +Organizations and governments have grappled with the feasibility and ethics of mitigating these +potential harms through content moderation, while at the same time trying not to lose users who feel +that their freedoms are being curtailed. +AI-assisted content moderation brings a level of speed and scale unmatched by manual moderation. A +transparency report from Google shows that over 90% of videos removed on YouTube between April +and June 2022 were reviewed as a result of automatic flagging [8]. However, these approaches have +implications for people’s future uses and attitudes towards online content sharing, so it is important +that the AI employed in these processes aligns with human values at multiple levels. +3.1 +Using the Framework +Organizational ⇒ Individual +The first issue comes from the organizational level, where there is a +major misalignment between businesses and individuals. Businesses that employ content moderation +are incentivized to maximize shareholder value, which leads to prioritizing profit over social good. +For example, companies often base their algorithm on “engagement”—the more likes, comments and +shares a topic or post receives, the more it will appear on people’s newsfeeds. Per profile as well, +3 + +companies then keep track of the user’s behavior and habits based on engagement to feed them what +they want to see. This way, users will spend more time on the site and generate more ad revenue for +the business to boost shareholder value. This however leads to echo chambers and polarization, as +users are not exposed to opinions that differ from theirs, ultimately affecting not only individuals and +families, but also entire nations, and even global discourse. The misalignment between organization +and individuals has already proven to be dangerous with cases like Myanmar’s attack on minorities +illustrating the potential consequences [9]. +National ⇒ Organizational ⇒ Individual +National regulations shape how organizations mod- +erate content, as organizations must build AI within the bounds of these regulations. A country’s +content moderation legislation is typically an expression of the cultural values of the majority of +its citizens, which is often similar to the cultural values of its leadership, though not always. While +these regulations are made by individual lawmakers and may express the values of many individual +citizens, these regulations also will affect both organizations and other individuals. For example, a +common good perspective might lean towards high content moderation for the sake of minimizing +social harm, but at the expense of individual freedom of expression. +The question then arises regarding the alignment of cultural values with AI content moderation. +We may be able to recognize where there are mis-alignments between national and organizational +values, which in turn affects individuals. For example, in the US, where individual freedoms is a +priority, there is very little content moderation regulation and it requires companies to only moderate +things such as illegally sharing copyrighted content and criminal activity such as sharing child sexual +abuse materials. Therefore, while companies may comply with every relevant government regulation, +there have nevertheless been harmful effects on society, showing how the US government content +moderation legislation is not aligned with societal needs. Cases like Myanmar also suggest that this +American legislation may not be aligned with global needs, as other countries are subject to these +same problems and are facing the repercussions of it. +Based on the above, it might seem that the first goal for AI alignment would be to align the national +and organizational levels (assuming that the organization is also aligned with individual well-being). +However, this is not enough—we must also consider whether these national values are aligned on +the global level, that is, whether they support global human flourishing. Content which promotes +division, subversion of governments, extremism, conspiracy theories, and other socially destructive +and anti-social behaviors can result in not only local or national problems, but, if these groups become +broadly empowered, global problems. And yet governing content moderation at the international level +is simply not feasible in our pluralistic world. However, this framework at least helps to diagnose the +problem, even if the problem cannot yet be solved. +Individual ⇒ Organizational ⇒ National ⇒ Global +The effects flow in both directions. Organi- +zations doing content moderation sometimes respond most to individual user feedback, a powerful +enough organization can have a hand in swaying national interests, and a nation or group of nations +can potentially change the international order, for example, as with the EU’s GDPR. +All in all, content moderation is a prime example of how value alignment is at work right now in +society. It may not be feasible to align all four levels quickly or easily, but with this framework we +can identify some causes of the complex mis-alignments that we are seeing now—and will see more +of in the future—and consider some possible beginnings of solutions. +4 +Conclusion +If we are to make good progress on the normative side of AI alignment, we must consider all levels: +Individual, Organizational, National, and Global, and understand how each works together, rather +than only aligning one or a few of the parts. Here we have presented a framework for considering +these issues. In Appendix A, we have provided an analysis of how this work relates to AI x-risk. +The versatility of the framework means that it can be applied to many other topics, including but +not limited to autonomous vehicles, AI-assisted clinical decision support systems, surveillance, and +criminal justice tools. In these hotly contested spaces with no clear answers, by analysing these +problems at four levels, we are able to see the many interacting parts at play, in order to create more +ethical and aligned AI. +4 + +References +[1] Iason Gabriel. "Artificial Intelligence, Values, and Alignment". In: Minds & Machines, 30:411–437, 2020. +[2] Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mané. "Concrete +Problems in AI Safety". In: arXiv preprint arXiv:1606.06565, 2016. +[3] Robert Wiblin and Keiran Harris. “Max Tegmark on how a ‘put-up-or-shut-up’ resolution led him to work +on AI and algorithmic news selection”. The 80,000 Hours Podcast, July 1st, 2022, minutes 1:13:13-1:51:01. URL: +https://80000hours.org/podcast/episodes/max-tegmark-ai-and-algorithmic-news-selection/. +[4] Dan Hendrycks, Collin Burns, Steven Basart, Andrew Critch, Jerry Li, Dawn Song, and Jacob Steinhardt. +"Aligning AI with Shared Human Values". In: arXiv preprint arXiv:2008.02275, 2021. +[5] Dan Hendrycks, Nicholas Carlini, John Schulman, and Jacob Steinhardt. "Unsolved Problems in ML Safety". +In: arXiv preprint arXiv:2109.13916v5, 2022. +[6] "THE 17 GOALS - Sustainable Development Goals". United Nations. URL: https://sdgs.un.org/ +goals. +[7] "General Data Protection Regulation". European Unions. URL: https://gdpr-info.eu. +[8] “YouTube Community Guidelines enforcement”. Google. URL: https://transparencyreport.google. +com/youtube-policy/removals. +[9] Nathaniel Persily and Joshua A. Tucker. "Social Media and Democracy: The State of the Field, Prospects for +Reform". Cambridge University Press, 2020. +A +X-Risk Analysis +AI X-Risk Analysis template from: Dan Hendrycks and Mantas Mazeika. "X-Risk Analysis for AI +Research". In: arXiv preprint arXiv:2206.05862v7, 2022. +A.1 +Long-Term Impact on Advanced AI Systems +In this section, we analyze how this work shapes the process that will lead to advanced AI systems +and how it steers the process in a safer direction. +1. Overview. How is this work intended to reduce existential risks from advanced AI systems? +Answer: Advanced AI systems will have to exist within the context of the world composed of +different nations, organizations, and individuals. In order to avoid global catastrophe, we must +look at all the levels. By aligning these and considering the ways in which they interact, we can be +more confident that advanced AI systems can be built to be compatible with the world, or at least +be more aware of the ways in which they can potentially cause conflict. In this way, developers +can have a better idea of how they aim to have the system operate in the world. But without doing +the former, there is no way for future advanced AI, including AGI, to be aligned in any sense on a +global scale, which opens the door for existential risk. These misalignments are already visible +when it comes to AI content moderation which has yielded innumerable ill social effects in the +hands of malicious, deluded, or merely ignorant actors. +2. Direct Effects. If this work directly reduces existential risks, what are the main hazards, vulnera- +bilities, or failure modes that it directly affects? +Answer: The work directly addresses the problem of existential risk by clarifying the precondi- +tions for any real solutions to the problems posed by X-risks. Proposed solutions which do not +consider how the individual, organization, national, and global levels integrate will be flawed, and +therefore not real solutions. This work highlights how we should go about directly mitigating +risks: first, by understanding how the levels align or misalign, and then considering how AI +solutions to risk are likely to exist within that multilevel structure. A failure mode of this is +that, as knowledge, it could be used for creating misalignment as well, allowing for optimizing +multilevel misalignment rather than alignment. Certain nations and groups have already exploited +AI moderated systems in this way, so this is hardly a new discovery on the negative side—it is +the positive side which seems to have remained implicit until now, and so, by making it explicit, +hopefully we can begin to recapture the escaped genie of misaligned AI and reverse its effects. +3. Diffuse Effects. If this work reduces existential risks indirectly or diffusely, what are the main +contributing factors that it affects? +5 + +Answer: The main effect of this paper is a diffuse reduction in X-risk, by enabling those who +generate other AI alignment X-risk solutions to better understand the multilevel nature of society +in which solutions must exist, and then how to integrate their solutions into that multilevel society. +If the ideas in this paper are ignored then AI alignment X-risk solutions are likely to be less +effective and less comprehensive, thus not solving the AI alignment problem as well as they could +have if the ideas in this paper had been more fully integrated with their work. It is not enough to +merely create alignment at one level, for example, that of government, since various governments +of the world will remain misaligned. Currently we see this between autocracies vs. democracies +and free nations vs. oppressive ones. AI alignment X-risk solutions which merely enable nations +to align their populaces with the government will only empower oppressive autocracies and make +global misalignment worse, not better. This is true for the other levels as well. Alignment has to +be consistent from top to bottom or the problem merely squeezes out into the other levels, like a +water balloon about to burst. +4. What’s at Stake? What is a future scenario in which this research direction could prevent the +sudden, large-scale loss of life? If not applicable, what is a future scenario in which this research +direction be highly beneficial? +Answer: +As discussed previously, this framework could help to prevent situations like what +happened in Myanmar as a result of AI-assisted social media content moderation. If the businesses +creating these systems are aligned with individual, national, and global needs, we can begin to fix +the problems of polarization, echo chambers, and remove the possibility for AI to be exploited the +way it was in Myanmar to target minorities. Without this work, a theoretical AGI could be built +in line with one country’s values but in direct conflict with another country’s values and cause +significant harm, or have to be restricted to a smaller domain, say just one country. However, if +the organizations in a country are aligned to one definition of success, and countries are united +on a definition of global flourishing, then it is possible to create an AGI that could provide much +greater social benefit. +5. Result Fragility. Do the findings rest on strong theoretical assumptions; are they not demonstrated +using leading-edge tasks or models; or are the findings highly sensitive to hyperparameters? +⊠ +6. Problem Difficulty. Is it implausible that any practical system could ever markedly outperform +humans at this task? +□ +7. Human Unreliability. Does this approach strongly depend on handcrafted features, expert +supervision, or human reliability? +□ +8. Competitive Pressures. Does work towards this approach strongly trade off against raw intelli- +gence, other general capabilities, or economic utility? +□ +A.2 +Safety-Capabilities Balance +In this section, we analyze how this work relates to general capabilities and how it affects the balance +between safety and hazards from general capabilities. +9. Overview. How does this improve safety more than it improves general capabilities? +Answer: The paper frames how any efforts to improve general capabilities should first consider +safety in the broader societal context and how to do so. Because this is a framework within which +thinking about general capabilities exists, on the downside, those capabilities could be directed +towards bad ends in a more comprehensive way—however, it seems that the nations who act +as spoilers in international affairs already understand this, and so it is the other nations of the +world, and organizations and individuals, who wish to improve the state of the world that need to +work harder to learn this lesson and act to align AI in a more comprehensive and integrated way. +Therefore this promotes safety more than hindering it. +10. Red Teaming. What is a way in which this hastens general capabilities or the onset of x-risks? +Answer: This hastens general abilities in AI only in the same way as any comprehensive model +of society might enable better thinking about the integration of technology into society. However, +it could perhaps be used by bad actors to either attempt to sow conflicting alignments across +various AIs, or subtly direct AI towards one very bad misalignment. If an AGI were to be built +in these ways, then safety would be harmed—but this appears to be almost the default mode for +current AI work (often confused, not aligned between levels, or intentionally abused by malicious +actors), and so by being more explicit about the true form of the problem, it would seem that the +6 + +likely direction of progress among those with good intent would be towards improvement and not +towards degradation. +11. General Tasks. Does this work advance progress on tasks that have been previously considered +the subject of usual capabilities research? +□ +12. General Goals. Does this improve or facilitate research towards general prediction, classification, +state estimation, efficiency, scalability, generation, data compression, executing clear instructions, +helpfulness, informativeness, reasoning, planning, researching, optimization, (self-)supervised +learning, sequential decision making, recursive self-improvement, open-ended goals, models +accessing the Internet, or similar capabilities? +⊠ +13. Correlation With General Aptitude. Is the analyzed capability known to be highly predicted by +general cognitive ability or educational attainment? +□ +14. Safety via Capabilities. Does this advance safety along with, or as a consequence of, advancing +other capabilities or the study of AI? +⊠ +A.3 +Elaborations and Other Considerations +15. Other. What clarifications or uncertainties about this work and x-risk are worth mentioning? +Answer: This paper is a description of and framework for the context in which AI-associated +risks appear. As a description and framework, it allows the problem to be understood in a new +way, and one which is hopefully enlightening to those seeking to solve the AI alignment problem. +Because the alignment problem runs on a continuous spectrum from contemporary problems +like content moderation, all the way to AGI and X-risks, this framework should provide helpful +insights to those seeking to create comprehensively safe AI. And for those who would seek to do +the opposite, they already know how to sow discord and evil. This framework is a tool for those +who seek to do good, better. The dual use is essentially useless for those motivated by malice, as it +is already well-known by them at a tactical, operational, and strategic level. It is those on the side +seeking the comprehensive good that should find this to be of most help, tactically, operationally, +and strategically, making AI safer for everyone. +7 + diff --git a/29E2T4oBgHgl3EQfNwZM/content/tmp_files/load_file.txt b/29E2T4oBgHgl3EQfNwZM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a365e426a59ac7aa219d124e14c72c0dc5c7b472 --- /dev/null +++ b/29E2T4oBgHgl3EQfNwZM/content/tmp_files/load_file.txt @@ -0,0 +1,255 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf,len=254 +page_content='A Multi-Level Framework for the AI Alignment Problem Betty Li Hou1, 2, Brian Patrick Green1 1 Markkula Center for Applied Ethics 2 New York University blh9134@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='edu, bpgreen@scu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='edu Abstract AI alignment considers how we can encode AI systems in a way that is compatible with human values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The normative side of this problem asks what moral values or principles, if any, we should encode in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' To this end, we present a framework to consider the question at four levels: Individual, Organizational, National, and Global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' We aim to illustrate how AI alignment is made up of value alignment problems at each of these levels, where values at each level affect the others and effects can flow in either direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' We outline key questions and considerations of each level and demonstrate an application of this framework to the topic of AI content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 1 Introduction AI is used on a global scale in a multitude of ways, from social media algorithms and cybersecurity to smart home devices and increasingly-more-autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' This poses risks of both direct and indirect negative effects on our political, economic, and social structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' With this new realm of technology, we must thoroughly understand and work to address the risks in order to navigate the space and use the technology wisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' This is the field of AI ethics, and here, specifically, AI safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='1 AI Alignment The AI alignment problem considers how we can encode AI systems in a way that is compatible with human moral values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The problem becomes complex when there are multiple values that we want to prioritize in a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' For example, we might want both speed and accuracy out of a system performing a morally relevant task, such as online content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' If these values are conflicting to any extent, then it is impossible to maximize for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' AI alignment becomes even more important when the systems operate at a scale where humans cannot feasibly evaluate every decision made to check whether it was performed in a responsible and ethical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The alignment problem has two parts [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The first is the technical aspect which focuses on how to formally encode values and principles into AI so that it does what it ought to do in a reliable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Cases of unintended negative side effects and reward hacking can result if this is not done properly [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The second part of the alignment problem is normative, which asks what moral values or principles, if any, we should encode in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' To this end, we present a framework to consider the question at four levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='1 While other notable recent papers have focused more on the content of the 1For another take on the problem, see the “multiscale alignment” section of Max Tegmark’s interview with the 80,000 Hours Podcast [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Tegmark’s framework does not yet seem to be published, so we cannot know in exactly what ways his and our frameworks are similar or different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' ML Safety Workshop, 36th Conference on Neural Information Processing Systems (NeurIPS 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='03740v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='CY] 10 Jan 2023 problem solution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=', 2021, [4] and Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=', 2022, [5]), this framework focuses on the social context in which those content-focused solutions must exist, and how that context has multiple layers across which solutions should be integrated and coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 2 Breaking Down the Alignment Problem AI alignment is made up of value alignment problems at multiple different levels, not just in the technology itself, how it is built, and the design methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' In order for AI to truly be aligned with human moral values, all levels must be aligned with each other as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The following is an approach to AI alignment in which the values at each level affect the others, with effects flowing both downwards and upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' At each level, there are key questions that need to be answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Figure 1: Four Levels of Alignment Individual & Familial On the individual level, the framework invites individuals and families to ask questions about values and flourishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' In our everyday actions, we are shaping our own definitions of individual flourishing—what makes life fulfilling and brings contentment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' We must consider what role models and lifestyles we seek to emulate, how we define success for ourselves, what sacrifices we are willing to make, and what ethical values we prioritize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Organizational The organizational level refers to corporations, state and local governments, uni- versities, churches, social movements, and various other groups in civil society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' When considering alignment at this level, we must determine what values the organization operates on, what values are instilled in its products and services, and what role the organization plays within society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' For institutions, important considerations are what constitutes success, what metrics are used to evaluate success, and how they are involved in the broader movements for AI alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' National The next level is the national level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Each nation has either implicitly or explicitly defined values that determine the country’s goals and objectives pertaining to AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' A country aiming to assert itself as a global power may invest resources into building a domestic AI industry, as well as regulate the usage of AI to moderate and nudge users’ behaviors towards particular views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' On the other hand, a country aiming to promote freedom may follow a decentralized approach to AI production, giving firms freedom and privacy while allowing for competition amongst firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Alternatively, countries may try to build an AI initiative in a way that not only ensures that they are aligned with moral values, but also encourages or requires other countries to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Global Globally, humankind must think about the kind of future we want to have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The recently articulated United Nations Sustainable Development Goals (SDGs) offer a good starting point, but these goals are merely the preconditions necessary for survival and flourishing, so they are not enough [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' A further step is needed to determine our common goals as a civilization, and more philosophically, the purpose of human existence, and how AI will fit into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Is it to survive, raise 2 Individual : How do we define success and flourishing for ourselves?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' What ethical values do we operate on?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Organizational : What are the core values of the organization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' : What role does the organization play within society?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=" National What are the country's values, priorities, and objectives?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' How does the nation affect and rely on other nations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Global .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' What should our common goals be as a civilization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' What does global flourishing look like and entail?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='children, live in society, seek the truth, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Related to this are the end goals of economic and political structures, as well as what powerful nations and corporations need to give up in order to attend to the needs of the poor and the earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='1 Putting the Levels Together All of these levels interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Because AI typically originates from the organizational level, often in profit driven corporations, the primary motivation is often simply to make money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' However, when put in the context of these other levels, further goals should become visible: 1) AI development should be aligned to individual and familial needs, 2) AI development should align with national interests, and 3) AI development should contribute to human survival and flourishing on the global level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' But other layers in the framework also interact with each other, through inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' For example, looking at the same organizational layer from the inbound perspective, individuals can choose whether or not to buy certain kinds of technologies, nations can pass laws and regulations to control what technology companies can do, and at the global level, international pressure (for example from the UN through ideas such as the Sustainable Development Goals) can also influence technology company behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Of note, these levels can have intermediate levels too, such as the European Union, which is above national but below global, and which has, through the General Data Protection Regulation (GDPR) [7], had a major influence on the internet, data, and through those, AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Examining the individual level, we have already seen how it influences and is influenced by the organizational level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The individual level can influence the national through elections, and the global through organizations such as the UN, although these influences are quite underdeveloped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Similarly, the global can influence individuals through international treaties, while nations obviously exert significant control over their citizens through laws and other behavioral expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Lastly, the national and global levels interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Nations influence the global state of the Earth, for example through war and other national policies with global effects (such as energy policies which can drive or mitigate climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=') The global level can exert power back, whether through the UN or other international expectations of national behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' To get a more practical view of the framework, we look at the problem of social media content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 3 Content Moderation as an Example A global debate has emerged on the risks posed by certain types of content on the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' User generated content is not subject to the same editorial controls as traditional media, which enables users to post content that could harm others, particularly children or vulnerable people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' This includes but is not limited to content promoting terrorism, child abuse material, hate speech, sexual content, and violent or extremist content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Yet at the same time, attempts to restrict this content can seem to some like it violates user freedom of expression and freedom to hear certain kinds of expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Organizations and governments have grappled with the feasibility and ethics of mitigating these potential harms through content moderation, while at the same time trying not to lose users who feel that their freedoms are being curtailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' AI-assisted content moderation brings a level of speed and scale unmatched by manual moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' A transparency report from Google shows that over 90% of videos removed on YouTube between April and June 2022 were reviewed as a result of automatic flagging [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' However, these approaches have implications for people’s future uses and attitudes towards online content sharing, so it is important that the AI employed in these processes aligns with human values at multiple levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='1 Using the Framework Organizational ⇒ Individual The first issue comes from the organizational level, where there is a major misalignment between businesses and individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Businesses that employ content moderation are incentivized to maximize shareholder value, which leads to prioritizing profit over social good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' For example, companies often base their algorithm on “engagement”—the more likes, comments and shares a topic or post receives, the more it will appear on people’s newsfeeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Per profile as well, 3 companies then keep track of the user’s behavior and habits based on engagement to feed them what they want to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' This way, users will spend more time on the site and generate more ad revenue for the business to boost shareholder value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' This however leads to echo chambers and polarization, as users are not exposed to opinions that differ from theirs, ultimately affecting not only individuals and families, but also entire nations, and even global discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The misalignment between organization and individuals has already proven to be dangerous with cases like Myanmar’s attack on minorities illustrating the potential consequences [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' National ⇒ Organizational ⇒ Individual National regulations shape how organizations mod- erate content, as organizations must build AI within the bounds of these regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' A country’s content moderation legislation is typically an expression of the cultural values of the majority of its citizens, which is often similar to the cultural values of its leadership, though not always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' While these regulations are made by individual lawmakers and may express the values of many individual citizens, these regulations also will affect both organizations and other individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' For example, a common good perspective might lean towards high content moderation for the sake of minimizing social harm, but at the expense of individual freedom of expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The question then arises regarding the alignment of cultural values with AI content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' We may be able to recognize where there are mis-alignments between national and organizational values, which in turn affects individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' For example, in the US, where individual freedoms is a priority, there is very little content moderation regulation and it requires companies to only moderate things such as illegally sharing copyrighted content and criminal activity such as sharing child sexual abuse materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Therefore, while companies may comply with every relevant government regulation, there have nevertheless been harmful effects on society, showing how the US government content moderation legislation is not aligned with societal needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Cases like Myanmar also suggest that this American legislation may not be aligned with global needs, as other countries are subject to these same problems and are facing the repercussions of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Based on the above, it might seem that the first goal for AI alignment would be to align the national and organizational levels (assuming that the organization is also aligned with individual well-being).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' However, this is not enough—we must also consider whether these national values are aligned on the global level, that is, whether they support global human flourishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Content which promotes division, subversion of governments, extremism, conspiracy theories, and other socially destructive and anti-social behaviors can result in not only local or national problems, but, if these groups become broadly empowered, global problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' And yet governing content moderation at the international level is simply not feasible in our pluralistic world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' However, this framework at least helps to diagnose the problem, even if the problem cannot yet be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Individual ⇒ Organizational ⇒ National ⇒ Global The effects flow in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Organi- zations doing content moderation sometimes respond most to individual user feedback, a powerful enough organization can have a hand in swaying national interests, and a nation or group of nations can potentially change the international order, for example, as with the EU’s GDPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' All in all, content moderation is a prime example of how value alignment is at work right now in society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' It may not be feasible to align all four levels quickly or easily, but with this framework we can identify some causes of the complex mis-alignments that we are seeing now—and will see more of in the future—and consider some possible beginnings of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 4 Conclusion If we are to make good progress on the normative side of AI alignment, we must consider all levels: Individual, Organizational, National, and Global, and understand how each works together, rather than only aligning one or a few of the parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Here we have presented a framework for considering these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' In Appendix A, we have provided an analysis of how this work relates to AI x-risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The versatility of the framework means that it can be applied to many other topics, including but not limited to autonomous vehicles, AI-assisted clinical decision support systems, surveillance, and criminal justice tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' In these hotly contested spaces with no clear answers, by analysing these problems at four levels, we are able to see the many interacting parts at play, in order to create more ethical and aligned AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 4 References [1] Iason Gabriel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' "Artificial Intelligence, Values, and Alignment".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' In: Minds & Machines, 30:411–437, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' [2] Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mané.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' "Concrete Problems in AI Safety".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' In: arXiv preprint arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='06565, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' [3] Robert Wiblin and Keiran Harris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' “Max Tegmark on how a ‘put-up-or-shut-up’ resolution led him to work on AI and algorithmic news selection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The 80,000 Hours Podcast, July 1st, 2022, minutes 1:13:13-1:51:01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' URL: https://80000hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='org/podcast/episodes/max-tegmark-ai-and-algorithmic-news-selection/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' [4] Dan Hendrycks, Collin Burns, Steven Basart, Andrew Critch, Jerry Li, Dawn Song, and Jacob Steinhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' "Aligning AI with Shared Human Values".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' In: arXiv preprint arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='02275, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' [5] Dan Hendrycks, Nicholas Carlini, John Schulman, and Jacob Steinhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' "Unsolved Problems in ML Safety".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' In: arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='13916v5, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' [6] "THE 17 GOALS - Sustainable Development Goals".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' United Nations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' URL: https://sdgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='org/ goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' [7] "General Data Protection Regulation".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' European Unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' URL: https://gdpr-info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' [8] “YouTube Community Guidelines enforcement”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' URL: https://transparencyreport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' com/youtube-policy/removals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' [9] Nathaniel Persily and Joshua A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Tucker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' "Social Media and Democracy: The State of the Field, Prospects for Reform".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Cambridge University Press, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' A X-Risk Analysis AI X-Risk Analysis template from: Dan Hendrycks and Mantas Mazeika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' "X-Risk Analysis for AI Research".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' In: arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='05862v7, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='1 Long-Term Impact on Advanced AI Systems In this section, we analyze how this work shapes the process that will lead to advanced AI systems and how it steers the process in a safer direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' How is this work intended to reduce existential risks from advanced AI systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Answer: Advanced AI systems will have to exist within the context of the world composed of different nations, organizations, and individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' In order to avoid global catastrophe, we must look at all the levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' By aligning these and considering the ways in which they interact, we can be more confident that advanced AI systems can be built to be compatible with the world, or at least be more aware of the ways in which they can potentially cause conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' In this way, developers can have a better idea of how they aim to have the system operate in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' But without doing the former, there is no way for future advanced AI, including AGI, to be aligned in any sense on a global scale, which opens the door for existential risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' These misalignments are already visible when it comes to AI content moderation which has yielded innumerable ill social effects in the hands of malicious, deluded, or merely ignorant actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Direct Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' If this work directly reduces existential risks, what are the main hazards, vulnera- bilities, or failure modes that it directly affects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Answer: The work directly addresses the problem of existential risk by clarifying the precondi- tions for any real solutions to the problems posed by X-risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Proposed solutions which do not consider how the individual, organization, national, and global levels integrate will be flawed, and therefore not real solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' This work highlights how we should go about directly mitigating risks: first, by understanding how the levels align or misalign, and then considering how AI solutions to risk are likely to exist within that multilevel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' A failure mode of this is that, as knowledge, it could be used for creating misalignment as well, allowing for optimizing multilevel misalignment rather than alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Certain nations and groups have already exploited AI moderated systems in this way, so this is hardly a new discovery on the negative side—it is the positive side which seems to have remained implicit until now, and so, by making it explicit, hopefully we can begin to recapture the escaped genie of misaligned AI and reverse its effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Diffuse Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' If this work reduces existential risks indirectly or diffusely, what are the main contributing factors that it affects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 5 Answer: The main effect of this paper is a diffuse reduction in X-risk, by enabling those who generate other AI alignment X-risk solutions to better understand the multilevel nature of society in which solutions must exist, and then how to integrate their solutions into that multilevel society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' If the ideas in this paper are ignored then AI alignment X-risk solutions are likely to be less effective and less comprehensive, thus not solving the AI alignment problem as well as they could have if the ideas in this paper had been more fully integrated with their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' It is not enough to merely create alignment at one level, for example, that of government, since various governments of the world will remain misaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Currently we see this between autocracies vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' democracies and free nations vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' oppressive ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' AI alignment X-risk solutions which merely enable nations to align their populaces with the government will only empower oppressive autocracies and make global misalignment worse, not better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' This is true for the other levels as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Alignment has to be consistent from top to bottom or the problem merely squeezes out into the other levels, like a water balloon about to burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' What’s at Stake?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' What is a future scenario in which this research direction could prevent the sudden, large-scale loss of life?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' If not applicable, what is a future scenario in which this research direction be highly beneficial?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Answer: As discussed previously, this framework could help to prevent situations like what happened in Myanmar as a result of AI-assisted social media content moderation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' If the businesses creating these systems are aligned with individual, national, and global needs, we can begin to fix the problems of polarization, echo chambers, and remove the possibility for AI to be exploited the way it was in Myanmar to target minorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Without this work, a theoretical AGI could be built in line with one country’s values but in direct conflict with another country’s values and cause significant harm, or have to be restricted to a smaller domain, say just one country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' However, if the organizations in a country are aligned to one definition of success, and countries are united on a definition of global flourishing, then it is possible to create an AGI that could provide much greater social benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Result Fragility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Do the findings rest on strong theoretical assumptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' are they not demonstrated using leading-edge tasks or models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' or are the findings highly sensitive to hyperparameters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' ⊠ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Problem Difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Is it implausible that any practical system could ever markedly outperform humans at this task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Human Unreliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Does this approach strongly depend on handcrafted features, expert supervision, or human reliability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Competitive Pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Does work towards this approach strongly trade off against raw intelli- gence, other general capabilities, or economic utility?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='2 Safety-Capabilities Balance In this section, we analyze how this work relates to general capabilities and how it affects the balance between safety and hazards from general capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' How does this improve safety more than it improves general capabilities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Answer: The paper frames how any efforts to improve general capabilities should first consider safety in the broader societal context and how to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Because this is a framework within which thinking about general capabilities exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' on the downside,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' those capabilities could be directed towards bad ends in a more comprehensive way—however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' it seems that the nations who act as spoilers in international affairs already understand this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' and so it is the other nations of the world,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' and organizations and individuals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' who wish to improve the state of the world that need to work harder to learn this lesson and act to align AI in a more comprehensive and integrated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Therefore this promotes safety more than hindering it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Red Teaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' What is a way in which this hastens general capabilities or the onset of x-risks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Answer: This hastens general abilities in AI only in the same way as any comprehensive model of society might enable better thinking about the integration of technology into society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' However, it could perhaps be used by bad actors to either attempt to sow conflicting alignments across various AIs, or subtly direct AI towards one very bad misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' If an AGI were to be built in these ways, then safety would be harmed—but this appears to be almost the default mode for current AI work (often confused, not aligned between levels, or intentionally abused by malicious actors), and so by being more explicit about the true form of the problem, it would seem that the 6 likely direction of progress among those with good intent would be towards improvement and not towards degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' General Tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Does this work advance progress on tasks that have been previously considered the subject of usual capabilities research?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' □ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' General Goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Does this improve or facilitate research towards general prediction, classification, state estimation, efficiency, scalability, generation, data compression, executing clear instructions, helpfulness, informativeness, reasoning, planning, researching, optimization, (self-)supervised learning, sequential decision making, recursive self-improvement, open-ended goals, models accessing the Internet, or similar capabilities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' ⊠ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Correlation With General Aptitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Is the analyzed capability known to be highly predicted by general cognitive ability or educational attainment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' □ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Safety via Capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Does this advance safety along with, or as a consequence of, advancing other capabilities or the study of AI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' ⊠ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content='3 Elaborations and Other Considerations 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' What clarifications or uncertainties about this work and x-risk are worth mentioning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Answer: This paper is a description of and framework for the context in which AI-associated risks appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' As a description and framework, it allows the problem to be understood in a new way, and one which is hopefully enlightening to those seeking to solve the AI alignment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' Because the alignment problem runs on a continuous spectrum from contemporary problems like content moderation, all the way to AGI and X-risks, this framework should provide helpful insights to those seeking to create comprehensively safe AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' And for those who would seek to do the opposite, they already know how to sow discord and evil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' This framework is a tool for those who seek to do good, better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' The dual use is essentially useless for those motivated by malice, as it is already well-known by them at a tactical, operational, and strategic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' It is those on the side seeking the comprehensive good that should find this to be of most help, tactically, operationally, and strategically, making AI safer for everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} +page_content=' 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf'} diff --git a/6tAzT4oBgHgl3EQf-P5_/content/tmp_files/2301.01931v1.pdf.txt b/6tAzT4oBgHgl3EQf-P5_/content/tmp_files/2301.01931v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8fa3548493ad4bc85e4d614b530ca32bacd74ed8 --- /dev/null +++ b/6tAzT4oBgHgl3EQf-P5_/content/tmp_files/2301.01931v1.pdf.txt @@ -0,0 +1,998 @@ +Reduced Deep Convolutional Activation Features (R-DeCAF) in +Histopathology Images to Improve the Classification +Performance for Breast Cancer Diagnosis +Bahareh Morovati, Reza Lashgari, Mojtaba Hajihasani and Hasti Shabani * +Institute of Medical Science and Technology, Shahid Beheshti University, Tehran, Iran +* Correspondence: ha_shabani@sbu.ac.ir + +Abstract: Breast cancer is the second most common cancer among women worldwide. Diagnosis of breast cancer by the +pathologists is a time-consuming procedure and subjective. Computer aided diagnosis frameworks are utilized to relieve +pathologist workload by classifying the data automatically, in which deep convolutional neural networks (CNNs) are +effective solutions. The features extracted from activation layer of pre-trained CNNs are called deep convolutional +activation features (DeCAF). In this paper, we have analyzed that all DeCAF features are not necessarily led to a higher +accuracy in the classification task and dimension reduction plays an important role. Therefore, different dimension +reduction methods are applied to achieve an effective combination of features by capturing the essence of DeCAF +features. To this purpose, we have proposed reduced deep convolutional activation features (R-DeCAF). In this +framework, pre-trained CNNs such as AlexNet, VGG-16 and VGG-19 are utilized in transfer learning mode as feature +extractors. DeCAF features are extracted from the first fully connected layer of the mentioned CNNs and support vector +machine has been used for binary classification. Among linear and nonlinear dimensionality reduction algorithms, linear +approaches such as principal component analysis (PCA) represent a better combination among deep features and lead +to a higher accuracy in the classification task using small number of features considering specific amount of cumulative +explained variance (CEV) of features. The proposed method is validated using experimental BreakHis dataset. +Comprehensive results show improvement in the classification accuracy up to 4.3% with less computational time. Best +achieved accuracy is 91.13% for 400× data with feature vector size (FVS) of 23 and CEV equals to 0.15 using pre-trained +AlexNet as feature extractor and PCA as feature reduction algorithm. +Keywords: Breast cancer; deep feature extraction; feature reduction; histopathology images; pre-trained convolutional +neural networks + +1. Introduction +Breast cancer (BC) is one of the leading causes of mortality in the world almost observed in women, but +it can occur in men, too. Diagnosis of BC ordinarily comprises of an initial detection by palpation and regular +check-ups by ultrasound imaging or mammography and diagnosis of possible malignant tissue growth is +tested by breast tissue biopsy [1]. According to the world health organization (WHO), BC is affecting large +number of women health [2]. Recent studies predict around 27 million new cases of BC by 2030 [3]. Early +detection of BC is essential for appropriate treatment and decreasing the mortality rate. However, BC +diagnosis may not be accurate enough as pathologist could only apply visual inspection of samples under +microscopes [4, 5]. According these challenges, computer aided diagnosis and automatic classification using +convolutional neural networks (CNNs) for image classification are an active research area to make a precise +diagnosis with less probability of misdiagnosis and fast detection process. +Current state-of-the-art investigations on BC detection confirm that CNNs are more reliable and faster +than the conventional hand-crafted features in the classification task [6]. However, estimated time to train +CNNs might be longer and it needs expertise to design such networks [6-8]. An applicable solution reported +in the literature is referred as deep convolutional activation feature (DeCAF) also known as deep features [6, + +8, 9]. These approaches reuse pre-trained CNNs to extract deep features and apply them to a classifier for +final decision. +The hand-crafted features in BC histopathological dataset (737 images) has been studied by Filipczuk et +al. [10] using circular Hough transform to segment the cell nuclei by circles. Their best result reached 98.51% +accuracy utilizing k-nearest neighbor (KNN) as a classifier [10]. However, the region of interest in the virtual +slides is not selected automatically and it is a time-consuming process. Additionally, the method cannot +guarantee a global optimum and elliptical segmentation requires more accurate model which is +computationally more demanding. In another work by Sharma and Mehra, hand-crafted features like color, +shape and texture extracted from BreakHis dataset and fed them to the conventional classifiers such as +support vector machine (SVM) and random forest (RF). They reported RF with 1000 number of trees could +achieve 90.33% accuracy for 40× data [11]. In addition, they have compared the hand-crafted features with +deep ones. The accuracy obtained for the classification of the deep features using VGG-16 network for 40× +data is 93.97%. They have reported that the performance of the hand-crafted features is not satisfactory since +it requires deep knowledge about the morphology of cancerous cells and deep features are a preferred +alternative. Alhindi et al. compared the local binary patterns (LBP), the histogram of oriented gradients +(HOG) as the hand-crafted features with deep features using the pre-trained VGG-19 for KIMIA Path960 +dataset [12]. The highest accuracy is 90.52% for LBP features and SVM classifier. It is worth mentioning that +feature vector size (FVS) of LBP is equal to 1182 and almost twice of the one of the extracted deep features. +To address deep features in histopathological images, Spanhole et al. extracted DeCAF from different +fully connected (FC) layers of pre-trained AlexNet with logistic regression classifier to diagnose BC using +BreakHis dataset [6]. The obtained results show that transfer learning is a viable alternative with 84.6% +accuracy for 40× data. Then, Deniz et al. developed a framework to take advantage of two pre-trained CNNs +for binary classification of BreakHis dataset. They have combined DeCAF features from AlexNet and VGG- +16 followed by SVM classifier and reached 84.87% accuracy [8]. In [13], Kumar et al. proposed a variant of +VGG-16, wherein all FC layers were removed and evaluated by different classifiers for CMT and BreakHis +datasets. The best reported accuracy is 97.01% for 200× data from BreakHis dataset, in which the FVS is 1472. +To overcome the lack of training dataset, dividing the histopathological image into non-overlapping or +random patches and providing them as the input to the pre-trained CNNs for feature extraction has been +studied [1, 6, 14]. However, extracting some patches can lead to uncertainty of the classification [15]. To +improve the accuracy of the classification, some approaches focused on training CNNs from scratch or fine- +tuning the pre-trained CNNs [15-19]. Some of these approaches have been reached to the higher performance +while experiencing a time-consuming procedure and arranging hyperparameters precisely. In some cases, +training the model or fine-tuning all the layers may not achieve a better performance compared to transfer +learning technique [7, 16, 17]. Additionally, transfer learning hits the spot either encountering the lack of +training dataset to train a deep model or adding a few training data to re-train the whole model [13, 14, 16]. +Dimension reduction or feature selection of deep features has attracted the attention of researchers +recently. Alinsaif et al. applied Infinite Latent Feature Selection (ILFS) method to select top ranked features +from pre-trained CNNs such as ResNet and DenseNet w/wo fine-tuning. The accuracy of binary +classification for BreakHis dataset with SVM classifier is reported 97.96% where FVS is 1300 [16]. Moreover, +Gupta et al. proposed extreme gradient boosting (XGboost) to reduce the number of features extracted from +ResNet and used information theoretic measure (ITS) to select the optimal number of layers. The accuracy is +reported 97.07±1.18% for 40× data where FVS is 500. Although, the accuracy decreased with fewer number +of features [20]. +In this study, dimensionality reduction is the main scope to investigate the influence on capturing +informative features with a smaller number of features. We have analyzed that all the deep features are not +necessarily led to a higher accuracy in the classification task and dimension reduction plays an important +role. We have proposed R-DeCAF features to capture the essence of the data with low computational time. +To achieve such a milestone, the weight of the pre-trained CNNs (AlexNet, VGG-16 and VGG-19) will be +kept in freeze mode and deep features extracted from the first FC layer, in which the size of the feature vector +is high, i.e., FVS = 4096. In order to reduce the size of the extracted deep features, different linear and + +nonlinear dimension reduction methods such as PCA, singular value decomposition (SVD), linear +discriminant analysis (LDA), kernel PCA (kPCA) and t-Distributed Stochastic Neighbor Embedding (t-SNE) +have been evaluated to generate R-DeCAF features. The comprehensive comparison of DeCAF and R- +DeCAF features (which are reduced by linear methods) classified by SVM with RBF kernel on BreakHis +dataset shows keeping less than 120 features not only improves the classification accuracy but also decreases +the computational time. +This paper is organized as follows: Section 2 describes the histopathological dataset; Section 3 illustrates +the problem formulation and provides details of the proposed model to extract R-DeCAF. In section 4, the +experimental case studies and comparative analysis of the obtained results are discussed. Finally, section 5 +represents the conclusion and the future research. +2. Histopathological database +The BreakHis database [21] developed in a laboratory (Pathological Anatomy and Cytopathology, +Parana, Brazil) and it is a publicly accessible histopathologic BC dataset which used in this work. This +selected dataset, includes microscopic histopathology images of BC, consists of 7,909 images of BC tissue +taken from 82 patients which is available in 40×, 100×, 200× and 400× magnification factors. This dataset +includes 2480 benign and 5429 malignant samples with the color image size of 700×460. In addition, benign +and malignant tumors are divided into subgroups. Samples of this dataset are collected by surgical open +biopsy (SOB) method and stained by Hematoxylin and Eosin method. Each image filename, includes stored +information about the image such as biopsy procedure method, magnification factor, type of cancer and its +subtypes, and patient identification. In Fig. 1, sample images of benign and malignant tumor from this +dataset at different magnification factors are shown. + +Figure 1. Sample images from BreakHis database in different magnification factors. First row belongs to the +same slide of the benign tumor and second row belongs to the same slide of the malignant tumor. +3. Methodology +This study involves deep feature extraction from BreakHis histopathological dataset and we have +shown that all the features extracted from the pre-trained CNNs cannot be effective in classifying the data. +Therefore, reducing the FVS of the extracted deep features to keep informative features and remove +unnecessary ones which cause misleading or do not play an important role in the classification, is the primary +goal of this study. Different contemporary pre-trained CNN models involving deep feature extraction from +BreakHis dataset are considered. Pre-trained CNNs have been trained by ImageNet dataset, which contains +more than 14 million natural images with 1000 categories. +The advantage of transfer learning technique is to avoid a time-consuming procedure for training a +CNN from scratch [6, 13, 16]. In other words, training and fine-tuning a CNN requires a huge amount of data +or a medium size of data, respectively. However, transfer learning involves different structures and it does +not need a huge amount of data. Actually, transfer learning is a method to transfer the knowledge learned +in one domain to a defined task for the purpose of classification or feature extraction. Thus, the goal is +transferring the knowledge from natural images to BC histopathological images and simplify the diagnosis + +process. As pre-trained CNNs are trained on a large dataset with significant number of classes and samples, +it can provide large networks with effective combination of features to classify the data [6, 13]. +Deep features are extracted automatically from the first FC layer of AlexNet, VGG-16 and VGG-19 +networks in transfer learning mode with considering the weights in freeze mode. Thus, freezing the weights +trained based on ImageNet dataset makes the model prepared to use all defined pre-trained weights. The +reason of considering the first FC layer i.e. FC6 of AlexNet, VGG-16 and VGG-19 is that FC6 layer provides +features more informative for an accurate classification [6, 22]. +3.1. Proposed framework for R-DeCAF +We reduce FVS of DeCAF features which is fixed to 4096 by applying appropriate dimension reduction +algorithms to generate R-DeCAF features, in which FVS is less than 120. According to the redundancy or +misleading features in a data [23], and based the analysis that we have done all DeCAF features are not +required in the classification task. Moreover, a high dimensional feature vector can dramatically impact the +performance of machine learning algorithms to fit on data and generally this can referred to as the “curse of +dimensionality” [24]. Therefore, we have proposed R-DeCAF features to capture the essence of the data with +low computational time by analyzing different dimensionality reduction techniques. The architecture of the +proposed framework is illustrated in Fig. 2. + +Figure 2. The diagram of the proposed framework. +3.2. Pre-trained CNNs +All three defined pre-trained CNNs which are AlexNet, VGG-16 and VGG-19 are studied as the basic +of our framework to extract deep features. AlexNet is known to be the primary profound CNN model +presented by Krizhevsky et al. [25]. This network contains five convolutional layers and three FC layers +where the number of neurons in the last layer is based on the number of classes of the data. The number of +neurons in first and second FC layers are 4096. The VGG-16 and VGG-19 CNNs with more layers are +proposed by Simonyan et al. in 2014 [26]. In these two CNNs, small filters of 3×3 are used for all the layers in +order to capture fine details in the images and control the number of parameters. VGG-19 has 19 weight +layers and VGG-16 has 16 weight layers [26]. It should be mentioned that all the input images are resized to +224×224 for the sake of convenience with CNN models in pytorch library in this work. +3.3. Feature Reduction Algorithms +This study analyzes different dimension reduction methods on DeCAF features categorized in two +groups; linear and nonlinear. The former includes PCA, SVD, and LDA, where the later contains kPCA and +t-SNE [23, 27]. PCA method is a linear and unsupervised algorithm, in which new features can be produced +by calculating a linear transformation. Eigenvectors and eigenvalues can be computed from the covariance + +Pre-trained CNNs: +Extracted +AlexNet +features +Dimension +VGG-16 +from first +reduction +VGG-19 +FC layermatrix of the data to determine the principal components (PC) of the data. PCA keeps the maximum +information of the data in the first PC and continues in descending order because principal directions and +corresponding PCs are considered as the directions of the maximum data variance [23, 24]. SVD is another +linear dimension reduction method which is appropriate for sparse data. SVD of a matrix is a factorization +of the main matrix into three matrices. In this method, the largest singular values are picked, where the +eigenvalues and eigenvectors are in descending order same as PCA method. Hence, the input matrix will +rebuild in low dimension [23]. LDA is another linear and supervised dimension reduction method which +focuses on two critical terms called “scatter between class” and “scatter within class”. The main aim is to +maximizing the “scatter between class” or separability of classes. Therefore, LDA can pick components +which separates the data classes in the best way. It should be mentioned that the number of +components/features in a reduced dimension can be equal or smaller than the number of classes-1 [23, 28]. +In the group of nonlinear dimension reduction methods, kPCA is one of the popular unsupervised +techniques. When the PCA method does not work well and the structure of the data is nonlinear, kPCA +method may perform better. In kPCA, the dimension of the original data can be reduced in a high +dimensional space with the advantage of “kernel trick”. In high dimensional space decision boundary +becomes linear. In this method the eigenvalues and eigenvectors of the kernel matrix are calculated based on +the reduced dimension set of eigenvectors selected in descending order. The product of the original matrix +and eigenvectors is calculated to rebuild the new reduced data [24, 27]. The nonlinear and unsupervised t- +SNE method is known as a common technique for data exploration and visualization. In this method, data +is mapped to a low dimension, such as 2 or 3 dimensions. t-SNE converts the high dimensional Euclidean +distances between pairwise data points 𝑥!, 𝑥" into conditional probability 𝑝"|! which shows the similarity of +the pairwise data points and a similar conditional probability in low dimensional counterparts 𝑦!, 𝑦" of the +high dimensional data points 𝑥!, 𝑥" defined by 𝑞"|!. The conditional probabilities 𝑝"|! and 𝑞"|! will be equal if +the data points 𝑦!, 𝑦" model the similarity between the data points in a high dimensional space [23, 29]. +3.4. Cumulative Explained Variance (CEV) and the size of the features +To reduce the size of the features, we have used eigenvalues and the corresponding cumulative +explained variance (CEV). Actually, determining the optimal number of PCs is a challenging and a critical +key in order to get an efficient performance and CEV is a way to solve this challenge. CEV is the accumulation +of variances to show the summation of variances of the new features i.e., PCs as the percentage of this +accumulated variance by the PC numbers [23]. Figure 3 displays CEV of DeCAF features extracted from the +first FC layer of pre-trained AlexNet, VGG-16 and VGG-19 which is related to the whole magnification data +of BreakHis dataset. As it can be seen, approximately with more than 2560 PCs, the CEV has changed +insignificantly. In other words, the first 2560 PCs contains 100% of cumulative variances where the first 512 +PCs covers 67% of variance of data. + +Figure 3. CEV of DeCAF features using pre-trained CNNs for the whole magnification data of BreakHis +dataset. + +1.0 +0.8 +0.6 +0.4 +0.2 +AlexNet +VGG-16 +VGG-19 +0.0 +0 +512 +1024 +1536 +2048 +2560 +3072 +3584 +4096It can be concluded that almost half of the transformed features i.e., PCs do not have important role in +the classification as a rule of thumb where Zhong et al. [9] took advantage of this simple rule. The main reason +is because of high correlation among extracted deep features. To investigate more in details, we have +considered a full range of CEV from 5% to 100%, in which 100% means we have used all PCs obtained from +DeCAF features to examine the classification accuracy. Here, the average accuracy is obtained for three +pretrained CNNs in 10 different splits of feature vectors into train and test datasets. Figure 4 shows these +results for the whole magnification data (7909 images) of BreakHis dataset. + +Figure 4. The average classification accuracy vs Cumulative Explained Variance (CEV) for the whole +magnification data of BreakHis dataset. +First, these results confirm that using CEV less than 100% but more than 15% not only keep the same +performance but also causes improvement in classifying deep features. Second, the results of Fig. 4 provide +more information that we need to feed the classifier with more effective and proper features rather than large +number of features. Therefore, a better accuracy can be achieved with a smaller number of features and the +content of features plays a crucial role in the classification task. We have shown by these results that +considering large number of features could not necessarily lead to a higher performance and all DeCAF +features extracted from pre-trained AlexNet, VGG-16, and VGG-19 are not compelling and informative in +the classification. Figure 4, discloses keeping only 20% to 25% of CEV makes more improvement compared +to 50% of CEV. These also reduced FVS significantly from 4096 to 63, 103, and 93 for the pre-trained AlexNet, +VGG-16, and VGG-19, respectively. More details of this investigation are presented in Table. 1. +3.5. Classifier +Here, the SVM algorithm has been selected for the classification as it has the ability to handle the high +dimensional data and nonlinear classification by using a kernel trick [13]. This technique is used to evaluate +the performance of DeCAF and R-DeCAF features in classification task to predict a sample is benign or +malignant. The trained SVM with RBF kernel is considered as the common kernels-based on a Grid search +among different kernels with the SVM parameter C=5. The defined dataset is divided into a training set (80%) +and test set (20%). The split method is used and the results are reported by taking an average of 10 different +splits. Comprehensive results are provided by pytorch and Scikitlearn libraries to validate the proposed +method. Since most of machine learning algorithms are sensitive to data scaling, in this manner we apply +Standard Scalar of Scikitlearn library to scale the feature vectors that are extracted. + + + + +Pre-trained CNN: AlexNet +Pre-trained CNN: VGG-16 +Pre-trainedCNN:VGG-19 +51015202530 3540455055 60 65707580 85 90 951004. Results and Discussion +The classification accuracy of DeCAF and R-DeCAF features are summarized in Table. 1. R-DeCAF +features obtained by three linear dimension reduction algorithms, i.e., PCA, SVD and LDA. First, we have +investigated the classification performance using DeCAF features by three mentioned CNNs in more details. +As you can see in third column of Table.1, the accuracy of VGG-16 and VGG-19 outperform AlexNet for 40×, +100×, and 200× data of BreakHis dataset but underperform for 400×. The reason can be found in both the +number of layers and 3×3 filters in VGG-16 and VGG-19 networks which can extract more details form +images. However, for 400× data such details from VGG-16 and VGG-19 networks are not necessary as the +magnification is higher and the images provide such details. Therefore, AlexNet is a better choice for high +magnification data. It is worth mentioning that the accuracy considering the whole magnification is better +for VGG-16 and VGG-19 networks as expected. In addition, the lowest accuracy is observed for a 40× data. +This might be because of the region of interest of 40× data as includes a higher complexity compared with +other magnification factors and carries more information which makes the accurate data classification more +difficult. The magnification factor effect on the classification accuracy depending on complexity level of BC +histopathological images. +Second, the classification performance using R-DeCAF features based on different dimension reduction +algorithms has been explored further. The results from applying PCA and SVD to generate R-DeCAF +features are provided almost the same improvement up to 4.3% compared to DeCAF features. For example, +the observed accuracy considering pre-trained AlexNet for whole magnification data are 85.95%, 90.24% and +90.18 for DeCAF features, R-DeCAF features using PCA, and R-DeCAF features using SVD, respectively. +Again, we can say that AlexNet is still a better choice for high magnification data even for R-DeCAF features. +The results obtained by LDA depict the classification accuracy has been decreased and this method is not +able to capture the essence of data caused by removing informative features. The main reason explaining the +low accuracy by LDA is that the number of features of original dataset is ignored and the obtained dimension +(FVS) will be less than the number of classes subtract one. Therefore, we will have only one feature for binary +classification using LDA technique [23]. +Table 1. The classification accuracy (%) of DeCAF and R-DeCAF features using different linear dimension +reduction methods (Best results are bolded). +Framework +Magnification +DeCAF +FVS = 4096 +R-DeCAF (reduced by linear methods) +FVS (CEV) +PCA +FVS (CEV) +SVD +LDA +AlexNet +(FC6), SVM +40× +84.38±1.6 +67 (0.25) +88.04±2.1 +67 (0.25) +88.02±2.1 +77.62±1.8 +100× +86.16±1.1 +45 (0.20) +89.54±1.0 +45 (0.20) +89.66±0.9 +79.87±2.3 +200× +87.69±1.4 +97 (0.30) +90.60±1.4 +135 (0.35) +90.50±1.3 +83.87±1.4 +400× +87.91±1.5 +23 (0.15) +91.13±1.4 +23 (0.15) +91.15±1.5 +82.58±1.5 +Whole mag. +85.95±0.8 +63 (0.20) +90.24±0.6 +63 (0.20) +90.18±0.6 +71.73±1.1 +VGG-16 +(FC6), SVM +40× +86.64±2.2 +90 (0.30) +89.82±1.7 +90 (0.30) +89.60±1.7 +82.23±0.9 +100× +89.52±1.2 +58 (0.25) +91.01±0.7 +206 (0.45) +91.01±1.2 +84.05±1.8 +200× +88.71±0.9 +84 (0.30) +90.78±1.4 +56 (0.25) +90.82±1.1 +82.68±0.9 +400× +85.60±1.9 +58 (0.25) +88.30±1.3 +58 (0.25) +88.21±1.3 +81.95±1.9 +Whole mag. +87.23±0.8 +103 (0.25) +90.61±0.9 +103 (0.25) +90.35±0.8 +73.10±1.0 +VGG-19 +(FC6), SVM +40× +85.09±1.8 +78 (0.30) +87.34±2.0 +51 (0.25) +87.14±1.6 +79.62±1.9 +100× +88.06±1.5 +118 (0.35) +90.19±1.3 +118 (0.35) +90.05±1.3 +81.10±1.3 +200× +88.51±1.0 +114 (0.35) +89.35±1.2 +114 (0.35) +89.60±1.6 +82.80±1.3 +400× +86.73±0.8 +59 (0.25) +88.60±0.8 +59 (0.25) +88.76±0.9 +82.03±1.9 +Whole mag. +86.68±0.5 +93 (0.25) +89.43±0.5 +93 (0.25) +89.37±0.5 +72.03±0.8 + +To evaluate our proposed method, we have computed the confusion matrix of the classification result +for DeCAF and R-DeCAF features. These matrixes are shown in Fig. 5, in which the results obtained by pre- +trained AlexNet for 400× data (1820 images, 588 benign and 1232 malignant) of BreakHis dataset. It can be +seen that the classification result of R-DeCAF features obtained by PCA outperforms the one of DeCAF +features. In more details, correctly predicted benign cases increased from 68% to 82%. This is very impressive + +result as our proposed method could increase the accuracy in benign class although the number of the data +in this class is more limited due to imbalanced BreakHis dataset. + +Figure 5. Confusion matrix for 400× data form BreakHis dataset to show the classification result of deep +features extracted from pre-trained AlexNet. (a). DeCAF features, (b). R-DeCAF features using PCA algorithm +(FVS=23, CEV=0.15). 0: Benign, 1: Malignant. +Moreover, we have validated the classification result of DeCAF and R-DeCAF features using other +metrics as shown in Table. 2. We have reported R-DeCAF features that reduced by PCA algorithm as we +have observed that the accuracy obtained by PCA outperforms other linear dimension reduction algorithms +(see Table. 1). Table. 2 shows that the precision and F1 score of R-DeCAF features have been improved +compared to the ones of DeCAF features, however recall is different and it has been decreased for R-DeCAF +features in some cases. +Since BreakHis dataset is imbalanced in which the number of the samples in the malignant class is +almost twice the one in the benign class and this ratio is almost the same for different magnifications, we +have addressed this issue by reproducing the results based on two strategies. First, we used weighted SVM +and the data is divided into train and test dataset with stratified k-fold (k=10). Second, we forced the data to +be balanced by randomly selected malignant samples to be the same as the number of benign samples in +each magnification factor and SVM is used. Both results show that the classification metrics in Table. 1 are +higher and the effect of imbalance data is not significant as the imbalance ratio of the data is not too high. +These investigations also confirm that the balance data could affect recall and improve it however its overall +improvement is less compared to Table. 2. +Table 2. The classification accuracy, precision, recall and F1 score (%) of DeCAF and R-DeCAF features +(reduced by PCA). +Framework +Magnification +DeCAF (FVS = 4096) +R-DeCAF (reduced by PCA) +Accuracy +Precision +Recall +F1 +FVS (CEV) +Accuracy +Precision +Recall +F1 +AlexNet +(FC6), SVM +40× +84.38±1.6 +85.20±2.1 +93.42±1.0 +89.10±1.2 +67 (0.25) +88.04±2.1 +89.61±2.7 +94.73±1.5 +91.54±1.6 +100× +86.16±1.1 +85.77±1.5 +95.95±0.8 +90.57±0.9 +45 (0.20) +89.54±1.0 +90.05±1.9 +95.52±1.3 +92.68±0.7 +200× +87.69±1.4 +87.63±1.5 +95.58±1.5 +91.42±0.9 +97 (0.30) +90.60±1.4 +91.11±1.8 +95.65±1.3 +93.31±1.0 +400× +87.91±1.5 +87.32±2.1 +96.02±1.2 +91.44±1.2 +23 (0.15) +91.13±1.4 +91.31±1.6 +95.96±0.9 +93.57±1.1 +Whole mag. +85.95±0.8 +86.98±0.9 +93.35±0.6 +90.05±0.5 +63 (0.20) +90.24±0.6 +91.24±0.6 +94.78±0.7 +92.97±0.5 +VGG-16 +(FC6), SVM +40× +86.64±2.2 +88.09±2.7 +93.17±1.9 +90.53±1.7 +90 (0.30) +89.82±1.7 +90.43±1.9 +95.28±1.8 +92.77±1.3 +100× +89.52±1.2 +89.21±1.4 +96.53±1.2 +92.72±0.8 +58 (0.25) +91.01±0.7 +92.15±1.1 +95.11±1.5 +93.59±0.6 +200× +88.71±0.9 +89.23±1.5 +95.07±1.6 +92.04±0.6 +84 (0.30) +90.78±1.4 +91.03±1.8 +96.04±1.1 +93.46±1.0 +400× +85.60±1.9 +85.43±2.2 +95.08±1.4 +89.98±1.5 +58 (0.25) +88.30±1.3 +88.92±1.8 +95.64±1.4 +91.67±1.0 +Whole mag. +87.23±0.8 +87.86±0.9 +94.40±0.6 +91.01±0.5 +103 (0.25) +90.61±0.9 +91.30±0.8 +95.18±0.8 +93.20±0.7 +VGG-19 +(FC6), SVM +40× +85.09±1.8 +86.00±2.4 +93.51±1.0 +89.58±1.3 +78 (0.30) +87.34±2.0 +87.01±2.2 +95.87±1.3 +91.21±1.5 +100× +88.06±1.5 +87.33±2.0 +96.86±0.9 +91.84±1.1 +118 (0.35) +90.19±1.3 +90.60±1.5 +95.81±1.0 +93.13±1.0 +200× +88.51±1.0 +88.33±1.0 +96.31±0.9 +92.14±0.6 +114 (0.35) +89.35±1.2 +90.10±1.6 +95.29±1.2 +92.61±0.8 +400× +86.73±0.8 +86.59±1.5 +95.35±1.3 +90.74±0.6 +59 (0.25) +88.60±0.8 +90.27±0.6 +93.36±1.4 +91.78±0.6 +Whole mag. +86.68±0.5 +87.39±0.5 +94.10±0.7 +90.62±0.5 +93 (0.25) +89.43±0.5 +89.93±0.6 +95.22±0.5 +92.50±0.4 + +0.8 +0 - +0.68 +0.32 +0.82 +0.18 +0.6 +0.4 +1 +0.031 +0.97 +0.04 +0.96 +0.2 +0 +1 +0 +1In addition, this study evaluates the performance of nonlinear dimension reduction methods, including +kPCA and t-SNE described in Section 3.3. The classification accuracy based on DeCAF and R-DeCAF features +is presented in Table. 3 in addition to FVS. Using kPCA algorithm, a different number of features have been +tested. However, the classification accuracy based on R-DeCAF features is not high enough. So, we have +considered the same number of features similar to PCA method. To apply t-SNE algorithm, it is highly +recommended to first use PCA method before decreasing the dimension to 2 or 4 features by t-SNE. +Therefore, we have reduced FVS in the same step as applying only PCA method on the feature vectors. Then, +t-SNE is implemented to reduce the number of features to 2. As it is clearly shown, nonlinear dimension +reduction methods are not effective to capture informative features from DeCAF features and classification +accuracy has decreased. However, linear approaches such as PCA could represent a better combination +among deep features and lead to a higher accuracy in the classification task. Nonlinear dimensionality +reduction techniques might be sensitive to the curse of dimensionality and this could be the reason of their +improper performance in our study. Hence, these methods are not able to guarantee better performance than +linear ones, such as PCA [23, 27]. Moreover, we can consider the presence of more complexity in R-DeCAF +features obtained by nonlinear dimension reduction methods which lead to lower classification accuracy. +Table 3. The classification accuracy (%) of DeCAF and R-DeCAF features using different nonlinear dimension +reduction methods (Best results are bolded). +Framework +Magnification +DeCAF +FVS = 4096 +R-DeCAF (reduced by nonlinear methods) +FVS (CEV) +KPCA +PCA+ t-SNE +AlexNet (FC6), +SVM +40× +84.38±1.6 +67 (0.25) +68.45±2.6 +65.94±5.7 +100× +86.16±1.1 +45 (0.20) +69.33±1.9 +68.85±1.5 +200× +87.69±1.4 +97 (0.30) +68.58±1.8 +65.23±4.8 +400× +87.91±1.5 +23 (0.15) +67.47±2.4 +67.03±4.7 +Whole mag. +85.95±0.8 +63 (0.20) +68.15±1.0 +62.96±6.0 +VGG-16 (FC6), +SVM +40× +86.64±2.2 +90 (0.30) +68.77±2.8 +63.51±11.2 +100× +89.52±1.2 +58 (0.25) +69.11±1.0 +68.94±3.2 +200× +88.71±0.9 +84 (0.30) +68.67±1.5 +63.57±10.9 +400× +85.60±1.9 +58 (0.25) +68.21±2.5 +64.78±6.9 +Whole mag. +87.23±0.8 +103 (0.25) +68.46±0.7 +66.15±2.7 +VGG-19 (FC6), +SVM +40× +85.09±1.8 +78 (0.30) +68.65±2.2 +55.93±13.20 +100× +88.06±1.5 +118 (0.35) +69.45±2.0 +63.86±7.8 +200× +88.51±1.0 +114 (0.35) +69.95±1.4 +68.39±3.0 +400× +86.73±0.8 +59 (0.25) +68.27±1.9 +66.73±5.4 +Whole mag. +86.68±0.5 +93 (0.25) +68.44±0.9 +66.96±2.5 + +Based on the analysis that we have done and the results reported in Table 1 and 3, we can conclude that +the accuracy of binary classification for BreakHis dataset will be enhanced using R-DeCAF features with +linear dimension reduction algorithms like PCA and SVD up to 4.3% in different magnification factors. Less +probability of overfitting and noise rejection capability of PCA algorithm and the benefits of sparse data +management by SVD algorithm [23, 30] are the reasons which make improvement in our R-DeCAF features. +This is an important finding in which there is a linear combination among deep features which could help +us to consider it in modifying networks to perform better. +Moreover, we have compared the performance of the proposed framework with the state-of-the-art +studies which is summarized in Table. 4. In our method, the results of R-DeCAF features obtained by PCA +algorithm have been reported. In the previous works, deep features are extracted from different pre-trained +CNNs followed by SVM classifier as our case to classify BreakHis dataset. FVS is also mentioned in Table. 4 +for a comprehensive analysis and comparison. As we can see, the results obtained from the proposed method +have sought to increase the accuracy compared to some approaches. As a case in point, in [8] and [14], the +classification of deep features which are extracted from pre-trained CNNs i.e., AlexNet, VGG-16, and VGG- +19 led to lower accuracy in comparison with this study. The higher accuracy obtained by Kumar et al. [13], +Gupta et al. [20], and Alinsaif et al. [16] while FVS is not comparable to our case which is almost less than + +120. In [13], a global average pooling is applied to five external convolutional layers of all five blocks of VGG- +16 and make a feature vector of 1472 after concatenation. This approach is a different from ours where we +have just extracted features form one layer (first FC layer). This declares that looking at the features form all +layers improves the result. The higher accuracy was reported in [20] where FVS is equal to 500. Although, +the accuracy decreased with fewer number of features. In addition, the authors in [16] could only keep the +classification accuracy unchanged by FVS equals to 1300. On the other hand, since we have only analyzed +three pre-trained models as feature extractors, we could not examine our proposed concept on the mentioned +works [16, 20] in which the CNNs used as a feature extractor are different. Moreover, we believe that our +proposed method is able to enhance the performance of transfer learning. In another study based on fine- +tuning pre-trained CNNs, the classification accuracy is reported as 80.80% for the 40× data of BreakHis +dataset [5]. Reducing FVS to less than 120, our proposed method could hit the spot in comparison with +previous works and classification accuracy has increased up to 4.3% simultaneously. +Table 4. Comparison of the classification accuracy obtained from the proposed method and previous +methods. +Existing +Methods +CNN +FVS +Classification Accuracy (%) +40× +100× +200× +400× +Whole +mag. +Bardou et al. [18] +new CNN +2000 +90.64 +89.58 +90.23 +75.96 +- +Deniz et al. [8] +AlexNet + +VGG-16 +4096 + +4096 +84.87±1.1 +89.21±1.4 +88.65±2.4 +86.75±4.2 +- +Gupta et al. [20] +ResNet +500 +97.07±1.2 +96.10±1.0 +94.69±1.2 +90.85±2.1 +- +Kumar et al. [13] +VGG-16 +1472 +94.11±1.8 +95.12±1.1 +97.01±1.1 +93.40±1.0 +- +Saxena et al. [14] +AlexNet +1526 +84.06 +87.54 +89.40 +85.16 +- +VGG-16 +3072 +86.36 +87.77 +86.80 +84.35 +- +VGG-19 +3072 +86.64 +88.17 +85.84 +81.67 +- +Alinsaif et al. [16] +DenseNet +1300 +- +- +- +- +97.96±0.6 +Proposed +AlexNet +23-97 +88.04±2.1 +89.54±1.0 +90.60±1.4 +91.13±1.4 +90.24±0.6 +VGG-16 +58-103 +89.82±1.7 +91.01±0.7 +90.78±1.4 +88.30±1.3 +90.61±0.9 +VGG-19 +59-118 +87.34±2.0 +90.19±1.3 +89.35±1.2 +88.60±0.8 +89.43±0.5 +5. Conclusions +This study proposes R-DeCAF features for BC detection using histopathological images and compares +with DeCAF features. To extract DeCAF features, three different pre-trained CNNs emerged as an +unsupervised feature extractor. A feature vector from the first FC layer of CNNs with FVS of 4096 has been +extracted. The results show that keeping all DeCAF features extracted from pre-trained AlexNet, VGG-16, +and VGG-19 is not effective in the classification task. Thus, various dimension reduction methods on DeCAF +features are evaluated to capture informative feature vectors and decrease the computational time too. Based +on the analysis that we have done considering about 15% to 35% of CEV of features in the new space with +FVS of less than 120 is sufficient and could significantly improve the accuracy up to 4.3% in the best case. +Evaluations show that linear dimensionality reduction algorithms could represent an effective combination +among deep features and lead to a higher accuracy in the classification task, however nonlinear approaches +fail. This is an important finding in which there is a linear combination among deep features which could +help us to consider it in modifying networks to perform better. Moreover, PCA performs better among +various linear dimension reduction methods. The best-achieved result for 400× data using pre-trained +AlexNet as the feature extractor is 91.13±1.4%. It should be noted that data augmentation and particular data +preprocessing are not required in the proposed model that is considered a fully automatic model for cancer +diagnosis. Moreover, magnification level of BreakHis dataset affect the classification accuracy as it depends +on the complexity level of histopathological images. As a future work, modification in deep CNN models +based on PCA algorithm to provide less features complexity and increase classification accuracy with more +reliable and informative features may break this curse. Additionally, examining more other pre-trained CNN + +models to extract deep features and applying this proposed method for performance enhancement will be +considered in the future work. + +References +[1] +T. Araújo, G. Aresta, E. Castro, J. Rouco, P. Aguiar, C. Eloy, A. Polónia, and A. Campilho, "Classification +of breast cancer histology images using convolutional neural networks," PLoS One, vol. 12, no. 6, p. +e0177544, 2017. +[2] +W. H. Organization, WHO position paper on mammography screening. World Health Organization, 2014. +[3] +P. Boyle and B. Levin, World cancer report 2008. IARC Press, International Agency for Research on +Cancer, 2008. +[4] +J. Arevalo, A. Cruz-Roa, and F. A. GONZÁLEZ O, "Histopathology image representation for automatic +analysis: A state-of-the-art review," Revista Med, vol. 22, no. 2, pp. 79-91, 2014. +[5] +S. Singh and R. Kumar, "Breast cancer detection from histopathology images with deep inception and +residual blocks," Multimedia Tools and Applications, vol. 81, no. 4, pp. 5849-5865, 2022. +[6] +F. A. Spanhol, L. S. Oliveira, P. R. Cavalin, C. Petitjean, and L. Heutte, "Deep features for breast cancer +histopathological image classification," in 2017 IEEE International Conference on Systems, Man, and +Cybernetics (SMC), 2017: IEEE, pp. 1868-1873. +[7] +R. Mehra, "Breast cancer histology images classification: Training from scratch or transfer learning?," ICT +Express, vol. 4, no. 4, pp. 247-254, 2018. +[8] +E. Deniz, A. Şengür, Z. Kadiroğlu, Y. Guo, V. Bajaj, and Ü. Budak, "Transfer learning based +histopathologic image classification for breast cancer detection," Health information science and systems, +vol. 6, no. 1, pp. 1-7, 2018. +[9] +G. Zhong, S. Yan, K. Huang, Y. Cai, and J. Dong, "Reducing and stretching deep convolutional activation +features for accurate image classification," Cognitive Computation, vol. 10, no. 1, pp. 179-186, 2018. +[10] +P. Filipczuk, T. Fevens, A. Krzyżak, and R. Monczak, "Computer-aided breast cancer diagnosis based on +the analysis of cytological images of fine needle biopsies," IEEE transactions on medical imaging, vol. 32, +no. 12, pp. 2169-2178, 2013. +[11] +S. Sharma and R. Mehra, "Conventional machine learning and deep learning approach for multi- +classification of breast cancer histopathology images—a comparative insight," Journal of digital imaging, +vol. 33, no. 3, pp. 632-654, 2020. +[12] +T. J. Alhindi, S. Kalra, K. H. Ng, A. Afrin, and H. R. Tizhoosh, "Comparing LBP, HOG and deep features +for classification of histopathology images," in 2018 international joint conference on neural networks +(IJCNN), 2018: IEEE, pp. 1-7. +[13] +A. Kumar, S. K. Singh, S. Saxena, K. Lakshmanan, A. K. Sangaiah, H. Chauhan, S. Shrivastava, and R. K. +Singh, "Deep feature learning for histopathological image classification of canine mammary tumors and +human breast cancer," Information Sciences, vol. 508, pp. 405-421, 2020. +[14] +S. Saxena, S. Shukla, and M. Gyanchandani, "Pre‐trained convolutional neural networks as feature +extractors for diagnosis of breast cancer using histopathology," International Journal of Imaging Systems +and Technology, vol. 30, no. 3, pp. 577-591, 2020. +[15] +P. Yamlome, A. D. Akwaboah, A. Marz, and M. Deo, "Convolutional neural network based breast cancer +histopathology image classification," in 2020 42nd Annual International Conference of the IEEE +Engineering in Medicine & Biology Society (EMBC), 2020: IEEE, pp. 1144-1147. +[16] +S. Alinsaif and J. Lang, "Histological image classification using deep features and transfer learning," in +2020 17th Conference on Computer and Robot Vision (CRV), 2020: IEEE, pp. 101-108. +[17] +S. Boumaraf, X. Liu, Y. Wan, Z. Zheng, C. Ferkous, X. Ma, Z. Li, and D. Bardou, "Conventional machine +learning versus deep learning for magnification dependent histopathological breast cancer image +classification: A comparative study with visual explanation," Diagnostics, vol. 11, no. 3, p. 528, 2021. +[18] +D. Bardou, K. Zhang, and S. M. Ahmad, "Classification of breast cancer based on histology images using +convolutional neural networks," Ieee Access, vol. 6, pp. 24680-24693, 2018. +[19] +M. Z. Alom, C. Yakopcic, M. Nasrin, T. M. Taha, and V. K. Asari, "Breast cancer classification from +histopathological images with inception recurrent residual convolutional neural network," Journal of digital +imaging, vol. 32, no. 4, pp. 605-617, 2019. + +[20] +V. Gupta and A. Bhavsar, "Partially-independent framework for breast cancer histopathological image +classification," in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition +Workshops, 2019, pp. 0-0. +[21] +"Breast Cancer Histopathological Database (BreakHis)." https://web.inf.ufpr.br/vri/databases/breast- +cancer-histopathological-database-breakhis/ (accessed. +[22] +R. F. Mansour, "Deep-learning-based automatic computer-aided diagnosis system for diabetic retinopathy," +Biomedical engineering letters, vol. 8, no. 1, pp. 41-57, 2018. +[23] +F. Anowar, S. Sadaoui, and B. Selim, "Conceptual and empirical comparison of dimensionality reduction +algorithms (pca, kpca, lda, mds, svd, lle, isomap, le, ica, t-sne)," Computer Science Review, vol. 40, p. +100378, 2021. +[24] +K. P. Murphy, Machine learning: a probabilistic perspective. MIT press, 2012. +[25] +A. Krizhevsky, I. Sutskever, and G. E. Hinton, "Imagenet classification with deep convolutional neural +networks," Advances in neural information processing systems, vol. 25, 2012. +[26] +K. Simonyan and A. Zisserman, "Very deep convolutional networks for large-scale image recognition," +arXiv preprint arXiv:1409.1556, 2014. +[27] +L. Van Der Maaten, E. Postma, and J. Van den Herik, "Dimensionality reduction: a comparative," J Mach +Learn Res, vol. 10, no. 66-71, p. 13, 2009. +[28] +A. Tharwat, T. Gaber, A. Ibrahim, and A. E. Hassanien, "Linear discriminant analysis: A detailed tutorial," +AI communications, vol. 30, no. 2, pp. 169-190, 2017. +[29] +L. Van der Maaten and G. Hinton, "Visualizing data using t-SNE," Journal of machine learning research, +vol. 9, no. 11, 2008. +[30] +S. Karamizadeh, S. M. Abdullah, A. A. Manaf, M. Zamani, and A. Hooman, "An overview of principal +component analysis," Journal of Signal and Information Processing, vol. 4, 2020. + + diff --git a/6tAzT4oBgHgl3EQf-P5_/content/tmp_files/load_file.txt b/6tAzT4oBgHgl3EQf-P5_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea688c859fbb88a5bb3b1059eadcda623b552060 --- /dev/null +++ b/6tAzT4oBgHgl3EQf-P5_/content/tmp_files/load_file.txt @@ -0,0 +1,1178 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf,len=1177 +page_content='Reduced Deep Convolutional Activation Features (R-DeCAF) in Histopathology Images to Improve the Classification Performance for Breast Cancer Diagnosis Bahareh Morovati, Reza Lashgari, Mojtaba Hajihasani and Hasti Shabani * Institute of Medical Science and Technology, Shahid Beheshti University, Tehran, Iran Correspondence: ha_shabani@sbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='ir Abstract: Breast cancer is the second most common cancer among women worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Diagnosis of breast cancer by the pathologists is a time-consuming procedure and subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Computer aided diagnosis frameworks are utilized to relieve pathologist workload by classifying the data automatically, in which deep convolutional neural networks (CNNs) are effective solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The features extracted from activation layer of pre-trained CNNs are called deep convolutional activation features (DeCAF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In this paper, we have analyzed that all DeCAF features are not necessarily led to a higher accuracy in the classification task and dimension reduction plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Therefore, different dimension reduction methods are applied to achieve an effective combination of features by capturing the essence of DeCAF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' To this purpose, we have proposed reduced deep convolutional activation features (R-DeCAF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In this framework, pre-trained CNNs such as AlexNet, VGG-16 and VGG-19 are utilized in transfer learning mode as feature extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' DeCAF features are extracted from the first fully connected layer of the mentioned CNNs and support vector machine has been used for binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Among linear and nonlinear dimensionality reduction algorithms, linear approaches such as principal component analysis (PCA) represent a better combination among deep features and lead to a higher accuracy in the classification task using small number of features considering specific amount of cumulative explained variance (CEV) of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The proposed method is validated using experimental BreakHis dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Comprehensive results show improvement in the classification accuracy up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='3% with less computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Best achieved accuracy is 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='13% for 400× data with feature vector size (FVS) of 23 and CEV equals to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='15 using pre-trained AlexNet as feature extractor and PCA as feature reduction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Keywords: Breast cancer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' deep feature extraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' feature reduction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' histopathology images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' pre-trained convolutional neural networks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Introduction Breast cancer (BC) is one of the leading causes of mortality in the world almost observed in women, but it can occur in men, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Diagnosis of BC ordinarily comprises of an initial detection by palpation and regular check-ups by ultrasound imaging or mammography and diagnosis of possible malignant tissue growth is tested by breast tissue biopsy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' According to the world health organization (WHO), BC is affecting large number of women health [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Recent studies predict around 27 million new cases of BC by 2030 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Early detection of BC is essential for appropriate treatment and decreasing the mortality rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' However, BC diagnosis may not be accurate enough as pathologist could only apply visual inspection of samples under microscopes [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' According these challenges, computer aided diagnosis and automatic classification using convolutional neural networks (CNNs) for image classification are an active research area to make a precise diagnosis with less probability of misdiagnosis and fast detection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Current state-of-the-art investigations on BC detection confirm that CNNs are more reliable and faster than the conventional hand-crafted features in the classification task [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' However, estimated time to train CNNs might be longer and it needs expertise to design such networks [6-8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' An applicable solution reported in the literature is referred as deep convolutional activation feature (DeCAF) also known as deep features [6, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' These approaches reuse pre-trained CNNs to extract deep features and apply them to a classifier for final decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The hand-crafted features in BC histopathological dataset (737 images) has been studied by Filipczuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [10] using circular Hough transform to segment the cell nuclei by circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Their best result reached 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='51% accuracy utilizing k-nearest neighbor (KNN) as a classifier [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' However, the region of interest in the virtual slides is not selected automatically and it is a time-consuming process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Additionally, the method cannot guarantee a global optimum and elliptical segmentation requires more accurate model which is computationally more demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In another work by Sharma and Mehra, hand-crafted features like color, shape and texture extracted from BreakHis dataset and fed them to the conventional classifiers such as support vector machine (SVM) and random forest (RF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' They reported RF with 1000 number of trees could achieve 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='33% accuracy for 40× data [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In addition, they have compared the hand-crafted features with deep ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The accuracy obtained for the classification of the deep features using VGG-16 network for 40× data is 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='97%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' They have reported that the performance of the hand-crafted features is not satisfactory since it requires deep knowledge about the morphology of cancerous cells and deep features are a preferred alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Alhindi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' compared the local binary patterns (LBP), the histogram of oriented gradients (HOG) as the hand-crafted features with deep features using the pre-trained VGG-19 for KIMIA Path960 dataset [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The highest accuracy is 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='52% for LBP features and SVM classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' It is worth mentioning that feature vector size (FVS) of LBP is equal to 1182 and almost twice of the one of the extracted deep features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' To address deep features in histopathological images, Spanhole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' extracted DeCAF from different fully connected (FC) layers of pre-trained AlexNet with logistic regression classifier to diagnose BC using BreakHis dataset [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The obtained results show that transfer learning is a viable alternative with 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='6% accuracy for 40× data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Then, Deniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' developed a framework to take advantage of two pre-trained CNNs for binary classification of BreakHis dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' They have combined DeCAF features from AlexNet and VGG- 16 followed by SVM classifier and reached 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='87% accuracy [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In [13], Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' proposed a variant of VGG-16, wherein all FC layers were removed and evaluated by different classifiers for CMT and BreakHis datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The best reported accuracy is 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='01% for 200× data from BreakHis dataset, in which the FVS is 1472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' To overcome the lack of training dataset, dividing the histopathological image into non-overlapping or random patches and providing them as the input to the pre-trained CNNs for feature extraction has been studied [1, 6, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' However, extracting some patches can lead to uncertainty of the classification [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' To improve the accuracy of the classification, some approaches focused on training CNNs from scratch or fine- tuning the pre-trained CNNs [15-19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Some of these approaches have been reached to the higher performance while experiencing a time-consuming procedure and arranging hyperparameters precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In some cases, training the model or fine-tuning all the layers may not achieve a better performance compared to transfer learning technique [7, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Additionally, transfer learning hits the spot either encountering the lack of training dataset to train a deep model or adding a few training data to re-train the whole model [13, 14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Dimension reduction or feature selection of deep features has attracted the attention of researchers recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Alinsaif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' applied Infinite Latent Feature Selection (ILFS) method to select top ranked features from pre-trained CNNs such as ResNet and DenseNet w/wo fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The accuracy of binary classification for BreakHis dataset with SVM classifier is reported 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='96% where FVS is 1300 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Moreover, Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' proposed extreme gradient boosting (XGboost) to reduce the number of features extracted from ResNet and used information theoretic measure (ITS) to select the optimal number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The accuracy is reported 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='07±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='18% for 40× data where FVS is 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Although, the accuracy decreased with fewer number of features [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In this study, dimensionality reduction is the main scope to investigate the influence on capturing informative features with a smaller number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' We have analyzed that all the deep features are not necessarily led to a higher accuracy in the classification task and dimension reduction plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' We have proposed R-DeCAF features to capture the essence of the data with low computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' To achieve such a milestone, the weight of the pre-trained CNNs (AlexNet, VGG-16 and VGG-19) will be kept in freeze mode and deep features extracted from the first FC layer, in which the size of the feature vector is high, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=', FVS = 4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In order to reduce the size of the extracted deep features, different linear and nonlinear dimension reduction methods such as PCA, singular value decomposition (SVD), linear discriminant analysis (LDA), kernel PCA (kPCA) and t-Distributed Stochastic Neighbor Embedding (t-SNE) have been evaluated to generate R-DeCAF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The comprehensive comparison of DeCAF and R- DeCAF features (which are reduced by linear methods) classified by SVM with RBF kernel on BreakHis dataset shows keeping less than 120 features not only improves the classification accuracy but also decreases the computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' This paper is organized as follows: Section 2 describes the histopathological dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Section 3 illustrates the problem formulation and provides details of the proposed model to extract R-DeCAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In section 4, the experimental case studies and comparative analysis of the obtained results are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Finally, section 5 represents the conclusion and the future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Histopathological database The BreakHis database [21] developed in a laboratory (Pathological Anatomy and Cytopathology, Parana, Brazil) and it is a publicly accessible histopathologic BC dataset which used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' This selected dataset, includes microscopic histopathology images of BC, consists of 7,909 images of BC tissue taken from 82 patients which is available in 40×, 100×, 200× and 400× magnification factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' This dataset includes 2480 benign and 5429 malignant samples with the color image size of 700×460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In addition, benign and malignant tumors are divided into subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Samples of this dataset are collected by surgical open biopsy (SOB) method and stained by Hematoxylin and Eosin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Each image filename, includes stored information about the image such as biopsy procedure method, magnification factor, type of cancer and its subtypes, and patient identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 1, sample images of benign and malignant tumor from this dataset at different magnification factors are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Sample images from BreakHis database in different magnification factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' First row belongs to the same slide of the benign tumor and second row belongs to the same slide of the malignant tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Methodology This study involves deep feature extraction from BreakHis histopathological dataset and we have shown that all the features extracted from the pre-trained CNNs cannot be effective in classifying the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Therefore, reducing the FVS of the extracted deep features to keep informative features and remove unnecessary ones which cause misleading or do not play an important role in the classification, is the primary goal of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Different contemporary pre-trained CNN models involving deep feature extraction from BreakHis dataset are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Pre-trained CNNs have been trained by ImageNet dataset, which contains more than 14 million natural images with 1000 categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The advantage of transfer learning technique is to avoid a time-consuming procedure for training a CNN from scratch [6, 13, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In other words, training and fine-tuning a CNN requires a huge amount of data or a medium size of data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' However, transfer learning involves different structures and it does not need a huge amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Actually, transfer learning is a method to transfer the knowledge learned in one domain to a defined task for the purpose of classification or feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Thus, the goal is transferring the knowledge from natural images to BC histopathological images and simplify the diagnosis process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' As pre-trained CNNs are trained on a large dataset with significant number of classes and samples, it can provide large networks with effective combination of features to classify the data [6, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Deep features are extracted automatically from the first FC layer of AlexNet, VGG-16 and VGG-19 networks in transfer learning mode with considering the weights in freeze mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Thus, freezing the weights trained based on ImageNet dataset makes the model prepared to use all defined pre-trained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The reason of considering the first FC layer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' FC6 of AlexNet, VGG-16 and VGG-19 is that FC6 layer provides features more informative for an accurate classification [6, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Proposed framework for R-DeCAF We reduce FVS of DeCAF features which is fixed to 4096 by applying appropriate dimension reduction algorithms to generate R-DeCAF features, in which FVS is less than 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' According to the redundancy or misleading features in a data [23], and based the analysis that we have done all DeCAF features are not required in the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Moreover, a high dimensional feature vector can dramatically impact the performance of machine learning algorithms to fit on data and generally this can referred to as the “curse of dimensionality” [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Therefore, we have proposed R-DeCAF features to capture the essence of the data with low computational time by analyzing different dimensionality reduction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The architecture of the proposed framework is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The diagram of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Pre-trained CNNs All three defined pre-trained CNNs which are AlexNet, VGG-16 and VGG-19 are studied as the basic of our framework to extract deep features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' AlexNet is known to be the primary profound CNN model presented by Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' This network contains five convolutional layers and three FC layers where the number of neurons in the last layer is based on the number of classes of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The number of neurons in first and second FC layers are 4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The VGG-16 and VGG-19 CNNs with more layers are proposed by Simonyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' in 2014 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In these two CNNs, small filters of 3×3 are used for all the layers in order to capture fine details in the images and control the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' VGG-19 has 19 weight layers and VGG-16 has 16 weight layers [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' It should be mentioned that all the input images are resized to 224×224 for the sake of convenience with CNN models in pytorch library in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Feature Reduction Algorithms This study analyzes different dimension reduction methods on DeCAF features categorized in two groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' linear and nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The former includes PCA, SVD, and LDA, where the later contains kPCA and t-SNE [23, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' PCA method is a linear and unsupervised algorithm, in which new features can be produced by calculating a linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Eigenvectors and eigenvalues can be computed from the covariance Pre-trained CNNs: Extracted AlexNet features Dimension VGG-16 from first reduction VGG-19 FC layermatrix of the data to determine the principal components (PC) of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' PCA keeps the maximum information of the data in the first PC and continues in descending order because principal directions and corresponding PCs are considered as the directions of the maximum data variance [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' SVD is another linear dimension reduction method which is appropriate for sparse data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' SVD of a matrix is a factorization of the main matrix into three matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In this method, the largest singular values are picked, where the eigenvalues and eigenvectors are in descending order same as PCA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Hence, the input matrix will rebuild in low dimension [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' LDA is another linear and supervised dimension reduction method which focuses on two critical terms called “scatter between class” and “scatter within class”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The main aim is to maximizing the “scatter between class” or separability of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Therefore, LDA can pick components which separates the data classes in the best way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' It should be mentioned that the number of components/features in a reduced dimension can be equal or smaller than the number of classes-1 [23, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In the group of nonlinear dimension reduction methods, kPCA is one of the popular unsupervised techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' When the PCA method does not work well and the structure of the data is nonlinear, kPCA method may perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In kPCA, the dimension of the original data can be reduced in a high dimensional space with the advantage of “kernel trick”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In high dimensional space decision boundary becomes linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In this method the eigenvalues and eigenvectors of the kernel matrix are calculated based on the reduced dimension set of eigenvectors selected in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The product of the original matrix and eigenvectors is calculated to rebuild the new reduced data [24, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The nonlinear and unsupervised t- SNE method is known as a common technique for data exploration and visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In this method, data is mapped to a low dimension, such as 2 or 3 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' t-SNE converts the high dimensional Euclidean distances between pairwise data points 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=', 𝑥" into conditional probability 𝑝"|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' which shows the similarity of the pairwise data points and a similar conditional probability in low dimensional counterparts 𝑦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=', 𝑦" of the high dimensional data points 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=', 𝑥" defined by 𝑞"|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='. The conditional probabilities 𝑝"|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' and 𝑞"|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' will be equal if the data points 𝑦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=', 𝑦" model the similarity between the data points in a high dimensional space [23, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Cumulative Explained Variance (CEV) and the size of the features To reduce the size of the features, we have used eigenvalues and the corresponding cumulative explained variance (CEV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Actually, determining the optimal number of PCs is a challenging and a critical key in order to get an efficient performance and CEV is a way to solve this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' CEV is the accumulation of variances to show the summation of variances of the new features i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=', PCs as the percentage of this accumulated variance by the PC numbers [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Figure 3 displays CEV of DeCAF features extracted from the first FC layer of pre-trained AlexNet, VGG-16 and VGG-19 which is related to the whole magnification data of BreakHis dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' As it can be seen, approximately with more than 2560 PCs, the CEV has changed insignificantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In other words, the first 2560 PCs contains 100% of cumulative variances where the first 512 PCs covers 67% of variance of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' CEV of DeCAF features using pre-trained CNNs for the whole magnification data of BreakHis dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='2 AlexNet VGG-16 VGG-19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='0 0 512 1024 1536 2048 2560 3072 3584 4096It can be concluded that almost half of the transformed features i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=', PCs do not have important role in the classification as a rule of thumb where Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [9] took advantage of this simple rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The main reason is because of high correlation among extracted deep features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' To investigate more in details, we have considered a full range of CEV from 5% to 100%, in which 100% means we have used all PCs obtained from DeCAF features to examine the classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Here, the average accuracy is obtained for three pretrained CNNs in 10 different splits of feature vectors into train and test datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Figure 4 shows these results for the whole magnification data (7909 images) of BreakHis dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The average classification accuracy vs Cumulative Explained Variance (CEV) for the whole magnification data of BreakHis dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' First, these results confirm that using CEV less than 100% but more than 15% not only keep the same performance but also causes improvement in classifying deep features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Second, the results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 4 provide more information that we need to feed the classifier with more effective and proper features rather than large number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Therefore, a better accuracy can be achieved with a smaller number of features and the content of features plays a crucial role in the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' We have shown by these results that considering large number of features could not necessarily lead to a higher performance and all DeCAF features extracted from pre-trained AlexNet, VGG-16, and VGG-19 are not compelling and informative in the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Figure 4, discloses keeping only 20% to 25% of CEV makes more improvement compared to 50% of CEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' These also reduced FVS significantly from 4096 to 63, 103, and 93 for the pre-trained AlexNet, VGG-16, and VGG-19, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' More details of this investigation are presented in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Classifier Here, the SVM algorithm has been selected for the classification as it has the ability to handle the high dimensional data and nonlinear classification by using a kernel trick [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' This technique is used to evaluate the performance of DeCAF and R-DeCAF features in classification task to predict a sample is benign or malignant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The trained SVM with RBF kernel is considered as the common kernels-based on a Grid search among different kernels with the SVM parameter C=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The defined dataset is divided into a training set (80%) and test set (20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The split method is used and the results are reported by taking an average of 10 different splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Comprehensive results are provided by pytorch and Scikitlearn libraries to validate the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Since most of machine learning algorithms are sensitive to data scaling, in this manner we apply Standard Scalar of Scikitlearn library to scale the feature vectors that are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Pre-trained CNN: AlexNet Pre-trained CNN: VGG-16 Pre-trainedCNN:VGG-19 51015202530 3540455055 60 65707580 85 90 951004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Results and Discussion The classification accuracy of DeCAF and R-DeCAF features are summarized in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' R-DeCAF features obtained by three linear dimension reduction algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=', PCA, SVD and LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' First, we have investigated the classification performance using DeCAF features by three mentioned CNNs in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' As you can see in third column of Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='1, the accuracy of VGG-16 and VGG-19 outperform AlexNet for 40×, 100×, and 200× data of BreakHis dataset but underperform for 400×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The reason can be found in both the number of layers and 3×3 filters in VGG-16 and VGG-19 networks which can extract more details form images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' However, for 400× data such details from VGG-16 and VGG-19 networks are not necessary as the magnification is higher and the images provide such details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Therefore, AlexNet is a better choice for high magnification data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' It is worth mentioning that the accuracy considering the whole magnification is better for VGG-16 and VGG-19 networks as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In addition, the lowest accuracy is observed for a 40× data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' This might be because of the region of interest of 40× data as includes a higher complexity compared with other magnification factors and carries more information which makes the accurate data classification more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The magnification factor effect on the classification accuracy depending on complexity level of BC histopathological images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Second, the classification performance using R-DeCAF features based on different dimension reduction algorithms has been explored further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The results from applying PCA and SVD to generate R-DeCAF features are provided almost the same improvement up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='3% compared to DeCAF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' For example, the observed accuracy considering pre-trained AlexNet for whole magnification data are 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='95%, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='24% and 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='18 for DeCAF features, R-DeCAF features using PCA, and R-DeCAF features using SVD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Again, we can say that AlexNet is still a better choice for high magnification data even for R-DeCAF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The results obtained by LDA depict the classification accuracy has been decreased and this method is not able to capture the essence of data caused by removing informative features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The main reason explaining the low accuracy by LDA is that the number of features of original dataset is ignored and the obtained dimension (FVS) will be less than the number of classes subtract one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Therefore, we will have only one feature for binary classification using LDA technique [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The classification accuracy (%) of DeCAF and R-DeCAF features using different linear dimension reduction methods (Best results are bolded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Framework Magnification DeCAF FVS = 4096 R-DeCAF (reduced by linear methods) FVS (CEV) PCA FVS (CEV) SVD LDA AlexNet (FC6), SVM 40× 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='38±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='6 67 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='25) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='04±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='1 67 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='25) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='02±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='1 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='62±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='8 100× 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='16±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='1 45 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='20) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='03±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='9 Whole mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='5 93 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='25) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='5 93 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='25) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='37±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='8 To evaluate our proposed method, we have computed the confusion matrix of the classification result for DeCAF and R-DeCAF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' These matrixes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 5, in which the results obtained by pre- trained AlexNet for 400× data (1820 images, 588 benign and 1232 malignant) of BreakHis dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' It can be seen that the classification result of R-DeCAF features obtained by PCA outperforms the one of DeCAF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In more details, correctly predicted benign cases increased from 68% to 82%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' This is very impressive result as our proposed method could increase the accuracy in benign class although the number of the data in this class is more limited due to imbalanced BreakHis dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Confusion matrix for 400× data form BreakHis dataset to show the classification result of deep features extracted from pre-trained AlexNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' DeCAF features, (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' R-DeCAF features using PCA algorithm (FVS=23, CEV=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 0: Benign, 1: Malignant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Moreover, we have validated the classification result of DeCAF and R-DeCAF features using other metrics as shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' We have reported R-DeCAF features that reduced by PCA algorithm as we have observed that the accuracy obtained by PCA outperforms other linear dimension reduction algorithms (see Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 2 shows that the precision and F1 score of R-DeCAF features have been improved compared to the ones of DeCAF features, however recall is different and it has been decreased for R-DeCAF features in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Since BreakHis dataset is imbalanced in which the number of the samples in the malignant class is almost twice the one in the benign class and this ratio is almost the same for different magnifications, we have addressed this issue by reproducing the results based on two strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' First, we used weighted SVM and the data is divided into train and test dataset with stratified k-fold (k=10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Second, we forced the data to be balanced by randomly selected malignant samples to be the same as the number of benign samples in each magnification factor and SVM is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Both results show that the classification metrics in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 1 are higher and the effect of imbalance data is not significant as the imbalance ratio of the data is not too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' These investigations also confirm that the balance data could affect recall and improve it however its overall improvement is less compared to Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The classification accuracy, precision, recall and F1 score (%) of DeCAF and R-DeCAF features (reduced by PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Framework Magnification DeCAF (FVS = 4096) R-DeCAF (reduced by PCA) Accuracy Precision Recall F1 FVS (CEV) Accuracy Precision Recall F1 AlexNet (FC6), SVM 40× 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='38±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='20±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='1 93.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='50±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='8 0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='2 0 1 0 1In addition, this study evaluates the performance of nonlinear dimension reduction methods, including kPCA and t-SNE described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The classification accuracy based on DeCAF and R-DeCAF features is presented in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 3 in addition to FVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Using kPCA algorithm, a different number of features have been tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' However, the classification accuracy based on R-DeCAF features is not high enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' So, we have considered the same number of features similar to PCA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' To apply t-SNE algorithm, it is highly recommended to first use PCA method before decreasing the dimension to 2 or 4 features by t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Therefore, we have reduced FVS in the same step as applying only PCA method on the feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Then, t-SNE is implemented to reduce the number of features to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' As it is clearly shown, nonlinear dimension reduction methods are not effective to capture informative features from DeCAF features and classification accuracy has decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' However, linear approaches such as PCA could represent a better combination among deep features and lead to a higher accuracy in the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Nonlinear dimensionality reduction techniques might be sensitive to the curse of dimensionality and this could be the reason of their improper performance in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Hence, these methods are not able to guarantee better performance than linear ones, such as PCA [23, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Moreover, we can consider the presence of more complexity in R-DeCAF features obtained by nonlinear dimension reduction methods which lead to lower classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The classification accuracy (%) of DeCAF and R-DeCAF features using different nonlinear dimension reduction methods (Best results are bolded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Framework Magnification DeCAF FVS = 4096 R-DeCAF (reduced by nonlinear methods) FVS (CEV) KPCA PCA+ t-SNE AlexNet (FC6), SVM 40× 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='38±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='6 67 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='25) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='45±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='94±5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='9 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='96±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='5 Based on the analysis that we have done and the results reported in Table 1 and 3, we can conclude that the accuracy of binary classification for BreakHis dataset will be enhanced using R-DeCAF features with linear dimension reduction algorithms like PCA and SVD up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='3% in different magnification factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Less probability of overfitting and noise rejection capability of PCA algorithm and the benefits of sparse data management by SVD algorithm [23, 30] are the reasons which make improvement in our R-DeCAF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' This is an important finding in which there is a linear combination among deep features which could help us to consider it in modifying networks to perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Moreover, we have compared the performance of the proposed framework with the state-of-the-art studies which is summarized in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In our method, the results of R-DeCAF features obtained by PCA algorithm have been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In the previous works, deep features are extracted from different pre-trained CNNs followed by SVM classifier as our case to classify BreakHis dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' FVS is also mentioned in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' 4 for a comprehensive analysis and comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' As we can see, the results obtained from the proposed method have sought to increase the accuracy compared to some approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' As a case in point, in [8] and [14], the classification of deep features which are extracted from pre-trained CNNs i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=', AlexNet, VGG-16, and VGG- 19 led to lower accuracy in comparison with this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The higher accuracy obtained by Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [13], Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [20], and Alinsaif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [16] while FVS is not comparable to our case which is almost less than 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In [13], a global average pooling is applied to five external convolutional layers of all five blocks of VGG- 16 and make a feature vector of 1472 after concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' This approach is a different from ours where we have just extracted features form one layer (first FC layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' This declares that looking at the features form all layers improves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The higher accuracy was reported in [20] where FVS is equal to 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Although, the accuracy decreased with fewer number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In addition, the authors in [16] could only keep the classification accuracy unchanged by FVS equals to 1300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' On the other hand, since we have only analyzed three pre-trained models as feature extractors, we could not examine our proposed concept on the mentioned works [16, 20] in which the CNNs used as a feature extractor are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Moreover, we believe that our proposed method is able to enhance the performance of transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' In another study based on fine- tuning pre-trained CNNs, the classification accuracy is reported as 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='80% for the 40× data of BreakHis dataset [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Reducing FVS to less than 120, our proposed method could hit the spot in comparison with previous works and classification accuracy has increased up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='3% simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Comparison of the classification accuracy obtained from the proposed method and previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Existing Methods CNN FVS Classification Accuracy (%) 40× 100× 200× 400× Whole mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Bardou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [18] new CNN 2000 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='64 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='58 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='23 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='96 Deniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [8] AlexNet + VGG-16 4096 + 4096 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='87±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='21±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='65±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='75±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='2 Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [20] ResNet 500 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='07±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='10±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='69±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='85±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='1 Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [13] VGG-16 1472 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='11±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='8 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='12±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='01±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='40±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='0 Saxena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [14] AlexNet 1526 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='06 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='54 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='40 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='16 VGG-16 3072 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='36 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='77 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='80 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='35 VGG-19 3072 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='64 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='17 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='84 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='67 Alinsaif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' [16] DenseNet 1300 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='96±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='6 Proposed AlexNet 23-97 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='04±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='54±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='60±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} 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59-118 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='34±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='19±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='35±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='60±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='8 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Conclusions This study proposes R-DeCAF features for BC detection using histopathological images and compares with DeCAF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' To extract DeCAF features, three different pre-trained CNNs emerged as an unsupervised feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' A feature vector from the first FC layer of CNNs with FVS of 4096 has been extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The results show that keeping all DeCAF features extracted from pre-trained AlexNet, VGG-16, and VGG-19 is not effective in the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Thus, various dimension reduction methods on DeCAF features are evaluated to capture informative feature vectors and decrease the computational time too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Based on the analysis that we have done considering about 15% to 35% of CEV of features in the new space with FVS of less than 120 is sufficient and could significantly improve the accuracy up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='3% in the best case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Evaluations show that linear dimensionality reduction algorithms could represent an effective combination among deep features and lead to a higher accuracy in the classification task, however nonlinear approaches fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' This is an important finding in which there is a linear combination among deep features which could help us to consider it in modifying networks to perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Moreover, PCA performs better among various linear dimension reduction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' The best-achieved result for 400× data using pre-trained AlexNet as the feature extractor is 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='13±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' It should be noted that data augmentation and particular data preprocessing are not required in the proposed model that is considered a fully automatic model for cancer diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Moreover, magnification level of BreakHis dataset affect the classification accuracy as it depends on the complexity level of histopathological images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' As a future work, modification in deep CNN models based on PCA algorithm to provide less features complexity and increase classification accuracy with more reliable and informative features may break this curse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQf-P5_/content/2301.01931v1.pdf'} +page_content=' Additionally, examining more other pre-trained CNN models to extract deep features and applying this proposed method for performance enhancement will be considered in the future work.' metadata={'source': 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[math.PR] 9 Jan 2023 +CONVERGENCE TOWARDS THE POPULATION CROSS-DIFFUSION +SYSTEM FROM STOCHASTIC MANY-PARTICLE SYSTEM +YUE LI, LI CHEN, AND ZHIPENG ZHANG +Abstract. In this paper, we derive rigorously a general cross-diffusion system from a moderate +interacting stochastic many-particle system in the whole space. The convergence is proved in the +sense of probability by introducing an intermediate particle system with a mollified interaction +potential, where the mollification is of algebraic scaling. The main idea of the proof is to study +the time evolution of a stopped process. Based on the global existence and uniform estimates of +solutions to the local (and non-local) cross-diffusion equation, we are able to obtain a Gr¨onwall type +estimate by using Taylor’s expansion around the limiting stochastic process. +1. +Introduction +In this paper, we give a rigorous justification for the mean-field limit from a moderate interacting +particle system to the population cross-diffusion system as the number of particles goes to infinity. +More precisely, we present the derivation of n-species cross-diffusion system as follows + + + + + + + +∂tui = div(ui∇Ui) + σi∆ui + div +� +ui +n +� +j=1 +∇f(aijuj) +� +, +ui(0) = u0 +i (x), +i = 1, . . . , n, +(1.1) +where σi > 0 are the constant diffusion coefficients, aij ∈ R describe the strength of cross-diffusion +effects, u = (u1, . . . , un) stands for the vector of population densities and Ui(x) = − 1 +2|x|2 represent +environment potentials. The transitions rates depend on the densities by a nonlinear term f. +The aim of this paper is to rigorously derive the system (1.1) from the following stochastic many- +particle system. This system describes the movements of n species of particles, with the particle +numbers Ni ∈ N (i = 1, . . . , n), according to the given law. +Without loss of generality, we let +N = Ni (i = 1, . . . , n). Let (Ω, F, (Ft≥0), P) be a complete filtered probability space. We consider +d-dimensional Ft-Brownian motions (W k +i (t))t≥0 (k = 1, . . . , N, i = 1, . . . , n) which are assumed to +be independent of each other. We assume that (ξk +i ) (k = 1, . . . , N, i = 1, . . . , n) are i.i.d. random +variables, independent of (W k +i (t))t≥0, and have common probability density function u0 +i . We use +the notation XN,k +η,i (t) to represent the k-th particle of i-th species and the dynamics of XN,k +η,i (t) are +governed by + + + + + + + +dXN,k +η,i += +� +− ∇Ui(XN,k +η,i ) − +n +� +j=1 +∇fγ +� 1 +N +N +� +l=1 +Bη +ij(XN,k +η,i − XN,l +η,j ) +�� +dt + +√ +2σidW k +i (t), +XN,k +η,i (0) = ξk +i , +i = 1, . . . , n, +k = 1, . . . , N, +(1.2) +2010 Mathematics Subject Classification. 35Q92, 35K45, 60J70, 82C22. +Key words and phrases. Stochastic particle systems; Cross-diffusion system; Mean-field limit; Population dynamics. +1 + +2 +YUE LI, LI CHEN, AND ZHIPENG ZHANG +where fγ is an approximation of f. +The moderate interaction potentials Bη +ij(x) = η−dBij(|x| +η ) +are used to approximate the Delta distribution, namely Bη +ij → aijδ0 as η → 0 in the sense of +distributions. In these representation, the functions Bij are chosen to be any smooth functions with +compact supports and +� +Rd Bij dx = aij. +The problem considered in this paper dedicates to the understanding of diffusion (and cross- +diffusion) effects on the microscopic level. It belongs to the research of mean-field limit for moderate +interacting particle system. There have been extensive studies of the mean-field limit problems +in the last decades. +Many important contributions have been made for problems with singular +interacting potentials such as the Coulomb potential in Keller-Segel systems. An extensive review +of this research field is out of the scope of this paper, we refer to [7, 8, 13, 2] for more detailed +summary on the results and methods. +The convergence of moderate interacting system was introduced and proved by Oelschl¨ager in +[10, 11, 12] in order to derive reaction-diffusion equations and the porous medium equation. The +authors [9] considered further the fluctuation of this problem. This idea has been used to derive +chemotaxis equation from an interacting stochastic many-particle system in [15]. The derivation of +cross-diffusion type systems has only been studied in the last few years. It is proved in [14] that +the hydrodynamic limit of the empirical densities of two types is the solution to the Maxwell-Stefan +equation. The authors in [6] derived the non-local Lotka-Volterra system with cross-diffusion from +particle system. The Shigesada-Kawasaki-Teramoto system was obtained from a microscopic many- +particle Markov process in [5]. Rigorous derivation of the degenerate parabolic-elliptic Keller-Segel +system from a moderate interacting stochastic particle system was given in [2]. There are very +few results for more than two species. In [4], the authors established the global existence of weak +solutions to cross-diffusion systems for an arbitrary number of competing population species. The +mean-field limit of a moderate interacting stochastic many-particle system for multiple population +species is obtained in [3] with the logarithmic scaling. +Furthermore, with the same scaling the +authors in [1] derived population cross-diffusion systems of Shigesada-Kawasaki-Teramoto type from +stochastic moderately interacting many-particle systems for multiple population species. +To precisely state the main results of this paper, we first give the general assumptions on f and +a list of possible choices of it. +Assumption 1.1. Let f ∈ W s+1,∞ +loc +(R; [0, ∞)) (s > d +2 + 2) satisfy +∥f∥C3([−M,M]) ≤ eM +for any +M ≥ 0; +and for a properly small ε which depends on n, σmin, A, Cs, ∥u0∥Hs(Rd) such that +∥f∥Cs+1(I) ≤ ε, +where I = [−2ACs∥u0∥Hs(Rd), 2ACs∥u0∥Hs(Rd)], σmin := mini=1,...,n σi, A := maxi,j=1,...,n ∥Bij∥L1(Rd) +and Cs is the constant of the continuous embedding Hs(Rd) ֒→ L∞(Rd). +Remark 1.2. It is easy to find the approximation fγ of f that satisfies +fγ = f on +� +− 1 +γ , 1 +γ +� +, + +MEAN-FIELD LIMIT +3 +where 1 +γ ≥ 2ACs∥u0∥Hs(Rd), and +∥fγ∥C3(R) ≤ ∥f∥C3([− 1 +γ , 1 +γ ]) ≤ e +1 +γ . +A typical example for f(v) is the linear function f(v) = v, which is exactly the case handled in +[3]. In this case, the requirement ∥f∥Cs+1(I) ≤ ε is equivalent to the smallness assumption of initial +data. Further examples are f(v) = vm for m ∈ (N ∩ (0, s]) ∪ (s, ∞). Or in case aij ≥ 0, possible +choices might be f(v) = (v + 1)m for m ∈ R, f(v) = ln(v + 1), or f(v) = ev. +In order to prove the limit from (1.2) to (1.1), we introduce an intermediate particle problem. +This problem is formally viewed as a mean field limit N → ∞ in the system (1.2) for fixed η, γ > 0, +namely + + + + + + + +d ¯Xk +η,i = +� +− ∇Ui( ¯Xk +η,i) − +n +� +j=1 +∇fγ(Bη +ij ∗ uη,j( ¯Xk +η,i)) +� +dt + +√ +2σidW k +i (t), +¯Xk +η,i(0) = ξk +i , +i = 1, . . . , n, +k = 1, . . . , N. +(1.3) +Here uη,j is the probability density function of ¯Xk +η,j and satisfies the following non-local cross- +diffusion system: + + + + + + + +∂tuη,i = div(uη,i∇Ui) + σi∆uη,i + div +� +uη,i +n +� +j=1 +∇fγ(Bη +ij ∗ uη,j) +� +, +uη,i(0) = u0 +i (x), +i = 1, . . . , n. +(1.4) +Similar to the proof of the wellposedness theory for non-local and local PDE problems given in [1] +we obtain that, under the assumption 1.1 and u0 ∈ Hs(Rd), the systems (1.1) and (1.4) possess a +unique solution uη and u respectively. Moreover, uη and u satisfy for any T > 0, +∥uη∥2 +L∞(0,T;Hs(Rd)) + σ∗∥∇uη∥2 +L2(0,T;Hs(Rd)) ≤ ∥u0∥2 +Hs(Rd), +(1.5) +∥u∥2 +L∞(0,T;Hs(Rd)) + σ∗∥∇u∥2 +L2(0,T;Hs(Rd)) ≤ ∥u0∥2 +Hs(Rd), +(1.6) +∥u − uη∥L∞(0,T;L2(Rd)) + ∥∇(u − uη)∥L2((0,T)×Rd) ≤ Cη, +(1.7) +where 0 < σ∗ < σmin and C is a positive constant which depends on T, σ∗, Cs, n, A and initial +data. +Based on these, we can also similarly obtain the wellposedness of SDE system (1.2) and the +McKean-Vlasov problem of (1.1), i.e. + + + + + + + +d ˆXk +i = +� +− ∇Ui( ˆXk +i ) − +n +� +j=1 +∇f(aijuj( ˆXk +i )) +� +dt + +√ +2σidW k +i (t), +ˆXk +i (0) = ξk +i , +i = 1, . . . , n, +k = 1, . . . , N, +(1.8) +where ui solves the limiting cross-diffusion system (1.1) and is the probability density function of +ˆXk +i . Namely, when additional +� +Rd |x|2|u0(x)|dx < ∞, there exist unique square-integrable adapted +stochastic processes with continuous paths, which are strong solutions to systems (1.3) and (1.8), +respectively. + +4 +YUE LI, LI CHEN, AND ZHIPENG ZHANG +Therefore, the estimate in (1.7) provides directly the following estimate: there exists C > 0 which +is independent of N and η such that +E +� +n +� +i=1 +sup +0≤t≤T +max +1≤k≤N | ¯Xk +η,i(t) − ˆXk +i (t)|2� +≤ Cη2. +(1.9) +The main result of this paper is the following +Theorem 1.3. Let the assumption 1.1 hold, u0 ∈ Hs(Rd), and +� +Rd |x|2|u0(x)|dx < ∞. Assume +that T > 0, η = N −β, 1 +γ = β log N, where β ∈ (0, +1 +2(5d+6)), then for any arbitrary λ > 0, it holds +sup +0≤t≤T +P +� +n +� +i=1 +max +1≤k≤N |XN,k +η,i (t) − ¯Xk +η,i(t)| > N −α� +≤ CN −λ, +where α < 1 +2 − β(2d + 2) and C is a positive constant independent of N. +Remark 1.4. The assumption 1 +γ = β log N is determined by the given exponential grow of f in +assumption 1.1. If we only assume the algebraic grow of f, then the assumption of 1 +γ can be relaxed +to algebraic scaling. The key contribution of the main result is the moderate interacting radius η +which required more delicate ideas than those have been used in [3]. +Combined with the estimate in (1.9), we obtain the mean field limit result on the trajectory level +and the propagation of chaos as a corollary +Corollary 1.5. Under the same assumptions as in theorem 1.3, we have for any ˜β < β +sup +0≤t≤T +P +� +n +� +i=1 +max +1≤k≤N |XN,k +η,i (t) − ˆXk +i (t)| > N − ˜β� +≤ CN −(β− ˜β). +Let l ∈ N and consider an l-tuple (XN,1 +η,i (t), . . . , XN,l +η,i (t)). We denote by P N,l +η,i (t) its joint distribution. +Then it holds that +P N,l +η,i (t) convergres weakly to P ⊗l +i (t) as N → ∞, +where Pi(t) is a measure which is absolutely continuous with respect to the Lebesgue measure and +has a probability density function ui(t, x). +Compared with the convergence result obtained in [1], where the propagation of chaos has been +obtained under the logarithmic moderate scaling η ∼ 1/ log N, one can clearly see that the main +result of this paper gives the same propagation of chaos result under the algebraic scaling η ∼ 1/N β. +This result is obtained through the convergence in the sense of probability on the particle level. +The benefit of algebraic scaling is that one can capture the singular interaction to some extend. +To overcome the difficulty originated from the singular interaction, a suitable stopped process is +established. Based on this, it is reduced to estimate the expectation of the stopped process with the +help of Markov’s inequality. A generalized version of Law of Large Numbers is the key point when +we study the expectation. Another difficulty is caused by the nonlinear term. Since the regularity +of f is not global, we have to find an approximate function fγ and give explicit scaling between γ +and N. Section 2 is dedicated to the proof of the main theorem. + +MEAN-FIELD LIMIT +5 +2. The proof of Theorem 1.3 +We prove the convergence in probability on the particle level. Because of the singular interaction, +one can not expect that under the algebraic scaling the convergence can be obtained in the expec- +tation sense. The convergence in probability means that one allows that the particle trajectories +are not always close, but the probability that they are not close is very low. Actually we can prove +that the probability has a arbitrary convergence rate. +For any κ ∈ N, we define a stopping time τα, a random variable Sκ +α and a set Bα +τα(ω) := inf +� +t ∈ (0, T) : +n +� +i=1 +max +1≤k≤N |XN,k +η,i (t) − ¯Xk +η,i(t)| ≥ N −α� +, +ω ∈ Ω, +Sk +α(t) := N ακ +n +� +i=1 +max +1≤k≤N |XN,k +η,i (t ∧ τα) − ¯Xk +η,i(t ∧ τα)|κ ≤ 1, +Bα(t) := {ω ∈ Ω : Sκ +α(t) = 1}. +By Markov’s inequality, it holds +P +� +n +� +i=1 +max +1≤k≤N |XN,k +η,i (t) − ¯Xk +η,i(t)| ≥ N −α� +≤P +� +n +� +i=1 +max +1≤k≤N |XN,k +η,i (t ∧ τα) − ¯Xk +η,i(t ∧ τα)| = N −α� +≤ E(Sκ +α(t)). +We notice that the introduction of parameter κ is to increase the convergence rate. Actually, the +above Markov’s inequality works for arbitrary κ. In order to complete the proof of Theorem 1.3, +we just need to show that for any λ > 0 and T > 0, it holds E(Sκ +α(t)) ≤ CN −λ, where the letter C +appeared in this section is a generic positive constant independent of N. To this end, we need the +following Law of Large Numbers, which can be found for example in [2]: +Lemma 2.1. For ϕij ∈ L∞(Rd), i, j = 1, . . . , n, we define for arbitrary θ ∈ (0, 1 +2) +AN,n +θ,ϕ (s) := +n� +i,j=1 +N +� +k=1 +� +ω ∈ Ω : +��� 1 +N +N +� +l=1 +ϕij( ¯Xk +η,i(s) − ¯Xl +η,j(s)) − ϕij ∗ uη,j( ¯Xk +η,i(s)) +��� > +1 +N θ +� +. +Then for any m ∈ N, it holds that +P(AN,n +θ,ϕ (s)) ≤ C(n, m)N 2m(θ− 1 +2 )+1� +sup +i,j +∥ϕij∥2m +L∞(Rd) + sup +i,j +∥ϕij ∗ uη,j∥2m +L∞((0,T)×Rd) +� +, +∀s ∈ [0, T]. +Next we study the time evolution of the cut-offed process Sκ +α(t). Notice that +|XN,k +η,i (t ∧ τα) − ¯Xk +η,i(t ∧ τα)|κ +≤C +� t∧τα +0 +|∇Ui(XN,k +η,i ) − ∇Ui( ¯Xk +η,i)|κ ds ++ C +� t∧τα +0 +��� +n +� +j=1 +∇fγ +� 1 +N +N +� +l=1 +Bη +ij(XN,k +η,i − XN,l +η,j ) +� +− +n +� +j=1 +∇fγ(Bη +ij ∗ uη,j( ¯Xk +η,i)) +��� +κ +ds +=:J1 +k,i(t) + J2 +k,i(t). + +6 +YUE LI, LI CHEN, AND ZHIPENG ZHANG +From the definition of Ui(x) = − |x|2 +2 , we get +E +� +N ακ +n +� +i=1 +max +1≤k≤N J1 +k,i(t) +� +≤C +� t +0 +E(Sκ +α(s)) ds. +(2.1) +For the second term J2 +k,i, +E +� +N ακ +n +� +i=1 +max +1≤k≤N J2 +k,i(t) +� +≤CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +��� +n +� +j=1 +f ′ +γ +� 1 +N +N +� +l=1 +Bη +ij(XN,k +η,i − XN,l +η,j ) +� +· 1 +N +N +� +l=1 +� +∇Bη +ij(XN,k +η,i − XN,l +η,j ) − ∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +���� +κ +ds +� ++ CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +��� +n +� +j=1 +� +f ′ +γ +� 1 +N +N +� +l=1 +Bη +ij(XN,k +η,i − XN,l +η,j ) +� +− f ′ +γ +� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +�� +· 1 +N +N +� +l=1 +∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +��� +κ +ds +� ++ CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +��� +n +� +j=1 +� +f ′ +γ +� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +� +− f ′ +γ(Bη +ij ∗ uη,j( ¯Xk +η,i)) +� +· 1 +N +N +� +l=1 +∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +��� +κ +ds +� ++ CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +��� +n +� +j=1 +f ′ +γ(Bη +ij ∗ uη,j( ¯Xk +η,i)) +· +� 1 +N +N +� +l=1 +∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − ∇Bη +ij ∗ uη,j( ¯Xk +η,i) +���� +κ +ds +� +=:J21 + J22 + J23 + J24. +(2.2) +The term J21 can be divided into two terms: +J21 ≤CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +��� +n +� +j=1 +� +f ′ +γ +� 1 +N +N +� +l=N +Bη +ij(XN,k +η,i − XN,l +η,j ) +� +− f ′ +γ +� 1 +N +N +� +l=N +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +�� +· 1 +N +N +� +l=1 +[∇Bη +ij(XN,k +η,i − XN,l +η,j ) − ∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j)] +��� +κ +ds +� ++ CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +��� +n +� +j=1 +f ′ +γ +� 1 +N +N +� +l=N +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +� +· 1 +N +N +� +l=1 +[∇Bη +ij(XN,k +η,i − XN,l +η,j ) − ∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j)] +��� +κ +ds +� + +MEAN-FIELD LIMIT +7 +=:J211 + J212. +(2.3) +The term J211 can be handled with +J211 ≤ C∥f ′′ +γ ∥κ +L∞(0,supi,j ∥Bη +ij∥L∞(Rd)) sup +i,j +∥∇Bη +ij∥κ +L∞(Rd) sup +i,j +∥D2Bη +ij∥κ +L∞(Rd) +· +� t∧τα +0 +E +� +N ακ +n +� +i=1 +max +1≤k≤N |XN,k +η,i − ¯Xk +η,i|2κ� +ds +≤ C∥f ′′ +γ ∥κ +L∞(0,supi,j ∥Bη +ij∥L∞(Rd)) +supi,j ∥∇Bij∥κ +L∞(Rd) +ηκ(d+1) +supi,j ∥D2Bij∥κ +L∞(Rd) +ηκ(d+2) +· N −ακ +� t∧τα +0 +E(Sκ +α(s)) ds +≤ CN κ[β(2d+4)−α] +� t +0 +E(Sκ +α(s)) ds, +(2.4) +where we have used the assumption η = N −β and the fact that ∥fγ∥C3(R) ≤ ∥f∥C3([−β log N,β log N]) ≤ +eβ log N ≤ N β. With the help of ∥Bη +ij ∗ uη,j∥L∞((0,T)×Rd) ≤ 2ACs∥u0∥Hs(Rd), we have +∥f ′ +γ∥L∞(0,supi,j ∥Bη +ij∗uη,j∥L∞((0,T )×Rd)) ≤ C. Therefore, +J212 ≤CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +��� +n +� +j=1 +� +f ′ +γ +� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +� +− f ′ +γ(Bη +ij ∗ uη,j( ¯Xk +η,i)) +� +· 1 +N +N +� +l=1 +[∇Bη +ij(XN,k +η,i − XN,l +η,j ) − ∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j)] +��� +κ +ds +� ++ CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +��� +n +� +j=1 +f ′ +γ(Bη +ij ∗ uη,j( ¯Xk +η,i)) +· 1 +N +N +� +l=1 +[∇Bη +ij(XN,k +η,i − XN,l +η,j ) − ∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j)] +��� +κ +ds +� +≤C∥f ′′ +γ ∥κ +L∞(0,supi,j ∥Bη +ij∥L∞(Rd)) +supi,j ∥D2Bij∥κ +L∞(Rd) +ηκ(d+2) +· E +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − Bη +ij ∗ uη,j( ¯Xk +η,i) +��� +κ +Sκ +α(s) ds +� ++ CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +n +� +j=1 +��� 1 +N +N +� +l=1 +D2Bη +ij( ¯Xk +η,i − ¯Xl +η,j)(XN,k +η,i − ¯Xk +η,i) +��� +κ +ds +� ++ CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +n +� +j=1 +��� 1 +N +N +� +l=1 +D2Bη +ij( ¯Xk +η,i − ¯Xl +η,j)(XN,l +η,j − ¯Xl +η,j) +��� +κ +ds +� ++ C sup +i,j +∥D3Bη +ij∥κ +L∞(Rd)E +� � t∧τα +0 +N ακ +n +� +i=1 +max +1≤k≤N |XN,k +η,i − ¯Xk +η,i|2κ ds +� +=:J2121 + J2122 + J2123 + CN κ[β(d+3)−α] +� t +0 +E(Sκ +α(s)) ds. +(2.5) + +8 +YUE LI, LI CHEN, AND ZHIPENG ZHANG +For J2121, we split the domain Ω = AN,n +θ1,Bη ∪ (AN,n +θ1,Bη)c and obtain +J2121 ≤∥f ′′ +γ ∥κ +L∞(0,supi,j ∥Bη +ij∥L∞(Rd)) +supi,j ∥D2Bij∥κ +L∞(Rd) +ηκ(d+2) +· E +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − Bη +ij ∗ uη,j( ¯Xk +η,i) +��� +κ +IAN,n +θ1,Bη Sκ +α(s) ds +� ++ ∥f ′′ +γ ∥κ +L∞(0,supi,j ∥Bη +ij∥L∞(Rd)) +supi,j ∥D2Bij∥κ +L∞(Rd) +ηκ(d+2) +· E +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − Bη +ij ∗ uη,j( ¯Xk +η,i) +��� +κ +I(AN,n +θ1,Bη )cSκ +α(s) ds +� +≤N κβ(d+3)� +sup +i,j +∥Bη +ij∥κ +L∞(Rd) + sup +i,j +∥Bη +ij ∗ uη,j∥κ +L∞((0,T)×Rd) +� � t +0 +P(AN,n +θ1,Bη) ds ++ N κ[β(d+3)−θ1] +� t∧τα +0 +E(Sκ +α(s)) ds +≤N κβ(2d+3) +� t +0 +P(AN,n +θ1,Bη) ds + N κ[β(d+3)−θ1] +� t +0 +E(Sκ +α(s)) ds. +(2.6) +For J2123, we split again the domain Ω = AN,n +0,|D2Bη| ∪ (AN,n +0,|D2Bη|)c and obtain +J2123 ≤E +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +n +� +j=1 +��� 1 +N +N +� +l=1 +|D2Bη +ij|( ¯Xk +η,i − ¯Xl +η,j) +��� +κ +sup +l +|XN,l +η,j − ¯Xl +η,j|κ ds +� +≤E +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +|D2Bη +ij|( ¯Xk +η,i − ¯Xl +η,j) − |D2Bη +ij| ∗ uη,j( ¯Xk +η,i) +��� +κ +IAN,n +0,|D2Bη|Sκ +α(s) ds +� ++ E +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +|D2Bη +ij|( ¯Xk +η,i − ¯Xl +η,j) − |D2Bη +ij| ∗ uη,j( ¯Xk +η,i) +��� +κ +I(AN,n +0,|D2Bη|)cSκ +α(s) ds +� ++ sup +i,j +∥|D2Bη +ij| ∗ uη,j∥κ +L∞((0,T)×Rd) +� t∧τα +0 +E(Sκ +α(s)) ds +≤CN κβ(d+2) +� t +0 +P(AN,n +0,|D2Bη|) ds + C +� t +0 +E(Sκ +α(s)) ds, +(2.7) +where we have used the result that ∥|D2Bη +ij| ∗ uη,j∥L∞((0,T)×Rd) ≤ C, more details can be found in +[2]. Similarly, we can derive that +J2122 ≤ CN κβ(d+2) +� t +0 +P(AN,n +0,D2Bη) ds + C +� t +0 +E(Sκ +α(s)) ds. +(2.8) +Combining (2.3)-(2.8), we have +J21 ≤C +� +1 + N κ[β(2d+4)−α] + N κ[β(d+3)−α] + N κ[β(d+3)−θ1]� � t +0 +E(Sκ +α(s)) ds ++ CN κβ(2d+3) +� t +0 +P(AN,n +θ1,Bη) ds + CN κβ(d+2) +� t +0 +P(AN,n +0,D2Bη) ds + CN κβ(d+2) +� t +0 +P(AN,n +0,|D2Bη|) ds. +(2.9) + +MEAN-FIELD LIMIT +9 +For J23, we split again the domain Ω = AN,n +θ2,Bη ∪ (AN,n +θ2,Bη)c and obtain +J23 ≤C sup +i,j +∥∇Bη +ij∥κ +L∞(Rd)∥f ′′ +γ ∥κ +L∞(0,supi,j ∥Bη +ij∥L∞(Rd)) +· E +� � t∧τα +0 +N ακ +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − Bη +ij ∗ uη,j( ¯Xk +η,i) +��� +κ +IAN,n +θ2,Bη ds +� ++ C sup +i,j +∥∇Bη +ij∥κ +L∞(Rd)∥f ′′ +γ ∥κ +L∞(0,supi,j ∥Bη +ij∥L∞(Rd)) +· E +� � t∧τα +0 +N ακ +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − Bη +ij ∗ uη,j( ¯Xk +η,i) +��� +κ +I(AN,n +θ2,Bη )c ds +� +≤CN κ[β(2d+2)+α] +� t +0 +P(AN,n +θ2,Bη) ds + CN κ[β(d+2)+α−θ2]. +(2.10) +For J24, we split again the domain Ω = AN,n +θ3,∇Bη ∪ (AN,n +θ3,∇Bη)c and obtain +J24 ≤CN ακE +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − ∇Bη +ij ∗ uη,j( ¯Xk +η,i) +��� +κ +IAN,n +θ3,∇Bη ds +� ++ CN ακE +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − ∇Bη +ij ∗ uη,j( ¯Xk +η,i) +��� +κ +I(AN,n +θ3,∇Bη )c ds +� +≤CN κ[β(d+1)+α] +� t +0 +P(AN,n +θ3,∇Bη) ds + CN κ(α−θ3). +(2.11) +The term J22 can be divided into two terms. +J22 ≤CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +n +� +j=1 +���f ′′ +γ +� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +� +· 1 +N +N +� +l=1 +[Bη +ij(XN,k +η,i − XN,l +η,j ) − Bη +ij( ¯Xk +η,i − ¯Xl +η,j)] +��� +κ��� 1 +N +N +� +l=1 +∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +��� +κ +ds +� ++ CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +∥f ′′′ +γ ∥κ +L∞(0,supj ∥Bη +ij∥L∞(Rd)) +· +n +� +j=1 +��� 1 +N +N +� +l=1 +[Bη +ij(XN,k +η,i − XN,l +η,j ) − Bη +ij( ¯Xk +η,i − ¯Xl +η,j)] +��� +2κ��� 1 +N +N +� +l=1 +∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +��� +κ +ds +� +=:J221 + J222. +(2.12) +For J222, +J222 ≤C∥f ′′′ +γ ∥κ +L∞(0,supi,j ∥Bη +ij∥L∞(Rd)) sup +i,j +∥∇Bη +ij∥3κ +L∞(Rd)E +� � t∧τα +0 +N ακ +n +� +i=1 +max +1≤k≤N |XN,k +η,i − ¯Xk +η,i|2κ ds +� +≤CN κ[β(3d+4)−α] +� t +0 +E(Sκ +α(s)) ds. +(2.13) + +10 +YUE LI, LI CHEN, AND ZHIPENG ZHANG +Now we focus on J221. +J221 ≤CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +n +� +j=1 +���f ′′ +γ +� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +���� +κ +· +��� 1 +N +N +� +l=1 +|∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j)| · |XN,l +η,j − ¯Xl +η,j| +��� +κ��� 1 +N +N +� +l=1 +∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +��� +κ +ds +� ++ C∥f ′′ +γ ∥κ +L∞(0,supi,j ∥Bη +ij∥L∞(Rd)) sup +i,j +∥∇Bη +ij∥κ +L∞(Rd) sup +i,j +∥D2Bη +ij∥κ +L∞(Rd) +· E +� � t∧τα +0 +N ακ +n +� +i=1 +max +1≤k≤N |XN,k +η,i − ¯Xk +η,i|2κ ds +� +≤CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +n +� +j=1 +���f ′′ +γ +� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) +� +− f ′′ +γ (Bη +ij ∗ uη,i( ¯Xk +η,i)) +��� +κ +· ∥∇Bη +ij∥2κ +L∞(Rd) max +1≤l≤N |XN,l +η,j − ¯Xl +η,j|κ ds +� ++ CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +n +� +j=1 +|f ′′ +γ (Bη +ij ∗ uη,j( ¯Xk +η,i))|κ +· +��� 1 +N +N +� +l=1 +|∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j)| − |∇Bη +ij| ∗ uη,j( ¯Xk +η,i) +��� +κ +max +1≤l≤N |XN,l +η,j − ¯Xl +η,j|κ∥∇Bη +ij∥κ +L∞(Rd) ds +� ++ CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +n +� +j=1 +|f ′′ +γ (Bη +ij ∗ uη,j( ¯Xk +η,i))|κ��|∇Bη +ij| ∗ uη,j( ¯Xk +η,i) +��κ +· max +1≤l≤N |XN,l +η,j − ¯Xl +η,j|κ��� 1 +N +N +� +l=1 +∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − ∇Bη +ij ∗ uη,j( ¯Xk +η,i) +��� +κ +ds +� ++ CE +� +N ακ +n +� +i=1 +max +1≤k≤N +� t∧τα +0 +n +� +j=1 +|f ′′ +γ (Bη +ij ∗ uη,j( ¯Xk +η,i))|κ��|∇Bη +ij| ∗ uη,j( ¯Xk +η,i) +��κ +· max +1≤l≤N |XN,l +η,j − ¯Xl +η,j|κ���∇Bη +ij ∗ uη,j( ¯Xk +η,i) +��� +κ +ds +� ++ C∥f ′′ +γ ∥κ +L∞(0,supi,j ∥Bη +ij∥L∞(Rd))N κ[β(d+1)+β(d+2)−α] +� t +0 +E(Sκ +α(s)) ds +=:J2211 + J2212 + J2213 + C +� t +0 +E(Sκ +α(s)) ds + CN κ[β(2d+4)−α] +� t +0 +E(Sκ +α(s)) ds. +(2.14) +For J2211, we split again the domain Ω = AN,n +θ4,Bη ∪ (AN,n +θ4,Bη)c and obtain +J2211 ≤CN κβ(2d+3)E +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − Bη +ij ∗ uη,j( ¯Xk +η,i) +��� +κ +IAN,n +θ4,Bη Sκ +α(s) ds +� ++ CN κβ(2d+3)E +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − Bη +ij ∗ uη,j( ¯Xk +η,i) +��� +κ +I(AN,n +θ4,Bη )cSκ +α(s) ds +� +≤CN κβ(3d+3) +� t +0 +P(AN,n +θ4,Bη) ds + CN κ[β(2d+3)−θ4] +� t +0 +E(Sκ +α(s)) ds. +(2.15) + +MEAN-FIELD LIMIT +11 +For J2212, we split again the domain Ω = AN,n +θ5,|∇Bη| ∪ (AN,n +θ5,|∇Bη|)c and obtain +J2212 +≤CN κβ(d+1)E +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +|∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j)| − |∇Bη +ij| ∗ uη,j( ¯Xk +η,i) +��� +κ +IAN,n +θ5,|∇Bη|Sκ +α(s) ds +� ++ CN κβ(d+1)E +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +|∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j)| − |∇Bη +ij| ∗ uη,j( ¯Xk +η,i) +��� +κ +I(AN,n +θ5,|∇Bη|)cSκ +α(s) ds +� +≤CN 2κβ(d+1) +� t +0 +P(AN,n +θ5,|∇Bη|) ds + CN κ[β(d+1)−θ5] +� t +0 +E(Sκ +α(s)) ds. +(2.16) +For J2213, we take θ6 = 0 and obtain +J2213 ≤CE +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − ∇Bη +ij ∗ uη,i( ¯Xk +η,i) +��� +κ +IAN,n +0,∇Bη Sκ +α(s) ds +� ++ CE +� � t∧τα +0 +n +� +i,j=1 +max +1≤k≤N +��� 1 +N +N +� +l=1 +∇Bη +ij( ¯Xk +η,i − ¯Xl +η,j) − ∇Bη +ij ∗ uη,i( ¯Xk +η,i) +��� +κ +I(AN,n +0,∇Bη )cSκ +α(s) ds +� +≤CN κβ(d+1) +� t +0 +P(AN,n +0,∇Bη) ds + C +� t +0 +E(Sκ +α(s)) ds. +(2.17) +From (2.12)-(2.17), it holds +J22 ≤C +� +1 + N κ[β(d+1)−θ5] + N κ[β(2d+3)−θ4] + N κ[β(3d+4)−α]� � t +0 +E(Sκ +α(s)) ds ++ CN κβ(3d+3) +� t +0 +P(AN,n +θ4,Bη) ds + CN κβ(2d+2) +� t +0 +P(AN,n +θ5,|∇Bη|) ds + CN κβ(d+1) +� t +0 +P(AN,n +0,∇Bη) ds. +(2.18) +Plugging (2.9)-(2.11) and (2.18) into (2.2), and combining (2.1) and Lemma 2.1, we get +E(Sκ +α(t)) ≤C(1 + N κ[β(2d+4)−α] + N κ[β(d+3)−α] + N κ[β(d+3)−θ1] ++ N κ[β(d+1)−θ5] + N κ[β(2d+3)−θ4] + N κ[β(3d+4)−α]) +� t +0 +E(Sκ +α(s)) ds ++ CN κβ(2d+3)+2m1(θ1− 1 +2+βd)+1 + CN κβ(d+2)+2m7[− 1 +2 +β(d+2)]+1 ++ CN κβ(d+2)+2m8[− 1 +2 +β(d+2)]+1 + CN κ[β(2d+2)+α]+2m2(θ2− 1 +2+βd)+1 ++ CN κ[β(d+1)+α]+2m3[θ3− 1 +2+β(d+1)]+1 + CN κβ(3d+3)+2m4(θ4− 1 +2+βd)+1 ++ CN κβ(2d+2)+2m5[θ5− 1 +2+β(d+1)]+1 + CN κβ(d+1)+2m6[− 1 +2+β(d+1)]+1 ++ CN κ[β(d+2)+α−θ2] + CN κ(α−θ3). +(2.19) +For a given β, we choose α such that +β(3d + 4) ≤ α, +(2.20) +which implies that all the coefficients of +� t +0 +E(Sκ +α(s)) ds are bounded. + +12 +YUE LI, LI CHEN, AND ZHIPENG ZHANG +To bound all the terms above, we need the following restrictions for θ1 - θ5. +β(d + 3) ≤ θ1 < 1 +2 − βd, +(2.21) +0 < 1 +2 − β(d + 2), +(2.22) +β(d + 2) + α < θ2 < 1 +2 − βd, +(2.23) +α < θ3 < 1 +2 − β(d + 1), +(2.24) +β(2d + 3) ≤ θ4 < 1 +2 − βd, +(2.25) +β(d + 1) ≤ θ5 < 1 +2 − β(d + 1). +(2.26) +By (2.20), (2.23) and (2.24), we derive that α should satisfy +β(3d + 4) ≤ α < 1 +2 − β(2d + 2). +Therefore, β has to satisfy +0 < β < +1 +2(5d + 6). +We take κ big enough to ensure that +N κ[β(d+2)+α−θ2] ≤ N −λ, +N κ(α−θ3) ≤ N −λ. +Then we choose m1 - m8 big enough such that +CN κβ(2d+3)+2m1(θ1− 1 +2 +βd)+1 + CN κβ(d+2)+2m7[− 1 +2+β(d+2)]+1 ++ CN κβ(d+2)+2m8[− 1 +2+β(d+2)]+1 + CN κ[β(2d+2)+α]+2m2(θ2− 1 +2 +βd)+1 ++ CN κ[β(d+1)+α]+2m3[θ3− 1 +2+β(d+1)]+1 + CN κβ(3d+3)+2m4(θ4− 1 +2+βd)+1 ++ CN κβ(2d+2)+2m5[θ5− 1 +2 +β(d+1)]+1 + CN κβ(d+1)+2m6[− 1 +2 +β(d+1)]+1 +≤CN −λ. +As a consequence, we infer that +E(Sκ +α(t)) ≤ C +� t +0 +E(Sκ +α(s)) ds + CN −λ. +By means of the Gr¨onwall inequality, we deduce that +sup +0≤t≤T +P +� +n +� +i=1 +max +1≤k≤N |XN,k +η,i (t) − ¯Xk +η,i(t)| > N −α� +≤ sup +0≤t≤T +E(Sκ +α(t)) ≤ CN −λ. +For any ˜α ≤ α, we have +sup +0≤t≤T +P +� +n +� +i=1 +max +1≤k≤N |XN,k +η,i (t) − ¯Xk +η,i(t)| > N −˜α� +≤ sup +0≤t≤T +P +� +n +� +i=1 +max +1≤k≤N |XN,k +η,i (t) − ¯Xk +η,i(t)| > N −α� +≤ CN −λ. + +MEAN-FIELD LIMIT +13 +Acknowledgements: Y. Li are supported by NSFC (Grant No. 12071212). Z. Zhang is sup- +ported by NSFC (Grant No. 12101305). +References +[1] L. Chen, E.S. Daus, A. Holzinger and A. J¨ungel, Rigorous derivation of population cross-diffusion systems from +moderately interacting particle systems, J. Nonlinear Sci., 2021 (31), no. 6, Paper No. 94, 38 pp. +[2] L. Chen, V. Gvozdik, A. Holzinger and Y. Li, Rigorous derivation of the degenerate parabolic-elliptic Keller-Segel +system from a moderately interacting stochastic particle system, draft. +[3] L, Chen, E.S. Daus and A. J¨ungel, Rigorous mena-field limit and cross-diffusion, Z. Angew. Math. Phys., 2019 +70:122. +[4] X. Chen, E.S. Daus and A. J¨ungel, Global existence analysis of cross-diffusion population systems for multiple +species, Arch. Rational Mech. Anal., 2018 (227), 715-747. +[5] E.S. Daus, L. Desvillettes and H. Dietert, About the entropic structure of detailed balanced multi-species cross- +diffusion equations, J. Differ. 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Fields, 1989 (82), 565-586. +[12] K. Oelschl¨ager, Large systems of intracting particles and the porous medium equation, J. Differ. Equ., 1990 (88), +294-346. +[13] S. Serfaty. Mean field limit for coulomb-type flows. Duke Mathematical Journal, 2020, 169(15): 2887C2935. +[14] I. Seo, Scaling limit of two-component interacting Brownian motions, Ann. Prob., 2018 (46), 2038-2063. +[15] A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many- +particle systems, SIAM J. Appl. Math, 2000 (61), 183-212. +Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China +Email address: liyue2011008@163.com +Lehrstuhl f¨ur Mathematik IV, Universit¨at Mannheim, Mannheim 68131, Germany +Email address: chen@math.uni-mannheim.de +Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China +Email address: zhangzhipeng@nju.edu.cn + diff --git a/7dE1T4oBgHgl3EQfnQQ6/content/tmp_files/load_file.txt b/7dE1T4oBgHgl3EQfnQQ6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3d79e5dea50f73c4b2b0bd9406c951047f81de2 --- /dev/null +++ b/7dE1T4oBgHgl3EQfnQQ6/content/tmp_files/load_file.txt @@ -0,0 +1,674 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf,len=673 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='03306v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='PR] 9 Jan 2023 CONVERGENCE TOWARDS THE POPULATION CROSS-DIFFUSION SYSTEM FROM STOCHASTIC MANY-PARTICLE SYSTEM YUE LI, LI CHEN, AND ZHIPENG ZHANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' In this paper, we derive rigorously a general cross-diffusion system from a moderate interacting stochastic many-particle system in the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The convergence is proved in the sense of probability by introducing an intermediate particle system with a mollified interaction potential, where the mollification is of algebraic scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The main idea of the proof is to study the time evolution of a stopped process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Based on the global existence and uniform estimates of solutions to the local (and non-local) cross-diffusion equation, we are able to obtain a Gr¨onwall type estimate by using Taylor’s expansion around the limiting stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Introduction In this paper, we give a rigorous justification for the mean-field limit from a moderate interacting particle system to the population cross-diffusion system as the number of particles goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' More precisely, we present the derivation of n-species cross-diffusion system as follows \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∂tui = div(ui∇Ui) + σi∆ui + div � ui n � j=1 ∇f(aijuj) � , ui(0) = u0 i (x), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , n, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1) where σi > 0 are the constant diffusion coefficients, aij ∈ R describe the strength of cross-diffusion effects, u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , un) stands for the vector of population densities and Ui(x) = − 1 2|x|2 represent environment potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The transitions rates depend on the densities by a nonlinear term f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The aim of this paper is to rigorously derive the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1) from the following stochastic many- particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' This system describes the movements of n species of particles, with the particle numbers Ni ∈ N (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , n), according to the given law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Without loss of generality, we let N = Ni (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Let (Ω, F, (Ft≥0), P) be a complete filtered probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' We consider d-dimensional Ft-Brownian motions (W k i (t))t≥0 (k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , N, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , n) which are assumed to be independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' We assume that (ξk i ) (k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , N, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , n) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' random variables, independent of (W k i (t))t≥0, and have common probability density function u0 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' We use the notation XN,k η,i (t) to represent the k-th particle of i-th species and the dynamics of XN,k η,i (t) are governed by \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 dXN,k η,i = � − ∇Ui(XN,k η,i ) − n � j=1 ∇fγ � 1 N N � l=1 Bη ij(XN,k η,i − XN,l η,j ) �� dt + √ 2σidW k i (t), XN,k η,i (0) = ξk i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , n, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , N, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='2) 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' 35Q92, 35K45, 60J70, 82C22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Stochastic particle systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Cross-diffusion system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Mean-field limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Population dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' 1 2 YUE LI, LI CHEN, AND ZHIPENG ZHANG where fγ is an approximation of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The moderate interaction potentials Bη ij(x) = η−dBij(|x| η ) are used to approximate the Delta distribution, namely Bη ij → aijδ0 as η → 0 in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' In these representation, the functions Bij are chosen to be any smooth functions with compact supports and � Rd Bij dx = aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The problem considered in this paper dedicates to the understanding of diffusion (and cross- diffusion) effects on the microscopic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' It belongs to the research of mean-field limit for moderate interacting particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' There have been extensive studies of the mean-field limit problems in the last decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Many important contributions have been made for problems with singular interacting potentials such as the Coulomb potential in Keller-Segel systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' An extensive review of this research field is out of the scope of this paper, we refer to [7, 8, 13, 2] for more detailed summary on the results and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The convergence of moderate interacting system was introduced and proved by Oelschl¨ager in [10, 11, 12] in order to derive reaction-diffusion equations and the porous medium equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The authors [9] considered further the fluctuation of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' This idea has been used to derive chemotaxis equation from an interacting stochastic many-particle system in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The derivation of cross-diffusion type systems has only been studied in the last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' It is proved in [14] that the hydrodynamic limit of the empirical densities of two types is the solution to the Maxwell-Stefan equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The authors in [6] derived the non-local Lotka-Volterra system with cross-diffusion from particle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The Shigesada-Kawasaki-Teramoto system was obtained from a microscopic many- particle Markov process in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system from a moderate interacting stochastic particle system was given in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' There are very few results for more than two species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' In [4], the authors established the global existence of weak solutions to cross-diffusion systems for an arbitrary number of competing population species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The mean-field limit of a moderate interacting stochastic many-particle system for multiple population species is obtained in [3] with the logarithmic scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Furthermore, with the same scaling the authors in [1] derived population cross-diffusion systems of Shigesada-Kawasaki-Teramoto type from stochastic moderately interacting many-particle systems for multiple population species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' To precisely state the main results of this paper, we first give the general assumptions on f and a list of possible choices of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Let f ∈ W s+1,∞ loc (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' [0, ∞)) (s > d 2 + 2) satisfy ∥f∥C3([−M,M]) ≤ eM for any M ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' and for a properly small ε which depends on n, σmin, A, Cs, ∥u0∥Hs(Rd) such that ∥f∥Cs+1(I) ≤ ε, where I = [−2ACs∥u0∥Hs(Rd), 2ACs∥u0∥Hs(Rd)], σmin := mini=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=',n σi, A := maxi,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=',n ∥Bij∥L1(Rd) and Cs is the constant of the continuous embedding Hs(Rd) ֒→ L∞(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' It is easy to find the approximation fγ of f that satisfies fγ = f on � − 1 γ , 1 γ � , MEAN-FIELD LIMIT 3 where 1 γ ≥ 2ACs∥u0∥Hs(Rd), and ∥fγ∥C3(R) ≤ ∥f∥C3([− 1 γ , 1 γ ]) ≤ e 1 γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' A typical example for f(v) is the linear function f(v) = v, which is exactly the case handled in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' In this case, the requirement ∥f∥Cs+1(I) ≤ ε is equivalent to the smallness assumption of initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Further examples are f(v) = vm for m ∈ (N ∩ (0, s]) ∪ (s, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Or in case aij ≥ 0, possible choices might be f(v) = (v + 1)m for m ∈ R, f(v) = ln(v + 1), or f(v) = ev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' In order to prove the limit from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='2) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1), we introduce an intermediate particle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' This problem is formally viewed as a mean field limit N → ∞ in the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='2) for fixed η, γ > 0, namely \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 d ¯Xk η,i = � − ∇Ui( ¯Xk η,i) − n � j=1 ∇fγ(Bη ij ∗ uη,j( ¯Xk η,i)) � dt + √ 2σidW k i (t), ¯Xk η,i(0) = ξk i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , n, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='3) Here uη,j is the probability density function of ¯Xk η,j and satisfies the following non-local cross- diffusion system: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∂tuη,i = div(uη,i∇Ui) + σi∆uη,i + div � uη,i n � j=1 ∇fγ(Bη ij ∗ uη,j) � , uη,i(0) = u0 i (x), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='4) Similar to the proof of the wellposedness theory for non-local and local PDE problems given in [1] we obtain that, under the assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1 and u0 ∈ Hs(Rd), the systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='4) possess a unique solution uη and u respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Moreover, uη and u satisfy for any T > 0, ∥uη∥2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Hs(Rd)) + σ∗∥∇uη∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Hs(Rd)) ≤ ∥u0∥2 Hs(Rd), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='5) ∥u∥2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Hs(Rd)) + σ∗∥∇u∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Hs(Rd)) ≤ ∥u0∥2 Hs(Rd), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='6) ∥u − uη∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='L2(Rd)) + ∥∇(u − uη)∥L2((0,T)×Rd) ≤ Cη, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='7) where 0 < σ∗ < σmin and C is a positive constant which depends on T, σ∗, Cs, n, A and initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Based on these, we can also similarly obtain the wellposedness of SDE system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='2) and the McKean-Vlasov problem of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 d ˆXk i = � − ∇Ui( ˆXk i ) − n � j=1 ∇f(aijuj( ˆXk i )) � dt + √ 2σidW k i (t), ˆXk i (0) = ξk i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , n, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , N, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='8) where ui solves the limiting cross-diffusion system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1) and is the probability density function of ˆXk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Namely, when additional � Rd |x|2|u0(x)|dx < ∞, there exist unique square-integrable adapted stochastic processes with continuous paths, which are strong solutions to systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' 4 YUE LI, LI CHEN, AND ZHIPENG ZHANG Therefore, the estimate in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='7) provides directly the following estimate: there exists C > 0 which is independent of N and η such that E � n � i=1 sup 0≤t≤T max 1≤k≤N | ¯Xk η,i(t) − ˆXk i (t)|2� ≤ Cη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='9) The main result of this paper is the following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Let the assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1 hold, u0 ∈ Hs(Rd), and � Rd |x|2|u0(x)|dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Assume that T > 0, η = N −β, 1 γ = β log N, where β ∈ (0, 1 2(5d+6)), then for any arbitrary λ > 0, it holds sup 0≤t≤T P � n � i=1 max 1≤k≤N |XN,k η,i (t) − ¯Xk η,i(t)| > N −α� ≤ CN −λ, where α < 1 2 − β(2d + 2) and C is a positive constant independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The assumption 1 γ = β log N is determined by the given exponential grow of f in assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' If we only assume the algebraic grow of f, then the assumption of 1 γ can be relaxed to algebraic scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The key contribution of the main result is the moderate interacting radius η which required more delicate ideas than those have been used in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Combined with the estimate in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='9), we obtain the mean field limit result on the trajectory level and the propagation of chaos as a corollary Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Under the same assumptions as in theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='3, we have for any ˜β < β sup 0≤t≤T P � n � i=1 max 1≤k≤N |XN,k η,i (t) − ˆXk i (t)| > N − ˜β� ≤ CN −(β− ˜β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Let l ∈ N and consider an l-tuple (XN,1 η,i (t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , XN,l η,i (t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' We denote by P N,l η,i (t) its joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Then it holds that P N,l η,i (t) convergres weakly to P ⊗l i (t) as N → ∞, where Pi(t) is a measure which is absolutely continuous with respect to the Lebesgue measure and has a probability density function ui(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Compared with the convergence result obtained in [1], where the propagation of chaos has been obtained under the logarithmic moderate scaling η ∼ 1/ log N, one can clearly see that the main result of this paper gives the same propagation of chaos result under the algebraic scaling η ∼ 1/N β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' This result is obtained through the convergence in the sense of probability on the particle level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The benefit of algebraic scaling is that one can capture the singular interaction to some extend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' To overcome the difficulty originated from the singular interaction, a suitable stopped process is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Based on this, it is reduced to estimate the expectation of the stopped process with the help of Markov’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' A generalized version of Law of Large Numbers is the key point when we study the expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Another difficulty is caused by the nonlinear term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Since the regularity of f is not global, we have to find an approximate function fγ and give explicit scaling between γ and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Section 2 is dedicated to the proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' MEAN-FIELD LIMIT 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='3 We prove the convergence in probability on the particle level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Because of the singular interaction, one can not expect that under the algebraic scaling the convergence can be obtained in the expec- tation sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' The convergence in probability means that one allows that the particle trajectories are not always close, but the probability that they are not close is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Actually we can prove that the probability has a arbitrary convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' For any κ ∈ N, we define a stopping time τα, a random variable Sκ α and a set Bα τα(ω) := inf � t ∈ (0, T) : n � i=1 max 1≤k≤N |XN,k η,i (t) − ¯Xk η,i(t)| ≥ N −α� , ω ∈ Ω, Sk α(t) := N ακ n � i=1 max 1≤k≤N |XN,k η,i (t ∧ τα) − ¯Xk η,i(t ∧ τα)|κ ≤ 1, Bα(t) := {ω ∈ Ω : Sκ α(t) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' By Markov’s inequality, it holds P � n � i=1 max 1≤k≤N |XN,k η,i (t) − ¯Xk η,i(t)| ≥ N −α� ≤P � n � i=1 max 1≤k≤N |XN,k η,i (t ∧ τα) − ¯Xk η,i(t ∧ τα)| = N −α� ≤ E(Sκ α(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' We notice that the introduction of parameter κ is to increase the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Actually, the above Markov’s inequality works for arbitrary κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' In order to complete the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='3, we just need to show that for any λ > 0 and T > 0, it holds E(Sκ α(t)) ≤ CN −λ, where the letter C appeared in this section is a generic positive constant independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' To this end, we need the following Law of Large Numbers, which can be found for example in [2]: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' For ϕij ∈ L∞(Rd), i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' , n, we define for arbitrary θ ∈ (0, 1 2) AN,n θ,ϕ (s) := n� i,j=1 N � k=1 � ω ∈ Ω : ��� 1 N N � l=1 ϕij( ¯Xk η,i(s) − ¯Xl η,j(s)) − ϕij ∗ uη,j( ¯Xk η,i(s)) ��� > 1 N θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Then for any m ∈ N, it holds that P(AN,n θ,ϕ (s)) ≤ C(n, m)N 2m(θ− 1 2 )+1� sup i,j ∥ϕij∥2m L∞(Rd) + sup i,j ∥ϕij ∗ uη,j∥2m L∞((0,T)×Rd) � , ∀s ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Next we study the time evolution of the cut-offed process Sκ α(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Notice that |XN,k η,i (t ∧ τα) − ¯Xk η,i(t ∧ τα)|κ ≤C � t∧τα 0 |∇Ui(XN,k η,i ) − ∇Ui( ¯Xk η,i)|κ ds + C � t∧τα 0 ��� n � j=1 ∇fγ � 1 N N � l=1 Bη ij(XN,k η,i − XN,l η,j ) � − n � j=1 ∇fγ(Bη ij ∗ uη,j( ¯Xk η,i)) ��� κ ds =:J1 k,i(t) + J2 k,i(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' 6 YUE LI, LI CHEN, AND ZHIPENG ZHANG From the definition of Ui(x) = − |x|2 2 , we get E � N ακ n � i=1 max 1≤k≤N J1 k,i(t) � ≤C � t 0 E(Sκ α(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1) For the second term J2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' E � N ακ n � i=1 max 1≤k≤N J2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i(t) � ≤CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 ��� n � j=1 f ′ γ � 1 N N � l=1 Bη ij(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ) � 1 N N � l=1 � ∇Bη ij(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ) − ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) ���� κ ds � + CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 ��� n � j=1 � f ′ γ � 1 N N � l=1 Bη ij(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ) � − f ′ γ � 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) �� 1 N N � l=1 ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) ��� κ ds � + CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 ��� n � j=1 � f ′ γ � 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) � − f ′ γ(Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i)) � 1 N N � l=1 ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) ��� κ ds � + CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 ��� n � j=1 f ′ γ(Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i)) � 1 N N � l=1 ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) − ∇Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ���� κ ds � =:J21 + J22 + J23 + J24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='2) The term J21 can be divided into two terms: J21 ≤CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 ��� n � j=1 � f ′ γ � 1 N N � l=N Bη ij(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ) � − f ′ γ � 1 N N � l=N Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) �� 1 N N � l=1 [∇Bη ij(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ) − ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)] ��� κ ds � + CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 ��� n � j=1 f ′ γ � 1 N N � l=N Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) � 1 N N � l=1 [∇Bη ij(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ) − ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)] ��� κ ds � MEAN-FIELD LIMIT 7 =:J211 + J212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='3) The term J211 can be handled with J211 ≤ C∥f ′′ γ ∥κ L∞(0,supi,j ∥Bη ij∥L∞(Rd)) sup i,j ∥∇Bη ij∥κ L∞(Rd) sup i,j ∥D2Bη ij∥κ L∞(Rd) � t∧τα 0 E � N ακ n � i=1 max 1≤k≤N |XN,k η,i − ¯Xk η,i|2κ� ds ≤ C∥f ′′ γ ∥κ L∞(0,supi,j ∥Bη ij∥L∞(Rd)) supi,j ∥∇Bij∥κ L∞(Rd) ηκ(d+1) supi,j ∥D2Bij∥κ L∞(Rd) ηκ(d+2) N −ακ � t∧τα 0 E(Sκ α(s)) ds ≤ CN κ[β(2d+4)−α] � t 0 E(Sκ α(s)) ds, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='4) where we have used the assumption η = N −β and the fact that ∥fγ∥C3(R) ≤ ∥f∥C3([−β log N,β log N]) ≤ eβ log N ≤ N β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' With the help of ∥Bη ij ∗ uη,j∥L∞((0,T)×Rd) ≤ 2ACs∥u0∥Hs(Rd), we have ∥f ′ γ∥L∞(0,supi,j ∥Bη ij∗uη,j∥L∞((0,T )×Rd)) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' J212 ≤CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 ��� n � j=1 � f ′ γ � 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) � − f ′ γ(Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i)) � 1 N N � l=1 [∇Bη ij(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ) − ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)] ��� κ ds � + CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 ��� n � j=1 f ′ γ(Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i)) 1 N N � l=1 [∇Bη ij(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ) − ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)] ��� κ ds � ≤C∥f ′′ γ ∥κ L∞(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='supi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥Bη ij∥L∞(Rd)) supi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥D2Bij∥κ L∞(Rd) ηκ(d+2) E � � t∧τα 0 n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j=1 max 1≤k≤N ��� 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) − Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ Sκ α(s) ds � + CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 n � j=1 ��� 1 N N � l=1 D2Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ ds � + CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 n � j=1 ��� 1 N N � l=1 D2Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) ��� κ ds � + C sup i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥D3Bη ij∥κ L∞(Rd)E � � t∧τα 0 N ακ n � i=1 max 1≤k≤N |XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i|2κ ds � =:J2121 + J2122 + J2123 + CN κ[β(d+3)−α] � t 0 E(Sκ α(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='5) 8 YUE LI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' LI CHEN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' AND ZHIPENG ZHANG For J2121,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' we split the domain Ω = AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη ∪ (AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη)c and obtain J2121 ≤∥f ′′ γ ∥κ L∞(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='supi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥Bη ij∥L∞(Rd)) supi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥D2Bij∥κ L∞(Rd) ηκ(d+2) E � � t∧τα 0 n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j=1 max 1≤k≤N ��� 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) − Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ IAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη Sκ α(s) ds � + ∥f ′′ γ ∥κ L∞(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='supi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥Bη ij∥L∞(Rd)) supi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥D2Bij∥κ L∞(Rd) ηκ(d+2) E � � t∧τα 0 n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j=1 max 1≤k≤N ��� 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) − Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ I(AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη )cSκ α(s) ds � ≤N κβ(d+3)� sup i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥Bη ij∥κ L∞(Rd) + sup i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j∥κ L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='T)×Rd) � � t 0 P(AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη) ds + N κ[β(d+3)−θ1] � t∧τα 0 E(Sκ α(s)) ds ≤N κβ(2d+3) � t 0 P(AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη) ds + N κ[β(d+3)−θ1] � t 0 E(Sκ α(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='6) For J2123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' we split again the domain Ω = AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='|D2Bη| ∪ (AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='|D2Bη|)c and obtain J2123 ≤E � N ακ n � i=1 max 1≤k≤N � t∧τα 0 n � j=1 ��� 1 N N � l=1 |D2Bη ij|( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) ��� κ sup l |XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j|κ ds � ≤E � � t∧τα 0 n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j=1 max 1≤k≤N ��� 1 N N � l=1 |D2Bη ij|( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) − |D2Bη ij| ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ IAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='|D2Bη|Sκ α(s) ds � + E � � t∧τα 0 n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j=1 max 1≤k≤N ��� 1 N N � l=1 |D2Bη ij|( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) − |D2Bη ij| ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ I(AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='|D2Bη|)cSκ α(s) ds � + sup i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥|D2Bη ij| ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j∥κ L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='T)×Rd) � t∧τα 0 E(Sκ α(s)) ds ≤CN κβ(d+2) � t 0 P(AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='|D2Bη|) ds + C � t 0 E(Sκ α(s)) ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='7) where we have used the result that ∥|D2Bη ij| ∗ uη,j∥L∞((0,T)×Rd) ≤ C, more details can be found in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Similarly, we can derive that J2122 ≤ CN κβ(d+2) � t 0 P(AN,n 0,D2Bη) ds + C � t 0 E(Sκ α(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='8) Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='3)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='8), we have J21 ≤C � 1 + N κ[β(2d+4)−α] + N κ[β(d+3)−α] + N κ[β(d+3)−θ1]� � t 0 E(Sκ α(s)) ds + CN κβ(2d+3) � t 0 P(AN,n θ1,Bη) ds + CN κβ(d+2) � t 0 P(AN,n 0,D2Bη) ds + CN κβ(d+2) � t 0 P(AN,n 0,|D2Bη|) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='9) MEAN-FIELD LIMIT 9 For J23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' we split again the domain Ω = AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη ∪ (AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη)c and obtain J23 ≤C sup i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥∇Bη ij∥κ L∞(Rd)∥f ′′ γ ∥κ L∞(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='supi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥Bη ij∥L∞(Rd)) E � � t∧τα 0 N ακ n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j=1 max 1≤k≤N ��� 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) − Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ IAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη ds � + C sup i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥∇Bη ij∥κ L∞(Rd)∥f ′′ γ ∥κ L∞(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='supi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥Bη ij∥L∞(Rd)) E � � t∧τα 0 N ακ n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j=1 max 1≤k≤N ��� 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) − Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ I(AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη )c ds � ≤CN κ[β(2d+2)+α] � t 0 P(AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη) ds + CN κ[β(d+2)+α−θ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='10) For J24, we split again the domain Ω = AN,n θ3,∇Bη ∪ (AN,n θ3,∇Bη)c and obtain J24 ≤CN ακE � � t∧τα 0 n � i,j=1 max 1≤k≤N ��� 1 N N � l=1 ∇Bη ij( ¯Xk η,i − ¯Xl η,j) − ∇Bη ij ∗ uη,j( ¯Xk η,i) ��� κ IAN,n θ3,∇Bη ds � + CN ακE � � t∧τα 0 n � i,j=1 max 1≤k≤N ��� 1 N N � l=1 ∇Bη ij( ¯Xk η,i − ¯Xl η,j) − ∇Bη ij ∗ uη,j( ¯Xk η,i) ��� κ I(AN,n θ3,∇Bη )c ds � ≤CN κ[β(d+1)+α] � t 0 P(AN,n θ3,∇Bη) ds + CN κ(α−θ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='11) The term J22 can be divided into two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' J22 ≤CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 n � j=1 ���f ′′ γ � 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) � 1 N N � l=1 [Bη ij(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ) − Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)] ��� κ��� 1 N N � l=1 ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) ��� κ ds � + CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 ∥f ′′′ γ ∥κ L∞(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='supj ∥Bη ij∥L∞(Rd)) n � j=1 ��� 1 N N � l=1 [Bη ij(XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ) − Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)] ��� 2κ��� 1 N N � l=1 ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) ��� κ ds � =:J221 + J222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='12) For J222, J222 ≤C∥f ′′′ γ ∥κ L∞(0,supi,j ∥Bη ij∥L∞(Rd)) sup i,j ∥∇Bη ij∥3κ L∞(Rd)E � � t∧τα 0 N ακ n � i=1 max 1≤k≤N |XN,k η,i − ¯Xk η,i|2κ ds � ≤CN κ[β(3d+4)−α] � t 0 E(Sκ α(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='13) 10 YUE LI, LI CHEN, AND ZHIPENG ZHANG Now we focus on J221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' J221 ≤CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 n � j=1 ���f ′′ γ � 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) ���� κ ��� 1 N N � l=1 |∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)| · |XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j| ��� κ��� 1 N N � l=1 ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) ��� κ ds � + C∥f ′′ γ ∥κ L∞(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='supi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥Bη ij∥L∞(Rd)) sup i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥∇Bη ij∥κ L∞(Rd) sup i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥D2Bη ij∥κ L∞(Rd) E � � t∧τα 0 N ακ n � i=1 max 1≤k≤N |XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='k η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i|2κ ds � ≤CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 n � j=1 ���f ′′ γ � 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) � − f ′′ γ (Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i)) ��� κ ∥∇Bη ij∥2κ L∞(Rd) max 1≤l≤N |XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j|κ ds � + CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 n � j=1 |f ′′ γ (Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i))|κ ��� 1 N N � l=1 |∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)| − |∇Bη ij| ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ max 1≤l≤N |XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j|κ∥∇Bη ij∥κ L∞(Rd) ds � + CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 n � j=1 |f ′′ γ (Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i))|κ��|∇Bη ij| ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��κ max 1≤l≤N |XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j|κ��� 1 N N � l=1 ∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) − ∇Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ ds � + CE � N ακ n � i=1 max 1≤k≤N � t∧τα 0 n � j=1 |f ′′ γ (Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i))|κ��|∇Bη ij| ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��κ max 1≤l≤N |XN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='l η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j|κ���∇Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ ds � + C∥f ′′ γ ∥κ L∞(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='supi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j ∥Bη ij∥L∞(Rd))N κ[β(d+1)+β(d+2)−α] � t 0 E(Sκ α(s)) ds =:J2211 + J2212 + J2213 + C � t 0 E(Sκ α(s)) ds + CN κ[β(2d+4)−α] � t 0 E(Sκ α(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='14) For J2211,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' we split again the domain Ω = AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη ∪ (AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη)c and obtain J2211 ≤CN κβ(2d+3)E � � t∧τα 0 n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j=1 max 1≤k≤N ��� 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) − Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ IAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη Sκ α(s) ds � + CN κβ(2d+3)E � � t∧τα 0 n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j=1 max 1≤k≤N ��� 1 N N � l=1 Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j) − Bη ij ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ I(AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη )cSκ α(s) ds � ≤CN κβ(3d+3) � t 0 P(AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='Bη) ds + CN κ[β(2d+3)−θ4] � t 0 E(Sκ α(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='15) MEAN-FIELD LIMIT 11 For J2212,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' we split again the domain Ω = AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='|∇Bη| ∪ (AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='|∇Bη|)c and obtain J2212 ≤CN κβ(d+1)E � � t∧τα 0 n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j=1 max 1≤k≤N ��� 1 N N � l=1 |∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)| − |∇Bη ij| ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ IAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='|∇Bη|Sκ α(s) ds � + CN κβ(d+1)E � � t∧τα 0 n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j=1 max 1≤k≤N ��� 1 N N � l=1 |∇Bη ij( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i − ¯Xl η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j)| − |∇Bη ij| ∗ uη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='j( ¯Xk η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='i) ��� κ I(AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='|∇Bη|)cSκ α(s) ds � ≤CN 2κβ(d+1) � t 0 P(AN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='n θ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='|∇Bη|) ds + CN κ[β(d+1)−θ5] � t 0 E(Sκ α(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='16) For J2213, we take θ6 = 0 and obtain J2213 ≤CE � � t∧τα 0 n � i,j=1 max 1≤k≤N ��� 1 N N � l=1 ∇Bη ij( ¯Xk η,i − ¯Xl η,j) − ∇Bη ij ∗ uη,i( ¯Xk η,i) ��� κ IAN,n 0,∇Bη Sκ α(s) ds � + CE � � t∧τα 0 n � i,j=1 max 1≤k≤N ��� 1 N N � l=1 ∇Bη ij( ¯Xk η,i − ¯Xl η,j) − ∇Bη ij ∗ uη,i( ¯Xk η,i) ��� κ I(AN,n 0,∇Bη )cSκ α(s) ds � ≤CN κβ(d+1) � t 0 P(AN,n 0,∇Bη) ds + C � t 0 E(Sκ α(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='17) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='12)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='17), it holds J22 ≤C � 1 + N κ[β(d+1)−θ5] + N κ[β(2d+3)−θ4] + N κ[β(3d+4)−α]� � t 0 E(Sκ α(s)) ds + CN κβ(3d+3) � t 0 P(AN,n θ4,Bη) ds + CN κβ(2d+2) � t 0 P(AN,n θ5,|∇Bη|) ds + CN κβ(d+1) � t 0 P(AN,n 0,∇Bη) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='18) Plugging (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='9)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='18) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='2), and combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='1, we get E(Sκ α(t)) ≤C(1 + N κ[β(2d+4)−α] + N κ[β(d+3)−α] + N κ[β(d+3)−θ1] + N κ[β(d+1)−θ5] + N κ[β(2d+3)−θ4] + N κ[β(3d+4)−α]) � t 0 E(Sκ α(s)) ds + CN κβ(2d+3)+2m1(θ1− 1 2+βd)+1 + CN κβ(d+2)+2m7[− 1 2 +β(d+2)]+1 + CN κβ(d+2)+2m8[− 1 2 +β(d+2)]+1 + CN κ[β(2d+2)+α]+2m2(θ2− 1 2+βd)+1 + CN κ[β(d+1)+α]+2m3[θ3− 1 2+β(d+1)]+1 + CN κβ(3d+3)+2m4(θ4− 1 2+βd)+1 + CN κβ(2d+2)+2m5[θ5− 1 2+β(d+1)]+1 + CN κβ(d+1)+2m6[− 1 2+β(d+1)]+1 + CN κ[β(d+2)+α−θ2] + CN κ(α−θ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='19) For a given β, we choose α such that β(3d + 4) ≤ α, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='20) which implies that all the coefficients of � t 0 E(Sκ α(s)) ds are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' 12 YUE LI, LI CHEN, AND ZHIPENG ZHANG To bound all the terms above, we need the following restrictions for θ1 - θ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' β(d + 3) ≤ θ1 < 1 2 − βd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='21) 0 < 1 2 − β(d + 2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='22) β(d + 2) + α < θ2 < 1 2 − βd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='23) α < θ3 < 1 2 − β(d + 1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='24) β(2d + 3) ≤ θ4 < 1 2 − βd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='25) β(d + 1) ≤ θ5 < 1 2 − β(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='26) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='20), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='23) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='24), we derive that α should satisfy β(3d + 4) ≤ α < 1 2 − β(2d + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Therefore, β has to satisfy 0 < β < 1 2(5d + 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' We take κ big enough to ensure that N κ[β(d+2)+α−θ2] ≤ N −λ, N κ(α−θ3) ≤ N −λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Then we choose m1 - m8 big enough such that CN κβ(2d+3)+2m1(θ1− 1 2 +βd)+1 + CN κβ(d+2)+2m7[− 1 2+β(d+2)]+1 + CN κβ(d+2)+2m8[− 1 2+β(d+2)]+1 + CN κ[β(2d+2)+α]+2m2(θ2− 1 2 +βd)+1 + CN κ[β(d+1)+α]+2m3[θ3− 1 2+β(d+1)]+1 + CN κβ(3d+3)+2m4(θ4− 1 2+βd)+1 + CN κβ(2d+2)+2m5[θ5− 1 2 +β(d+1)]+1 + CN κβ(d+1)+2m6[− 1 2 +β(d+1)]+1 ≤CN −λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' As a consequence, we infer that E(Sκ α(t)) ≤ C � t 0 E(Sκ α(s)) ds + CN −λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' By means of the Gr¨onwall inequality, we deduce that sup 0≤t≤T P � n � i=1 max 1≤k≤N |XN,k η,i (t) − ¯Xk η,i(t)| > N −α� ≤ sup 0≤t≤T E(Sκ α(t)) ≤ CN −λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' For any ˜α ≤ α, we have sup 0≤t≤T P � n � i=1 max 1≤k≤N |XN,k η,i (t) − ¯Xk η,i(t)| > N −˜α� ≤ sup 0≤t≤T P � n � i=1 max 1≤k≤N |XN,k η,i (t) − ¯Xk η,i(t)| > N −α� ≤ CN −λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' MEAN-FIELD LIMIT 13 Acknowledgements: Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Li are supported by NSFC (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' 12071212).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Zhang is sup- ported by NSFC (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' 12101305).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' Chen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' 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+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content=' China Email address: zhangzhipeng@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE1T4oBgHgl3EQfnQQ6/content/2301.03306v1.pdf'} diff --git a/8dFST4oBgHgl3EQfaDjo/content/tmp_files/2301.13794v1.pdf.txt b/8dFST4oBgHgl3EQfaDjo/content/tmp_files/2301.13794v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c89403e4c0f22af097db2422441bd710b3e7e1be --- /dev/null +++ b/8dFST4oBgHgl3EQfaDjo/content/tmp_files/2301.13794v1.pdf.txt @@ -0,0 +1,1426 @@ +arXiv:2301.13794v1 [econ.TH] 31 Jan 2023 +Auctions with Tokens∗ +Andrea Canidio † +First version: September 30, 2021. This version: February 1, 2023. Please check +here for the latest version. +Abstract +I study mechanism design with blockchain-based tokens, that is, tokens that can be used +within a mechanism but can also be saved and traded outside of the mechanism. I do so by +considering a repeated, private-value auction, in which the auctioneer accepts payments in +a blockchain-based token he creates and initially owns. I show that the present-discounted +value of the expected revenues is the same as in a standard auction with dollars, but these +revenues accrue earlier and are less variable. I then introduce non-contractible effort and +the possibility of misappropriating revenues. +I compare the auction with tokens to an +auction with dollars in which the auctioneer can also issue financial securities. An auction +with tokens is preferred when there are sufficiently severe contracting frictions, while the +opposite is true when contracting frictions are low. +JEL classification: D44, E42, L86 +Keywords: Mechanism design, Auctions, Blockchain, Cryptocurrencies, Tokens, Private +Money +1 +Introduction +Blockchain protocols resemble the mechanisms studied in mechanism design. Like in +these mechanisms, each blockchain protocol needs to generate incentive-compatible +∗I’m grateful to John Asker, Rainer Böhme, Sylvain Chassang, Lin William Cong, Andreas +Park and conference participants at the Crypto Asset Lab conference 2021, CSH Workshop on +Decentralized Finance 2022, ASSA 2023, seminars at CREST, CY University (Thema) for their +insightful comments and suggestions. +†IMT school of advanced studies, Lucca, Italy; andrea.canidio@imtlucca.it +1 + +1 Introduction +2 +behavior, so to achieve a given aggregate outcome. Furthermore, each blockchain +protocol is associated with a specific blockchain-based token, which is necessary +to use the protocol, usually as its internal currency.1 +The use of tokens within +blockchain protocols is, therefore, similar to the use of “virtual money” in several +mechanisms. For example, several business schools allocate students to MBA classes +by distributing “points” to students, who then use them to bid for classes.2 Simi- +larly, the organization Feeding America distributes food to various local food banks +via auctions in which a virtual currency is used.3 Unlike these virtual currencies, +however, the tokens associated with blockchain protocols exist also outside of their +respective protocol: they can be held without being used and can be exchanged on +financial markets. They are, therefore, a new financial instrument, which opens sev- +eral issues not usually considered in mechanisms design: will some of these tokens +be held and not used in the mechanism? And how will this affect the mechanism +itself? And the revenues earned by the designer? +Studying blockchain protocols as mechanisms, therefore, required a theory of +mechanism design that incorporates blockchain-based tokens. This paper makes the +first step in this direction by considering a specific mechanism: an auction. Auc- +tions are the most widely studied and better-understood mechanism and therefore +constitute an ideal starting point in this research agenda. They are also effective at +allocating a given good to the bidder who values it the most, something that, as we +will see, remains true with the introduction of tokens. The role of tokens within the +mechanism is, therefore, limited, which allows me to focus on the most novel aspect +of the model: the fact that these tokens can be held and can be traded outside of +the mechanism. +I consider a finite sequence of private-value auctions in which multiple objects +1 The fact that the token is the internal currency is very explicit in protocols creating decen- +tralized marketplaces, for example, for buying and selling computer storage space (see Filecoin, +Storij, Sia) or CPU cycles (see the Golem network). It is also the case in decentralized computing +platforms such as Ethereum, in which users pay miners/validators for executing smart contracts. It +also extends to cryptocurrencies such as Bitcoin, in which users pay miners to process transactions +in Bitcoins. +2 See Hylland and Zeckhauser (1979), Budish (2011), Budish et al. (2017), He et al. (2018). +3 See Prendergast (2017) and Prendergast (forthcoming). Note that, in the existing literature +studying the use of virtual currencies within mechanisms, the objective of the auctioneer is not to +maximize revenues. + +1 Introduction +3 +are sold (one per period).4 In every period, risk-neutral bidders draw their valuation +for the object sold from an i.i.d. distribution. Then they submit bids. Given the +profile of bids, the auction format determines the winning bidder and the payment +of each player. The good is then consumed within the period. Each auction is, +therefore, a simple static auction, repeated multiple times (that is, there is no con- +nection between auctions in different periods). As a part of the auction design, the +auctioneer can decide to accept payments in dollars or in a blockchain-based token +that he creates and initially owns. If he chooses the latter option, the auctioneer +also commits to a monetary policy: a set of rules determining how the stock of +tokens evolves. The auctioneer earns revenues by selling newly created tokens to +bidders and re-selling tokens he received as payment. Tokens can be held without +being used for bidding and can be traded on a financial market where their value is +determined as an equilibrium outcome.5 +I show that in the auction with tokens when the realized valuations in a given +period are low (relative to the future expected valuations), bidders might purchase +tokens for speculation, not bidding. The reason is that when valuations are low, the +demand for tokens for bidding is low, creating an arbitrage opportunity for bidders: +they may purchase tokens, not use them for bidding in that period, and sell them +(or use them) in future periods. The speculative demand for tokens increases the +price for tokens in a given period and, as a consequence, drives the auctioneer’s +expected revenues for that period above those of the auction with dollars. At the +same time, today’s speculators will compete with the auctioneer on tomorrow’s +market for tokens, pushing the auctioneer’s future expected revenues below those of +the auction with dollars. Also, the speculative demand in a given period depends +on the expected future valuations. Hence, speculation transforms future uncertain +revenues into present certain revenues +4 The companion short paper Canidio (forthcoming) considers the case of an infinitely repeated +auction. The main issue there is the emergence of financial bubbles on tokens. +5 Note that creating new tokens and trading them on a financial market is quite easy to do. +There are several simple tutorials explaining how to create blockchain-based tokens (I invite the +reader to search “how to create an ERC-20 Token”, where ERC-20 is the simplest type of token on +the Ethereum blockchain). Also, after the creation of a new token, anyone can then use a protocol +such as Uniswap to create a decentralized financial market for exchanging the new token against, +for example, a stablecoin (i.e., a blockchain-based token with constant value relative to the dollar). + +1 Introduction +4 +I show that there is a form of revenue equivalence: the present-discounted value +of the expected revenues is the same in all auction formats (with or without tokens). +However, in each specific period, the revenues in the auction with tokens may be +different from those in the auction with dollars. More precisely, revenues accrue +earlier and are less variable in the auction with tokens than in the auction with +dollars. How exactly depends on the specific monetary policy announced by the +auctioneer. +A particularly relevant case is a policy in which all tokens used to +pay the auctioneer are then destroyed. In this case, all revenues are earned in the +first period, when the auctioneer sells the initial stock of tokens. Furthermore, the +present-discounted value of the expected revenues from period-2 onward is earned +with probability 1.6 +Hence, by designing an appropriate auction with tokens, the auctioneer can fully +front-load his revenues and eliminate (almost) all risk. +Quite immediately, it is +possible to achieve the same outcome by holding an auction with dollars in which +the auctioneer also sells equity (i.e., transferring to investors his future cash flow). +The first result is, therefore, an equivalence result: absent costs of issuing tokens or +equity, the optimal auction with tokens is equivalent to issuing equity. +I then extend the model by introducing two frictions (i) costly non-contractible +effort and (ii) revenue misappropriation, i.e., the possibility that the auctioneer can +hide revenues by “running away with the till” (at a cost). Intuitively, when issuing +tokens, the auctioneer commits to deliver the object in the future. This commitment +may be imperfect whenever the auctioneer can shirk, exert low effort, and provide +an object of lower value to the token holders. Alternatively, the auctioneer can hold +an auction with dollars while simultaneously pledging future cash flows to investors +via a traditional financial instrument. The critical observation is that everything +that limits the auctioneer’s ability to commit the object also limits his ability to +commit the revenues generated by the sale of the object. But the possibility of +revenue misappropriation only affects traditional financial instruments because it +reduces the auctioneer’s ability to commit revenues. +6 The reason is that, in the model, the first sale of tokens happens after the period-1 valuations +are drawn, which therefore is the only risk faced by the auctioneer. There is a straightforward +extension of the model in which the auctioneer can also sell tokens before period-1 valuations are +drawn, in which case the auctioneer can eliminate all risks. + +1 Introduction +5 +Hence, whether to issue tokens or a traditional financial instrument hinges on a +trade-off. On the one hand, the possibility of misappropriating revenues is a concern +only when issuing a traditional financial instrument. On the other hand, depend- +ing on the contracting environment, a traditional financial instrument may specify +contingent payments that tokens cannot replicate. I show that if the cost of hiding +revenue is sufficiently low, tokens are the preferred financial instrument because the +possibility of revenue misappropriation is absent. If instead, running away is suffi- +ciently costly, a traditional financial instrument is the preferred means of financing. +Note that the cost of revenue misappropriation proxies contract incompleteness be- +cause it determines the set of incentive-compatible financial instruments. Hence, +the result is that if the contracting space is limited, issuing tokens is preferred, +while when the contracting space is sufficiently large, then a traditional financial +instrument is preferred. +This paper makes several contributions. First, it shows that destroying (or burn- +ing) tokens may be the optimal monetary policy. Burning has been widely discussed +within the Ethereum community in relation to the implementation of EIP-1559, a +new way to calculate transaction fees according to which a fraction of the fees paid +by users are burned rather than transferred to miners/validators.7 This paper also +contributed to the discussion on whether tokens are securities. The tokens in the +model provide access to a service and are, therefore, “utility tokens.” The logic be- +hind their pricing and valuation is different from that of equity (see, in particular, +Prat et al., 2019). Because of that, some argue that utility tokens are not securi- +ties. However, the benchmark model (i.e., the one without frictions) shows that +in equilibrium, the value of these tokens is identical to that of equity. Finally, this +paper also provides a rationale for using tokens over equity. When the contracting +space is sufficiently restricted, then tokens are preferred to equity because revenue +misappropriation is not an issue with tokens, but it is with equity. +7 See the original proposal here https://eips.ethereum.org/EIPS/eip-1559. +For an economic +analysis of EIP-1559, see Roughgarden (2020). The original motivation for burning is to prevent +miners/validators from manipulating how fees are calculated. There is, however, a heated debate +on whether tokens should be burned rather than automatically forwarded to a dedicated fund. + +1 Introduction +6 +Literature review +The literature studying the use of tokens within mechanisms includes papers in +which the mechanism encompasses the entire economy. +The goal is these pa- +pers is to establish whether certain centralized mechanisms can be implemented +in a decentralized way using tokens (see, for example, Ostroy and Starr, 1974, +Kocherlakota, 1998). Similarly, some of the early papers in monetary theory con- +sidered general-equilibrium models in which there is at least an equilibrium with +money (see Samuelson, 1958, and Townsend, 1980). Because the equilibrium with +money is Pareto superior to that without money, again, we can think of money +as allowing the decentralized implementation of some (usually constrained) optimal +allocation. Also related are models in which money has value (also in finite time) +because of exogenous reasons. For example, in Starr (1974), a government creates +money and establishes that taxes need to be paid using money. Likewise, in the +cash-in-advance model of Clower (1967), fiat money needs to be acquired one pe- +riod in advance. Note that in these models, the optimal monetary policy is typically +time inconsistent, and hence the issue of commitment to a monetary policy arises +(see, in particular, Lucas Jr and Stokey, 1983). +The advent of blockchain and blockchain-based tokens provided a new impulse +to the above literature (for an overview, see Townsend, 2020). The reason is that +blockchain-based smart contracts can be used to generate commitment, for exam- +ple, to perform payments based on contingencies. Several authors have therefore +studied how blockchain-based smart contracts can be used to implement various +types of mechanisms (see, Holden and Malani, 2019, Gans, 2019, Lee et al., 2021, +Brzustowski et al., 2021, Townsend and Zhang, forthcoming).8 With this respect, +note that, in the model presented here, the auction itself could be a traditional, +centralized auction or a decentralized one (via a smart contract). Hence, this paper +does not contribute to the study of how auctions (or other mechanisms) can be +implemented using blockchain. However, the issue of implementation is relevant for +the equilibrium of the game (of which the auction is a part). The reason is that, +8 There is also a small but growing literature using insights from mechanism to propose im- +provements to blockchain protocols. See, for example, Gans and Holden (2022a), Gans and Holden +(2022b), Capponi and Jia (2021), Canidio and Danos (2022). + +1 Introduction +7 +in equilibrium, the auction with tokens is equivalent to an auction with dollars in +which the auctioneer also issues equity. +Several papers study theoretically firms’ incentives to issue blockchain-based +tokens, which can represent a pre-sale of a given unit of future output, the only cur- +rency that the firm will accept in the future, or a claim on future revenues or profits. +Some of these papers showed that, in the presence of network externalities, selling +tokens helps avoid coordination failures (Sockin and Xiong, 2018, Cong et al., 2021, +Bakos and Halaburda, 2018, and Li and Mann, 2018). Other papers focused on the +sale of tokens as an innovative way to raise capital and finance the development of a +product or a platform (Catalini and Gans, 2018, Malinova and Park, 2018, Canidio, +2018, Bakos and Halaburda, 2019, Goldstein et al., 2019, Cong et al., 2020, Canidio, +2020, Gryglewicz et al., 2021, Garratt and van Oordt, 2021, Chod and Lyandres, +2021). In the model considered here, the auctioneer has no financing need, and +there are no network externalities. Hence, tokens are sold purely to earn a profit. +Nonetheless, there is a connection with the above literature because by issuing to- +kens, the auctioneer can manipulate the time profile and riskiness of his revenues. +Both these elements are important in determining the incentives to invest and create +new ventures. +I will frequently refer to two important results in auction theory. The first is the +revenue equivalence theorem, which states:9 +Assume each of a given number of risk-neutral potential buyers of an +object has a privately-known signal independently drawn from a common, +strictly-increasing, atomless distribution. Then any auction mechanism +in which (i) the object always goes to the buyer with the highest signal, +and (ii) any bidder with the lowest-feasible signal expects zero surplus, +yields the same expected revenue (and results in each bidder making the +same expected payment as a function of her signal). +For our purposes, the above statement implies that all common auction formats +(i.e., first-price, second-price, all-pay, ...) generate the same expected payment from +9 Vickrey (1961) developed some special case of the revenue equivalence theorem. The statement +presented here is taken from Klemperer (1999), and summarizes results in Myerson (1981) and +Riley and Samuelson (1981). For a more general formulation, see Milgrom and Segal (2002). + +2 The model +8 +bidders and hence the same expected revenues for the auctioneer. The second result +is the design of optimal auctions (see, again, Myerson, 1981, Bulow and Roberts, +1989, Bulow and Klemperer, 1996 and Klemperer, 1999). In particular, in the model +I will assume that the distribution of valuations is such that all common auction +formats with a reservation price of zero maximize the auctioneer’s expected revenues. +2 +The model +I consider a single-object private-value auction repeated T ≥ 1 times. There are +n ≥ 2 ex-ante identical bidders and an auctioneer. Bidders are risk-neutral and cash +abundant in the sense that their cash constraint is never binding. The auctioneer’s +per-period utility function is U(), assumed concave. There is a common discount +factor β ∈ (0, 1). Bidders and the auctioneer can hold a risk-free asset yielding a +per-period gross return of R ≥ 1. For ease of derivations, I assume that R = 1 +β.10 +The auctioneer’s initial assets are w1 ≥ 0. +For the moment, I assume that the +auctioneer can only save by investing in the risk-free asset, and hence wt ≥ 0 for all +t ≤ T, where wt are the asset owned by the auctioneer at the beginning of a given +period t. Section 3.3 discusses what happens when the auctioneer can also issue +equity. +In period 0, the auctioneer decides whether to accept payments in fiat currency +(for simplicity, dollars) or in tokens. +When the auction format requires the use +of tokens, the auctioneer creates an initial stock of tokens M1 and announces a +monetary policy, that is, how the stock of tokens will evolve (see below). Then, +in every period t ∈ {1, ..., T}, the auctioneer sells a single object according to the +auction format specified initially. +At the beginning of each period t ≥ 1, each bidder draws a valuation vi,t > +0 from a continuous and atomless distribution with c.d.f F(v), p.d.f. +f(v) and +support [v, v]. Each vi,t is bidder i’s private information, but the distribution F(v) is +common knowledge. The auction is in private value so that each bidder’s valuation is +independent of the other bidders’ valuations. To avoid uninteresting complications, +10 Hence, R is the steady-state rate of return of the Ramsey-Cass-Koopmans growth model (with +no population growth or exogenous productivity growth). This assumption is not essential for the +results but simplifies the derivations. + +2 The model +9 +I assume that vf(v) ≥ 1 − F(v) for all v ∈ [v, v].11 Each object sold has zero value +to the auctioneer. +If the auctioneer uses dollars, the sequence of events in each period is stan- +dard: after drawing the valuations, each bidder sends a message mi,t ∈ R+ to the +auctioneer, interpreted as his bid. As a function of the messages received and the +auction format initially announced, the auctioneer determines who is the winner +and payments bi,t ≤ mi,t for each bidder. Each bidder then pays bi,t dollars to the +auctioneer, and the winner receives and consumes the object. The period-t payoff of +the winning bidder is vi,t − bi,t; the period-t payoffs of all other bidders is −bi,t. The +auctioneer’s period-t revenues are �n +i bi,t. At this point, the auctioneer invests wt+1 +R +in the risk-free asset (which become wt+1 in the following period), and consumes +wt + �n +i bi,t − wt+1 +R . +If the auctioneer instead uses tokens, then the timeline of each period t ∈ +{1, ..., T} is the following: +• Again, at the start of a period, each bidder draws a valuation vi,t from the +distribution F(v), and then sends a message mi,t ∈ R+ to the auctioneer, +interpreted as his bid in dollars. Note that, at this point, both the auctioneer +and bidders may own tokens that they accumulated from previous periods. +Call At ≥ 0 the tokens owned by the auctioneer at the beginning of the +period, and ai,t ≥ 0 the tokens owned by bidder i. By assumption, A1 = M1 +and ai,1 = 0 for all i ≤ n. +• As a function of the messages received and the auction format initially an- +nounced, the auctioneer determines who is the winner and payments bi,t ≤ mi,t +for each bidder (implicitly a function of all messages received). This payment +is expressed in dollars, but needs to be settled using tokens.12 +• A frictionless, anonymous financial market for tokens opens, in which both +the auctioneer and bidders participate. All market participants are price tak- +11 As we will see, under this assumption, the revenue-maximizing reservation price in the auc- +tion with dollars is zero, and calculating the revenues from the optimal auction with dollars is +straightforward. +12 The fact that bids and payments are expressed in dollars is for convenience, as it allows me to +use the same notation as in the auction with dollars. The results are identical whenever bids are +expressed in tokens. + +2 The model +10 +ers. Call pt the equilibrium price for tokens; call qi,t the equilibrium demand +for tokens of bidder i; call Qt the equilibrium demand for tokens of the auc- +tioneer. Both qi,t and Qt could be positive or negative: if negative, the bid- +der/auctioneer is a net seller of tokens in equilibrium; if positive, he is a net +buyer. Furthermore, feasibility implies At + � +i ai,t = Qt + � +i qi,t. Also, be- +cause tokens will be used to pay the auctioneer (see the next point), it must +be that qi,t ≥ bi,t +pt − ai,t. +• Then, each bidder sends bi,t +pt tokens to the auctioneer. The winner receives the +object and consumes it. At this point, each bidder owns ai,t +qi,t − bi,t +pt tokens, +and the auctioneer owns At + Qt + � +i +bi,t +pt tokens. +• The winning bidder enjoys a per-period payoff equal to the value of the object +minus the expenditure in tokens vi −pt ·qi,t. Similarly, the losing bidders enjoy +per-period payoffs equal to −ptqi,t, and the auctioneer’s revenues are −ptQt. +Again, the auctioneer chooses wt+1 and then consumes wt − ptQt + wt+1 +R . +• The stock of tokens changes according to the monetary policy announced by +the auctioneer. Here I restrict my attention to two time-varying monetary- +policy parameters: a uniform increase (or decrease) of all tokens by the same +factor τt ≥ −1, and an increase (or decrease) of only the tokens used for +bidding by a time-varying factor σt ≥ −1. As a result, at the beginning of the +subsequent period, each bidder i owns ai,t+1 = (1 + τt)(ai,t + qi,t − bi,t +pt ), while +the auctioneer owns At+1 = (1+τt)(At+Qt +(1+σt) � +i +bi,t +pt ). Hence, the total +stock of tokens at the beginning of period t + 1 is Mt+1 ≡ At+1 + �n +i ai,t+1. +The monetary policy considered here may sound far-fetched, but it is very simple +to implement with blockchain-based tokens. In particular, the fact that the tokens +used for bidding may grow at a different rate than the other tokens is inspired by +staking. In staking, those who “lock” some tokens (or, more in general, do not use +them) are rewarded with additional tokens. Here the staking reward is positive if +σt < 0, in which case those who do not use the tokens receive an additional reward +relative to those who use them. +The above auction with tokens is as close as possible to a traditional auction +with dollars: bids and payments are expressed in dollars but need to be settled + +3 Equilibrium +11 +using tokens. However, other assumptions are possible. For example, bidders may +be required to bid by submitting tokens, which could then be partially returned to +the bidders after the winner is determined.13 Also, here I consider only two possible +monetary policies, but many more are possible. The bottom line is that the above is +the least complex auction with tokens and not the most general auction with tokens. +3 +Equilibrium +I first consider the auction with dollars and then the auction with tokens. +3.1 +Auction with dollars +When the auction uses dollars, all the standard results from auction theory apply.14 +Quite immediately, in every period, the revenue equivalence theorem holds: all +standard auction formats generate the same expected revenues. +Also, given our +assumption on the distribution of valuations, revenues are maximized when the +reservation price is zero. By considering a second-price auction, in every period, +expected revenues are +k ≡ E[vMax−1,t]. +where vMax,t ≡ maxi{vi,t} is the realized highest valuation in period t, and vMax−1,t ≡ +maxi̸=Max{vi,t} is the realized second-highest valuation in period t. Also, each bid- +der’s expected payoff from the auction is +g ≡ E[max{vi − vMax−1 +t +, 0}] +13 In this case, bidders need to purchase tokens before bidding, which means that the equilibrium +price of tokens may reveal some information relative to the realized distribution of valuations. +Hence, the equilibrium on the market for tokens should be a rational expectation equilibrium, +which opens several additional complications, including the existence of the equilibrium. +14 See, for example, Klemperer (1999), in particular Section 4 (for the revenue equivalence theo- +rem) and Appendix B (how to calculate the optimal reservation price). + +3 Equilibrium +12 +Hence, from period-1 viewpoint, the present-discounted value of the expected rev- +enues of the auction with dollars are: +ΠUSD = k +T +� +t=1 +βt−1 = 1 − βT +1 − β k +and the present discounted value of participating in the auction as a bidder is +uUSD = g +T +� +t=1 +βt−1 = g1 − βT +1 − β +For a given auction format, the auctioneer’s utility is +UUSD = +max +w2≥0,...,wT ≥0,wT +1=0 +� T +� +t=1 +βt−1EU +� n +� +i +bi,t + wt − wt+1 +R +� +, +� +, +where EUt() is the auctioneer’s expected utility in period t. Note that, if the auc- +tioneer is risk averse (i.e. his utility function is strictly concave), not all common +auction formats maximize expected utility. The reason is that the variance of the +revenues also matters. Nonetheless, for our purposes, it is enough to establish an +upper bound to the auctioneer’s utility, as the next lemma does (its proof is omit- +ted). +Lemma 1. Consider a given auction format, a given level of initial assets w1, and +realized period-1 revenues �n +i bi,1. Define w∗ +2, ..., w∗ +T as the unconstrained optimal +sequence of assets in the absence of risk, that is +{w∗ +2, ..., w∗ +T } ≡ argmaxw2,...,wT +1=0 +� +U +� n +� +i +bi,1 + w1 − w2 +R +� ++ +T +� +t=2 +βT−1U +� +k + wt − wt+1 +R +�� +. +It must be that +UUSD ≤ E +� +U +� n +� +i +bi,1 + w1 − w∗ +2 +R +� ++ +T +� +t=2 +βT−1U +� +k + w∗ +t − w∗ +t+1 +R +�� +where the expectation is taken over period-1 revenues �n +i bi,1. The above inequality +is strict if U() is strictly concave. + +3 Equilibrium +13 +The above lemma says that, for every possible auction format, the auctioneer’s +utility must be lower than the utility when all risks after period 1 are eliminated +and credit constraints are removed, allowing him to borrow at the risk-free rate. +The inequality must be strict if his utility is strictly concave. Quite intuitively, if +the auctioneer is risk averse, he would rather receive the largest possible expected +revenues k with probability 1 in each period rather than be exposed to the variability +of these revenues. Furthermore, the fact that utility is strictly concave also implies +that the auctioneer benefits from smoothing consumption across periods. Because +achieving optimal consumption smoothing may require borrowing, the auctioneer +benefits when borrowing constraints are removed (independently of whether risk is +also removed). +3.2 +Auction with tokens +I start by deriving the price of tokens in a given period t as a function of the +following-period expected price of tokens pe +t+1 and the realization of bids. +Lemma 2. Consider a given pe +t+1 and a given realization of period-t valuations (and +hence given profile of bids mi,t, ..., mn,t and payments bi,t, ..., bn,t). The demand for +tokens in period t is +� +i bi,t +pt ++ St +where St ≥ 0 is the speculative demand for tokens, that is, the demand for tokens +not used for paying the auctioneer in period t, and is defined as +St = max +� +Mt − +�n +i bi,t +β(1 + τt)pe +t+1 +, 0 +� +. +The equilibrium period-t price is: +pt = max +�� +i bi,t +Mt +, β(1 + τt)pe +t+1 +� +. +(1) +The most important observation is that some tokens may be purchased not for +bidding but for speculative purposes. This happens in equilibrium when the realized +distribution of valuations is such that total payments to the auctioneer are low. In + +3 Equilibrium +14 +this case, if the demand for tokens was determined exclusively by the tokens used for +bidding, we would have pt < β(1 + τt)pe +t+1 which cannot be an equilibrium because +the return on investing in tokens would be strictly greater than that of investing in +the risk-free asset (remember that the risk-free asset generates a present-discounted +return equal to βR = 1 and that tokens held for speculation grow by a factor 1 + τt +between periods). The possibility of purchasing tokens for speculation implies that +the period-t price of tokens has a lower bound at β(1 + τt)pe +t+1. +The above lemma can be used to derive the bidders’ incentives to bid for a given +sequence of token prices. If pt ≥ β(1 + τ)pe +t+1 then bidders do not want to hold +any token between the two periods, hence ai,t+1 = 0. If instead pt = β(1 + τ)pe +t+1, +then holding tokens for speculation generates zero utility. +Bidders are therefore +indifferent between holding any amount of tokens between period t and period t+1, +including zero tokens. It follows that, in both cases, bidder i’s utility as a function +of pt, pe +t+1 and the profile of bids is: + + + +vi,t + ptai,t − bi,t + βut+1(0) +if i is the winner +ptai,t − bi,t + βut+1(0) +otherwise, +(2) +where ut+1(ai,t+1) is the expected continuation utility from period t + 1 onward, as +a function of the tokens owned at the start of period t + 1. It immediately follows +that, for a given auction format, the bidders’ incentives to bid are the same with or +without tokens. Because of this, the revenue equivalence theorem holds also here: +any standard auction format with a reservation price of zero maximizes the expected +payment to the auctioneer, which is E[� +i bi,t] = k. As a consequence, the bidders’ +expected payoff from participating in the auction is, again, g. +Importantly, (2) also implies that the bidders’ payoffs depend both on the ex- +pected payoff from participating in the auction g, and on the expected value of the +tokens held at the beginning of the period pe +t · ai,t.15 I can therefore take the expec- +tation of (2), use the fact that the expected payoff from participating in the auction +15 It is useful to think of each bidder selling all their tokens and earning ptai,t, while simultane- +ously purchasing tokens to bid bi,t. The expected cost of the bid is part of the expected payoff +from the auction. + +3 Equilibrium +15 +is g, and write bidder i’s expected continuation utility as: +ut(ai,t) = pe +tai,t + g +T +� +t=1 +βt−1 = pe +tai,t + g1 − βT +1 − β +Note that because ai,1 = 0, then a bidder’s expected continuation utility from period +1 viewpoint is the same as in the auction with dollars. But because ai,t could be +positive for t ≥ 2, then the time profile of the bidder’s utility may differ in the +auction with tokens from the auction with dollars. +Having determined the bidders’ expected utility, I can now derive the auctioneer’s +expected revenues, as the next proposition does. +Proposition 1 (Expected revenues). Consider a given sequence of equilibrium ex- +pected prices (i.e., prices such that equation 1 holds). At the beginning of each period +t ≤ T, the present-discounted value of the auctioneer’s future expected revenues is +ΠTokens,t(At) = 1 − βT +1 − β k − pe +t(Mt − At). +The key observation is that the speculative demand for tokens in period t in- +creases the price of tokens and the auctioneer’s revenues in that period. At the +same time, speculators compete against the auctioneer on the market for tokens in +period t + 1. The proposition shows that, in expected terms, the two effects cancel +out. As a consequence, the present discounted value of the auctioneer’s expected +revenues at the beginning of the game is ΠTokens,1(M1) = 1−βT +1−β k, as in the auction +with dollars. In any subsequent period, the auctioneers’ continuation revenues may +be lower than those in the auction with dollars by an amount equal to the value of +the tokens not held by the auctioneer in that period. Finally, note that the above +proposition highlights the effect of the speculative demand for tokens on the time +profile of the auctioneer’s revenues. The speculative demand for tokens, however, +has an additional effect on the variability of these revenues. The reason is that the +speculative demand for tokens depends on the future expected price and therefore +transforms uncertain future revenues into certain present revenues. +Crucially, the incentive to purchase tokens for speculation—and, as a conse- +quence, the time profile and variability of revenues—depends on the monetary pol- + +3 Equilibrium +16 +icy specified by the auctioneer. If σt is sufficiently large for all t ≤ T, then in every +period, the auctioneer creates and keeps for himself a large number of tokens. As a +consequence of the resulting inflation, the speculative demand for tokens is, there- +fore, zero, and, like in the auction with tokens, the auctioneer earns � +i bi,t dollars +in every period. As σt decreases, the incentive to purchase tokens for speculation +increases, causing revenues to accrue earlier and to be less uncertain. +The case +σt = −1 is therefore particularly relevant: bidders purchase all tokens in period 1, +and progressively use them to pay the auctioneer, who then destroys these tokens. +The next proposition derives the auctioneer’s revenues for this case. +Proposition 2 (Revenues when σt = −1 for all t ≤ T). If σt = −1 for all t ≤ T, +then the auctioneer earns revenues exclusively in period 1, which are +p1M1 = +� +i +bi,t + β1 − βT−1 +1 − β +k +(3) +Quite intuitively, when the auctioneer destroys all tokens received as payment, +then he earns revenues only in period 1. The revenues pertaining to period 1 are +subject to risk, and the present-discounted value of the expected revenues pertaining +to all following periods is earned with probability 1.16 Note also that because all +revenues are earned in period 1 and all risk past period 1 is eliminated, the auctioneer +can achieve optimal consumption smoothing by simply saving. Hence, for every +possible realization of � +i bi,t, the auctioneer achieves utility: +U +� n +� +i +bi,1 + Rw1 − w∗ +2 +� ++ +T +� +t=2 +βT−1U(k + Rw∗ +t − w∗ +t+1) +where w∗ +2, ..., w∗ +T are defined in Lemma 1. Lemma 1 then immediately implies the +following corollary: +Corollary 1. The auction with tokens with σt = −1 for all t ≤ T is the preferred +auction with tokens, strictly so if U() is strictly concave. It is also preferred to the +16 There is a straightforward extension of the model in which the auctioneer is allowed to sell +tokens already in period 0. In that case, also the variability of period-1 revenues is eliminated +because the auctioneer earns the total present-discounted value of the expected revenues in period +0 with probability 1. + +4 Extension: pledging an object vs. pledging cash +17 +auction with dollars, strictly so if U() is strictly concave. +3.3 +Discussion: traditional financial instruments. +A natural question is whether the optimal auction with tokens can be replicated by +holding an auction with dollars while simultaneously issuing a traditional financial +instrument. +The answer is, quite clearly, yes. To see this, suppose that the auctioneer holds an +auction with dollars and also issues equity in period 1: investors pay the auctioneer +for the right to receive all future cash flows. To ease comparison with the auction +with tokens, assume that equity is sold to investors at the end of period 1, just before +consumption occurs. Furthermore, suppose the auctioneer sells 1 unit of equity. +In equilibrium, investors must be indifferent between purchasing equity or not, +which implies that the price at which the auctioneer can sell equity is +P1 = β +1 +1 − β k. +The auctioneer’s period-1 revenues (from the sale of the object and equity) are +� +i +bi,1 + β +1 +1 − β k. +Because the auctioneer does not earn revenues in any subsequent period, the equi- +librium is, therefore, identical to the optimal auction with tokens. +4 +Extension: pledging an object vs. pledging cash +When issuing tokens, the auctioneer pledges to deliver the object to investors in the +future. When issuing equity, instead, the auctioneer pledges to deliver the revenues +earned via the sale of the object to investors in the future. In the model discussed +above, the absence of any friction makes the two identical under an appropriate +monetary policy. However, in a more realistic environment, pledging revenues may +be easier or more difficult than pledging an object. In particular, note that any fric- +tion affecting the ability of the auctioneer to pledge the object (e.g., non-contractible + +4 Extension: pledging an object vs. pledging cash +18 +effort) will necessary also affect the ability to pledge revenues that are generated +from the sale of the object. But there may be frictions that affect the ability to +pledge revenues but not the object—for example, when cash is easier to hide than +the object—therefore making revenues harder to pledge. At the same time, tradi- +tional financial instruments can be very flexible and specify contingent payments in +a way that cannot be replicated with tokens. +To explore the interaction of these two effects, here I modify the above model +by introducing both effort and the possibility of misappropriating revenues. The +auction runs for T = 2 periods, and the risk-free asset has a return of 1 so that +R = 1. The auctioneer is also risk-neutral and cash-abundant. Differently from +the above model, here the auctioneer is less patient than the investors: whereas +investors do not discount the future, the auctioneer’s discount factor is β < 1.17 +The timing of the first period is identical to the model presented earlier. Unlike +the model presented earlier, at the beginning of the second period, the auctioneer +exerts effort e ≥ 0 to improve the quality of the object sold. Effort is observable +(but non-contractible) and shifts uniformly the distribution of valuations, so that +when bidder i draws vi,2 from the distribution F(x), his utility from consuming +the object is vi,2 + e.18 +Effort has a quadratic cost equal to +e2 +2 . +Furthermore, +after period-2 revenues are realized but before payments to investors are made, the +auctioneer can misappropriate revenues. More precisely, by paying a cost c, he can +consume revenues that should otherwise be forwarded to investors. A familiar micro- +foundation for this is the possibility for the auctioneer to “run away with the till” at +a cost c, which therefore measures the extent to which revenues can be pledged to +investors. +First best +In the first best, effort is such that its marginal cost equals its marginal +benefit, and hence e∗∗ = 1. +At the same time, because the auctioneer is more +impatient than investors, in period 1 investors will pay the auctioneer an amount +17 Clearly, the results derived earlier are robust to this change in assumptions: if fully front +loading revenues by setting σt = −1 for all t is optimal when the auctioneer is as patient as +investors, it is also optimal when the auctioneer is more impatient than investors. +18 Note that, because e is observable, the auction is still in private value despite the valuations +having a “common” component. + +4 Extension: pledging an object vs. pledging cash +19 +equal to the expected future revenues and will earn back these revenues in period +2. As a consequence, the auctioneer earns all his revenues in period 1. +Auction with dollars with outside investors. +Suppose now that the auction uses +dollars and that the auctioneer sells a contingent security to risk-neutral investors, +who also do not discount the future. Assume that effort is not contractible, but +revenues are. The security specifies a payment to the auctioneer in period 1 y1, +and a contingent period-2 payment from the auctioneer to investors y2(� +i b2,i + e) +subject to the investors’ rationality constraint: +E[y2( +� +i +b2,i + e)] = y1. +Note that the possibility of “running away with the till” implies that, in period 2, +the auctioneer should always earn at least � bi,2 + 1 − c, or else he would prefer to +run away. The incentive compatibility constraint is, therefore: +� +i +b2,i + e − y2( +� +i +b2,i + e) ≥ +� +i +bi,2 + e − c +∀ +� +i +b2,i, e +or +y2( +� +i +b2,i + e) ≤ c +∀ +� +i +b2,i, e. +The first important observation is that if c = 0 (i.e., misappropriation of rev- +enues is costless), then the incentive compatibility constraint immediately implies +y2(� +i b2,i + e) = 0: no outside investment is possible. The choice of effort solves: +max +e≥0 +� +k + e − e2 +2 +� +, +with solution e∗ = 1, which is efficient. However, the time profile of revenues is not +efficient because the auctioneer earns his revenues (and consumes) in each period. +If instead c is sufficiently large, then the first best can be achieved. To see this, +note that if the auctioneer sets the optimal level of effort e∗∗ = 1, then revenues +should always be larger than v + 1. The contract between the auctioneer and the +investor can therefore impose the punishment c whenever revenues are below v + 1. + +4 Extension: pledging an object vs. pledging cash +20 +If this punishment is sufficiently large, an efficient level of effort is achieved. The +revenues generated are transferred to the investor, so that E[y2(� +i b2,i + e)] = y1 = +k + 1, therefore, implementing the first best. +Auction with tokens +Without loss of generality, I normalize the initial stock +of tokens so that M1 = 1. In the second period of the auction with tokens, the +speculative demand is zero for any level of effort. Hence, optimal period-2 effort +solves: +max +e2≥0 +� +A2 +k + e2 +M2 +− e2 +2 +2 +� +s.t. M2 = A2 + S1(1 + τ). +Define +α ≡ A2 +M2 +as the fraction of the total stock of tokens held by the auctioneer at the beginning +of period 2. The equilibrium effort is +e∗ +2(α) = α. +Hence, the period-2 expected price is +pe +2 = k + α +M2 +, +and the expected value of the tokens held by investors at the beginning of period 2 +is +S1(1 + τ)pe +2 = (1 − α)(k + α). +(4) +For future reference, note that α may have an ambiguous effect on the expected +value of tokens held by investors, because higher α implies higher effort and higher +expected revenue, but also that a lower fraction of these revenues is earned by +investors. +More precisely, 4 is strictly increasing in α for α < +1−k +2 +and strictly +decreasing otherwise. In particular, it is always strictly decreasing for α > 1/2. +The parameter α depends on the period-1 speculative demand S1, which itself +depends on period-1 realized valuations and the monetary policy, as the next lemma + +4 Extension: pledging an object vs. pledging cash +21 +shows. +Lemma 3. For given � +i bi,1 we have: +α = min +� +1, +� +k2 + 4 � +i bi,1(1 + σ) − k +2 +� +Hence, α depends on σ but not τ. Furthermore, for given � +i bi,1, α is zero at +σ = −1, it is strictly increasing in σ, and it reaches 1 whenever � +i bi,1 < +k+1 +1+σ. +Hence, for any given � +i bi,1, α can take any value between 0 and 1 as a function of +σ (which is chosen by the auctioneer). More important, for σ such that v < k+1 +1+σ the +period-1 speculative demand for tokens is always zero and α = 1 for all realizations +of � +i bi,1. In this case, the equilibrium is identical to that of the auction with dollars +when c = 0 (i.e., without outside investors). +Given this, I can write the auctioneer payoff from period-1 viewpoint. I use the +fact that, in equilibrium, the expected value of the tokens held by investors at the +beginning of period 2 (c.f., equation 4) must equal the extra revenues earned by +the auctioneer in period 1. I can therefore write the auctioneer’s period-1 expected +utility as +k + E[(1 − α)(k + α)] + β +� +E[α(k + e(α))] − E[e(α)2] +2 +� +where the expectations are taken with respect to the realization of period-1 valua- +tions. Taking first order conditions with respect to σ yields:19 +E +� +(1 − k − 2α + β(k + α)) ∂α +∂σ +� +The important observation is that, as long as β < 1 the above expression is negative +at σ such that v = k+1 +1+σ, that is, σ such that α = 1 with probability 1. The opti- +mal σ therefore implies an intermediate expected α, balancing the tension between +inducing effort and front-loading revenues. Because the auction with tokens with +v = k+1 +1+σ is equivalent to the auction with dollars with c = 0, this observations also +19 By the envelope theorem, the effect of σ on the optimal period-2 effort can be ignored. + +5 Conclusions. +22 +implies that the optimal auction with tokens is strictly preferred to the auction with +dollars with c = 0. +The following proposition summarizes these observations. +Proposition 3. If c is sufficiently low (i.e., revenue misappropriation is easy), +then the auctioneer is better off holding an auction with tokens. If c is sufficiently +large (i.e., revenue misappropriation is very costly), then the auctioneer is better off +holding an auction with dollars and issuing a traditional financial instrument. +The key takeaway is that the possibility of misappropriating revenues affects the +auction with dollars but not the auction with tokens. Despite this, if the contract- +ing space is sufficiently large, then the auction with dollars with an appropriately +designed contingent security is preferred to the auction with tokens. If instead the +contracting space is restricted, then the auction with tokens is preferred. +5 +Conclusions. +By issuing tokens and accepting them as payment, an auctioneer earns revenues +earlier and is exposed to less risk relative to a standard, repeated auction in which +bidders pay in dollars. In particular, if the auctioneer commits to destroying all +tokens received as payment, he earns all his revenues in period 1 and eliminates +all risk from period 2 onward. The same outcome can be achieved in a traditional +auction with dollars in which the auctioneer issues equity. +The reason for this equivalence is that the auctioneer has full commitment, both +to deliver the object to token holders and to deliver the revenues generated by +the sale of the object to equity holders. +I move away from this assumption by +introducing effort and the possibility of misappropriating revenues. I show that, in +this case, the equilibrium in the auction with tokens may differ from that in the +auction with dollars in which the auctioneer can issue a financial security. On the +one hand, the possibility of misappropriating revenues is a friction that does not +matter when issuing tokens. On the other hand, a financial security can specify +contingent payments in a way that cannot be replicated with tokens. I show that if +the contracting frictions are severe, then the auction with tokens is preferred, while +the opposite is true if the contracting frictions are mild or absent. + +A Mathematical derivations +23 +I considered issuing tokens and issuing a contingent security as alternative ways +to raise funds. But they need not be. For example, an auctioneer may issue tokens, +sell some on the open market (to be used for bidding), while simultaneously signing +an investment contract with an investor for the delivery of additional tokens in the +future. This investment contract may also specify penalties in case, for example, +the value of the tokens (which depends on the value of the object) falls below a +certain threshold. This punishment, however, needs to be incentive compatible if +the auctioneer can run away to escape it. Exploring this intriguing possibility is left +for future work. +I assumed that the market for tokens is frictionless. Hence, holding tokens is +inconvenient only because exchanging them for consumption may require waiting +one period. However, this exchange generates no cost. A more realistic view is that +the market for tokens has frictions, and these frictions further reduce the benefit of +using tokens. In an even more realistic model, these frictions would depend on the +volume of transactions, which itself is a function of the value of the object being +exchanged. This extension is also left for future work. +A +Mathematical derivations +Proof of Lemma 2. Consider period t. Take the total payments to the auctioneer +� +i bi,t and the expected future price pe +t+1 as given. Note that a bidder can spend 1 +dollar to purchase 1 +pt tokens in period t, these tokens then multiply by 1+τt and can +be sold in period t+1, for a present-discounted return of +β(1+τt)pe +t+1 +pt +. Alternatively, he +can invest the same amount of money in the risk-free asset for a present-discounted +return of βR = 1 in the following period. It follows that there can be an equilibrium +in the market for tokens if and only if pt ≥ β(1 + τt)pe +t+1. +If pt > β(1 + τt)pe +t+1, no tokens are purchased and then brought to the next +period. The demand for tokens is given by the tokens used for bidding +� +i bi,t +pt +. The +supply of tokens is Mt, and hence the equilibrium price is +pt = +� +i bi,t +Mt + +A Mathematical derivations +24 +This is indeed an equilibrium if +� +i bi,t +Mt +> β(1 + τt)pe +t+1, that is, if the realized bids +are sufficiently high relative to the future expected price. +If instead +� +i bi,t +Mt +≤ β(1 + τt)pe +t+1, then tokens may be purchased but not used +for bidding. I call this demand the speculative demand for tokens St. The total +demand for tokens is now St + +� +i bi,t +pt +, and the equilibrium price is +pt = +� +i bi,t +Mt − St += β(1 + τt)pe +t+1 +which pins down the speculative demand for tokens: +St = max +� +Mt − +�n +i bi,t +β(1 + τt)pe +t+1 +, 0 +� +Proof of Proposition 1. To start, note the following two facts: +1. in each period the auctioneer will liquidate all his tokens. If the price in a +given period is such that pt > β(1 + τt)pe +t+1, then the auctioneer is better off +selling tokens in period t at a higher price than in period t+1 at a lower price. +If instead pt = β(1 + τt)pe +t+1 and the auctioneer is risk averse, then again he +prefers to earn revenues in period t than to wait one period and earn the same +expected revenues (but this time being exposed to risk). If pt = β(1 + τt)pe +t+1 +and the auctioneer is risk neutral then his expected revenues are the same +whether he holds tokens between periods (i.e., he acts as a speculator) or not. +Without loss of generality, we can assume that he sells all his tokens in period +t also in this case; +2. an implication of the above fact is that, for given pe +t+1, At+1 is independent of +At. That is because At+1 depends on the monetary policy parameters σt, τt +and on the tokens received for payment in period t, which depend exclusively +on pe +t+1 and the realization of valuation in period t. +I can therefore write the auctioneer’s expected continuation revenues in period +t as: +Πt(At) = pe +tAt + βE[pt+1At+1] + β2Πt+2(At+2) + +A Mathematical derivations +25 +where the expectations are taken at the beginning of period t, before the valuations +are realized (note: I use Πt() to indicate unconditional expected revenues. I will +write the expectation explicitly if, instead, the expectation is conditional). +Consider now a given pe +t+1. The realization of valuations in period t may be such +that St = 0 and hence pt = +� +i bi,t +Mt . In this case, At+1 = Mt+1, so that, conditional +on St = 0: +E[Πt(At)|St = 0] =E[pt|St = 0]At + βE[pt+1At+1|St = 0] + β2E[Πt+2(At+2)|St = 0] +=E +�� +i bi,t +Mt +|St = 0 +� +(Mt − Mt + At) + βpe +t+1E[Mt+1|St = 0] + β2E[Πt+2(At+2)|St = 0] +=E[ +� +i +bi,t|St = 0] − E[pt|St = 0](Mt − At) + βpe +t+1E[Mt+1|St = 0] + β2E[Πt+2(At+2)|St = 0] +If instead St > 0, then pt = β(1 + τ)pe +t+1 and At+1 = Mt+1 − St(1 + τ) ≥ 0. In this +case, conditional on St > 0: +E[Πt(At)|St > 0] = E[pt|St > 0]At + βE[pt+1At+1|St > 0] + β2E[Πt+2(At+2)|St > 0] += E[pt|St > 0]At + βpe +t+1(E[Mt+1|St > 0] − E[St|St > 0](1 + τ)) + β2E[Πt+2(At+2)|St > 0] += E[pt|St > 0]At + βpe +t+1E[Mt+1|St > 0] − βpe +t+1(1 + τt) +� +Mt − E[� +i bi,t|St > 0] +βpe +t+1(1 + τt) +� ++ β2E[Πt+2(At+2)|St > 0] += E +�� +i +bi,t|St > 0 +� +− E[pt|St = 0](Mt − At) + βpe +t+1E[Mt+1|St > 0] + β2E[Πt+2(At+2)|St > 0] +where I used the definition of St as well as the fact that E[pt|St > 0] = βpe +t+1(1+τt). +The above derivations then imply that, for a given sequence of expected equilib- +rium prices, the unconditional expected revenues are +Πt(At) = k−pe +t(Mt−At)+βpe +t+1E[Mt+1]+β2Πt+2(At+2) = k−pe +t(Mt−At)+βΠt+1(Mt+1) +(5) +where the last equality exploits the fact that, for given equilibrium expected prices, +At+2 is independent of At+1. The statement then follows by iterating the above +equation. +Proof of Proposition 2. The statement follows from Proposition 1, and by noting +that all revenues accrue in period 1, after the period-1 valuations are drawn. It is +nonetheless instructive to present a direct proof of the statement. + +A Mathematical derivations +26 +Because all tokens earned by the auctioneer are burned, in equilibrium, the +speculative demand for tokens is strictly positive in every period t ≤ T and for +every possible realization of valuations. +Because of this, in every period t and +every realization of period t < T valuations, equilibrium prices must be such that +bidders are indifferent between holding tokens between any two periods, that is +pt = β · (1 + τt) · pe +t+1. +This observation has two implications. First, I can write the sequence of equi- +librium prices as p1, pe +2 = p1/(β(1 + τ1)), ... pe +t = p1/ +� +βt−1 �t−1 +s=1(1 + τs) +� +for t ≤ T. +Second, in every period t ≥ 2 the supply of tokens is St−1(1+τt−1). By equating +this supply with the demand for tokens, I can write +pt = +� bi,t +St−1(1 + τt−1) − St +so that +St−1 = +� bi,t +pt(1 + τt−1) + +St +(1 + τt−1) +(6) +Note that the speculative demand in period T is zero. Hence, for given � bi,T I +can write +E[ST−1] = +k +pT(1 + τT−1) +E[ST−2] = +k +pT−1(1 + τT−2) + +k +pT(1 + τT−2)(1 + τT−1) +... +S1 = k +T +� +s=2 +1 +ps +�s−1 +j=1(1 + τj) +Using the fact that pe +s = p1/ +� +βt−1 �s−1 +j=1(1 + τj) +� +, the above expression becomes: +S1 = k +p1 +t−1 +� +s=1 +βs. +Finally, use the fact that the initial supply of tokens is M1, so that the period-1 +market clearing price is +p1 = +� bi,1 +M1 − S1 +, + +A Mathematical derivations +27 +to get the expression: +p1M1 = +� +bi,1 + k +t−1 +� +s=1 +βs +(7) +Proof of Lemma 3. By using the definition of α, I can write +α = +(1 − S1)(1 + σ)(1 + τ) +(1 − S1)(1 + σ)(1 + τ) + S1(1 + τ) +(8) +Next, I use the above expression to write +pe +2 = +k + α +(1 − S1)(1 + σ)(1 + τ) + S1(1 + τ) = +(k + α)α +(1 − S1)(1 + σ)(1 + τ) +(9) +I use the definition of S1 and the above expression to obtain: +S1 = max +� +0, 1 − +� +i bi,1 +pe +2(1 + τ) +� += max +� +0, 1 − +� +i bi,1 +(k + α)α(1 − S1)(1 + σ) +� +1 − S1 = min +� +1, +� +i bi,1 +(k + α)α(1 − S1)(1 + σ) +� +Suppose α < 1 so that 0 < S1 ≤ 1. Then α must be such that +α = +� +� +� +� +�k +2 +�2 ++ ( +� +i +bi,1)(1 + σ) − k +2 +this can be the solution as long as α < 1, or k +1 > � +i bi,1(1+σ). If instead α = 1, +then S1 = 0 and it must be that +k + 1 < +� +i +bi,1(1 + σ) + +A Mathematical derivations +28 +Putting everything together, we have that +α = min + + +1, +� +� +� +� +�k +2 +�2 ++ +� +i +bi,1(1 + σ) − k +2 + + + . +References +Bakos, Y. and H. Halaburda (2018). 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Counterspeculation, auctions, and competitive sealed tenders. +The Journal of finance 16(1), 8–37. + diff --git a/8dFST4oBgHgl3EQfaDjo/content/tmp_files/load_file.txt b/8dFST4oBgHgl3EQfaDjo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..502dc45e0484e00fefc63c8bb0946bbf123a26ad --- /dev/null +++ b/8dFST4oBgHgl3EQfaDjo/content/tmp_files/load_file.txt @@ -0,0 +1,758 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf,len=757 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='13794v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='TH] 31 Jan 2023 Auctions with Tokens∗ Andrea Canidio † First version: September 30, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' This version: February 1, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Please check here for the latest version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Abstract I study mechanism design with blockchain-based tokens, that is, tokens that can be used within a mechanism but can also be saved and traded outside of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I do so by considering a repeated, private-value auction, in which the auctioneer accepts payments in a blockchain-based token he creates and initially owns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I show that the present-discounted value of the expected revenues is the same as in a standard auction with dollars, but these revenues accrue earlier and are less variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I then introduce non-contractible effort and the possibility of misappropriating revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I compare the auction with tokens to an auction with dollars in which the auctioneer can also issue financial securities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' An auction with tokens is preferred when there are sufficiently severe contracting frictions, while the opposite is true when contracting frictions are low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' JEL classification: D44, E42, L86 Keywords: Mechanism design, Auctions, Blockchain, Cryptocurrencies, Tokens, Private Money 1 Introduction Blockchain protocols resemble the mechanisms studied in mechanism design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Like in these mechanisms, each blockchain protocol needs to generate incentive-compatible ∗I’m grateful to John Asker, Rainer Böhme, Sylvain Chassang, Lin William Cong, Andreas Park and conference participants at the Crypto Asset Lab conference 2021, CSH Workshop on Decentralized Finance 2022, ASSA 2023, seminars at CREST, CY University (Thema) for their insightful comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' †IMT school of advanced studies, Lucca, Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='canidio@imtlucca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='it 1 1 Introduction 2 behavior, so to achieve a given aggregate outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Furthermore, each blockchain protocol is associated with a specific blockchain-based token, which is necessary to use the protocol, usually as its internal currency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='1 The use of tokens within blockchain protocols is, therefore, similar to the use of “virtual money” in several mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' For example, several business schools allocate students to MBA classes by distributing “points” to students, who then use them to bid for classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='2 Simi- larly, the organization Feeding America distributes food to various local food banks via auctions in which a virtual currency is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='3 Unlike these virtual currencies, however, the tokens associated with blockchain protocols exist also outside of their respective protocol: they can be held without being used and can be exchanged on financial markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' They are, therefore, a new financial instrument, which opens sev- eral issues not usually considered in mechanisms design: will some of these tokens be held and not used in the mechanism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' And how will this affect the mechanism itself?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' And the revenues earned by the designer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Studying blockchain protocols as mechanisms, therefore, required a theory of mechanism design that incorporates blockchain-based tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' This paper makes the first step in this direction by considering a specific mechanism: an auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Auc- tions are the most widely studied and better-understood mechanism and therefore constitute an ideal starting point in this research agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' They are also effective at allocating a given good to the bidder who values it the most, something that, as we will see, remains true with the introduction of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The role of tokens within the mechanism is, therefore, limited, which allows me to focus on the most novel aspect of the model: the fact that these tokens can be held and can be traded outside of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I consider a finite sequence of private-value auctions in which multiple objects 1 The fact that the token is the internal currency is very explicit in protocols creating decen- tralized marketplaces, for example, for buying and selling computer storage space (see Filecoin, Storij, Sia) or CPU cycles (see the Golem network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' It is also the case in decentralized computing platforms such as Ethereum, in which users pay miners/validators for executing smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' It also extends to cryptocurrencies such as Bitcoin, in which users pay miners to process transactions in Bitcoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 2 See Hylland and Zeckhauser (1979), Budish (2011), Budish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' (2017), He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 3 See Prendergast (2017) and Prendergast (forthcoming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Note that, in the existing literature studying the use of virtual currencies within mechanisms, the objective of the auctioneer is not to maximize revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 1 Introduction 3 are sold (one per period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='4 In every period, risk-neutral bidders draw their valuation for the object sold from an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Then they submit bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Given the profile of bids, the auction format determines the winning bidder and the payment of each player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The good is then consumed within the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Each auction is, therefore, a simple static auction, repeated multiple times (that is, there is no con- nection between auctions in different periods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' As a part of the auction design, the auctioneer can decide to accept payments in dollars or in a blockchain-based token that he creates and initially owns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If he chooses the latter option, the auctioneer also commits to a monetary policy: a set of rules determining how the stock of tokens evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The auctioneer earns revenues by selling newly created tokens to bidders and re-selling tokens he received as payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Tokens can be held without being used for bidding and can be traded on a financial market where their value is determined as an equilibrium outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='5 I show that in the auction with tokens when the realized valuations in a given period are low (relative to the future expected valuations), bidders might purchase tokens for speculation, not bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The reason is that when valuations are low, the demand for tokens for bidding is low, creating an arbitrage opportunity for bidders: they may purchase tokens, not use them for bidding in that period, and sell them (or use them) in future periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The speculative demand for tokens increases the price for tokens in a given period and, as a consequence, drives the auctioneer’s expected revenues for that period above those of the auction with dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' At the same time, today’s speculators will compete with the auctioneer on tomorrow’s market for tokens, pushing the auctioneer’s future expected revenues below those of the auction with dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Also, the speculative demand in a given period depends on the expected future valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, speculation transforms future uncertain revenues into present certain revenues 4 The companion short paper Canidio (forthcoming) considers the case of an infinitely repeated auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The main issue there is the emergence of financial bubbles on tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 5 Note that creating new tokens and trading them on a financial market is quite easy to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' There are several simple tutorials explaining how to create blockchain-based tokens (I invite the reader to search “how to create an ERC-20 Token”, where ERC-20 is the simplest type of token on the Ethereum blockchain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Also, after the creation of a new token, anyone can then use a protocol such as Uniswap to create a decentralized financial market for exchanging the new token against, for example, a stablecoin (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', a blockchain-based token with constant value relative to the dollar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 1 Introduction 4 I show that there is a form of revenue equivalence: the present-discounted value of the expected revenues is the same in all auction formats (with or without tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' However, in each specific period, the revenues in the auction with tokens may be different from those in the auction with dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' More precisely, revenues accrue earlier and are less variable in the auction with tokens than in the auction with dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' How exactly depends on the specific monetary policy announced by the auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' A particularly relevant case is a policy in which all tokens used to pay the auctioneer are then destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In this case, all revenues are earned in the first period, when the auctioneer sells the initial stock of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Furthermore, the present-discounted value of the expected revenues from period-2 onward is earned with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='6 Hence, by designing an appropriate auction with tokens, the auctioneer can fully front-load his revenues and eliminate (almost) all risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Quite immediately, it is possible to achieve the same outcome by holding an auction with dollars in which the auctioneer also sells equity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', transferring to investors his future cash flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The first result is, therefore, an equivalence result: absent costs of issuing tokens or equity, the optimal auction with tokens is equivalent to issuing equity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I then extend the model by introducing two frictions (i) costly non-contractible effort and (ii) revenue misappropriation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', the possibility that the auctioneer can hide revenues by “running away with the till” (at a cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Intuitively, when issuing tokens, the auctioneer commits to deliver the object in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' This commitment may be imperfect whenever the auctioneer can shirk, exert low effort, and provide an object of lower value to the token holders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Alternatively, the auctioneer can hold an auction with dollars while simultaneously pledging future cash flows to investors via a traditional financial instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The critical observation is that everything that limits the auctioneer’s ability to commit the object also limits his ability to commit the revenues generated by the sale of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' But the possibility of revenue misappropriation only affects traditional financial instruments because it reduces the auctioneer’s ability to commit revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 6 The reason is that, in the model, the first sale of tokens happens after the period-1 valuations are drawn, which therefore is the only risk faced by the auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' There is a straightforward extension of the model in which the auctioneer can also sell tokens before period-1 valuations are drawn, in which case the auctioneer can eliminate all risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 1 Introduction 5 Hence, whether to issue tokens or a traditional financial instrument hinges on a trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' On the one hand, the possibility of misappropriating revenues is a concern only when issuing a traditional financial instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' On the other hand, depend- ing on the contracting environment, a traditional financial instrument may specify contingent payments that tokens cannot replicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I show that if the cost of hiding revenue is sufficiently low, tokens are the preferred financial instrument because the possibility of revenue misappropriation is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If instead, running away is suffi- ciently costly, a traditional financial instrument is the preferred means of financing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Note that the cost of revenue misappropriation proxies contract incompleteness be- cause it determines the set of incentive-compatible financial instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, the result is that if the contracting space is limited, issuing tokens is preferred, while when the contracting space is sufficiently large, then a traditional financial instrument is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' This paper makes several contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' First, it shows that destroying (or burn- ing) tokens may be the optimal monetary policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Burning has been widely discussed within the Ethereum community in relation to the implementation of EIP-1559, a new way to calculate transaction fees according to which a fraction of the fees paid by users are burned rather than transferred to miners/validators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='7 This paper also contributed to the discussion on whether tokens are securities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The tokens in the model provide access to a service and are, therefore, “utility tokens.” The logic be- hind their pricing and valuation is different from that of equity (see, in particular, Prat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Because of that, some argue that utility tokens are not securi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' However, the benchmark model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', the one without frictions) shows that in equilibrium, the value of these tokens is identical to that of equity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Finally, this paper also provides a rationale for using tokens over equity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' When the contracting space is sufficiently restricted, then tokens are preferred to equity because revenue misappropriation is not an issue with tokens, but it is with equity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 7 See the original proposal here https://eips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='org/EIPS/eip-1559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' For an economic analysis of EIP-1559, see Roughgarden (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The original motivation for burning is to prevent miners/validators from manipulating how fees are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' There is, however, a heated debate on whether tokens should be burned rather than automatically forwarded to a dedicated fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 1 Introduction 6 Literature review The literature studying the use of tokens within mechanisms includes papers in which the mechanism encompasses the entire economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The goal is these pa- pers is to establish whether certain centralized mechanisms can be implemented in a decentralized way using tokens (see, for example, Ostroy and Starr, 1974, Kocherlakota, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Similarly, some of the early papers in monetary theory con- sidered general-equilibrium models in which there is at least an equilibrium with money (see Samuelson, 1958, and Townsend, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Because the equilibrium with money is Pareto superior to that without money, again, we can think of money as allowing the decentralized implementation of some (usually constrained) optimal allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Also related are models in which money has value (also in finite time) because of exogenous reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' For example, in Starr (1974), a government creates money and establishes that taxes need to be paid using money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Likewise, in the cash-in-advance model of Clower (1967), fiat money needs to be acquired one pe- riod in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Note that in these models, the optimal monetary policy is typically time inconsistent, and hence the issue of commitment to a monetary policy arises (see, in particular, Lucas Jr and Stokey, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The advent of blockchain and blockchain-based tokens provided a new impulse to the above literature (for an overview, see Townsend, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The reason is that blockchain-based smart contracts can be used to generate commitment, for exam- ple, to perform payments based on contingencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Several authors have therefore studied how blockchain-based smart contracts can be used to implement various types of mechanisms (see, Holden and Malani, 2019, Gans, 2019, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', 2021, Brzustowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', 2021, Townsend and Zhang, forthcoming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='8 With this respect, note that, in the model presented here, the auction itself could be a traditional, centralized auction or a decentralized one (via a smart contract).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, this paper does not contribute to the study of how auctions (or other mechanisms) can be implemented using blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' However, the issue of implementation is relevant for the equilibrium of the game (of which the auction is a part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The reason is that, 8 There is also a small but growing literature using insights from mechanism to propose im- provements to blockchain protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' See, for example, Gans and Holden (2022a), Gans and Holden (2022b), Capponi and Jia (2021), Canidio and Danos (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 1 Introduction 7 in equilibrium, the auction with tokens is equivalent to an auction with dollars in which the auctioneer also issues equity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Several papers study theoretically firms’ incentives to issue blockchain-based tokens, which can represent a pre-sale of a given unit of future output, the only cur- rency that the firm will accept in the future, or a claim on future revenues or profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Some of these papers showed that, in the presence of network externalities, selling tokens helps avoid coordination failures (Sockin and Xiong, 2018, Cong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', 2021, Bakos and Halaburda, 2018, and Li and Mann, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Other papers focused on the sale of tokens as an innovative way to raise capital and finance the development of a product or a platform (Catalini and Gans, 2018, Malinova and Park, 2018, Canidio, 2018, Bakos and Halaburda, 2019, Goldstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', 2019, Cong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', 2020, Canidio, 2020, Gryglewicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', 2021, Garratt and van Oordt, 2021, Chod and Lyandres, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In the model considered here, the auctioneer has no financing need, and there are no network externalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, tokens are sold purely to earn a profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Nonetheless, there is a connection with the above literature because by issuing to- kens, the auctioneer can manipulate the time profile and riskiness of his revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Both these elements are important in determining the incentives to invest and create new ventures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I will frequently refer to two important results in auction theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The first is the revenue equivalence theorem, which states:9 Assume each of a given number of risk-neutral potential buyers of an object has a privately-known signal independently drawn from a common, strictly-increasing, atomless distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Then any auction mechanism in which (i) the object always goes to the buyer with the highest signal, and (ii) any bidder with the lowest-feasible signal expects zero surplus, yields the same expected revenue (and results in each bidder making the same expected payment as a function of her signal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' For our purposes, the above statement implies that all common auction formats (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', first-price, second-price, all-pay, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=') generate the same expected payment from 9 Vickrey (1961) developed some special case of the revenue equivalence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The statement presented here is taken from Klemperer (1999), and summarizes results in Myerson (1981) and Riley and Samuelson (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' For a more general formulation, see Milgrom and Segal (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 2 The model 8 bidders and hence the same expected revenues for the auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The second result is the design of optimal auctions (see, again, Myerson, 1981, Bulow and Roberts, 1989, Bulow and Klemperer, 1996 and Klemperer, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In particular, in the model I will assume that the distribution of valuations is such that all common auction formats with a reservation price of zero maximize the auctioneer’s expected revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 2 The model I consider a single-object private-value auction repeated T ≥ 1 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' There are n ≥ 2 ex-ante identical bidders and an auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Bidders are risk-neutral and cash abundant in the sense that their cash constraint is never binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The auctioneer’s per-period utility function is U(), assumed concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' There is a common discount factor β ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Bidders and the auctioneer can hold a risk-free asset yielding a per-period gross return of R ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' For ease of derivations, I assume that R = 1 β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='10 The auctioneer’s initial assets are w1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' For the moment, I assume that the auctioneer can only save by investing in the risk-free asset, and hence wt ≥ 0 for all t ≤ T, where wt are the asset owned by the auctioneer at the beginning of a given period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='3 discusses what happens when the auctioneer can also issue equity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In period 0, the auctioneer decides whether to accept payments in fiat currency (for simplicity, dollars) or in tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' When the auction format requires the use of tokens, the auctioneer creates an initial stock of tokens M1 and announces a monetary policy, that is, how the stock of tokens will evolve (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Then, in every period t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', T}, the auctioneer sells a single object according to the auction format specified initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' At the beginning of each period t ≥ 1, each bidder draws a valuation vi,t > 0 from a continuous and atomless distribution with c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='f F(v), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' f(v) and support [v, v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Each vi,t is bidder i’s private information, but the distribution F(v) is common knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The auction is in private value so that each bidder’s valuation is independent of the other bidders’ valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' To avoid uninteresting complications, 10 Hence, R is the steady-state rate of return of the Ramsey-Cass-Koopmans growth model (with no population growth or exogenous productivity growth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' This assumption is not essential for the results but simplifies the derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 2 The model 9 I assume that vf(v) ≥ 1 − F(v) for all v ∈ [v, v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='11 Each object sold has zero value to the auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If the auctioneer uses dollars, the sequence of events in each period is stan- dard: after drawing the valuations, each bidder sends a message mi,t ∈ R+ to the auctioneer, interpreted as his bid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' As a function of the messages received and the auction format initially announced, the auctioneer determines who is the winner and payments bi,t ≤ mi,t for each bidder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Each bidder then pays bi,t dollars to the auctioneer, and the winner receives and consumes the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The period-t payoff of the winning bidder is vi,t − bi,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' the period-t payoffs of all other bidders is −bi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The auctioneer’s period-t revenues are �n i bi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' At this point, the auctioneer invests wt+1 R in the risk-free asset (which become wt+1 in the following period), and consumes wt + �n i bi,t − wt+1 R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If the auctioneer instead uses tokens, then the timeline of each period t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', T} is the following: Again, at the start of a period, each bidder draws a valuation vi,t from the distribution F(v), and then sends a message mi,t ∈ R+ to the auctioneer, interpreted as his bid in dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Note that, at this point, both the auctioneer and bidders may own tokens that they accumulated from previous periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Call At ≥ 0 the tokens owned by the auctioneer at the beginning of the period, and ai,t ≥ 0 the tokens owned by bidder i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' By assumption, A1 = M1 and ai,1 = 0 for all i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' As a function of the messages received and the auction format initially an- nounced, the auctioneer determines who is the winner and payments bi,t ≤ mi,t for each bidder (implicitly a function of all messages received).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' This payment is expressed in dollars, but needs to be settled using tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='12 A frictionless, anonymous financial market for tokens opens, in which both the auctioneer and bidders participate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' All market participants are price tak- 11 As we will see, under this assumption, the revenue-maximizing reservation price in the auc- tion with dollars is zero, and calculating the revenues from the optimal auction with dollars is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 12 The fact that bids and payments are expressed in dollars is for convenience, as it allows me to use the same notation as in the auction with dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The results are identical whenever bids are expressed in tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 2 The model 10 ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Call pt the equilibrium price for tokens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' call qi,t the equilibrium demand for tokens of bidder i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' call Qt the equilibrium demand for tokens of the auc- tioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Both qi,t and Qt could be positive or negative: if negative, the bid- der/auctioneer is a net seller of tokens in equilibrium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' if positive, he is a net buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Furthermore, feasibility implies At + � i ai,t = Qt + � i qi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Also, be- cause tokens will be used to pay the auctioneer (see the next point), it must be that qi,t ≥ bi,t pt − ai,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Then, each bidder sends bi,t pt tokens to the auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The winner receives the object and consumes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' At this point, each bidder owns ai,t +qi,t − bi,t pt tokens, and the auctioneer owns At + Qt + � i bi,t pt tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The winning bidder enjoys a per-period payoff equal to the value of the object minus the expenditure in tokens vi −pt ·qi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Similarly, the losing bidders enjoy per-period payoffs equal to −ptqi,t, and the auctioneer’s revenues are −ptQt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Again, the auctioneer chooses wt+1 and then consumes wt − ptQt + wt+1 R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The stock of tokens changes according to the monetary policy announced by the auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Here I restrict my attention to two time-varying monetary- policy parameters: a uniform increase (or decrease) of all tokens by the same factor τt ≥ −1, and an increase (or decrease) of only the tokens used for bidding by a time-varying factor σt ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' As a result, at the beginning of the subsequent period, each bidder i owns ai,t+1 = (1 + τt)(ai,t + qi,t − bi,t pt ), while the auctioneer owns At+1 = (1+τt)(At+Qt +(1+σt) � i bi,t pt ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, the total stock of tokens at the beginning of period t + 1 is Mt+1 ≡ At+1 + �n i ai,t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The monetary policy considered here may sound far-fetched, but it is very simple to implement with blockchain-based tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In particular, the fact that the tokens used for bidding may grow at a different rate than the other tokens is inspired by staking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In staking, those who “lock” some tokens (or, more in general, do not use them) are rewarded with additional tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Here the staking reward is positive if σt < 0, in which case those who do not use the tokens receive an additional reward relative to those who use them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The above auction with tokens is as close as possible to a traditional auction with dollars: bids and payments are expressed in dollars but need to be settled 3 Equilibrium 11 using tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' However, other assumptions are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' For example, bidders may be required to bid by submitting tokens, which could then be partially returned to the bidders after the winner is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='13 Also, here I consider only two possible monetary policies, but many more are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The bottom line is that the above is the least complex auction with tokens and not the most general auction with tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 3 Equilibrium I first consider the auction with dollars and then the auction with tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='1 Auction with dollars When the auction uses dollars, all the standard results from auction theory apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='14 Quite immediately, in every period, the revenue equivalence theorem holds: all standard auction formats generate the same expected revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Also, given our assumption on the distribution of valuations, revenues are maximized when the reservation price is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' By considering a second-price auction, in every period, expected revenues are k ≡ E[vMax−1,t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' where vMax,t ≡ maxi{vi,t} is the realized highest valuation in period t, and vMax−1,t ≡ maxi̸=Max{vi,t} is the realized second-highest valuation in period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Also, each bid- der’s expected payoff from the auction is g ≡ E[max{vi − vMax−1 t , 0}] 13 In this case, bidders need to purchase tokens before bidding, which means that the equilibrium price of tokens may reveal some information relative to the realized distribution of valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, the equilibrium on the market for tokens should be a rational expectation equilibrium, which opens several additional complications, including the existence of the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 14 See, for example, Klemperer (1999), in particular Section 4 (for the revenue equivalence theo- rem) and Appendix B (how to calculate the optimal reservation price).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 3 Equilibrium 12 Hence, from period-1 viewpoint, the present-discounted value of the expected rev- enues of the auction with dollars are: ΠUSD = k T � t=1 βt−1 = 1 − βT 1 − β k and the present discounted value of participating in the auction as a bidder is uUSD = g T � t=1 βt−1 = g1 − βT 1 − β For a given auction format, the auctioneer’s utility is UUSD = max w2≥0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=',wT ≥0,wT +1=0 � T � t=1 βt−1EU � n � i bi,t + wt − wt+1 R � , � , where EUt() is the auctioneer’s expected utility in period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Note that, if the auc- tioneer is risk averse (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' his utility function is strictly concave), not all common auction formats maximize expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The reason is that the variance of the revenues also matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Nonetheless, for our purposes, it is enough to establish an upper bound to the auctioneer’s utility, as the next lemma does (its proof is omit- ted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Consider a given auction format, a given level of initial assets w1, and realized period-1 revenues �n i bi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Define w∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', w∗ T as the unconstrained optimal sequence of assets in the absence of risk, that is {w∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', w∗ T } ≡ argmaxw2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=',wT +1=0 � U � n � i bi,1 + w1 − w2 R � + T � t=2 βT−1U � k + wt − wt+1 R �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' It must be that UUSD ≤ E � U � n � i bi,1 + w1 − w∗ 2 R � + T � t=2 βT−1U � k + w∗ t − w∗ t+1 R �� where the expectation is taken over period-1 revenues �n i bi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The above inequality is strict if U() is strictly concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 3 Equilibrium 13 The above lemma says that, for every possible auction format, the auctioneer’s utility must be lower than the utility when all risks after period 1 are eliminated and credit constraints are removed, allowing him to borrow at the risk-free rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The inequality must be strict if his utility is strictly concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Quite intuitively, if the auctioneer is risk averse, he would rather receive the largest possible expected revenues k with probability 1 in each period rather than be exposed to the variability of these revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Furthermore, the fact that utility is strictly concave also implies that the auctioneer benefits from smoothing consumption across periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Because achieving optimal consumption smoothing may require borrowing, the auctioneer benefits when borrowing constraints are removed (independently of whether risk is also removed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='2 Auction with tokens I start by deriving the price of tokens in a given period t as a function of the following-period expected price of tokens pe t+1 and the realization of bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Consider a given pe t+1 and a given realization of period-t valuations (and hence given profile of bids mi,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', mn,t and payments bi,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', bn,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The demand for tokens in period t is � i bi,t pt + St where St ≥ 0 is the speculative demand for tokens, that is, the demand for tokens not used for paying the auctioneer in period t, and is defined as St = max � Mt − �n i bi,t β(1 + τt)pe t+1 , 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The equilibrium period-t price is: pt = max �� i bi,t Mt , β(1 + τt)pe t+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' (1) The most important observation is that some tokens may be purchased not for bidding but for speculative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' This happens in equilibrium when the realized distribution of valuations is such that total payments to the auctioneer are low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In 3 Equilibrium 14 this case, if the demand for tokens was determined exclusively by the tokens used for bidding, we would have pt < β(1 + τt)pe t+1 which cannot be an equilibrium because the return on investing in tokens would be strictly greater than that of investing in the risk-free asset (remember that the risk-free asset generates a present-discounted return equal to βR = 1 and that tokens held for speculation grow by a factor 1 + τt between periods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The possibility of purchasing tokens for speculation implies that the period-t price of tokens has a lower bound at β(1 + τt)pe t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The above lemma can be used to derive the bidders’ incentives to bid for a given sequence of token prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If pt ≥ β(1 + τ)pe t+1 then bidders do not want to hold any token between the two periods, hence ai,t+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If instead pt = β(1 + τ)pe t+1, then holding tokens for speculation generates zero utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Bidders are therefore indifferent between holding any amount of tokens between period t and period t+1, including zero tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' It follows that, in both cases, bidder i’s utility as a function of pt, pe t+1 and the profile of bids is: \uf8f1 \uf8f2 \uf8f3 vi,t + ptai,t − bi,t + βut+1(0) if i is the winner ptai,t − bi,t + βut+1(0) otherwise, (2) where ut+1(ai,t+1) is the expected continuation utility from period t + 1 onward, as a function of the tokens owned at the start of period t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' It immediately follows that, for a given auction format, the bidders’ incentives to bid are the same with or without tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Because of this, the revenue equivalence theorem holds also here: any standard auction format with a reservation price of zero maximizes the expected payment to the auctioneer, which is E[� i bi,t] = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' As a consequence, the bidders’ expected payoff from participating in the auction is, again, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Importantly, (2) also implies that the bidders’ payoffs depend both on the ex- pected payoff from participating in the auction g, and on the expected value of the tokens held at the beginning of the period pe t · ai,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='15 I can therefore take the expec- tation of (2), use the fact that the expected payoff from participating in the auction 15 It is useful to think of each bidder selling all their tokens and earning ptai,t, while simultane- ously purchasing tokens to bid bi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The expected cost of the bid is part of the expected payoff from the auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 3 Equilibrium 15 is g, and write bidder i’s expected continuation utility as: ut(ai,t) = pe tai,t + g T � t=1 βt−1 = pe tai,t + g1 − βT 1 − β Note that because ai,1 = 0, then a bidder’s expected continuation utility from period 1 viewpoint is the same as in the auction with dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' But because ai,t could be positive for t ≥ 2, then the time profile of the bidder’s utility may differ in the auction with tokens from the auction with dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Having determined the bidders’ expected utility, I can now derive the auctioneer’s expected revenues, as the next proposition does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Proposition 1 (Expected revenues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Consider a given sequence of equilibrium ex- pected prices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', prices such that equation 1 holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' At the beginning of each period t ≤ T, the present-discounted value of the auctioneer’s future expected revenues is ΠTokens,t(At) = 1 − βT 1 − β k − pe t(Mt − At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The key observation is that the speculative demand for tokens in period t in- creases the price of tokens and the auctioneer’s revenues in that period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' At the same time, speculators compete against the auctioneer on the market for tokens in period t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The proposition shows that, in expected terms, the two effects cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' As a consequence, the present discounted value of the auctioneer’s expected revenues at the beginning of the game is ΠTokens,1(M1) = 1−βT 1−β k, as in the auction with dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In any subsequent period, the auctioneers’ continuation revenues may be lower than those in the auction with dollars by an amount equal to the value of the tokens not held by the auctioneer in that period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Finally, note that the above proposition highlights the effect of the speculative demand for tokens on the time profile of the auctioneer’s revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The speculative demand for tokens, however, has an additional effect on the variability of these revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The reason is that the speculative demand for tokens depends on the future expected price and therefore transforms uncertain future revenues into certain present revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Crucially, the incentive to purchase tokens for speculation—and, as a conse- quence, the time profile and variability of revenues—depends on the monetary pol- 3 Equilibrium 16 icy specified by the auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If σt is sufficiently large for all t ≤ T, then in every period, the auctioneer creates and keeps for himself a large number of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' As a consequence of the resulting inflation, the speculative demand for tokens is, there- fore, zero, and, like in the auction with tokens, the auctioneer earns � i bi,t dollars in every period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' As σt decreases, the incentive to purchase tokens for speculation increases, causing revenues to accrue earlier and to be less uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The case σt = −1 is therefore particularly relevant: bidders purchase all tokens in period 1, and progressively use them to pay the auctioneer, who then destroys these tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The next proposition derives the auctioneer’s revenues for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Proposition 2 (Revenues when σt = −1 for all t ≤ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If σt = −1 for all t ≤ T, then the auctioneer earns revenues exclusively in period 1, which are p1M1 = � i bi,t + β1 − βT−1 1 − β k (3) Quite intuitively, when the auctioneer destroys all tokens received as payment, then he earns revenues only in period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The revenues pertaining to period 1 are subject to risk, and the present-discounted value of the expected revenues pertaining to all following periods is earned with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='16 Note also that because all revenues are earned in period 1 and all risk past period 1 is eliminated, the auctioneer can achieve optimal consumption smoothing by simply saving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, for every possible realization of � i bi,t, the auctioneer achieves utility: U � n � i bi,1 + Rw1 − w∗ 2 � + T � t=2 βT−1U(k + Rw∗ t − w∗ t+1) where w∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', w∗ T are defined in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Lemma 1 then immediately implies the following corollary: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The auction with tokens with σt = −1 for all t ≤ T is the preferred auction with tokens, strictly so if U() is strictly concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' It is also preferred to the 16 There is a straightforward extension of the model in which the auctioneer is allowed to sell tokens already in period 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In that case, also the variability of period-1 revenues is eliminated because the auctioneer earns the total present-discounted value of the expected revenues in period 0 with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 4 Extension: pledging an object vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' pledging cash 17 auction with dollars, strictly so if U() is strictly concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='3 Discussion: traditional financial instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' A natural question is whether the optimal auction with tokens can be replicated by holding an auction with dollars while simultaneously issuing a traditional financial instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The answer is, quite clearly, yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' To see this, suppose that the auctioneer holds an auction with dollars and also issues equity in period 1: investors pay the auctioneer for the right to receive all future cash flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' To ease comparison with the auction with tokens, assume that equity is sold to investors at the end of period 1, just before consumption occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Furthermore, suppose the auctioneer sells 1 unit of equity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In equilibrium, investors must be indifferent between purchasing equity or not, which implies that the price at which the auctioneer can sell equity is P1 = β 1 1 − β k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The auctioneer’s period-1 revenues (from the sale of the object and equity) are � i bi,1 + β 1 1 − β k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Because the auctioneer does not earn revenues in any subsequent period, the equi- librium is, therefore, identical to the optimal auction with tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 4 Extension: pledging an object vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' pledging cash When issuing tokens, the auctioneer pledges to deliver the object to investors in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' When issuing equity, instead, the auctioneer pledges to deliver the revenues earned via the sale of the object to investors in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In the model discussed above, the absence of any friction makes the two identical under an appropriate monetary policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' However, in a more realistic environment, pledging revenues may be easier or more difficult than pledging an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In particular, note that any fric- tion affecting the ability of the auctioneer to pledge the object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', non-contractible 4 Extension: pledging an object vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' pledging cash 18 effort) will necessary also affect the ability to pledge revenues that are generated from the sale of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' But there may be frictions that affect the ability to pledge revenues but not the object—for example, when cash is easier to hide than the object—therefore making revenues harder to pledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' At the same time, tradi- tional financial instruments can be very flexible and specify contingent payments in a way that cannot be replicated with tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' To explore the interaction of these two effects, here I modify the above model by introducing both effort and the possibility of misappropriating revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The auction runs for T = 2 periods, and the risk-free asset has a return of 1 so that R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The auctioneer is also risk-neutral and cash-abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Differently from the above model, here the auctioneer is less patient than the investors: whereas investors do not discount the future, the auctioneer’s discount factor is β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='17 The timing of the first period is identical to the model presented earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Unlike the model presented earlier, at the beginning of the second period, the auctioneer exerts effort e ≥ 0 to improve the quality of the object sold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Effort is observable (but non-contractible) and shifts uniformly the distribution of valuations, so that when bidder i draws vi,2 from the distribution F(x), his utility from consuming the object is vi,2 + e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='18 Effort has a quadratic cost equal to e2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Furthermore, after period-2 revenues are realized but before payments to investors are made, the auctioneer can misappropriate revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' More precisely, by paying a cost c, he can consume revenues that should otherwise be forwarded to investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' A familiar micro- foundation for this is the possibility for the auctioneer to “run away with the till” at a cost c, which therefore measures the extent to which revenues can be pledged to investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' First best In the first best, effort is such that its marginal cost equals its marginal benefit, and hence e∗∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' At the same time, because the auctioneer is more impatient than investors, in period 1 investors will pay the auctioneer an amount 17 Clearly, the results derived earlier are robust to this change in assumptions: if fully front loading revenues by setting σt = −1 for all t is optimal when the auctioneer is as patient as investors, it is also optimal when the auctioneer is more impatient than investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 18 Note that, because e is observable, the auction is still in private value despite the valuations having a “common” component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 4 Extension: pledging an object vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' pledging cash 19 equal to the expected future revenues and will earn back these revenues in period 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' As a consequence, the auctioneer earns all his revenues in period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Auction with dollars with outside investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Suppose now that the auction uses dollars and that the auctioneer sells a contingent security to risk-neutral investors, who also do not discount the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Assume that effort is not contractible, but revenues are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The security specifies a payment to the auctioneer in period 1 y1, and a contingent period-2 payment from the auctioneer to investors y2(� i b2,i + e) subject to the investors’ rationality constraint: E[y2( � i b2,i + e)] = y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Note that the possibility of “running away with the till” implies that, in period 2, the auctioneer should always earn at least � bi,2 + 1 − c, or else he would prefer to run away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The incentive compatibility constraint is, therefore: � i b2,i + e − y2( � i b2,i + e) ≥ � i bi,2 + e − c ∀ � i b2,i, e or y2( � i b2,i + e) ≤ c ∀ � i b2,i, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The first important observation is that if c = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', misappropriation of rev- enues is costless), then the incentive compatibility constraint immediately implies y2(� i b2,i + e) = 0: no outside investment is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The choice of effort solves: max e≥0 � k + e − e2 2 � , with solution e∗ = 1, which is efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' However, the time profile of revenues is not efficient because the auctioneer earns his revenues (and consumes) in each period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If instead c is sufficiently large, then the first best can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' To see this, note that if the auctioneer sets the optimal level of effort e∗∗ = 1, then revenues should always be larger than v + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The contract between the auctioneer and the investor can therefore impose the punishment c whenever revenues are below v + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 4 Extension: pledging an object vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' pledging cash 20 If this punishment is sufficiently large, an efficient level of effort is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The revenues generated are transferred to the investor, so that E[y2(� i b2,i + e)] = y1 = k + 1, therefore, implementing the first best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Auction with tokens Without loss of generality, I normalize the initial stock of tokens so that M1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In the second period of the auction with tokens, the speculative demand is zero for any level of effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, optimal period-2 effort solves: max e2≥0 � A2 k + e2 M2 − e2 2 2 � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' M2 = A2 + S1(1 + τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Define α ≡ A2 M2 as the fraction of the total stock of tokens held by the auctioneer at the beginning of period 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The equilibrium effort is e∗ 2(α) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, the period-2 expected price is pe 2 = k + α M2 , and the expected value of the tokens held by investors at the beginning of period 2 is S1(1 + τ)pe 2 = (1 − α)(k + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' (4) For future reference, note that α may have an ambiguous effect on the expected value of tokens held by investors, because higher α implies higher effort and higher expected revenue, but also that a lower fraction of these revenues is earned by investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' More precisely, 4 is strictly increasing in α for α < 1−k 2 and strictly decreasing otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In particular, it is always strictly decreasing for α > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The parameter α depends on the period-1 speculative demand S1, which itself depends on period-1 realized valuations and the monetary policy, as the next lemma 4 Extension: pledging an object vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' pledging cash 21 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' For given � i bi,1 we have: α = min � 1, � k2 + 4 � i bi,1(1 + σ) − k 2 � Hence, α depends on σ but not τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Furthermore, for given � i bi,1, α is zero at σ = −1, it is strictly increasing in σ, and it reaches 1 whenever � i bi,1 < k+1 1+σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, for any given � i bi,1, α can take any value between 0 and 1 as a function of σ (which is chosen by the auctioneer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' More important, for σ such that v < k+1 1+σ the period-1 speculative demand for tokens is always zero and α = 1 for all realizations of � i bi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In this case, the equilibrium is identical to that of the auction with dollars when c = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', without outside investors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Given this, I can write the auctioneer payoff from period-1 viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I use the fact that, in equilibrium, the expected value of the tokens held by investors at the beginning of period 2 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', equation 4) must equal the extra revenues earned by the auctioneer in period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I can therefore write the auctioneer’s period-1 expected utility as k + E[(1 − α)(k + α)] + β � E[α(k + e(α))] − E[e(α)2] 2 � where the expectations are taken with respect to the realization of period-1 valua- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Taking first order conditions with respect to σ yields:19 E � (1 − k − 2α + β(k + α)) ∂α ∂σ � The important observation is that, as long as β < 1 the above expression is negative at σ such that v = k+1 1+σ, that is, σ such that α = 1 with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The opti- mal σ therefore implies an intermediate expected α, balancing the tension between inducing effort and front-loading revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Because the auction with tokens with v = k+1 1+σ is equivalent to the auction with dollars with c = 0, this observations also 19 By the envelope theorem, the effect of σ on the optimal period-2 effort can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 5 Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 22 implies that the optimal auction with tokens is strictly preferred to the auction with dollars with c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The following proposition summarizes these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If c is sufficiently low (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', revenue misappropriation is easy), then the auctioneer is better off holding an auction with tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If c is sufficiently large (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', revenue misappropriation is very costly), then the auctioneer is better off holding an auction with dollars and issuing a traditional financial instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The key takeaway is that the possibility of misappropriating revenues affects the auction with dollars but not the auction with tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Despite this, if the contract- ing space is sufficiently large, then the auction with dollars with an appropriately designed contingent security is preferred to the auction with tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If instead the contracting space is restricted, then the auction with tokens is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 5 Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' By issuing tokens and accepting them as payment, an auctioneer earns revenues earlier and is exposed to less risk relative to a standard, repeated auction in which bidders pay in dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In particular, if the auctioneer commits to destroying all tokens received as payment, he earns all his revenues in period 1 and eliminates all risk from period 2 onward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The same outcome can be achieved in a traditional auction with dollars in which the auctioneer issues equity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The reason for this equivalence is that the auctioneer has full commitment, both to deliver the object to token holders and to deliver the revenues generated by the sale of the object to equity holders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I move away from this assumption by introducing effort and the possibility of misappropriating revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I show that, in this case, the equilibrium in the auction with tokens may differ from that in the auction with dollars in which the auctioneer can issue a financial security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' On the one hand, the possibility of misappropriating revenues is a friction that does not matter when issuing tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' On the other hand, a financial security can specify contingent payments in a way that cannot be replicated with tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I show that if the contracting frictions are severe, then the auction with tokens is preferred, while the opposite is true if the contracting frictions are mild or absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' A Mathematical derivations 23 I considered issuing tokens and issuing a contingent security as alternative ways to raise funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' But they need not be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' For example, an auctioneer may issue tokens, sell some on the open market (to be used for bidding), while simultaneously signing an investment contract with an investor for the delivery of additional tokens in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' This investment contract may also specify penalties in case, for example, the value of the tokens (which depends on the value of the object) falls below a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' This punishment, however, needs to be incentive compatible if the auctioneer can run away to escape it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Exploring this intriguing possibility is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I assumed that the market for tokens is frictionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, holding tokens is inconvenient only because exchanging them for consumption may require waiting one period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' However, this exchange generates no cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' A more realistic view is that the market for tokens has frictions, and these frictions further reduce the benefit of using tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In an even more realistic model, these frictions would depend on the volume of transactions, which itself is a function of the value of the object being exchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' This extension is also left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' A Mathematical derivations Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Consider period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Take the total payments to the auctioneer � i bi,t and the expected future price pe t+1 as given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Note that a bidder can spend 1 dollar to purchase 1 pt tokens in period t, these tokens then multiply by 1+τt and can be sold in period t+1, for a present-discounted return of β(1+τt)pe t+1 pt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Alternatively, he can invest the same amount of money in the risk-free asset for a present-discounted return of βR = 1 in the following period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' It follows that there can be an equilibrium in the market for tokens if and only if pt ≥ β(1 + τt)pe t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If pt > β(1 + τt)pe t+1, no tokens are purchased and then brought to the next period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The demand for tokens is given by the tokens used for bidding � i bi,t pt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The supply of tokens is Mt, and hence the equilibrium price is pt = � i bi,t Mt A Mathematical derivations 24 This is indeed an equilibrium if � i bi,t Mt > β(1 + τt)pe t+1, that is, if the realized bids are sufficiently high relative to the future expected price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If instead � i bi,t Mt ≤ β(1 + τt)pe t+1, then tokens may be purchased but not used for bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I call this demand the speculative demand for tokens St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The total demand for tokens is now St + � i bi,t pt , and the equilibrium price is pt = � i bi,t Mt − St = β(1 + τt)pe t+1 which pins down the speculative demand for tokens: St = max � Mt − �n i bi,t β(1 + τt)pe t+1 , 0 � Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' To start, note the following two facts: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' in each period the auctioneer will liquidate all his tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If the price in a given period is such that pt > β(1 + τt)pe t+1, then the auctioneer is better off selling tokens in period t at a higher price than in period t+1 at a lower price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If instead pt = β(1 + τt)pe t+1 and the auctioneer is risk averse, then again he prefers to earn revenues in period t than to wait one period and earn the same expected revenues (but this time being exposed to risk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If pt = β(1 + τt)pe t+1 and the auctioneer is risk neutral then his expected revenues are the same whether he holds tokens between periods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=', he acts as a speculator) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Without loss of generality, we can assume that he sells all his tokens in period t also in this case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' an implication of the above fact is that, for given pe t+1, At+1 is independent of At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' That is because At+1 depends on the monetary policy parameters σt, τt and on the tokens received for payment in period t, which depend exclusively on pe t+1 and the realization of valuation in period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I can therefore write the auctioneer’s expected continuation revenues in period t as: Πt(At) = pe tAt + βE[pt+1At+1] + β2Πt+2(At+2) A Mathematical derivations 25 where the expectations are taken at the beginning of period t, before the valuations are realized (note: I use Πt() to indicate unconditional expected revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I will write the expectation explicitly if, instead, the expectation is conditional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Consider now a given pe t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The realization of valuations in period t may be such that St = 0 and hence pt = � i bi,t Mt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In this case, At+1 = Mt+1, so that, conditional on St = 0: E[Πt(At)|St = 0] =E[pt|St = 0]At + βE[pt+1At+1|St = 0] + β2E[Πt+2(At+2)|St = 0] =E �� i bi,t Mt |St = 0 � (Mt − Mt + At) + βpe t+1E[Mt+1|St = 0] + β2E[Πt+2(At+2)|St = 0] =E[ � i bi,t|St = 0] − E[pt|St = 0](Mt − At) + βpe t+1E[Mt+1|St = 0] + β2E[Πt+2(At+2)|St = 0] If instead St > 0, then pt = β(1 + τ)pe t+1 and At+1 = Mt+1 − St(1 + τ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' In this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' conditional on St > 0: E[Πt(At)|St > 0] = E[pt|St > 0]At + βE[pt+1At+1|St > 0] + β2E[Πt+2(At+2)|St > 0] = E[pt|St > 0]At + βpe t+1(E[Mt+1|St > 0] − E[St|St > 0](1 + τ)) + β2E[Πt+2(At+2)|St > 0] = E[pt|St > 0]At + βpe t+1E[Mt+1|St > 0] − βpe t+1(1 + τt) � Mt − E[� i bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='t|St > 0] βpe t+1(1 + τt) � + β2E[Πt+2(At+2)|St > 0] = E �� i bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='t|St > 0 � − E[pt|St = 0](Mt − At) + βpe t+1E[Mt+1|St > 0] + β2E[Πt+2(At+2)|St > 0] where I used the definition of St as well as the fact that E[pt|St > 0] = βpe t+1(1+τt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The above derivations then imply that, for a given sequence of expected equilib- rium prices, the unconditional expected revenues are Πt(At) = k−pe t(Mt−At)+βpe t+1E[Mt+1]+β2Πt+2(At+2) = k−pe t(Mt−At)+βΠt+1(Mt+1) (5) where the last equality exploits the fact that, for given equilibrium expected prices, At+2 is independent of At+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The statement then follows by iterating the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The statement follows from Proposition 1, and by noting that all revenues accrue in period 1, after the period-1 valuations are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' It is nonetheless instructive to present a direct proof of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' A Mathematical derivations 26 Because all tokens earned by the auctioneer are burned, in equilibrium, the speculative demand for tokens is strictly positive in every period t ≤ T and for every possible realization of valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Because of this, in every period t and every realization of period t < T valuations, equilibrium prices must be such that bidders are indifferent between holding tokens between any two periods, that is pt = β · (1 + τt) · pe t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' This observation has two implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' First, I can write the sequence of equi- librium prices as p1, pe 2 = p1/(β(1 + τ1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' pe t = p1/ � βt−1 �t−1 s=1(1 + τs) � for t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Second, in every period t ≥ 2 the supply of tokens is St−1(1+τt−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' By equating this supply with the demand for tokens, I can write pt = � bi,t St−1(1 + τt−1) − St so that St−1 = � bi,t pt(1 + τt−1) + St (1 + τt−1) (6) Note that the speculative demand in period T is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Hence, for given � bi,T I can write E[ST−1] = k pT(1 + τT−1) E[ST−2] = k pT−1(1 + τT−2) + k pT(1 + τT−2)(1 + τT−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' S1 = k T � s=2 1 ps �s−1 j=1(1 + τj) Using the fact that pe s = p1/ � βt−1 �s−1 j=1(1 + τj) � , the above expression becomes: S1 = k p1 t−1 � s=1 βs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Finally, use the fact that the initial supply of tokens is M1, so that the period-1 market clearing price is p1 = � bi,1 M1 − S1 , A Mathematical derivations 27 to get the expression: p1M1 = � bi,1 + k t−1 � s=1 βs (7) Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' By using the definition of α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I can write α = (1 − S1)(1 + σ)(1 + τ) (1 − S1)(1 + σ)(1 + τ) + S1(1 + τ) (8) Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' I use the above expression to write pe 2 = k + α (1 − S1)(1 + σ)(1 + τ) + S1(1 + τ) = (k + α)α (1 − S1)(1 + σ)(1 + τ) (9) I use the definition of S1 and the above expression to obtain: S1 = max � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 1 − � i bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='1 pe 2(1 + τ) � = max � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' 1 − � i bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='1 (k + α)α(1 − S1)(1 + σ) � 1 − S1 = min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' � i bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content='1 (k + α)α(1 − S1)(1 + σ) � Suppose α < 1 so that 0 < S1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Then α must be such that α = � � � � �k 2 �2 + ( � i bi,1)(1 + σ) − k 2 this can be the solution as long as α < 1, or k +1 > � i bi,1(1+σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' If instead α = 1, then S1 = 0 and it must be that k + 1 < � i bi,1(1 + σ) A Mathematical derivations 28 Putting everything together, we have that α = min \uf8f1 \uf8f2 \uf8f31, � � � � �k 2 �2 + � i bi,1(1 + σ) − k 2 \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' References Bakos, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Halaburda (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The role of cryptographic tokens and icos in fostering platform 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Bulow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Roberts (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The simple economics of optimal auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Journal of political economy 97(5), 1060–1090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Canidio, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Mann (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Digital tokens and platform building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Working paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Lucas Jr, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Townsend, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Distributed Ledgers: Design and Regulation of Financial Infrastructure and Payment Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' MIT Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Vickrey, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' Counterspeculation, auctions, and competitive sealed tenders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} +page_content=' The Journal of finance 16(1), 8–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFST4oBgHgl3EQfaDjo/content/2301.13794v1.pdf'} diff --git a/8tE5T4oBgHgl3EQfQw5Q/content/tmp_files/2301.05515v1.pdf.txt b/8tE5T4oBgHgl3EQfQw5Q/content/tmp_files/2301.05515v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6eae56f6482df52b0d1626bc8b54956ff6c60e55 --- /dev/null +++ b/8tE5T4oBgHgl3EQfQw5Q/content/tmp_files/2301.05515v1.pdf.txt @@ -0,0 +1,466 @@ + + + + + +Transfer of multi-DNA patches by colloidal stamping +Rawan Khalaf,a† Andrea Viamonte,a Etienne Ducrot,*a Rémi Mérindol*b and Serge Ravaine*a +Patchy particles have received great attention due to their ability to develop directional and selective interactions and serve +as building units for the self-assembly of innovative colloidal molecules and crystalline structures. However, synthesizing +particles with multiple dissimilar patches is still highly challenging and lacks efficient methods, these building blocks would +open paths towards a broader range of ordered materials and their inherent properties. Herein, we describe a new approach +to pattern functional DNA patches at the surface of particles, by use of colloidal stamps. DNA inks are transferred only at +the contact zones between the target particle and the stamps thanks to selective strand-displacement reactions. The +produced DNA-patchy particles are ideal candidates to act as advanced precision/designer building blocks to self-assemble +the next generation of colloidal materials. +Introduction +Over the past decades, scientists have aspired to fabricate +functional materials by colloidal self-assembly.1 Although many +beautiful examples of self-assembled colloidal molecules2 or +colloidal crystals3 from particles with well-defined shapes and +composition have been reported so far, colloidal systems +cannot be targeted towards most of the sophisticated +structures that Nature built. Indeed, the latter require encoding +the building units with information to guide their self-assembly +by programming their geometry as well as the directionality, +valence, range of their pair interactions. Several strategies have +been developed to address this challenge, including the +attachment of molecules that recognize one another onto the +surface of particles.4 Among the wide range of binding groups +that have been employed, synthetic DNA strands have been +proven to be very versatile and promising as a tremendous +number of orthogonal interactions can be programmed based +on the design of nucleotide sequences, giving access to highly +specific programmable interactions. DNA-coated particles have +thus been extensively employed as building blocks for the self- +assembly of clusters with precise symmetries5 and crystalline +lattices.6 To further control both the valence of the particles and +the directionality of the bonds they form with their partners, a +number of groups have recently proposed strategies to +regioselectively pattern particles with DNA patches.7 Sleiman et +al. successfully transferred DNA motifs from a parent 3D DNA +template to gold8 and polymeric nanoparticles.9 Two-10 and +three11-dimensional DNA origami structures were used as +stamping platforms to transfer DNA inks onto gold +nanoparticles. In both cases, the printed nanoparticles were +released from the frame by a strand displacement reaction.12 +In order to create micron-sized particles with several dissimilar +patches, we translate this strategy by using colloidal particles +coated with DNA inks as stamps. The colloidal stamps can +assemble with support colloids via DNA hybridization. The +injection of eject strands allowed us to transfer the DNA inks at +the contact zones between support and stamp particles (Fig. 1a) +leading to patchy particles. We also take benefit of packing +constraints to control the number of stamp particles that can +park around the support ones5a, which finally defines the +number of transferred patches. +Experimental +Synthesis of DNA-coated particles +Azidated 3-(trimethoxysilyl)propyl methacrylate (TPM) particles +were prepared through the azidation of chlorine groups present +at the surface of particles previously synthesized according to +the protocol developed by Wang et al.6e (see ESI). After +synthesis the particles were imaged by TEM and SEM. Fig. S1 +shows that they are spherical and monodisperse in size. Their +surface is smooth, which has been shown to be required to +allow an homogeneous distribution of DNA strands during the +former step.13 In order to functionalize monodisperse +polystyrene (PS) particles with azide groups, we followed the +protocol developed by Oh et al.,6d which relies on the physical +entrapment of an azidated PS-b-PEO copolymer (PS-b-PEO-N3). +For this, the PS particles are swollen with tetrahydrofuran (THF) +to allow the PS block of the copolymer to penetrate in the PS +particles (see ESI). After evaporation of the THF, the PS block of +the copolymer is physically trapped in the PS particles while the +PEO block and the terminal azide group form a brush at the +surface, swollen by water and exposed to the surrounding +media. Azidated particles were further functionalized with DNA +following the protocol described by Wang et al.6e that ensures +a dense surface coverage of the colloids with DNA (see ESI). The +process +relies +on +the +strain-promoted +azide-alkyne +cycloaddition (SPAAC) to graft DNA strands end-functionalized +with a dibenzocyclooctyne moiety (DBCO) onto azide +functionalized particles. + +Characterization +Transmission electron microscope (TEM) images were taken +using a Hitachi H600 microscope operating at an acceleration +voltage of 75 kV. The samples for TEM observation were +supported on conventional carbon-coated copper grids. +Scanning electron microscope (SEM) images were taken using a +Hitachi S4500 microscope at an accelerating voltage of 5 kV. +Confocal fluorescence microscopy images were taken using a +Leica SP2 confocal laser scanning microscope as well as a ZEISS +LSM980 equipped with an Airyscan detector. + + + +Results and discussion +DNA strands A and B (Table S1) have first been grafted onto +azidated TPM and PS particles, respectively. The coated +particles are referred to as TPMA and PSB. We then +functionalized/inked the stamp TPMA particles with the Ink565 +(see Table S1 for details) by adding a large excess of ink to a +suspension of particles (Fig. 1b). The ink consists in two +hybridized DNA strands, T-X-A* and X*565-B*, the latter being +modified with the fluorescent dye Atto565. As the domain A* is +complementary to the sequence A at the surface of the TPM +particles, the ink sticks on their surface due to the formation of +A/A* duplexes. The particles and strands were maintained in a +buffer enriched in magnesium ions and at low temperature in +order to strengthen the DNA duplexes and prevent strand +migration (see ESI). The excess of ink was subsequently washed +away by centrifugation/dispersion steps. As Ink565 is also +complementary to strands B, the TPMA~Ink565 stamp particles +were mixed with a 40:1 excess of PSB particles to form +preferentially small clusters with only one TPMA particle at the +core and PSB satellites (Fig. 1c). This ultimately maximizes the +number of one-patch PS particles produced in the process. +When the assembly is completed, Y*488-B* strands (labelled +with Alexa488) are added to hybridize with the remaining B +strands available on the PSB particles and passivate their surface +(Fig. 1d). Fig. 2a shows that clusters made of one TPMA particle +surrounded by PSB particles were obtained. A detailed analysis +reveals that the number of PSB particles in the clusters varies +from 1 to 7. Their relative amounts have been determined by +statistical analysis performed over 100 clusters (Fig. 2c-i). Some +rare clusters (~5 %) formed of one PSB particle in contact with +two TPMA particles could be observed as well as a large amount +of free PSA particles. Thanks to the density difference between +TPM (1.2 g.cm-3 14) and PS (1.06 g.cm-3), we successfully +removed most of these free PS particles (Fig. 2b) by +sedimentation in a PBS based buffer solution of intermediate +density (1.07 g.cm-3) prepared by mixing H2O and D2O (see ESI). +The relative proportions of the different clusters remained +unchanged during this purification stage, as indicated in Fig. 2c- +i, proving that the clusters are sufficiently robust and do not +break during centrifugation. The final step to form patchy +particles consists in disassembling the clusters formed by the + + + +Fig. 1 General scheme and key steps. a) General schematic representation of the preparation of particles with DNA patches by colloidal stamping following sequentially the +subsequent steps. b) Inking: the ink, formed by the association of T-X-A* and X*-B*, is hybridized at the surface of the bare stamp particle decorated with an A* DNA brush. c) +Assembly: in the contact zone, formation of duplexes between the support particle decorated by B strands and the stamp thanks to the B* domain exposed by the ink. d) Passivation: +the surface of the support particle, outside of the contact zone is passivated by hybridization between the B strands of the surface and Y*-B* strands. e) Eject: strand displacement +reaction to separate the stamp from the support particle, leading to the formation of a X*-B* patch at the contact zone and the recovery of patchy particles exposing the stand Y* +on the surface with patches of X*. f) Schematic representation of the DNA strands and assemblies used in this study. + +a) +Ink +Passivation +Eject +Stamp +Assembly +★NVV +Support +b) +c) +(p +e) +Contact +zone +Inking +Assembly +Passivation +Eject +f) +A +B +T-X-A* +X*-B* +Ink +Y*-B* +X*-T* +Fig. 2 Confocal fluorescence microscopy images of clusters obtained by incubating +1.6 µm TPMA~Ink565 and 1.5 µm PSB particles in a 1:40 ratio a) before and b) after +purification. c-i) Zoom on the different clusters containing 1 to 7 PS particles. Their +relative amounts (in %) in the sample before/after purification are given at the top right +corner of each image. j) Confocal fluorescence microscopy image of the patchy PS +particles resulting from the strand-displacement disassembly of TPMA~Ink565~PSB +clusters. Scale bar of 10 µm for a, b and j and 5 µm for c to i. +stamp and support particles leaving the fluorescent part of the +ink on the support particle only at the contact point between +them. To do so, we injected the EjectX strand which binds to the +toehold T of Ink565 and replaces the strand X*565-B*. This breaks +the duplex X/X* that was holding the stamp and support +particles and results in the release of the support particle, which +still carries the red fluorescent strands X*565-B* at the former +contact point with the stamp (Fig. 1e and S2). Fig. 2j and Movie +S1 show that PS particles with one red fluorescent patch are +mostly obtained, validating the developed strategy. +Some non-patchy PS particles that were not completely +removed by centrifugation and a few two-patch particles +(~5 %), which result from the disassembly of the clusters in +which one PS particle is in contact with two TPM particles are +also observed. +We firstly extended our strategy to prepare particles with +multiple identical patches precisely located at their surface. To +do so, we prepared clusters with different controlled +morphologies by the random parking5a of an excess of large PS +spheres functionalized with DNA strands A and Ink565 on smaller +TPM particles functionalized with DNA strands B. Due to packing +constraints inherent in the ratio of radii between large and small +spheres, only a fixed number of large spheres can park, leading +to a population of clusters with well-defined coordination. +When the assembly is completed, the Y*488-B* strands, +complementary to DNA strands B, are added to passivate the +surface of TPMB particles. Fig. 3a-c shows confocal images of the +DNA-colloidal clusters obtained when TPM particles with a +diameter of 1.1, 1.6, and 2 µm are employed, respectively. +Different clusters made of one TPM core and different numbers +of PS satellites are observed (Fig. 3d-g and S3). The relative +proportions of each type of clusters obtained for different +values of the size ratio α of PS/TPM, are listed in Table 1. One +can note that when α is 4.39, clusters made of one or two PS +particles attached to one TPM sphere are mainly formed. +Decreasing α to 3.02 and 2.42 led to the formation of higher +proportions of clusters containing 3 and 4 PS particles, as +expected. +Fig. 3 Confocal fluorescence microscopy images of the clusters obtained by incubating +TPMB and PSA~Ink565 in a 1:40 ratio followed by the addition of the Y*488-B* passivation +strand. The diameter of the TPM particles is: a) 1.1 µm (α = 4.39), b) and d-g) 1.6 µm (α += 3.02), c) 2 µm (α = 2.42). h) Confocal fluorescence microscopy images (Alexa488, green +channel; Atto565, red channel) with transmission microscopy (grey channel) of +PSA~Ink565~TPMB & Y*488-B* (α = 3.02) clusters after strand displacement reaction using +EjectX strand. i-l) Zoom on the patchy TPM particles with an increasing number of patches +obtained after strand displacement reaction using EjectX (α = 3.02). Scale bar of 10 µm +for a to h and 2 µm for i to l. +Table 1 Compositions of the batches resulting from the mixing of TPMB of +different sizes with PSA~Ink565 in a 1:40 number ratio determined by statistical +analysis of confocal fluorescence images over about 100 clusters. + +α + + + + + +4.39 +3 +39 +50 +8 +0 +3.02 +0 +20 +29 +44 +7 +2.42 +0 +3 +27 +39 +31 + +After injection of the eject strands EjectX in the clusters +suspension, non-fluorescent PS particles and TPM particles with +red fluorescent patches are observed, proving the transfer of +the fluorescent DNA from PSA~Ink565 (Fig. 3h and S3). More +precisely, Fig. 3i-l show that 1.6 µm TPM particles with one to +four patches are obtained. Similar results were obtained with +1.1 µm and 2 µm TPM particles (Fig. S5), validating the strategy +based on the combination of colloidal parking and colloidal +stamping. + +3/2d) +9/6 +el +12/15 +30/32 +6 +TE/EE +h) +5/4 +3/2b1 +h) +00 +88 +(p +e +9) +i) +k) +Lastly, we further extended our strategy to prepare particles +with multiple dissimilar patches. We first divided the PS +particles into two batches, and coated one batch with Ink565 and +the other with Ink647 (Table S1). The two batches were then +mixed together and TPMB particles functionalized with B +strands +were +added +in +a +number +ratio +TPMB:PSA~Ink565:PSA~Ink647 of 1:20:20. The sample was kept in +the fridge at 4 ℃ for 24 h to maximize the formation of clusters. +Then, Y*488-B* strands were added to hybridize with the B +strands outside of the contact zones and passivate the surface +of the TPMB particles. + +Fig. 4 Confocal fluorescence microscopy images of the clusters obtained by incubating +TPMB, PSA~Ink565 and PSA~Ink647 in a 1:20:20 number ratio followed by the addition of the +passivation strand Y*488-B*. The diameter of the TPM particles is 1.6 µm (α = 3.02). +k) Confocal fluorescence microscopy image of the patchy particles obtained after +injection of EjectX and EjectZ. The inset shows a TPM particle with one red and one blue +fluorescent patches. l) Confocal fluorescence microscopy image of a cluster made of one +TPM particle surrounded by one PSA~Ink488, one PSA~Ink565 and one PSA~Ink647 particles. +m) Confocal fluorescence microscopy image of the patchy 1.6 µm TPM particles obtained +after injection of EjectX, EjectY and EjectZ. Scale bar of 10 µm for a, 5 µm for b to m, and +2 µm for k inset. +Different clusters made of one TPM core and different numbers +of PSA~Ink565 and PSA~Ink647 are observed (Fig. 4a-j). The relative +proportions of each type of clusters are listed in Table 2. After +injection of the eject strands EjectX and EjectZ in the clusters +suspension, non-fluorescent PS particles and TPM particles with +red and/or blue fluorescent patches are observed, proving the +transfer of the fluorescent DNA from PSA~Ink565 and PSA~Ink647 +(Fig. 4k and S6 and Movie S2). When we worked with three +batches of PSA particles coated with Ink488, Ink565 and Ink647, +respectively, and mixed them with TPMB particles in a number +ratio TPMB:PSA~Ink488:PSA~Ink565:PSA~Ink647 of 1:13:13:13, we +observed the formation of a few clusters made of one TPM +particles surrounded by varying numbers of PSA~Ink488, +PSA~Ink565 and PSA~Ink647 particles (Fig. 4l). After injection of +EjectX, EjectY and EjectZ, TPM particles with red and/or blue +and/or green fluorescent patches are observed, proving once +again the efficiency of our approach (Fig. 4m and Movie S3). + +Table 2 Compositions of the batches resulting from the mixing of TPMB with PSA~Ink565 +and PSA~Ink647 in a 1:20:20 number ratio determined by statistical analysis of confocal +fluorescence images over about 100 clusters. + + + + + + + + + + +7 +22 +26 +8 +19 +13 +0 +2 +2 +1 +Conclusions +In conclusion, we have synthesized micron-sized particles with +one or several identical and distinct DNA patches by combining +colloidal parking and the transfer of DNA strands at the contact +zones with colloidal stamps thanks to strand-displacement +reactions. Our strategy is versatile and can be extended to a +gallery of hard particles and soft systems whose shape and size +could be independently varied, opening the way to the +synthesis of new DNA-patchy building blocks and the +comprehensive study of their assembly into novel structures, +such as alternating polymers or rings, dendrimers or gyroid +crystals. +Author Contributions +R. Mérindol, E. Ducrot and S. Ravaine conceived this research. +R. Mérindol and E. Ducrot designed the experimental process +and revised the manuscript. R. Khalaf performed most of the +experiments. A. Viamonte helped R. Khalaf to perform some of +the experiments. R. Khalaf, E. Ducrot and S. Ravaine performed the +data analysis. S. Ravaine wrote the manuscript. +Conflicts of interest +There are no conflicts to declare. +Acknowledgements +The authors thank S. Yao and I-S. Jo for the functionalization of +the copolymer. R. Khalaf thanks the French Ministry of Higher +Education, Research and Innovation and Campus France for her +PhD grant. This work was supported by the Agence Nationale de +la Recherche (POESY project, ANR-18-CE09-0019). We +acknowledge funding from IdEx Bordeaux, a program of the +French government managed by the Agence Nationale de la +Recherche (ANR-10-IDEX-03-02), and from Région Nouvelle- +Aquitaine (AAPR 2020-2019-8330510). +References +1 (a) X. Bouju, E. Duguet, F. Gauffre, C. R. Henry, M. L. Kahn, P. +Mélinon and S. Ravaine, Adv. Mater., 2018, 30, 1706558; (b) +G. M. Whitesides and M. Boncheva, Proc. Natl. Acad. Sci. USA, + +b) +c) +(p +e +f +g) +h) +k) +m)02002, 99, 4769–4774; (c) D. Frenkel, Science, 2002, 296, 65- +66. +2 (a) R. Mérindol, E. Duguet and S. Ravaine, Chem. Asian J., +2019, 14, 3232-3239; (b) W. Li, H. Palis, R. Mérindol, J. +Majimel, S. Ravaine and E. Duguet, Chem. Soc. Rev., 2020, 49, +1955-1976. +3 (a) H. Zheng and S. Ravaine, Crystals, 2016, 6, 54; (b) Z. 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Bhan, R. W. Verweij, H. C. Frijters, Z. +Gong, A. D. Hollingsworth, S. Sacanna and D. J. Kraft, +Langmuir, 2017, 33, 8174–8180. + + diff --git a/8tE5T4oBgHgl3EQfQw5Q/content/tmp_files/load_file.txt b/8tE5T4oBgHgl3EQfQw5Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..010af1aff27a0fe396f1345dfa2e90513397317e --- /dev/null +++ b/8tE5T4oBgHgl3EQfQw5Q/content/tmp_files/load_file.txt @@ -0,0 +1,528 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf,len=527 +page_content='Transfer of multi-DNA patches by colloidal stamping Rawan Khalaf,a† Andrea Viamonte,a Etienne Ducrot,*a Rémi Mérindol*b and Serge Ravaine*a Patchy particles have received great attention due to their ability to develop directional and selective interactions and serve as building units for the self-assembly of innovative colloidal molecules and crystalline structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' However, synthesizing particles with multiple dissimilar patches is still highly challenging and lacks efficient methods, these building blocks would open paths towards a broader range of ordered materials and their inherent properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Herein, we describe a new approach to pattern functional DNA patches at the surface of particles, by use of colloidal stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' DNA inks are transferred only at the contact zones between the target particle and the stamps thanks to selective strand-displacement reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The produced DNA-patchy particles are ideal candidates to act as advanced precision/designer building blocks to self-assemble the next generation of colloidal materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Introduction Over the past decades, scientists have aspired to fabricate functional materials by colloidal self-assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='1 Although many beautiful examples of self-assembled colloidal molecules2 or colloidal crystals3 from particles with well-defined shapes and composition have been reported so far, colloidal systems cannot be targeted towards most of the sophisticated structures that Nature built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Indeed, the latter require encoding the building units with information to guide their self-assembly by programming their geometry as well as the directionality, valence, range of their pair interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Several strategies have been developed to address this challenge, including the attachment of molecules that recognize one another onto the surface of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='4 Among the wide range of binding groups that have been employed, synthetic DNA strands have been proven to be very versatile and promising as a tremendous number of orthogonal interactions can be programmed based on the design of nucleotide sequences, giving access to highly specific programmable interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' DNA-coated particles have thus been extensively employed as building blocks for the self- assembly of clusters with precise symmetries5 and crystalline lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='6 To further control both the valence of the particles and the directionality of the bonds they form with their partners, a number of groups have recently proposed strategies to regioselectively pattern particles with DNA patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='7 Sleiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' successfully transferred DNA motifs from a parent 3D DNA template to gold8 and polymeric nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='9 Two-10 and three11-dimensional DNA origami structures were used as stamping platforms to transfer DNA inks onto gold nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' In both cases, the printed nanoparticles were released from the frame by a strand displacement reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='12 In order to create micron-sized particles with several dissimilar patches, we translate this strategy by using colloidal particles coated with DNA inks as stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The colloidal stamps can assemble with support colloids via DNA hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The injection of eject strands allowed us to transfer the DNA inks at the contact zones between support and stamp particles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 1a) leading to patchy particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' We also take benefit of packing constraints to control the number of stamp particles that can park around the support ones5a, which finally defines the number of transferred patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Experimental Synthesis of DNA-coated particles Azidated 3-(trimethoxysilyl)propyl methacrylate (TPM) particles were prepared through the azidation of chlorine groups present at the surface of particles previously synthesized according to the protocol developed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='6e (see ESI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' After synthesis the particles were imaged by TEM and SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' S1 shows that they are spherical and monodisperse in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Their surface is smooth, which has been shown to be required to allow an homogeneous distribution of DNA strands during the former step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='13 In order to functionalize monodisperse polystyrene (PS) particles with azide groups, we followed the protocol developed by Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=',6d which relies on the physical entrapment of an azidated PS-b-PEO copolymer (PS-b-PEO-N3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' For this, the PS particles are swollen with tetrahydrofuran (THF) to allow the PS block of the copolymer to penetrate in the PS particles (see ESI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' After evaporation of the THF, the PS block of the copolymer is physically trapped in the PS particles while the PEO block and the terminal azide group form a brush at the surface, swollen by water and exposed to the surrounding media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Azidated particles were further functionalized with DNA following the protocol described by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='6e that ensures a dense surface coverage of the colloids with DNA (see ESI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The process relies on the strain-promoted azide-alkyne cycloaddition (SPAAC) to graft DNA strands end-functionalized with a dibenzocyclooctyne moiety (DBCO) onto azide functionalized particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Characterization Transmission electron microscope (TEM) images were taken using a Hitachi H600 microscope operating at an acceleration voltage of 75 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The samples for TEM observation were supported on conventional carbon-coated copper grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Scanning electron microscope (SEM) images were taken using a Hitachi S4500 microscope at an accelerating voltage of 5 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Confocal fluorescence microscopy images were taken using a Leica SP2 confocal laser scanning microscope as well as a ZEISS LSM980 equipped with an Airyscan detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Results and discussion DNA strands A and B (Table S1) have first been grafted onto azidated TPM and PS particles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The coated particles are referred to as TPMA and PSB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' We then functionalized/inked the stamp TPMA particles with the Ink565 (see Table S1 for details) by adding a large excess of ink to a suspension of particles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The ink consists in two hybridized DNA strands, T-X-A* and X*565-B*, the latter being modified with the fluorescent dye Atto565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' As the domain A* is complementary to the sequence A at the surface of the TPM particles, the ink sticks on their surface due to the formation of A/A* duplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The particles and strands were maintained in a buffer enriched in magnesium ions and at low temperature in order to strengthen the DNA duplexes and prevent strand migration (see ESI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The excess of ink was subsequently washed away by centrifugation/dispersion steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' As Ink565 is also complementary to strands B, the TPMA~Ink565 stamp particles were mixed with a 40:1 excess of PSB particles to form preferentially small clusters with only one TPMA particle at the core and PSB satellites (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' This ultimately maximizes the number of one-patch PS particles produced in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' When the assembly is completed, Y*488-B* strands (labelled with Alexa488) are added to hybridize with the remaining B strands available on the PSB particles and passivate their surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 2a shows that clusters made of one TPMA particle surrounded by PSB particles were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' A detailed analysis reveals that the number of PSB particles in the clusters varies from 1 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Their relative amounts have been determined by statistical analysis performed over 100 clusters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 2c-i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Some rare clusters (~5 %) formed of one PSB particle in contact with two TPMA particles could be observed as well as a large amount of free PSA particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Thanks to the density difference between TPM (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='2 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='cm-3 14) and PS (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='06 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='cm-3), we successfully removed most of these free PS particles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 2b) by sedimentation in a PBS based buffer solution of intermediate density (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='07 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='cm-3) prepared by mixing H2O and D2O (see ESI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The relative proportions of the different clusters remained unchanged during this purification stage, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 2c- i, proving that the clusters are sufficiently robust and do not break during centrifugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The final step to form patchy particles consists in disassembling the clusters formed by the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 1 General scheme and key steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' a) General schematic representation of the preparation of particles with DNA patches by colloidal stamping following sequentially the subsequent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' b) Inking: the ink, formed by the association of T-X-A* and X*-B*, is hybridized at the surface of the bare stamp particle decorated with an A* DNA brush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' c) Assembly: in the contact zone, formation of duplexes between the support particle decorated by B strands and the stamp thanks to the B* domain exposed by the ink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' d) Passivation: the surface of the support particle, outside of the contact zone is passivated by hybridization between the B strands of the surface and Y*-B* strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' e) Eject: strand displacement reaction to separate the stamp from the support particle, leading to the formation of a X*-B* patch at the contact zone and the recovery of patchy particles exposing the stand Y* on the surface with patches of X*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' f) Schematic representation of the DNA strands and assemblies used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' a) Ink Passivation Eject Stamp Assembly ★NVV Support b) c) (p e) Contact zone Inking Assembly Passivation Eject f) A B T-X-A* X*-B* Ink Y*-B* X*-T* Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 2 Confocal fluorescence microscopy images of clusters obtained by incubating 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='6 µm TPMA~Ink565 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='5 µm PSB particles in a 1:40 ratio a) before and b) after purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' c-i) Zoom on the different clusters containing 1 to 7 PS particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Their relative amounts (in %) in the sample before/after purification are given at the top right corner of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' j) Confocal fluorescence microscopy image of the patchy PS particles resulting from the strand-displacement disassembly of TPMA~Ink565~PSB clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Scale bar of 10 µm for a, b and j and 5 µm for c to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' stamp and support particles leaving the fluorescent part of the ink on the support particle only at the contact point between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' To do so, we injected the EjectX strand which binds to the toehold T of Ink565 and replaces the strand X*565-B*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' This breaks the duplex X/X* that was holding the stamp and support particles and results in the release of the support particle, which still carries the red fluorescent strands X*565-B* at the former contact point with the stamp (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 1e and S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 2j and Movie S1 show that PS particles with one red fluorescent patch are mostly obtained, validating the developed strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Some non-patchy PS particles that were not completely removed by centrifugation and a few two-patch particles (~5 %), which result from the disassembly of the clusters in which one PS particle is in contact with two TPM particles are also observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' We firstly extended our strategy to prepare particles with multiple identical patches precisely located at their surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' To do so, we prepared clusters with different controlled morphologies by the random parking5a of an excess of large PS spheres functionalized with DNA strands A and Ink565 on smaller TPM particles functionalized with DNA strands B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Due to packing constraints inherent in the ratio of radii between large and small spheres, only a fixed number of large spheres can park, leading to a population of clusters with well-defined coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' When the assembly is completed, the Y*488-B* strands, complementary to DNA strands B, are added to passivate the surface of TPMB particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 3a-c shows confocal images of the DNA-colloidal clusters obtained when TPM particles with a diameter of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='6, and 2 µm are employed, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Different clusters made of one TPM core and different numbers of PS satellites are observed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 3d-g and S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The relative proportions of each type of clusters obtained for different values of the size ratio α of PS/TPM, are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' One can note that when α is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='39, clusters made of one or two PS particles attached to one TPM sphere are mainly formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Decreasing α to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='02 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='42 led to the formation of higher proportions of clusters containing 3 and 4 PS particles, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 3 Confocal fluorescence microscopy images of the clusters obtained by incubating TPMB and PSA~Ink565 in a 1:40 ratio followed by the addition of the Y*488-B* passivation strand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The diameter of the TPM particles is: a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='1 µm (α = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='39), b) and d-g) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='6 µm (α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='02), c) 2 µm (α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' h) Confocal fluorescence microscopy images (Alexa488, green channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Atto565, red channel) with transmission microscopy (grey channel) of PSA~Ink565~TPMB & Y*488-B* (α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='02) clusters after strand displacement reaction using EjectX strand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' i-l) Zoom on the patchy TPM particles with an increasing number of patches obtained after strand displacement reaction using EjectX (α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Scale bar of 10 µm for a to h and 2 µm for i to l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Table 1 Compositions of the batches resulting from the mixing of TPMB of different sizes with PSA~Ink565 in a 1:40 number ratio determined by statistical analysis of confocal fluorescence images over about 100 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' α 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='39 3 39 50 8 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='02 0 20 29 44 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='42 0 3 27 39 31 After injection of the eject strands EjectX in the clusters suspension, non-fluorescent PS particles and TPM particles with red fluorescent patches are observed, proving the transfer of the fluorescent DNA from PSA~Ink565 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 3h and S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' More precisely, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 3i-l show that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='6 µm TPM particles with one to four patches are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Similar results were obtained with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='1 µm and 2 µm TPM particles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' S5), validating the strategy based on the combination of colloidal parking and colloidal stamping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 3/2d) 9/6 el 12/15 30/32 6 TE/EE h) 5/4 3/2b1 h) 00 88 (p e 9) i) k) Lastly, we further extended our strategy to prepare particles with multiple dissimilar patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' We first divided the PS particles into two batches, and coated one batch with Ink565 and the other with Ink647 (Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The two batches were then mixed together and TPMB particles functionalized with B strands were added in a number ratio TPMB:PSA~Ink565:PSA~Ink647 of 1:20:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The sample was kept in the fridge at 4 ℃ for 24 h to maximize the formation of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Then, Y*488-B* strands were added to hybridize with the B strands outside of the contact zones and passivate the surface of the TPMB particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 4 Confocal fluorescence microscopy images of the clusters obtained by incubating TPMB, PSA~Ink565 and PSA~Ink647 in a 1:20:20 number ratio followed by the addition of the passivation strand Y*488-B*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The diameter of the TPM particles is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='6 µm (α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' k) Confocal fluorescence microscopy image of the patchy particles obtained after injection of EjectX and EjectZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The inset shows a TPM particle with one red and one blue fluorescent patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' l) Confocal fluorescence microscopy image of a cluster made of one TPM particle surrounded by one PSA~Ink488, one PSA~Ink565 and one PSA~Ink647 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' m) Confocal fluorescence microscopy image of the patchy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content='6 µm TPM particles obtained after injection of EjectX, EjectY and EjectZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Scale bar of 10 µm for a, 5 µm for b to m, and 2 µm for k inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Different clusters made of one TPM core and different numbers of PSA~Ink565 and PSA~Ink647 are observed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 4a-j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' The relative proportions of each type of clusters are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' After injection of the eject strands EjectX and EjectZ in the clusters suspension, non-fluorescent PS particles and TPM particles with red and/or blue fluorescent patches are observed, proving the transfer of the fluorescent DNA from PSA~Ink565 and PSA~Ink647 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 4k and S6 and Movie S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' When we worked with three batches of PSA particles coated with Ink488, Ink565 and Ink647, respectively, and mixed them with TPMB particles in a number ratio TPMB:PSA~Ink488:PSA~Ink565:PSA~Ink647 of 1:13:13:13, we observed the formation of a few clusters made of one TPM particles surrounded by varying numbers of PSA~Ink488, PSA~Ink565 and PSA~Ink647 particles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 4l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' After injection of EjectX, EjectY and EjectZ, TPM particles with red and/or blue and/or green fluorescent patches are observed, proving once again the efficiency of our approach (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 4m and Movie S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Table 2 Compositions of the batches resulting from the mixing of TPMB with PSA~Ink565 and PSA~Ink647 in a 1:20:20 number ratio determined by statistical analysis of confocal fluorescence images over about 100 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' 7 22 26 8 19 13 0 2 2 1 Conclusions In conclusion, we have synthesized micron-sized particles with one or several identical and distinct DNA patches by combining colloidal parking and the transfer of DNA strands at the contact zones with colloidal stamps thanks to strand-displacement reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Our strategy is versatile and can be extended to a gallery of hard particles and soft systems whose shape and size could be independently varied, opening the way to the synthesis of new DNA-patchy building blocks and the comprehensive study of their assembly into novel structures, such as alternating polymers or rings, dendrimers or gyroid crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Author Contributions R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Mérindol, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Ducrot and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Ravaine conceived this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Mérindol and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Ducrot designed the experimental process and revised the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Khalaf performed most of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Viamonte helped R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Khalaf to perform some of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Khalaf, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Ducrot and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Ravaine performed the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Ravaine wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Conflicts of interest There are no conflicts to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Acknowledgements The authors thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Yao and I-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Jo for the functionalization of the copolymer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Khalaf thanks the French Ministry of Higher Education, Research and Innovation and Campus France for her PhD grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' This work was supported by the Agence Nationale de la Recherche (POESY project, ANR-18-CE09-0019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' We acknowledge funding from IdEx Bordeaux, a program of the French government managed by the Agence Nationale de la Recherche (ANR-10-IDEX-03-02), and from Région Nouvelle- Aquitaine (AAPR 2020-2019-8330510).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' References 1 (a) X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Bouju, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Duguet, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Gauffre, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Henry, M.' metadata={'source': 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+page_content=' Sacanna and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} +page_content=' Kraft, Langmuir, 2017, 33, 8174–8180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE5T4oBgHgl3EQfQw5Q/content/2301.05515v1.pdf'} diff --git a/9NE1T4oBgHgl3EQfUQM_/content/tmp_files/2301.03087v1.pdf.txt b/9NE1T4oBgHgl3EQfUQM_/content/tmp_files/2301.03087v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9eff1bfa53716b510faa423da7ef86d2e9025a0d --- /dev/null +++ b/9NE1T4oBgHgl3EQfUQM_/content/tmp_files/2301.03087v1.pdf.txt @@ -0,0 +1,1830 @@ +Bivariate binomial conditionals distributions with positive and negative correlations: +A statistical study +Indranil Ghosh1, Filipe Marques2, Subrata Chakraborty3 +1University of North Carolina, Wilmington, USA +2Universidade Nova de Lisboa, Portugal +3Dibrugarh University, Assam, India +Corresponding author email address: ghoshi@uncw.edu +Abstract +In this article, we discuss a bivariate distribution whose conditionals are univariate bino- +mial distributions and the marginals are not binomial that exhibits negative correlation. Some +useful structural properties of this distribution namely marginals, moments, generating func- +tions, stochastic ordering are investigated. Simple proofs of negative correlation, marginal over- +dispersion, distribution of sum and conditional given the sum are also derived. The distribution +is shown to be a member of the multi-parameter exponential family and some natural but useful +consequences are also outlined. The proposed distribution tends to a recently investigated con- +ditional Poisson distribution studied by Ghosh et al. (2020). Finally, the distribution is fitted +to two bivariate count data sets with an inherent negative correlation to illustrate its suitability. +Keywords: Bivariate binomial distribution, Conditional specification, Negative and positive cor- +relation, Conditional failure rate, Limiting distribution. +1 +Introduction +The study of correlation between binomial random variables is an important statistical problem +with a lot of theoretical and practical applications and is not new in the literature, see Biswas et al. +(2022) and the references cited therein. There are also some attempts finding bivariate binomial +distributions in other directions. Hamdan et al. (1971) introduced a bivariate binomial distribution +(which is indeed, a bivariate compound Poisson distribution). Hamdan et al. (1976) studied the +joint distribution of the total numbers of occurrences of binary characters A and B, given three +independent samples in which both characters, A but not B, and B but not A, are observed— +effectively derived a bivariate binomial distribution. A symmetric bivariate binomial distribution +was proposed by Le (1984) to analyze clustered samples in medical research. Papageorgiou et al. +(1994) examined mixtures of bivariate binomial distributions which were derived from bivariate- +compounded Poisson distribution. Ling et al. (1989) discussed bivariate binomial distributions from +extension of classes of univariate discrete distributions of order k. Takeuchi et al. (1987) obtained +the sum of 0−1 random variables in the multivariate setup. A bivariate generalization of the three +parameter quasi-binomial distribution of Consul (1979) has been obtained by Mishra et al. (1996). +Crowder et al. +(1989) carried out Bayesian inference, and they defined the bivariate binomial +distribution in a different sense. They defined a two-fold binomial model like X1|m ∼ Bin (m, p) and +X2|X1; m ∼ Bin (X1, q) . For some discussions on bivariate binomial distributions see Kocherlakota +et al. (1992) and Johnson et al. (1997). However, none of the above cited references considered +1 +arXiv:2301.03087v1 [stat.ME] 8 Jan 2023 + +the construction and study of a bivariate distribution such that both the conditionals are binomial +with respective parameters, but the joint distribution may not necessarily be a bivariate binomial +in its traditional sense. Arnold et al. (1999) came up with the construction of a bivariate discrete +distribution starting from two conditional distributions that are binomial. In fact this idea is first +developed in Arnold et al. (1991). It appears that the resulting bivariate discrete model (albeit +ubiquitous normalizing constant) has the salient feature of exhibiting both positive and negative +correlation—this property is not enjoyed by many of the existing bivariate binomial models. In this +article, we explore some useful structural properties of the bivariate discrete distribution originally +proposed by Arnold et al. (1991,1999) and discusses its applicability in modeling bivariate discrete +data exhibiting either negative correlation. The rest of the paper is organized as follows. In Section +2, we introduce the bi-variate binomial conditionals distribution which was described in Arnold +et al. (1991,1999) and provide the expression for associated marginal p.m.f.’s, of X and Y. In +Section 3, we provide several useful structural properties of this distribution. Section 4 provides +the estimation of parameters using sample proportions and also under the maximum likelihood +method. In section 5, we discuss copula-based simulation. A real data application for the BBCD +is presented in section 6. Finally, some concluding remarks are provided in Section 7. +2 +Bivariate binomial conditional distributions +Let us assume the following: +• X|Y = y ∼ binomial (n1, p1(y)) , for each fixed Y = y. +• Y |X = x ∼ binomial (n2, p2(y)) , for each fixed X = x. +According to [2, 1], the associated joint p.m.f. will be +P (X = x, Y = y) = KB (n1, n2, p1, p2, t) +�n1 +x +��n2 +y +� +px +1py +2 (1 − p1)n1−x (1 − p2)n2−y txy, +(1) +where p1 ∈ (0, 1), and p2 ∈ (0, 1), and t > 0, and x = 0, 1, 2, · · · , n1; +y = 0, 1, 2, · · · , n2; and +KB (n1, n2, p1, p2, t) is the normalizing constant and +K−1 +B += K−1 +B (n1, n2, p1, p2, t) = +n1 +� +x=0 +n2 +� +y=0 +�n1 +x +��n2 +y +� +px +1py +2 (1 − p1)n1−x (1 − p2)n2−y txy. +We will denote (henceforth, in short) the bivariate binomial conditionals distribution of the pair +(X, Y ) with the p.m.f. in (1) as BBCD (n1, n2, p1, p2, t) . We list some useful results related to the +normalizing constant that will be utilized later on in deriving some structural properties. +(i) +K−1 +B (n1, n2, p1, p2, 1) += +n1 +� +x=0 +n2 +� +y=0 +�n1 +x +��n2 +y +� +px +1py +2 (1 − p1)n1−x (1 − p2)n2−y += +1. +2 + +(ii) +K−1 +B (n1, n2, p1, p2, t) += +K−1 +B (n2, n1, p2, p1, t) . +(iii) +K−1 +B (n1, n2, p1, 1, t) += +n1 +� +x=0 +�n1 +x +� +px +1 (1 − p1)n1−x pn2 +2 (1 − p2)n2−n2txn2 += +pn2 +2 (1 − p1 + tn2p1)n1 . +(iv) +K−1 +B (n1, n2, 1, p2, t) = pn1 +1 (1 − p2 + tn1p2)n2 . +(v) +0 < K−1 +B (n1, n2, p1, p2, t) +≤ +min{pn1 +1 (1 − p2 + p2tn1)n2 , pn2 +2 (1 − p1 + p1tn2)n1} +≤ +tn1n2. +(vi) +d +dp1 +K−1 +B (n1, n2, p1, 1, t) += +n1 +� +x=0 +n2 +� +y=0 +�n1 +x +��n2 +y +�� +xpx−1 +1 +(1 − p1)n1−x + (n1 − x) px +1 (1 − p1)n1−x−1 +� +py +2 (1 − p2)n2−y txy += p−1 +1 +n1 +� +x=0 +n2 +� +y=0 +x +�n1 +x +��n2 +y +� +px +1py +2 (1 − p1)n1−x (1 − p2)n2−y txy ++ (1 − p2)−1 +n1 +� +x=0 +n2 +� +y=0 +(n1 − x) +�n1 +x +��n2 +y +� +px +1py +2 (1 − p1)n1−x (1 − p2)n2−y txy += p−1 +1 +E(X) +KB ++ (1 − p1)−1 E (n − X) +KB +. +Note that the above result immediately implies that +d +dp1 +log K−1 +B += E(X) +p1 ++ E (n − X) +1 − p1 +. +Next, observe that for t = 1, this reduces to +0 = E(X) +p1 ++ E (n − X) +1 − p1 +=⇒ E (n − X) = +� +p−1 +1 +− 1 +� +E(X). +(vii) Again, by re-writing the expression for KB (n1, n2, p1, p2, t) as +K−1 +B (n1, n2, p1, 1, t) = (1 − p1)n1 (1 − p2)n2 +n1 +� +x=0 +n2 +� +y=0 +�n1 +x +� � +p1 +1 − p1 +�x �n2 +y +� � +p2 +1 − p2 +�y +txy, +3 + +and by writing +p1 +1 − p1 += q1, +p2 +1 − p2 += q2, and S (n1, n2, q1, q2, t) = +n1 +� +x=0 +n2 +� +y=0 +�n1 +x +� +qx +1 +�n2 +y +� +qy +2txy, +we have +K−1 +B (n1, n2, p1, 1, t) = (1 − p1)n1 (1 − p2)n2 S (n1, n2, q1, q2, t) . +Note that +1. The marginal p.m.f. of X will be +P (X = x) += +KB (n1, n2, p1, 1, t) +�n1 +x +� +px +1 (1 − p1)n1−x +n2 +� +y=0 +�n2 +y +� +(txp2)y (1 − p2)n2−y += +KB (n1, n2, p1, 1, t) +�n1 +x +� +px +1 (1 − p1)n1−x +� +1 − p2 + txp2 +�n2 +, +(2) +for x = 0, 1, 2, · · · , n1. +2. Similarly, the marginal p.m.f. of Y will be +P (Y = y) = KB (n1, n2, p1, 1, t) +�n2 +y +� +py +2 (1 − p2)n2−y +� +1 − p1 + typ1 +�n1 +, +(3) +for y = 0, 1, 2, · · · , n2. +3. For fixed t ∈ (0, 1), +P (X = x, Y = y|n1, n2, p1, p2, t) = P (X = y, Y = x|n2, n1, p2, p1, t) . +Thus, if p1 = p2 = p, say then P (X = x, Y = y|n1, n2, p, p, t) = P (X = y, Y = x|n2, n1, p, p, t) . +Furthermore, if n1 = n2 = nsay then P (X = x, Y = y|n, t) = P (X = y, Y = x|n, t) . +Some representative p.m.f. plots for varying parameter choices are provided in Figure 1. +3 +Structural properties +Note that since, X|Y = y ∼ binomial +� +n1, +typ1 +1−p1+typ1 +� +, +E (X|Y = y) = n1 +� +typ1 +1 − p1 + typ1 +� +, +and +V ar (X|Y = y) = n1 +� +typ1 +1 − p1 + typ1 +� � +1 − +typ1 +1 − p1 + typ1 +� +. +4 + +n1 = n2 = 10, p1 = 0.5, p2 = 0.5 +(i) t = 0.95 +(ii) t = 0.5 +(iii) t = 0.05 +0 +5 +10 +0 +5 +10 +0.00 +0.02 +0.04 +0 +5 +10 +0 +5 +10 +0.00 +0.02 +0.04 +0.06 +0 +5 +10 +0 +5 +10 +0.00 +0.05 +0.10 +n1 = n2 = 20, p1 = 0.1, p2 = 0.1 +(iv) t = 0.95 +(v) t = 0.5 +(vi) t = 0.05 +0 +5 +10 +0 +5 +10 +0.00 +0.02 +0.04 +0.06 +0.08 +0 +5 +10 +0 +5 +10 +0.00 +0.05 +0.10 +0 +5 +10 +0 +5 +10 +0.00 +0.05 +0.10 +0.15 +n1 = n2 = 30, p1 = 0.9, p2 = 0.9 +(vii) t = 0.95 +(viii) t = 0.5 +(ix) t = 0.05 +0 +10 +20 +30 +0 +10 +20 +30 +0.00 +0.01 +0.02 +0 +10 +20 +30 +0 +10 +20 +30 +0.00 +0.05 +0.10 +0 +10 +20 +30 +0 +10 +20 +30 +0.00 +0.05 +0.10 +n1 = n2 = 15, p1 = 0.3, p2 = 0.7 +(x) t = 0.95 +(xi) t = 0.5 +(xii) t = 0.05 +0 +5 +10 +15 +0 +5 +10 +15 +0.00 +0.02 +0.04 +0 +5 +10 +15 +0 +5 +10 +15 +0.00 +0.05 +0.10 +0.15 +0.20 +0 +5 +10 +15 +0 +5 +10 +15 +0.00 +0.05 +0.10 +0.15 +0.20 +Figure 1: Examples of p.m.f. plots for the BBCD distribution +5 + +Consequently, +E (X) += +EY (X|Y = y) += +KB (n1, n2, p1, p2, t) +n2 +� +y=0 +� +n1 +� +typ1 +1 − p1 + typ1 +� ��n2 +y +� +py +2 (1 − p2)n2−y += +n1KB (n1, n2, p1, p2, t) +n2 +� +y=0 +�n2 +y +� +(tp2)y (1 − p2)n2−y +� n1−1 +� +j=0 +�n1 − 1 +j +� +(typ1)j (1 − p1)n1−1−j +� += +n1p1KB (n1, n2, p1, p2, t) +n1−1 +� +j=0 +�n1 − 1 +j +� +pj +1 (1 − p1)n1−1−j +� n2 +� +y=0 +�n2 +y +� � +p2tj+1�y (1 − p2)n2−y +� += +n1p1KB (n1, n2, p1, p2, t) +n1−1 +� +j=0 +�n1 − 1 +j +� +pj +1 (1 − p1)n1−1−j +� +1 − p2 + p2tj+1 +�n2 +. +(4) +Alternatively, we can re-write (4) as +E (X) += +KB (n1, n2, p1, p2, t) +n1 +� +x=0 +x +�n1 +x +� +px +1 (1 − p1)n1−x [1 − p2 + p2tx]n2 += +KB (n1, n2, p1, p2, t) +n1 +� +x=0 +x +�n1 +x +� +px +1 (1 − p1)n1−x wx, +(5) +wx = [1 − p2 + p2tx]n2 . Clearly, +w(x) += +1, +for +t = 1, +< +1, +for +t < 1, +> +1, +for +t > 1. +Hence, for 0 < t ≤ 1, E(X) ≤ KB (n1, n2, p1, p2, t) n1p1. And for t > 1, E(X) > KB (n1, n2, p1, p2, t) n1p1. +Similarly we can show that +E (Y ) += +n2p2KB (n1, n2, p1, p2, t) +n2−1 +� +j=0 +�n2 − 1 +j +� +pj +2 (1 − p2)n2−1−j +� +1 − p1 + p1tj+1 +�n1 +. +(6) +Theorem 1. If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then the correlation between X and Y is < 0, > +0, = 0 respectively for t < 0, > 0, = 0. +Proof. +6 + +We consider the case when t < 0 to show negative correlation. +E(X)E(Y ) += +� +n1p1KB (n1, n2, p1, p2, t) +n1−1 +� +i=0 +�n1 − 1 +i +� +pi +1 (1 − p1)n1−1−j +� +1 − p2 + p2tj+1 +�n2� +× +� +n1p1KB (n1, n2, p1, p2, t) +n2−1 +� +j=0 +�n2 − 1 +j +� +pj +2 (1 − p2)n2−1−j +� +1 − p1 + p1tj+1 +�n1� += +� +KB (n1, n2, p1, p2, t) +�2� n2−1 +� +j=0 +�n2 − 1 +j +� +pj +2 (1 − p2)n2−1−j +� +1 − p1 + p1tj+1 +�n1�� +× +� +n1n2p1p2t +n1−1 +� +i=0 +�n1 − 1 +i +� +(tp1)i (1 − p1)n1−1−j +� +1 − p2 + p2tj+1 +�n2−1 �1 − p2 + p2ti+1 +ti+1 +� � +> E (XY ) +� +KB (n1, n2, p1, p2, t) +n2−1 +� +j=0 +�n2 − 1 +j +� +pj +2 (1 − p2)n2−1−j � +1 − p1 + p1tj+1�n1 +� +[Since 1 − p2 + p2ti+1 > ti+1 for t < 1]. +Again for t < 1, +n2−1 +� +j=0 +�n2 − 1 +j +� +pj +2 (1 − p2)n2−1−j � +1 − p1 + p1tj+1�n1 += +n1 +� +i=0 +�n1 +i +� +pi +1 (1 − p1)n1−i +n2−1 +� +j=0 +�n2 − 1 +j +� +pj +2 (1 − p2)n2−1−j tij+i +> +n1 +� +i=0 +�n1 +i +� +pi +1 (1 − p1)n1−i +n2−1 +� +j=0 +�n2 − 1 +j +� +pj +2 (1 − p2)n2−1−j tij since ti < 1 += K−1 +B (n1, n2 − 1, p1, p2, t) . +Therefore +E(X)E(Y ) +> +E (XY ) K−1 +B (n1, n2 − 1, p1, p2, t) +K−1 +B (n1, n2, p1, p2, t) +> +E (XY ) , Since K−1 +B (n1, n2, p1, p2, t) is a decreasing function of n2. +Hence Cov(X, Y ) = E(XY ) − E(X)E(Y ) < 0. +Theorem 2. If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then the joint the joint factorial moment will be +given by +E +� +X(r)Y(s) +� += +t−rsn1(r)n2(s)pr +1ps +2 (1 − p1)n1−r (1 − p2)n2−s +(1 − p1)n1 (1 − p2)n2 S +� +n1, n2, +p1 +1−p1 , +p2 +1−p2 , t +� +×S +� +n1 − r, n2 − s, tsp1 +1 − p1 +, trp2 +1 − p2 +, t +� +. +7 + +Proof. Simple and thus excluded. +Observe that for t = 1, E +� +X(r)Y(s) +� += n1(r)n2(s)pr +1ps +2. Putting r = 1, s = 1 we get +E (XY ) += +t−1n1(r)n2(s)p1p2 (1 − p1)−1 (1 − p2)−1 +S +� +n1, n2, +p1 +1−p1 , +p2 +1−p2 , t +� +×S +� +n1 − 1, n2 − 1, +tp1 +1 − p1 +, +tp2 +1 − p2 +, t +� +. +Theorem 3. If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then the joint probability generating function +(p.g.f.) will be given by +GX,Y (s1, s2) += +E +� +sX +1 sY +2 +� += +�n1 +x=0 +�n2 +y=0 sx +1sy +2 +�n1 +x +��n2 +y +� +px +1py +2 (1 − p1)n1−x (1 − p2)n2−y txy +�n1 +x=0 +�n2 +y=0 sx +1sy +2 +�n1 +x +��n2 +y +� +px +1py +2 (1 − p1)n1−x (1 − p2)n2−y txy += +�n1 +x=0 +�n2 +y=0 +� +s1p1 +1−p1 +�x � +s2p2 +1−p2 +�y +txy +�n1 +x=0 +�n2 +y=0 +� +p1 +1−p1 +�x � +p2 +1−p2 +�y +txy += +S +� +n1, n2, s1p1 +1−p1 , s2p2 +1−p2 , t +� +S +� +n1, n2, +p1 +1−p1 , +p2 +1−p2 , t +�. +(7) +for 0 < s1 < 1, +0 < s2 < 1. +Therefore, the joint moment generating function (m.g.f.) of (X, Y ) will be +MX,Y (t1, t2) = +S +� +n1, n2, exp(t1)p1 +1−p1 +, exp(t1)p2 +1−p2 +, t +� +S +� +n1, n2, +p1 +1−p1 , +p2 +1−p2 , t +� +, +for |t1| < 1, +|t2| < 1. +In particular, for t = 1, S (n1, n2, q1, q2, 1) = (1 + q1)n1 (1 + q2)n2 . Then, (7) reduces to +GX,Y (s1, s2) += +� +1 + s1p1 +1−p1 +�n1 � +1 + s2p2 +1−p2 +�n2 +� +1 + +p1 +1−p1 +�n1 � +1 + +p2 +1−p2 +�n2 += +(1 + s1p1 − p1)n1 (1 + s2p2 − p2)n2 . +Theorem 4. If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then +8 + +P (X = m, Y = n) += +tnp1 (1 − p1)−1 +� +n1 − m + 1 +m +� +× P (X = m − 1, Y = n) , +for +m ≥ 1 += +tmp2 (1 − p2)−1 +� +n2 − n + 1 +n +� +× P (X = m, Y = n − 1) , +for +n ≥ 1 += +p1p2P (X = m − 1, Y = n − 1) +� +n1 − m + 1 +m +� +× +� +n2 − n + 1 +n +� +, +for +(m, n) ≥ 1. +Proof. Simple and thus excluded. +Theorem 5. (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , belongs to three parameter exponential family. +Proof. +The joint p.m.f. +of (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , will be a member of the 3- +parameter exponential family if its p.m.f. can be expressed in the form +h (x, y) = exp +� +� +3 +� +j=1 +δj (p1, p2, t) Wj(x, y) − M (p1, p2, t) +� +� . +(8) +From (1), it is easy to observe that the proposed distribution belongs to the exponential family by +rewriting the p.m.f. as +P (X = x, Y = y) = exp exp +� +3 +� +j=1 +δj (p1, p2, t) Wj(x, y) − M (p1, p2, t) +� +, +(9) +and identifying +M (p1, p2, t) = log KB (n1, n2, p1, p2, t) + n1 log (1 − p1) + n2 log (1 − p2) + log +��n1 +x +��n2 +y +�� +; +W1(x, y) = x; W2(x, y) = y; W3(x, y) = xy; +δ1 (p1, p2, t) = log +p1 +1 − p1 +; δ2 (p1, p2, t) = log +p2 +1 − p2 +; δ3 (p1, p2, t) = log t. +Thus, based on a sample of size m from BBCD, (� x, � y, � xy) is complete sufficient for (p1, p2, t) . +Distributions belonging to exponential family enjoys many properties. For example mean, vari- +ance, co-variance and moment generating functions can be easily derived using differentiation’s of +M (p1, p2, t). Moreover using Lehmann-Scheffe (Lehman and Scheffe (1982)) result, it may be pos- +sible to derive UMVUE of the parameters, provided we get hold of function of T that is unbiased +for the parameter. Even otherwise one can derive MVUE implementing bias correction. +Theorem 6. If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then +9 + +P (X < Y ) += +[KB (n1, n2, p1, p2, t)]2 +� min{n1,n2−1} +� +j=0 +j +� +x=0 +�� n2 +j + 1 +� +pj+1 +2 +(1 − p2)n2−j−1 +� � +1 − p1 + p1tj+1�n1 +× +�n1 +x +� +px +1 (1 − p1)n1−x [1 − p2 + p2tx]n2 +� +. +Proof. Observe that +P (X < Y ) += +min{n1,n2−1} +� +j=0 +P (Y = j + 1) P (X ≤ j) += [KB (n1, n2, p1, p2, t)]2 +� min{n1,n2−1} +� +j=0 +j +� +x=0 +�� n2 +j + 1 +� +pj+1 +2 +(1 − p2)n2−j−1 +� � +1 − p1 + p1tj+1�n1 +× +�n1 +x +� +px +1 (1 − p1)n1−x [1 − p2 + p2tx]n2 +� +. +Hence, the proof. +Note: Since 0 < t ≤ 1, we may obtain a upper bound inequality of P (X < Y ) by setting t = 1, +which will be as follows: +P (X < Y ) +≤ [KB (n1, n2, p1, p2, t)]2 +� min{n1,n2−1} +� +j=0 +j +� +x=0 +�� n2 +j + 1 +� +pj+1 +2 +(1 − p2)n2−j−1 +� +{1 − p1 + p1}n1 +× +�n1 +x +� +px +1 (1 − p1)n1−x +� +. +Note that R = P (X < Y ) is known as the stress- strength reliability in engineering where +the random variables X and Y respectively represent the stress and strength associated with a +system. This measure is also useful in a probabilistic assessment of inequality in two phenomena +X and Y. While in most cases X and Y are assumed to be independent in real life there may be +dependence between X and Y. Stress-strength reliability with both variables having independent +binomial distribution was discussed in Becker et al. (2002). As such the result of Theorem 6 has +the potential to be used in such contexts. +As can be clearly seen that even though there is no closed form for the above expression of the +covariance, this can be computed by taking a large number of terms in the series above. For that +the command NSum of Mathematica package can be used in order to get a value close to the exact. +Theorem 7. If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) then we have the following results. +10 + +(a) +P (X = x|X + Y = u) = +�n1 +x +�� n2 +u−x +� � +p1(1−p2) +p2(1−p1) +�x +t−x2 +�u +x=0 +�n1 +x +�� n2 +u−x +� � +p1(1−p2) +p2(1−p1) +�x +t−x2 , +u = 0, 1, · · · , n1 + n2. Here, max{0, u − n2} ≤ x min{u, n1}, we write as +P (X = x|X + Y = u) ∝ +�n1 +x +�� n2 +u − x +� �p1 (1 − p2) +p2 (1 − p1) +�x +t−x2�n1 + n2 +u +� � p1 (1 − p2) +txp2 (1 − p1) +�x +. +Observe that for t = 1, we get +P (X = x|X + Y = u) ∝ +�n1 +x +�� n2 +u − x +� �p1 (1 − p2) +p2 (1 − p1) +�x �n1 + n2 +u +� �p1 (1 − p2) +p2 (1 − p1) +�x +, +which is the p.m.f. of extended hypergeometric distribution of [12]. Again, it reduces to a +classical hypergeometric distribution when p1 = p2. +(b) The regression of X on Y is given by E (X|Y = y) = n1 +� +p1ty +1−p1+p1ty +� +, and the regression of +Y on X is given by E (Y |X = x) = n2 +� +p2tx +1−p2+p2tx +� +. +Proof. Part (a) is straight forward. Proof of part(b) can be obtained immediately by the information +that states that both the conditionals are binomial with respective parameters. Precisely, +• Since, X|Y = y ∼ binomial (q1qy +3) , therefore, the regression of X on Y will be obtained as +X = E (X|Y = y) = n1 +� +p1ty +1−p1+p1ty +� +. +• Similarly, since, Y |X = x ∼ binomial (q2qx +3) , therefore, the regression of Y on X will be +obtained as Y = E (Y |X = x) = n2 +� +p2tx +1−p2+p2tx +� +. +Theorem 8. If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then we have the following stochastic ordering +results related to the bivariate binomial conditional distribution in (1). +(a) If p1 > p2, then for any 0 < t < 1, and for fixed n1 and n2, X is stochastically larger than Y. +This also implies that under this parametric restriction, Y is smaller than X in the hazard +rate order, mean residual life order, and the likelihood ratio order. +(b) If p1 < p2, and for fixed n1 and n2, then for any 0 < t < 1, Y is stochastically larger than X. +This also implies that under this parametric restriction, X is smaller than Y in the hazard +rate order, mean residual life order, and the likelihood ratio order. +(c) If p1 < p2, and for fixed n1 and n2, then for any 0 < t < min{p1, p2} < 1, Y is stochastically +larger than X. This also implies that under this parametric restriction, X is smaller than Y +in the hazard rate order, mean residual life order, and the likelihood ratio order. +11 + +(d) If p1 > p2, and for fixed n1 and n2, then for any 1 > t > max{p1, p2}, X is stochastically +larger than Y. This also implies that under this parametric restriction, Y is smaller than X +in the hazard rate order, mean residual life order, and the likelihood ratio order. +Proof. The proof is quite simple and hence, the details avoided. +Theorem 9. Limiting distribution: If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then for 0 < t < 1, +lim +n1→∞ lim +n2→∞ BBCD (n1, n2, p1, p2, t) D∼ BPD (λ1, λ2, t) , +where as n1 → ∞, and n2 → ∞, and p1, p2 are small such that n1p1 = λ1, and n2p2 = λ2 are both +finite positive. For details on the BPD distribution, see Ghosh et al. (2020). +Proof. The proof is straightforward and hence, the details avoided. +Theorem 10. Suppose (X, Y ) ∼ BBCD (q1, q2, q3) . Let M = max{X, Y }, and U = min{X, Y }. +Then the p.m.f. of M will be +gM(m) = (1 − p1)n1 KB (n1, n2, p1, p2, t) +�n1 +m +� � +p1 +1 − p1 +�m +[1 − p2 + tmp2]n2 +Proof. +P(M ≤ m) += +P (X ≤ m, Y ≤ m) += +m +� +i=0 +m +� +j=0 +P (X = i, Y = j) += +(1 − p1)n1 (1 − p2)n2 KB (n1, n2, p1, p2, t) +m +� +i=0 +m +� +j=0 +�n1 +i +��n2 +j +� � +p1 +1 − p1 +�i � +p2 +1 − p2 +�j +tij += +(1 − p1)n1 KB (n1, n2, p1, p2, t) +m +� +i=0 +�n1 +i +� � +p1 +1 − p1 +�i � +1 − p2 + tip2 +�n2 +the p.m.f. of M can be easily seen as +gM(m) = (1 − p1)n1 KB (n1, n2, p1, p2, t) +�n1 +m +� � +p1 +1 − p1 +�m +[1 − p2 + tmp2]n2 +In fact we can find the following in general result of survival function which is as follows: +P (U > u) += +P (X > u, Y > u) += +n1−1−u +� +i=0 +n2−1−u +� +j=0 +P (X = u + i + 1, Y = u + j + 1) += +KB (n1, n2, p1, p2, t) +n1−1−u +� +i=0 +n2−1−u +� +j=0 +� +n1 +u + i + 1 +�� +n2 +u + j + 1 +� +pu+i+1 +1 +pu+i+1 +2 +(1 − p1)n1−u−i−1 (1 − p2)n1−u−i−1 t(u+i+1)2. +12 + +4 +Statistical Inference +4.1 +From sample proportions +From the pmf of BBCD in (1) we can write +f0,0 += +KB (n1, n2, p1, p2, t) (1 − p1)n1 (1 − p2)n2 , +f0,1 += +KB (n1, n2, p1, p2, t) n2 (1 − p1)n1 (1 − p2)n2−1 , +f1,0 += +KB (n1, n2, p1, p2, t) n1 (1 − p1)n1−1 (1 − p2)n2 , +f1,1 += +KB (n1, n2, p1, p2, t) n1n2 (1 − p1)n1−1 (1 − p2)n2−1 t. +(10) +From the above three equations we get +f0,0 +f0,1 += +1 − p2 +n2p2 +, +f0,0 +f1,0 += +1 − p1 +n1p1 +, +f1,1 +f0,1 += +n1t +1 − p1 +. +(11) +Solving for parameters we get +�p2 = +f0,1 +f0,1 + n2f0,0 +, +�p1 = +f1,0 +f1,0 + n2f0,0 +, +�t = 1 − �p1 +n1 +f1,1 +f0,1 +. +4.2 +Maximum Likelihood Estimation +In this subsection, we consider the maximum likelihood estimation of the unknown parameters +n1, n2, p1, p2 and t of the BBCD distribution based on a observed sample of size m, C = +((x1, y1), · · · , (xm, ym)). For ∆ = (n1, n2, p1, p2, t), the log-likelihood function is given by +ℓ (∆|C) += +mKG (q1, q2, q3) + +m +� +i=1 +� +log +��n1 +xi +�� ++ log +��n2 +yi +��� ++ log(p1) +m +� +i=1 +xi + log(p2) +m +� +i=1 +yi +log(1 − p1) +m +� +i=1 +(n − xi) + log(1 − p2) +m +� +i=1 +(n − yi) + log(t) +m +� +i=1 +xiyi. +The maximum likelihood estimators of the unknown parameters can be obtained by maximization +of the log-likelihood function, with respect to ∆, but it is quite difficult or even impossible, in this +case, to obtained explicit forms for the estimators, even when the values of n1 and n2 are assumed +to be known. To overcome this problem there are several numerical methods that may be used, +for example, based on the Newton-Raphson method or on the expectation–maximization (EM) +13 + +algorithm. In this work, we have decided to used Mathematica software and the NMaximize function +to obtain the parameters estimates. This allowed us to make the estimation of the parameters p1, +p2, t assuming the values of n1 and n2 known, or even to make the estimation of all the parameters +using as assumptions: n1 ∈ N, n2 ∈ N and p1, p2, t ∈ (0, 1). +5 +Simulation +Since it is not easy to simulate form the BBCD distribution we consider, in this section, the Gibbs +sampler method, see Gelfand (2000). Using the conditional distributions +X|Y = y ∼ binomial +� +n1, +typ1 +1 − p1 + typ1 +� +and Y |X = x ∼ binomial +� +n2, +txp2 +1 − p2 + txp2 +� +, +the implementation of this method is quite straightforward and the code, for the Mathematica +software, to simulated 500 points is provided in Figure 2. +Figure 2: Gibbs sampler method for the BBCD distribution +In Table 1, we assess the performance of Gibbs sampler method to generate samples from the +BBCD distribution in two scenarios and also the precision of the maximum likelihood estimates for +the parameters n1, n2, p1, p2 and t. For a matter of simplicity and to reduce the computational +time of the maximum likelihood estimations we consider n1 = n2 = n. From Table 1 it is possible +to observe that, in the first scenario, n = 10, p1 = 0.5, p2 = 0.9 and t = 0.8, for samples obtained +using Gibbs sampling method of sizes big enough, for example N ≥ 1000, it is possible to obtain +reasonable maximum likelihood estimates of the parameters. In this first scenario we have t = 0.8, +close to 1, which points to a low correlation between the variables. In the second scenario, n = 25, +p1 = 0.1, p2 = 0.2 and t = 0.1, the correlation is stronger since t = 0.1 and close to 0, and bigger +samples are needed in order to obtain a fair agreement between the exact and estimated values of +the parameters. In Figure 3, we present the p.m.f.s of the empirical (in gray) and fitted BBCD +(in black) distributions, in the two scenarios considered in Table 1, taking N = 1000 for the first +scenario and N = 100000 for the second scenario. This Figure shows the good fit between the +empirical and the fitted BBCD distributions. +14 + +data=[[xl=Floor[nl*p],x2=Floor[n2★p2]]]; +points = 5oo; +Do +xl=RandomVariate[BinomialDistribution[nl,(t^x2*pl)/(1-pl+t^x2*pl)]]; +x2=RandomVariate[BinomialDistribution[n2,(t^xl*p2)/(1-p2+t^xl*p2)]]; +data = Append[data, [xl, x2]], [points]]n +p1 +p2 +t +N +ˆn +ˆp1 +ˆp2 +ˆt +10 +0.5 +0.9 +0.8 +100 +10 +0.37303 +0.88436 +0.83132 +250 +10 +0.40986 +0.89467 +0.84159 +500 +10 +0.53465 +0.89819 +0.79151 +1000 +10 +0.47977 +0.89905 +0.80359 +2500 +10 +0.49307 +0.89952 +0.80356 +5000 +10 +0.49949 +0.89978 +0.79913 +25 +0.1 +0.2 +0.1 +100 +23 +0.09688 +0.23313 +0.30804 +250 +29 +0.08049 +0.17482 +0.22896 +500 +23 +0.07646 +0.22170 +0.18795 +1000 +29 +0.08789 +0.17109 +0.15450 +2500 +29 +0.08449 +0.17278 +0.11737 +5000 +26 +0.10164 +0.19089 +0.10958 +10000 +26 +0.10164 +0.19089 +0.10958 +50000 +24 +0.10345 +0.20834 +0.10305 +100000 +25 +0.09989 +0.20001 +0.09996 +Table 1: Maximum likelihood estimates of n, p1, p2 and t for samples of sizes N from a BBCD +distribution using Gibbs sampler method +6 +Real data application +We consider the data about seeds and plants grown in Rao (1990) and also in Table 1 in Laksh- +minarayana et al. (1999) and also in Ghosh et al. (2020). The data set reports the number of +seeds and plants grown over a plot of size five square feet. In Table 2 descriptive measures of the +observed data is presented while in Figure 4 we present the histogram of the data. +min Q0.25 Median Mean Q0.75 +max +X1 +0 +1.000 +2.000 +1.692 2.000 5.000 +X2 +0 +1.000 +2.000 +2.013 3.000 5.000 +Correlation +-0.0938 +Table 2: Descriptive measures of the data of seeds and plants grown +In Table 3, we present the estimated values of p1, p2 and t, for different choices of n1 = n2 = n +together with the sample and population correlations and with the p-value of χ2 goodness-of-fit +test for the BBCD distribution. In this table the row in text bold presents the values obtained by +maximum likelihood estimation for all the parameters n, p1, p2 and t. In rest of the rows, the ones +that are not in bold, we assume a specific value for n and only the parameters p1, p2 and t are +estimated. +We may observe that for n ≥ 14 the sample and population correlations are similar and also +that the p-value of the χ2 goodness-of-fit test for the BBCD distribution is close to 1, suggesting +an excellent fit of the BBCD distribution to the data. It is also interesting to note that when +15 + +(i) +(ii) +0 +5 +10 +0 +5 +10 +0.00 +0.05 +0.10 +0.15 +0 +5 +10 +0 +5 +10 +0.00 +0.05 +0.10 +0.15 +Figure 3: p.m.f.s of the empirical (in gray) and of the BBCD distributions (in black), for (i) a +sample of size 1000 and n = 10, p1 = 0.5, p2 = 0.9, t = 0.8, and for (ii) a sample of size 100000 and +n = 25, p1 = 0.1, p2 = 0.2, t = 0.1 +0 +2 +4 +6 +0 +2 +4 +6 +0 +20 +40 +60 +80 +Figure 4: Histogram of the data of seeds and plants grown +Correlation +ˆn +ˆp1 +ˆp2 +ˆt +BBCD +data +p-value +8 +0.238 +0.277 +0.926 +-0.107 +-0.094 +0.001 +10 +0.189 +0.220 +0.934 +-0.101 +-0.094 +0.363 +14 +0.134 +0.156 +0.943 +-0.092 +-0.094 +0.788 +15 +0.125 +0.146 +0.942 +-0.0943 +-0.094 +0.852 +20 +0.0935 +0.109 +0.946 +-0.0911 +-0.094 +0.931 +30 +0.0621 +0.0727 +0.949 +-0.0881 +-0.094 +0.987 +40 +0.0465 +0.0545 +0.951 +-0.0867 +-0.094 +0.989 +50 +0.0372 +0.0435 +0.952 +-0.0859 +-0.094 +0.989 +100 +0.0186 +0.0217 +0.954 +-0.0842 +-0.094 +0.978 +Table 3: Analysis of the fit of the BBCD distribution to the data of seeds and plants grown +n increases the values of n1 × p1, n2 × p2 and t converge to the values obtained in Ghosh et al. +(2020) for the same application and, respectively, for the parameters λ1, λ2 and λ3 of the bivariate +Poisson distribution as established in Theorem 9. In Figure 5, the p.m.f.s of the empirical and of +the fitted BBCD distributions are presented for the case n = 14 supporting the good fit of the +BBCD distribution to the data. +16 + +0 +2 +4 +6 +0 +2 +4 +6 +0.00 +0.02 +0.04 +0.06 +0.08 +Figure 5: p.m.f.s of the empirical (in gray) and of the BBCD distributions (in black) +7 +Conclusion: +In this paper, we have studied a bivariate binomial distribution via conditional specification orig- +inally proposed by Arnold et al. (1991, 1999). The model is useful for bivariate-dependent count +data when negative correlation structure is observed. The flexibility and the importance of the +model are discussed. One real data example is provided to illustrate the importance of the pro- +posed model. This example deals with negative correlation and over-dispersed marginals. It is +envisaged that the BBCD studied here will be a viable alternative to the existing bivariate count +data dealing with the kind of data sets considered here in various other real life scenarios. A mul- +tivariate extension of the BBCD will be explored in a separate article. However, appropriate real +life scenarios must be found for its possible application albeit computational complexity that is +expected for higher dimensions. +References +[1] Arnold, B. C., & Strauss, D. J. (1991). Bivariate distributions with conditionals in prescribed +exponential families. Journal of the Royal Statistical Society: Series B, 53(2), 365-375. +[2] Arnold, B.C., Castillo, E., & Sarabia, J.M. (1999). Conditional Specification of Statistical +Models. Springer, New York. +[3] Aitken, A.C., & Gonin, H.T. (1935). On fourfold sampling with and without replacement. +Proceedings of the Royal Society of Edinburgh. 55, 114-125. +[4] Becker, N. G., & Utev, S. (2002). 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Hardle, editors, Copulae in Mathematical and Quantitative +Finance, 231-249. +18 + +[23] Nikoloulopoulos, A. K. (2013b). On the estimation of normal copula discrete regression mod- +els using the continuous extension and simulated likelihood. In P. Jaworski, F. Durante, +and W. Hardle, editors, Copulae in Mathematical and Quantitative Finance, 231-249. A. K. +Nikoloulopoulos. Journal of Statistical Planning and Inference, 143(11):1923-1937. +[24] Ong, S.H., & Ng, C.M. (2013). A bivariate generalization of the non-central negative binomial +distribution. Communications in Statistics - Simulation and Computation, 42, 570-585. +[25] Papageorgiou, H., & David, K. M. (1994). On countable mixtures of bivariate binomial distri- +butions. Biometrical journal, 36(5), 581-601. +[26] Rao, S. (1990). Experimental studies on the yield of groundnuts in coastal region. Technical +Report, Andhra University, Visakhapatnam. +[27] Sun, K., & Basu, A. P. (1995). 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Univariate case. +Annals of the Institute of Statistical Mathematics, 39(1), 85-102. +19 + diff --git a/9NE1T4oBgHgl3EQfUQM_/content/tmp_files/load_file.txt b/9NE1T4oBgHgl3EQfUQM_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5957a8b610f0641adcb6f0962512c8dfebace85d --- /dev/null +++ b/9NE1T4oBgHgl3EQfUQM_/content/tmp_files/load_file.txt @@ -0,0 +1,699 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf,len=698 +page_content='Bivariate binomial conditionals distributions with positive and negative correlations: A statistical study Indranil Ghosh1, Filipe Marques2, Subrata Chakraborty3 1University of North Carolina, Wilmington, USA 2Universidade Nova de Lisboa, Portugal 3Dibrugarh University, Assam, India Corresponding author email address: ghoshi@uncw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='edu Abstract In this article, we discuss a bivariate distribution whose conditionals are univariate bino- mial distributions and the marginals are not binomial that exhibits negative correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Some useful structural properties of this distribution namely marginals, moments, generating func- tions, stochastic ordering are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Simple proofs of negative correlation, marginal over- dispersion, distribution of sum and conditional given the sum are also derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' The distribution is shown to be a member of the multi-parameter exponential family and some natural but useful consequences are also outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' The proposed distribution tends to a recently investigated con- ditional Poisson distribution studied by Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Finally, the distribution is fitted to two bivariate count data sets with an inherent negative correlation to illustrate its suitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Keywords: Bivariate binomial distribution, Conditional specification, Negative and positive cor- relation, Conditional failure rate, Limiting distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 1 Introduction The study of correlation between binomial random variables is an important statistical problem with a lot of theoretical and practical applications and is not new in the literature, see Biswas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (2022) and the references cited therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' There are also some attempts finding bivariate binomial distributions in other directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Hamdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1971) introduced a bivariate binomial distribution (which is indeed, a bivariate compound Poisson distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Hamdan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1976) studied the joint distribution of the total numbers of occurrences of binary characters A and B, given three independent samples in which both characters, A but not B, and B but not A, are observed— effectively derived a bivariate binomial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' A symmetric bivariate binomial distribution was proposed by Le (1984) to analyze clustered samples in medical research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Papageorgiou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1994) examined mixtures of bivariate binomial distributions which were derived from bivariate- compounded Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1989) discussed bivariate binomial distributions from extension of classes of univariate discrete distributions of order k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Takeuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1987) obtained the sum of 0−1 random variables in the multivariate setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' A bivariate generalization of the three parameter quasi-binomial distribution of Consul (1979) has been obtained by Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Crowder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1989) carried out Bayesian inference, and they defined the bivariate binomial distribution in a different sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' They defined a two-fold binomial model like X1|m ∼ Bin (m, p) and X2|X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' m ∼ Bin (X1, q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' For some discussions on bivariate binomial distributions see Kocherlakota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1992) and Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' However, none of the above cited references considered 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='03087v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='ME] 8 Jan 2023 the construction and study of a bivariate distribution such that both the conditionals are binomial with respective parameters, but the joint distribution may not necessarily be a bivariate binomial in its traditional sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Arnold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1999) came up with the construction of a bivariate discrete distribution starting from two conditional distributions that are binomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In fact this idea is first developed in Arnold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' It appears that the resulting bivariate discrete model (albeit ubiquitous normalizing constant) has the salient feature of exhibiting both positive and negative correlation—this property is not enjoyed by many of the existing bivariate binomial models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In this article, we explore some useful structural properties of the bivariate discrete distribution originally proposed by Arnold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1991,1999) and discusses its applicability in modeling bivariate discrete data exhibiting either negative correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In Section 2, we introduce the bi-variate binomial conditionals distribution which was described in Arnold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1991,1999) and provide the expression for associated marginal p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.’s, of X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In Section 3, we provide several useful structural properties of this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Section 4 provides the estimation of parameters using sample proportions and also under the maximum likelihood method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In section 5, we discuss copula-based simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' A real data application for the BBCD is presented in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Finally, some concluding remarks are provided in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 2 Bivariate binomial conditional distributions Let us assume the following: X|Y = y ∼ binomial (n1, p1(y)) , for each fixed Y = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Y |X = x ∼ binomial (n2, p2(y)) , for each fixed X = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' According to [2, 1], the associated joint p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' will be P (X = x, Y = y) = KB (n1, n2, p1, p2, t) �n1 x ��n2 y � px 1py 2 (1 − p1)n1−x (1 − p2)n2−y txy, (1) where p1 ∈ (0, 1), and p2 ∈ (0, 1), and t > 0, and x = 0, 1, 2, · · · , n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' y = 0, 1, 2, · · · , n2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' and KB (n1, n2, p1, p2, t) is the normalizing constant and K−1 B = K−1 B (n1, n2, p1, p2, t) = n1 � x=0 n2 � y=0 �n1 x ��n2 y � px 1py 2 (1 − p1)n1−x (1 − p2)n2−y txy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' We will denote (henceforth, in short) the bivariate binomial conditionals distribution of the pair (X, Y ) with the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' in (1) as BBCD (n1, n2, p1, p2, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' We list some useful results related to the normalizing constant that will be utilized later on in deriving some structural properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (i) K−1 B (n1, n2, p1, p2, 1) = n1 � x=0 n2 � y=0 �n1 x ��n2 y � px 1py 2 (1 − p1)n1−x (1 − p2)n2−y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 2 (ii) K−1 B (n1, n2, p1, p2, t) = K−1 B (n2, n1, p2, p1, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (iii) K−1 B (n1, n2, p1, 1, t) = n1 � x=0 �n1 x � px 1 (1 − p1)n1−x pn2 2 (1 − p2)n2−n2txn2 = pn2 2 (1 − p1 + tn2p1)n1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (iv) K−1 B (n1, n2, 1, p2, t) = pn1 1 (1 − p2 + tn1p2)n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (v) 0 < K−1 B (n1, n2, p1, p2, t) ≤ min{pn1 1 (1 − p2 + p2tn1)n2 , pn2 2 (1 − p1 + p1tn2)n1} ≤ tn1n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (vi) d dp1 K−1 B (n1, n2, p1, 1, t) = n1 � x=0 n2 � y=0 �n1 x ��n2 y �� xpx−1 1 (1 − p1)n1−x + (n1 − x) px 1 (1 − p1)n1−x−1 � py 2 (1 − p2)n2−y txy = p−1 1 n1 � x=0 n2 � y=0 x �n1 x ��n2 y � px 1py 2 (1 − p1)n1−x (1 − p2)n2−y txy + (1 − p2)−1 n1 � x=0 n2 � y=0 (n1 − x) �n1 x ��n2 y � px 1py 2 (1 − p1)n1−x (1 − p2)n2−y txy = p−1 1 E(X) KB + (1 − p1)−1 E (n − X) KB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Note that the above result immediately implies that d dp1 log K−1 B = E(X) p1 + E (n − X) 1 − p1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Next, observe that for t = 1, this reduces to 0 = E(X) p1 + E (n − X) 1 − p1 =⇒ E (n − X) = � p−1 1 − 1 � E(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (vii) Again, by re-writing the expression for KB (n1, n2, p1, p2, t) as K−1 B (n1, n2, p1, 1, t) = (1 − p1)n1 (1 − p2)n2 n1 � x=0 n2 � y=0 �n1 x � � p1 1 − p1 �x �n2 y � � p2 1 − p2 �y txy, 3 and by writing p1 1 − p1 = q1, p2 1 − p2 = q2, and S (n1, n2, q1, q2, t) = n1 � x=0 n2 � y=0 �n1 x � qx 1 �n2 y � qy 2txy, we have K−1 B (n1, n2, p1, 1, t) = (1 − p1)n1 (1 − p2)n2 S (n1, n2, q1, q2, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Note that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' The marginal p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' of X will be P (X = x) = KB (n1, n2, p1, 1, t) �n1 x � px 1 (1 − p1)n1−x n2 � y=0 �n2 y � (txp2)y (1 − p2)n2−y = KB (n1, n2, p1, 1, t) �n1 x � px 1 (1 − p1)n1−x � 1 − p2 + txp2 �n2 , (2) for x = 0, 1, 2, · · · , n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Similarly, the marginal p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' of Y will be P (Y = y) = KB (n1, n2, p1, 1, t) �n2 y � py 2 (1 − p2)n2−y � 1 − p1 + typ1 �n1 , (3) for y = 0, 1, 2, · · · , n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' For fixed t ∈ (0, 1), P (X = x, Y = y|n1, n2, p1, p2, t) = P (X = y, Y = x|n2, n1, p2, p1, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Thus, if p1 = p2 = p, say then P (X = x, Y = y|n1, n2, p, p, t) = P (X = y, Y = x|n2, n1, p, p, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Furthermore, if n1 = n2 = nsay then P (X = x, Y = y|n, t) = P (X = y, Y = x|n, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Some representative p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' plots for varying parameter choices are provided in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 3 Structural properties Note that since, X|Y = y ∼ binomial � n1, typ1 1−p1+typ1 � , E (X|Y = y) = n1 � typ1 1 − p1 + typ1 � , and V ar (X|Y = y) = n1 � typ1 1 − p1 + typ1 � � 1 − typ1 1 − p1 + typ1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 4 n1 = n2 = 10, p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='5, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='5 (i) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='95 (ii) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='5 (iii) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0 5 10 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='04 0 5 10 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='06 0 5 10 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='10 n1 = n2 = 20, p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='1, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='1 (iv) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='95 (v) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='5 (vi) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0 5 10 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='08 0 5 10 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='10 0 5 10 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='15 n1 = n2 = 30, p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='9, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='9 (vii) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='95 (viii) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='5 (ix) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0 10 20 30 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='02 0 10 20 30 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='10 0 10 20 30 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='10 n1 = n2 = 15, p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='3, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='7 (x) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='95 (xi) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='5 (xii) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0 5 10 15 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='04 0 5 10 15 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='20 0 5 10 15 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='20 Figure 1: Examples of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' plots for the BBCD distribution 5 Consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' E (X) = EY (X|Y = y) = KB (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' t) n2 � y=0 � n1 � typ1 1 − p1 + typ1 � ��n2 y � py 2 (1 − p2)n2−y = n1KB (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' t) n2 � y=0 �n2 y � (tp2)y (1 − p2)n2−y � n1−1 � j=0 �n1 − 1 j � (typ1)j (1 − p1)n1−1−j � = n1p1KB (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' t) n1−1 � j=0 �n1 − 1 j � pj 1 (1 − p1)n1−1−j � n2 � y=0 �n2 y � � p2tj+1�y (1 − p2)n2−y � = n1p1KB (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' t) n1−1 � j=0 �n1 − 1 j � pj 1 (1 − p1)n1−1−j � 1 − p2 + p2tj+1 �n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (4) Alternatively, we can re-write (4) as E (X) = KB (n1, n2, p1, p2, t) n1 � x=0 x �n1 x � px 1 (1 − p1)n1−x [1 − p2 + p2tx]n2 = KB (n1, n2, p1, p2, t) n1 � x=0 x �n1 x � px 1 (1 − p1)n1−x wx, (5) wx = [1 − p2 + p2tx]n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Clearly, w(x) = 1, for t = 1, < 1, for t < 1, > 1, for t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Hence, for 0 < t ≤ 1, E(X) ≤ KB (n1, n2, p1, p2, t) n1p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' And for t > 1, E(X) > KB (n1, n2, p1, p2, t) n1p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Similarly we can show that E (Y ) = n2p2KB (n1, n2, p1, p2, t) n2−1 � j=0 �n2 − 1 j � pj 2 (1 − p2)n2−1−j � 1 − p1 + p1tj+1 �n1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (6) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then the correlation between X and Y is < 0, > 0, = 0 respectively for t < 0, > 0, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 6 We consider the case when t < 0 to show negative correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' E(X)E(Y ) = � n1p1KB (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' t) n1−1 � i=0 �n1 − 1 i � pi 1 (1 − p1)n1−1−j � 1 − p2 + p2tj+1 �n2� × � n1p1KB (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' t) n2−1 � j=0 �n2 − 1 j � pj 2 (1 − p2)n2−1−j � 1 − p1 + p1tj+1 �n1� = � KB (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' t) �2� n2−1 � j=0 �n2 − 1 j � pj 2 (1 − p2)n2−1−j � 1 − p1 + p1tj+1 �n1�� × � n1n2p1p2t n1−1 � i=0 �n1 − 1 i � (tp1)i (1 − p1)n1−1−j � 1 − p2 + p2tj+1 �n2−1 �1 − p2 + p2ti+1 ti+1 � � > E (XY ) � KB (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' t) n2−1 � j=0 �n2 − 1 j � pj 2 (1 − p2)n2−1−j � 1 − p1 + p1tj+1�n1 � [Since 1 − p2 + p2ti+1 > ti+1 for t < 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Again for t < 1, n2−1 � j=0 �n2 − 1 j � pj 2 (1 − p2)n2−1−j � 1 − p1 + p1tj+1�n1 = n1 � i=0 �n1 i � pi 1 (1 − p1)n1−i n2−1 � j=0 �n2 − 1 j � pj 2 (1 − p2)n2−1−j tij+i > n1 � i=0 �n1 i � pi 1 (1 − p1)n1−i n2−1 � j=0 �n2 − 1 j � pj 2 (1 − p2)n2−1−j tij since ti < 1 = K−1 B (n1, n2 − 1, p1, p2, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Therefore E(X)E(Y ) > E (XY ) K−1 B (n1, n2 − 1, p1, p2, t) K−1 B (n1, n2, p1, p2, t) > E (XY ) , Since K−1 B (n1, n2, p1, p2, t) is a decreasing function of n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Hence Cov(X, Y ) = E(XY ) − E(X)E(Y ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then the joint the joint factorial moment will be given by E � X(r)Y(s) � = t−rsn1(r)n2(s)pr 1ps 2 (1 − p1)n1−r (1 − p2)n2−s (1 − p1)n1 (1 − p2)n2 S � n1, n2, p1 1−p1 , p2 1−p2 , t � ×S � n1 − r, n2 − s, tsp1 1 − p1 , trp2 1 − p2 , t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Simple and thus excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Observe that for t = 1, E � X(r)Y(s) � = n1(r)n2(s)pr 1ps 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Putting r = 1, s = 1 we get E (XY ) = t−1n1(r)n2(s)p1p2 (1 − p1)−1 (1 − p2)−1 S � n1, n2, p1 1−p1 , p2 1−p2 , t � ×S � n1 − 1, n2 − 1, tp1 1 − p1 , tp2 1 − p2 , t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then the joint probability generating function (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=') will be given by GX,Y (s1, s2) = E � sX 1 sY 2 � = �n1 x=0 �n2 y=0 sx 1sy 2 �n1 x ��n2 y � px 1py 2 (1 − p1)n1−x (1 − p2)n2−y txy �n1 x=0 �n2 y=0 sx 1sy 2 �n1 x ��n2 y � px 1py 2 (1 − p1)n1−x (1 − p2)n2−y txy = �n1 x=0 �n2 y=0 � s1p1 1−p1 �x � s2p2 1−p2 �y txy �n1 x=0 �n2 y=0 � p1 1−p1 �x � p2 1−p2 �y txy = S � n1, n2, s1p1 1−p1 , s2p2 1−p2 , t � S � n1, n2, p1 1−p1 , p2 1−p2 , t �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (7) for 0 < s1 < 1, 0 < s2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Therefore, the joint moment generating function (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=') of (X, Y ) will be MX,Y (t1, t2) = S � n1, n2, exp(t1)p1 1−p1 , exp(t1)p2 1−p2 , t � S � n1, n2, p1 1−p1 , p2 1−p2 , t � , for |t1| < 1, |t2| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In particular, for t = 1, S (n1, n2, q1, q2, 1) = (1 + q1)n1 (1 + q2)n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Then, (7) reduces to GX,Y (s1, s2) = � 1 + s1p1 1−p1 �n1 � 1 + s2p2 1−p2 �n2 � 1 + p1 1−p1 �n1 � 1 + p2 1−p2 �n2 = (1 + s1p1 − p1)n1 (1 + s2p2 − p2)n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then 8 P (X = m, Y = n) = tnp1 (1 − p1)−1 � n1 − m + 1 m � × P (X = m − 1, Y = n) , for m ≥ 1 = tmp2 (1 − p2)−1 � n2 − n + 1 n � × P (X = m, Y = n − 1) , for n ≥ 1 = p1p2P (X = m − 1, Y = n − 1) � n1 − m + 1 m � × � n2 − n + 1 n � , for (m, n) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Simple and thus excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , belongs to three parameter exponential family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' The joint p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' of (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , will be a member of the 3- parameter exponential family if its p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' can be expressed in the form h (x, y) = exp � � 3 � j=1 δj (p1, p2, t) Wj(x, y) − M (p1, p2, t) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (8) From (1), it is easy to observe that the proposed distribution belongs to the exponential family by rewriting the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' as P (X = x, Y = y) = exp exp � 3 � j=1 δj (p1, p2, t) Wj(x, y) − M (p1, p2, t) � , (9) and identifying M (p1, p2, t) = log KB (n1, n2, p1, p2, t) + n1 log (1 − p1) + n2 log (1 − p2) + log ��n1 x ��n2 y �� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' W1(x, y) = x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' W2(x, y) = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' W3(x, y) = xy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' δ1 (p1, p2, t) = log p1 1 − p1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' δ2 (p1, p2, t) = log p2 1 − p2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' δ3 (p1, p2, t) = log t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Thus, based on a sample of size m from BBCD, (� x, � y, � xy) is complete sufficient for (p1, p2, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Distributions belonging to exponential family enjoys many properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' For example mean, vari- ance, co-variance and moment generating functions can be easily derived using differentiation’s of M (p1, p2, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Moreover using Lehmann-Scheffe (Lehman and Scheffe (1982)) result, it may be pos- sible to derive UMVUE of the parameters, provided we get hold of function of T that is unbiased for the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Even otherwise one can derive MVUE implementing bias correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then 9 P (X < Y ) = [KB (n1, n2, p1, p2, t)]2 � min{n1,n2−1} � j=0 j � x=0 �� n2 j + 1 � pj+1 2 (1 − p2)n2−j−1 � � 1 − p1 + p1tj+1�n1 × �n1 x � px 1 (1 − p1)n1−x [1 − p2 + p2tx]n2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Observe that P (X < Y ) = min{n1,n2−1} � j=0 P (Y = j + 1) P (X ≤ j) = [KB (n1, n2, p1, p2, t)]2 � min{n1,n2−1} � j=0 j � x=0 �� n2 j + 1 � pj+1 2 (1 − p2)n2−j−1 � � 1 − p1 + p1tj+1�n1 × �n1 x � px 1 (1 − p1)n1−x [1 − p2 + p2tx]n2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Hence, the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Note: Since 0 < t ≤ 1, we may obtain a upper bound inequality of P (X < Y ) by setting t = 1, which will be as follows: P (X < Y ) ≤ [KB (n1, n2, p1, p2, t)]2 � min{n1,n2−1} � j=0 j � x=0 �� n2 j + 1 � pj+1 2 (1 − p2)n2−j−1 � {1 − p1 + p1}n1 × �n1 x � px 1 (1 − p1)n1−x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Note that R = P (X < Y ) is known as the stress- strength reliability in engineering where the random variables X and Y respectively represent the stress and strength associated with a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' This measure is also useful in a probabilistic assessment of inequality in two phenomena X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' While in most cases X and Y are assumed to be independent in real life there may be dependence between X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Stress-strength reliability with both variables having independent binomial distribution was discussed in Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' As such the result of Theorem 6 has the potential to be used in such contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' As can be clearly seen that even though there is no closed form for the above expression of the covariance, this can be computed by taking a large number of terms in the series above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' For that the command NSum of Mathematica package can be used in order to get a value close to the exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) then we have the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 10 (a) P (X = x|X + Y = u) = �n1 x �� n2 u−x � � p1(1−p2) p2(1−p1) �x t−x2 �u x=0 �n1 x �� n2 u−x � � p1(1−p2) p2(1−p1) �x t−x2 , u = 0, 1, · · · , n1 + n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Here, max{0, u − n2} ≤ x min{u, n1}, we write as P (X = x|X + Y = u) ∝ �n1 x �� n2 u − x � �p1 (1 − p2) p2 (1 − p1) �x t−x2�n1 + n2 u � � p1 (1 − p2) txp2 (1 − p1) �x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Observe that for t = 1, we get P (X = x|X + Y = u) ∝ �n1 x �� n2 u − x � �p1 (1 − p2) p2 (1 − p1) �x �n1 + n2 u � �p1 (1 − p2) p2 (1 − p1) �x , which is the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' of extended hypergeometric distribution of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Again, it reduces to a classical hypergeometric distribution when p1 = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (b) The regression of X on Y is given by E (X|Y = y) = n1 � p1ty 1−p1+p1ty � , and the regression of Y on X is given by E (Y |X = x) = n2 � p2tx 1−p2+p2tx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Part (a) is straight forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Proof of part(b) can be obtained immediately by the information that states that both the conditionals are binomial with respective parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Precisely, Since, X|Y = y ∼ binomial (q1qy 3) , therefore, the regression of X on Y will be obtained as X = E (X|Y = y) = n1 � p1ty 1−p1+p1ty � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Similarly, since, Y |X = x ∼ binomial (q2qx 3) , therefore, the regression of Y on X will be obtained as Y = E (Y |X = x) = n2 � p2tx 1−p2+p2tx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then we have the following stochastic ordering results related to the bivariate binomial conditional distribution in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (a) If p1 > p2, then for any 0 < t < 1, and for fixed n1 and n2, X is stochastically larger than Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' This also implies that under this parametric restriction, Y is smaller than X in the hazard rate order, mean residual life order, and the likelihood ratio order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (b) If p1 < p2, and for fixed n1 and n2, then for any 0 < t < 1, Y is stochastically larger than X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' This also implies that under this parametric restriction, X is smaller than Y in the hazard rate order, mean residual life order, and the likelihood ratio order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (c) If p1 < p2, and for fixed n1 and n2, then for any 0 < t < min{p1, p2} < 1, Y is stochastically larger than X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' This also implies that under this parametric restriction, X is smaller than Y in the hazard rate order, mean residual life order, and the likelihood ratio order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 11 (d) If p1 > p2, and for fixed n1 and n2, then for any 1 > t > max{p1, p2}, X is stochastically larger than Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' This also implies that under this parametric restriction, Y is smaller than X in the hazard rate order, mean residual life order, and the likelihood ratio order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' The proof is quite simple and hence, the details avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Limiting distribution: If (X, Y ) ∼ BBCD (n1, n2, p1, p2, t) , then for 0 < t < 1, lim n1→∞ lim n2→∞ BBCD (n1, n2, p1, p2, t) D∼ BPD (λ1, λ2, t) , where as n1 → ∞, and n2 → ∞, and p1, p2 are small such that n1p1 = λ1, and n2p2 = λ2 are both finite positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' For details on the BPD distribution, see Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' The proof is straightforward and hence, the details avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Suppose (X, Y ) ∼ BBCD (q1, q2, q3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Let M = max{X, Y }, and U = min{X, Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Then the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' of M will be gM(m) = (1 − p1)n1 KB (n1, n2, p1, p2, t) �n1 m � � p1 1 − p1 �m [1 − p2 + tmp2]n2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' P(M ≤ m) = P (X ≤ m, Y ≤ m) = m � i=0 m � j=0 P (X = i, Y = j) = (1 − p1)n1 (1 − p2)n2 KB (n1, n2, p1, p2, t) m � i=0 m � j=0 �n1 i ��n2 j � � p1 1 − p1 �i � p2 1 − p2 �j tij = (1 − p1)n1 KB (n1, n2, p1, p2, t) m � i=0 �n1 i � � p1 1 − p1 �i � 1 − p2 + tip2 �n2 the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' of M can be easily seen as gM(m) = (1 − p1)n1 KB (n1, n2, p1, p2, t) �n1 m � � p1 1 − p1 �m [1 − p2 + tmp2]n2 In fact we can find the following in general result of survival function which is as follows: P (U > u) = P (X > u, Y > u) = n1−1−u � i=0 n2−1−u � j=0 P (X = u + i + 1, Y = u + j + 1) = KB (n1, n2, p1, p2, t) n1−1−u � i=0 n2−1−u � j=0 � n1 u + i + 1 �� n2 u + j + 1 � pu+i+1 1 pu+i+1 2 (1 − p1)n1−u−i−1 (1 − p2)n1−u−i−1 t(u+i+1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 12 4 Statistical Inference 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='1 From sample proportions From the pmf of BBCD in (1) we can write f0,0 = KB (n1, n2, p1, p2, t) (1 − p1)n1 (1 − p2)n2 , f0,1 = KB (n1, n2, p1, p2, t) n2 (1 − p1)n1 (1 − p2)n2−1 , f1,0 = KB (n1, n2, p1, p2, t) n1 (1 − p1)n1−1 (1 − p2)n2 , f1,1 = KB (n1, n2, p1, p2, t) n1n2 (1 − p1)n1−1 (1 − p2)n2−1 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (10) From the above three equations we get f0,0 f0,1 = 1 − p2 n2p2 , f0,0 f1,0 = 1 − p1 n1p1 , f1,1 f0,1 = n1t 1 − p1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (11) Solving for parameters we get �p2 = f0,1 f0,1 + n2f0,0 , �p1 = f1,0 f1,0 + n2f0,0 , �t = 1 − �p1 n1 f1,1 f0,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='2 Maximum Likelihood Estimation In this subsection, we consider the maximum likelihood estimation of the unknown parameters n1, n2, p1, p2 and t of the BBCD distribution based on a observed sample of size m, C = ((x1, y1), · · · , (xm, ym)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' For ∆ = (n1, n2, p1, p2, t), the log-likelihood function is given by ℓ (∆|C) = mKG (q1, q2, q3) + m � i=1 � log ��n1 xi �� + log ��n2 yi ��� + log(p1) m � i=1 xi + log(p2) m � i=1 yi log(1 − p1) m � i=1 (n − xi) + log(1 − p2) m � i=1 (n − yi) + log(t) m � i=1 xiyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' The maximum likelihood estimators of the unknown parameters can be obtained by maximization of the log-likelihood function, with respect to ∆, but it is quite difficult or even impossible, in this case, to obtained explicit forms for the estimators, even when the values of n1 and n2 are assumed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' To overcome this problem there are several numerical methods that may be used, for example, based on the Newton-Raphson method or on the expectation–maximization (EM) 13 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In this work, we have decided to used Mathematica software and the NMaximize function to obtain the parameters estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' This allowed us to make the estimation of the parameters p1, p2, t assuming the values of n1 and n2 known, or even to make the estimation of all the parameters using as assumptions: n1 ∈ N, n2 ∈ N and p1, p2, t ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 5 Simulation Since it is not easy to simulate form the BBCD distribution we consider, in this section, the Gibbs sampler method, see Gelfand (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Using the conditional distributions X|Y = y ∼ binomial � n1, typ1 1 − p1 + typ1 � and Y |X = x ∼ binomial � n2, txp2 1 − p2 + txp2 � , the implementation of this method is quite straightforward and the code, for the Mathematica software, to simulated 500 points is provided in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Figure 2: Gibbs sampler method for the BBCD distribution In Table 1, we assess the performance of Gibbs sampler method to generate samples from the BBCD distribution in two scenarios and also the precision of the maximum likelihood estimates for the parameters n1, n2, p1, p2 and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' For a matter of simplicity and to reduce the computational time of the maximum likelihood estimations we consider n1 = n2 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' From Table 1 it is possible to observe that, in the first scenario, n = 10, p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='5, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='9 and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='8, for samples obtained using Gibbs sampling method of sizes big enough, for example N ≥ 1000, it is possible to obtain reasonable maximum likelihood estimates of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In this first scenario we have t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='8, close to 1, which points to a low correlation between the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In the second scenario, n = 25, p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='1, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='2 and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='1, the correlation is stronger since t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='1 and close to 0, and bigger samples are needed in order to obtain a fair agreement between the exact and estimated values of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In Figure 3, we present the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='s of the empirical (in gray) and fitted BBCD (in black) distributions, in the two scenarios considered in Table 1, taking N = 1000 for the first scenario and N = 100000 for the second scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' This Figure shows the good fit between the empirical and the fitted BBCD distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 14 data=[[xl=Floor[nl*p],x2=Floor[n2★p2]]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' points = 5oo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Do xl=RandomVariate[BinomialDistribution[nl,(t^x2*pl)/(1-pl+t^x2*pl)]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' x2=RandomVariate[BinomialDistribution[n2,(t^xl*p2)/(1-p2+t^xl*p2)]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' data = Append[data, [xl, x2]], [points]]n p1 p2 t N ˆn ˆp1 ˆp2 ˆt 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='20001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='09996 Table 1: Maximum likelihood estimates of n, p1, p2 and t for samples of sizes N from a BBCD distribution using Gibbs sampler method 6 Real data application We consider the data about seeds and plants grown in Rao (1990) and also in Table 1 in Laksh- minarayana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1999) and also in Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' The data set reports the number of seeds and plants grown over a plot of size five square feet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In Table 2 descriptive measures of the observed data is presented while in Figure 4 we present the histogram of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' min Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='25 Median Mean Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='75 max X1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='692 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='000 X2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='013 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='000 Correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='0938 Table 2: Descriptive measures of the data of seeds and plants grown In Table 3, we present the estimated values of p1, p2 and t, for different choices of n1 = n2 = n together with the sample and population correlations and with the p-value of χ2 goodness-of-fit test for the BBCD distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In this table the row in text bold presents the values obtained by maximum likelihood estimation for all the parameters n, p1, p2 and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In rest of the rows, the ones that are not in bold, we assume a specific value for n and only the parameters p1, p2 and t are estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' We may observe that for n ≥ 14 the sample and population correlations are similar and also that the p-value of the χ2 goodness-of-fit test for the BBCD distribution is close to 1, suggesting an excellent fit of the BBCD distribution to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' It is also interesting to note that when 15 (i) (ii) 0 5 10 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='15 0 5 10 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='15 Figure 3: p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='s of the empirical (in gray) and of the BBCD distributions (in black), for (i) a sample of size 1000 and n = 10, p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='5, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='9, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='8, and for (ii) a sample of size 100000 and n = 25, p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='1, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='2, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='1 0 2 4 6 0 2 4 6 0 20 40 60 80 Figure 4: Histogram of the data of seeds and plants grown Correlation ˆn ˆp1 ˆp2 ˆt BBCD data p-value 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='277 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='926 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='978 Table 3: Analysis of the fit of the BBCD distribution to the data of seeds and plants grown n increases the values of n1 × p1, n2 × p2 and t converge to the values obtained in Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (2020) for the same application and, respectively, for the parameters λ1, λ2 and λ3 of the bivariate Poisson distribution as established in Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' In Figure 5, the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='s of the empirical and of the fitted BBCD distributions are presented for the case n = 14 supporting the good fit of the BBCD distribution to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 16 0 2 4 6 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='08 Figure 5: p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='s of the empirical (in gray) and of the BBCD distributions (in black) 7 Conclusion: In this paper, we have studied a bivariate binomial distribution via conditional specification orig- inally proposed by Arnold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1991, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' The model is useful for bivariate-dependent count data when negative correlation structure is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' The flexibility and the importance of the model are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' One real data example is provided to illustrate the importance of the pro- posed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' This example deals with negative correlation and over-dispersed marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' It is envisaged that the BBCD studied here will be a viable alternative to the existing bivariate count data dealing with the kind of data sets considered here in various other real life scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' A mul- tivariate extension of the BBCD will be explored in a separate article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' However, appropriate real life scenarios must be found for its possible application albeit computational complexity that is expected for higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' References [1] Arnold, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=', & Strauss, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Bivariate distributions with conditionals in prescribed exponential families.' metadata={'source': 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Quantitative Finance, 231-249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Nikoloulopoulos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Journal of Statistical Planning and Inference, 143(11):1923-1937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' [24] Ong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content='H.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=', & David, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' On countable mixtures of bivariate binomial distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Biometrical journal, 36(5), 581-601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' [26] Rao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Experimental studies on the yield of groundnuts in coastal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Technical Report, Andhra University, Visakhapatnam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' [27] Sun, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=', & Basu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' A characterization of a bivariate binomial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Statistics & probability letters, 23(4), 307-311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' [28] Takeuchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=', & Takemura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' On sum of 0 − 1 random variables I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Univariate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' Annals of the Institute of Statistical Mathematics, 39(1), 85-102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQfUQM_/content/2301.03087v1.pdf'} diff --git a/BtE4T4oBgHgl3EQf5Q6j/content/tmp_files/2301.05322v1.pdf.txt b/BtE4T4oBgHgl3EQf5Q6j/content/tmp_files/2301.05322v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a80be14bf37a7c1ae579ffb9038d67a3ac9b0b80 --- /dev/null +++ b/BtE4T4oBgHgl3EQf5Q6j/content/tmp_files/2301.05322v1.pdf.txt @@ -0,0 +1,891 @@ + + + +A tale of the scattering lifetime and the mean free path + +Pedro Contreras*,1 and Dianela Osorio 2 + +1 Department of Physics, University of the Andes, Me rida, Venezuela. ORCID: 0000-0002-3394-1195 +2 Department of Brain and Behavioral Sciences, University of Pavia, Pavia, Italy. ORCID: 0000-0002-8171-0703 + +*Corresponding author (pedrocontre@gmail.com) + +ABSTRACT + +The idea of applying the scattering lifetime calculated from the imaginary part of the zero temperature elastic scatter- +ing cross-section to study a hidden self-consistent damping in two spaces of importance for non-equilibrium statistical +mechanics is proposed. It is discussed its relation with the classical phase space from statistical mechanics and the +configuration space from nonrelativistic quantum mechanics. This idea is contrasted with the mean free path values +in three elastic collision regimes. The main exercise is to study the behavior of a self-consistent probabilistic distribu- +tion function in a space we have called the reduced phase space, since it is related to the scattering lifetime. This exer- +cise has been solved in two unconventional superconductors, for which several calculations are discussed. One of them +is to obtain the scattering phase shift from the inverse strength of an atomic potential and the other is to build several +phases with different nodal configuration of the superconducting order parameter and show that the imaginary self- +consistent part of the scattering cross-section is always positive for two compounds: the triplet strontium ruthenate +and the singlet doped with strontium lanthanum cuprate when three models of superconducting order parameters +are used: the quasi-point, the point and the line nodal cases. We finally compare the frequency dispersion in the anom- +alous skin effect with singular shapes of the Fermi surface with the frequency dispersion in the scattering lifetime and +their respective mean free paths. This idea is useful because it intuitively explores the nonlocality of this type of hidden +self-consistent damping for those incoherent fermionic quasiparticles. + +Keywords: Reduced phase space; Configuration space; Classical phase space; Mean free path; Collision lifetime; +Damping; Non-equilibrium statistical mechanics. + + + + + +1. Introduction + +This work is aimed at phenomenologically understanding the role of two parameters widely used in non-equilibrium +statistical mechanics, the mean free path “l” and the scattering lifetime “𝜏”. One calculated and the other used in the +study of the elastic scattering cross-section “”, Both parameters are inversely proportional to “” [1,2,3] (see also Fig. +1 for a graphical abstract) in two unconventional superconductors (strontium ruthenate [4,5] and doped with stron- +tium lanthanum cuprate [6,7,8]) where unconventional superconductivity is suppressed by a nonmagnetic potential +following the Larkin equation [9]. These compounds possess different nodal structures that belong to different point +group representations. In addition, both compounds have similar crystal structures although they have different stoi- +chiometric/doped composition of the nonmagnetic “strontium” in their elementary crystal cells. + +We illustrate the idea by showing some data calculated self-consistently and address several macroscopic properties +that appear numerically, scanning the behavior of the inverse collision lifetime “𝜏-1”. It is formalized and explored what +we call “the reduced phase space” (RPS), used in this particular case for dressed fermion quasiparticles that are called +incoherent carriers following a dependence on the doping concentration (see for example [15]). All this is made with +a first neighbors tight-binding procedure. These incoherent carriers obey the Fermi-Dirac statistics and their scatter- +ing lifetime strongly depends on the Fermi energy value and the anisotropic Fermi surface average. + +Understand the input frequency window that are needed for the calculations in the reduced phase space is crucial and +plays a fundamental role since the study of the imaginary part of the scattering cross-section is a well-established + +Contreras and Osorio + + +2 +methodology [16,17] and itis an instructive computational tool that helps to understand the numerical relation be- +tween the macroscopic and microscopic interpretations of different physical phenomena when nonmagnetic disorder +is added for the two crystals in their superconducting phases. + + +FIGURE 1: AN INFOGRAPHIC TALE OF THE 2 PHYSICAL PARAMETERS +The numerical disorder is added with the help of two parameters [18]: The dimensionless collision parameter 𝑐 = + 1 (𝜋 𝑁𝐹 𝑈0) +⁄ + where U0 is an impurity atomic potential and NF is the density of states at the Fermi level. The other +parameter is the amount of doping Γ+ = 𝑛𝑖𝑚𝑝 +(𝜋2 𝑁𝐹) +⁄ + where nimp is the impurity concentration. The reduced phase +space (RPS) found, maps a self-consistent distribution probability function always positive for the dressed fermion +quasiparticles (incoherent carriers) in the two mentioned compounds in their superconducting phase as it will be +shown below. + +On the other hand, the non-equilibrium statistical mechanics makes use of the parameters “l” and “𝜏”; for example, for +a gas of dressed Fermi quasiparticles. The play between these two parameters, makes it possible to move from a com- +plete description of a non-equilibrium state to an abbreviated description using a single distribution function of one +quasiparticle as the one we have obtained [19]. Collision elastic regimes for fermionic dressed quasiparticles depend- +ing on the type of collision in the function ℑ [𝜔̃( + 𝑖 0+)] due to nonmagnetic impurities are three [20]: + + +The unitary collision regime with a maximum in ℑ [𝜔̃( + 𝑖 0+)] at zero frequency where holds the relation +𝜔 𝜏 (𝜔̃) ∼ 1 and the mean free path is “𝑙” with 𝑙 ∼ 𝑎 and is obtained from l kF ~ l a-1 ~ 1. “𝜔̃ – is the self- +consistent frequency”, “is the real frequency”, “kF is the Fermi momentum” and “a – is the constant lattice +parameter”. + +The intermediate collision limit with a nonzero minimum in the imaginary function at the center of the distribution +function and two maxima at real frequencies different from zero, where the inequalities 𝜔 ≤ 𝜏(𝜔̃)−1 and 𝑙 ≥ 𝑎 +take place. + +The hydrodynamic (Born) collision scattering with a null imaginary function at zero frequency and two maxima in +the imaginary part at finite real frequencies following the inequalities 𝜔 ≪ 𝜏−1 and 𝑙 ≫ 𝑎 and where self-con- +sistency can be neglected at very low frequencies. + +Nonrelativistic configuration space +Infographic Tale +0 +Reduced +phase +space +Sr and t +5[a(o +i0+) +C and I+ +Self Consistent Tool +V +Classical phase spaceContreras and Osorio + + +3 +In this work, the physical parametrization of the RPS is made with the help of five physical parameters: the supercon- +ducting energy gap at zero temperature “0 (meV)”, the inverse of the scattering strength “c” (dimensionless parame- +ter), the concentration of non-magnetic impurities “+(meV)”, the Fermi energy of the dressed quasiparticles (inco- +herent carriers) “F (meV)” and the first neighbor hoping tight-binding parameter “t (meV)”. Therefore, this is a tight- +binding case that generalizes the isotropic case [18,21] adding numerical anisotropy and dispersion in energy (see Fig. +1 for a graphical abstract). The idea of using four physical parameters self-consistently (0, F, c and +) as a modeling +tool in disordered HTSC was introduced and pointed out by Profs. J. Carbotte and E. Schachinger using isotropic Fermi +surfaces in a series of works (check [18,22] and references therein). +The body of this manuscript is as follows. Section 2 introduces the reduced phase space. Section 3 analyzes the sign of +the imaginary self-consistent function and the meaning of a hidden damping, additionally links the reduced phase +space with the phase spaces of nonequilibrium statistical mechanics and configuration space of nonrelativistic quan- +tum mechanism; and finally; uses numerical values from the self-consistent procedure to build several phenomeno- +logically disordered phase diagrams for the strontium doped La2-xSrxCuO4, and the triplet Sr2RuO4. Section 4 calculates +the values for the scattering phase-shift in these compounds using the RPS analysis of the previous section. Section 5 +compares briefly the frequency, mean free path and collision scattering lifetime of these two unconventional super- +conductors with those used in the anomalous skin effect with singular shapes in the Fermi surface for normal metals, +and shortly addresses the difficult mathematical issue of nonlocality in “l” and “𝜏”. Finally, conclusions and recommen- +dations are given. + +2. The role of the “Reduced Phase Space” between non-equilibrium Statistical Mechanics +and nonrelativistic Quantum Mechanics +The two dimensional self-consistent reduced phase space (RPS) for dressed fermions (incoherent carriers) is built +with the pair of coordinates (ℜ(𝜔̃), ℑ(𝜔̃)) and has the following properties: + +Property 1: “The reduced phase space (RPS) in the unitary, intermedium and Born limits has two axis: the real +axis ℛ [𝜔̃( + 𝑖 0+)] =  and the imaginary axis ℑ [𝜔̃( + 𝑖 0+)]. It serves to map a distribution function of +dressed fermion quasiparticles, therefor is a fermionic space (also could be called incoherent phase space). + +Property 2: “Unconventional superconductors [17,23] can be also defined as those with nodes/quasinodal regions +around the Fermi surface with an order parameter that has a spin paired dependence (singlet or triplet). This +property allows to build self-consistently different macroscopic phases as happen for the isotope 3He. + +Property 3: “The real part ℛ [𝜔̃( + 𝑖 0+)] belongs to the x interval ∈ (−∞,+∞), and the imaginary part only +to the positive y axis ∈ (0, +∞) with the function ℑ [𝜔̃( + 𝑖 0+)] > 0 always”. + +Property 4: “The reduced phase space resembles a space where damping is contained in the self-consistent imag- +inary part of the elastic scattering cross-section following a relationship that holds between the damping and +the imaginary part: 𝛾 = −ℑ [𝜔̃( + 𝑖 0+)]”. +The units for the input and output parameters in the reduced phase space are the rationalized Planck units where +always hold that ℏ = kB = c = 1 and input and output units are in in milielectronvolts (meV). + +Finally, if is incorporated the tight-binding method (TB) [24] into the dispersion law, the order parameter and the +Fermi surface average, considering the group symmetry properties (such as parity and time reversal symmetries), the +RPS opens a window to understand some macroscopic properties in these two compounds. Worthy to notice, that the +use of the tight-binding enriches but also complicates the computational level of the self-consistent procedure to find +the fermionic reduced phase space distribution function, making it more computing demanding. + + + + + + +Contreras and Osorio + + +4 +3. The sign of the imaginary elastic cross-section for dressed fermion quasiparticles + +The inverse of the scattering lifetime is given in normal metals and unconventional superconductors by the following +expresion 𝜏−1() = 2  [𝜔̃( + 𝑖 0+)] [16,17]. In general, the mathematical treatment of an external constant po- +tential “U0” using the elastic scattering theory in nonrelativistic quantum mechanics is a complicated subject [25]. In +this work, the real part is given in the RPS with the coordinate ℜ(𝜔̃) = . The imaginary term in the RPS is repre- +sented by the function  [𝜔̃( )] = (2𝜏)−1[𝜔̃( )] with a hidden self-consistent damping 𝛾 = − [𝜔̃( + 𝑖 0+)]. + +Now, let us bring to the attention some examples that address this issue. In first instance, to describe “self-consistent +damping” in the classical phase space of the non-equilibrium statistical mechanics (NESM), we need the time-depend- +ent distribution function “f(t)” using the -approximation in the Boltzmann equation, where the partial derivative re- +spect to time refers to the collision of dressed fermion quasiparticles [26] with (𝜕 𝑓 +𝜕 𝑡 +⁄ +)𝑐𝑜𝑙𝑙 = − (𝑓 − 𝑓0) 𝜏 +⁄ . If the +distribution function goes rapidly to an equilibrium situation denoted by the function 𝑓0 , the previous expression can +be approximated by + +( 𝜕 𝑓 +𝜕 𝑡 +⁄ +)𝑐𝑜𝑙𝑙 + 2  [𝜔̃( + 𝑖 0+)](𝑓 − 𝑓0) = 0, (1) + +with a hidden self-consistent collision “coll” behavior and a damping 𝛾 = −  [𝜔̃( + 𝑖 0+)] = −(2𝜏)−1[𝜔̃( )]. The +solution of Eq. 1 for 𝑓(t) will depend on the whole set of TB parameters 0, F, c, t and +. + +A second example, comes from the configuration space in non-relativistic quantum mechanics (NRQM) [27,28]. If the +equation for the time dependent probability density 𝒲(𝑡) is obtained with a wave function containing an extra expo- +nential term which describes some damping at the quasi-stationary level. This can happen for one dressed quasiparti- +cle inside an isotropic or anisotropic Fermi reservoir as suggested in [27]. The wave function will contain quasi-sta- +tionary levels of the form 𝜓𝜔(𝑡) ∼ 𝑒−𝑖 +ℏ(𝜖−𝑖Γ)𝑡. For fermionic quasiparticles is known that Γ ℏ +⁄ = (2 𝜏)−1 [29] with a +probability density 𝒲(𝑡) = |𝜓𝜔(𝑡)|2 = 𝒲0𝑒−2 Γ ℏ +⁄ 𝑡 where 𝒲0 denotes the equilibrium case. + +For 𝒲(𝑡) in the configuration space [28], the following equation holds 𝜕 𝒲(𝑡) +𝜕 𝑡 +⁄ += − 2 Γ ℏ +⁄ 𝒲(𝑡) [27]. If we again +look at Eq. 1 and rearrange this new expression as a partial differential equation with Γ ℏ +⁄ = (2 𝜏)−1 = +  [𝜔̃( + 𝑖 0+)], we obtain + +(𝜕 𝒲(𝑡) +𝜕 𝑡 +⁄ +)𝑞𝑠𝑑 + 2  [𝜔̃( + 𝑖 0+)] 𝒲(𝑡) = 0, +(2) + +where now “qsd” means quasi-stationary damping, and the time partial derivative refers to quasi-stationary levels +such as those that can be originated in an unconventional superconductor with strontium from the influence of its +nonmagnetic atomic potential U0. Equations 1 and 2 are identical although refer to different physical processes (colli- +sion and damping). However, Eq. 2 resembles the -approximation in the kinetic Boltzmann equation, but for NRQM. +Henceforth, we can define a hidden damping from Eq. 2 as being given by a coefficient 𝛾 = −  [𝜔̃( + 𝑖 0+)] where +on the self-consistent mechanism depends how long will survive the dressed quasiparticle (incoherent state) around +the atomic potential. We control the physical phases in the RPS by learning how to use properly the five parameters: +the number of dressed fermions, the hoping, the strength of the scattering, the zero superconducting gap and the dis- +order. + +Now is clear that this analogy links the quasi-stationary probability density 𝒲(𝑡) on the configuration space [28] and +the quasi-stationary distribution function 𝑓(t) on the phase space [27], one being a classical phenomenon, the other a +quantum one (see Fig. 1). We now understand why is called a “reduced phase space”. The answer we find is that the +“lifetime” is the only output parameter, and the “mean free path” has to be given by the strength “c” of the strontium +atomic potential as an input dimensionless number, and looking and the distribution functions obtained from the im- +aginary part, several phases can be predicted. + +Contreras and Osorio + + +5 +Non-equilibrium “classical or quantum” statistical mechanics refers also to phenomena where the damping is hidden +self-consistently in the distribution probability function 𝑓(t) or the quasi-stationary probability density 𝒲(𝑡), near +the equilibrium and with a coefficient +𝛾 [𝜔̃( + 𝑖 0+)] = −  [𝜔̃( + 𝑖 0+)] < 0. (3) +Relation (3) means that the imaginary part of the elastic scattering cross-section is always positive defined and can +open the possibility for the quasi-nodal points in the OP such as the ones in the Miyake-Narikiyo model [12] where +four superconducting isolate quasinodal points are symmetrically distributed in the first Brillouin zone. A second +condition in the zero temperature imaginary elastic cross-section is derived from the first + [𝜔̃( + 𝑖 0+)] > 0 . (4) +In order to validate relation (4) in the case of the two unconventional superconductors, we discuss several calculations +in detail. +We begin with Table 1 showing a few points of the whole set of data calculated self-consistently to obtain the Miyake- +Narikiyo tiny gap [30] in the unitary collision regime with the five input values 0 = 1.0 meV, F = -0.4 meV, c = 0, t = 0.4 +meV and += 0.05 meV. As can be seen from the second column in Table 1 with values taken from the self-consistent +solution for the function  [𝜔̃( + 𝑖 0+)], the numbers that represent the tiny gap are close to zero but always pos- +itive (since 1 meV = 10-3 eV), so the values of the imaginary self-consistent elastic scattering cross-section are never +zero or negative in our calculations when the Fermi energy is negative and very small (F = - 0.4 meV). The smallest +number obtained self-consistently is shadowed gray in the second column of Table 1. +To complement this, some numbers for the case where Sr2RuO4 has point nodes is also showed in the third column +of Table 1 [31]. The parameter for the Fermi energy is now bigger and close to the zero value (F = - 0.04 meV), but +the other four TB parameters remain equal to those used in the quasinodal case. For the node points situation (Fig. +2), there are not small values in the imaginary part as seen in the third column of Table 1 and in Fig. 2, with the +minimum of the imaginary function shadowed gray for a dilute coalescent + = 0.05 meV. +At this point is good to remember that the Fermi-Dirac distribution describes the function of dressed electrons and +holes on the quasi-stationary quantum energy levels (n and where n = 0,1,2…) with 𝑓𝑛 = 1 +(𝑒 +− +𝜀𝑛−𝜀𝑓 +𝑘𝐵 𝑇 + 1) +⁄ +. Therefore, +it is important to recall that the Fermi energy F enters as a parameter in the function fn, and that the consequence of +increasing the number of dressed fermion quasiparticles in the system results in an increase of the Fermi energy [33] +as we do to obtain point-nodes in strontium ruthenate [31]. Despite strontium ruthenate continues to be part of an +intense discussion with respect to its OP as expressed recently in [37], for the point nodes triplet model in the unitary +collision regime, Fig. 2 shows the behavior of the function  [𝜔̃( + 𝑖 0+)] with parameters: 0 = 1.0 meV, F = -0.04 +meV, c = 0, t = 0.4 meV and varying +≈ (0.05-0.40) meV from dilute to optimal [31]. From Fig. 2, it can be observed for +example, that only for += 0.05 meV there is a noticeable change in slope around the frequency value of 1.4 meV (Tc +for this compound when samples are clear is around 1.5 Kelvin). The other dressed curves show a smooth minimum +displaced to higher frequencies [31]. +The case involving the HTSC La2-xSrxCuO4 is more difficult to obtain numerically because the real frequency window +should suffix to locate the normal state-superconducting transition point; and in addition; we cannot extend this pro- +cedure to the antiferromagnetic phase. This is due to the existence of gap values that strongly depend on disorder [32], +and this kind of numerical calculation is a difficult task since it depends on the Fermi energy value (the number of +dressed quasiparticles) and is very computing demanding task, with real frequencies in a window of ±120 meV to +describe properly the whole behavior of the imaginary elastic cross-section part (details of the last statement to be +published by the authors in a separate manuscript). + +Contreras and Osorio + + +6 +One of the peculiarities with the compound La2-xSrxCuO4 is that Tc depends on both the concentration of doped ions +and the number of CuO2 layers and makes the use of this procedure a computational challenge where the initial fre- +quency values are not always stable to obtain the hidden self-consistency. Similitudes and differences of the two com- +pounds using this approach with a small frequency window is given in [34,35]. We think of a model composed by a +gas of fermionic dressed quasiparticles which obey the Fermi liquid behavior [36]. +For La2-xSrxCuO4 we show Table 2 and Table 3 with some numerical results from [20] for a zero superconducting gap +with the value 0 =33.9 meV, F = -0.4 meV, c = 0, t = 0.4 meV and += 0.05 meV using a linear nodal OP model [10,11]. +Notice in Table 2, that the box shaded gray represents the minimum value for the imaginary self-consistent function, +which is given in Fig. 3 in orange color and represents a coalescent phase where the nonmagnetic strontium atoms +stick together in a metallic region and get the quasi-momentum transferred from the dressed Fermi quasiparticles, +but only for a very dilute doping with + ≈ (0.01 - 0.05) meV represented in Fig. 3 with the yellow and orange curves +[20]. +In the same Fig. 3 is observed a very small displacement of the minimum in the imaginary function  [𝜔̃( + 𝑖 0+)] +when frequency values are increased. This behavior is notorious in the other compound strontium ruthenate and the +varying parameter becomes the zero temperature gap as was obtained in [38]. But to slightly notice the same behavior +in the doped lanthanum, for now, we show some values taken from Fig. 3 in the third column of Table 3, where we have +also shadowed some numerical fluctuations in the real frequency values in gray color at the point where the transition +occurs, when scanning the function from dilute to optimal values of the doping +. +If the dressed fermionic quasiparticles momentum is transferred to the strontium atoms in the crystal lattice, sticking +together in a coalescing metallic state with an almost constant scattering lifetime for the whole set of real frequencies, +it allows to adjust non-equilibrium low temperature data fairly well using the same normal state scattering lifetime, +but only if the impurity concentration is low enough with +≈ (0.01 -0.05) meV. This hypothesis was firstly proposed +in [39]. In addition, we were able to fit ultrasound and electronic heat transport data for bulk crystals of strontium +ruthenate at very low temperatures with a constant lifetime by properly averaging the kinetic coeficients using tight +binding parameters, and making use of the three sheets of the Fermi surface, thanks to what, a self-consistency pro- +cedure wasn’t required [40,41]. +In Fig. 4 we give an intuitive sketch located inside the dashed blue rectangle built from Fig. 3 on how looks like the +superconducting part of the phase diagram in the reduced phase space for La2-xSrxCuO4. We could make it, interpreting +the results from the imaginary part of the zero scattering cross-section and the doping + is scanned from light to +optimal values in the unitary collision regime [20]. At this point, we remind that all calculations were possible thanks +to the fact that we added Edwards disorder. A review of the work in this direction with the original references can be +found in [42]. +The use of the time dependence (non-equilibrium processes) in both functions 𝑓(t) and 𝒲(𝑡) mentioned in the previ- +ous section is crucial to understand the physical picture underlying this approach, that comes from a well-established +methodology as the elastic cross-section analysis [16,17,18,39] when we look at the numbers obtained in the reduced +phase space for the lifetime considering the unitary limit. This remark gives the title of this manuscript. +TABLE 1: SMALLEST VALUES OF THE IMAGINARY ELASTIC SCATTERING CROSS-SECTION FOR THE MIYAKE-NARIKIYO QUASI-POINTS [30] +AND THE POINT NODES [31] OP. THE PARAMETERS USED ARE GIVEN IN THE MAIN TEXT, + = 0.05 MILIELECTRONVOLTS + = ℜ(𝜔̃) +(meV) +8.51e-001 +8.61e-001 + +8.71e-001 +8.81e-001 +8.91e-001 +9.01e-001 +9.11e-001 +9.21e-001 +9.31e-001 + + ℑ(𝜔̃) +Quasi- +point +nodes +8.63e-008 +3.49e-008 +3.54e-004 +1.59e-005 +6.25e-007 +2.21e-008 +5.65e-004 +1.87e-005 +6.74e-007 + +Contreras and Osorio + + +7 + + ℑ(𝜔̃) +Point no- +des +3.43e-001 +3.43e-001 +3.43e-001 +3.43e-001 +3.43e-001 +3.43e-001 +3.43e-001 +3.42e-001 +3.42e-001 + +TABLE 2: SMALLEST VALUES OF THE IMAGINARY ELASTIC SCATTERING CROSS-SECTION FOR THE LINE NODES OP IN THE UNITARY LIMIT +WITH A ZERO GAP 0 =33.94 MEV AND COALESCENT (DILUTE) DOPING + = 0.05 MEV. + = ℜ(𝜔̃) +(meV) +33.66 + +33.71 +33.78 +33.81 +33.86 +33.91 +33.96 +34.01 +34.10 + + ℑ(𝜔̃) +line nodes +6.06e-002 +5.97e-002 +5.86e-002 +5.75e-002 +5.61e-002 +5.47e-002 +5.56e-002 + +5.98e-002 +6.33e-002 + +TABLE 3: DISPLACEMENT IN THE VALUES OF THE REAL AND IMAGINARY PARTS OF THE ELASTIC SCATTERING CROSS-SECTION OBSERVED +FOR THE SINGLET LINEAR OP WHEN THE ZERO SUPERCONDUCTING GAP IS 0 = 33.94 MEV AND DOPING GOES FROM VERY DILUTE TO AN +OPTIMAL VALUE. +Γ+ (meV) +0.01 +0.05 + +0.10 + +0.15 +0.20 + = ℜ(𝜔̃) +(meV) +33.950 +33.910 +33.900 +33.901 +33.801 + ℑ(𝜔̃) +Line nodes + (meV) +9.63e-003 +5.47e-002 +1.19e-001 +1.89e-001 +2.63e-001 + + +FIGURE 2: POINTS NODES IN THE TRIPLET MODEL WHEN THE FERMI ENERGY IS VERY CLOSE TO ZERO. DATA IN TABLE 1 +COMES FROM THE BLACK CURVE CALCULATED IN [31]. + +1.6 +r+=0.05meV +C=0.0 +1.4 +『+=0.10meV +[+=0.15 meV +Im[(w + io + )](meV) +1.2 +F+=0.20meV +r+=0.25meV +1.0 +r+=0.30 meV +F+=0.35 meV +0.8 +r+=0.40 meV +0.6 +0.4 +0.2 +0.0 +1 +1 +1 +.4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +w(meV)Contreras and Osorio + + +8 + +FIGURE 3: IMAGINARY PART OF THE ELASTIC SCATTERING CROSS-SECTION IN THE UNITARY LIMIT FOR LINE NODES. DATA +IN TABLE 2 COMES FROM THE ORANGE CURVE [20]. + +FIGURE 4: SUPERCONDUCTING PART OF THE PHASE DIAGRAM FOR STRONTIUM DOPED LANTHANUM IS SKETCHED INSIDE +THE BLUE DASHED RECTANGLE FROM THE ANALYSIS OF FIG. 3 IN THE REDUCED PHASE SPACE. + +4. The scattering phase shift 0 versus the inverse scattering strength c + +Since we used the RPS to numerically calculate self-consistently and study the behavior of several families of positive +fermionic distribution functions depending on disorder and scattering strength, that we called in first instance “Wig- +ner macroscopic probabilistic distributions” [43,44] and where the energy is conserved in the three collision regimes, +i.e., the unitary, the intermediate and the Born cases. Therefore, we can calculated an important property, “the scat- +tering phase-shift” for the two compounds using the equation cot−1 𝑐 = cot−1(𝜋 𝑁𝐹 𝑈0 )−1 = 𝛿0 [22] and the results +obtained from the set of distribution functions when considering different scattering regimes. Henceforth, we build +Table 4 that relates the inverse nonmagnetic dimensionless strength c which the phase shift 𝛿0. + +As we can observe from the second column in Table 4, numerically this model shows that the HTSC unconventional +superconductor La2-xSrxCuO4 has a major diversity of phase-shift values than the triplet superconductor strontium +ruthenate. This happens when the numerical calculation is performed for the TB values mentioned in section 2 for +both compounds since the singlet compound can be numerically found in more regimes, i.e., the unitary, the interme- +diate and the hydrodynamic limits [20], meanwhile the triplet model remains most of the time in the unitary and + +3.5 +C=0.0 +[+=0.01 meV +[+=0.05meV +3.0 +[+=0.10 meV +Im[w(w + io +)l(meV) +T+=0.15 meV +2.5 +「+=0.20meV +2.0 +1.5 +1.0 +0.5 +0.0 +=40 +-30 +-20 +-10 +0 +10 +20 +30 +40 +w(meV)Reduced phase Space sketch of the phase diagram +in rationaliged Planck units for Laz-xSr.CuO : +(Λaw) +considering the unitary collision regime with ++i0+)] +a gero gap o = 33. 9 mel +Antiferro +Superconducting phase with optimal +Magnetic +impurities concentrationI +)ml +insulating +phase +strange/incoherent/dressed)inthe +with +phase reduced space +27 +a broken +lattice +symmetry +Normal state +Superconducting coalescencephase with dilute +phase +impurities concentration+ +R[a(o + io+)] = (mev)Contreras and Osorio + + +9 +intermediate limits. + +TABLE 4: CALCULATION OF THE PHASE SHIFT FOR BOTH COMPOUNDS USING DIFFERENT REGIMES FOR ELASTIC COLLISIONS +Strontium ruthenate +c values observed from the imaginary part +self-consistently (0.0 for the unitary limit +and 0.4 for the intermediate scattering +limit) [20] +𝛿0 values in degrees calculated for the phase shift +from the previous column: 90.00° for the unitary re- +gime and 68.20° for the intermediate scattering +limit. +Doped strontium +lanthanum cuprate +c values observed self-consistently (0.0 for +the unitary limit , 0.2 for the intermediate +limit, and 0.4 for the Born regime) [30] +𝛿0 values in degrees found for the phase shift from +the previous column: 90° for the unitary, 78.70° for +the intermediate and 68.20° for the hydrodynamic +limit. + +5. Frequency dispersion relations for the anomalous skin effect versus the elastic self-con- +sistent scattering lifetime + +Finally, in order to gain additional credibility in the use of the RPS approach with respect to the Boltzmann kinetic +equation; we conclude with a very short analysis by contrasting frequency values with the anomalous skin effect [45] +and the examples discussed in previous sections. We first, give a brief introduction to the anomalous skin effect and +after that we assemble Table 5 to summarize section five. + +5.1 Differences between normal and anomalous skin effects: +In the anomalous skin effect, the equation for the metallic impedance changes and the electronic mean free path “l” +starts to play a role. Let us, summarize the main differences between the normal and anomalous skin effect briefly to +start with [46]. In the normal skin effect, the metallic impedance “" has the equation Re() –i Im () composed by +equal real resistive and imaginary reactive terms where Re  = Im  = √2 𝜋𝜔 (𝜎 𝑐2) +⁄ +. The physical behavior of an +external electromagnetic field (EMF) on the metal surface is to penetrate it and decay as ~ 𝑒− 𝑥 𝛿 +⁄ with an effective +penetration depth of the EMF given by 𝛿𝑛𝑜𝑟𝑚𝑎𝑙 = 𝑐 +√2𝜋𝜔𝜎 +⁄ + which does not depend on the mean free path [46]. +However, normal metals have a high conductivity “”when normalis small, but at low temperatures the mean free path +“l” becomes larger and the Ohm law in the local form 𝑗 = 𝜎 𝐸 cannot be applied. Thus, it is used a non-local equation +(*) 𝑗 (𝑟) = ∫ 𝑘𝑖𝑘(𝑟, 𝑟´)𝐸𝑘(𝑟´) 𝑑𝑟´ where the anomalous skin effect is defined by saying that the kernel of the equation +(*) depends on the mean free path “𝑘𝑖𝑘(𝑟, 𝑟´) ~ l” [47]. As a consequence, the external electric field is non-uniform, +and since the normal skin effect can be derived from the kinetic equation only if the electric field is assumed uniform, +the kinetic equation in the diffusive limit for a non-equilibrium fermionic distribution function has to be solved [47]. +The main qualitative difference between normal and anomalous skin effects in the impedance equation is given by the +square root of three in the imaginary part of the impedance: Re() – √3 i Im (). Additionally, the depth penetration +has a mean free path dependence given by 𝛿𝑎𝑛𝑜𝑚𝑎𝑙𝑜𝑢𝑠 = √𝑐2 𝑙 +3 +√4𝜋𝜔𝑎𝜎 +⁄ + with 𝑎 ~ 1, and this dependence between the +mean free path “l” and the anomalous penetration depthis used to plot “". Otherwise, normal and anomalous skin +effects can be differentiate sketching (Re  versus 𝜎1/2, where two regions are well defined [46]. One of them where +the inverse resistive impedance has an approximate linear dependence on the square root of the conductivity that is +the normal skin effect and another where the resistive impedance is constant and is called the anomalous skin effect +[46]. + + +Contreras and Osorio + + +10 +5.2 Geometrical interpretation of the singular behavior in the anomalous skin effect +To describe the anomalous skin effect in geometrical terms, we say that the anomalous skin effect happens if the fer- +mionic quasiparticles lye in a belt of the Fermi surface with two geometrical conditions: First, 𝒏 . 𝒗(𝒑) = 0 where n +is a vector normal to the metallic surface; and; Second 𝜀(𝑝) − 𝜀𝐹 = 0 [48] . The singularities in 𝜀(𝑝) will become im- +portant for the anomalous skin effect region when the radius ℓ 𝛿𝑛𝑜𝑟𝑚𝑎𝑙 +⁄ +⋙ 1 and that happens when the dispersion +law for fermionic quasiparticles has terms of the type 0 ~ −𝜖𝐹 + |𝑝𝑥|𝑣 +(higher order terms in momentum) [48] +which is possible if the fermionic quasiparticles obey a non-quadratic energy spectrum. In that case 𝒏 . 𝒗(𝒑) = 0 is not +the equation of a plane in the phase space and the belt is not a planar curve [48]. In this case the geometry of the belt +is a strong function of the geometry of the Fermi surface and the direction of the vector n. As a consequence of this, +the type of connectivity changes in two different ways: either a closed loop can appear or disappear in the belt (O-type +singularity), or a bridge between two loops can rupture or rejoin (X-type singularity) [49]. As a consequence, non- +equilibrium “kinetic” characteristics of a metal such as the anomalous skin effect, or the sound absorption have singu- +larities of the “0” or “X” types and the change in the shape of the belt gives "local information" about the Fermi surface. +[48,49] called the p-point responsible for this type of change in connectivity “a critical point pc”, and showing that they +are located “along curves of parabolic points”. Therefore, the singularities of 0 and X types can only occur only for those +metals whose Fermi surfaces have parabolic points (called also zero curvature lines [49]). +If the metal is isotropic, then there will be an effective conductivity given by the equation 𝜎𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 = 𝑖𝑎 𝜎 (|𝑘| ℓ) +⁄ + with +𝑎 ~ 1 because the number of fermion quasiparticles that participate in the anomalous skin effect is approximated by +𝑛𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 ~ 𝑛 𝛿 ℓ +⁄ [46]. Thus, one can say that the effective conductivity when the Fermi surfaces are isotropic de- +pends on the mean free path as 1 ℓ +⁄ , i.e., 𝜎𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 ∝ 𝜎 (|𝑘| ℓ) +⁄ + and depends “only” on the characteristics of the fer- +mionic spectrum [46]. To finalize this brief summary, it is important to mention that the diffusive reflection in the +anomalous skin effect is given by including the term 𝒗 . 𝜕𝑓 +𝜕 𝒓 +⁄ + in the Boltzmann kinetic equation, where 𝒗 = 𝑓(𝒑) is +the velocity of a fermion quasiparticle with p a quasi-momentum in the crystal lattice [47], that can be mitted only in +the case when the mean free path is much smaller than the distances along which the electric field changes signifi- +cantly, in other words, nonlocality is neglected and the skin effect is in the normal regime when the resistive and +reactive parts of the impedance are equal and the conductivity does not depend on the mean free path. +5.3 Anomalous singular skin effect “l” versus incoherent (dressed) “1/ (2" +A link with the previous sections arises naturally because we seek an analogy between the phase and the configuration +spaces and the existence of a kernel in the integro-diferential equations that include nonlocality of the kinetic param- +eters l and . As pointed out in [47] “to find out the explicit form of the kernel k (ik) the kinetic equation for the non- +equilibrium part of the electron distribution function must be solved”. Table 5 shows several frequency dependent dis- +persion relations for “l” and “ “by comparing the two effects: the anomalous skin effect in normal metals with Fermi +surfaces with parabolic points [48], with the reduced phase space for unconventional superconductors. In [48], it was +found theoretically the impedance in the hydrodynamic limit 𝜔 𝜏 ≪ 1 for the anomalous skin effect in thin metallic +films by giving some examples using complicated 3D Fermi surfaces to average the conductivity and the impedance. +We found that the real part of the impedance strongly depends on two parameters: the mean free path and the shape +of the belts on each Fermi surface studied (the shape of the singular belts makes used of the topological generalized +Lifshitz transitions [47]). + +It was noticed in [48] that by doing an appropriate integration, two physical behaviors can be distinguished in the +anomalous real part of the impedance (one of them called a singular behavior, check Fig. 6 in ref. [48] and Table 1 in +ref. [47] for the type of singular points [49] and the impedance dependence on the mean free path). Hence, it was +stated in [47] that the solution for the impedance and conductivity depend sensitively on the ratio of spatial and tem- +poral dispersions of the kinetic parameters “l” and “”. Therefore, we state in this work, that the solution for the im- +aginary function  [𝜔̃( + 𝑖 0+)] (or the inverse scattering lifetime) depends sensitively on the ratio of spatial and + +Contreras and Osorio + + +11 +temporal dispersions for “l” and “” as well, since for the analysis of the previous sections, we needed the unitary +collision limit where the mean free path l ∼ a, being a the lattice parameter, but requiring this time a self-consistent +calculation of the inverse scattering lifetime 1/𝜏 (𝜔̃), although it might be no obvious in this case, because the exist- +ence of the kernel is not clear. +In addition, the frequency window required in the reduced phase space for the two unconventional superconductors +happens if 𝜔 ~ 1 𝜏 +⁄ ∼ 4 Δ0 (around 4 meV for strontium ruthenate and 120 meV for the doped with strontium lan- +thanum cuprate). Moreover, the tight-binding parameters (𝑡, 𝜖𝐹) influence strongly the Fermi surface averages and +their values are able to distinguish different OP physical phases as it was done by comparing the singular belts in +the anomalous skin effect [37]. Therefore, the relation dispersion in the scattering lifetime that holds for the unitary +collision regime in the reduced phase space might be written as stated in the introduction +𝜔 𝜏 (𝜔̃(ω)) ~ 1 (5). +To conclude, it is important to recall that recently, the anomalous skin effect with this type of anomalies in the Fermi +surface has gained attention among the research community. Mainly for microwave applications [50,51] and in the +study of nonlocality phenomena in solids, as recently was theoretically and experimental realized for the compound +PdCoO2 [52,53]. +TABLE 5 DISPERSIONS FOR THE ANOMALOUS SKIN EFFECT VERSUS THE TWO UNCONVENTIONAL SUPERCONDUCTORS +Kinetical Physics +Condensed Matter +Phenomena. +To study in: +Theoretical methods of so- +lution +Temporal dispersion re- +lation for the scattering +lifetime +Spatial dispersion rela- +tion for the quasiparti- +cles +Anomalous skin ef- +fect and surface +impedance with +Fermion qua- +siparticles. +Normal metal thin samples. +Kinetic Boltzmann eq. in +the approximation. +𝜔 𝜏 ≪ 1 +Hydrodynamic limit +𝑙 ≫ 𝛿 +𝜹 is the anomalous skin +depth, the mean free +path is l +Strange metallic +phase in two un- +conventional su- +perconductors +Superconducting ceramic +thin samples for the doped +HTSC and crystal bulb sam- +ples for the ruthenate +Numerical self-consistent +equation  [𝜔̃( + 𝑖 0+)] +in the reduced phase +space. +𝜔 𝜏 (𝜔̃() ) ∼ 1 +Unitary limit + +𝑙 ∼ 𝑎 +a is the lattice parame- +ter + +5. Conclusion and recommendations +This work was aimed at introducing with some numerical examples the importance of two physical parameters, the +mean free path and scattering lifetime, both widely used in non-equilibrium statistical mechanics and a brief analysis +of what we have called the reduced phase space for the real and imaginary parts of the elastic scattering cross-section, +using two unconventional superconductors in the unitary limit as examples, when the fermionic quasiparticles are +dressed by a non-magnetic impurity potential, for three cases of the order parameter, the quasi/nodes, point nodes +and line nodes using a 2D anisotropic TB self-consistent parametrization with nearest neighbor hoping. + +Despite, we focused our study to the unitary regime, we took into account a discussion involving three scattering +regimes in the imaginary part of the elastic cross-section. We have defined a “hidden damping parameter” +  [𝜔̃( + 𝑖 0+)] in “the imaginary part of the elastic scattering cross-section”, being the last always positive, +i.e., " [𝜔̃( + 𝑖 0+)] > 0” obtained using a self-consistent numerical procedure. Therefore, that kind of self-con- +sistent hidden behavior might be of interest for researchers who study the statistical physics of non-equilibrium +phenomena (classical or quantum) from a macroscopic point of view. + +To conclude, several examples were analyzed in sections 2 to 5. Sometimes using tables and figures from numerical + +Contreras and Osorio + + +12 +calculations, but also giving analogies between the classical phase space of the non-equilibrium statistical mechan- +ics, the configuration space of nonrelativistic quantum mechanics, and the reduced phase space (see Fig. 1 for a +graphic summary). The study of the imaginary part of the elastic cross-section not only is important for these two +models of unconventional superconductors with strontium, but also is of interest for the study of fermionic and +bosonic trapped gases at very low temperatures as it has been addressed in reference [54]. + +6. CRediT authorship contribution statement +P. Contreras: Conceptualization, Methodology, Software, Investigation, Validation, Writing – origi- +nal draft, Supervision, Writing – review & editing. +Dianela Osorio: Methodology, Data curation, Software, Visualization, Investigation, Validation, +Writing – review & editing. +7. Declaration of competing interest +The authors declare that they have no known competing financial interests or personal relation- +ships that could have appeared to influence the work reported in this paper. +8. Acknowledgements +This research did not receive any specific grant from funding agencies in the public, commercial, +or not-for-profit sectors. 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Usp. 51 pp. 603 DOI: +10.1070/PU2008v051n06ABEH006548 + diff --git a/BtE4T4oBgHgl3EQf5Q6j/content/tmp_files/load_file.txt b/BtE4T4oBgHgl3EQf5Q6j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbf8d8009fafb017a192209a769c3f2d21cd9846 --- /dev/null +++ b/BtE4T4oBgHgl3EQf5Q6j/content/tmp_files/load_file.txt @@ -0,0 +1,834 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf,len=833 +page_content='A tale of the scattering lifetime and the mean free path Pedro Contreras*,1 and Dianela Osorio 2 1 Department of Physics, University of the Andes, Me rida, Venezuela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' ORCID: 0000-0002-3394-1195 2 Department of Brain and Behavioral Sciences, University of Pavia, Pavia, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' ORCID: 0000-0002-8171-0703 Corresponding author (pedrocontre@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='com) ABSTRACT The idea of applying the scattering lifetime calculated from the imaginary part of the zero temperature elastic scatter- ing cross-section to study a hidden self-consistent damping in two spaces of importance for non-equilibrium statistical mechanics is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' It is discussed its relation with the classical phase space from statistical mechanics and the configuration space from nonrelativistic quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' This idea is contrasted with the mean free path values in three elastic collision regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The main exercise is to study the behavior of a self-consistent probabilistic distribu- tion function in a space we have called the reduced phase space, since it is related to the scattering lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' This exer- cise has been solved in two unconventional superconductors, for which several calculations are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='One ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='singlet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='doped ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='strontium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='lanthanum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='cuprate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='when ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='three ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='superconducting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='order ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='used: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='quasi-point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' the point and the line nodal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' We finally compare the frequency dispersion in the anom- alous skin effect with singular shapes of the Fermi surface with the frequency dispersion in the scattering lifetime and their respective mean free paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' This idea is useful because it intuitively explores the nonlocality of this type of hidden self-consistent damping for those incoherent fermionic quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Keywords: Reduced phase space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Configuration space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Classical phase space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Mean free path;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Collision lifetime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Damping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Non-equilibrium statistical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Introduction This work is aimed at phenomenologically understanding the role of two parameters widely used in non-equilibrium statistical mechanics, the mean free path “l” and the scattering lifetime “𝜏”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' One calculated and the other used in the study of the elastic scattering cross-section “\uf073”, Both parameters are inversely proportional to “\uf073” [1,2,3] (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 1 for a graphical abstract) in two unconventional superconductors (strontium ruthenate [4,5] and doped with stron- tium lanthanum cuprate [6,7,8]) where unconventional superconductivity is suppressed by a nonmagnetic potential following the Larkin equation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' These compounds possess different nodal structures that belong to different point group representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In addition, both compounds have similar crystal structures although they have different stoi- chiometric/doped composition of the nonmagnetic “strontium” in their elementary crystal cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' We illustrate the idea by showing some data calculated self-consistently and address several macroscopic properties that appear numerically, scanning the behavior of the inverse collision lifetime “𝜏-1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' It is formalized and explored what we call “the reduced phase space” (RPS), used in this particular case for dressed fermion quasiparticles that are called incoherent carriers following a dependence on the doping concentration (see for example [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' All this is made with a first neighbors tight-binding procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' These incoherent carriers obey the Fermi-Dirac statistics and their scatter- ing lifetime strongly depends on the Fermi energy value and the anisotropic Fermi surface average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Understand the input frequency window that are needed for the calculations in the reduced phase space is crucial and plays a fundamental role since the study of the imaginary part of the scattering cross-section is a well-established Contreras and Osorio 2 methodology [16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='17] and itis an instructive computational tool that helps to understand the numerical relation be- tween the macroscopic and microscopic interpretations of different physical phenomena when nonmagnetic disorder is added for the two crystals in their superconducting phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' FIGURE 1: AN INFOGRAPHIC TALE OF THE 2 PHYSICAL PARAMETERS The numerical disorder is added with the help of two parameters [18]: The dimensionless collision parameter 𝑐 = 1 (𝜋 𝑁𝐹 𝑈0) ⁄ where U0 is an impurity atomic potential and NF is the density of states at the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The other parameter is the amount of doping Γ+ = 𝑛𝑖𝑚𝑝 (𝜋2 𝑁𝐹) ⁄ where nimp is the impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The reduced phase space (RPS) found, maps a self-consistent distribution probability function always positive for the dressed fermion quasiparticles (incoherent carriers) in the two mentioned compounds in their superconducting phase as it will be shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' On the other hand, the non-equilibrium statistical mechanics makes use of the parameters “l” and “𝜏”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' for example, for a gas of dressed Fermi quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The play between these two parameters, makes it possible to move from a com- plete description of a non-equilibrium state to an abbreviated description using a single distribution function of one quasiparticle as the one we have obtained [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Collision elastic regimes for fermionic dressed quasiparticles depend- ing on the type of collision in the function ℑ [𝜔̃(\uf077 + 𝑖 0+)] due to nonmagnetic impurities are three [20]: The unitary collision regime with a maximum in ℑ [𝜔̃(\uf077 + 𝑖 0+)] at zero frequency where holds the relation 𝜔 𝜏 (𝜔̃) ∼ 1 and the mean free path is “𝑙” with 𝑙 ∼ 𝑎 and is obtained from l kF ~ l a 1 ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' “𝜔̃ – is the self consistent frequency”, “\uf077\uf020\uf02d\uf020is the real frequency”, “kF is the Fermi momentum” and “a – is the constant lattice parameter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The intermediate collision limit with a nonzero minimum in the imaginary function at the center of the distribution function and two maxima at real frequencies different from zero, where the inequalities 𝜔 ≤ 𝜏(𝜔̃)−1 and 𝑙 ≥ 𝑎 take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The hydrodynamic (Born) collision scattering with a null imaginary function at zero frequency and two maxima in the imaginary part at finite real frequencies following the inequalities 𝜔 ≪ 𝜏−1 and 𝑙 ≫ 𝑎 and where self con sistency can be neglected at very low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Nonrelativistic configuration space Infographic Tale 0 Reduced phase space Sr and t 5[a(o +i0+) C and I+ Self Consistent Tool V Classical phase spaceContreras and Osorio 3 In this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' the physical parametrization of the RPS is made with the help of five physical parameters: the supercon- ducting energy gap at zero temperature “\uf0440 (meV)”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' the inverse of the scattering strength “c” (dimensionless parame- ter),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' the concentration of non-magnetic impurities “\uf047+(meV)”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' the Fermi energy of the dressed quasiparticles (inco- herent carriers) “\uf065F (meV)” and the first neighbor hoping tight-binding parameter “t (meV)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Therefore, this is a tight- binding case that generalizes the isotropic case [18,21] adding numerical anisotropy and dispersion in energy (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 1 for a graphical abstract).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The idea of using four physical parameters self-consistently (\uf0440, \uf065F, c and \uf047+) as a modeling tool in disordered HTSC was introduced and pointed out by Profs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Carbotte and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Schachinger using isotropic Fermi surfaces in a series of works (check [18,22] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The body of this manuscript is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Section 2 introduces the reduced phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Section 3 analyzes the sign of the imaginary self-consistent function and the meaning of a hidden damping, additionally links the reduced phase space with the phase spaces of nonequilibrium statistical mechanics and configuration space of nonrelativistic quan- tum mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' and finally;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' uses numerical values from the self-consistent procedure to build several phenomeno- logically disordered phase diagrams for the strontium doped La2-xSrxCuO4, and the triplet Sr2RuO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Section 4 calculates the values for the scattering phase-shift in these compounds using the RPS analysis of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Section 5 compares briefly the frequency, mean free path and collision scattering lifetime of these two unconventional super- conductors with those used in the anomalous skin effect with singular shapes in the Fermi surface for normal metals, and shortly addresses the difficult mathematical issue of nonlocality in “l” and “𝜏”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Finally, conclusions and recommen- dations are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The role of the “Reduced Phase Space” between non-equilibrium Statistical Mechanics and nonrelativistic Quantum Mechanics The two dimensional self-consistent reduced phase space (RPS) for dressed fermions (incoherent carriers) is built with the pair of coordinates (ℜ(𝜔̃), ℑ(𝜔̃)) and has the following properties: \uf0b7 Property 1: “The reduced phase space (RPS) in the unitary, intermedium and Born limits has two axis: the real axis ℛ [𝜔̃(\uf077 + 𝑖 0+)] = \uf077 and the imaginary axis ℑ [𝜔̃(\uf077 + 𝑖 0+)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' It serves to map a distribution function of dressed fermion quasiparticles, therefor is a fermionic space (also could be called incoherent phase space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' \uf0b7 Property 2: “Unconventional superconductors [17,23] can be also defined as those with nodes/quasinodal regions around the Fermi surface with an order parameter that has a spin paired dependence (singlet or triplet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' This property allows to build self-consistently different macroscopic phases as happen for the isotope 3He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' \uf0b7 Property 3: “The real part ℛ [𝜔̃(\uf077 + 𝑖 0+)] belongs to the x interval ∈ (−∞,+∞), and the imaginary part only to the positive y axis ∈ (0, +∞) with the function ℑ [𝜔̃(\uf077 + 𝑖 0+)] > 0 always”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' \uf0b7 Property 4: “The reduced phase space resembles a space where damping is contained in the self-consistent imag- inary part of the elastic scattering cross-section following a relationship that holds between the damping and the imaginary part: 𝛾 = −ℑ [𝜔̃(\uf077 + 𝑖 0+)]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The units for the input and output parameters in the reduced phase space are the rationalized Planck units where always hold that ℏ = kB = c = 1 and input and output units are in in milielectronvolts (meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Finally, if is incorporated the tight-binding method (TB) [24] into the dispersion law, the order parameter and the Fermi surface average, considering the group symmetry properties (such as parity and time reversal symmetries), the RPS opens a window to understand some macroscopic properties in these two compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Worthy to notice, that the use of the tight-binding enriches but also complicates the computational level of the self-consistent procedure to find the fermionic reduced phase space distribution function, making it more computing demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Contreras and Osorio 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The sign of the imaginary elastic cross-section for dressed fermion quasiparticles The inverse of the scattering lifetime is given in normal metals and unconventional superconductors by the following expresion 𝜏−1(\uf077) = 2 \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In general, the mathematical treatment of an external constant po- tential “U0” using the elastic scattering theory in nonrelativistic quantum mechanics is a complicated subject [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In this work, the real part is given in the RPS with the coordinate ℜ(𝜔̃) = \uf077.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The imaginary term in the RPS is repre- sented by the function \uf0c1 [𝜔̃(\uf077 )] = (2𝜏)−1[𝜔̃(\uf077 )] with a hidden self-consistent damping 𝛾 = −\uf0c1 [𝜔̃(\uf077 + 𝑖 0+)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Now, let us bring to the attention some examples that address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In first instance, to describe “self-consistent damping” in the classical phase space of the non-equilibrium statistical mechanics (NESM), we need the time-depend- ent distribution function “f(t)” using the \uf074-approximation in the Boltzmann equation, where the partial derivative re- spect to time refers to the collision of dressed fermion quasiparticles [26] with (𝜕 𝑓 𝜕 𝑡 ⁄ )𝑐𝑜𝑙𝑙 = − (𝑓 − 𝑓0) 𝜏 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' If the distribution function goes rapidly to an equilibrium situation denoted by the function 𝑓0 , the previous expression can be approximated by ( 𝜕 𝑓 𝜕 𝑡 ⁄ )𝑐𝑜𝑙𝑙 + 2 \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)](𝑓 − 𝑓0) = 0, (1) with a hidden self-consistent collision “coll” behavior and a damping 𝛾 = − \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] = −(2𝜏)−1[𝜔̃(\uf077 )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 1 for 𝑓(t) will depend on the whole set of TB parameters \uf0440, \uf065F, c, t and \uf047+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' A second example, comes from the configuration space in non-relativistic quantum mechanics (NRQM) [27,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' If the equation for the time dependent probability density 𝒲(𝑡) is obtained with a wave function containing an extra expo- nential term which describes some damping at the quasi-stationary level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' This can happen for one dressed quasiparti- cle inside an isotropic or anisotropic Fermi reservoir as suggested in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The wave function will contain quasi-sta- tionary levels of the form 𝜓𝜔(𝑡) ∼ 𝑒−𝑖 ℏ(𝜖−𝑖Γ)𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' For fermionic quasiparticles is known that Γ ℏ ⁄ = (2 𝜏)−1 [29] with a probability density 𝒲(𝑡) = |𝜓𝜔(𝑡)|2 = 𝒲0𝑒−2 Γ ℏ ⁄ 𝑡 where 𝒲0 denotes the equilibrium case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' For 𝒲(𝑡) in the configuration space [28], the following equation holds 𝜕 𝒲(𝑡) 𝜕 𝑡 ⁄ = − 2 Γ ℏ ⁄ 𝒲(𝑡) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' If we again look at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 1 and rearrange this new expression as a partial differential equation with Γ ℏ ⁄ = (2 𝜏)−1 = \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)], we obtain (𝜕 𝒲(𝑡) 𝜕 𝑡 ⁄ )𝑞𝑠𝑑 + 2 \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] 𝒲(𝑡) = 0, (2) where now “qsd” means quasi-stationary damping, and the time partial derivative refers to quasi-stationary levels such as those that can be originated in an unconventional superconductor with strontium from the influence of its nonmagnetic atomic potential U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Equations 1 and 2 are identical although refer to different physical processes (colli- sion and damping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' However, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 2 resembles the \uf074-approximation in the kinetic Boltzmann equation, but for NRQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Henceforth, we can define a hidden damping from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 2 as being given by a coefficient 𝛾 = − \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] where on the self-consistent mechanism depends how long will survive the dressed quasiparticle (incoherent state) around the atomic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' We control the physical phases in the RPS by learning how to use properly the five parameters: the number of dressed fermions, the hoping, the strength of the scattering, the zero superconducting gap and the dis- order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Now is clear that this analogy links the quasi-stationary probability density 𝒲(𝑡) on the configuration space [28] and the quasi-stationary distribution function 𝑓(t) on the phase space [27], one being a classical phenomenon, the other a quantum one (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' We now understand why is called a “reduced phase space”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The answer we find is that the “lifetime” is the only output parameter, and the “mean free path” has to be given by the strength “c” of the strontium atomic potential as an input dimensionless number, and looking and the distribution functions obtained from the im- aginary part, several phases can be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Contreras and Osorio 5 Non-equilibrium “classical or quantum” statistical mechanics refers also to phenomena where the damping is hidden self-consistently in the distribution probability function 𝑓(t) or the quasi-stationary probability density 𝒲(𝑡), near the equilibrium and with a coefficient 𝛾 [𝜔̃(\uf077 + 𝑖 0+)] = − \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' (3) Relation (3) means that the imaginary part of the elastic scattering cross-section is always positive defined and can open the possibility for the quasi-nodal points in the OP such as the ones in the Miyake-Narikiyo model [12] where four superconducting isolate quasinodal points are symmetrically distributed in the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' A second condition in the zero temperature imaginary elastic cross-section is derived from the first \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' (4) In order to validate relation (4) in the case of the two unconventional superconductors, we discuss several calculations in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' We begin with Table 1 showing a few points of the whole set of data calculated self-consistently to obtain the Miyake- Narikiyo tiny gap [30] in the unitary collision regime with the five input values \uf0440 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 meV, \uf065F = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 meV, c = 0, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 meV and \uf047+= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' As can be seen from the second column in Table 1 with values taken from the self-consistent solution for the function \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)], the numbers that represent the tiny gap are close to zero but always pos- itive (since 1 meV = 10-3 eV), so the values of the imaginary self-consistent elastic scattering cross-section are never zero or negative in our calculations when the Fermi energy is negative and very small (\uf065F = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The smallest number obtained self-consistently is shadowed gray in the second column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' To complement this, some numbers for the case where Sr2RuO4 has point nodes is also showed in the third column of Table 1 [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The parameter for the Fermi energy is now bigger and close to the zero value (\uf065F = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='04 meV), but the other four TB parameters remain equal to those used in the quasinodal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' For the node points situation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 2), there are not small values in the imaginary part as seen in the third column of Table 1 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 2, with the minimum of the imaginary function shadowed gray for a dilute coalescent \uf047+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' At this point is good to remember that the Fermi-Dirac distribution describes the function of dressed electrons and holes on the quasi-stationary quantum energy levels (\uf065n and where n = 0,1,2…) with 𝑓𝑛 = 1 (𝑒 − 𝜀𝑛−𝜀𝑓 𝑘𝐵 𝑇 + 1) ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Therefore, it is important to recall that the Fermi energy \uf065F enters as a parameter in the function fn, and that the consequence of increasing the number of dressed fermion quasiparticles in the system results in an increase of the Fermi energy [33] as we do to obtain point-nodes in strontium ruthenate [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Despite strontium ruthenate continues to be part of an intense discussion with respect to its OP as expressed recently in [37], for the point nodes triplet model in the unitary collision regime, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 2 shows the behavior of the function \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] with parameters: \uf0440 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 meV, \uf065F = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='04 meV, c = 0, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 meV and varying \uf047+≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='40) meV from dilute to optimal [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 2, it can be observed for example, that only for \uf047+= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05 meV there is a noticeable change in slope around the frequency value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 meV (Tc for this compound when samples are clear is around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='5 Kelvin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The other dressed curves show a smooth minimum displaced to higher frequencies [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The case involving the HTSC La2-xSrxCuO4 is more difficult to obtain numerically because the real frequency window should suffix to locate the normal state-superconducting transition point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' and in addition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' we cannot extend this pro- cedure to the antiferromagnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' This is due to the existence of gap values that strongly depend on disorder [32], and this kind of numerical calculation is a difficult task since it depends on the Fermi energy value (the number of dressed quasiparticles) and is very computing demanding task, with real frequencies in a window of ±120 meV to describe properly the whole behavior of the imaginary elastic cross-section part (details of the last statement to be published by the authors in a separate manuscript).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Contreras and Osorio 6 One of the peculiarities with the compound La2-xSrxCuO4 is that Tc depends on both the concentration of doped ions and the number of CuO2 layers and makes the use of this procedure a computational challenge where the initial fre- quency values are not always stable to obtain the hidden self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Similitudes and differences of the two com- pounds using this approach with a small frequency window is given in [34,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' We think of a model composed by a gas of fermionic dressed quasiparticles which obey the Fermi liquid behavior [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' For La2-xSrxCuO4 we show Table 2 and Table 3 with some numerical results from [20] for a zero superconducting gap with the value \uf0440 =33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='9 meV, \uf065F = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 meV, c = 0, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 meV and \uf047+= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05 meV using a linear nodal OP model [10,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Notice in Table 2, that the box shaded gray represents the minimum value for the imaginary self-consistent function, which is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 3 in orange color and represents a coalescent phase where the nonmagnetic strontium atoms stick together in a metallic region and get the quasi-momentum transferred from the dressed Fermi quasiparticles, but only for a very dilute doping with \uf047+ ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='01 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05) meV represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 3 with the yellow and orange curves [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In the same Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 3 is observed a very small displacement of the minimum in the imaginary function \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] when frequency values are increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' This behavior is notorious in the other compound strontium ruthenate and the varying parameter becomes the zero temperature gap as was obtained in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' But to slightly notice the same behavior in the doped lanthanum, for now, we show some values taken from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 3 in the third column of Table 3, where we have also shadowed some numerical fluctuations in the real frequency values in gray color at the point where the transition occurs, when scanning the function from dilute to optimal values of the doping \uf047+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' If the dressed fermionic quasiparticles momentum is transferred to the strontium atoms in the crystal lattice, sticking together in a coalescing metallic state with an almost constant scattering lifetime for the whole set of real frequencies, it allows to adjust non-equilibrium low temperature data fairly well using the same normal state scattering lifetime, but only if the impurity concentration is low enough with \uf047+≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05) meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' This hypothesis was firstly proposed in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In addition, we were able to fit ultrasound and electronic heat transport data for bulk crystals of strontium ruthenate at very low temperatures with a constant lifetime by properly averaging the kinetic coeficients using tight binding parameters, and making use of the three sheets of the Fermi surface, thanks to what, a self-consistency pro- cedure wasn’t required [40,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 4 we give an intuitive sketch located inside the dashed blue rectangle built from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 3 on how looks like the superconducting part of the phase diagram in the reduced phase space for La2-xSrxCuO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' We could make it, interpreting the results from the imaginary part of the zero scattering cross-section and the doping \uf047+ is scanned from light to optimal values in the unitary collision regime [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' At this point, we remind that all calculations were possible thanks to the fact that we added Edwards disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' A review of the work in this direction with the original references can be found in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The use of the time dependence (non-equilibrium processes) in both functions 𝑓(t) and 𝒲(𝑡) mentioned in the previ- ous section is crucial to understand the physical picture underlying this approach, that comes from a well-established methodology as the elastic cross-section analysis [16,17,18,39] when we look at the numbers obtained in the reduced phase space for the lifetime considering the unitary limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' This remark gives the title of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' TABLE 1: SMALLEST VALUES OF THE IMAGINARY ELASTIC SCATTERING CROSS-SECTION FOR THE MIYAKE-NARIKIYO QUASI-POINTS [30] AND THE POINT NODES [31] OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' THE PARAMETERS USED ARE GIVEN IN THE MAIN TEXT, \uf047+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05 MILIELECTRONVOLTS \uf077 = ℜ(𝜔̃) (meV) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='51e-001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='61e-001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='71e 001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='81e 001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='91e 001 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='01e 001 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='11e 001 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='21e 001 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='31e 001 \uf028\uf032\uf074\uf02d\uf031\uf020\uf029\uf020\uf03d\uf020 ℑ(𝜔̃)\uf020\uf020 Quasi point nodes 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='63e 008 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='49e 008 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='54e 004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='59e 005 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='25e 007 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='21e 008 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='65e 004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='87e 005 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='74e 007 Contreras and Osorio 7 \uf028\uf032\uf074\uf02d\uf031\uf020\uf029\uf020\uf03d\uf020 ℑ(𝜔̃)\uf020 Point no des 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='43e 001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='43e 001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='43e 001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='43e 001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='43e 001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='43e 001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='43e 001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='42e 001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='42e 001 TABLE 2: SMALLEST VALUES OF THE IMAGINARY ELASTIC SCATTERING CROSS-SECTION FOR THE LINE NODES OP IN THE UNITARY LIMIT WITH A ZERO GAP \uf0440 =33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='94 MEV AND COALESCENT (DILUTE) DOPING \uf047+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05 MEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' \uf077 = ℜ(𝜔̃) (meV) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='66 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='71 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='78 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='81 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='86 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='91 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='96 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='01 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='10 \uf028\uf032\uf074\uf02d\uf031\uf020\uf029\uf020\uf03d\uf020 ℑ(𝜔̃)\uf020\uf020 line nodes 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='06e 002 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='97e 002 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='86e 002 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='75e 002 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='61e 002 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='47e 002 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='56e 002 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='98e 002 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='33e 002 TABLE 3: DISPLACEMENT IN THE VALUES OF THE REAL AND IMAGINARY PARTS OF THE ELASTIC SCATTERING CROSS-SECTION OBSERVED FOR THE SINGLET LINEAR OP WHEN THE ZERO SUPERCONDUCTING GAP IS \uf0440 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='94 MEV AND DOPING GOES FROM VERY DILUTE TO AN OPTIMAL VALUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Γ+ (meV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='20 \uf077 = ℜ(𝜔̃) (meV) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='950 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='910 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='900 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='901 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='801 \uf028\uf032\uf074\uf02d\uf031\uf020\uf029\uf020\uf03d\uf020 ℑ(𝜔̃)\uf020 Line nodes (meV) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='63e 003 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='47e 002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='19e 001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='89e 001 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='63e 001 FIGURE 2: POINTS NODES IN THE TRIPLET MODEL WHEN THE FERMI ENERGY IS VERY CLOSE TO ZERO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' DATA IN TABLE 1 COMES FROM THE BLACK CURVE CALCULATED IN [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='6 r+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05meV C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 『+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='10meV [+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='15 meV Im[(w + io + )](meV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='2 F+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='20meV r+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='25meV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 r+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='30 meV F+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='35 meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='8 r+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='40 meV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 -3 -2 -1 0 1 2 3 4 w(meV)Contreras and Osorio 8 FIGURE 3: IMAGINARY PART OF THE ELASTIC SCATTERING CROSS-SECTION IN THE UNITARY LIMIT FOR LINE NODES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' DATA IN TABLE 2 COMES FROM THE ORANGE CURVE [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' FIGURE 4: SUPERCONDUCTING PART OF THE PHASE DIAGRAM FOR STRONTIUM DOPED LANTHANUM IS SKETCHED INSIDE THE BLUE DASHED RECTANGLE FROM THE ANALYSIS OF FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 3 IN THE REDUCED PHASE SPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The scattering phase shift \uf0640 versus the inverse scattering strength c Since we used the RPS to numerically calculate self-consistently and study the behavior of several families of positive fermionic distribution functions depending on disorder and scattering strength, that we called in first instance “Wig- ner macroscopic probabilistic distributions” [43,44] and where the energy is conserved in the three collision regimes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=', the unitary, the intermediate and the Born cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Therefore, we can calculated an important property, “the scat- tering phase-shift” for the two compounds using the equation cot−1 𝑐 = cot−1(𝜋 𝑁𝐹 𝑈0 )−1 = 𝛿0 [22] and the results obtained from the set of distribution functions when considering different scattering regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Henceforth, we build Table 4 that relates the inverse nonmagnetic dimensionless strength c which the phase shift 𝛿0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' As we can observe from the second column in Table 4, numerically this model shows that the HTSC unconventional superconductor La2-xSrxCuO4 has a major diversity of phase-shift values than the triplet superconductor strontium ruthenate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' This happens when the numerical calculation is performed for the TB values mentioned in section 2 for both compounds since the singlet compound can be numerically found in more regimes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=', the unitary, the interme- diate and the hydrodynamic limits [20], meanwhile the triplet model remains most of the time in the unitary and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='5 C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 [+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='01 meV [+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='05meV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 [+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='10 meV Im[w(w + io +)l(meV) T+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='15 meV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='5 「+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='20meV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 =40 -30 -20 -10 0 10 20 30 40 w(meV)Reduced phase Space sketch of the phase diagram in rationaliged Planck units for Laz-xSr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='CuO : (Λaw) considering the unitary collision regime with +i0+)] a gero gap o = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 9 mel Antiferro Superconducting phase with optimal Magnetic impurities concentrationI )ml insulating phase strange/incoherent/dressed)inthe with phase reduced space 27 a broken lattice symmetry Normal state Superconducting coalescencephase with dilute phase impurities concentration+ R[a(o + io+)] = (mev)Contreras and Osorio 9 intermediate limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' TABLE 4: CALCULATION OF THE PHASE SHIFT FOR BOTH COMPOUNDS USING DIFFERENT REGIMES FOR ELASTIC COLLISIONS Strontium ruthenate c values observed from the imaginary part self-consistently (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 for the unitary limit and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 for the intermediate scattering limit) [20] 𝛿0 values in degrees calculated for the phase shift from the previous column: 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='00° for the unitary re- gime and 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='20° for the intermediate scattering limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Doped strontium lanthanum cuprate c values observed self-consistently (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='0 for the unitary limit , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='2 for the intermediate limit, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='4 for the Born regime) [30] 𝛿0 values in degrees found for the phase shift from the previous column: 90° for the unitary, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='70° for the intermediate and 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='20° for the hydrodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Frequency dispersion relations for the anomalous skin effect versus the elastic self-con- sistent scattering lifetime Finally, in order to gain additional credibility in the use of the RPS approach with respect to the Boltzmann kinetic equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' we conclude with a very short analysis by contrasting frequency values with the anomalous skin effect [45] and the examples discussed in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' We first, give a brief introduction to the anomalous skin effect and after that we assemble Table 5 to summarize section five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='1 Differences between normal and anomalous skin effects: In the anomalous skin effect, the equation for the metallic impedance changes and the electronic mean free path “l” starts to play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Let us, summarize the main differences between the normal and anomalous skin effect briefly to start with [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In the normal skin effect, the metallic impedance “\uf07a" has the equation \uf07a\uf020\uf03d\uf020Re(\uf07a) –i Im (\uf07a) composed by equal real resistive and imaginary reactive terms where Re \uf07a = Im \uf07a = √2 𝜋𝜔 (𝜎 𝑐2) ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The physical behavior of an external electromagnetic field (EMF) on the metal surface is to penetrate it and decay as ~ 𝑒− 𝑥 𝛿 ⁄ with an effective penetration depth of the EMF given by 𝛿𝑛𝑜𝑟𝑚𝑎𝑙 = 𝑐 √2𝜋𝜔𝜎 ⁄ which does not depend on the mean free path [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' However, normal metals have a high conductivity “\uf073”\uf020when \uf064normal\uf020is small, but at low temperatures the mean free path “l” becomes larger and the Ohm law in the local form 𝑗 = 𝜎 𝐸 cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Thus, it is used a non-local equation (*) 𝑗 (𝑟) = ∫ 𝑘𝑖𝑘(𝑟, 𝑟´)𝐸𝑘(𝑟´) 𝑑𝑟´ where the anomalous skin effect is defined by saying that the kernel of the equation (*) depends on the mean free path “𝑘𝑖𝑘(𝑟, 𝑟´) ~ l” [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' As a consequence, the external electric field is non-uniform, and since the normal skin effect can be derived from the kinetic equation only if the electric field is assumed uniform, the kinetic equation in the diffusive limit for a non-equilibrium fermionic distribution function has to be solved [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The main qualitative difference between normal and anomalous skin effects in the impedance equation is given by the square root of three in the imaginary part of the impedance: \uf07a\uf020\uf03d\uf020Re(\uf07a) – √3 i Im (\uf07a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Additionally, the depth penetration has a mean free path dependence given by 𝛿𝑎𝑛𝑜𝑚𝑎𝑙𝑜𝑢𝑠 = √𝑐2 𝑙 3 √4𝜋𝜔𝑎𝜎 ⁄ with 𝑎 ~ 1, and this dependence between the mean free path “l” and the anomalous penetration depth\uf020is used to plot “\uf07a".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Otherwise, normal and anomalous skin effects can be differentiate sketching (Re \uf07a\uf029\uf02d\uf031 versus 𝜎1/2, where two regions are well defined [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' One of them where the inverse resistive impedance has an approximate linear dependence on the square root of the conductivity that is the normal skin effect and another where the resistive impedance is constant and is called the anomalous skin effect [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Contreras and Osorio 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='2 Geometrical interpretation of the singular behavior in the anomalous skin effect To describe the anomalous skin effect in geometrical terms, we say that the anomalous skin effect happens if the fer- mionic quasiparticles lye in a belt of the Fermi surface with two geometrical conditions: First, 𝒏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 𝒗(𝒑) = 0 where n is a vector normal to the metallic surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Second 𝜀(𝑝) − 𝜀𝐹 = 0 [48] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The singularities in 𝜀(𝑝) will become im- portant for the anomalous skin effect region when the radius ℓ 𝛿𝑛𝑜𝑟𝑚𝑎𝑙 ⁄ ⋙ 1 and that happens when the dispersion law for fermionic quasiparticles has terms of the type 0 ~ −𝜖𝐹 + |𝑝𝑥|𝑣 +(higher order terms in momentum) [48] which is possible if the fermionic quasiparticles obey a non-quadratic energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In that case 𝒏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 𝒗(𝒑) = 0 is not the equation of a plane in the phase space and the belt is not a planar curve [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In this case the geometry of the belt is a strong function of the geometry of the Fermi surface and the direction of the vector n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' As a consequence of this, the type of connectivity changes in two different ways: either a closed loop can appear or disappear in the belt (O-type singularity), or a bridge between two loops can rupture or rejoin (X-type singularity) [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' As a consequence, non- equilibrium “kinetic” characteristics of a metal such as the anomalous skin effect, or the sound absorption have singu- larities of the “0” or “X” types and the change in the shape of the belt gives "local information" about the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' [48,49] called the p-point responsible for this type of change in connectivity “a critical point pc”, and showing that they are located “along curves of parabolic points”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Therefore, the singularities of 0 and X types can only occur only for those metals whose Fermi surfaces have parabolic points (called also zero curvature lines [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' If the metal is isotropic, then there will be an effective conductivity given by the equation 𝜎𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 = 𝑖𝑎 𝜎 (|𝑘| ℓ) ⁄ with 𝑎 ~ 1 because the number of fermion quasiparticles that participate in the anomalous skin effect is approximated by 𝑛𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 ~ 𝑛 𝛿 ℓ ⁄ [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Thus, one can say that the effective conductivity when the Fermi surfaces are isotropic de- pends on the mean free path as 1 ℓ ⁄ , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=', 𝜎𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 ∝ 𝜎 (|𝑘| ℓ) ⁄ and depends “only” on the characteristics of the fer- mionic spectrum [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' To finalize this brief summary, it is important to mention that the diffusive reflection in the anomalous skin effect is given by including the term 𝒗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 𝜕𝑓 𝜕 𝒓 ⁄ in the Boltzmann kinetic equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' where 𝒗 = 𝑓(𝒑) is the velocity of a fermion quasiparticle with p a quasi-momentum in the crystal lattice [47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' that can be mitted only in the case when the mean free path is much smaller than the distances along which the electric field changes signifi- cantly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' in other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' nonlocality is neglected and the skin effect is in the normal regime when the resistive and reactive parts of the impedance are equal and the conductivity does not depend on the mean free path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='3 Anomalous singular skin effect “l” versus incoherent (dressed) “1/ (2\uf074\uf029" A link with the previous sections arises naturally because we seek an analogy between the phase and the configuration spaces and the existence of a kernel in the integro-diferential equations that include nonlocality of the kinetic param- eters l and \uf074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' As pointed out in [47] “to find out the explicit form of the kernel k (ik) the kinetic equation for the non- equilibrium part of the electron distribution function must be solved”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Table 5 shows several frequency dependent dis- persion relations for “l” and “\uf074 “by comparing the two effects: the anomalous skin effect in normal metals with Fermi surfaces with parabolic points [48], with the reduced phase space for unconventional superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In [48], it was found theoretically the impedance in the hydrodynamic limit 𝜔 𝜏 ≪ 1 for the anomalous skin effect in thin metallic films by giving some examples using complicated 3D Fermi surfaces to average the conductivity and the impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' We found that the real part of the impedance strongly depends on two parameters: the mean free path and the shape of the belts on each Fermi surface studied (the shape of the singular belts makes used of the topological generalized Lifshitz transitions [47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' It was noticed in [48] that by doing an appropriate integration, two physical behaviors can be distinguished in the anomalous real part of the impedance (one of them called a singular behavior, check Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 6 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' [48] and Table 1 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' [47] for the type of singular points [49] and the impedance dependence on the mean free path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Hence, it was stated in [47] that the solution for the impedance and conductivity depend sensitively on the ratio of spatial and tem- poral dispersions of the kinetic parameters “l” and “\uf074”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' we state in this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' that the solution for the im- aginary function \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] (or the inverse scattering lifetime) depends sensitively on the ratio of spatial and Contreras and Osorio 11 temporal dispersions for “l” and “\uf074” as well,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' since for the analysis of the previous sections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' we needed the unitary collision limit where the mean free path l ∼ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' being a the lattice parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' but requiring this time a self-consistent calculation of the inverse scattering lifetime 1/𝜏 (𝜔̃),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' although it might be no obvious in this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' because the exist- ence of the kernel is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' In addition, the frequency window required in the reduced phase space for the two unconventional superconductors happens if 𝜔 ~ 1 𝜏 ⁄ ∼ 4 Δ0 (around 4 meV for strontium ruthenate and 120 meV for the doped with strontium lan- thanum cuprate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Moreover, the tight-binding parameters (𝑡, 𝜖𝐹) influence strongly the Fermi surface averages and their values are able to distinguish different OP physical phases as it was done by comparing the singular belts in the anomalous skin effect [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Therefore, the relation dispersion in the scattering lifetime that holds for the unitary collision regime in the reduced phase space might be written as stated in the introduction 𝜔 𝜏 (𝜔̃(ω)) ~ 1 (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' To conclude, it is important to recall that recently, the anomalous skin effect with this type of anomalies in the Fermi surface has gained attention among the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Mainly for microwave applications [50,51] and in the study of nonlocality phenomena in solids, as recently was theoretically and experimental realized for the compound PdCoO2 [52,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' TABLE 5 DISPERSIONS FOR THE ANOMALOUS SKIN EFFECT VERSUS THE TWO UNCONVENTIONAL SUPERCONDUCTORS Kinetical Physics Condensed Matter Phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' To study in: Theoretical methods of so- lution Temporal dispersion re- lation for the scattering lifetime Spatial dispersion rela- tion for the quasiparti- cles Anomalous skin ef- fect and surface impedance with Fermion qua- siparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Normal metal thin samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Kinetic Boltzmann eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' in the \uf074\uf02dapproximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 𝜔 𝜏 ≪ 1 Hydrodynamic limit 𝑙 ≫ 𝛿 𝜹 is the anomalous skin depth, the mean free path is l Strange metallic phase in two un- conventional su- perconductors Superconducting ceramic thin samples for the doped HTSC and crystal bulb sam- ples for the ruthenate Numerical self-consistent equation \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] in the reduced phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 𝜔 𝜏 (𝜔̃(\uf077) ) ∼ 1 Unitary limit 𝑙 ∼ 𝑎 a is the lattice parame- ter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Conclusion and recommendations This work was aimed at introducing with some numerical examples the importance of two physical parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' the mean free path and scattering lifetime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' both widely used in non-equilibrium statistical mechanics and a brief analysis of what we have called the reduced phase space for the real and imaginary parts of the elastic scattering cross-section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' using two unconventional superconductors in the unitary limit as examples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' when the fermionic quasiparticles are dressed by a non-magnetic impurity potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' for three cases of the order parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' the quasi/nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' point nodes and line nodes using a 2D anisotropic TB self-consistent parametrization with nearest neighbor hoping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Despite, we focused our study to the unitary regime, we took into account a discussion involving three scattering regimes in the imaginary part of the elastic cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' We have defined a “hidden damping parameter” \uf067\uf020\uf03d\uf020\uf02d \uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] in “the imaginary part of the elastic scattering cross-section”, being the last always positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=', "\uf0c1 [𝜔̃(\uf077 + 𝑖 0+)] > 0” obtained using a self-consistent numerical procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Therefore, that kind of self-con- sistent hidden behavior might be of interest for researchers who study the statistical physics of non-equilibrium phenomena (classical or quantum) from a macroscopic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' To conclude, several examples were analyzed in sections 2 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Sometimes using tables and figures from numerical Contreras and Osorio 12 calculations, but also giving analogies between the classical phase space of the non-equilibrium statistical mechan- ics, the configuration space of nonrelativistic quantum mechanics, and the reduced phase space (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 1 for a graphic summary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' The study of the imaginary part of the elastic cross-section not only is important for these two models of unconventional superconductors with strontium, but also is of interest for the study of fermionic and bosonic trapped gases at very low temperatures as it has been addressed in reference [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' CRediT authorship contribution statement P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Contreras: Conceptualization, Methodology, Software, Investigation, Validation, Writing – origi- nal draft, Supervision, Writing – review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Dianela Osorio: Methodology, Data curation, Software, Visualization, Investigation, Validation, Writing – review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Declaration of competing interest The authors declare that they have no known competing financial interests or personal relation- ships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' Acknowledgements This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQf5Q6j/content/2301.05322v1.pdf'} +page_content=' We thank an anonymous reviewer of this work to help us to clarify and expand the meaning of section 5 and an 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a/DtE3T4oBgHgl3EQfVAog/content/tmp_files/2301.04455v1.pdf.txt b/DtE3T4oBgHgl3EQfVAog/content/tmp_files/2301.04455v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f58290775284ac1ed2817b21485ff4c97d71604d --- /dev/null +++ b/DtE3T4oBgHgl3EQfVAog/content/tmp_files/2301.04455v1.pdf.txt @@ -0,0 +1,607 @@ +UTILIZING TECHNICAL DATA TO DISCOVER SIMILAR +COMPANIES IN DHAKA STOCK EXCHANGE +Tashreef Muhammad +Department of Computer Science and Engineering +Southeast University +Dhaka, Bangladesh +tashreef.muhammad@seu.edu.bd +Tahsin Aziz +Department of Computer Science and Engineering +Ahsanullah University of Science and Technology +Dhaka, Bangladesh +tahsinaziz.cse@aust.edu +Mohammad Shafiul Alam +Department of Computer Science and Engineering +Ahsanullah University of Science and Technology +Dhaka, Bangladesh +shafiul.cse@aust.edu +January 12, 2023 +ABSTRACT +Stock market investment have been an ideal form of investment for many years. Investing capitals +smartly in stock market yields high profit returns. But there are many companies available in a market. +Currently there are more than 345 active companies who have stocks in Dhaka Stock Exchange +(DSE). Analyzing all these companies is quite impossible. However, many companies tend to move +together. This study aims at finding which companies in DSE have a close connection and move +alongside each other. By analyzing this relation, the investors and traders will be able to analyze a +lot of companies’ statistics from a calculating just a handful number of companies. The conducted +experiment yielded promising results. It was found that though the system was not given anything +other than technical data, it was able to identify companies that show domain specific outcomes. +In other words, a relation between technical data and fundamental data was discovered from the +conducted experiment. +Keywords Correlation Matrix · Dhaka Stock Exchange · Company Network · Technical Data +1 +Introduction +Stock markets provide an easy way for one to earn some fresh money. Through buying a stock of a company, the +stockholder earns the title of owning a certain amount of that company. It makes the stockholder eligible to receive +profit based on companies’ income. It is actually quite similar to owning that business, only that the stockholder +does not have to maintain all the tedious office work. However, it also mandates the stockholder to buy stock of +companies that will gain profit, and more importantly do not get loss. To make such decisions, one needs to analyze +different companies. There are couple of hundred of companies in different stock markets each and only in Dhaka +Stock Exchange there are more than 345 companies. Analyzing all companies is not a very feasible solution. But, it has +been seen that a number of companies behave similarly. Hence, this study is an approach to try finding companies that +tend to move together when it comes to price. +arXiv:2301.04455v1 [q-fin.ST] 11 Jan 2023 + +A PREPRINT - JANUARY 12, 2023 +1.1 +Fundamental and Technical Data +There are some specific understanding that help understand which companies usually move together. Usually, the data +that helps identify such relation are the one’s known as “Fundamental Data”. Fundamental data are those data that are +related to the company itself, but is not portrayed as direct visible price that a stock contains on a specific moment. For +example, it is known to many investors, that the “Insurance” category companies tend to move together in DSE. But in +the conducted study, our concentration was on “Technical Data” which is the direct raw data visible for a stock in +market. Ironically, through the conducted experiments, it was found that the known hypothesis from “Fundamental +Data” that the insurance companies move together, was actually supported through “Technical Data”. +1.2 +Motivation and Contribution +Many studies have been conducted throughout the world on stock markets. However, on a scale to it, researches done on +Dhaka Stock Exchange is very little. In addition, most of the conducted researches so far in the field is on predicting the +closing price of stocks. For any investor or trader, knowing or predicting the price of stocks might be helpful, but other +information also carry significant importance. Hence, this study pursues to find one such knowledge, that is to discover +connection between companies through their technical data. The significant contribution of the study can be enlisted as: +• Find correlation between DSE companies based on technical data only +• Discover evidence of technical data supporting hypothesis coming from fundamental data +• Find more specific real-world explainable connection between different companies +• Develop a visualization of the connection using graph to help visualize the system of companies in DSE +The rest of this paper is divided into some specific sections. Section 2 discusses about studies on this field that have +already been conducted. Section 3 described how the data was collected and processed. It also describes how processed +data was converted into a correlation matrix to find correlated companies. Then it discusses on how the visualization +of the system was done using tools of graph theory. In Section 4, the discussion is on the found results and some +assumptions from the found results are discussed. Finally, in Section 5 the conclusion is drawn with some references to +future work. +2 +Related Works +Most of the conducted researches in the field of stock markets are on predicting stock prices. Research related to stock +price prediction using prediction techniques like neural networks has been ongoing for more than thirty years [1]. +Among several research works that have been conducted to predict stock price using Convolutional Neural Networks +(CNNs) [2, 3, 4, 5, 6, 7] have shown good performance. Since stock prices are time series data, they have property pf +sequence data. Vanilla Recurrent Neural Networks (RNN) and Long-Short Term Memory (LSTM) models have been +utilized [4, 5, 6] for predicting stock prices as well. Transformer based models for stock price prediction is also picking +up pace. It has already been used to forecasting S&P volatility [8]. Transformer models have also been used on natural +language data collected from social media related to stock price forecasting [9]. +A number of researchers have used a variety of Artificial Intelligence (AI) techniques in stock price prediction [10]. +The so-called evolutionary and bio-inspired algorithms lead the deployment of meta-heuristics and AI-based techniques +such as Genetic Algorithm, Artificial Bee Colony, Ant Colony, Fish Swarm optimization, Particle Swarm Optimization +and the like [11]. Techniques of time series analysis like Box Jenkins method have also been used in some studies [12]. +This paper is on data from Dhaka Stock Exchange. Kamruzzaman et al. [13] published a study that uses Box-Jenkins +methodology and applied Autoregressive Integrated Moving Average (ARIMA) to find interval forecasts of market +return of DSE with 95% confidence level. Maksuda et al. [14] predicted the DSE Broad Index (DSEX) using a +multi-layer feed-forward neural network and report satisfactory performance. Mujibur et al. [15] deployed ARIMA, +an artificial neural network, linear model, Holt-Winters model, and Holt-Winters exponential smoothing model on as +many as 35 stocks of DSE and report the artificial neural network to perform relatively better compared to the other +techniques. A recent study have also been conducted on DSE to predict stock prices using transformer based model +[16]. Alavi et al. [17] utilized different machine learning models for predicting the future using some factors. They also +constructed a profit based ranking of different organizations based on observed accuracy and error rate. +There have been other applications of networks in stock market related works as well. Minjun Kim and Hiroki Sayama +[18] used network science for forecasting stock prices. Using correlation to analyze stock market network is not +2 + +A PREPRINT - JANUARY 12, 2023 +Data Collection +January 01, 2013 - July 13, 2022 +Pruning +Taking Companies who are active during +the whole specified timeline +Closing Price Only +Only consider the closing price per date +and removing open, high, low, volume +Calculate Returns +Calculate closing price returns from the +collected closing price +Value Matrix +Each row contains data for one company, +and each column represent the closing +price return for each day +Correlation Matrix +Process the closing price return matrix +and construct the correlation matrix +Figure 1: Construction of Correlation Matrix for the Experiment +completely new. Quite the similar task was done by Wenyue Sun et al. [19] in 2015 but with a completely different +stock market based data and a completely different goal in mind. Piotr Szczepocki [20] used time-varying beta to study +on Warsaw and Mansoor Momeni et al. [21] used k-means algorithm on Tehran Stock Exchange (TSE) to try and group +similar companies. +A very recent survey on graph based works on stock markets were published by [22] that contains a very comprehensive +collection of how graph-based approaches are being used in the field of stock markets. They have a completely dedicated +section on discussion regarding using graphs for clustering companies. +The research gap that this study aims to overcome is developing a field for DSE companies to be classified in some +clusters or groups. Stock data vary a lot from time to time, and also from place to place. The objective was to develop a +very simple approach from which without much of a calculation a good analysis on companies of DSE can be found. +3 +Experimental Setup +3.1 +Dataset Overview +For the purpose of this experiment, adjusted closing price of 386 companies and three (3) market indices (00DS30, +00DSES, 00DSEX) were initially collected [23]. The time line of collected data was from January 01, 2013 to July 13, +2022. The nature of collected data was End of Day (EoD) format and thus we had one row of data for each date for a +specific company. Each row contained the date, opening price, highest price, lowest price, closing price and volume. +For the experiment, only closing price was taken into consideration. +Later some pruning was done. Companies who did not have started at or before January 01, 2013 were removed. +Similarly, companies that were closed before July 13, 2022 was moved out. Market indices are average of the market +and they show relation between almost all the members. Hence, they were also removed. After all these pruning, there +were 347 companies left for the experiment to be conducted. +3.2 +Constructing Correlation Matrix +From the selected 347 companies, first the closing price return was calculated. Closing price itself may vary a lot, but +by calculating the difference, more knowledge can be gained about movement correspondence. Then a matrix was +created where each column represented the closing price of each company for a certain date. After that, using all the +data of different dates, the correlation matrix was created. Hence, the closing price return of each company from the +timeline January 01, 2013 to July 13, 2022 was used to construct the correlation matrix. It was later on used for finding +closely related companies. The whole process can be seen expressed as a diagram in Figure 1. +Pearson correlation coefficient formula was used for calculating correlation between two companies. Let us consider +two companies are expressed using X and Y . The closing price return on ith day for X company expressed as Xi. +Similarly, for ith day Yi represents closing price return of that day. i represents the different dates in the data and +i ∈ [1, n]. Then, the Pearson correlation can be expressed as, +3 + +A PREPRINT - JANUARY 12, 2023 +ρX,Y = cov(X, Y ) +σXσY +(1) +where, +• cov(X, Y ) = +�n +i=1(Xi− ¯ +X)(Yi− ¯Y ) +n +• σX = Standard Deviation of X +• σY = Standard Deviation of Y +Using these tools the correlation between different companies were calculated and expressed through the correlation +matrix. The values of Pearson correlation co-efficient lies between [−1.0, 1.0] where, +• −1 = Complete Negative Correlation +• 0 = No Correlation +• 1 = Complete Positive Correlation +3.3 +Constructing Network +The objective of this study is to find companies that are interrelated to each other through the value found correlation +matrix. It can be considered like formulating a network between different companies, where each company is connected +by the value of their correlation coefficient. The stronger the correlation, the stronger is the connection between them. +Now if it is considered that each company is a node and the correlation between them, is an edge that connects them, a +graph can be constructed. +For the purpose of simplification we only took edge values of positive correlation and values greater than 0.5. In that +case the number of companies was reduced to 278 and the number of found edges was 1393. Using these data a network +was constructed through Gephi [24] and it can be seen in Figure 2. +It can be seen easily from Figure 2 that not all the companies in DSE have similar connections. In fact there are some +visible cliques in the network. It can easily be assumed from the network that the companies forming close clusters tend +to move quite alike. The found result through correlation matrix is quite large and hence this visual representation in +Figure 2 provides a very concise and clear understanding of the findings of this study. +4 +Result Analysis +The performance of this study cannot be quantified using any particular metrics. The application of this study is purely +evaluated through real-life intuition of the findings. Hence, some specific findings are highlighted in this section to +verify the findings of the study with real-life scenarios. +4.1 +APEXFOODS, APEXSPINN, APEXFOOT are close +In general APEXFOODS, APEXSPINN and APEXFOOT are from different sectors, But it can be seen that they are +showing good interrelation with each other. It is mainly because they are part of same mother company. Though the +fact that they are all part of the same company was not present in the data, still it was possible to get the interrelation +between them. +4.2 +Insurance Companies are Closely Connected +It is a common knowledge to traders of DSE that insurance companies tend to be very much interrelated. But during the +process, there were no given data about which companies were insurance types. However, the found result easily shows +that insurance companies are very much tightly clustered with each other. +4.3 +Bank and Financial Institutions +It can be seen that banks and financial institutions are very close to each other. Also, from Figure 2 it can be seen that +banks have more active relation with the whole network than financial institutions. The banks can also be seen to be +connected to many distant companies in the network as well. +4 + +A PREPRINT - JANUARY 12, 2023 +1JANATAMF +1STPRIMFMF +AAMRANET +AAMRATECH +ABB1STMF +ABBANK +ACFL +ACI +ACIFORMULA +ACMELAB +ACTIVEFINE +ADVENT +AFCAGRO +AFTABAUTO +AGNISYSL +AGRANINS +AIL +ALARABANK +ALIF +AMBEEPHA +AMCL(PRAN) +ANLIMAYARN +APEXFOODS +APEXFOOT +APEXSPINN +APEXTANRY +APOLOISPAT +ARAMIT +ARGONDENIM +ASIAINS +ASIAPACINS +ATCSLGF +AZIZPIPES +BANGAS +BARKAPOWER +BATBC +BAYLEASING +BBS +BBSCABLES +BDAUTOCA +BDCOM +BDFINANCE +BDLAMPS +BENGALWTL +BERGERPBL +BEXIMCO +BGIC +BIFC +BNICL +BRACBANK +BSCCL +BSRMLTD +BSRMSTEEL +BXPHARMA +CAPMBDBLMF +CAPMIBBLMF +CENTRALINS +CENTRALPHL +CITYBANK +CITYGENINS +CONFIDCEM +CONTININS +COPPERTECH +CVOPRL +DBH +DBH1STMF +DESCO +DESHBANDHU +DHAKABANK +DHAKAINS +DOREENPWR +DSHGARME +DUTCHBANGL +EASTERNINS +EASTLAND +EASTRNLUB +EBL1STMF +EBLNRBMF +ECABLES +EHL +ESQUIRENIT +ETL +EXIM1STMF +EXIMBANK +FAMILYTEX +FARCHEM +FAREASTFIN +FAREASTLIF +FASFIN +FBFIF +FEDERALINS +FEKDIL +FINEFOODS +FIRSTFIN +FIRSTSBANK +FUWANGCER +FUWANGFOOD +GBBPOWER +GEMINISEA +GENNEXT +GHAIL +GHCL +GLOBALINS +GOLDENSON +GPHISPAT +GQBALLPEN +GREENDELT +GSPFINANCE +HAKKANIPUL +HEIDELBCEM +HFL +IBP +ICB3RDNRB +ICBAGRANI1 +ICBEPMF1S1 +ICBSONALI1 +IDLC +IFADAUTOS +IFIC +IFIC1STMF +IFILISLMF1 +ILFSL +INTRACO +IPDC +ISLAMIBANK +ISLAMICFIN +ISLAMIINS +ISNLTD +ITC +JAMUNABANK +JAMUNAOIL +JANATAINS +JMISMDL +KARNAPHULI +KBPPWBIL +KDSALTD +KEYACOSMET +KOHINOOR +KPCL +KPPL +KTL +LANKABAFIN +LEGACYFOOT +LHBL +LIBRAINFU +LINDEBD +MAKSONSPIN +MALEKSPIN +MARICO +MEGCONMILK +MEGHNACEM +MEGHNALIFE +MEGHNAPET +MERCANBANK +MERCINS +METROSPIN +MHSML +MIDASFIN +MITHUNKNIT +MJLBD +MLDYEING +MONNOAGML +MONNOCERA +MPETROLEUM +NAHEEACP +NAVANACNG +NBL +NCCBANK +NCCBLMF1 +NFML +NHFIL +NITOLINS +NORTHERN +NORTHRNINS +NPOLYMER +NTLTUBES +NURANI +OAL +OIMEX +OLYMPIC +ONEBANKLTD +ORIONINFU +ORIONPHARM +PADMALIFE +PADMAOIL +PARAMOUNT +PDL +PENINSULA +PEOPLESINS +PF1STMF +PHARMAID +PHENIXINS +PHOENIXFIN +PHPMF1 +PIONEERINS +POPULAR1MF +POWERGRID +PRAGATIINS +PRAGATILIF +PREMIERBAN +PREMIERLEA +PRIME1ICBA +PRIMEBANK +PRIMEFIN +PRIMELIFE +PRIMETEX +PROGRESLIF +PROVATIINS +PURABIGEN +QUASEMIND +QUEENSOUTH +RAKCERAMIC +RANFOUNDRY +RDFOOD +REGENTTEX +RELIANCINS +REPUBLIC +RINGSHINE +RNSPIN +RSRMSTEEL +RUNNERAUTO +RUPALIBANK +RUPALIINS +RUPALILIFE +SAIFPOWER +SAIHAMCOT +SAIHAMTEX +SALAMCRST +SAMATALETH +SANDHANINS +SAPORTL +SEAPEARL +SEMLFBSLGF +SEMLIBBLSF +SEMLLECMF +SHAHJABANK +SHASHADNIM +SHURWID +SILCOPHL +SILVAPHL +SIMTEX +SINGERBD +SINOBANGLA +SKTRIMS +SONALIANSH +SONARBAINS +SOUTHEASTB +SPCL +SQUARETEXT +SQURPHARMA +SSSTEEL +STANDARINS +STANDBANKL +STYLECRAFT +SUMITPOWER +SUNLIFEINS +TAKAFULINS +TALLUSPIN +TITASGAS +TOSRIFA +TRUSTB1MF +TUNGHAI +UCB +UNIONCAP +UNITEDFIN +UNITEDINS +UPGDCL +USMANIAGL +UTTARABANK +UTTARAFIN +VAMLBDMF1 +VAMLRBBF +VFSTDL +WATACHEM +WMSHIPYARD +YPL +ZAHEENSPIN +ZAHINTEX +Figure 2: Network of Companies based on their closing price correlation +4.4 +Mutual Funds are Separated +The mutual funds also show some level of difference from the rest of the networks and closeness among themselves. +The graph in Figure 2 also shows that there can be made two parts of even among the mutual funds. +4.5 +Model Error Difference Calculation +From the Figure 2 it can be seen that GLOBALINS is closely related to DHAKAINS but has a very far relation with +TITASGAS. Using the model proposed by Tashreef et al. [16] first we train a model using GLOBALINS and then apply +that model on DHKAINS and TITASGAS. Because of the relation that is seen, TITASGAS prediction error should be +much higher in contrast to DHAKINS prediction error. +After conducting the experiment, it was seen that truly the error for DHAKAINS and GLOBALINS was almost +identical and very close, though GLOBALINS does have relatively smaller error. After all, the model was trained using +GLOBALINS. But it still performed almost as well for DHAKAINS. However, the error value was much higher when +the same model tried to forecast closing price for TITASGAS. The calculated Root Mean Squared Error (RMSE) and +5 + +A PREPRINT - JANUARY 12, 2023 +Trading Code +Error Value +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +RMSE +MAE +DHAKAINS +GLOBALINS +TITASGAS +Figure 3: Forecasting Error Values for Three Different Companies +Mean Absolute Error (MAE) values for the three companies DHAKAINS, GLOBALINS and TITASGAS can be seen +in Table 1. From Figure 3 the relation between the forecasting error is very prominent. +Table 1: Forecasting Error After Applying Machine Learning Model +Trading Code +RMSE +MAE +DHAKAINS +4.41E-02 +3.48E-02 +GLOBALINS +4.19E-02 +3.29E-02 +TITASGAS +1.07E-01 +9.14E-02 +5 +Conclusion +The aspects of this study is endless. Through analyzing domain data it can be made conclusive that the results that +was found were quite accurate. Next, these values can be utilized by the investors to make better decision on which +companies are similar that in the long run can help them invest in DSE. Also, the graph and data analysis might help +find more connections present in Dhaka Stock Exchange that are yet not very possible to detect among the scattered +fundamental data. Further study of DSE based on domain data will be greatly influenced by the study that has been +conducted here. The most mention-able contribution of this study is the formation of a system that allots technical data +from the market to present the fundamental relation of different companies. Being much more easily organized than the +fundamental data, it will create many options for analysis of the Dhaka Stock Exchange, thus stock markets in total. +References +[1] Eberhard Schöneburg. Stock price prediction using neural networks: A project report. Neurocomputing, 2(1):17– +27, 1990. +[2] Avraam Tsantekidis, Nikolaos Passalis, Anastasios Tefas, Juho Kanniainen, Moncef Gabbouj, and Alexandros +Iosifidis. Forecasting stock prices from the limit order book using convolutional neural networks. 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Hasan. Indices prediction of bangladeshi stock by using time +series forecasting and performance analysis. In 2019 International Conference on Electrical, Computer and +Communication Engineering (ECCE), pages 1–5, 2019. +[16] Tashreef Muhammad, Anika Bintee Aftab, Md Ahsan, Maishameem Meherin Muhu, Muhammad Ibrahim, +Shahidul Islam Khan, Mohammad Shafiul Alam, et al. Transformer-based deep learning model for stock price +prediction: A case study on bangladesh stock market. arXiv preprint arXiv:2208.08300, 2022. +[17] Muhaddid Alavi, Selina Sharmin, Ashraf Uddin, Tanvir Ahammad, and Fatema Siddika. Profitable ranking of +stock market organizations using different machine learning models. In 2021 IEEE 9th Region 10 Humanitarian +Technology Conference (R10-HTC), pages 1–6. IEEE, 2021. +[18] Minjun Kim and Hiroki Sayama. Predicting stock market movements using network science: an information +theoretic approach. 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In International AAAI Conference on Weblogs and Social Media, 2009. +7 + diff --git a/DtE3T4oBgHgl3EQfVAog/content/tmp_files/load_file.txt b/DtE3T4oBgHgl3EQfVAog/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9e11fb7206eed14f8bfa15ef0ede455f6eeb9df --- /dev/null +++ b/DtE3T4oBgHgl3EQfVAog/content/tmp_files/load_file.txt @@ -0,0 +1,552 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf,len=551 +page_content='UTILIZING TECHNICAL DATA TO DISCOVER SIMILAR COMPANIES IN DHAKA STOCK EXCHANGE Tashreef Muhammad Department of Computer Science and Engineering Southeast University Dhaka, Bangladesh tashreef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='muhammad@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='bd Tahsin Aziz Department of Computer Science and Engineering Ahsanullah University of Science and Technology Dhaka, Bangladesh tahsinaziz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='cse@aust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='edu Mohammad Shafiul Alam Department of Computer Science and Engineering Ahsanullah University of Science and Technology Dhaka, Bangladesh shafiul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='cse@aust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='edu January 12, 2023 ABSTRACT Stock market investment have been an ideal form of investment for many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Investing capitals smartly in stock market yields high profit returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' But there are many companies available in a market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Currently there are more than 345 active companies who have stocks in Dhaka Stock Exchange (DSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Analyzing all these companies is quite impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' However, many companies tend to move together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' This study aims at finding which companies in DSE have a close connection and move alongside each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' By analyzing this relation, the investors and traders will be able to analyze a lot of companies’ statistics from a calculating just a handful number of companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The conducted experiment yielded promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' It was found that though the system was not given anything other than technical data, it was able to identify companies that show domain specific outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' In other words, a relation between technical data and fundamental data was discovered from the conducted experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Keywords Correlation Matrix · Dhaka Stock Exchange · Company Network · Technical Data 1 Introduction Stock markets provide an easy way for one to earn some fresh money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Through buying a stock of a company, the stockholder earns the title of owning a certain amount of that company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' It makes the stockholder eligible to receive profit based on companies’ income.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' It is actually quite similar to owning that business, only that the stockholder does not have to maintain all the tedious office work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' However, it also mandates the stockholder to buy stock of companies that will gain profit, and more importantly do not get loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' To make such decisions, one needs to analyze different companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' There are couple of hundred of companies in different stock markets each and only in Dhaka Stock Exchange there are more than 345 companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Analyzing all companies is not a very feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' But, it has been seen that a number of companies behave similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Hence, this study is an approach to try finding companies that tend to move together when it comes to price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='04455v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='ST] 11 Jan 2023 A PREPRINT - JANUARY 12, 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='1 Fundamental and Technical Data There are some specific understanding that help understand which companies usually move together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Usually, the data that helps identify such relation are the one’s known as “Fundamental Data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Fundamental data are those data that are related to the company itself, but is not portrayed as direct visible price that a stock contains on a specific moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' For example, it is known to many investors, that the “Insurance” category companies tend to move together in DSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' But in the conducted study, our concentration was on “Technical Data” which is the direct raw data visible for a stock in market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Ironically, through the conducted experiments, it was found that the known hypothesis from “Fundamental Data” that the insurance companies move together, was actually supported through “Technical Data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='2 Motivation and Contribution Many studies have been conducted throughout the world on stock markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' However, on a scale to it, researches done on Dhaka Stock Exchange is very little.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' In addition, most of the conducted researches so far in the field is on predicting the closing price of stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' For any investor or trader, knowing or predicting the price of stocks might be helpful, but other information also carry significant importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Hence, this study pursues to find one such knowledge, that is to discover connection between companies through their technical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The significant contribution of the study can be enlisted as: Find correlation between DSE companies based on technical data only Discover evidence of technical data supporting hypothesis coming from fundamental data Find more specific real-world explainable connection between different companies Develop a visualization of the connection using graph to help visualize the system of companies in DSE The rest of this paper is divided into some specific sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Section 2 discusses about studies on this field that have already been conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Section 3 described how the data was collected and processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' It also describes how processed data was converted into a correlation matrix to find correlated companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Then it discusses on how the visualization of the system was done using tools of graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' In Section 4, the discussion is on the found results and some assumptions from the found results are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Finally, in Section 5 the conclusion is drawn with some references to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 2 Related Works Most of the conducted researches in the field of stock markets are on predicting stock prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Research related to stock price prediction using prediction techniques like neural networks has been ongoing for more than thirty years [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Among several research works that have been conducted to predict stock price using Convolutional Neural Networks (CNNs) [2, 3, 4, 5, 6, 7] have shown good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Since stock prices are time series data, they have property pf sequence data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Vanilla Recurrent Neural Networks (RNN) and Long-Short Term Memory (LSTM) models have been utilized [4, 5, 6] for predicting stock prices as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Transformer based models for stock price prediction is also picking up pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' It has already been used to forecasting S&P volatility [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Transformer models have also been used on natural language data collected from social media related to stock price forecasting [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' A number of researchers have used a variety of Artificial Intelligence (AI) techniques in stock price prediction [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The so-called evolutionary and bio-inspired algorithms lead the deployment of meta-heuristics and AI-based techniques such as Genetic Algorithm, Artificial Bee Colony, Ant Colony, Fish Swarm optimization, Particle Swarm Optimization and the like [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Techniques of time series analysis like Box Jenkins method have also been used in some studies [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' This paper is on data from Dhaka Stock Exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Kamruzzaman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' [13] published a study that uses Box-Jenkins methodology and applied Autoregressive Integrated Moving Average (ARIMA) to find interval forecasts of market return of DSE with 95% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Maksuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' [14] predicted the DSE Broad Index (DSEX) using a multi-layer feed-forward neural network and report satisfactory performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Mujibur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' [15] deployed ARIMA, an artificial neural network, linear model, Holt-Winters model, and Holt-Winters exponential smoothing model on as many as 35 stocks of DSE and report the artificial neural network to perform relatively better compared to the other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' A recent study have also been conducted on DSE to predict stock prices using transformer based model [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Alavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' [17] utilized different machine learning models for predicting the future using some factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' They also constructed a profit based ranking of different organizations based on observed accuracy and error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' There have been other applications of networks in stock market related works as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Minjun Kim and Hiroki Sayama [18] used network science for forecasting stock prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Using correlation to analyze stock market network is not 2 A PREPRINT - JANUARY 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 2023 Data Collection January 01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 2013 - July 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 2022 Pruning Taking Companies who are active during the whole specified timeline Closing Price Only Only consider the closing price per date and removing open,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' high,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' low,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' volume Calculate Returns Calculate closing price returns from the collected closing price Value Matrix Each row contains data for one company,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' and each column represent the closing price return for each day Correlation Matrix Process the closing price return matrix and construct the correlation matrix Figure 1: Construction of Correlation Matrix for the Experiment completely new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Quite the similar task was done by Wenyue Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' [19] in 2015 but with a completely different stock market based data and a completely different goal in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Piotr Szczepocki [20] used time-varying beta to study on Warsaw and Mansoor Momeni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' [21] used k-means algorithm on Tehran Stock Exchange (TSE) to try and group similar companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' A very recent survey on graph based works on stock markets were published by [22] that contains a very comprehensive collection of how graph-based approaches are being used in the field of stock markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' They have a completely dedicated section on discussion regarding using graphs for clustering companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The research gap that this study aims to overcome is developing a field for DSE companies to be classified in some clusters or groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Stock data vary a lot from time to time, and also from place to place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The objective was to develop a very simple approach from which without much of a calculation a good analysis on companies of DSE can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 3 Experimental Setup 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='1 Dataset Overview For the purpose of this experiment, adjusted closing price of 386 companies and three (3) market indices (00DS30, 00DSES, 00DSEX) were initially collected [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The time line of collected data was from January 01, 2013 to July 13, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The nature of collected data was End of Day (EoD) format and thus we had one row of data for each date for a specific company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Each row contained the date, opening price, highest price, lowest price, closing price and volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' For the experiment, only closing price was taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Later some pruning was done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Companies who did not have started at or before January 01, 2013 were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Similarly, companies that were closed before July 13, 2022 was moved out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Market indices are average of the market and they show relation between almost all the members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Hence, they were also removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' After all these pruning, there were 347 companies left for the experiment to be conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='2 Constructing Correlation Matrix From the selected 347 companies, first the closing price return was calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Closing price itself may vary a lot, but by calculating the difference, more knowledge can be gained about movement correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Then a matrix was created where each column represented the closing price of each company for a certain date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' After that, using all the data of different dates, the correlation matrix was created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Hence, the closing price return of each company from the timeline January 01, 2013 to July 13, 2022 was used to construct the correlation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' It was later on used for finding closely related companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The whole process can be seen expressed as a diagram in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Pearson correlation coefficient formula was used for calculating correlation between two companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Let us consider two companies are expressed using X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The closing price return on ith day for X company expressed as Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Similarly, for ith day Yi represents closing price return of that day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' i represents the different dates in the data and i ∈ [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Then, the Pearson correlation can be expressed as, 3 A PREPRINT - JANUARY 12, 2023 ρX,Y = cov(X, Y ) σXσY (1) where, cov(X, Y ) = �n i=1(Xi− ¯ X)(Yi− ¯Y ) n σX = Standard Deviation of X σY = Standard Deviation of Y Using these tools the correlation between different companies were calculated and expressed through the correlation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The values of Pearson correlation co-efficient lies between [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='0] where, −1 = Complete Negative Correlation 0 = No Correlation 1 = Complete Positive Correlation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='3 Constructing Network The objective of this study is to find companies that are interrelated to each other through the value found correlation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' It can be considered like formulating a network between different companies, where each company is connected by the value of their correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The stronger the correlation, the stronger is the connection between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Now if it is considered that each company is a node and the correlation between them, is an edge that connects them, a graph can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' For the purpose of simplification we only took edge values of positive correlation and values greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' In that case the number of companies was reduced to 278 and the number of found edges was 1393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Using these data a network was constructed through Gephi [24] and it can be seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' It can be seen easily from Figure 2 that not all the companies in DSE have similar connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' In fact there are some visible cliques in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' It can easily be assumed from the network that the companies forming close clusters tend to move quite alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The found result through correlation matrix is quite large and hence this visual representation in Figure 2 provides a very concise and clear understanding of the findings of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 4 Result Analysis The performance of this study cannot be quantified using any particular metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The application of this study is purely evaluated through real-life intuition of the findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Hence, some specific findings are highlighted in this section to verify the findings of the study with real-life scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='1 APEXFOODS, APEXSPINN, APEXFOOT are close In general APEXFOODS, APEXSPINN and APEXFOOT are from different sectors, But it can be seen that they are showing good interrelation with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' It is mainly because they are part of same mother company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Though the fact that they are all part of the same company was not present in the data, still it was possible to get the interrelation between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='2 Insurance Companies are Closely Connected It is a common knowledge to traders of DSE that insurance companies tend to be very much interrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' But during the process, there were no given data about which companies were insurance types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' However, the found result easily shows that insurance companies are very much tightly clustered with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='3 Bank and Financial Institutions It can be seen that banks and financial institutions are very close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Also, from Figure 2 it can be seen that banks have more active relation with the whole network than financial institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The banks can also be seen to be connected to many distant companies in the network as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 4 A PREPRINT - JANUARY 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='1JANATAMF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='1STPRIMFMF ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='ZAHEENSPIN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='ZAHINTEX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='Figure 2: Network of Companies based on their closing price correlation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='4 Mutual Funds are Separated The mutual funds also show some level of difference from the rest of the networks and closeness among themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The graph in Figure 2 also shows that there can be made two parts of even among the mutual funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='5 Model Error Difference Calculation From the Figure 2 it can be seen that GLOBALINS is closely related to DHAKAINS but has a very far relation with TITASGAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Using the model proposed by Tashreef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' [16] first we train a model using GLOBALINS and then apply that model on DHKAINS and TITASGAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Because of the relation that is seen, TITASGAS prediction error should be much higher in contrast to DHAKINS prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' After conducting the experiment, it was seen that truly the error for DHAKAINS and GLOBALINS was almost identical and very close, though GLOBALINS does have relatively smaller error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' After all, the model was trained using GLOBALINS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' But it still performed almost as well for DHAKAINS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' However, the error value was much higher when the same model tried to forecast closing price for TITASGAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The calculated Root Mean Squared Error (RMSE) and 5 A PREPRINT - JANUARY 12, 2023 Trading Code Error Value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='125 RMSE MAE DHAKAINS GLOBALINS TITASGAS Figure 3: Forecasting Error Values for Three Different Companies Mean Absolute Error (MAE) values for the three companies DHAKAINS, GLOBALINS and TITASGAS can be seen in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' From Figure 3 the relation between the forecasting error is very prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Table 1: Forecasting Error After Applying Machine Learning Model Trading Code RMSE MAE DHAKAINS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='41E-02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='48E-02 GLOBALINS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='19E-02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='29E-02 TITASGAS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='07E-01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content='14E-02 5 Conclusion The aspects of this study is endless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Through analyzing domain data it can be made conclusive that the results that was found were quite accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Next, these values can be utilized by the investors to make better decision on which companies are similar that in the long run can help them invest in DSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Also, the graph and data analysis might help find more connections present in Dhaka Stock Exchange that are yet not very possible to detect among the scattered fundamental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Further study of DSE based on domain data will be greatly influenced by the study that has been conducted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' The most mention-able contribution of this study is the formation of a system that allots technical data from the market to present the fundamental relation of different companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Being much more easily organized than the fundamental data, it will create many options for analysis of the Dhaka Stock Exchange, thus stock markets in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' References [1] Eberhard Schöneburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Stock price prediction using neural networks: A project report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Neurocomputing, 2(1):17– 27, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' [2] Avraam Tsantekidis, Nikolaos Passalis, Anastasios Tefas, Juho Kanniainen, Moncef Gabbouj, and Alexandros Iosifidis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' Forecasting stock prices from the limit order book using convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' In 2017 IEEE 19th Conference on Business Informatics (CBI), volume 1, pages 7–12.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' In International AAAI Conference on Weblogs and Social Media, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} +page_content=' 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE3T4oBgHgl3EQfVAog/content/2301.04455v1.pdf'} diff --git a/HdFAT4oBgHgl3EQftx6O/content/tmp_files/2301.08666v1.pdf.txt b/HdFAT4oBgHgl3EQftx6O/content/tmp_files/2301.08666v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d6752a0ceccd93b334451b816e19ccf3e456612 --- /dev/null +++ b/HdFAT4oBgHgl3EQftx6O/content/tmp_files/2301.08666v1.pdf.txt @@ -0,0 +1,631 @@ +arXiv:2301.08666v1 [econ.TH] 20 Jan 2023 +On Sufficientarianism +Christopher P. Chambers∗ +Siming Ye† +January 23, 2023 +Abstract +We introduce a generalization of the concept of sufficientarianism, intended (but not limited) to +study multiple consumption goods. We discuss three properties that uniquely pin down our family +and, up to continuity requirements, endogenize the sufficientarian threshold. Two, anonymity and +reinforcement, are standard. The third, sufficientarianmonotonicity, generalizes classical monotonicity +and requires that for any two lists of bundles in which individuals are ordered similarly by how much +they receive, the worse of the two is indifferent to the new list of bundles considered by taking the +componentwise minimum of the two bundles for each agent. +1 +Introduction +Consider a society in which all individuals have access to enough clean water, food, and education. +Independently of whether any of them are wealthy or how the individuals obtained their positions, such +an environment arguably dominates a society in which some of the individuals lack access to clean water, +and some lack access to food. In this paper, we study a new form of sufficientarianism, an interesting +∗Corresponding author. Department of Economics, Georgetown University, ICC 580 37th and O Streets NW, Washington +DC 20057. E-mail: Christopher.Chambers@georgetown.edu. +†Department of Economics, Georgetown University, ICC 580 37th and O Streets NW, Washington DC 20057. E-mail: +sy677@georgetown.edu. +1 + +criterion of distributive justice recently introduced to the economics literature by Alcantud et al. (2022) +and Bossert et al. (2022a,b). Sufficientarianism’s goal is to ensure each individual has a “sufficient” level +of well-being. This work departs from the concept of “well-being” as the primitive, directly working +instead with economic observables. +The basic idea of sufficientarianism is that potential societies are compared according to the number +of individuals meeting an exogenously specified threshold of well-being. The more agents who reach the +level of well-being, the better. A sufficientarian ranking satisfies a weak version of the Pareto principle +(if all agents weakly gain, then society weakly gains), but not the typical stronger version. +Our goal in this work is twofold: first, to study a concept which could be applied to directly +observable and measurable bundles of goods, rather than levels of well-being. Say, the amount of food +or clean water. Second, we provide a novel axiomatization of sufficientarianism. Previous works base +their results on a principle called “prioritarian threshold,” as well as other ethical axioms related to those +in Mariotti and Veneziani (2009, 2012, 2013); Alcantud (2013); Lombardi et al. (2016). By contrast, our +novel axiom is not ethical in nature. However, we believe the ordinal content of our axioms captures the +spirit of sufficientarianism in a particularly straightforward way, and up to technical details, allows us +to endogenize the prioritarian threshold. We believe our axiomatization is novel even in the previously +studied single-dimensional context. +To understand the multicommodity aspect of our work, consider a model of food and clean water. We +can imagine a society consisting of three individuals, Alice, Bob and Charles. Suppose (for the sake of +argument) that it is determined that each individual needs 2500 daily calories of food and four daily liters +of water: these are the sufficientarian thresholds for these consumption goods. The presumption here is +that any individual who does not reach both of these thresholds cannot thrive. Let us now consider several +ways of allocating food and clean water. Writing 푥퐴 = (1500, 5) to denote a bundle where Alice gets 1500 +calories and 5 liters of water, we can now compare 푥퐴 = (1500, 5), 푥퐵 = (3000, 30), 푥퐶 = (5000, 50) +2 + +to 푦퐴 = (2500, 5), 푦퐵 = (2500, 5), 푦퐶 = (2500, 5). According to our sufficientarian thresholds, the 푦 +allocation dominates the 푥 allocation for the simple reason that the 푥 allocation does not give Alice enough +to eat (she does not meet the threshold of 2500 calories). This is so even though Bob and Charles have +orders of magnitude more water under the 푥 allocation. In contrast, all individuals (exactly) meet the +sufficientarian threshold in the 푦 allocation, so it is deemed more appropriate. +Food and water are obviously not the only commodities of interest. For example, in the survey +Deaton (2008), people report their life satisfaction according to various components of well-being, +including health, family financial status and human and political rights. Veenhoven (1991) argues that it +is only when people’s basic needs are met that an increase in income contributes to people’s evaluation +of life. Cutler et al. (2006) and Case and Deaton (2020) further enhanced the argument in Veenhoven +(1991) by studying the deaths of despair, namely the fast rising death rates among Americans that were +from drug overdoses, suicide and alcoholic liver disease. Their work links the deaths of less-educated +Americans to the social and economics conditions, emphasizing a strong correlation between education +and drug-related and alcohol-related mortality. Case and Deaton (2022) further points out that the rising +despair and deteriorating mental health conditions are making this group of citizens more volatile to +shocks such as COVID. Any one of these criteria (educaton, drug use, mental health) are criteria which +we may want to directly employ in our sufficientarian ranking. +We understand the sufficientarian ranking in a very simple light: it is to be understood as a kind of +pre-ranking. It allows us to discard social profiles of alternatives which a society should never consider +unless necessary. However, two profiles which are rendered indifferent by the sufficientarian ranking may +be distinguished by some other, more finely tuned ranking, at a later stage. Our judgment as to which of +these should be preferred by society is silent. +We establish that sufficientariansm emerges from a nearly classical collection of axioms. Two, +anonymity and reinforcement, are well understood. On reinforcement, see for example Smith (1973) +3 + +or Young (1974). +Our third property is a strengthening of monotonicity (Pareto), which we deem +sufficientarian monotonicity. +To understand sufficientarian monotonicity, first consider a reformulation of monotonicity. It states +that for two potential societies with the same individuals, if everybody receives at least as much of each +good in the second of the two societies, then society as a whole is made at least as well off in the second +society. A reformulation of this property is possible: for any pair of societies with the same individuals, +one deemed at least as good as the other, the worse of the two is additionally at least as good as the society +constructed by assigning to each individual the minimum level of each good she receives across the two +societies. +Our contribution is to strengthen this notion. Imagine two societies, each with the same set of agents. +Each society features an unambiguous ordering of the agents in terms of consumption: for any pair of +agents, one of them must get at least as much of each good as the other. Thus, there is an objective +ranking of agents in terms of wealth in either society. Suppose this ordering of agents remains essentially +unchanged across the two societies: for any pair of agents, it is impossible that the first is richer in one +society and poorer in the other. We call such a pair of societies comonotonic. Our strengthening of +the classical monotonicity property described above requires that if the two societies are comonotonic, +then the worse of the two is actually socially indifferent to the one assigning to each individual the +minimum level of each good she receives across the two societies. For lack of a better word, we call this +sufficientarian monotonicity. +Sufficientarian monotonicity captures what we believe is the essence of sufficientarianism. For an +illustration, imagine again a group consisting of three agents, Alice, Bob, and Charles, and suppose (for +simplicity), that consumption space is single-dimensional. A vector (푎, 푏, 푐) specifies the amount that +each individual gets of the good. Imagine comparing 푥 = (1, 4, 2) and 푦 = (0, 5, 5). First, observe that +these two profiles are comonotonic: the ranking induced by these two profiles agrees weakly. It is true +4 + +that 푥(푏) > 푥(푐), while 푦(푐) ≥ 푦(푏), but for our definition of comonotonicity, this does not matter. The +only profiles which would be precluded would feature strict reversals. For example, 푧 = (1, 2, 3) and +푤 = (2, 1, 7) are not comonotonic because 푧(푏) > 푧(푎) and 푤(푎) > 푤(푏). +Sufficientarian monotonicity states that if 푥 is deemed at least as good as 푦, that is, 푥 ⪰ 푦, then it +follows that the new profile, which results by replacing each individual’s consumption by the minimum +that they get across the two profiles, must be indifferent to 푦. Specifically (푥 ∧ 푦) = (0, 4, 2) ∼ 푦. +What does this have to do with sufficientarianism? For a sufficientarian rule and comonotonic +profiles, the agents reaching the sufficient level for the worse of the two must be a subset of those reaching +the sufficient level for the better of the two. And consequently, the pointwise minimum profile ensures +that all of the individuals reaching the sufficient level under the worse of the two profiles continue to do +under the minimum profile. +Return to our example. If 푥 is deemed at least as good as 푦, this is consistent with a sufficientarian +ranking counting the number of agents who get at least 1: 푥 has three such agents, and 푦 has two; namely +agents 푏 and 푐. Under (푥 ∧ 푦) = (0, 4, 2), the same agents get at least 1 as they did under 푦, namely 푏 and +푐. +Of course, not all sufficientarian rules will feature 푥 ⪰ 푦. Take a rule that counts the number of +agents receiving at least 4. Then under 푥, there is one such agent 푏, and under 푦, there are two. So in this +case 푦 ⪰ 푥; and again 푥 ∧ 푦 = (0, 4, 2) features only one agent receiving at least 4, again 푏. +Let us return to the non-comonotonic profiles 푧 = (1, 2, 3) and 푤 = (2, 1, 7). A similar property +would not obtain here. Imagine the sufficientarian rule with level 2. Then according to this rule, both 푧 +and 푤 have two agents meeting the sufficient level: for 푧 it is 푏 and 푐 and for 푤 it is 푎 and 푏. However, +these sets are not nested. Consequently, in taking the pointwise minimum, 푧 ∧ 푤 = (1, 1, 3), only one +agent remains above the sufficient level, agent 푐: in this case, 푧 ∧ 푤 is deemed strictly worse than either 푧 +or 푤. The restriction to comonotonic profiles is thus revealed as a method to ensure that no matter which +5 + +sufficient level is chosen, the set of agents meeting a sufficient level will be nested. +Our work deviates from previous work, even in the single dimensional case, as it allows for what we +might think of as “strict” sufficientarianism. This is a technical point, but it is worth mentioning. Previous +works considered sufficientarian rules to be those for which there is some level of well-being, and the +goal was to maximize the number of agents meeting at least that level of well-being. Our three axioms +also permit rules which have the goal of maximizing the number of agents meeting strictly more than a +given level. The classical formulation of the rules would emerge with a simple order-theoretic continuity +axiom. +Finally, our work applies in multi-dimensional space, in fact; it is broad enough to apply to any +semilattice, where the infimum of any pair of alternatives always exists. We model the notion of a sufficient +set in a fairly general way, but which is consistent with the spirit of earlier models of sufficientarianism. +The paper proceeds as follows. First, we investigate some related literature; then we proceed directly +to the model and main results. Finally, Section 3 concludes. +1.1 +Related Literature +Aside from the previously mentioned work on sufficientarianism, Alcantud et al. (2022); Bossert et al. +(2022a,b), the sufficientarian monotonicity axiom is closely related to a collection of axioms employed +in characterizing similar representations. In the case of 푋 = Z+, and under the hypothesis of anonymity, +any profile (푁, 푥) can be understood as a downward closed set: 퐴(푁, 푥) ≡ {(푦1, 푦2) ∈ Z2 ++ : |{푖 ∈ 푁 : +푥푖 ≥ 푦1}| ≥ 푦2}. In this formulation, the axiom is essentially the axiom of “meet-separability” found in +Miller (2008); Chambers and Miller (2014a,b, 2018). In those works, it is explained that the main idea is +originally due to Kreps (1979) in his study of preference for flexibility over menus. Related is the work +of Hougaard and Keiding (1998); Christensen et al. (1999), where meet-separability appears in a cardinal +6 + +form. +2 +The model and results +Let 푋 be a set of alternatives. We endow 푋 with a partial order ≤.1 +The partial order ≤ represents an unambiguous comparative notion of dominance. As a motivating +example, we have in mind a multi-dimensional commodity space, say 푋 = Z푚 ++ , where 푥 ≥ 푦 if 푥푖 ≥ 푦푖 for +all 푖 = 1, . . . , 푚. +The strict part of ≤ will be written <.2 +We need a language for talking about the greatest lower bound of a pair of alternatives. For example, +if 푋 = Z2 ++, then the bundle (1, 1) is the largest bundle which is dominated by both (2, 1) and (1, 2). The +meet of two elements 푥, 푦 ∈ 푋 is its greater lower bound when it exists, and is denoted 푥 ∧ 푦. Formally, +we assume that (푋, ≤) is a meet-semilattice: any pair of points possesses a meet. We do not necessarily +want or need to assume that every pair of alternatives has a least upper bound. The reason will become +clear with the following examples. Say (푋, ≤) is a meet-semilattice if every pair of elements has a meet. +Say a meet-semilattice is complete if every nonempty subset 퐴 ⊆ 푋 has a greatest lower bound. +With an abuse of notation, for any set 푁 and any 푥, 푦 ∈ 푋 푁, we let (푥 ∧ 푦) ∈ 푋 푁 denote the pointwise +meet across 푁: (푥 ∧ 푦)푖 = 푥푖 ∧ 푦푖. Similarly, we say that 푥 ≥ 푦 if for all 푖 ∈ 푁, 푥푖 ≥ 푦푖. The terminology +here is standard, see, e.g., Davey and Priestley (2002). +1A partial order is a binary relation ≤ satisfying the following three properties: +1. Reflexivity: 푥 ≤ 푥 +2. Antisymmetry: If 푥 ≤ 푦 and 푥 ≠ 푦, then 푦 ≤ 푥 must not hold. +3. Transitivity: If 푥 ≤ 푦 and 푦 ≤ 푧, then 푥 ≤ 푧 +2So, for example in the case of 푋 = Z푚 ++ , this relation dictates that 푥 > 푦 if 푥 ≥ 푦 and 푥 ≠ 푦; e.g. (3, 2) > (2, 2). +7 + +Following are examples of meet-semilattices which are relevant for our framework. +Example 1. Our motivating example, already mentioned is 푋 = Z푚 ++ , or 푋 = R푚 ++ , with its usual pointwise +order ≤. In this case, each pair 푥, 푦 has a least upper bound, but consider a slightly modified example +whereby consumption capacity constraints require that 푧 ∈ 푋 only if �푚 +푖=1 푧푖 ≤ 푤, where 푤 > 0 represents +wealth, or some storage cost. In this case, we could take 푋 = {푥 ∈ Z푚 ++ : �푚 +푖=1 푥푖 ≤ 푤}, where ≤ is the +restriction of the standard partial order on Z푚 ++ to 푋. Observe that (푋, ≤) is a meet-semilattice, but a +pair of elements typically does not possess a least upper bound. For example, if 푚 = 2 and 푤 = 2, then +푥 = (2, 0) and 푥′ = (0, 2) have no least upper bound: these are each already maximal with respect to ≤ +in 푋. +Example 2. Let 푍 be a set of discrete goods, and let 푋 ⊆ 2푍 be any collection of packages of goods for +which 퐴 ∈ 푋 and 퐵 ⊆ 퐴 implies 퐵 ∈ 푋. Again this is a simple model of an environment with capacity +constraints. Let ≤ coincide with set inclusion, so that 퐴 ≤ 퐵 if and only if 퐴 ⊆ 퐵. Observe then that 푋 is +a meet-semilattice: if each of 퐴, 퐵 are members of 푋, then so is 퐴 ∩ 퐵. However, in general, least upper +bounds may not exist. Consider for example 푍 = {푎, 푏} and 푋 = {∅, {푎}, {푏}}. Then the pair {푎} and +{푏} have no least upper bound. +Example 3. Denote by Δ(R) the set of Borel probability distributions on R, and let ≤ coincide with +the first order stochastic dominance relation, so that 푝 ≤ 푞 if 푞 first order stochastically dominates +푝.3 +It is well-known that ≤ is a lattice, and hence a meet-semilattice, see e.g. +Remark 1.A.18 of +Shaked and Shanthikumar (2007). 4 +Example 4. For a set 푆 of states of the world, one can let 푋 be either the set of partitions, the set of +3We say that 푞 first order stochastically dominates 푞 if for every 훼 ∈ R, 푞({푥 : 푥 ≥ 훼}) ≥ 푝({푥 : 푥 ≥ 훼}). +4In fact, many partial orders on the set Δ(R) have a lattice structure, including second order stochastic dominance. See e.g. +Müller and Scarsini (2006). +8 + +algebras, or the set of 휎-algebras, where Π ≤ Π′ if and only if Π refines Π′. Then the pair (푋, ≤) +constitutes a meet-semilattice. Such lattices arise frequently in the theory of information econmomics. +Example 5. For a set 푌 of alternatives, let W(푌) denote the set of weak orders (complete and transitive +relations) on 푌, ordered by reverse set inclusion. So for two weak orders 푅 and 푅′, 푅 ⪰ 푅′ means that +푅 ⊆ 푅′, or 푥 푅 푦 implies 푥 푅′ 푦. Then (W(푌), ⪰) is a meet-semilattice, whereby 푅 ∧ 푅′ is the transitive +closure of 푅 ∪ 푅′. See for example Day and McMorris (2003), Table 5.1. Similarly, one could consider +the set of pre-orders (reflexive and transitive relations) with the same ordering ⪰. +Let 푀 a set of potential agents, where |푀| ≥ 2. Let N denote the set of nonempty and finite subsets +of 푀. A society is a tuple (푁, 푥), where 푥 ∈ 푋 푁. A social ranking, or simply a ranking, associates +with each 푁 ∈ N a weak order ⪰푁 on 푋 푁.5 For two societies (푁, 푥), (푁′, 푦), where 푁′ ∩ 푁 = ∅, let +(푁 ∪ 푁′, (푥, 푦)) denote the society where (푥, 푦)푁 = 푥 and (푥, 푦)푁′ = 푦. For a bijection 휎 : 푁′ → 푁 +between 푁 and 푁′, and 푥 ∈ 푋 푁, 푥 ◦ 휎 ∈ 푋 푁′ is defined as (푥 ◦ 휎)휎(푖) = 푥푖. +For a society (푁, 푥), say that 푥 is a ≤-chain if for all 푖, 푗 ∈ 푛, 푥푖 ≥ 푥 푗 or 푥 푗 ≥ 푥푖. We say that +two societies (푁, 푥) and (푁, 푦) with the same set of agents 푁 are ≤-comonotonic if each of 푥 and 푦 +are ≤-chains, and that for all 푖, 푗 ∈ 푁 푥푖 > 푥 푗 implies 푦푖 ≥ 푦 푗. Comonotonicity is a natural way of +controlling for objective levels of wealth: it allows us to consider two profiles where we can meaningfully +talk about “rich” and “poor” agents (the chain assumption), and the rich and poor agents do not change +across profiles. +Comonotonicity requirements on bundles being compared are common in the theory of inequality +measurement. Comonotonicity ensures that the ranking of agents by wealth does not change across two +profiles. Thus, it is the appropriate concept for controlling for the distribution of wealth. See for example +the classic work by Weymark (1980). More closely related to our paper is Gajdos and Weymark (2005), +5A weak order ⪰ is a binary relation which is complete and transitive. +9 + +where a similar multidimensional notion of comonotonicity is described. +2.1 +Axioms +In this section, we provide three axioms that exhibit the main characteristics of sufficientarianism. +Anonymity: Let 푁, 푁′ ∈ N for which |푁| = |푁′|, and let (푁, 푥), (푁, 푦) be societies. Let 휎 : 푁′ → 푁 +be a bijection. Then (푁, 푥) ⪰푁 (푁, 푦) if and only if (푁′, 푥 ◦ 휎) ⪰푁′ (푁′, 푦 ◦ 휎). +This axiom is classical, see e.g. Moulin (1988). Sufficientarian social welfare rankings should be +insensitive to individual identities. +Reinforcement/consistency: Let 푁, 푁′ ∈ N where 푁 ∩ 푁′ = ∅. Let 푥, 푦 ∈ 푋 푁 and 푥′, 푦′ ∈ 푋 푁′ for +which 푥 ⪰푁 푦 and 푥′ ⪰푁′ 푦′. Then (푁 ∪ 푁′, (푥, 푥′)) ⪰푁∪푁′ (푁 ∪ 푁′, (푦, 푦′)), where the ranking is strict +if either of the original rankings is strict. +This property is a standard consistency condition in social choice studdies with variable populations +Young (1974); Smith (1973). +Recalling the ∧-notation for members of 푋 푁, we now introduce our main concept: +Sufficientarian-monotonicity: Let 푁 ∈ N and let 푥, 푦 ∈ 푋 푁. Then (푁, 푥) ⪰푁 (푁, 푦) implies +(푁, 푦) ⪰푁 (푁, 푥 ∧ 푦). If 푥, 푦 are additionally ≤-comonotonic chains, then (푁, 푦) ∼푁 (푁, 푥 ∧ 푦). +Sufficientarian-monotonicity demonstrates a core principle of sufficientarianism. One part of it +merely states that for any two profiles, the profile which results by replacing each agents consumption +by the greatest lower bound of her consumption across the two profiles cannot be better than any of the +original two profiles. The intuition for this part is straightforward: this new profile gives each agent less +than she would have received from either of the original profiles, and consequently it cannot be ranked as +high. +The second part of the property claims that, if we control for wealth, so that the ranking of agents +10 + +from richest to poorest is the same across the two profiles, then the new profile must actually be deemed +equivalent to the worse of the original two profiles. It cannot be deemed strictly worse than the worse of +the original two profiles. +To understand this property, suppose that we know that ⪰푁 is in fact a sufficientarian ranking. Then if +the ranking of agents from richest to poorest is the same across the two profiles, it follows that everybody +who has a sufficient amount for the worse of the two profiles also has a sufficient amount for the better +of the two. Consequently, they must also have a sufficient amount under the newly constructed profile, +as the sufficient sets are closed under taking greatest lower bounds. And everybody who did not have a +sufficient amount under the worse of the original two profiles can also not have a sufficient amount under +the newly constructed profile. +A +B +C +D +E +F +20 +25 +30 +35 +40 +Agents +Wealth Level +Society 1 +Society 2 +훽 +Figure 1: Comparison of societies +To understand this intuition, consider Figure 1. +The figure considers 푋 = R+, and a society +consisting of agents 푁 = {퐴, 퐵, 퐶, 퐷, 퐸, 퐹}. Any individual reaching 훽 is deemed to have a sufficient +11 + +amount. Observe that the two profiles (labelled “Society 1” and “Society 2” here) are comonotonic, 퐴 is +the wealthiest agent and 퐹 the poorest. Agents 퐷 and 퐸 have the same wealth in Society 1, but this does +not contradict comonotonicity. Observe that for society 1, only 퐴 and 퐵 reach 훽, but for society 2, 퐴, 퐵, +and 퐶 all reach it. So, according to the ranking, society 2 is deemed better than society 1. +Figure 1 also demonstrates the meet of the two profiles: this consists of the elements which are +circled. Observe that the meet profile is also comonotonic with respect to society 1 and society 2. Further, +for the meet profile, only agents 퐴 and 퐵 reach 훽, in accordance with our axiom: the meet profile is +socially indifferent to society 1, which was the worse of society 1 and 2. +The axiom is intended to be understood as a property a natural “pre-ranking” would satisfy. To +understand what we mean, if we were to deem (푁, 푥) ∼푁 (푁, 푦), we would not necessarily argue that 푥 +and 푦 are socially indifferent. Rather, we would only argue that our ranking lacks the power to make any +finer distinctions. However, if (푁, 푥) and (푁, 푦) are both available, and (푁, 푥) ≻푁 (푁, 푦), then we argue +that (푁, 푦) should never be selected. The sufficientarian monotonicity property is compatible with this +value judgment. Sufficientarian monotonicity also claims that if (푁, 푥) ∼푁 (푁, 푦), then we also should +not rule out (푁, 푥 ∧ 푦) so long as 푥 and 푦 are comonotonic. We believe this is intuitive as well: even +though 푥 ∧푦 is dominated by each of 푥 and 푦, we see no particular grounds to rule it out except for Paretian +reasons: but we are not studying a Paretian ranking. We may wish, in fact, to retain 푥 ∧ 푦 in our choice +set in order to achieve other equity objectives, or perhaps to trade off against other relevant attributes not +considered by the model. +Compare sufficientarian monotonicity to the classical notion of weak monotonicity, usually inter- +preted as a Pareto property: +Monotonicity: Let 푁 ∈ N and 푥, 푦 ∈ 푋 푁 such that 푥 ≥ 푦. Then (푁, 푥) ⪰푁 (푁, 푦). +Sufficientarian monotonicity is an apparently innocuous strengthening of monotonicity. To under- +stand this, witness the following proposition. +12 + +Proposition 1. For a complete binary relation ⪰ on 푋 푁, the following are equivalent: +1. 푥 ⪰ 푦 implies 푦 ⪰ 푥 ∧ 푦 +2. 푥 ≥ 푦 implies 푥 ⪰ 푦. +Proof. Suppose the second of the two properties is satisfied, and let 푥 ⪰ 푦. Because 푥 ≥ 푥 ∧ 푦, the +second property implies 푥 ⪰ 푥 ∧ 푦 (this would have held also in case 푦 ≻ 푥). Conversely, suppose the first +property is satisfied and let 푥 ≥ 푦. By means of contradiction (and invoking completeness), suppose that +푦 ≻ 푥. Then by the first property, 푥 ⪰ 푦 ∧ 푥 = 푦, a contradiction. +□ +Sufficientarian monotonicity therefore is a strengthening of the classical notion, whereby the strength- +ening applies only to comonotonic chains. +Finally, we discuss a strengthening of sufficientarian monotonicity which applies to complete semi- +lattices. Observe that for any 푁 ∈ N and any finite 퐴 ⊆ 푋 푁 and 푦 ∈ 푋 푁 for which all pairs from +퐴 ∪ {푦} are comonotonic, sufficientarian monotonicity and transitivity of ⪰푁 imply that if for all 푥 ∈ 퐴, +(푁, 푥) ⪰푁 (푁, 푦), then (푁, 푦) ∼푁 (푁, 푦 ∧ � 퐴).6 +The following is a “continuous” analogue of sufficientarian monotonicity applying for infinite sets +퐴. For simplicity, the following property is stated for complete meet-semilattices only. +Strong sufficientarian-monotonicity: Suppose that 푋 is complete, and let ∅ ≠ 퐴 ⊆ 푋 푁, 푦 ∈ 푋 푁. +Suppose that for all 푥 ∈ 퐴, (푁, 푥) ⪰푁 (푁, 푦). Then (푁, 푦) ⪰푁 (푁, 푦 ∧ � 퐴), in addition, (푁, 푦) ∼푁 +(푁, 푦 ∧ (� 퐴)) if all pairs from 퐴 ∪ {푦} are comonotonic. +6This follows since for any set 퐴 ⊆ 푋 푁 for which all pairs are comonotonic, � 퐴 is comonotonic with respect to every +푥 ∈ 퐴. So let 퐴 be such a set and let 푥 ∈ 퐴. Let 푖, 푗 ∈ 푁; we want to claim that (� 퐴)푖 and (� 퐴) 푗 are ordered. If there +is 푦 ∈ 퐴 for which 푦푖 > 푦 푗, then for every 푥 ∈ 퐴, 푥푖 ≥ 푥 푗, so that (� 퐴)푖 ≥ (� 퐴) 푗; similarly if there is 푦 ∈ 퐴 for which +푦 푗 > 푦푖. Finally if 푦푖 = 푦 푗 for all 푦 ∈ 퐴, then (� 퐴)푖 = (� 퐴) 푗. To see that � 퐴 exhibits comonotonicity with respect to 푥, +if 푥푖 > 푥 푗, we know that (� 퐴)푖 ≥ (� 퐴) 푗 (we just showed this). Conversely if (� 퐴)푖 > (� 퐴) 푗, then there exists 푦 ∈ 퐴 for +which 푦푖 > 푦 푗 (otherwise 푦 푗 ≥ 푦푖 for all 푦 ∈ 퐴 and so (� 퐴) 푗 ≥ (� 퐴)푖, and so 푥푖 ≥ 푥 푗. +13 + +2.2 +On sufficient sets and multidimensional sufficientarianism +Finally, we introduce our version of sufficientarianism. Departing from previous literature on the critical- +level sufficientarian threshold, we represent the “sufficient set” of agent’s consumption by a mathematical +object called a filter. We say that a nonempty subset F ⊆ 푋 is a filter if +1. For all 푥 ∈ F and all 푦 ∈ 푋, if 푥 ≤ 푦, then 푦 ∈ F +2. For all 푥, 푦 ∈ F , 푥 ∧ 푦 ∈ F . +Say that a social ranking is sufficientarian if there exists a filter F such that for all 푁 ∈ N and all +푥, 푦 ∈ 푋 푁, (푁, 푥) ⪰푁 (푁, 푦) iff |{푖 ∈ 푁 : 푥푖 ∈ F }| ≥ |{푖 ∈ 푁 : 푦푖 ∈ F }|. +This notion is a modest generalization of the notion described in previous literature;the generalization +is largely technical in nature. To this end, a slightly more restrictive notion in line with previous work is +the following. +A social ranking is threshold sufficientarian if there exists 훽 ∈ 푋 such that for all 푁 ∈ N and all +푥, 푦 ∈ 푋 푁, (푁, 푥) ⪰푁 (푁, 푦) iff |{푖 ∈ 푁 : 푥푖 ≥ 훽}| ≥ |{푖 ∈ 푁 : 푦푖 ≥ 훽}|. Here, 훽 is called the +sufficientarian threshold. +Every threshold sufficientarian ranking is sufficientarian with filter F = {푥 : 푥 ≥ 훽}. Filters of this +form are called principal filters in the mathematics literature. The distinction can be gleaned from the +case of [0, 1] with its usual order ≤. Principal filters in [0, 1] are all sets of the form [훽, 1]; these are +the ones considered in Alcantud et al. (2022). Non-principal filters also include sets of the form (훽, 1]. +Roughly, a non-principal filter need not be “closed below,” in an order-theoretic sense. +A filter is a useful model of such a sufficient set. If 푥 obtains a sufficient level and 푦 ≥ 푥, then 푦 +obtains a sufficient level. Further, and following our motivation of commodity space, if each of 푥, 푦 ∈ 푋 +are sufficient, then for each of 푥 and 푦, there is a sufficient amount of each commodity. Therefore, there +must be a sufficient amount of each commodity under 푥 ∧ 푦. +14 + +2.3 +Main result +Following is our main result. +Theorem 1. A social ranking is sufficientarian iff it satisfies anonymity, reinforcement, and sufficientarian +monotonicity. In addition, if 푋 is complete, the ranking also satisfies strong-sufficientarian monotonicity +iff it is threshold sufficientarian. +Remark 1. In the statement of Theorem 1, we could replace sufficientarian monotonicity with a weaker +version whereby it is only required to hold for 푁 ∈ N with |푁| = 2. Similarly, reinforcement could +be replaced with the statement that if for all 푁 and all 푖 ∈ 푁, we have ({푖}, 푥푖) ⪰{푖} ({푖}, 푦푖), then +(푁, 푥) ⪰푁 (푁, 푦), with a strict ranking if any initial ranking is strict. +Remark 2. For many semilattices, all sufficientarian rules are threshold sufficientarian. For example, +for any finite semilattice, or for our canonical example of nonnegative consumption bundles in discrete +consumption space: 푋 = Z푘 ++ with the usual pointwise order. To see why, say that a semilattice has the +descending chain condition (DCC) if for any sequence 푥1 ≥ 푥2 ≥ 푥3 . . ., there is some 푛 for which for +all 푚 ≥ 푛, 푥푚 = 푥푛. This property is equivalent to the property that all nonempty subsets of 푋 have an +element which is not dominated below ((Davey and Priestley, 2002) Lemma 2.39). For any semilattice, +all filters are principal iff DCC is satisfied.7 So in such environments, sufficientarianism coincides with +threshold sufficientarianism. +7This is well-known. Suppose that a semilattice does not have the DCC, and let 푥1 > 푥2 > 푥3 . . . be a strictly decreasing +sequence. Define F = � +푛{푥 ∈ 푋 : 푥 ≥ 푥푛} and observe F is a filter but cannot be principal: if F were principal, there would +exist 훽 ∈ F for which for all 푥 ∈ F , 푥 ≥ 훽. But then 훽 ∈ {푥 ∈ 푋 : 푥 ≥ 푥푛} for some 푛,and hence 훽 ≥ 푥푛 > 푥푛+1, a contradiction. +Conversely, if all filters are principal, then for any decreasing sequence 푥1 ≥ 푥2 . . ., again F = � +푛{푥 ∈ 푋 : 푥 ≥ 푥푛} is a +filter, and there is 훽 ∈ F for which 푥 ≥ 훽 iff 푥 ∈ F . Consequently, there exists 푛 such that 훽 ≥ 푥푛, and so for any 푚 ≥ 푛, +푥푚 ≥ 훽 ≥ 푥푛, so that 푥푚 = 푥푛. +15 + +Remark 3. Alternatively, suppose (푋, ≤) is a semilattice. Then there is another semilattice (푋′, ≤′) and +a one-to-one function 푓 : 푋 → 푋′ for which 푥 ≥ 푦 iff 푓 (푥) ≥′ 푓 (푦), and 푓 (푥 ∧ 푦) = 푓 (푥) ∧′ 푓 (푦), where +for every ≤-filter F ⊆ 푋 there is 훽′ ∈ 푋′ such that F = {푥 ∈ 푋 : 푓 (푥) ≥′ 훽′}. That is, any sufficientarian +rule results from a threshold sufficientarian rule on a larger set. Sufficientarian rules which are not +threshold sufficientarian then have a threshold in the larger set 푋′.8 +The proof relies on the following Lemma, which we believe is standard.9 +Lemma 1. For any anonymous social ranking and any 푁 ∈ N, if 휎 : 푁 → 푁 is a bijection, then for any +푥 ∈ 푋 푁, (푁, 푥) ∼푁 (푁, 푥 ◦ 휎). +Proof. Let 푘 be the order of permutation 휎, so that 휎푘 = id, the identity permutation. +Dropping +the 푁 from our notation, suppose by means of contradiction and without loss of generality that (using +completeness of ⪰) 푥 ≻ 푥 ◦ 휎. Then by anonymity, it follows that 푥 ◦ 휎 ≻ 푥 ◦ 휎2. By induction, for all +푙 = 1, . . . , 푘 − 1, 푥 ◦ 휎푙 ≻ 푥 ◦ 휎푙+1. Consequently we have 푥 ≻ 푥 ◦ 휎 . . . ≻ 푥 ◦ 휎푘 = 푥, a violation of +transitivity. +□ +Of Theorem 1. We first show that the representation is implied by the axioms. +8The argument is as follows. Let 푋 ′ = 퐹(푋) denote the set of ≤ filters on 푋, ordered so that F ≥′ F ′ if F ⊆ F ′. Define +푓 (푥) = {푦 : 푦 ≥ 푥}. Clearly 푥 ≥ 푥′ implies that if 푦 ≥ 푥, then 푦 ≥ 푥′, so 푓 (푥) ⊆ 푓 (푥′), and so 푓 (푥) ≥′ 푓 (푥′). Similarly, if +푓 (푥) ≥′ 푓 (푥′), then 푓 (푥) ⊆ 푓 (푥′), so in particular 푥 ≥ 푥′. Now, observe that 푋 itself is a ≤-filter and that the intersection of +an arbitrary collection of filters containing a given collection of filters is also a filter, hence (푋 ′, ≥′) is a complete semilattice: +�′ +푥′∈퐴′ 푥′ is simply the smallest filter containing all filters 푥′ ∈ 퐴′. Further, finite meets are preserved: 푓 (푥) ∧′ 푓 (푥′) is the +smallest filter containing both 푓 (푥) and 푓 (푥′); equivalently, having each of 푥 and 푥′ as members. Observe that a filter has 푥 +and 푥′ as members if and only if 푥 ∧ 푥′ is a member, so that 푓 (푥 ∧ 푥′) = 푓 (푥) ∧′ 푓 (푥′). Finally, for any filter F ⊆ 푋, it is clear +that 푥 ∈ F if and only if 푓 (푥) ⊆ F , or 푓 (푥) ≥′ F , let 훽′ = F and we are done. It is not necessarily true that arbitrary meets +are preserved under 푓 , nor is it necessarily true that all ≤′-filters are principal. +9This lemma relies on transitivity, but would clearly also hold under the weaker requirement that there are no strict cycles. +16 + +First, we consider the case of any 푁 with cardinality one. By anonymity, the ranking of members of +푋 does not depend on 푁, so we refer to it without loss of generality as ⪰, without referencing the subscript +푁 (we will do this generally when necessary). +We claim that there are at most two equivalence classes of ⪰. Suppose by means of contradiction +that there are more than two, and let 푥, 푦, 푧 ∈ 푋 for which 푥 ≻ 푦 ≻ 푧. Consider any 푁′ with two agents; +by anonymity it does not matter which one we choose and we again without loss refer to the ranking as ⪰. +When we write (푎, 푏), this references a situation giving the first agent 푎 ∈ 푋, and the second agent 푏 ∈ 푋. +We claim that we may without loss assume that 푥 > 푧. This follows as Proposition 1 demonstrates +that 푎 ≥ 푏 implies 푎 ⪰ 푏 (monotonicity is satisfied), so that 푥 ≻ 푦 ≻ 푧 ≥ 푥∧푦∧푧 implies 푥 ≻ 푦 ≻ 푥∧푦∧푧, +so we may replace 푧 by 푥 ∧ 푦 ∧ 푧 if necessary. Since 푥 ≻ 푥 ∧ 푦 ∧ 푧, we know that 푥 ≠ 푥 ∧ 푦 ∧ 푧 and so +푥 > 푥 ∧ 푦 ∧ 푧. +So without loss, assume that 푥 > 푧 and 푥 ≻ 푦 ≻ 푧. +Observe that the pairs (푥, 푧) and (푦, 푦) are ≤-comonotonic. +Case 1: (푥, 푧) ⪰ (푦, 푦). +Then by sufficientarian monotonicity, (푦, 푦) ∼ (푥 ∧ 푦, 푦 ∧ 푧). Now, 푦 ≻ 푧 and since 푧 ≥ 푦 ∧ 푧, by +monotonicity, we have 푧 ⪰ 푦 ∧ 푧, so that by transitivity, 푦 ≻ 푦 ∧ 푧. Further, by monotonicity, 푦 ⪰ 푥 ∧ 푦. +By reinforcement, we thus have (푦, 푦) ≻ (푥 ∧ 푦, 푦 ∧ 푧), contradicting (푦, 푦) ∼ (푥 ∧ 푦, 푦 ∧ 푧). +Case 2: (푦, 푦) ≻ (푥, 푧) +Then by sufficientarian-monotonicity, (푥, 푧) ∼ (푥 ∧ 푦, 푦 ∧ 푧). Recall that 푥 ≻ 푦. Further, 푦 ≥ 푥 ∧ 푦, +so that 푦 ⪰ 푥 ∧ 푦 (monotonicity) whereby transitivity implies that 푥 ≻ 푥 ∧ 푦. Further, 푧 ⪰ 푦 ∧ 푧 as +푧 ≥ 푦 ∧ 푧, so by reinforcement, we obtain (푥, 푧) ≻ (푥 ∧ 푦, 푦 ∧ 푧), a contradiction to (푥, 푧) ∼ (푥 ∧ 푦, 푦 ∧ 푧). +Therefore, ⪰ as restricted to 푋 has at most two indifference classes. If there is only one, it is easy to +use reinforcement to show that ⪰푁 coincides with total indifference everywhere, and thus corresponds to +the filter F = 푋. +17 + +Otherwise, let F denote all of the elements in the higher of the two indifference classes; so that 푥 ∼ 푦 +for all 푥, 푦 ∈ F , 푧 ∼ 푤 for all 푧, 푤 ∉ F , and 푎 ≻ 푏 for all 푎 ∈ F and 푏 ∉ F . +We claim that F is a filter. To see this, first let 푥 ∈ F and let 푦 ≥ 푥. Then if 푦 ∉ F , we must have +푥 ≻ 푦, which contradicts monotonicity. Now, suppose that 푥, 푦 ∈ F . Then 푥 ⪰ 푦, so that 푦 ∼ 푥 ∧ 푦 by +sufficientarian-monotonicity; consequently, 푥 ∧ 푦 ∈ F . +In case 푋 is also complete and satisfies strong sufficientarian monotonicity, let 푦 ∈ F . Then for all +푥 ∈ F , 푥 ⪰ 푦, so 푦 ∼ 푦 ∧ (∧F ); conclude that 훽 ≡ ∧F ∈ F and observe that by definition, 푥 ∈ F if and +only if 푥 ≥ 훽; hence F is a principal filter. +Finally, we claim that our ranking is sufficientarian with filter F . To see this, let 푁 be arbitrary and +let 푥, 푦 ∈ 푋 푁. +First, suppose that |{푖 ∈ 푁 : 푥푖 ∈ F }| ≥ |{푖 ∈ 푁 : 푦푖 ∈ F }| +Let us without loss, by anonymity, suppose that 푁 = {1, . . . , 푛}. +Observe that by Lemma 1, we may without loss assume that (푥1, . . . , 푥푛) is such that 푥푖 ⪰ 푥푖+1 for all +푖, and similarly for 푦. +Now, since 푥1 ⪰ 푥2 ⪰ . . . 푥푛 and 푦1 ⪰ . . . 푦푛, and by the hypothesis that |{푖 ∈ 푁 : 푥푖 ∈ F }| ≥ |{푖 ∈ +푁 : 푦푖 ∈ F }|, we may conclude for each 푖 ∈ 푁 that 푥푖 ⪰ 푦푖 (recall that F was the set of objects in the +higher indifference class) and so 푥 ⪰푁 푦 follows by reinforcement. +Likewise if |{푖 ∈ 푁 : 푥푖 ∈ F }| > |{푖 ∈ 푁 : 푦푖 ∈ F }|, then 푥 ≻푁 푦 using the same argument and the +fact that there is 푖 ∈ 푁 for which 푥푖 ≻ 푦푖. +Now we establish that the axioms are satisfied by the representation. +It is trivial to see that a sufficientarian social ranking is anonymous and satisfies reinforcement. +Verifying the first part of sufficientarian monotonicity (the part which is equivalent to monotonicity) is +similarly striaghtforward. +To verify that it satisfies the second part of sufficientarian monotonicity, let F be the associated filter +18 + +and take any 푁 ∈ N and 푥, 푦 ∈ 푋 푁 which are comonotonic ≤-chains. +Suppose that (푁, 푥) ⪰푁 (푁, 푦). Then |{푖 ∈ 푁 : 푥푖 ∈ F }| ≥ |{푖 ∈ 푁 : 푦푖 ∈ F }|. +We claim that if 푖 ∈ 푁 is such that 푦푖 ∈ F , then 푥푖 ∈ F as well; that is, {푖 ∈ 푁 : 푦푖 ∈ F } ⊆ +{푖 ∈ 푁 : 푥푖 ∈ F }. By means of contradiction, suppose there is 푖 ∈ 푁 such that 푦푖 ∈ F and 푥푖 ∉ F . +Clearly, for any 푗 ∈ 푁, if 푦 푗 ≥ 푦푖, then 푦 푗 ∈ F as well; and for any 푘 ∈ 푁, if 푥푘 ≤ 푥푖, then 푥푘 ∉ F . +Therefore we must conclude that for any 푗 ∈ 푁 \ {푖}, if 푥 푗 ∈ F , it follows that 푥 푗 > 푥푖 (by the fact +that 푥 is an ≤-chain), from which we conclude by comonotonicity that 푦 푗 ≥ 푦푖, so that 푦 푗 ∈ F . This +implies that { 푗 : 푥 푗 ∈ F } ⊆ { 푗 : 푦 푗 ∈ F } \ {푖}. Since 푦푖 ∈ F , this contradicts the hypothesis that +|{푖 ∈ 푁 : 푥푖 ∈ F }| ≥ |{푖 ∈ 푁 : 푦푖 ∈ F }|. +The result now follows, as whenever 푦푖 ∈ F , we have 푥푖 ∈ F , and thus by the filter property +(푥푖 ∧ 푦푖) ∈ F . Similarly, if 푦푖 ∉ F , (푥푖 ∧ 푦푖) ∉ F as (푥푖 ∧ 푦푖) ≤ 푦푖. So, (푥푖 ∧ 푦푖) ∈ F if and only if 푦푖 ∈ F , +from which we conclude that (푁, 푦) ∼푁 (푁, 푥 ∧ 푦). +For the case of a threshold sufficientarian rule on a complete semilattice, observe that the same +argument allows us to establish that if (푁, 푥) ⪰푁 (푁, 푦) for all 푥 ∈ 퐴, then for all 푥 ∈ 퐴, {푖 ∈ 푁 : 푦푖 ≥ +훽} ⊆ {푖 ∈ 푁 : 푥푖 ≥ 훽}. Consequently, for any 푖 ∈ 푁, 푦푖 ≥ 훽 implies ∧푥∈퐴푥푖 ≥ 훽. The argument then +parallels that of the filter case. +□ +2.4 +Independence of the axioms +We illustrate by examples that the three axioms are independent. These examples will illustrate the general +structure of rules satisfying these axioms. +Anonymity +Start with dropping anonymity. Many rules satisfy both reinforcement and sufficientarian monotonicity. +19 + +Start with an obvious generalization: Say that a rule is weighted sufficientarian if there is a function +휆 : 푀 → R+ and a filter F ⊆ 푋 such that for all 푁 ∈ N, and all 푥, 푦 ∈ 푋 푁, 푥 ⪰푁 푦 if and only +if � +푖∈푁 휆푖1{푥푖∈F } ≥ � +푖∈푁 휆푖1{푦푖∈F }.10 A related generalization, which would feature each individual +possessing their own filter F푖, generally fails sufficientarian monotonicity. +Another class of solutions would are lexicographic dictatorships, consisting of a linear order on 푀 +and for each 푖 ∈ 푀, a relation ⪰푖 for which ≥⊆⪰푖, so that for any 푁 ∈ N, and any 푥, 푦 ∈ 푋 푁, 푥 ⪰푁 푦 if +and only if 푥푖 ⪰푖 푦푖 for the highest priority 푖 ∈ 푁 according to the linear order. +Still other, hybrid, variations of these two methods would satisfy the remaining axioms. We leave +this to future research. +Reinforcment +Rules violating reinforcement are similarly many. A particularly canonical example would obtain in the +case of 푋 = [0, 1], whereby for all 푁 ∈ N and all 푥, 푦 ∈ 푋 푁, 푥 ⪰푁 푦 if and only if min푖 푥푖 ≥ min푖 푦푖. For +a more general set 푋, one would construct a ≤-chain C with the property that for every 푥 ∈ 푋, there is a +uniquely ≤-maximal element of C. Then define, for any 푁 ∈ N and 푥, 푦 ∈ 푋 푁, 푥 ⪰푁 푦 if and only if said +maximal element for � +푖∈푁 푥푖 ≤-dominates the maximal element for � +푖∈푁 푦푖. However, even in the case of +푋 = [0, 1], many other rules are easy to describe. For any 푁 ∈ N and 푥 ∈ 푋 푁, and any 푛 ∈ {1, . . . , |푁|}, +let 푥∗(푛) denote the 푛-th highest value of 푥. Then define 푈푁 : 푋 푁 → R as 푈푁(푥) = inf푛 푛푥∗(푛). For any +푥, 푦 ∈ 푋 푁, assign 푥 ⪰푁 푦 if and only if 푈푁(푥) ≥ 푈푁(푦). We leave the study of this class of rules to +future research. +Sufficientarian Monotonicity +Rules violating sufficientarian monotonicity constitute much of social choice. A particularly interesting +one is the trigonometry rule on 푋 = [0, 1], whereby for any 푁 ∈ N and 푥, 푦 ∈ 푋 푁, 푥 ⪰푁 푦 if and only if +10Or just an ordinal “qualitative measure” ranking the sets. +20 + +� +푖∈푁 sin(cos(푥푖)) ≥ � +푖∈푁 sin(cos(푦푖)). As far as we are aware, this rule is novel. +3 +Conclusion +In this work, we have studied the concept of sufficientarianism, applied to a multidimensional framework. +Sufficientarianism is to be understood as a “pre-ranking,” imposed to rule out profiles that should never +be chosen in the presence of other feasible profiles. +There are several possible extensions for our work. As a first point, though our work is technically +a variable population work, we only ever consider a fixed population in comparing two profiles; we are +essentially doing head-counting. A work that allows one to consider multidimensional profiles across +different populations would be similarly interesting; see, for example, Bossert et al. (2022a,b). +Second, sufficientarianism lends itself to a natural cardinal ranking, whereby we only count the +number of people who reach a given threshold. One could consider imposing a random threshold and +studying the expected number of people to reach the threshold. In the case of 푋 = R+, we could clearly +end up with the set of all (linear) rankings satisfying a weak form of monotonicity in doing so. But in +higher dimensional environments, we simply have not studied what this approach would generate. It is +conceivable that the sufficientarian rankings generate as their expectation some type of supermodular and +increasing rankings. We leave this to future research. +Of course, weakening axioms, especially that of completeness of the social ranking, might prove +interesting as well. +Finally, we have motivated our concept of sufficientarianism as a kind of “pre-ranking,” to be imposed +prior to other, more finely-grained rankings. We do not know what happens when taking this literally, and +imposing a Paretian ranking “on top” of a sufficientarian ranking. For example, given a sufficientarian +21 + +ranking ⪰ and a Paretian ranking ⪰∗, this approach suggests ruling out any (푁, 푥) for which either there +exists a feasible 푦 for which (푁, 푦) ≻ (푁, 푥) or for which there exists a feasible 푦 for which (푁, 푦) ∼ (푁, 푥) +and (푁, 푦) ≻∗ (푁, 푥). Again, we leave this to future research. +22 + +References +Alcantud, J. C. R. (2013). Liberal approaches to ranking infinite utility streams: when can we avoid +interference? Social Choice and Welfare 41(2), 381–396. +Alcantud, J. C. R., M. Mariotti, and R. Veneziani (2022). Sufficientarianism. Theoretical Economics 17(4), +1529–1557. +Bossert, W., S. Cato, and K. Kamaga (2022a). Critical-level sufficientarianismError. Journal of Political +Philosophy 30, 434–461. +Bossert, W., S. Cato, and K. Kamaga (2022b). 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Introduction to Lattices and Order (2 ed.). Cambridge University +Press. +Day, W. H. and F. R. McMorris (2003). Axiomatic consensus theory in group choice and biomathematics. +SIAM. +Deaton, A. (2008). Income, health, and well-being around the world: Evidence from the gallup world +poll. Journal of Economic Perspectives 22, 53–72. +Gajdos, T. and J. A. Weymark (2005). Multidimensional generalized gini indices. Economic Theory 26, +471–496. +Hougaard, J. L. and H. Keiding (1998). On the functional form of an efficiency index. Journal of +Productivity Analysis 9(2), 103–111. +Kreps, D. M. (1979). A representation theorem for “preference for flexibility". Econometrica 47(3), +565–577. +Lombardi, M., K. Miyagishima, and R. Veneziani (2016). Liberal egalitarianism and the harm principle. +Economic Journal 126(597), 2173–2196. +Mariotti, M. and R. Veneziani (2009). ‘non-interference’implies equality. Social Choiceand Welfare32(1), +123–128. +Mariotti, M. and R. Veneziani (2012). Allocating chances of success in finite and infinite societies: The +utilitarian criterion. Journal of Mathematical Economics 48(4), 226–236. +24 + +Mariotti, M. and R. Veneziani (2013). On the impossibility of complete non-interference in paretian +social judgements. Journal of Economic Theory 148(4), 1689–1699. +Miller, A. D. (2008). Group identification. Games and Economic Behavior 63(1), 188–202. +Moulin, H. (1988). Axioms of Cooperative Decision Making. Econometric Society Monographs. Cam- +bridge University Press. +Müller, A. and M. Scarsini (2006). Stochastic order relations and lattices of probability measures. SIAM +Journal on Optimization 16(4), 1024–1043. +Shaked, M. and J. G. Shanthikumar (2007). Stochastic orders. Springer. +Smith, J. H. (1973). Aggregation of preferences with variable electorate. Econometrica 41(6), 1027–1041. +Veenhoven, R. (1991, 02). Is happiness relative? Social Indicators Research 24, 1–34. +Weymark, J. A. (1980). Generalized gini inequality indices. Mathematical Social Sciences 1, 409–430. +Young, H. P. (1974). A note on preference aggregation. Econometrica 42(6), 1129–1131. +25 + diff --git a/HdFAT4oBgHgl3EQftx6O/content/tmp_files/load_file.txt b/HdFAT4oBgHgl3EQftx6O/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9965e2fb115b4af6d82ee90a73373f4fab400475 --- /dev/null +++ b/HdFAT4oBgHgl3EQftx6O/content/tmp_files/load_file.txt @@ -0,0 +1,657 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf,len=656 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='08666v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='TH] 20 Jan 2023 On Sufficientarianism Christopher P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Chambers∗ Siming Ye† January 23, 2023 Abstract We introduce a generalization of the concept of sufficientarianism, intended (but not limited) to study multiple consumption goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We discuss three properties that uniquely pin down our family and, up to continuity requirements, endogenize the sufficientarian threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Two, anonymity and reinforcement, are standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The third, sufficientarianmonotonicity, generalizes classical monotonicity and requires that for any two lists of bundles in which individuals are ordered similarly by how much they receive, the worse of the two is indifferent to the new list of bundles considered by taking the componentwise minimum of the two bundles for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 1 Introduction Consider a society in which all individuals have access to enough clean water, food, and education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Independently of whether any of them are wealthy or how the individuals obtained their positions, such an environment arguably dominates a society in which some of the individuals lack access to clean water, and some lack access to food.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' In this paper, we study a new form of sufficientarianism, an interesting ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Department of Economics, Georgetown University, ICC 580 37th and O Streets NW, Washington DC 20057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' E-mail: Christopher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='Chambers@georgetown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' †Department of Economics, Georgetown University, ICC 580 37th and O Streets NW, Washington DC 20057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' E-mail: sy677@georgetown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 1 criterion of distributive justice recently introduced to the economics literature by Alcantud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' (2022) and Bossert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' (2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Sufficientarianism’s goal is to ensure each individual has a “sufficient” level of well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' This work departs from the concept of “well-being” as the primitive, directly working instead with economic observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The basic idea of sufficientarianism is that potential societies are compared according to the number of individuals meeting an exogenously specified threshold of well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The more agents who reach the level of well-being, the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A sufficientarian ranking satisfies a weak version of the Pareto principle (if all agents weakly gain, then society weakly gains), but not the typical stronger version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Our goal in this work is twofold: first, to study a concept which could be applied to directly observable and measurable bundles of goods, rather than levels of well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Say, the amount of food or clean water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Second, we provide a novel axiomatization of sufficientarianism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Previous works base their results on a principle called “prioritarian threshold,” as well as other ethical axioms related to those in Mariotti and Veneziani (2009, 2012, 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Alcantud (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Lombardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' By contrast, our novel axiom is not ethical in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' However, we believe the ordinal content of our axioms captures the spirit of sufficientarianism in a particularly straightforward way, and up to technical details, allows us to endogenize the prioritarian threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We believe our axiomatization is novel even in the previously studied single-dimensional context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' To understand the multicommodity aspect of our work, consider a model of food and clean water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We can imagine a society consisting of three individuals, Alice, Bob and Charles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Suppose (for the sake of argument) that it is determined that each individual needs 2500 daily calories of food and four daily liters of water: these are the sufficientarian thresholds for these consumption goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The presumption here is that any individual who does not reach both of these thresholds cannot thrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Let us now consider several ways of allocating food and clean water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Writing 푥퐴 = (1500, 5) to denote a bundle where Alice gets 1500 calories and 5 liters of water, we can now compare 푥퐴 = (1500, 5), 푥퐵 = (3000, 30), 푥퐶 = (5000, 50) 2 to 푦퐴 = (2500, 5), 푦퐵 = (2500, 5), 푦퐶 = (2500, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' According to our sufficientarian thresholds, the 푦 allocation dominates the 푥 allocation for the simple reason that the 푥 allocation does not give Alice enough to eat (she does not meet the threshold of 2500 calories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' This is so even though Bob and Charles have orders of magnitude more water under the 푥 allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' In contrast, all individuals (exactly) meet the sufficientarian threshold in the 푦 allocation, so it is deemed more appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Food and water are obviously not the only commodities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For example, in the survey Deaton (2008), people report their life satisfaction according to various components of well-being, including health, family financial status and human and political rights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Veenhoven (1991) argues that it is only when people’s basic needs are met that an increase in income contributes to people’s evaluation of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Cutler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' (2006) and Case and Deaton (2020) further enhanced the argument in Veenhoven (1991) by studying the deaths of despair, namely the fast rising death rates among Americans that were from drug overdoses, suicide and alcoholic liver disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Their work links the deaths of less-educated Americans to the social and economics conditions, emphasizing a strong correlation between education and drug-related and alcohol-related mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Case and Deaton (2022) further points out that the rising despair and deteriorating mental health conditions are making this group of citizens more volatile to shocks such as COVID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Any one of these criteria (educaton, drug use, mental health) are criteria which we may want to directly employ in our sufficientarian ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We understand the sufficientarian ranking in a very simple light: it is to be understood as a kind of pre-ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' It allows us to discard social profiles of alternatives which a society should never consider unless necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' However, two profiles which are rendered indifferent by the sufficientarian ranking may be distinguished by some other, more finely tuned ranking, at a later stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Our judgment as to which of these should be preferred by society is silent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We establish that sufficientariansm emerges from a nearly classical collection of axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Two, anonymity and reinforcement, are well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' On reinforcement, see for example Smith (1973) 3 or Young (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Our third property is a strengthening of monotonicity (Pareto), which we deem sufficientarian monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' To understand sufficientarian monotonicity, first consider a reformulation of monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' It states that for two potential societies with the same individuals, if everybody receives at least as much of each good in the second of the two societies, then society as a whole is made at least as well off in the second society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A reformulation of this property is possible: for any pair of societies with the same individuals, one deemed at least as good as the other, the worse of the two is additionally at least as good as the society constructed by assigning to each individual the minimum level of each good she receives across the two societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Our contribution is to strengthen this notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Imagine two societies, each with the same set of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Each society features an unambiguous ordering of the agents in terms of consumption: for any pair of agents, one of them must get at least as much of each good as the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Thus, there is an objective ranking of agents in terms of wealth in either society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Suppose this ordering of agents remains essentially unchanged across the two societies: for any pair of agents, it is impossible that the first is richer in one society and poorer in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We call such a pair of societies comonotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Our strengthening of the classical monotonicity property described above requires that if the two societies are comonotonic, then the worse of the two is actually socially indifferent to the one assigning to each individual the minimum level of each good she receives across the two societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For lack of a better word, we call this sufficientarian monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Sufficientarian monotonicity captures what we believe is the essence of sufficientarianism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For an illustration, imagine again a group consisting of three agents, Alice, Bob, and Charles, and suppose (for simplicity), that consumption space is single-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A vector (푎, 푏, 푐) specifies the amount that each individual gets of the good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Imagine comparing 푥 = (1, 4, 2) and 푦 = (0, 5, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' First, observe that these two profiles are comonotonic: the ranking induced by these two profiles agrees weakly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' It is true 4 that 푥(푏) > 푥(푐), while 푦(푐) ≥ 푦(푏), but for our definition of comonotonicity, this does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The only profiles which would be precluded would feature strict reversals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For example, 푧 = (1, 2, 3) and 푤 = (2, 1, 7) are not comonotonic because 푧(푏) > 푧(푎) and 푤(푎) > 푤(푏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Sufficientarian monotonicity states that if 푥 is deemed at least as good as 푦, that is, 푥 ⪰ 푦, then it follows that the new profile, which results by replacing each individual’s consumption by the minimum that they get across the two profiles, must be indifferent to 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Specifically (푥 ∧ 푦) = (0, 4, 2) ∼ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' What does this have to do with sufficientarianism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For a sufficientarian rule and comonotonic profiles, the agents reaching the sufficient level for the worse of the two must be a subset of those reaching the sufficient level for the better of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' And consequently, the pointwise minimum profile ensures that all of the individuals reaching the sufficient level under the worse of the two profiles continue to do under the minimum profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Return to our example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' If 푥 is deemed at least as good as 푦, this is consistent with a sufficientarian ranking counting the number of agents who get at least 1: 푥 has three such agents, and 푦 has two;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' namely agents 푏 and 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Under (푥 ∧ 푦) = (0, 4, 2), the same agents get at least 1 as they did under 푦, namely 푏 and 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Of course, not all sufficientarian rules will feature 푥 ⪰ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Take a rule that counts the number of agents receiving at least 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then under 푥, there is one such agent 푏, and under 푦, there are two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' So in this case 푦 ⪰ 푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' and again 푥 ∧ 푦 = (0, 4, 2) features only one agent receiving at least 4, again 푏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Let us return to the non-comonotonic profiles 푧 = (1, 2, 3) and 푤 = (2, 1, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A similar property would not obtain here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Imagine the sufficientarian rule with level 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then according to this rule, both 푧 and 푤 have two agents meeting the sufficient level: for 푧 it is 푏 and 푐 and for 푤 it is 푎 and 푏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' However, these sets are not nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Consequently, in taking the pointwise minimum, 푧 ∧ 푤 = (1, 1, 3), only one agent remains above the sufficient level, agent 푐: in this case, 푧 ∧ 푤 is deemed strictly worse than either 푧 or 푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The restriction to comonotonic profiles is thus revealed as a method to ensure that no matter which 5 sufficient level is chosen, the set of agents meeting a sufficient level will be nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Our work deviates from previous work, even in the single dimensional case, as it allows for what we might think of as “strict” sufficientarianism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' This is a technical point, but it is worth mentioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Previous works considered sufficientarian rules to be those for which there is some level of well-being, and the goal was to maximize the number of agents meeting at least that level of well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Our three axioms also permit rules which have the goal of maximizing the number of agents meeting strictly more than a given level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The classical formulation of the rules would emerge with a simple order-theoretic continuity axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Finally, our work applies in multi-dimensional space, in fact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' it is broad enough to apply to any semilattice, where the infimum of any pair of alternatives always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We model the notion of a sufficient set in a fairly general way, but which is consistent with the spirit of earlier models of sufficientarianism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The paper proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' First, we investigate some related literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' then we proceed directly to the model and main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Finally, Section 3 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='1 Related Literature Aside from the previously mentioned work on sufficientarianism, Alcantud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Bossert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' (2022a,b), the sufficientarian monotonicity axiom is closely related to a collection of axioms employed in characterizing similar representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' In the case of 푋 = Z+, and under the hypothesis of anonymity, any profile (푁, 푥) can be understood as a downward closed set: 퐴(푁, 푥) ≡ {(푦1, 푦2) ∈ Z2 + : |{푖 ∈ 푁 : 푥푖 ≥ 푦1}| ≥ 푦2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' In this formulation, the axiom is essentially the axiom of “meet-separability” found in Miller (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Chambers and Miller (2014a,b, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' In those works, it is explained that the main idea is originally due to Kreps (1979) in his study of preference for flexibility over menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Related is the work of Hougaard and Keiding (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' (1999), where meet-separability appears in a cardinal 6 form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 2 The model and results Let 푋 be a set of alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We endow 푋 with a partial order ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='1 The partial order ≤ represents an unambiguous comparative notion of dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' As a motivating example, we have in mind a multi-dimensional commodity space, say 푋 = Z푚 + , where 푥 ≥ 푦 if 푥푖 ≥ 푦푖 for all 푖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' , 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The strict part of ≤ will be written <.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='2 We need a language for talking about the greatest lower bound of a pair of alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For example, if 푋 = Z2 +, then the bundle (1, 1) is the largest bundle which is dominated by both (2, 1) and (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The meet of two elements 푥, 푦 ∈ 푋 is its greater lower bound when it exists, and is denoted 푥 ∧ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Formally, we assume that (푋, ≤) is a meet-semilattice: any pair of points possesses a meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We do not necessarily want or need to assume that every pair of alternatives has a least upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The reason will become clear with the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Say (푋, ≤) is a meet-semilattice if every pair of elements has a meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Say a meet-semilattice is complete if every nonempty subset 퐴 ⊆ 푋 has a greatest lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' With an abuse of notation, for any set 푁 and any 푥, 푦 ∈ 푋 푁, we let (푥 ∧ 푦) ∈ 푋 푁 denote the pointwise meet across 푁: (푥 ∧ 푦)푖 = 푥푖 ∧ 푦푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Similarly, we say that 푥 ≥ 푦 if for all 푖 ∈ 푁, 푥푖 ≥ 푦푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The terminology here is standard, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=', Davey and Priestley (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 1A partial order is a binary relation ≤ satisfying the following three properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Reflexivity: 푥 ≤ 푥 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Antisymmetry: If 푥 ≤ 푦 and 푥 ≠ 푦, then 푦 ≤ 푥 must not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Transitivity: If 푥 ≤ 푦 and 푦 ≤ 푧, then 푥 ≤ 푧 2So, for example in the case of 푋 = Z푚 + , this relation dictates that 푥 > 푦 if 푥 ≥ 푦 and 푥 ≠ 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' (3, 2) > (2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 7 Following are examples of meet-semilattices which are relevant for our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Our motivating example, already mentioned is 푋 = Z푚 + , or 푋 = R푚 + , with its usual pointwise order ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' In this case, each pair 푥, 푦 has a least upper bound, but consider a slightly modified example whereby consumption capacity constraints require that 푧 ∈ 푋 only if �푚 푖=1 푧푖 ≤ 푤, where 푤 > 0 represents wealth, or some storage cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' In this case, we could take 푋 = {푥 ∈ Z푚 + : �푚 푖=1 푥푖 ≤ 푤}, where ≤ is the restriction of the standard partial order on Z푚 + to 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Observe that (푋, ≤) is a meet-semilattice, but a pair of elements typically does not possess a least upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For example, if 푚 = 2 and 푤 = 2, then 푥 = (2, 0) and 푥′ = (0, 2) have no least upper bound: these are each already maximal with respect to ≤ in 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Let 푍 be a set of discrete goods, and let 푋 ⊆ 2푍 be any collection of packages of goods for which 퐴 ∈ 푋 and 퐵 ⊆ 퐴 implies 퐵 ∈ 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Again this is a simple model of an environment with capacity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Let ≤ coincide with set inclusion, so that 퐴 ≤ 퐵 if and only if 퐴 ⊆ 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Observe then that 푋 is a meet-semilattice: if each of 퐴, 퐵 are members of 푋, then so is 퐴 ∩ 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' However, in general, least upper bounds may not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Consider for example 푍 = {푎, 푏} and 푋 = {∅, {푎}, {푏}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then the pair {푎} and {푏} have no least upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Denote by Δ(R) the set of Borel probability distributions on R, and let ≤ coincide with the first order stochastic dominance relation, so that 푝 ≤ 푞 if 푞 first order stochastically dominates 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='3 It is well-known that ≤ is a lattice, and hence a meet-semilattice, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='18 of Shaked and Shanthikumar (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 4 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For a set 푆 of states of the world, one can let 푋 be either the set of partitions, the set of 3We say that 푞 first order stochastically dominates 푞 if for every 훼 ∈ R, 푞({푥 : 푥 ≥ 훼}) ≥ 푝({푥 : 푥 ≥ 훼}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 4In fact, many partial orders on the set Δ(R) have a lattice structure, including second order stochastic dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Müller and Scarsini (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 8 algebras, or the set of 휎-algebras, where Π ≤ Π′ if and only if Π refines Π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then the pair (푋, ≤) constitutes a meet-semilattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Such lattices arise frequently in the theory of information econmomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For a set 푌 of alternatives, let W(푌) denote the set of weak orders (complete and transitive relations) on 푌, ordered by reverse set inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' So for two weak orders 푅 and 푅′, 푅 ⪰ 푅′ means that 푅 ⊆ 푅′, or 푥 푅 푦 implies 푥 푅′ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then (W(푌), ⪰) is a meet-semilattice, whereby 푅 ∧ 푅′ is the transitive closure of 푅 ∪ 푅′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' See for example Day and McMorris (2003), Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Similarly, one could consider the set of pre-orders (reflexive and transitive relations) with the same ordering ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Let 푀 a set of potential agents, where |푀| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Let N denote the set of nonempty and finite subsets of 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A society is a tuple (푁, 푥), where 푥 ∈ 푋 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A social ranking, or simply a ranking, associates with each 푁 ∈ N a weak order ⪰푁 on 푋 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='5 For two societies (푁, 푥), (푁′, 푦), where 푁′ ∩ 푁 = ∅, let (푁 ∪ 푁′, (푥, 푦)) denote the society where (푥, 푦)푁 = 푥 and (푥, 푦)푁′ = 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For a bijection 휎 : 푁′ → 푁 between 푁 and 푁′, and 푥 ∈ 푋 푁, 푥 ◦ 휎 ∈ 푋 푁′ is defined as (푥 ◦ 휎)휎(푖) = 푥푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For a society (푁, 푥), say that 푥 is a ≤-chain if for all 푖, 푗 ∈ 푛, 푥푖 ≥ 푥 푗 or 푥 푗 ≥ 푥푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We say that two societies (푁, 푥) and (푁, 푦) with the same set of agents 푁 are ≤-comonotonic if each of 푥 and 푦 are ≤-chains, and that for all 푖, 푗 ∈ 푁 푥푖 > 푥 푗 implies 푦푖 ≥ 푦 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Comonotonicity is a natural way of controlling for objective levels of wealth: it allows us to consider two profiles where we can meaningfully talk about “rich” and “poor” agents (the chain assumption), and the rich and poor agents do not change across profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Comonotonicity requirements on bundles being compared are common in the theory of inequality measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Comonotonicity ensures that the ranking of agents by wealth does not change across two profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Thus, it is the appropriate concept for controlling for the distribution of wealth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' See for example the classic work by Weymark (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' More closely related to our paper is Gajdos and Weymark (2005), 5A weak order ⪰ is a binary relation which is complete and transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 9 where a similar multidimensional notion of comonotonicity is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='1 Axioms In this section, we provide three axioms that exhibit the main characteristics of sufficientarianism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Anonymity: Let 푁, 푁′ ∈ N for which |푁| = |푁′|, and let (푁, 푥), (푁, 푦) be societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Let 휎 : 푁′ → 푁 be a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then (푁, 푥) ⪰푁 (푁, 푦) if and only if (푁′, 푥 ◦ 휎) ⪰푁′ (푁′, 푦 ◦ 휎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' This axiom is classical, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Moulin (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Sufficientarian social welfare rankings should be insensitive to individual identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Reinforcement/consistency: Let 푁, 푁′ ∈ N where 푁 ∩ 푁′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Let 푥, 푦 ∈ 푋 푁 and 푥′, 푦′ ∈ 푋 푁′ for which 푥 ⪰푁 푦 and 푥′ ⪰푁′ 푦′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then (푁 ∪ 푁′, (푥, 푥′)) ⪰푁∪푁′ (푁 ∪ 푁′, (푦, 푦′)), where the ranking is strict if either of the original rankings is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' This property is a standard consistency condition in social choice studdies with variable populations Young (1974);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Smith (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Recalling the ∧-notation for members of 푋 푁, we now introduce our main concept: Sufficientarian-monotonicity: Let 푁 ∈ N and let 푥, 푦 ∈ 푋 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then (푁, 푥) ⪰푁 (푁, 푦) implies (푁, 푦) ⪰푁 (푁, 푥 ∧ 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' If 푥, 푦 are additionally ≤-comonotonic chains, then (푁, 푦) ∼푁 (푁, 푥 ∧ 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Sufficientarian-monotonicity demonstrates a core principle of sufficientarianism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' One part of it merely states that for any two profiles, the profile which results by replacing each agents consumption by the greatest lower bound of her consumption across the two profiles cannot be better than any of the original two profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The intuition for this part is straightforward: this new profile gives each agent less than she would have received from either of the original profiles, and consequently it cannot be ranked as high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The second part of the property claims that, if we control for wealth, so that the ranking of agents 10 from richest to poorest is the same across the two profiles, then the new profile must actually be deemed equivalent to the worse of the original two profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' It cannot be deemed strictly worse than the worse of the original two profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' To understand this property, suppose that we know that ⪰푁 is in fact a sufficientarian ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then if the ranking of agents from richest to poorest is the same across the two profiles, it follows that everybody who has a sufficient amount for the worse of the two profiles also has a sufficient amount for the better of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Consequently, they must also have a sufficient amount under the newly constructed profile, as the sufficient sets are closed under taking greatest lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' And everybody who did not have a sufficient amount under the worse of the original two profiles can also not have a sufficient amount under the newly constructed profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A B C D E F 20 25 30 35 40 Agents Wealth Level Society 1 Society 2 훽 Figure 1: Comparison of societies To understand this intuition, consider Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The figure considers 푋 = R+, and a society consisting of agents 푁 = {퐴, 퐵, 퐶, 퐷, 퐸, 퐹}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Any individual reaching 훽 is deemed to have a sufficient 11 amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Observe that the two profiles (labelled “Society 1” and “Society 2” here) are comonotonic, 퐴 is the wealthiest agent and 퐹 the poorest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Agents 퐷 and 퐸 have the same wealth in Society 1, but this does not contradict comonotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Observe that for society 1, only 퐴 and 퐵 reach 훽, but for society 2, 퐴, 퐵, and 퐶 all reach it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' So, according to the ranking, society 2 is deemed better than society 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Figure 1 also demonstrates the meet of the two profiles: this consists of the elements which are circled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Observe that the meet profile is also comonotonic with respect to society 1 and society 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Further, for the meet profile, only agents 퐴 and 퐵 reach 훽, in accordance with our axiom: the meet profile is socially indifferent to society 1, which was the worse of society 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The axiom is intended to be understood as a property a natural “pre-ranking” would satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' To understand what we mean, if we were to deem (푁, 푥) ∼푁 (푁, 푦), we would not necessarily argue that 푥 and 푦 are socially indifferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Rather, we would only argue that our ranking lacks the power to make any finer distinctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' However, if (푁, 푥) and (푁, 푦) are both available, and (푁, 푥) ≻푁 (푁, 푦), then we argue that (푁, 푦) should never be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The sufficientarian monotonicity property is compatible with this value judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Sufficientarian monotonicity also claims that if (푁, 푥) ∼푁 (푁, 푦), then we also should not rule out (푁, 푥 ∧ 푦) so long as 푥 and 푦 are comonotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We believe this is intuitive as well: even though 푥 ∧푦 is dominated by each of 푥 and 푦, we see no particular grounds to rule it out except for Paretian reasons: but we are not studying a Paretian ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We may wish, in fact, to retain 푥 ∧ 푦 in our choice set in order to achieve other equity objectives, or perhaps to trade off against other relevant attributes not considered by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Compare sufficientarian monotonicity to the classical notion of weak monotonicity, usually inter- preted as a Pareto property: Monotonicity: Let 푁 ∈ N and 푥, 푦 ∈ 푋 푁 such that 푥 ≥ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then (푁, 푥) ⪰푁 (푁, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Sufficientarian monotonicity is an apparently innocuous strengthening of monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' To under- stand this, witness the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 12 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For a complete binary relation ⪰ on 푋 푁, the following are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 푥 ⪰ 푦 implies 푦 ⪰ 푥 ∧ 푦 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 푥 ≥ 푦 implies 푥 ⪰ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Suppose the second of the two properties is satisfied, and let 푥 ⪰ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Because 푥 ≥ 푥 ∧ 푦, the second property implies 푥 ⪰ 푥 ∧ 푦 (this would have held also in case 푦 ≻ 푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Conversely, suppose the first property is satisfied and let 푥 ≥ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' By means of contradiction (and invoking completeness), suppose that 푦 ≻ 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then by the first property, 푥 ⪰ 푦 ∧ 푥 = 푦, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' □ Sufficientarian monotonicity therefore is a strengthening of the classical notion, whereby the strength- ening applies only to comonotonic chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Finally, we discuss a strengthening of sufficientarian monotonicity which applies to complete semi- lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Observe that for any 푁 ∈ N and any finite 퐴 ⊆ 푋 푁 and 푦 ∈ 푋 푁 for which all pairs from 퐴 ∪ {푦} are comonotonic, sufficientarian monotonicity and transitivity of ⪰푁 imply that if for all 푥 ∈ 퐴, (푁, 푥) ⪰푁 (푁, 푦), then (푁, 푦) ∼푁 (푁, 푦 ∧ � 퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='6 The following is a “continuous” analogue of sufficientarian monotonicity applying for infinite sets 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For simplicity, the following property is stated for complete meet-semilattices only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Strong sufficientarian-monotonicity: Suppose that 푋 is complete, and let ∅ ≠ 퐴 ⊆ 푋 푁, 푦 ∈ 푋 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Suppose that for all 푥 ∈ 퐴, (푁, 푥) ⪰푁 (푁, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then (푁, 푦) ⪰푁 (푁, 푦 ∧ � 퐴), in addition, (푁, 푦) ∼푁 (푁, 푦 ∧ (� 퐴)) if all pairs from 퐴 ∪ {푦} are comonotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 6This follows since for any set 퐴 ⊆ 푋 푁 for which all pairs are comonotonic, � 퐴 is comonotonic with respect to every 푥 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' So let 퐴 be such a set and let 푥 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Let 푖, 푗 ∈ 푁;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' we want to claim that (� 퐴)푖 and (� 퐴) 푗 are ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' If there is 푦 ∈ 퐴 for which 푦푖 > 푦 푗, then for every 푥 ∈ 퐴, 푥푖 ≥ 푥 푗, so that (� 퐴)푖 ≥ (� 퐴) 푗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' similarly if there is 푦 ∈ 퐴 for which 푦 푗 > 푦푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Finally if 푦푖 = 푦 푗 for all 푦 ∈ 퐴, then (� 퐴)푖 = (� 퐴) 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' To see that � 퐴 exhibits comonotonicity with respect to 푥, if 푥푖 > 푥 푗, we know that (� 퐴)푖 ≥ (� 퐴) 푗 (we just showed this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Conversely if (� 퐴)푖 > (� 퐴) 푗, then there exists 푦 ∈ 퐴 for which 푦푖 > 푦 푗 (otherwise 푦 푗 ≥ 푦푖 for all 푦 ∈ 퐴 and so (� 퐴) 푗 ≥ (� 퐴)푖, and so 푥푖 ≥ 푥 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='2 On sufficient sets and multidimensional sufficientarianism Finally, we introduce our version of sufficientarianism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Departing from previous literature on the critical- level sufficientarian threshold, we represent the “sufficient set” of agent’s consumption by a mathematical object called a filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We say that a nonempty subset F ⊆ 푋 is a filter if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For all 푥 ∈ F and all 푦 ∈ 푋, if 푥 ≤ 푦, then 푦 ∈ F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For all 푥, 푦 ∈ F , 푥 ∧ 푦 ∈ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Say that a social ranking is sufficientarian if there exists a filter F such that for all 푁 ∈ N and all 푥, 푦 ∈ 푋 푁, (푁, 푥) ⪰푁 (푁, 푦) iff |{푖 ∈ 푁 : 푥푖 ∈ F }| ≥ |{푖 ∈ 푁 : 푦푖 ∈ F }|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' This notion is a modest generalization of the notion described in previous literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='the generalization is largely technical in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' To this end, a slightly more restrictive notion in line with previous work is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A social ranking is threshold sufficientarian if there exists 훽 ∈ 푋 such that for all 푁 ∈ N and all 푥, 푦 ∈ 푋 푁, (푁, 푥) ⪰푁 (푁, 푦) iff |{푖 ∈ 푁 : 푥푖 ≥ 훽}| ≥ |{푖 ∈ 푁 : 푦푖 ≥ 훽}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Here, 훽 is called the sufficientarian threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Every threshold sufficientarian ranking is sufficientarian with filter F = {푥 : 푥 ≥ 훽}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Filters of this form are called principal filters in the mathematics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The distinction can be gleaned from the case of [0, 1] with its usual order ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Principal filters in [0, 1] are all sets of the form [훽, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' these are the ones considered in Alcantud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Non-principal filters also include sets of the form (훽, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Roughly, a non-principal filter need not be “closed below,” in an order-theoretic sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A filter is a useful model of such a sufficient set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' If 푥 obtains a sufficient level and 푦 ≥ 푥, then 푦 obtains a sufficient level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Further, and following our motivation of commodity space, if each of 푥, 푦 ∈ 푋 are sufficient, then for each of 푥 and 푦, there is a sufficient amount of each commodity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Therefore, there must be a sufficient amount of each commodity under 푥 ∧ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='3 Main result Following is our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A social ranking is sufficientarian iff it satisfies anonymity, reinforcement, and sufficientarian monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' In addition, if 푋 is complete, the ranking also satisfies strong-sufficientarian monotonicity iff it is threshold sufficientarian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' In the statement of Theorem 1, we could replace sufficientarian monotonicity with a weaker version whereby it is only required to hold for 푁 ∈ N with |푁| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Similarly, reinforcement could be replaced with the statement that if for all 푁 and all 푖 ∈ 푁, we have ({푖}, 푥푖) ⪰{푖} ({푖}, 푦푖), then (푁, 푥) ⪰푁 (푁, 푦), with a strict ranking if any initial ranking is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For many semilattices, all sufficientarian rules are threshold sufficientarian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For example, for any finite semilattice, or for our canonical example of nonnegative consumption bundles in discrete consumption space: 푋 = Z푘 + with the usual pointwise order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' To see why, say that a semilattice has the descending chain condition (DCC) if for any sequence 푥1 ≥ 푥2 ≥ 푥3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=', there is some 푛 for which for all 푚 ≥ 푛, 푥푚 = 푥푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' This property is equivalent to the property that all nonempty subsets of 푋 have an element which is not dominated below ((Davey and Priestley, 2002) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For any semilattice, all filters are principal iff DCC is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='7 So in such environments, sufficientarianism coincides with threshold sufficientarianism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 7This is well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Suppose that a semilattice does not have the DCC, and let 푥1 > 푥2 > 푥3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' be a strictly decreasing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Define F = � 푛{푥 ∈ 푋 : 푥 ≥ 푥푛} and observe F is a filter but cannot be principal: if F were principal, there would exist 훽 ∈ F for which for all 푥 ∈ F , 푥 ≥ 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' But then 훽 ∈ {푥 ∈ 푋 : 푥 ≥ 푥푛} for some 푛,and hence 훽 ≥ 푥푛 > 푥푛+1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Conversely, if all filters are principal, then for any decreasing sequence 푥1 ≥ 푥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=', again F = � 푛{푥 ∈ 푋 : 푥 ≥ 푥푛} is a filter, and there is 훽 ∈ F for which 푥 ≥ 훽 iff 푥 ∈ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Consequently, there exists 푛 such that 훽 ≥ 푥푛, and so for any 푚 ≥ 푛, 푥푚 ≥ 훽 ≥ 푥푛, so that 푥푚 = 푥푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 15 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Alternatively, suppose (푋, ≤) is a semilattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then there is another semilattice (푋′, ≤′) and a one-to-one function 푓 : 푋 → 푋′ for which 푥 ≥ 푦 iff 푓 (푥) ≥′ 푓 (푦), and 푓 (푥 ∧ 푦) = 푓 (푥) ∧′ 푓 (푦), where for every ≤-filter F ⊆ 푋 there is 훽′ ∈ 푋′ such that F = {푥 ∈ 푋 : 푓 (푥) ≥′ 훽′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' That is, any sufficientarian rule results from a threshold sufficientarian rule on a larger set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Sufficientarian rules which are not threshold sufficientarian then have a threshold in the larger set 푋′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='8 The proof relies on the following Lemma, which we believe is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='9 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For any anonymous social ranking and any 푁 ∈ N, if 휎 : 푁 → 푁 is a bijection, then for any 푥 ∈ 푋 푁, (푁, 푥) ∼푁 (푁, 푥 ◦ 휎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Let 푘 be the order of permutation 휎, so that 휎푘 = id, the identity permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Dropping the 푁 from our notation, suppose by means of contradiction and without loss of generality that (using completeness of ⪰) 푥 ≻ 푥 ◦ 휎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then by anonymity, it follows that 푥 ◦ 휎 ≻ 푥 ◦ 휎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' By induction, for all 푙 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' , 푘 − 1, 푥 ◦ 휎푙 ≻ 푥 ◦ 휎푙+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Consequently we have 푥 ≻ 푥 ◦ 휎 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' ≻ 푥 ◦ 휎푘 = 푥, a violation of transitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' □ Of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We first show that the representation is implied by the axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 8The argument is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Let 푋 ′ = 퐹(푋) denote the set of ≤ filters on 푋, ordered so that F ≥′ F ′ if F ⊆ F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Define 푓 (푥) = {푦 : 푦 ≥ 푥}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Clearly 푥 ≥ 푥′ implies that if 푦 ≥ 푥, then 푦 ≥ 푥′, so 푓 (푥) ⊆ 푓 (푥′), and so 푓 (푥) ≥′ 푓 (푥′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Similarly, if 푓 (푥) ≥′ 푓 (푥′), then 푓 (푥) ⊆ 푓 (푥′), so in particular 푥 ≥ 푥′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Now, observe that 푋 itself is a ≤-filter and that the intersection of an arbitrary collection of filters containing a given collection of filters is also a filter, hence (푋 ′, ≥′) is a complete semilattice: �′ 푥′∈퐴′ 푥′ is simply the smallest filter containing all filters 푥′ ∈ 퐴′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Further, finite meets are preserved: 푓 (푥) ∧′ 푓 (푥′) is the smallest filter containing both 푓 (푥) and 푓 (푥′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' equivalently, having each of 푥 and 푥′ as members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Observe that a filter has 푥 and 푥′ as members if and only if 푥 ∧ 푥′ is a member, so that 푓 (푥 ∧ 푥′) = 푓 (푥) ∧′ 푓 (푥′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Finally, for any filter F ⊆ 푋, it is clear that 푥 ∈ F if and only if 푓 (푥) ⊆ F , or 푓 (푥) ≥′ F , let 훽′ = F and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' It is not necessarily true that arbitrary meets are preserved under 푓 , nor is it necessarily true that all ≤′-filters are principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 9This lemma relies on transitivity, but would clearly also hold under the weaker requirement that there are no strict cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 16 First, we consider the case of any 푁 with cardinality one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' By anonymity, the ranking of members of 푋 does not depend on 푁, so we refer to it without loss of generality as ⪰, without referencing the subscript 푁 (we will do this generally when necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We claim that there are at most two equivalence classes of ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Suppose by means of contradiction that there are more than two, and let 푥, 푦, 푧 ∈ 푋 for which 푥 ≻ 푦 ≻ 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Consider any 푁′ with two agents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' by anonymity it does not matter which one we choose and we again without loss refer to the ranking as ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' When we write (푎, 푏), this references a situation giving the first agent 푎 ∈ 푋, and the second agent 푏 ∈ 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We claim that we may without loss assume that 푥 > 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' This follows as Proposition 1 demonstrates that 푎 ≥ 푏 implies 푎 ⪰ 푏 (monotonicity is satisfied), so that 푥 ≻ 푦 ≻ 푧 ≥ 푥∧푦∧푧 implies 푥 ≻ 푦 ≻ 푥∧푦∧푧, so we may replace 푧 by 푥 ∧ 푦 ∧ 푧 if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Since 푥 ≻ 푥 ∧ 푦 ∧ 푧, we know that 푥 ≠ 푥 ∧ 푦 ∧ 푧 and so 푥 > 푥 ∧ 푦 ∧ 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' So without loss, assume that 푥 > 푧 and 푥 ≻ 푦 ≻ 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Observe that the pairs (푥, 푧) and (푦, 푦) are ≤-comonotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Case 1: (푥, 푧) ⪰ (푦, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then by sufficientarian monotonicity, (푦, 푦) ∼ (푥 ∧ 푦, 푦 ∧ 푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Now, 푦 ≻ 푧 and since 푧 ≥ 푦 ∧ 푧, by monotonicity, we have 푧 ⪰ 푦 ∧ 푧, so that by transitivity, 푦 ≻ 푦 ∧ 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Further, by monotonicity, 푦 ⪰ 푥 ∧ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' By reinforcement, we thus have (푦, 푦) ≻ (푥 ∧ 푦, 푦 ∧ 푧), contradicting (푦, 푦) ∼ (푥 ∧ 푦, 푦 ∧ 푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Case 2: (푦, 푦) ≻ (푥, 푧) Then by sufficientarian-monotonicity, (푥, 푧) ∼ (푥 ∧ 푦, 푦 ∧ 푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Recall that 푥 ≻ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Further, 푦 ≥ 푥 ∧ 푦, so that 푦 ⪰ 푥 ∧ 푦 (monotonicity) whereby transitivity implies that 푥 ≻ 푥 ∧ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Further, 푧 ⪰ 푦 ∧ 푧 as 푧 ≥ 푦 ∧ 푧, so by reinforcement, we obtain (푥, 푧) ≻ (푥 ∧ 푦, 푦 ∧ 푧), a contradiction to (푥, 푧) ∼ (푥 ∧ 푦, 푦 ∧ 푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Therefore, ⪰ as restricted to 푋 has at most two indifference classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' If there is only one, it is easy to use reinforcement to show that ⪰푁 coincides with total indifference everywhere, and thus corresponds to the filter F = 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 17 Otherwise, let F denote all of the elements in the higher of the two indifference classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' so that 푥 ∼ 푦 for all 푥, 푦 ∈ F , 푧 ∼ 푤 for all 푧, 푤 ∉ F , and 푎 ≻ 푏 for all 푎 ∈ F and 푏 ∉ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We claim that F is a filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' To see this, first let 푥 ∈ F and let 푦 ≥ 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then if 푦 ∉ F , we must have 푥 ≻ 푦, which contradicts monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Now, suppose that 푥, 푦 ∈ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then 푥 ⪰ 푦, so that 푦 ∼ 푥 ∧ 푦 by sufficientarian-monotonicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' consequently, 푥 ∧ 푦 ∈ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' In case 푋 is also complete and satisfies strong sufficientarian monotonicity, let 푦 ∈ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then for all 푥 ∈ F , 푥 ⪰ 푦, so 푦 ∼ 푦 ∧ (∧F );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' conclude that 훽 ≡ ∧F ∈ F and observe that by definition, 푥 ∈ F if and only if 푥 ≥ 훽;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' hence F is a principal filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Finally, we claim that our ranking is sufficientarian with filter F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' To see this, let 푁 be arbitrary and let 푥, 푦 ∈ 푋 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' First, suppose that |{푖 ∈ 푁 : 푥푖 ∈ F }| ≥ |{푖 ∈ 푁 : 푦푖 ∈ F }| Let us without loss, by anonymity, suppose that 푁 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' , 푛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Observe that by Lemma 1, we may without loss assume that (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' , 푥푛) is such that 푥푖 ⪰ 푥푖+1 for all 푖, and similarly for 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Now, since 푥1 ⪰ 푥2 ⪰ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 푥푛 and 푦1 ⪰ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 푦푛, and by the hypothesis that |{푖 ∈ 푁 : 푥푖 ∈ F }| ≥ |{푖 ∈ 푁 : 푦푖 ∈ F }|, we may conclude for each 푖 ∈ 푁 that 푥푖 ⪰ 푦푖 (recall that F was the set of objects in the higher indifference class) and so 푥 ⪰푁 푦 follows by reinforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Likewise if |{푖 ∈ 푁 : 푥푖 ∈ F }| > |{푖 ∈ 푁 : 푦푖 ∈ F }|, then 푥 ≻푁 푦 using the same argument and the fact that there is 푖 ∈ 푁 for which 푥푖 ≻ 푦푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Now we establish that the axioms are satisfied by the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' It is trivial to see that a sufficientarian social ranking is anonymous and satisfies reinforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Verifying the first part of sufficientarian monotonicity (the part which is equivalent to monotonicity) is similarly striaghtforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' To verify that it satisfies the second part of sufficientarian monotonicity, let F be the associated filter 18 and take any 푁 ∈ N and 푥, 푦 ∈ 푋 푁 which are comonotonic ≤-chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Suppose that (푁, 푥) ⪰푁 (푁, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then |{푖 ∈ 푁 : 푥푖 ∈ F }| ≥ |{푖 ∈ 푁 : 푦푖 ∈ F }|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We claim that if 푖 ∈ 푁 is such that 푦푖 ∈ F , then 푥푖 ∈ F as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' that is, {푖 ∈ 푁 : 푦푖 ∈ F } ⊆ {푖 ∈ 푁 : 푥푖 ∈ F }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' By means of contradiction, suppose there is 푖 ∈ 푁 such that 푦푖 ∈ F and 푥푖 ∉ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Clearly, for any 푗 ∈ 푁, if 푦 푗 ≥ 푦푖, then 푦 푗 ∈ F as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' and for any 푘 ∈ 푁, if 푥푘 ≤ 푥푖, then 푥푘 ∉ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Therefore we must conclude that for any 푗 ∈ 푁 \\ {푖}, if 푥 푗 ∈ F , it follows that 푥 푗 > 푥푖 (by the fact that 푥 is an ≤-chain), from which we conclude by comonotonicity that 푦 푗 ≥ 푦푖, so that 푦 푗 ∈ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' This implies that { 푗 : 푥 푗 ∈ F } ⊆ { 푗 : 푦 푗 ∈ F } \\ {푖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Since 푦푖 ∈ F , this contradicts the hypothesis that |{푖 ∈ 푁 : 푥푖 ∈ F }| ≥ |{푖 ∈ 푁 : 푦푖 ∈ F }|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The result now follows, as whenever 푦푖 ∈ F , we have 푥푖 ∈ F , and thus by the filter property (푥푖 ∧ 푦푖) ∈ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Similarly, if 푦푖 ∉ F , (푥푖 ∧ 푦푖) ∉ F as (푥푖 ∧ 푦푖) ≤ 푦푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' So, (푥푖 ∧ 푦푖) ∈ F if and only if 푦푖 ∈ F , from which we conclude that (푁, 푦) ∼푁 (푁, 푥 ∧ 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For the case of a threshold sufficientarian rule on a complete semilattice, observe that the same argument allows us to establish that if (푁, 푥) ⪰푁 (푁, 푦) for all 푥 ∈ 퐴, then for all 푥 ∈ 퐴, {푖 ∈ 푁 : 푦푖 ≥ 훽} ⊆ {푖 ∈ 푁 : 푥푖 ≥ 훽}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Consequently, for any 푖 ∈ 푁, 푦푖 ≥ 훽 implies ∧푥∈퐴푥푖 ≥ 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' The argument then parallels that of the filter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='4 Independence of the axioms We illustrate by examples that the three axioms are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' These examples will illustrate the general structure of rules satisfying these axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Anonymity Start with dropping anonymity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Many rules satisfy both reinforcement and sufficientarian monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 19 Start with an obvious generalization: Say that a rule is weighted sufficientarian if there is a function 휆 : 푀 → R+ and a filter F ⊆ 푋 such that for all 푁 ∈ N, and all 푥, 푦 ∈ 푋 푁, 푥 ⪰푁 푦 if and only if � 푖∈푁 휆푖1{푥푖∈F } ≥ � 푖∈푁 휆푖1{푦푖∈F }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content='10 A related generalization, which would feature each individual possessing their own filter F푖, generally fails sufficientarian monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Another class of solutions would are lexicographic dictatorships, consisting of a linear order on 푀 and for each 푖 ∈ 푀, a relation ⪰푖 for which ≥⊆⪰푖, so that for any 푁 ∈ N, and any 푥, 푦 ∈ 푋 푁, 푥 ⪰푁 푦 if and only if 푥푖 ⪰푖 푦푖 for the highest priority 푖 ∈ 푁 according to the linear order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Still other, hybrid, variations of these two methods would satisfy the remaining axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We leave this to future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Reinforcment Rules violating reinforcement are similarly many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A particularly canonical example would obtain in the case of 푋 = [0, 1], whereby for all 푁 ∈ N and all 푥, 푦 ∈ 푋 푁, 푥 ⪰푁 푦 if and only if min푖 푥푖 ≥ min푖 푦푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For a more general set 푋, one would construct a ≤-chain C with the property that for every 푥 ∈ 푋, there is a uniquely ≤-maximal element of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then define, for any 푁 ∈ N and 푥, 푦 ∈ 푋 푁, 푥 ⪰푁 푦 if and only if said maximal element for � 푖∈푁 푥푖 ≤-dominates the maximal element for � 푖∈푁 푦푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' However, even in the case of 푋 = [0, 1], many other rules are easy to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For any 푁 ∈ N and 푥 ∈ 푋 푁, and any 푛 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' , |푁|}, let 푥∗(푛) denote the 푛-th highest value of 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Then define 푈푁 : 푋 푁 → R as 푈푁(푥) = inf푛 푛푥∗(푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For any 푥, 푦 ∈ 푋 푁, assign 푥 ⪰푁 푦 if and only if 푈푁(푥) ≥ 푈푁(푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We leave the study of this class of rules to future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Sufficientarian Monotonicity Rules violating sufficientarian monotonicity constitute much of social choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A particularly interesting one is the trigonometry rule on 푋 = [0, 1], whereby for any 푁 ∈ N and 푥, 푦 ∈ 푋 푁, 푥 ⪰푁 푦 if and only if 10Or just an ordinal “qualitative measure” ranking the sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 20 � 푖∈푁 sin(cos(푥푖)) ≥ � 푖∈푁 sin(cos(푦푖)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' As far as we are aware, this rule is novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 3 Conclusion In this work, we have studied the concept of sufficientarianism, applied to a multidimensional framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Sufficientarianism is to be understood as a “pre-ranking,” imposed to rule out profiles that should never be chosen in the presence of other feasible profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' There are several possible extensions for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' As a first point, though our work is technically a variable population work, we only ever consider a fixed population in comparing two profiles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' we are essentially doing head-counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A work that allows one to consider multidimensional profiles across different populations would be similarly interesting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' see, for example, Bossert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' (2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Second, sufficientarianism lends itself to a natural cardinal ranking, whereby we only count the number of people who reach a given threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' One could consider imposing a random threshold and studying the expected number of people to reach the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' In the case of 푋 = R+, we could clearly end up with the set of all (linear) rankings satisfying a weak form of monotonicity in doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' But in higher dimensional environments, we simply have not studied what this approach would generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' It is conceivable that the sufficientarian rankings generate as their expectation some type of supermodular and increasing rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We leave this to future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Of course, weakening axioms, especially that of completeness of the social ranking, might prove interesting as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Finally, we have motivated our concept of sufficientarianism as a kind of “pre-ranking,” to be imposed prior to other, more finely-grained rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' We do not know what happens when taking this literally, and imposing a Paretian ranking “on top” of a sufficientarian ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' For example, given a sufficientarian 21 ranking ⪰ and a Paretian ranking ⪰∗, this approach suggests ruling out any (푁, 푥) for which either there exists a feasible 푦 for which (푁, 푦) ≻ (푁, 푥) or for which there exists a feasible 푦 for which (푁, 푦) ∼ (푁, 푥) and (푁, 푦) ≻∗ (푁, 푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Again, we leave this to future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 22 References Alcantud, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Liberal approaches to ranking infinite utility streams: when can we avoid interference?' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' A note on preference aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' Econometrica 42(6), 1129–1131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFAT4oBgHgl3EQftx6O/content/2301.08666v1.pdf'} diff --git a/J9E1T4oBgHgl3EQfGQOR/content/tmp_files/2301.02912v1.pdf.txt b/J9E1T4oBgHgl3EQfGQOR/content/tmp_files/2301.02912v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..72de11d7809bd6a0022135760c8deb4fa414ee8e --- /dev/null +++ b/J9E1T4oBgHgl3EQfGQOR/content/tmp_files/2301.02912v1.pdf.txt @@ -0,0 +1,695 @@ +arXiv:2301.02912v1 [q-fin.MF] 7 Jan 2023 +HEDGING OF EUROPEAN TYPE CONTINGENT CLAIMS IN +DISCRETE TIME BINOMIAL MARKET MODELS +JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA +Abstract. We consider a discrete-time binomial model of a market consisting of m ≥ 1 +risky securities and one bond. For a European type contingent claim we give an explicit +formula for the minimum-cost maximal hedging strategy. +1. The main result +In this note we consider a discrete-time binomail model for a market with m risky securities +S1, . . . , Sm and one bond S0 with return R > 0. Time has values k = 0, . . . , n, and we write +Si(k) for the price of the i-th security at time k. The model comes with a choice of numbers +0 < Di < R < Ui for each 1 ≤ i ≤ m. To describe the random process of the values of Si, +suppose that the prices of S0, . . . , Sm are known at time k < n. Their values at time k + 1 +is determined as follows. +(a) For the bond process, +S0(k + 1) = S0(k) · R. +(b) For the remaining securities, flip m coins and according to the results set +Si(k + 1) = Si(k) · Ui +or +Si(k + 1) = Si(k) · Di +The coins are not assumed to be independent, nor do the flips at time k and time k′ ̸= k. +We consider a European contingent claim X with pay-off at time n (of maturity) given by +(1.1) +F = +� m +� +i=0 +γiSi(n) − K +�+ +, +where γ1, . . . , γn ≥ 0, K ≥ 0 and x+ def += max{x, 0} for any real number x. +It is known that the set of rational values of X at time k, (i.e its no-arbitrage price range at +time k, forms an open interval whose upper bound we denote by Cmax(X, k). In [1, Section +6A eqns. (6.1) and (6.2)] we have shown that Cmax(X, k) can be expressed solely by means +of the prices of S0, . . . , Sm at time k and the parameters of the model (see Proposition 2.6 +below). +A minimum cost maximal hedging strategy for X consist of a choice, at each time +k = 0, . . . , n − 1, of numbers α0(k), . . . , αm(k) which minimize (the cost of the hedging +1 + +portfolio) +(1.2) +Vα(k) = +m +� +i=0 +αi(k)Si(k) +subject to the (maximal-hedging) condition that at time k + 1 the value of this portfolio +satisfies +(1.3) +m +� +i=0 +αi(k) · Si(k + 1) ≥ Cmax(X, k + 1). +In particular the value of the portfolio Vα(k) acquired at time k is guaranteed to exceed the +value of the option X at time k + 1. Notice that the values chosen for αi(k) depend on the +“state of the world” at time k, and in particular on the prices of S0, . . . , Sm at time k. In [1, +Proposition 4.2] we showed that a minimum cost maximal hedging strategy exists and that +its set up cost at each time k is exactly Cmax(X, k), namely the maximal rational value of X +at time k. +The purpose of this note is to give an explicit formula for the values of α0(k), . . . , αm(k). In +the remainder of this section we describe this formula. +For every 1 ≤ i ≤ m set +(1.4) +bi = R − Di +Ui − Di +. +If necessary, reorder the securities S1, . . . , Sm so that b1, . . . , bm is non-increasing, namely +(1.5) +b1 ≥ b2 ≥ · · · ≥ bm. +Notice that 0 < bi < 1 for all i. Define for any 0 ≤ j ≤ m +(1.6) +qj = + + + +1 − b1 +j = 0 +bj − bj+1 +1 ≤ j ≤ m − 1 +bm +j = m. +Define for 0 ≤ i ≤ m and for 0 ≤ j ≤ m numbers χi(j) as follows +(1.7) +χ0(j) = R +and +χi(j) = +� +Ui +i ≤ j +Di +i > j. +We denote +Pk(m) = {0, . . . , m}k = {(j1, . . . , jk) : 0 ≤ ji ≤ m}. +For any J ∈ Pk(m) set +qJ = +� +j∈J +qj +and +χi(J) = +� +j∈J +χi(j). +Assume that at time 0 ≤ k ≤ n − 1, the prices S0(k), . . . , Si(m) are known. Define for any +0 ≤ j ≤ m +(1.8) +Yj(k) = Rk+1−n +� +J∈Pn−k−1(m) +qJ +� m +� +i=0 +γiχi(J)χi(j)Si(k) − K +�+ +. +2 + +Finally, define the following (m + 1) × (m + 1) matrices +Q = + + +1 +0 +0 +· · · · · · +0 +0 +−1 +1 +0 +· · · · · · +0 +0 +0 +−1 +1 +· · · · · · +0 +0 +... +... +... +· · · · · · +... +0 +0 +0 +· · · · · · +1 +0 +0 +0 +0 +· · · · · · +−1 +1 + + +N = + + +1 +D1 +D2 +· · · +Dm +0 +U1 − D1 +0 +. . . +0 +0 +0 +U2 − D2 +... +... +... +... +0 +0 +. . . +Um − Dm + + +T = + + +RS0(k) +S1(k) +... +Sm(k) + + +Observe that the matrix N only depend on the parameters of the model, and is easily seen +to be invertible. Only the matrix T depends on the state of the world at time k, and since +Si(k) > 0 it is clearly invertible. Also, Q is invertible. +Theorem 1.1. With the set up and notation above, at time 0 ≤ k ≤ n − 1 the portfolio +Vα(k) = �m +i=0 αi(k) · Si(k) of a minimum cost maximal hedge (1.2), (1.3) for a European +contingent claim X in (1.1) is given by + + +α0(k) +α1(k) +... +αm(k) + + = T −1 · N−1 · Q · + + +Y0(k) +Y1(k) +... +Ym(k) + + +Moreover, Vα(k) = Cmax(F, k). +REMARK: The formula given for Yj(k) has computational complexity O((m + 1)n−k) (the +number of terms in the sum). This is exponential in n. As discussed in [1], the action of the +symmetric group Sn−k−1 on Pn−k−1(m) gives a formula for Yj(k) whose complexity is only +O((n − k)m+1), polynomial in n. +REMARK: As is the case in [1], the function h(x) = x+ can be replaced with any convex +function. Thus, our results apply to several contingent claims other than European basket +call options. The reader is referred to [1] for details. +2. Formalisation of the model and proof of the main result +2.1. Single time-step. Each step of the model consists of flipping m coins. +A natural +sample space for this experiment is the set +L = {0, 1}m +3 + +of all the sequences of length m consisting of 0’s and 1’s. We denote its elements by λ = +(λ(1), . . . , λ(m)) and view L as a subset of Rm. Let ℓi denote the (random variable of the) +result of the i-th coin, namely ℓi : L → R is the projection to the i-th factor: +ℓi : λ �→ λ(i). +Observe that ℓi is the restriction to L of the linear projection function πi : Rm → R. +Let ψi be the “price jump” of the i-th security. One checks that ψ0 = R and that ψi(λ) = +Di + (Ui − Di)λ(i) for 1 ≤ i ≤ m, namely +(2.1) +ψ0 = R +and +ψi = Di + (Ui − Di)ℓi. +Thus, for every 1 ≤ i ≤ m, ψi is the restriction to L of an affine function fi : Rm → R where +fi(x1, . . . , xm) = Di + (Ui − Di)xi. +Let us consider probability measures on L on the σ-algebra ℘(L) of all the subsets of L. +These are equivalent to probability density functions p: L → R and we will abuse notation +and write p for both the density function and the probability measure it induces. +The +requirement that p is risk-neural in a single-step model is the condition +Ep(ψi) = R, +(1 ≤ i ≤ m). +By the linearity of the expectation and the definition (1.4), this is equivalent to +Ep(ℓi) = bi. +Throughout we assume that b1, . . . , bm is non-increasing (1.5). +Definition 2.1. Let P(L, b) denote the set of all probability density functions p: L → R +such that Ep(ℓi) = bi for all 1 ≤ i ≤ m. +Define ρ0, . . . ρm ∈ L as follows +(2.2) +ρj = (1, . . . , 1 +� �� � +j times +, 0, . . . , 0). +Thus, ρj describes the event of a run of j heads followed by a run of n − j tails. Use (1.6) +to define a probability density function q: L → R by +(2.3) +q(λ) = +� +qj +if λ = ρj for some 0 ≤ j ≤ m +0 +otherwise +We call q the upper supermodular vertex of P(L, b). +Compare with [1, Appendix A +equation (A.6)] where we denoted ρj by µj. One checks, see [1, Appendix A], that indeed +q ∈ P(L, b). In particular +(2.4) +Eq(ψi) = Di + (Ui − Di)Eq(ℓi) = Di + (Ui − Di)bi = R. +There is a canonical bijection of L with the set ℘({1, . . . , m}). This gives rise to a partial +order ⪯ on L induced by the partial order ⊆ on ℘({1, . . . , m}). Thus, +(2.5) +λ ⪯ λ′ ⇐⇒ supp(λ) ⊆ supp(λ′). +4 + +Union and intersection of sets render L a lattice with join ∨ and meet ∧, +λ ∨ λ′ += +(max{λ(1), λ′(1)}, . . . , max{λ(m), λ′(m)}) +(2.6) +λ ∧ λ′ += +(min{λ(1), λ′(1)}, . . . , min{λ(m), λ′(m)}) +The concept of submodular functions is originally due to Lovász in [2]. In this note, its +variant, supermodular functions [3], is used. +Definition 2.2. A function f : L → R is called supermodular if for any λ, λ′ ∈ L +f(λ ∨ λ′) + f(λ ∧ λ′) ≥ f(λ) + f(λ′). +It is called modular if equality holds. +Proposition 2.3. +(i) A linear combination with non-negative coefficients of supermodular +functions is supermodular. +(ii) Let g : Rm → R be an affine function. Then g|L is modular. +(iii) Let g : Rm → R be an affine function of the form g(x1, . . . , xm) = �m +i=1 aixi + b where +a1, . . . , am ≥ 0. If h: R → R is convex then the restriction of h◦g to L is supermodular. +Proof. Part (i) is [3, Proposition 2.2.5(a)]. Part (ii) follows from [3, Theorem 2.2.3] (and is +straightforward). Part (iii) follows from [3, Theorem 2.2.6(a)]. +□ +The crucial property of the upper supermodlar vertex q (2.3) is given by the following result. +Proposition 2.4 ([1, Theorem A.5(i)]). Let f : L → R be supermodular. Let q be the upper +supermodular vertex in P(L, b). Then +sup +p∈P (L,b) +Ep(f) = +max +p∈P (L,b) Ep(f) = Eq(f). +2.2. Multi time-step model. For the n-step model one performs n iterations (not neces- +sarily independent) of the experiment of flipping m coins. Thus, the natural sample space +for an n-step discrete time binomial market model is Ln. We equip it with the σ-algebra +F = ℘(Ln) of all subsets of Ln. +The “state of the world” at time 0 ≤ k ≤ n is described by a k-tuple (λ1, . . . , λk) ∈ Lk. +Thus, the set of the states of the world at time k is naturally identified with Lk. We obtain +a partition {ω × Ln−k : ω ∈ Lk} of Ln which generates a sub-σ-algebra Fk. +The price jump of the i-th security at time 1 ≤ k ≤ n, where 1 ≤ i ≤ m, is the random +variable Ψi(k): Ln → R +Ψi(k): (λ1, . . . , λn) �→ ψi(λk). +Of course, Ψ0(k): Ln → R is the constant function (random variable) with value R. +The random process Si(0), . . . , Si(n) of the prices of the i-th security at time 0 ≤ k ≤ n are +random variables Si(k): Ln → R. Clearly, they are given by +(2.7) +Si(k) = Si(0) · Ψi(1) · · ·Ψi(k) +5 + +Where Si(0) > 0 are constant (the initial prices of the securities at time 0). Clearly, the value +of Si(k) at ω ∈ Ln depend only on the first k entries of ω. Hence, Si(k) are Fk-measurable +random variables, namely their values only depend on the state of the world at time k. We +will therefore abuse notation and regard Si(k) as functions with domain Lk. +We now fix γ0, . . . , γm where γi ≥ 0 for all 1 ≤ i ≤ m and fix some K and set +(2.8) +F = +� m +� +i=0 +γiSi(n) − K +�+ +. +This random variable is the pay-off of the European contingent claim which is the subject +of study of this note. +Recall the elements ρj ∈ L from (2.2). Observe that by definition of ψi (2.1) and of χi (1.7) +we have +(2.9) +ψi(ρj) = χi(j), +(0 ≤ i, j ≤ m). +Proposition 2.5. Consider some ω = (λ1, . . . , λk) ∈ Lk, a state of the world at time +0 ≤ k ≤ n, and some J = (j1, . . . , jn−k) ∈ Pn−k(m). Set τ = (ρj1, . . . , ρjn−k) ∈ Ln−k. Then +F(ωτ) = +� m +� +i=0 +γiSi(k)(ω)χi(J) − K +�+ +. +Proof. By the definition of Si(n) (2.7) +F(ωτ) = +� m +� +i=0 +γiSi(0) · +n +� +p=1 +Ψi(p)(ωτ) − K +�+ += +� m +� +i=0 +γiSi(0) · +k +� +p=1 +ψi(λp) · +n +� +p=k+1 +ψi(ρjp) − K +�+ += +� m +� +i=0 +γiSi(k)(ω) · χi(J) − K +�+ +. +□ +For every 0 ≤ k ≤ n we will denote by +Cmax(F, k) +and +Cmin(F, k) +the upper and lower bounds of the rational values of F at time k. Of course, these numbers +depend only on the state of the world at time k, and so Cmax / min(F, k) are Fk-measurable +random variables (on Ln). +Proposition 2.6. With the pay-off F (2.8) of European basket call, at time 0 ≤ k ≤ n +Cmax(F, k) = Rk−n +� +J∈Pn−k(m) +qJ +� m +� +i=0 +γiSi(k)χi(J) − K +�+ +. +6 + +Proof. This is an immediate consequence of [1, Example 7.3 and Section 6A eqns. (6.1) and +(6.2)] and Proposition 2.5. We note that in [1] the elements ρj ∈ L are denoted µj and +Pn−k(m) is denoted In−k. +□ +Notice that Cmax(F, k) is an Fk measurable random variable on Ln. Hence, it will be conve- +nient to think of it as a function with domain Lk. +Proposition 2.7. Consider some 0 ≤ k ≤ n − 1 and some ω ∈ Lk representing the state of +the world at time k. Then the function f : L → R defined by +f(λ) = Cmax(F, k + 1)(ωλ) +is supermodular. Moreover, with respect to the upper supermodular vertex (2.3) +Eq(f) = R · Cmax(F, k)(ω). +Proof. It follows from Proposition 2.6 and since Si(k + 1) = Si(k) · Ψi(k + 1) that +f(λ) = Rk+1−n +� +J∈Pn−k−1(m) +qJ +� m +� +i=0 +γiSi(k)(ω) · ψi(λ) · χi(J) − K +�+ +. +Proposition 2.3(i),(iii) implies that f is supermodular since h(x) = x+ is convex and since +ψi = Di + (Ui − Di)ℓi is an affine function with non-negative coefficients and since R > 0 +and γi, Si(k), χi(J) ≥ 0 for all 1 ≤ i ≤ m. Moreover, by (2.9) +Eq(f) = +� +λ∈L +q(λ)f(λ) += +m +� +j=0 +q(ρj)f(ρj) += +m +� +j=0 +qj +� +J∈Pn−k−1(m) +qJRk+1−n +� m +� +i=0 +γiSi(k)(ω) · ψi(ρj) · χi(J) − K +�+ += +m +� +j=0 +qj +� +J∈Pn−k−1(m) +qJRk+1−n +� m +� +i=0 +γiSi(k)(ω) · χi(J)χi(j) − K +�+ += R · Rk−n +� +J∈Pn−k(m) +qJ +� m +� +i=0 +γiSi(k)(ω) · χi(J) − K +�+ += R · Cmax(F, k)(ω). +□ +Consider some 0 ≤ k ≤ n − 1 and recall Yj(k) from (1.8) where 0 ≤ j ≤ m. Observe that +Yj(k) is a function of the random variables Si(k), so Yj(k) is a random variable (on Ln) whose +values depend only on the state of the world at time k, namely Yj(k) is Fk-measurable. +7 + +Proposition 2.8. For any 0 ≤ j ≤ m and any ω = (λ1, . . . , λk) ∈ Lk +Cmax(F, k + 1)(ωρj) = Yj(ω). +Proof. This follows immediately from Propositions 2.6 and equation (2.9). +□ +Proof of Theorem 1.1. Fix some 0 ≤ k ≤ n−1 and some state of the world ω = (λ1, . . . , λk) ∈ +Lk at time k. Any subsequent state of the world at time k+1 has the form ωλ for λ ∈ L. Our +goal is to find numbers α0(k)(ω), . . . , αm(k)(ω), which for simplicity we denote by α0, . . . , αm, +which fulfil the inequality (1.3), namely for every λ ∈ L +m +� +i=0 +αiSi(k + 1)(ωλ) ≥ Cmax(F, k + 1)(ωλ) +and which minimize +(2.10) +Vα(k)(ω) = +m +� +i=0 +αiSi(k)(ω). +We rewrite the first inequality as a set of inequalities (indexed by λ ∈ L) +(2.11) +m +� +i=0 +αiSi(k)(ω) · ψi(λ) +� +�� +� +Φα(λ) +≥ Cmax(F, k + 1)(ωλ) +� +�� +� +Ξ(λ) +(λ ∈ L). +We obtain two functions Φα : L → R and Ξ: L → R, and (2.11) is the inequality +Φα ≥ Ξ. +The first step of the proof is to show that the following system of m + 1 linear equations +with the m + 1 unknowns α0, . . . , αm has a unique solution +m +� +i=0 +αiSi(k)(ω) · ψi(ρj) = Cmax(F, k + 1)(ωρj), +(0 ≤ j ≤ m). +Notice that these equations are obtained by imposing equalities in the inequalities (2.11) for +λ = ρ0, . . . , ρm. By (2.9) and by Proposition 2.8, this is the system of equations +(2.12) +m +� +i=0 +αiSi(k)(ω) · χi(j) = Yj(ω), +(0 ≤ j ≤ m). +Write α• for the column vector (α0, . . . , αm) and Y• for the column vector (Y0(ω), . . . , Ym(ω)). +Then this system of m + 1 linear equations is Mα• = Y• where M is the (m + 1) × (m + 1) +matrix +(Mj,i) = (Si(k)(ω) · χi(j)) = + + +1 +χ1(0) +· · · +χm(0) +1 +χ1(1) +· · · +χm(1) +... +... +... +1 +χ1(m) +· · · +χm(m) + + +� +�� +� +M′ +· + + +RS0(k)(ω) +S1(k)(ω) +... +Sm(k)(ω) + + +� +�� +� +T +. +8 + +Clearly, T is invertible since R, Si(k)(ω) > 0. Observe that for any 1 ≤ j ≤ m and any +0 ≤ i ≤ m − 1 +χj(i) − χj(i + 1) = +� +0 +i ̸= j +Ui − Di +i = j +By inspection we get +Q · M′ = + + +1 +D1 +D2 +· · · +Dm +0 +U1 − D1 +0 +· · · +0 +0 +0 +U2 − D2 +· · · +0 +... +... +0 +0 +0 +· · · +Um − Dm + + +� +�� +� +N +. +Clearly, Q and N are invertible, hence so is M′. It follows that M = M′T is invertible, so +the system (2.12) has a unique solution given by +α• = T −1N−1Q · Y•. +This gives the values of αi(k)(ω) stated in the theorem. It remains to show that these αi +solve all the inequalities in (2.11) (one for each λ ∈ L) and minimize (2.10) and Vα(k)(ω) = +Cmax(F, k)(ω). +Claim 1: α0, . . . , αm solve the inequalities (2.11). +Proof: Suppose that not all these inequalities are solved, namely Φα(λ) > Ξ(λ) for some +λ ∈ L. Since α0, . . . , αm solve the equations (2.12), we certainly get Φα(ρi) = Ξ(ρi) for all +0 ≤ i ≤ m. +Among all λ ∈ L for which Φ(λ) > Ξ(λ) choose one with maximal possible j such that +ρj ⪯ λ (2.5). Observe that j < m because if j = m then ρm ⪯ λ implies that λ = ρm which +we have seen is impossible. Set λ′ = λ ∨ ρj+1 (2.6). By the maximality of j we get that +λ ∧ ρj+1 = ρj. Observe that Φα is an affine function, hence it is modular by Proposition +2.3(ii), so +Φ(λ′) + Φ(ρj) = Φ(λ) + Φ(ρj+1). +By Proposition 2.7 Ξ is supermodular, so +Ξ(λ′) + Ξ(ρj) ≥ Ξ(λ) + Ξ(ρj+1). +Subtracting the first equality from the second inequality, and recalling that Φ(ρi) = Ξ(ρi) +for all i, we get +Ξ(λ′) − Φ(λ′) ≥ Ξ(λ) − Ψ(λ) > 0. +So Φ(λ′) > Ξ(λ′) and ρj+1 ⪯ λ′ which contradicts the maximality of j. +q.e.d +Recall that any β0, . . . , βm define Vβ(k)(ω) in (2.10). +Claim 2: For any β0, . . . , βm for which the inequalities (2.11) hold, +Vβ(k)(ω) ≥ Cmax(F, k)(ω). +In addition, α0, . . . , αm attain this lower bound, namely +Vα(k)(ω) = Cmax(F, k)(ω). +9 + +Proof: Suppose βi solve the inequalities (2.11). It follows from (2.4) that +(2.13) +Eq(Φβ) = +m +� +i=0 +βiSi(k)(ω) · Eq(ψi) = R · +m +� +i=0 +βiSi(k)(ω) = RVβ(k)(ω). +Proposition 2.7 implies that Eq(Ξ) = RCmax(F, k)(ω). Since βi solve the inequalities (2.11), +this means Φβ ≥ Ξ. By the monotonicity of the expectation, Eq(Φβ) ≥ Eq(Ξ), and since +R > 0, it follows that Vβ(k)(ω) ≥ Cmax(F, k)(ω) as needed. +By construction Φα(ρj) = Ξ(ρj) for all j = 0, . . . , m. Since q is supported on ρ0, . . . , ρm, we +deduce from (2.13) that +RVα(k)(ω) = Eq(Φα) = Eq(Ξ) = R · Cmax(F, k)(ω). +Hence Vα(k)(ω) = Cmax(F, k)(ω). +q.e.d +The theorem follows from Claims 1 and 2. +□ +References +[1] Jarek Kędra, Assaf Libman, and Victoria Steblovskaya. Pricing and hedging contingent claims in a +multi-asset binomial market. arXiv:2106.13283, 2021. +[2] L. Lovász. Submodular functions and convexity. In Mathematical programming: the state of the +art (Bonn, 1982), pages 235–257. Springer, Berlin, 1983. +[3] David Simchi-Levi, Xin Chen, and Julien Bramel. The logic of logistics. Springer Series in Operations +Research and Financial Engineering. Springer, New York, third edition, 2014. Theory, algorithms, and +applications for logistics management. +University of Aberdeen and University of Szczecin +Email address: kedra@abdn.ac.uk +University of Aberdeen +Email address: a.libman@abdn.ac.uk +Bentley University +Email address: vsteblovskay@bentley.edu +10 + diff --git a/J9E1T4oBgHgl3EQfGQOR/content/tmp_files/load_file.txt b/J9E1T4oBgHgl3EQfGQOR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f5973d9b4d9528a4a4bdc384c0f2928a6b1b2d2a --- /dev/null +++ b/J9E1T4oBgHgl3EQfGQOR/content/tmp_files/load_file.txt @@ -0,0 +1,479 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf,len=478 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='02912v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='MF] 7 Jan 2023 HEDGING OF EUROPEAN TYPE CONTINGENT CLAIMS IN DISCRETE TIME BINOMIAL MARKET MODELS JAREK KĘDRA, ASSAF LIBMAN, AND VICTORIA STEBLOVSKAYA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' We consider a discrete-time binomial model of a market consisting of m ≥ 1 risky securities and one bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' For a European type contingent claim we give an explicit formula for the minimum-cost maximal hedging strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' The main result In this note we consider a discrete-time binomail model for a market with m risky securities S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , Sm and one bond S0 with return R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Time has values k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , n, and we write Si(k) for the price of the i-th security at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' The model comes with a choice of numbers 0 < Di < R < Ui for each 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' To describe the random process of the values of Si, suppose that the prices of S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , Sm are known at time k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Their values at time k + 1 is determined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' (a) For the bond process, S0(k + 1) = S0(k) · R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' (b) For the remaining securities, flip m coins and according to the results set Si(k + 1) = Si(k) · Ui or Si(k + 1) = Si(k) · Di The coins are not assumed to be independent, nor do the flips at time k and time k′ ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' We consider a European contingent claim X with pay-off at time n (of maturity) given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='1) F = � m � i=0 γiSi(n) − K �+ , where γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , γn ≥ 0, K ≥ 0 and x+ def = max{x, 0} for any real number x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' It is known that the set of rational values of X at time k, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='e its no-arbitrage price range at time k, forms an open interval whose upper bound we denote by Cmax(X, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' In [1, Section 6A eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2)] we have shown that Cmax(X, k) can be expressed solely by means of the prices of S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , Sm at time k and the parameters of the model (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='6 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' A minimum cost maximal hedging strategy for X consist of a choice, at each time k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , n − 1, of numbers α0(k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , αm(k) which minimize (the cost of the hedging 1 portfolio) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2) Vα(k) = m � i=0 αi(k)Si(k) subject to the (maximal-hedging) condition that at time k + 1 the value of this portfolio satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='3) m � i=0 αi(k) · Si(k + 1) ≥ Cmax(X, k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' In particular the value of the portfolio Vα(k) acquired at time k is guaranteed to exceed the value of the option X at time k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Notice that the values chosen for αi(k) depend on the “state of the world” at time k, and in particular on the prices of S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , Sm at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' In [1, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2] we showed that a minimum cost maximal hedging strategy exists and that its set up cost at each time k is exactly Cmax(X, k), namely the maximal rational value of X at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' The purpose of this note is to give an explicit formula for the values of α0(k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , αm(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' In the remainder of this section we describe this formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' For every 1 ≤ i ≤ m set (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='4) bi = R − Di Ui − Di .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' If necessary, reorder the securities S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , Sm so that b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , bm is non-increasing, namely (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='5) b1 ≥ b2 ≥ · · · ≥ bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Notice that 0 < bi < 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Define for any 0 ≤ j ≤ m (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='6) qj = \uf8f1 \uf8f2 \uf8f3 1 − b1 j = 0 bj − bj+1 1 ≤ j ≤ m − 1 bm j = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Define for 0 ≤ i ≤ m and for 0 ≤ j ≤ m numbers χi(j) as follows (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='7) χ0(j) = R and χi(j) = � Ui i ≤ j Di i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' We denote Pk(m) = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , m}k = {(j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , jk) : 0 ≤ ji ≤ m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' For any J ∈ Pk(m) set qJ = � j∈J qj and χi(J) = � j∈J χi(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Assume that at time 0 ≤ k ≤ n − 1, the prices S0(k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , Si(m) are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Define for any 0 ≤ j ≤ m (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='8) Yj(k) = Rk+1−n � J∈Pn−k−1(m) qJ � m � i=0 γiχi(J)χi(j)Si(k) − K �+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 2 Finally, define the following (m + 1) × (m + 1) matrices Q = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 0 0 · · · · · 0 0 −1 1 0 · · · · · 0 0 0 −1 1 · · · · · 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' · · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 0 0 0 · · · · · 1 0 0 0 0 · · · · · −1 1 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb N = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 D1 D2 · · Dm 0 U1 − D1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 0 0 0 U2 − D2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Um − Dm \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb T = \uf8ee \uf8ef\uf8ef\uf8f0 RS0(k) S1(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Sm(k) \uf8f9 \uf8fa\uf8fa\uf8fb Observe that the matrix N only depend on the parameters of the model, and is easily seen to be invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Only the matrix T depends on the state of the world at time k, and since Si(k) > 0 it is clearly invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Also, Q is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' With the set up and notation above, at time 0 ≤ k ≤ n − 1 the portfolio Vα(k) = �m i=0 αi(k) · Si(k) of a minimum cost maximal hedge (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='3) for a European contingent claim X in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='1) is given by \uf8ee \uf8ef\uf8ef\uf8f0 α0(k) α1(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' αm(k) \uf8f9 \uf8fa\uf8fa\uf8fb = T −1 · N−1 · Q · \uf8ee \uf8ef\uf8ef\uf8f0 Y0(k) Y1(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Ym(k) \uf8f9 \uf8fa\uf8fa\uf8fb Moreover, Vα(k) = Cmax(F, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' REMARK: The formula given for Yj(k) has computational complexity O((m + 1)n−k) (the number of terms in the sum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' This is exponential in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' As discussed in [1], the action of the symmetric group Sn−k−1 on Pn−k−1(m) gives a formula for Yj(k) whose complexity is only O((n − k)m+1), polynomial in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' REMARK: As is the case in [1], the function h(x) = x+ can be replaced with any convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Thus, our results apply to several contingent claims other than European basket call options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' The reader is referred to [1] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Formalisation of the model and proof of the main result 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Single time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Each step of the model consists of flipping m coins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' A natural sample space for this experiment is the set L = {0, 1}m 3 of all the sequences of length m consisting of 0’s and 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' We denote its elements by λ = (λ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , λ(m)) and view L as a subset of Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Let ℓi denote the (random variable of the) result of the i-th coin, namely ℓi : L → R is the projection to the i-th factor: ℓi : λ �→ λ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Observe that ℓi is the restriction to L of the linear projection function πi : Rm → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Let ψi be the “price jump” of the i-th security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' One checks that ψ0 = R and that ψi(λ) = Di + (Ui − Di)λ(i) for 1 ≤ i ≤ m, namely (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='1) ψ0 = R and ψi = Di + (Ui − Di)ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Thus, for every 1 ≤ i ≤ m, ψi is the restriction to L of an affine function fi : Rm → R where fi(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , xm) = Di + (Ui − Di)xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Let us consider probability measures on L on the σ-algebra ℘(L) of all the subsets of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' These are equivalent to probability density functions p: L → R and we will abuse notation and write p for both the density function and the probability measure it induces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' The requirement that p is risk-neural in a single-step model is the condition Ep(ψi) = R, (1 ≤ i ≤ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' By the linearity of the expectation and the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='4), this is equivalent to Ep(ℓi) = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Throughout we assume that b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , bm is non-increasing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Let P(L, b) denote the set of all probability density functions p: L → R such that Ep(ℓi) = bi for all 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Define ρ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' ρm ∈ L as follows (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2) ρj = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , 1 � �� � j times , 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Thus, ρj describes the event of a run of j heads followed by a run of n − j tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Use (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='6) to define a probability density function q: L → R by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='3) q(λ) = � qj if λ = ρj for some 0 ≤ j ≤ m 0 otherwise We call q the upper supermodular vertex of P(L, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Compare with [1, Appendix A equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='6)] where we denoted ρj by µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' One checks, see [1, Appendix A], that indeed q ∈ P(L, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' In particular (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='4) Eq(ψi) = Di + (Ui − Di)Eq(ℓi) = Di + (Ui − Di)bi = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' There is a canonical bijection of L with the set ℘({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , m}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' This gives rise to a partial order ⪯ on L induced by the partial order ⊆ on ℘({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , m}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Thus, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='5) λ ⪯ λ′ ⇐⇒ supp(λ) ⊆ supp(λ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 4 Union and intersection of sets render L a lattice with join ∨ and meet ∧, λ ∨ λ′ = (max{λ(1), λ′(1)}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , max{λ(m), λ′(m)}) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='6) λ ∧ λ′ = (min{λ(1), λ′(1)}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , min{λ(m), λ′(m)}) The concept of submodular functions is originally due to Lovász in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' In this note, its variant, supermodular functions [3], is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' A function f : L → R is called supermodular if for any λ, λ′ ∈ L f(λ ∨ λ′) + f(λ ∧ λ′) ≥ f(λ) + f(λ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' It is called modular if equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' (i) A linear combination with non-negative coefficients of supermodular functions is supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' (ii) Let g : Rm → R be an affine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Then g|L is modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' (iii) Let g : Rm → R be an affine function of the form g(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , xm) = �m i=1 aixi + b where a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , am ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' If h: R → R is convex then the restriction of h◦g to L is supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Part (i) is [3, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='5(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Part (ii) follows from [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='3] (and is straightforward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Part (iii) follows from [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='6(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' □ The crucial property of the upper supermodlar vertex q (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='3) is given by the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='4 ([1, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='5(i)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Let f : L → R be supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Let q be the upper supermodular vertex in P(L, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Then sup p∈P (L,b) Ep(f) = max p∈P (L,b) Ep(f) = Eq(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Multi time-step model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' For the n-step model one performs n iterations (not neces- sarily independent) of the experiment of flipping m coins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Thus, the natural sample space for an n-step discrete time binomial market model is Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' We equip it with the σ-algebra F = ℘(Ln) of all subsets of Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' The “state of the world” at time 0 ≤ k ≤ n is described by a k-tuple (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , λk) ∈ Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Thus, the set of the states of the world at time k is naturally identified with Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' We obtain a partition {ω × Ln−k : ω ∈ Lk} of Ln which generates a sub-σ-algebra Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' The price jump of the i-th security at time 1 ≤ k ≤ n, where 1 ≤ i ≤ m, is the random variable Ψi(k): Ln → R Ψi(k): (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , λn) �→ ψi(λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Of course, Ψ0(k): Ln → R is the constant function (random variable) with value R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' The random process Si(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , Si(n) of the prices of the i-th security at time 0 ≤ k ≤ n are random variables Si(k): Ln → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Clearly, they are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='7) Si(k) = Si(0) · Ψi(1) · · ·Ψi(k) 5 Where Si(0) > 0 are constant (the initial prices of the securities at time 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Clearly, the value of Si(k) at ω ∈ Ln depend only on the first k entries of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Hence, Si(k) are Fk-measurable random variables, namely their values only depend on the state of the world at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' We will therefore abuse notation and regard Si(k) as functions with domain Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' We now fix γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , γm where γi ≥ 0 for all 1 ≤ i ≤ m and fix some K and set (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='8) F = � m � i=0 γiSi(n) − K �+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' This random variable is the pay-off of the European contingent claim which is the subject of study of this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Recall the elements ρj ∈ L from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Observe that by definition of ψi (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='1) and of χi (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='7) we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='9) ψi(ρj) = χi(j), (0 ≤ i, j ≤ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Consider some ω = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , λk) ∈ Lk, a state of the world at time 0 ≤ k ≤ n, and some J = (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , jn−k) ∈ Pn−k(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Set τ = (ρj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , ρjn−k) ∈ Ln−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Then F(ωτ) = � m � i=0 γiSi(k)(ω)χi(J) − K �+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' By the definition of Si(n) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='7) F(ωτ) = � m � i=0 γiSi(0) · n � p=1 Ψi(p)(ωτ) − K �+ = � m � i=0 γiSi(0) · k � p=1 ψi(λp) · n � p=k+1 ψi(ρjp) − K �+ = � m � i=0 γiSi(k)(ω) · χi(J) − K �+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' □ For every 0 ≤ k ≤ n we will denote by Cmax(F, k) and Cmin(F, k) the upper and lower bounds of the rational values of F at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Of course, these numbers depend only on the state of the world at time k, and so Cmax / min(F, k) are Fk-measurable random variables (on Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' With the pay-off F (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='8) of European basket call, at time 0 ≤ k ≤ n Cmax(F, k) = Rk−n � J∈Pn−k(m) qJ � m � i=0 γiSi(k)χi(J) − K �+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' This is an immediate consequence of [1, Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='3 and Section 6A eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='2)] and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' We note that in [1] the elements ρj ∈ L are denoted µj and Pn−k(m) is denoted In−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' □ Notice that Cmax(F, k) is an Fk measurable random variable on Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Hence, it will be conve- nient to think of it as a function with domain Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Consider some 0 ≤ k ≤ n − 1 and some ω ∈ Lk representing the state of the world at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Then the function f : L → R defined by f(λ) = Cmax(F, k + 1)(ωλ) is supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Moreover, with respect to the upper supermodular vertex (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='3) Eq(f) = R · Cmax(F, k)(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='6 and since Si(k + 1) = Si(k) · Ψi(k + 1) that f(λ) = Rk+1−n � J∈Pn−k−1(m) qJ � m � i=0 γiSi(k)(ω) · ψi(λ) · χi(J) − K �+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='3(i),(iii) implies that f is supermodular since h(x) = x+ is convex and since ψi = Di + (Ui − Di)ℓi is an affine function with non-negative coefficients and since R > 0 and γi, Si(k), χi(J) ≥ 0 for all 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Moreover, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='9) Eq(f) = � λ∈L q(λ)f(λ) = m � j=0 q(ρj)f(ρj) = m � j=0 qj � J∈Pn−k−1(m) qJRk+1−n � m � i=0 γiSi(k)(ω) · ψi(ρj) · χi(J) − K �+ = m � j=0 qj � J∈Pn−k−1(m) qJRk+1−n � m � i=0 γiSi(k)(ω) · χi(J)χi(j) − K �+ = R · Rk−n � J∈Pn−k(m) qJ � m � i=0 γiSi(k)(ω) · χi(J) − K �+ = R · Cmax(F, k)(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' □ Consider some 0 ≤ k ≤ n − 1 and recall Yj(k) from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='8) where 0 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Observe that Yj(k) is a function of the random variables Si(k), so Yj(k) is a random variable (on Ln) whose values depend only on the state of the world at time k, namely Yj(k) is Fk-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 7 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' For any 0 ≤ j ≤ m and any ω = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , λk) ∈ Lk Cmax(F, k + 1)(ωρj) = Yj(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' This follows immediately from Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='6 and equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Fix some 0 ≤ k ≤ n−1 and some state of the world ω = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , λk) ∈ Lk at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Any subsequent state of the world at time k+1 has the form ωλ for λ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Our goal is to find numbers α0(k)(ω), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , αm(k)(ω), which for simplicity we denote by α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , αm, which fulfil the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='3), namely for every λ ∈ L m � i=0 αiSi(k + 1)(ωλ) ≥ Cmax(F, k + 1)(ωλ) and which minimize (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='10) Vα(k)(ω) = m � i=0 αiSi(k)(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' We rewrite the first inequality as a set of inequalities (indexed by λ ∈ L) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='11) m � i=0 αiSi(k)(ω) · ψi(λ) � �� � Φα(λ) ≥ Cmax(F, k + 1)(ωλ) � �� � Ξ(λ) (λ ∈ L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' We obtain two functions Φα : L → R and Ξ: L → R, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='11) is the inequality Φα ≥ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' The first step of the proof is to show that the following system of m + 1 linear equations with the m + 1 unknowns α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , αm has a unique solution m � i=0 αiSi(k)(ω) · ψi(ρj) = Cmax(F, k + 1)(ωρj), (0 ≤ j ≤ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Notice that these equations are obtained by imposing equalities in the inequalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='11) for λ = ρ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , ρm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='9) and by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='8, this is the system of equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='12) m � i=0 αiSi(k)(ω) · χi(j) = Yj(ω), (0 ≤ j ≤ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Write α• for the column vector (α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , αm) and Y• for the column vector (Y0(ω), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , Ym(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Then this system of m + 1 linear equations is Mα• = Y• where M is the (m + 1) × (m + 1) matrix (Mj,i) = (Si(k)(ω) · χi(j)) = \uf8ee \uf8ef\uf8ef\uf8f0 1 χ1(0) · · χm(0) 1 χ1(1) · · χm(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 1 χ1(m) · · χm(m) \uf8f9 \uf8fa\uf8fa\uf8fb � �� � M′ \uf8ee \uf8ef\uf8ef\uf8f0 RS0(k)(ω) S1(k)(ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Sm(k)(ω) \uf8f9 \uf8fa\uf8fa\uf8fb � �� � T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 8 Clearly, T is invertible since R, Si(k)(ω) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Observe that for any 1 ≤ j ≤ m and any 0 ≤ i ≤ m − 1 χj(i) − χj(i + 1) = � 0 i ̸= j Ui − Di i = j By inspection we get Q · M′ = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 D1 D2 · · Dm 0 U1 − D1 0 · · 0 0 0 U2 − D2 · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 0 0 0 · · Um − Dm \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb � �� � N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Clearly, Q and N are invertible, hence so is M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' It follows that M = M′T is invertible, so the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='12) has a unique solution given by α• = T −1N−1Q · Y•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' This gives the values of αi(k)(ω) stated in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' It remains to show that these αi solve all the inequalities in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='11) (one for each λ ∈ L) and minimize (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='10) and Vα(k)(ω) = Cmax(F, k)(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Claim 1: α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , αm solve the inequalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proof: Suppose that not all these inequalities are solved, namely Φα(λ) > Ξ(λ) for some λ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Since α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , αm solve the equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='12), we certainly get Φα(ρi) = Ξ(ρi) for all 0 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Among all λ ∈ L for which Φ(λ) > Ξ(λ) choose one with maximal possible j such that ρj ⪯ λ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Observe that j < m because if j = m then ρm ⪯ λ implies that λ = ρm which we have seen is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Set λ′ = λ ∨ ρj+1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' By the maximality of j we get that λ ∧ ρj+1 = ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Observe that Φα is an affine function, hence it is modular by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='3(ii), so Φ(λ′) + Φ(ρj) = Φ(λ) + Φ(ρj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='7 Ξ is supermodular, so Ξ(λ′) + Ξ(ρj) ≥ Ξ(λ) + Ξ(ρj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Subtracting the first equality from the second inequality, and recalling that Φ(ρi) = Ξ(ρi) for all i, we get Ξ(λ′) − Φ(λ′) ≥ Ξ(λ) − Ψ(λ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' So Φ(λ′) > Ξ(λ′) and ρj+1 ⪯ λ′ which contradicts the maximality of j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='d Recall that any β0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , βm define Vβ(k)(ω) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Claim 2: For any β0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , βm for which the inequalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='11) hold, Vβ(k)(ω) ≥ Cmax(F, k)(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' In addition, α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , αm attain this lower bound, namely Vα(k)(ω) = Cmax(F, k)(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' 9 Proof: Suppose βi solve the inequalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='4) that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='13) Eq(Φβ) = m � i=0 βiSi(k)(ω) · Eq(ψi) = R · m � i=0 βiSi(k)(ω) = RVβ(k)(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='7 implies that Eq(Ξ) = RCmax(F, k)(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Since βi solve the inequalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='11), this means Φβ ≥ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' By the monotonicity of the expectation, Eq(Φβ) ≥ Eq(Ξ), and since R > 0, it follows that Vβ(k)(ω) ≥ Cmax(F, k)(ω) as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' By construction Φα(ρj) = Ξ(ρj) for all j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Since q is supported on ρ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' , ρm, we deduce from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='13) that RVα(k)(ω) = Eq(Φα) = Eq(Ξ) = R · Cmax(F, k)(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Hence Vα(k)(ω) = Cmax(F, k)(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='d The theorem follows from Claims 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' □ References [1] Jarek Kędra, Assaf Libman, and Victoria Steblovskaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Pricing and hedging contingent claims in a multi-asset binomial market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='13283, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Lovász.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Submodular functions and convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' In Mathematical programming: the state of the art (Bonn, 1982), pages 235–257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Springer, Berlin, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' [3] David Simchi-Levi, Xin Chen, and Julien Bramel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' The logic of logistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Springer Series in Operations Research and Financial Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Springer, New York, third edition, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' Theory, algorithms, and applications for logistics management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content=' University of Aberdeen and University of Szczecin Email address: kedra@abdn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='uk University of Aberdeen Email address: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='libman@abdn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='uk Bentley University Email address: vsteblovskay@bentley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} +page_content='edu 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfGQOR/content/2301.02912v1.pdf'} diff --git a/K9FIT4oBgHgl3EQfaiv2/content/tmp_files/2301.11258v1.pdf.txt b/K9FIT4oBgHgl3EQfaiv2/content/tmp_files/2301.11258v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b1ccf24cac056fedc990998d3bbfa9fc19a8c2f --- /dev/null +++ b/K9FIT4oBgHgl3EQfaiv2/content/tmp_files/2301.11258v1.pdf.txt @@ -0,0 +1,374 @@ +Quantum Test of the Local Position Invariance with Internal +Clock Interferometry +Zhifan Zhou∗ +Joint Quantum Institute, National Institute of Standards +and Technology and the University of Maryland, +College Park, Maryland 20742 USA +(Dated: 1/26/2023) +1 +arXiv:2301.11258v1 [quant-ph] 26 Jan 2023 + +Abstract +Current attempts to test local position invariance (LPI) compare different clock transition rates +with classically exchanged signals. We propose an experimental scheme for the quantum test of +LPI: an internal atomic clock interferometer comprising two interfering clocks within one atom. +We prepare the atom in a superposition of two clock states and one ground state, which evolves +coherently along two quantum clock oscillations into stable internal Ramsey interference patterns. +The interference pattern with the shared ground state shows a visibility modulation, which can be +interpreted as the beating of the individual clock oscillations and a direct consequence of comple- +mentarity. Upon the interferometer experiencing a different gravitational potential, LPI predicts +that both clock tick rates will change proportionally, while quantum complementarity indicates +that the visibility modulation should modify accordingly. This change is deemed insignificant for +the first period of visibility modulation but can be stacked up until the limit of the system co- +herence time. Since no splitting or recombining is involved, the system coherence time can be as +long as the trap lifetime or the clock state lifetime. The required resolution to observe the visi- +bility modulation is within reach of the state-of-art optical clocks’ sensitivities. This experimental +scheme is feasible in different scenarios, still or with speed, and may shed new light on studying +the quantum effect of time and general relativity. +Local position invariance (LPI), together with the weak equivalence principle (WEP) +and local Lorentz invariance (LLI), services as a prominent part of the Einstein equivalence +principle (EEP). A standard test of LPI typically involves comparing different clock transi- +tion rates at the same location [1–5] or the changed transition rates with the same clock at +different locations [6–10]. Recent years have witnessed more involved clock transitions, in- +cluding the two transitions within the same atom [11, 12]. Yet the development of quantum +interferometry has brought new insight into the interrogation of EEP in the regime in which +quantum mechanics (QM) becomes relevant [13–18]. Motivated by the subtle role that the +quantum effect may play, we have sent two different clock transitions that coherently inter- +fere with each other within a single atom. Our proposed internal clock interferometry can +serve as a new tool for studying the interplay of general relativity (GR) and QM. +In the standard test, LPI is done by comparing the rates of classic clocks within the +gravitational fields by classical exchange of electromagnetic signals. Hence, no quantum +∗ Email: zhifan@umd.edu +2 + +interferometric visibility is observed, resulting from the which-path witness and complemen- +tarity. In the quantum test of LPI, the two clock branches accumulate different phases with +different evolving rates. From the view of quantum mechanics, paths become distinguish- +able, resulting in visibility modulation. Consequently, whereas standard clock comparisons +test the classical LPI, internal clock interferometry probes the quantum aspects of LPI. For +example, under different gravitational potentials, the absolute difference of the clock tran- +sition frequencies will be changed, and whether the visibility modulation changes with the +fundamental frequency difference will tell whether quantum complementarity holds in this +gravitational redshift regime. On the other hand, standard two-slit interference stands at +the heart of standard quantum mechanics, where the paths are not sensitive to redshift and +gravitational fields. Internal clock interferometry can be treated as a varied version of two- +slit interference with paths coupled with gravitational fields. The more involved clock states +will lead the platform for testing three-slit interference under the influence of gravitational +effects [19–22]. +It is worthwhile noting the decades of progress in precise atomic clocks [5, 6, 23, 24], +optical clocks in space [25–27], and matter-wave interferometry [28–31]. Specifically, we note +the recent proposals towards observing visibility modulation with the spatial superposition +of clock wave packets [13, 14, 17, 32]. The proposal has been realized with a prototype +experiment with an atom chip system [15, 18, 33] involving an emulated time lag. +In our experiment, internal atomic clock interferometry - atoms in superpositions of two +clock states - involve an internal Ramsey scheme (Fig.1). We investigate that the visibility +of population oscillation depends on the relative clock rotation, the proper time differential +between each branch of the clock evolution. Our proposal shows that a differential clock +reading affects the visibility of internal clock interferometry; specifically, the visibility equals +the scalar product of the interfering clock states. +In principle, any system evolving with two well-defined periods can be used as internal +clock interferometry. Here we used a quantum three-level system. The candidates include +neutral atoms, ions, and nuclear clocks. In our proposed experiment, the internal clock +interferometer - clocks that interfere coherently within one atom - are prepared as a coherent +superposition of a three-level system. To maximize the interferometric effect, the population +ratios among the two excited clock states and the ground state are prepared as 0.25, 0.25, +and 0.5. As the clock transitions at each branch differ, atoms accumulate phases at different +rates. Thus each branch measures proper time and contains which-way information, leading +3 + +to visibility modulation. +To examine the coherent interference of internal clock superposition, we let the two clock +transitions be freely involved and closed by another π/2 pulse to create internal fringes. +Because the phase of π/2 is tunable to reveal the Ramsey fringe, the phase can be scanned +to test the system’s coherence. We note that during the interrogation process, the atom is +trapped and experiences no splitting or recombining processes. Hence, the system coherence +time is not influenced by the splitting and recombining atomic wave packets in standard atom +interferometers. The upper limit will be the clock state lifetime or the trap lifetime. +We now show that visibility modulation is observed as a function of interrogation time. +We treat the |1⟩ to |3⟩ transition as clock 2 and the |1⟩ to |2⟩ transition as clock 1. After +the initialization, the atom is equally split into two branches. Clock 2 has a larger transition +frequency than clock 1, and this branch accumulates phase faster than the other. The phase +difference will be reflected in the output as the population of state |1⟩. In the extreme case, +the two clocks are orthogonal to each other - for example, one is in the state of |1⟩-|3⟩, and +the other is in the state of |1⟩+|2⟩. The visibility of the state |1⟩ population will drop to +0. This is contrary to a standard matter-wave interferometer, where the visibility is always +high. A revival of the visibility is seen when the differential rotation angle is involved further +than π and reaches 2π. The periodicity of this visibility modulation will be 1/∆f. The beat +frequency out of the clock interference is ∆f = f2 −f1. The population of state |1⟩ oscillates +with the function as: +P = 1 +2[1 + cos(∆ft/2)cos(ωt)], +(1) +The visibility reads: +V = |cos(∆ft/2)|. +(2) +To obtain a more general view of the effect, we studied the dependence of the interference +pattern visibility on different gravitational potentials by putting the interferometer further +away. For example, one is located on the earth’s surface, and the other one’s position is +changed by ∆h (Fig.2). The clock transition frequencies and the beat frequency are shifted +by the fractions of δf1/f1 = g∆h/c2, δf2/f2 = g∆h/c2, and δ(f2 − f1)/(f2 − f1) = g∆h/c2. +The essence of the which-path witness is that visibility is complementary to the clock’s +relative rotation. In Fig.2, we chose to work with the first oscillation period to highlight +the visibility’s dependence on the proper clock rotation’s change. +The time change for +reaching the minimal visibility for the first period is insignificant as 1/∆f × g∆h/c2. To +4 + +further increase this signal to make the change from the gravitational field resolvable, we can +extend the interrogation time until the limit of the clock state lifetime or the trap lifetime, +as shown in Fig.3. The enhancement factor is τ × ∆f. Then the total time shift signal at +the limit of system coherence time is 1/∆f × g∆h/c2 × τ × ∆f = τ × g∆h/c2. Here we use +the system coherence time 1s as one example. The visibility modulation shift for the lifting +height of 1m will be 1.1 × 10−16, well within reach of the state-of-art clock technique. +The challenge of realizing such an internal clock interferometry is having two clock states +with long clock lifetime as shown in [11, 12]. +The other challenge for neutral atoms or +molecules is the trapping for both clock states at magic wavelengths [11]. The charged ion +and nuclear clocks provide appropriate candidates for providing parallel clock transitions +with long clock lifetime [34–38]. +In conclusion, this internal clock interferometry scheme uses an interferometer to measure +the relative frequency difference. Thus, it can suppress systematic common noise. Mean- +while, it doesn’t suffer from the recombining issue like the usual atom interferometers do, +so the setup can be located anywhere with any speed, considering the significant progress +with the compact atomic clock in space [25–27]. This internal atom interferometer can serve +as a precise quantum sensor for tests of local position invariance and measurements of the +possible time variation of fundamental constants such as fine-structure constant [11, 12]. +This work was supported by the Air Force Office of Scientific Research (FA9550- 16-1- +0423). We acknowledge Marianna S. Safronova for the fruitful discussions. +[1] T. M. Fortier, N. Ashby, J. C. Bergquist, M. J. 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Thirolf, Nuclear clocks +for testing fundamental physics, Quantum Science and Technology 6, 034002 (2021). +11 + diff --git a/K9FIT4oBgHgl3EQfaiv2/content/tmp_files/load_file.txt b/K9FIT4oBgHgl3EQfaiv2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1f56da4922c12de996de2f9b9c117b4841b6c5d4 --- /dev/null +++ b/K9FIT4oBgHgl3EQfaiv2/content/tmp_files/load_file.txt @@ -0,0 +1,569 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf,len=568 +page_content='Quantum Test of the Local Position Invariance with Internal Clock Interferometry Zhifan Zhou∗ Joint Quantum Institute, National Institute of Standards and Technology and the University of Maryland, College Park, Maryland 20742 USA (Dated: 1/26/2023) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='11258v1 [quant-ph] 26 Jan 2023 Abstract Current attempts to test local position invariance (LPI) compare different clock transition rates with classically exchanged signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' We propose an experimental scheme for the quantum test of LPI: an internal atomic clock interferometer comprising two interfering clocks within one atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' We prepare the atom in a superposition of two clock states and one ground state, which evolves coherently along two quantum clock oscillations into stable internal Ramsey interference patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The interference pattern with the shared ground state shows a visibility modulation, which can be interpreted as the beating of the individual clock oscillations and a direct consequence of comple- mentarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Upon the interferometer experiencing a different gravitational potential, LPI predicts that both clock tick rates will change proportionally, while quantum complementarity indicates that the visibility modulation should modify accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' This change is deemed insignificant for the first period of visibility modulation but can be stacked up until the limit of the system co- herence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Since no splitting or recombining is involved, the system coherence time can be as long as the trap lifetime or the clock state lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The required resolution to observe the visi- bility modulation is within reach of the state-of-art optical clocks’ sensitivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' This experimental scheme is feasible in different scenarios, still or with speed, and may shed new light on studying the quantum effect of time and general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Local position invariance (LPI), together with the weak equivalence principle (WEP) and local Lorentz invariance (LLI), services as a prominent part of the Einstein equivalence principle (EEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' A standard test of LPI typically involves comparing different clock transi- tion rates at the same location [1–5] or the changed transition rates with the same clock at different locations [6–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Recent years have witnessed more involved clock transitions, in- cluding the two transitions within the same atom [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Yet the development of quantum interferometry has brought new insight into the interrogation of EEP in the regime in which quantum mechanics (QM) becomes relevant [13–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Motivated by the subtle role that the quantum effect may play, we have sent two different clock transitions that coherently inter- fere with each other within a single atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Our proposed internal clock interferometry can serve as a new tool for studying the interplay of general relativity (GR) and QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' In the standard test, LPI is done by comparing the rates of classic clocks within the gravitational fields by classical exchange of electromagnetic signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Hence, no quantum ∗ Email: zhifan@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='edu 2 interferometric visibility is observed, resulting from the which-path witness and complemen- tarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' In the quantum test of LPI, the two clock branches accumulate different phases with different evolving rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' From the view of quantum mechanics, paths become distinguish- able, resulting in visibility modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Consequently, whereas standard clock comparisons test the classical LPI, internal clock interferometry probes the quantum aspects of LPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' For example, under different gravitational potentials, the absolute difference of the clock tran- sition frequencies will be changed, and whether the visibility modulation changes with the fundamental frequency difference will tell whether quantum complementarity holds in this gravitational redshift regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' On the other hand, standard two-slit interference stands at the heart of standard quantum mechanics, where the paths are not sensitive to redshift and gravitational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Internal clock interferometry can be treated as a varied version of two- slit interference with paths coupled with gravitational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The more involved clock states will lead the platform for testing three-slit interference under the influence of gravitational effects [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' It is worthwhile noting the decades of progress in precise atomic clocks [5, 6, 23, 24], optical clocks in space [25–27], and matter-wave interferometry [28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Specifically, we note the recent proposals towards observing visibility modulation with the spatial superposition of clock wave packets [13, 14, 17, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The proposal has been realized with a prototype experiment with an atom chip system [15, 18, 33] involving an emulated time lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' In our experiment, internal atomic clock interferometry - atoms in superpositions of two clock states - involve an internal Ramsey scheme (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' We investigate that the visibility of population oscillation depends on the relative clock rotation, the proper time differential between each branch of the clock evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Our proposal shows that a differential clock reading affects the visibility of internal clock interferometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' specifically, the visibility equals the scalar product of the interfering clock states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' In principle, any system evolving with two well-defined periods can be used as internal clock interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Here we used a quantum three-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The candidates include neutral atoms, ions, and nuclear clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' In our proposed experiment, the internal clock interferometer - clocks that interfere coherently within one atom - are prepared as a coherent superposition of a three-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' To maximize the interferometric effect, the population ratios among the two excited clock states and the ground state are prepared as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='25, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' As the clock transitions at each branch differ, atoms accumulate phases at different rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Thus each branch measures proper time and contains which-way information, leading 3 to visibility modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' To examine the coherent interference of internal clock superposition, we let the two clock transitions be freely involved and closed by another π/2 pulse to create internal fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Because the phase of π/2 is tunable to reveal the Ramsey fringe, the phase can be scanned to test the system’s coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' We note that during the interrogation process, the atom is trapped and experiences no splitting or recombining processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Hence, the system coherence time is not influenced by the splitting and recombining atomic wave packets in standard atom interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The upper limit will be the clock state lifetime or the trap lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' We now show that visibility modulation is observed as a function of interrogation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' We treat the |1⟩ to |3⟩ transition as clock 2 and the |1⟩ to |2⟩ transition as clock 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' After the initialization, the atom is equally split into two branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Clock 2 has a larger transition frequency than clock 1, and this branch accumulates phase faster than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The phase difference will be reflected in the output as the population of state |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' In the extreme case, the two clocks are orthogonal to each other - for example, one is in the state of |1⟩-|3⟩, and the other is in the state of |1⟩+|2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The visibility of the state |1⟩ population will drop to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' This is contrary to a standard matter-wave interferometer, where the visibility is always high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' A revival of the visibility is seen when the differential rotation angle is involved further than π and reaches 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The periodicity of this visibility modulation will be 1/∆f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The beat frequency out of the clock interference is ∆f = f2 −f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The population of state |1⟩ oscillates with the function as: P = 1 2[1 + cos(∆ft/2)cos(ωt)], (1) The visibility reads: V = |cos(∆ft/2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' (2) To obtain a more general view of the effect, we studied the dependence of the interference pattern visibility on different gravitational potentials by putting the interferometer further away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' For example, one is located on the earth’s surface, and the other one’s position is changed by ∆h (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The clock transition frequencies and the beat frequency are shifted by the fractions of δf1/f1 = g∆h/c2, δf2/f2 = g∆h/c2, and δ(f2 − f1)/(f2 − f1) = g∆h/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The essence of the which-path witness is that visibility is complementary to the clock’s relative rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='2, we chose to work with the first oscillation period to highlight the visibility’s dependence on the proper clock rotation’s change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The time change for reaching the minimal visibility for the first period is insignificant as 1/∆f × g∆h/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' To 4 further increase this signal to make the change from the gravitational field resolvable, we can extend the interrogation time until the limit of the clock state lifetime or the trap lifetime, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The enhancement factor is τ × ∆f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Then the total time shift signal at the limit of system coherence time is 1/∆f × g∆h/c2 × τ × ∆f = τ × g∆h/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Here we use the system coherence time 1s as one example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The visibility modulation shift for the lifting height of 1m will be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='1 × 10−16, well within reach of the state-of-art clock technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The challenge of realizing such an internal clock interferometry is having two clock states with long clock lifetime as shown in [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The other challenge for neutral atoms or molecules is the trapping for both clock states at magic wavelengths [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The charged ion and nuclear clocks provide appropriate candidates for providing parallel clock transitions with long clock lifetime [34–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' In conclusion, this internal clock interferometry scheme uses an interferometer to measure the relative frequency difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Thus, it can suppress systematic common noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Mean- while, it doesn’t suffer from the recombining issue like the usual atom interferometers do, so the setup can be located anywhere with any speed, considering the significant progress with the compact atomic clock in space [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' This internal atom interferometer can serve as a precise quantum sensor for tests of local position invariance and measurements of the possible time variation of fundamental constants such as fine-structure constant [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' This work was supported by the Air Force Office of Scientific Research (FA9550- 16-1- 0423).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' We acknowledge Marianna S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Safronova for the fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' M.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Campbell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Thomsen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Zelevinsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Boyd, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Ye, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Baillard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Fouch´e, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Le Targat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Brusch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Lemonde, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Takamoto, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Hong, 5 a b Clock2 Clock1 1 3 2 c Evolving time t1 t2 t3 t4 t5 Clock2 Clock1 Optical ������������/2 Clock rotation Imaging Optical ������������/2 Time |Ψ|2 ������������������������������������������������������������������������������������ Coherent splitter Coherent recombiner State preparation Ψ d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' 1: Experimental sequence of the internal clock interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' (a) The system consists of two clock states and one ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' (b) Detailed sequence (not to scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The clock interferometer is initialized by a pair of optical π/2 pulses, after which the two clocks are ticking at different rates, and the relative rotation is accumulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The other pair of optical π/2 pulses are applied to close the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Last, the population distribution of these three states is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' (c) Evolution in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Each clock wave packet shows a one-handed clock, in which the hand corresponds to a vector in the equatorial plane of the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' When the clock readings (the position of the clock hand) in the two clock wave packets are the same, the fringe visibility is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' When the clock readings are opposite (orthogonal), there is a ”which path” witness, and the fringe visibility is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' (d) the visibility evolution in time, synchronized with (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Katori, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Flambaum, New limits on coupling of fundamental constants to gravity using 87Sr optical lattice clocks, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' 100, 140801 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Gu´ena, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Abgrall, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Rovera, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Rosenbusch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Tobar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Laurent, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Clairon, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Bize, Improved tests of local position invariance using 87Rb and 133Cs fountains, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' 109, 080801 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Ludlow, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Boyd, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Ye, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Peik, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Schmidt, Optical atomic clocks, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' 87, 637 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' 6 Gravitational potential 1 Gravitational potential 2 a b c d |3〉 |1〉 |2〉 Δf |3〉 |1〉 |2〉 Δf’ f2 f1 f2-Δf2 f1-Δf1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' 2: (a) When the experiment is done at the earth’s surface, the beat frequency is the difference ∆f = f2 − f1 between the two clock transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' (b) The population of the shared state shows a visibility oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' (c) When the experiment is done in a different gravitational potential, for example, changed by one meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The time dilation is ∆f f = g∆h c2 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='8×1 9×1016 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='1 × 10−16 (to lowest order in c−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' (d) The visibility oscillation frequency will experience a shift due to the different gravitation potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The shifted amounts are for illustration purposes, with ∆f changed by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Chou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Hume, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Rosenband, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Wineland, Optical clocks and relativity, Science 329, 1630 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Delva, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Puchades, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Sch¨onemann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Dilssner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Courde, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Bertone, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Gonzalez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Hees, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Le Poncin-Lafitte, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Meynadier, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Prieto-Cerdeira, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Sohet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Ventura- Traveset, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Wolf, Gravitational redshift test using eccentric Galileo satellites, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='5/ f f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='5/ f 2/ f 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='5/ f 3/ f First Interval of Evolving Time 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='9 1 Visibility 1000/ f 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='5/ f 1001/ f 1001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='5/ f 1002/ f 1002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='5/ f 1003/ f Second Interval of Evolving Time FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' 3: The entire evolving time of the internal clock interferometry can be as long as the trap lifetime, which lasts tens of seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The detect resolution is not limited by the first dip of the interference visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Instead, the signal can be stacked up to multiple accumulating regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' In the figure, the red and blue dash lines represent the fractional shift of 4 × 10−4, which is negligible for the first several periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' After stacking for 1,000 periods, the fractional shift is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' The number is not scaled with the real fractional shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' 121, 231101 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FIT4oBgHgl3EQfaiv2/content/2301.11258v1.pdf'} +page_content=' [8] S.' 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Thyagarajan,1, ∗ S. Kanhirathingal,1, 2 B.L. Brock,1, 3 Juliang Li,1, 4 M.P. Blencowe,1 and A.J. Rimberg1, † +1Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA +2Rigetti Computing, Berkeley, California 94710, USA +3Department of Applied Physics, Yale University, New Haven, Connecticut, USA +4High Energy Physics Divison, Argonne National Laboratory, +9700 South Cass Avenue, Argonne, IL 60439, USA +(Dated: January 3, 2023) +Operating the cavity-embedded Cooper pair transistor (cCPT) in the Kerr bistable regime, we +demonstrate single-shot resolution between two charge states that are 0.09e apart. The measurement +is performed with 94% fidelity in a duration of 3 µs. The drive power at which the measurement +is performed corresponds to only 20 intracavity photons on average in the high oscillation ampli- +tude state of the cCPT, which is orders-of-magnitude smaller than that in rf-SETs. We find that +the limiting factor for this mode of operation of the cCPT is the spontaneous fluctuation-induced +switching between the two metastable oscillation amplitude states. We present empirical data on +the variation of the switching dynamics with drive parameters and cCPT DC bias. +I. +INTRODUCTION +Fast detection of charge on the order of a fraction of +an electron has long been an important task. Versatile +devices such as the quantum point contact and the sin- +gle electron transistor (SET) have been used to measure +electron lifetimes in a single electron trap [1], to map +electric fields with 100 nm spatial resolution [2], to ob- +serve macroscopic charge quantization [3], and to study +quasiparticle and electron tunneling events in real-time +[4, 5]. More recently, they have been used in the search +for Majorana zero modes in nanowires [6], and could po- +tentially be used to detect dark matter [7, 8]. Such fast, +ultrasensitive electrometers are instrumental in the read- +out of silicon-based spin qubits [9, 10] where the magnetic +moment of a single spin is too small to detect directly, +and is instead converted to a spin state dependent charge +which can be read out. Dispersive charge sensors oper- +ating on the supercurrent branch of the Josephson junc- +tions based inductive-SET (L-SET) [11] and the single +Cooper-pair box [12] are not shackled by the electron shot +noise which limits the operation of the rf-SETs [13]. The +cavity-embedded Cooper pair transistor (cCPT) used in +this work has previously been shown to achieve a charge +sensitivity of 14 µe/ +√ +Hz operating as a dispersive sensor +in the linear regime with a single intracavity photon on +average [14], close to the theoretical quantum limit for +this device [15]. +The cCPT is also a rich nonlinear system whose Hamil- +tonian contains a Kerr nonlinearity [16], and an emergent +parametric amplifier term when the flux line of the sys- +tem is driven at twice the resonance frequency. The Kerr +term opens up the possibility of more sensitive charge de- +tection than was achieved in the linear regime [17]. Such +∗ Bhargava.Thyagarajan.gr@dartmouth.edu +† Alexander.J.Rimberg@dartmouth.edu +a Kerr cavity coupled to a mechanical resonator was pro- +posed [18] and demonstrated [19] to achieve an order of +magnitude better cooling of the phonon mode compared +to a linear cavity. The Kerr nonlinearity is well known to +produce bifurcations in the system response [20]. Bifur- +cation amplifiers [21–23] based on a large change in the +response at a bifurcation edge have been used to read out +the state of superconducting qubits [24]. Nanomechani- +cal devices based on the bifurcation under a parametric +drive have been used to sense charges of 9e at room tem- +perature [25]. Similar devices have demonstrated charge +sensing of the order of 70e by manipulating the topology +of the bifurcation diagram [26]. +Here, using the bifurcation between a bistable and a +monostable region induced by the Kerr nonlinearity of +the cCPT, we demonstrate single-shot readout of 0.09e of +charge in 3 µs with 94% fidelity, using fewer than 25 intra- +cavity photons. Such low power operation ensures mini- +mal back-action on the system being measured [27], and +also aids in the integration of these cCPT detectors with +state-of-the-art first stage amplifiers such as the TWPAs +[28] without overwhelming them beyond their compres- +sion point. Such fast high fidelity readout is compara- +ble to the current state-of-the-art for semiconductor spin +qubits [29, 30]. +In Sec. II we present a semi-classical analysis of the +nonlinear cCPT and propose a scheme for it to function +as a sensitive charge state discriminator operating in the +bistable regime. In Sec. III we experimentally study the +hysteresis in the cCPT response in the bistable regime +to characterize the extent of the bistability as a function +of the drive detuning and strength. We then implement +a charge sensing protocol, and observe the presence of +fluctuation-induced spontaneous transitions between the +bistable states, which we study as a function of drive pa- +rameters and cCPT DC bias. Lastly, we characterize our +charge sensing protocol and demonstrate the optimum +high-fidelity, fast charge state readout possible with this +device. In Sec. IV we conclude by discussing some possi- +arXiv:2301.00110v1 [quant-ph] 31 Dec 2022 + +2 +ble improvements to this work. Details of the heterodyne +measurement scheme employed in this work and the mi- +crowave circuitry used in the dilution refrigerator are in +Appendix A, and some experimental considerations for +the charge sensing scheme used in Sec. III are detailed +in Appendix B. +0.5 +1 +-30 +-20 +-10 +0 +10 +100 +200 +300 +FIG. 1. (a) Schematic of the cCPT. (b) Variation of the Kerr +coefficient K as a function of gate, ng, and flux, Φext, over +the operational bias range of the cCPT simulated using the +extracted values of EJ and EC [16]. (c) Simulated response +of the cCPT for different drive powers, Pin. The red curve +is for a drive strength Pin ≪ P (c) +in . +The blue curve is for +Pin > P (c) +in +and the green is for Pin ≫ P (c) +in . Above P (c) +in , we +see bistability across a range of detunings indicated by the +corresponding shaded region. The △’s represent the stable +high oscillation amplitude state, the ▽’s represent the stable +low oscillation amplitude states, and the �’s represent the +unstable state. +The solid lines indicate monostability. +(d) +Simulated magnitude of the reflection coefficient for the drive +powers in (c). (e) Simulated phase of the reflection coefficient +for the drive powers in (c). All simulations in (c), (d) and (e) +were for a cCPT DC bias (ng, Φext) = (0, 0) with K/2π = −470 +kHz, and nominal damping rates for this bias point [16]. +II. +THEORETICAL DESCRIPTION +The cCPT, schematically depicted in Fig. 1(a), con- +sists of two parts: (i) the cavity, which is a λ/4 supercon- +ducting coplanar waveguide (CPW) with its end shorted +to the ground plane, and (ii) a Cooper pair transistor +(CPT) across the center line and ground plane of the +CPW at its voltage anti-node. In this geometry, the cav- +ity and CPT form a shared SQUID loop, which couples +them together. +When the CPT remains in its ground +state, it modifies the effective potential of the cavity, +such that the cCPT behaves as a nonlinear oscillator +whose resonant frequency can be tuned using the effective +gate charge, ng = CgVg +e +, and the magnetic flux thread- +ing the SQUID loop, Φext. Here, Vg is the external DC +voltage applied to the CPT island through the gate ca- +pacitance Cg. Along with the fabrication details for the +cCPT device used in this work, a detailed characteriza- +tion at low drive amplitudes where the nonlinearities do +not contribute substantially to the dynamics has been +carried out in Ref. [16]. Notably, the Josephson energy, +EJ, and the charging energy, EC, were estimated to be +EJ/h = 14.8 GHz and EC/h = 54.1 GHz respectively. +Finally, to drive and measure the cCPT, a probe trans- +mission line is coupled to the CPW through a coupling +capacitor Cc. +For an input drive close to resonance, under the rotat- +ing wave approximation, the Hamiltonian for the cCPT +is given by [15, 16] +H = ̵hω0(ng,Φext)a†a + 1 +2 +̵hK(ng,Φext)a†2a2, +(1) +where a(a†) are the annihilation (creation) operators +for the cavity mode, ω0(ng,Φext) and K(ng,Φext) are +the resonant frequency of the linear cCPT Hamiltonian +and the Kerr coefficient respectively. +The variation of +K(ng,Φext) over the operational range of the cCPT de- +vice used in this work is shown in Fig. 1(b). The Kerr co- +efficient changes sign with flux, attaining extremum val- +ues at half-integer multiples of Φ0, and passing through +zero close to Φext = 0.25Φ0. Kerr-free cavities have been +used to increase the dynamic range of parametric ampli- +fiers [31, 32]. +We use input-output theory [33] to model the dynamics +of the cavity mode. The quantum Langevin equation for +a gives +˙a = 1 +i̵h[a,H] − [a,a†](κtot +2 a − √κextain(t) − √κintbin(t)) += −(i(ω0 + Ka†a) + κtot +2 )a + √κextain(t) + √κintbin(t), +(2) +and a corresponding equation for a†, where κext is the ex- +ternal damping rate due to the coupling of the resonator +to the probe transmission line with the input bath op- +erator ain(t), and κint is the internal damping rate as- +sociated with the coupling of the resonator to an inter- +nal loss channel with input operator bin(t). +The total + +(c) 60 +C +in +in +(c) +P +in +400 +40 +<△00 +0 +U +20 +-30 +-20 +-10 +0 +10 +V +/2π(MHz0 - +-0.5 +-0.5 +0 +extH/ +0 +-500 +0.5(b) +0.5500 +Z3 +damping rate of the cavity is κtot = κext + κint. When +the input tone is a pure sine wave at frequency ωd of +the form ⟨ain⟩ = αine−iωdt, the steady state response of +the cavity is at this drive frequency. For this coherent +drive, using the semi-classical approximation, we make +the ansatz ⟨a⟩ = αe−iωdt, with ⟨˙a⟩ = −iωdαe−iωdt and the +average intracavity occupation number n = ∣α∣2 = ⟨a†a⟩. +Plugging this ansatz into the expectation value of Eq.(2) +we obtain +[−i(∆ − K∣α∣2) + κtot +2 ]α = √κextαin, +(3) +where we have defined the detuning ∆ = ωd − ω0. Using +Eq.(3) and the input-output relation aout(t) = ain(t) − +√κexta(t) [33], [34] we find the reflection coefficient +S11(∆) to be +S11(∆) = (αout +αin +) +∗ += (∆ − K∣α∣2) − i(κint − κext)/2 +(∆ − K∣α∣2) − i(κint + κext)/2 +, +(4) +where aout(t) is the transmission output bath operator. +Also, from Eq.(3) and the corresponding equation for α∗, +n = ∣α∣2 satisfies the cubic equation +K2n3 − 2K∆n2 + (∆2 + κ2 +tot +4 )n = κext +Pin +̵hωd +, +(5) +where Pin = nin̵hωd is the power of the input drive tone +incident on the cCPT, and nin = ∣αin∣2 is the input pho- +ton flux. As illustrated in Figs. 1(c-e), at very low drive +strengths this cubic equation has only one real root and +the oscillator exhibits only monostable behaviour across +all detunings. As the drive strength is increased beyond +a critical power P (c) +in += +√ +3 +9 +κ3 +tot +∣K∣κext ̵hω(c) +d , the oscillator sys- +tem undergoes a bifurcation, and exhibits bistability for +a range of detunings. Here, ω(c) +d +is the drive frequency +corresponding to a detuning of ∆c = sgn(K) +√ +3 +2 κtot and +(∆c,P (c) +in ) corresponds to a spinode point in the param- +eter space of the input drive [35]. In the bistable region, +two of the three real solutions of the cubic Eq.(5) corre- +spond to high and low oscillation amplitude states with +corresponding values of S11 from Eq.(4), while the third +is an unstable, experimentally inaccessible state [36]. +The variation of the resonant frequency of the cCPT +with the gate, ng, as illustrated in Fig. 2(a), forms the +basis for a sensitive dispersive charge detector. Operat- +ing in the single-photon, linear regime, this device was +demonstrated to have a charge sensitivity of 14 µe/ +√ +Hz +[14]. Fig. 2(b) uses Eq.(4) to simulate the reflected phase +as a function of drive frequency, ωd, for two gate values +separated by δng corresponding to a resonant frequency +shift δω0. The S11 for the two gate values are denoted +by pink and green curves respectively, both in the linear +(Pin ≪ P (c) +in ), single photon regime (dashed lines); and +-0.5 +0 +0.5 +5.77 +5.78 +5.79 +5.8 +5.81 +data +theory +-125 +-120 +-115 +-110 +-100 +-80 +-60 +-40 +-20 +FIG. 2. (a) Experimentally measured resonant frequency of +the linear cCPT (Pin ≪ P (c) +in ), ω0, as a function of the gate +charge on the cCPT, ng, at a fixed flux bias Φext = 0.15Φ0 +(triangles). +The line is the theoretically expected resonant +frequency for the junction parameters of this device. (b) Sim- +ulations illustrating the larger separation in reflected phase, +δS11 (δS(Kerr) +11 +) when operating at Pin > P (c) +in +(solid lines) com- +pared to Pin ≪ P (c) +in +(dashed lines). (c) Phase of the reflected +signal for a forward (solid line) and reverse (dashed line) tri- +angular ramp of the drive amplitude, Pin. The input power +is ramped between -140 dBm and -109 dBm in increasingly +longer times from 2 µs to 28 µs from red to blue curves. The +cCPT was biased at (ng, Φext) = (0, 0) and the detuning was +∆/2π = −9.5 MHz. +with a drive power Pin > P (c) +in +(solid lines). At a given +ωd, ng can hence be inferred from the measured S11. For +Pin > P (c) +in , these simulations describe what we would ob- +serve in the absence of spontaneous transitions between +the high and the low oscillation amplitude states in the +bistable region. In the absence of these transitions, while +ramping the drive detuning from a large blue-detuned +value (with respect to linear resonance, ω0, ∆ > 0), to a + +(b) ++Sng +(c +Phase(Sii(w)) +in +Kerr)1 +in +I +- +(pink) +green) +03 +Drive Freq (wd)4 +red-detuned value (∆ < 0), we expect to stay in the high +oscillation amplitude state until we reach the bifurcation +detuning further from ω0 for the green curve in Fig. 1(c- +e). We refer to this as the lower bifurcation point, while +referring to the bifurcation detuning closer to ω0 as the +upper bifurcation point. Upon crossing the lower bifurca- +tion point, an abrupt jump from the high to the low oscil- +lation amplitude state is expected, with a corresponding +large change in S11, as illustrated in Fig. 2(b). For an +appropriate drive frequency denoted by the dashed black +line, the same separation in gate charge, δng, produces a +larger difference in the reflected phase between the pink +and green curves, δS(Kerr) +11 +, than the δS11 while operat- +ing in the linear regime. Conversely, δS(Kerr) +11 +continues +to remain large as the green and pink curves are brought +together by reducing δng, whereas δS11 undergoes sub- +stantial reduction while doing so. The sensitivity of the +charge detector is the smallest δng that produces a δS11 +which can be detected with a signal-to-noise-ratio (SNR) +of 1 [14]. Given that the noise in the measurement is lim- +ited by the amplifier chain in the experimental setup [14], +the larger S(Kerr) +11 +for smaller δng promises a lower, much +improved charge sensitivity for the device operating in +this regime. +III. +EXPERIMENTS +In this section, we first describe the results of a tri- +angular input power ramp in order to understand the +extent of the bistable region with respect to the cCPT +drive parameters at a given bias point. We then outline +the protocol we use in order to perform charge sensing +based on the bifurcation described above. Contrary to +the sharp jump in S(kerr) +11 +at a precise value of the detun- +ing described in Sec. II, we see a non-zero probability of +obtaining a value on either end of the step illustrated in +Fig. 2(b) for a range of detunings. We discuss the re- +sults of this protocol for a range of cCPT bias points and +drive parameters. From this, we glean the optimal condi- +tions for charge sensing and finally perform an optimized +single-shot measurement. +In order to study the extent of the bistability, we bias +the cCPT at (ng,Φext) = (0, 0), and drive it at a fixed +detuning ∆/2π = −9.5 MHz with a triangular amplitude +ramp in the forward and the reverse direction to check +for hysteresis. This is the bias point at which we expect +minimum fluctuation in the resonant frequency of the +cCPT due to charge and flux noise [16, 37]. We perform +a heterodyne measurement to obtain the phase of the +reflected signal over the course of the ramp. The RF cir- +cuitry used in the experiments described here is detailed +in Appendix A. Fig. 2(c) plots the observed hysteresis in +the phase of S11 for different ramp rates, each averaged +over 5000 repetitions of the ramp. For fast ramps, we see +that we obtain a value for the reflected phase correspond- +ing to the low oscillation amplitude state for the forward +ramp, and a value that corresponds to the high oscilla- +tion amplitude state during the reverse ramp. However, +as the ramp time is increased, we observe that the spac- +ing between the observed phase during the forward and +the reverse ramps reduces, and for this cCPT bias point, +saturates to the values represented by the blue curves, +corresponding to ramp times of ∼ 25 µs. This is because, +when given enough time to do so, the oscillator system +undergoes fluctuation-induced spontaneous switching be- +tween the high and low oscillation amplitude states over +the course of a ramp. +This yields a weighted average +value for the phase at each Pin value over 5000 repeti- +tions of the pulse sequence. The weights depend on the +average lifetimes of the high and the low oscillation am- +plitude states at the chosen cCPT bias and the drive +parameters. We see less variation in the shape of the for- +ward and reverse ramp curves for the larger ramp times +in Fig. 2(c). This provides a rough estimate of ∼ 25 µs +for the average lifetimes of these bistable states. This is +a sign of spontaneous transitions between the high and +low oscillation amplitude states for a range of input drive +strengths, and will be detrimental to the charge sensing +scheme described above which counts on the sharp jump +from one oscillation amplitude state to the other at pre- +cisely a bifurcation point. +A similar reduction in the +area enclosed between the curves corresponding to the +forward and reverse ramps for longer ramps was recently +observed for a nonlinear semiconductor microcavity [38]. +While performing the charge sensing measurement, we +choose an input drive strength which gives rise to a re- +gion of bistability (Pin > P (c) +in ) for the chosen cCPT DC +bias (ng,Φext) with a corresponding K < 0. In order to +deterministically initialize the oscillator in the high oscil- +lation amplitude state, we perform a linear ramp on the +detuning of the drive tone from a blue- to a red-detuning +as illustrated in Fig. 3(a). More details on the initializa- +tion section (shaded pink) of this protocol are provided +in Appendix B. Once initialized, we measure and average +the phase of the reflected signal for a time tacq. Perform- +ing this measurement Ntot = 20,000 times, we obtain a +double Gaussian histogram as illustrated in the inset of +Fig. 3(b), and extract the probability of the high oscil- +lation amplitude state, P(ωd), as the ratio of the area of +the left Gaussian to the total area of the histogram. We +perform this measurement for different detunings at the +end of the initialization step of the pulse in Fig. 3(a), +and plot the obtained probability of being in the high +oscillation amplitude state for each detuning, obtaining +the S-curves in Fig. 3(c). We fit sigmoids of the form +P(ωd) = +1 +1 + exp{− 4.3944(ωd−∆0) +γ +} +, +(6) +where ∆0 is the center of the sigmoid, and the numerical +factor in the exponential ensures γ is its width between +P(ωd) = 0.1 and P(ωd) = 0.9. +As described earlier in Sec. II, we ideally expect an +abrupt step in P(ωd) from 1 → 0 at the lower bifurcation + +5 +5.775 5.78 5.785 5.79 5.795 +5.8 5.805 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.5 +0.53 +0.56 +0.59 +0.62 +0.65 +0.68 +0.71 +-100 +0 +100 +0 +1000 +2000 +FIG. 3. (a) Charge sensing protocol. Detuning of the input +pulse tone used to initiate and readout the oscillation state +in the charge sensing experiment described in the text with +representative values for the durations and detunings of the +different sections. +The pink area depicts the initialization +segment to initialize the oscillator in the high oscillation am- +plitude state. The phase is measured and averaged during the +green segment, for a time tacq. We wait a time tdown = 5 µs +between consecutive pulses and set flatch = 0. (b) Schematic +illustrating sigmoid S-curves for two different cCPT gate bi- +ases illustrated in pink and green, with the black arrow de- +noting the maximum contrast between the two. +The inset +shows a representative histogram of the reflected phase, φ, +upon running the above pulse sequence Ntot = 20, 000 times. +The two Gaussian distributions correspond to the oscillator +being in the high (left) and low (right) oscillation amplitude +states respectively, with the solid lines representing a double +Gaussian fit. (c) Obtained S-curves (�’s) for different ng val- +ues at Φext = 0.06Φ0 and an input drive power Pin = −128 +dBm. The averaging time, tacq = 3 µs. The solid lines are sig- +moid fits to Eq. (6). The horizontal error bars represent the +standard deviation of the resonant frequency fluctuations due +to charge and flux noise at the cCPT DC bias point [16, 39]. +point for our ramp protocol in Fig. 3(a). However, from +Fig. 3(c), we clearly do not see an abrupt step in P(ωd) +at just the bifurcation point, but a gradual change in +its value across a range of detunings, whose behavior for +different cCPT bias points and drive parameters we will +now study. +For systems where the ratio +∣K∣ +κtot < 1 [40], close to a +bifurcation, the switching between these two metastable +oscillation amplitude states is described by a quantum +activation model which predicts fluctuation-induced es- +cape over a metapotential barrier [35, 41]. This has been +demonstrated to accurately model the switching between +these states in nanomechanical systems [42, 43], Joseph- +son bifurcation amplifiers [23], and in Josephson junction +array devices [44, 45]. For systems with Kerr strengths +comparable to the cavity linewidth, a quantum calcula- +tion is required to accurately model this switching [40]. +We discuss some of the possible sources of these fluctua- +tions in Sec. IV. +From a charge sensing point of view, we want the S- +curves for two cCPT gate biases separated by a given +δng to have a large separation between their centers, +∆0, while the widths of these sigmoids, γ, should remain +small. Additionally, in order to perform single-shot mea- +surements separating the oscillation state using a thresh- +old phase value at the middle of the two Gaussian peaks +in the inset in Fig. 3(b), we need to minimize the overlap +between the Gaussians. +-800 +-700 +-600 +-500 +-400 +-300 +-200 +0 +20 +40 +60 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +5.76 +5.77 +5.78 +5.79 +5.8 +0 +0.06 +0.11 +0.15 +0.18 +0.21 +FIG. 4. (a) S-curve centers, ∆0, vs gate charge, ng, for dif- +ferent different flux biases, Φext, for Pin = −128 dBm. The +error bars are smaller than the markers. (b) S-curve widths, +γ, vs K at the cCPT DC bias points in (a), for different drive +strengths. The error bars are the 95% confidence intervals to +the sigmoid fits. +Fig. 4(a) shows the variation of the centers of the sig- +moid, ∆0, vs cCPT gate bias, ng, for different cCPT flux +biases, Φext. For each flux bias, the largest separations +between the centers of two S-curves are observed for large +gate biases. Given that we work close to ng = 0.71, the +largest separation is for flux values close to Φext = 0. This + +tacq +△/2π +4c +tstab +fiatch +-6 +0.5 +1 +1.5 +Time (μs)1-2 +-4 +-6a(b) +(pink) +(green) +0.8 +(ng + ong, ext +Contrast +(P3) +0.6 +P +0.4 +0.2 +green) +0 +Wd/2π(GHz6 +is related to both the large variation in the ground state +energy of the cCPT at these DC bias points, and con- +sequently the linear resonance frequency, ω0, as in Fig. +2(a) [16], as well as the variation of the metapotential +landscape in the bistable region that limits the extent of +switching between the two oscillation amplitude states at +a given set of drive parameters. The separation between +the ∆0 for two distinct cCPT bias points is also found to +be largest at low input powers, Pin (data not shown). Fig. +4(b) shows the variation of the width of the sigmoids, γ, +plotted against the theoretically computed value of the +Kerr coefficient, K, from Fig. +1(b) at different cCPT +bias points (ng,Φext), for different Pin. The cCPT DC +bias points we are interested in based on the separation +of the centers of the sigmoids in Fig. 4(a) correspond to +K/2π = −600 to -800 kHz, and we see γ is much smaller +for lower Pin at these bias points. This is related both +to the reduction in width of the bistable region with de- +creasing Pin, as seen from Eq.(5), and to the reduction +in barrier height of the metapotential with increasing Pin +[40, 46]. This reduced barrier height enables switching +between the two oscillation amplitude states within ex- +perimental timescales over a wider range of detunings. +The other consideration in demonstrating single-shot +readout is the resolution of the two Gaussians in the in- +set of Fig. 3(b). Fig. 5(a) shows the separation between +the centers of the two Gaussians for all the cCPT DC +bias points in Fig. 4(a) for a range of Pin. The small- +est Kerr strengths, ∣K∣, and the lowest drive strengths, +Pin, give us maximum separation. We understand this +as follows. The detuning corresponding to the maximum +oscillator response is either negative (spring softening) or +positive (spring hardening), based on sgn(K). The de- +gree of softening of the oscillation response curve (Fig. +1(c)) and hence the reflection coefficient, S11, illustrated +in Fig. 1(d-e) depends on ∣K∣. The low oscillation am- +plitude response is close to zero at all detunings in the +bistable region, with the corresponding S11 close to 1. +Meanwhile, the high oscillation amplitude increases from +close to zero at the upper bifurcation point to a maxi- +mum value at the lower bifurcation point, with a slope +inversely proportional to ∣K∣, with similar behavior for +S11. At detunings close to the upper bifurcation point +around which we see non-zero probability for both high +and low oscillation amplitude states within experimental +lifetimes, the high amplitude oscillation response and the +corresponding Phase(S11) at a given detuning assume a +finite non-zero value whose magnitude depends inversely +on ∣K∣, as seen in Fig. +5(a), such that the difference +in Phase(S11) between the high and the zero phase low +amplitude state also depends inversely on ∣K∣. +To understand the variation with Pin, we compare the +blue and the green curves in Figs. 1(c-e), which are both +for the same K. We see that the slope of the amplitude +response of both bistable states is nearly independent of +Pin, but close to the upper bifurcation point, the corre- +sponding S11 yields smaller separation between the high +and low oscillation amplitude states for higher Pin. Fig. +-800 +-700 +-600 +-500 +-400 +-300 +-200 +0 +50 +100 +-130.5 +-128 +-125 +-122 +-119 +-116 +0 +2 +4 +6 +8 +10 +80 +85 +90 +95 +100 +FIG. 5. (a) Measured separation between peaks of Gaussians +as in inset of Fig. 3(b) vs K for different Pin. (b) Histogram +count N (0.62) (N (0.71)) of the phase of the reflected signal for +optimal charge sensing. The data is for Ntot = 20, 000 trials +each at (ng, Φext) = (0.62, 0) (blue) and (ng, Φext) = (0.71, 0) +(red) respectively, while driving the cCPT with an input tone +at ωd/2π = 5.8013 GHz with Pin = −128 dBm. The dashed +line denotes the threshold phase, φth, used to discriminate the +charge state in a single-shot. (c) Measurement fidelity as a +function of averaging time tacq for the drive conditions in (b). +5(a) hence suggests that we work at low ∣K∣ and at low +Pin in order to observe maximum separation between the +reflected phase of the high and the low oscillation am- +plitude states. While we saw that the latter condition +also yields S-curves with the smallest widths (Fig. 4(b)), +Fig. +4(a) shows that the Φext corresponding to small +∣K∣ values correspond to poor separation between ∆0 for +two cCPT ng values, which is contradictory to our goal. +The pursuit of low ∣K∣ and low Pin suggests operating +the cCPT on the cusp of bistability where a large gain in +the dispersive readout is expected at a certain detuning +[17]. However, operation of the device studied here is not +possible in that regime as discussed further in Sec. IV. + +X（6。) +3000 +(0.71, +N(0.62) +2000 +1000 +0 +-40 +-20 +20 +40 +60 +O3000 +N(0.71) +2000 +1000 +0(b) +xt)=(0.62.0)7 +With these considerations, we obtain a maximum con- +trast of 96.61% in the S-curves between when the cCPT +is biased at (ng,Φext) = (0.62,0) and at (0.71,0), and +driven at ωd/2π = 5.8013 GHz with Pin = −128 dBm. For +this drive strength, using an averaging time, tacq = 3 µs, +Fig. +5(b) shows the obtained histograms with counts +N (0.62)(φ) and N (0.71)(φ) respectively. The separation +between the Gaussian peak centers is 36○ for this bias +point and Pin, and the width of each Gaussian is 12○ for +this tacq. Using a threshold value φth at the center of +the two Gaussian peaks as denoted by the dashed line, +we assign a charge state to each histogram data point. +Defining the fidelity F (0.62) = 1 − +1 +Ntot ∑180 +φ=φth N (0.62)(φ) +and F (0.71) = 1 − +1 +Ntot ∑φth +φ=−180 N (0.71)(φ), we obtain an +average fidelity F = 94.59 %. The similarity between the +obtained fidelity and the measured maximum contrast +which is agnostic to the overlap of the Gaussians caused +by the amplifier noise shows that for this tacq, the limiting +factor of our measurement is not the signal-to-noise ratio +of the amplifier chain, but is the broadening of the S- +curves caused by fluctuation-induced switching between +the metastable oscillation states. Finally, it is worth not- +ing that using the above DC bias and drive parameters in +Eq. (5) along with damping rates κint, κext extracted as +described in [16, 39], we find that the intracavity photon +number, n = 8.1 at ng = 0.62, and n = 20.94 at ng = 0.71 +respectively in the high oscillation amplitude state. At +ng = 0.71, for the optimal drive tone, the oscillator resides +predominantly in the low oscillation amplitude state with +an intracavity occupation on the order of 0.2 photons. +These are orders of magnitude fewer photons than used +by devices such as the rf-SETs [13, 47]. +IV. +DISCUSSION +The cCPT operating in the Kerr bistable regime is thus +sensitive to changes in its electrostatic environment that +produce a shift of δng = 0.09e and we have demonstrated +real-time single-shot high fidelity detection of this charge +difference in 3 µs. This corresponds to a charge sensi- +tivity per unit bandwidth = δng +√tacq = 155.89 µe/ +√ +Hz. +The bandwidth of this electrometer is set by κtot/2π ≈ 1.5 +MHz. This readout is performed with only a few tens of +intracavity photons, which is several orders of magnitude +smaller than in other state-of-the-art devices [13, 48]. An +application of the cCPT as a charge sensor is to detect the +state of a quantum-dot spin qubit using spin-to-charge +conversion [9, 10, 29]. The backaction by the charge sen- +sor on the system being measured is proportional to the +number of intracavity photons [49], making such low cav- +ity number operation desirable [27]. +Using techniques +such as defining the SET in the Si substrate [50, 51], and +extending the cCPT island [5] in order to increase the δng +induced on the cCPT island in the event of a spin tunnel- +ing out of a quantum dot, we could work at larger Pin for +the same Φext and corresponding K, while still achiev- +ing sufficient contrast and comparable fidelity for much +smaller tacq. For the more relaxed δng requirement, low +power operation with smaller tacq without compromising +fidelity would be possible at other Φext corresponding to +smaller ∣K∣ and larger phase separation between the high +and the low oscillation amplitude states as in Fig. 5(a), +while still retaining a large contrast value. +The major limitation to this mode of operation of the +cCPT as a charge sensor are the spontaneous fluctuation- +induced transitions between the high and low oscillation +amplitude states in the bistable regime. The metapoten- +tial landscape governing these transitions depends on Pin +and K [40, 46]. The ∣K∣/κtot range of the cCPT lies in +the interesting ‘mesoscopic’ region where quantum effects +begin to become important [40]. Mapping the metapo- +tential and corresponding switching rates between the +high and low oscillation amplitude state for such a device +could guide understanding of single-photon Kerr devices +[52] which have been proposed as single-photon sources +[53], to generate ultra-fast pulses [54] and to be used to +implement quantum non-demolition measurements [55]. +For a given metapotential, the intensity of the fluc- +tuations present in the system is the other factor that +affects the switching rates and hence the width of the S- +curves, γ. Given that thermal activation is unlikely since +̵hωd > kBT, one commonly studied source of fluctuations +is the dephasing of the oscillator caused by the modula- +tion of the resonant frequency [36], or equivalently, of the +detuning of the drive. The phase noise of the signal gen- +erator is typically 1/f in nature and is quite small in the +frequency range relevant for escape dynamics such as ob- +served in Fig. 2(c). The resonant frequency fluctuations +for systems such as the cCPT has been studied in de- +tail [39], and characterized [16]. The resonant frequency +fluctuations due to charge noise arising from fluctuating +two-level systems close to the cCPT island, and mag- +netic flux noise arising due to unpaired surface spins are +both 1/f in character, and should not be relevant to the +switching dynamics either. However, the frequency mod- +ulation due to white photon shot noise [49] induced Kerr +shift considered in Ref. [39] would explain the increase +in the width of the S-curves at larger ∣K∣ as in Fig. 4(b), +working in tandem with the reduced barrier metapoten- +tial barrier at larger ∣K∣. A careful study showing a direct +correlation between the switching rates of the cCPT and +Pin would confirm this hypothesis, since the frequency +independent power spectral density of the photon shot +noise depends on the average cavity occupation, n. +Another avenue to increase the sensitivity of the de- +vice would be to reduce the quasiparticle poisoning (QP) +in the device. We observe substantial switching out of +the even to the odd manifold of the CPT (ng → 1 − ng) +[56] for ∣ng∣ > 0.71 [16]. As illustrated in Fig. 2(a), the +cCPT resonance frequency, ω0, is most sensitive to ng +close to ∣ng∣ = 1, and employing techniques such as effec- +tive shielding from Cooper-pair-breaking, quasiparticle +generating radiation [57, 58] could greatly enhance the +performance of the cCPT. + +8 +Still using this inherent Kerr nonlinearity, but by driv- +ing the cCPT with a Pin ∼ P (c) +in +close to but before +the onset of bistability where +dS11 +d∆ +→ ∞ for some ∆, +we should be able to realize a large gain in the charge +sensitivity [17]. The presence of gate and flux noise in- +duced resonance frequency fluctuations [16, 39] make it +hard to operate at the precise ∆ where this enhancement +is expected, but using the resonance frequency stabiliz- +ing feedback scheme demonstrated in [37] should enable +such operation. Enhanced cooling of a nanomechanical +resonator coupled to a nonlinear cavity operating in this +regime has been shown [19]. +The cCPT Hamiltonian also has other nonlinear terms +such as those of a degenerate parametric amplifier which +can be driven into resonance using an appropriate flux +pump at close to 2ω0. The amplitude of the parametric +oscillations induced [59–61] depends on the gate bias of +the cCPT [62], and can be used as a charge sensor, similar +to the qubit state detector operating on this principle +[63]. +ACKNOWLEDGMENTS +We thank Ethan Williams, Hui Wang, Archana Ka- +mal, Chandrasekhar Ramanathan, William Braasch and +Hory Mohammadpour for helpful discussions and useful +feedback on the manuscript. This work was supported +by the NSF under Grants No. DMR-1807785 (S. K., B. +T., B. L. B., and A. R) and DMR-1507383 (M. P. B.), +and by a Google research award (S. K.). +Appendix A: Experimental setup +Fig. 6 shows the rf circuitry used in the experiments +described in this work. The input tone from a Keysight +N5183B signal generator is mixed with an intermedi- +ate frequency tone from a Tektronix arbitrary waveform +generator whose amplitude envelope can be ramped, or +whose frequency can be chirped for the charge sensing +protocol (see Fig. +3(a)). +This signal passes through +various stages of attenuation in the dilution refrigera- +tor before driving the cCPT, which is mounted inside a +magnetic shield at the mixing chamber stage of the re- +frigerator. The reflected signal goes through a circulator +to a traveling wave parametric amplifier (TWPA) [28] +which serves as the first stage amplifier. The signal is +then amplified by a Low Noise Factory LNF LNC4 8C +high electron mobility transistor (HEMT) and a room +temperature low noise field effect transistor (FET). This +signal is then mixed down to an intermediate frequency +of 21 MHz, filtered, digitally sampled, and demodulated +to extract the phase. +The TWPA has an average gain of 18 dB over the op- +erating bandwidth of the cCPT, ensuring that the added +noise of the amplifier chain is dominated by the noise +added by the TWPA. The added noise density referred +300 K +20 +dB +4.2 K +20 +dB +1.8 K +cCPT +30 mK +700 mK +20 +dB +TWPA +HEMT +FET +FET +R +L +I +R +L +I +AWG +LPF +BPF +ADC +FIG. 6. Microwave circuitry used in Sec. III. +to the input of the amplifier chain is separately measured +to be ∼4.67 photons/Hz (noise temperature of 1.28 K), +close to the quantum limit of 1 photon/Hz [49] for the +phase insensitive TWPA. +Appendix B: Charge sensing protocol initialization +Here, we elaborate upon the initialization section of +the charge sensing protocol illustrated in Fig. 3(a). In +order to initialize the oscillator in the high oscillation +amplitude state, we start from a detuning in the monos- +table region on the positive detuning side in Fig. 1(c-e), +and ramp the detuning by framp/2π = −41 MHz in a time +tr = 530 ns. +The final detuning is close to a bifurca- +tion edge, denoted by the black dashed line in Fig. 2(b). +The oscillator is driven with this constant tone for a time +tstab = 4.9 µs, during which time fluctuations could cause +a transition to the low oscillation amplitude state. These +values for tr and tstab were settled upon after performing +QuTiP [64] simulations using a master equation solver for +the exact input tone in Fig. 3(a), and seeing the system +through a transient evolution period to the steady state. + +9 +This value of tstab is also close to the nominal value of +5/(κtot/2π) over which transients of oscillating systems +are expected to decay, even in the region where switch- +ing between high and low oscillation amplitude states is +observed, where the oscillator dynamics are considerably +slowed [40]. The detuning could then be ramped to a +slightly larger blue-detuning, flatch, to reduce the proba- +bility of a switching event during the measurement time, +hence ‘latching’ the oscillator in the oscillation amplitude +state attained at the end of the stabilization time [23]. +However, unlike the systems studied in Refs. 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Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Dartmouth College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Hanover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' New Hampshire 03755,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' USA 2Rigetti Computing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' California 94710,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' USA 3Department of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Yale University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' New Haven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Connecticut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' USA 4High Energy Physics Divison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Argonne National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 9700 South Cass Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Argonne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' IL 60439,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' USA (Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2023) Operating the cavity-embedded Cooper pair transistor (cCPT) in the Kerr bistable regime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' we demonstrate single-shot resolution between two charge states that are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='09e apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The measurement is performed with 94% fidelity in a duration of 3 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The drive power at which the measurement is performed corresponds to only 20 intracavity photons on average in the high oscillation ampli- tude state of the cCPT, which is orders-of-magnitude smaller than that in rf-SETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We find that the limiting factor for this mode of operation of the cCPT is the spontaneous fluctuation-induced switching between the two metastable oscillation amplitude states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We present empirical data on the variation of the switching dynamics with drive parameters and cCPT DC bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' INTRODUCTION Fast detection of charge on the order of a fraction of an electron has long been an important task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Versatile devices such as the quantum point contact and the sin- gle electron transistor (SET) have been used to measure electron lifetimes in a single electron trap [1], to map electric fields with 100 nm spatial resolution [2], to ob- serve macroscopic charge quantization [3], and to study quasiparticle and electron tunneling events in real-time [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' More recently, they have been used in the search for Majorana zero modes in nanowires [6], and could po- tentially be used to detect dark matter [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Such fast, ultrasensitive electrometers are instrumental in the read- out of silicon-based spin qubits [9, 10] where the magnetic moment of a single spin is too small to detect directly, and is instead converted to a spin state dependent charge which can be read out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Dispersive charge sensors oper- ating on the supercurrent branch of the Josephson junc- tions based inductive-SET (L-SET) [11] and the single Cooper-pair box [12] are not shackled by the electron shot noise which limits the operation of the rf-SETs [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The cavity-embedded Cooper pair transistor (cCPT) used in this work has previously been shown to achieve a charge sensitivity of 14 µe/ √ Hz operating as a dispersive sensor in the linear regime with a single intracavity photon on average [14], close to the theoretical quantum limit for this device [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The cCPT is also a rich nonlinear system whose Hamil- tonian contains a Kerr nonlinearity [16], and an emergent parametric amplifier term when the flux line of the sys- tem is driven at twice the resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The Kerr term opens up the possibility of more sensitive charge de- tection than was achieved in the linear regime [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Such ∗ Bhargava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='Thyagarajan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='gr@dartmouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='edu † Alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='Rimberg@dartmouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='edu a Kerr cavity coupled to a mechanical resonator was pro- posed [18] and demonstrated [19] to achieve an order of magnitude better cooling of the phonon mode compared to a linear cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The Kerr nonlinearity is well known to produce bifurcations in the system response [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Bifur- cation amplifiers [21–23] based on a large change in the response at a bifurcation edge have been used to read out the state of superconducting qubits [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Nanomechani- cal devices based on the bifurcation under a parametric drive have been used to sense charges of 9e at room tem- perature [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Similar devices have demonstrated charge sensing of the order of 70e by manipulating the topology of the bifurcation diagram [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Here, using the bifurcation between a bistable and a monostable region induced by the Kerr nonlinearity of the cCPT, we demonstrate single-shot readout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='09e of charge in 3 µs with 94% fidelity, using fewer than 25 intra- cavity photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Such low power operation ensures mini- mal back-action on the system being measured [27], and also aids in the integration of these cCPT detectors with state-of-the-art first stage amplifiers such as the TWPAs [28] without overwhelming them beyond their compres- sion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Such fast high fidelity readout is compara- ble to the current state-of-the-art for semiconductor spin qubits [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' II we present a semi-classical analysis of the nonlinear cCPT and propose a scheme for it to function as a sensitive charge state discriminator operating in the bistable regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' III we experimentally study the hysteresis in the cCPT response in the bistable regime to characterize the extent of the bistability as a function of the drive detuning and strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We then implement a charge sensing protocol, and observe the presence of fluctuation-induced spontaneous transitions between the bistable states, which we study as a function of drive pa- rameters and cCPT DC bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Lastly, we characterize our charge sensing protocol and demonstrate the optimum high-fidelity, fast charge state readout possible with this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' IV we conclude by discussing some possi- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='00110v1 [quant-ph] 31 Dec 2022 2 ble improvements to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Details of the heterodyne measurement scheme employed in this work and the mi- crowave circuitry used in the dilution refrigerator are in Appendix A, and some experimental considerations for the charge sensing scheme used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' III are detailed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 1 30 20 10 0 10 100 200 300 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (a) Schematic of the cCPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (b) Variation of the Kerr coefficient K as a function of gate, ng, and flux, Φext, over the operational bias range of the cCPT simulated using the extracted values of EJ and EC [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (c) Simulated response of the cCPT for different drive powers, Pin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The red curve is for a drive strength Pin ≪ P (c) in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The blue curve is for Pin > P (c) in and the green is for Pin ≫ P (c) in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Above P (c) in , we see bistability across a range of detunings indicated by the corresponding shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The △’s represent the stable high oscillation amplitude state, the ▽’s represent the stable low oscillation amplitude states, and the �’s represent the unstable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The solid lines indicate monostability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (d) Simulated magnitude of the reflection coefficient for the drive powers in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (e) Simulated phase of the reflection coefficient for the drive powers in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' All simulations in (c), (d) and (e) were for a cCPT DC bias (ng, Φext) = (0, 0) with K/2π = −470 kHz, and nominal damping rates for this bias point [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' THEORETICAL DESCRIPTION The cCPT, schematically depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 1(a), con- sists of two parts: (i) the cavity, which is a λ/4 supercon- ducting coplanar waveguide (CPW) with its end shorted to the ground plane, and (ii) a Cooper pair transistor (CPT) across the center line and ground plane of the CPW at its voltage anti-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' In this geometry, the cav- ity and CPT form a shared SQUID loop, which couples them together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' When the CPT remains in its ground state, it modifies the effective potential of the cavity, such that the cCPT behaves as a nonlinear oscillator whose resonant frequency can be tuned using the effective gate charge, ng = CgVg e , and the magnetic flux thread- ing the SQUID loop, Φext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Here, Vg is the external DC voltage applied to the CPT island through the gate ca- pacitance Cg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Along with the fabrication details for the cCPT device used in this work, a detailed characteriza- tion at low drive amplitudes where the nonlinearities do not contribute substantially to the dynamics has been carried out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Notably, the Josephson energy, EJ, and the charging energy, EC, were estimated to be EJ/h = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='8 GHz and EC/h = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='1 GHz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Finally, to drive and measure the cCPT, a probe trans- mission line is coupled to the CPW through a coupling capacitor Cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' For an input drive close to resonance, under the rotat- ing wave approximation, the Hamiltonian for the cCPT is given by [15, 16] H = ̵hω0(ng,Φext)a†a + 1 2 ̵hK(ng,Φext)a†2a2, (1) where a(a†) are the annihilation (creation) operators for the cavity mode, ω0(ng,Φext) and K(ng,Φext) are the resonant frequency of the linear cCPT Hamiltonian and the Kerr coefficient respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The variation of K(ng,Φext) over the operational range of the cCPT de- vice used in this work is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The Kerr co- efficient changes sign with flux, attaining extremum val- ues at half-integer multiples of Φ0, and passing through zero close to Φext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='25Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Kerr-free cavities have been used to increase the dynamic range of parametric ampli- fiers [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We use input-output theory [33] to model the dynamics of the cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The quantum Langevin equation for a gives ˙a = 1 i̵h[a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='H] − [a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='a†](κtot 2 a − √κextain(t) − √κintbin(t)) = −(i(ω0 + Ka†a) + κtot 2 )a + √κextain(t) + √κintbin(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (2) and a corresponding equation for a†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' where κext is the ex- ternal damping rate due to the coupling of the resonator to the probe transmission line with the input bath op- erator ain(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' and κint is the internal damping rate as- sociated with the coupling of the resonator to an inter- nal loss channel with input operator bin(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The total (c) 60 C in in (c) P in 400 40 <△00 0 U 20 30 20 10 0 10 V /2π(MHz0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 0 extH/ 0 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5(b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5500 Z3 damping rate of the cavity is κtot = κext + κint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' When the input tone is a pure sine wave at frequency ωd of the form ⟨ain⟩ = αine−iωdt, the steady state response of the cavity is at this drive frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' For this coherent drive, using the semi-classical approximation, we make the ansatz ⟨a⟩ = αe−iωdt, with ⟨˙a⟩ = −iωdαe−iωdt and the average intracavity occupation number n = ∣α∣2 = ⟨a†a⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Plugging this ansatz into the expectation value of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (2) we obtain [−i(∆ − K∣α∣2) + κtot 2 ]α = √κextαin, (3) where we have defined the detuning ∆ = ωd − ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (3) and the input-output relation aout(t) = ain(t) − √κexta(t) [33], [34] we find the reflection coefficient S11(∆) to be S11(∆) = (αout αin ) ∗ = (∆ − K∣α∣2) − i(κint − κext)/2 (∆ − K∣α∣2) − i(κint + κext)/2 , (4) where aout(t) is the transmission output bath operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Also, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (3) and the corresponding equation for α∗, n = ∣α∣2 satisfies the cubic equation K2n3 − 2K∆n2 + (∆2 + κ2 tot 4 )n = κext Pin ̵hωd , (5) where Pin = nin̵hωd is the power of the input drive tone incident on the cCPT, and nin = ∣αin∣2 is the input pho- ton flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' As illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 1(c-e), at very low drive strengths this cubic equation has only one real root and the oscillator exhibits only monostable behaviour across all detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' As the drive strength is increased beyond a critical power P (c) in = √ 3 9 κ3 tot ∣K∣κext ̵hω(c) d , the oscillator sys- tem undergoes a bifurcation, and exhibits bistability for a range of detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Here, ω(c) d is the drive frequency corresponding to a detuning of ∆c = sgn(K) √ 3 2 κtot and (∆c,P (c) in ) corresponds to a spinode point in the param- eter space of the input drive [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' In the bistable region, two of the three real solutions of the cubic Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (5) corre- spond to high and low oscillation amplitude states with corresponding values of S11 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (4), while the third is an unstable, experimentally inaccessible state [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The variation of the resonant frequency of the cCPT with the gate, ng, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2(a), forms the basis for a sensitive dispersive charge detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Operat- ing in the single-photon, linear regime, this device was demonstrated to have a charge sensitivity of 14 µe/ √ Hz [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2(b) uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (4) to simulate the reflected phase as a function of drive frequency, ωd, for two gate values separated by δng corresponding to a resonant frequency shift δω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The S11 for the two gate values are denoted by pink and green curves respectively, both in the linear (Pin ≪ P (c) in ), single photon regime (dashed lines);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='77 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='78 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='79 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='81 data theory 125 120 115 110 100 80 60 40 20 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (a) Experimentally measured resonant frequency of the linear cCPT (Pin ≪ P (c) in ), ω0, as a function of the gate charge on the cCPT, ng, at a fixed flux bias Φext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='15Φ0 (triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The line is the theoretically expected resonant frequency for the junction parameters of this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (b) Sim- ulations illustrating the larger separation in reflected phase, δS11 (δS(Kerr) 11 ) when operating at Pin > P (c) in (solid lines) com- pared to Pin ≪ P (c) in (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (c) Phase of the reflected signal for a forward (solid line) and reverse (dashed line) tri- angular ramp of the drive amplitude, Pin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The input power is ramped between -140 dBm and -109 dBm in increasingly longer times from 2 µs to 28 µs from red to blue curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The cCPT was biased at (ng, Φext) = (0, 0) and the detuning was ∆/2π = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' with a drive power Pin > P (c) in (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' At a given ωd, ng can hence be inferred from the measured S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' For Pin > P (c) in , these simulations describe what we would ob- serve in the absence of spontaneous transitions between the high and the low oscillation amplitude states in the bistable region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' In the absence of these transitions, while ramping the drive detuning from a large blue-detuned value (with respect to linear resonance, ω0, ∆ > 0), to a (b) +Sng (c Phase(Sii(w)) in Kerr)1 in I (pink) green) 03 Drive Freq (wd)4 red-detuned value (∆ < 0), we expect to stay in the high oscillation amplitude state until we reach the bifurcation detuning further from ω0 for the green curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 1(c- e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We refer to this as the lower bifurcation point, while referring to the bifurcation detuning closer to ω0 as the upper bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Upon crossing the lower bifurca- tion point, an abrupt jump from the high to the low oscil- lation amplitude state is expected, with a corresponding large change in S11, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' For an appropriate drive frequency denoted by the dashed black line, the same separation in gate charge, δng, produces a larger difference in the reflected phase between the pink and green curves, δS(Kerr) 11 , than the δS11 while operat- ing in the linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Conversely, δS(Kerr) 11 continues to remain large as the green and pink curves are brought together by reducing δng, whereas δS11 undergoes sub- stantial reduction while doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The sensitivity of the charge detector is the smallest δng that produces a δS11 which can be detected with a signal-to-noise-ratio (SNR) of 1 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Given that the noise in the measurement is lim- ited by the amplifier chain in the experimental setup [14], the larger S(Kerr) 11 for smaller δng promises a lower, much improved charge sensitivity for the device operating in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' EXPERIMENTS In this section, we first describe the results of a tri- angular input power ramp in order to understand the extent of the bistable region with respect to the cCPT drive parameters at a given bias point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We then outline the protocol we use in order to perform charge sensing based on the bifurcation described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Contrary to the sharp jump in S(kerr) 11 at a precise value of the detun- ing described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' II, we see a non-zero probability of obtaining a value on either end of the step illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2(b) for a range of detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We discuss the re- sults of this protocol for a range of cCPT bias points and drive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' From this, we glean the optimal condi- tions for charge sensing and finally perform an optimized single-shot measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' In order to study the extent of the bistability, we bias the cCPT at (ng,Φext) = (0, 0), and drive it at a fixed detuning ∆/2π = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 MHz with a triangular amplitude ramp in the forward and the reverse direction to check for hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This is the bias point at which we expect minimum fluctuation in the resonant frequency of the cCPT due to charge and flux noise [16, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We perform a heterodyne measurement to obtain the phase of the reflected signal over the course of the ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The RF cir- cuitry used in the experiments described here is detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2(c) plots the observed hysteresis in the phase of S11 for different ramp rates, each averaged over 5000 repetitions of the ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' For fast ramps, we see that we obtain a value for the reflected phase correspond- ing to the low oscillation amplitude state for the forward ramp, and a value that corresponds to the high oscilla- tion amplitude state during the reverse ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' However, as the ramp time is increased, we observe that the spac- ing between the observed phase during the forward and the reverse ramps reduces, and for this cCPT bias point, saturates to the values represented by the blue curves, corresponding to ramp times of ∼ 25 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This is because, when given enough time to do so, the oscillator system undergoes fluctuation-induced spontaneous switching be- tween the high and low oscillation amplitude states over the course of a ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This yields a weighted average value for the phase at each Pin value over 5000 repeti- tions of the pulse sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The weights depend on the average lifetimes of the high and the low oscillation am- plitude states at the chosen cCPT bias and the drive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We see less variation in the shape of the for- ward and reverse ramp curves for the larger ramp times in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This provides a rough estimate of ∼ 25 µs for the average lifetimes of these bistable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This is a sign of spontaneous transitions between the high and low oscillation amplitude states for a range of input drive strengths, and will be detrimental to the charge sensing scheme described above which counts on the sharp jump from one oscillation amplitude state to the other at pre- cisely a bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' A similar reduction in the area enclosed between the curves corresponding to the forward and reverse ramps for longer ramps was recently observed for a nonlinear semiconductor microcavity [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' While performing the charge sensing measurement, we choose an input drive strength which gives rise to a re- gion of bistability (Pin > P (c) in ) for the chosen cCPT DC bias (ng,Φext) with a corresponding K < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' In order to deterministically initialize the oscillator in the high oscil- lation amplitude state, we perform a linear ramp on the detuning of the drive tone from a blue- to a red-detuning as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' More details on the initializa- tion section (shaded pink) of this protocol are provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Once initialized, we measure and average the phase of the reflected signal for a time tacq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Perform- ing this measurement Ntot = 20,000 times, we obtain a double Gaussian histogram as illustrated in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(b), and extract the probability of the high oscil- lation amplitude state, P(ωd), as the ratio of the area of the left Gaussian to the total area of the histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We perform this measurement for different detunings at the end of the initialization step of the pulse in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(a), and plot the obtained probability of being in the high oscillation amplitude state for each detuning, obtaining the S-curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We fit sigmoids of the form P(ωd) = 1 1 + exp{− 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='3944(ωd−∆0) γ } , (6) where ∆0 is the center of the sigmoid, and the numerical factor in the exponential ensures γ is its width between P(ωd) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='1 and P(ωd) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' As described earlier in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' II, we ideally expect an abrupt step in P(ωd) from 1 → 0 at the lower bifurcation 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='775 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='78 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='785 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='79 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='795 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71 100 0 100 0 1000 2000 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (a) Charge sensing protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Detuning of the input pulse tone used to initiate and readout the oscillation state in the charge sensing experiment described in the text with representative values for the durations and detunings of the different sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The pink area depicts the initialization segment to initialize the oscillator in the high oscillation am- plitude state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The phase is measured and averaged during the green segment, for a time tacq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We wait a time tdown = 5 µs between consecutive pulses and set flatch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (b) Schematic illustrating sigmoid S-curves for two different cCPT gate bi- ases illustrated in pink and green, with the black arrow de- noting the maximum contrast between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The inset shows a representative histogram of the reflected phase, φ, upon running the above pulse sequence Ntot = 20, 000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The two Gaussian distributions correspond to the oscillator being in the high (left) and low (right) oscillation amplitude states respectively, with the solid lines representing a double Gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (c) Obtained S-curves (�’s) for different ng val- ues at Φext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='06Φ0 and an input drive power Pin = −128 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The averaging time, tacq = 3 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The solid lines are sig- moid fits to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The horizontal error bars represent the standard deviation of the resonant frequency fluctuations due to charge and flux noise at the cCPT DC bias point [16, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' point for our ramp protocol in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' However, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(c), we clearly do not see an abrupt step in P(ωd) at just the bifurcation point, but a gradual change in its value across a range of detunings, whose behavior for different cCPT bias points and drive parameters we will now study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' For systems where the ratio ∣K∣ κtot < 1 [40], close to a bifurcation, the switching between these two metastable oscillation amplitude states is described by a quantum activation model which predicts fluctuation-induced es- cape over a metapotential barrier [35, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This has been demonstrated to accurately model the switching between these states in nanomechanical systems [42, 43], Joseph- son bifurcation amplifiers [23], and in Josephson junction array devices [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' For systems with Kerr strengths comparable to the cavity linewidth, a quantum calcula- tion is required to accurately model this switching [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We discuss some of the possible sources of these fluctua- tions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' From a charge sensing point of view, we want the S- curves for two cCPT gate biases separated by a given δng to have a large separation between their centers, ∆0, while the widths of these sigmoids, γ, should remain small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Additionally, in order to perform single-shot mea- surements separating the oscillation state using a thresh- old phase value at the middle of the two Gaussian peaks in the inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(b), we need to minimize the overlap between the Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 800 700 600 500 400 300 200 0 20 40 60 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='77 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='78 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='79 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='21 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (a) S-curve centers, ∆0, vs gate charge, ng, for dif- ferent different flux biases, Φext, for Pin = −128 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The error bars are smaller than the markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (b) S-curve widths, γ, vs K at the cCPT DC bias points in (a), for different drive strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The error bars are the 95% confidence intervals to the sigmoid fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 4(a) shows the variation of the centers of the sig- moid, ∆0, vs cCPT gate bias, ng, for different cCPT flux biases, Φext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' For each flux bias, the largest separations between the centers of two S-curves are observed for large gate biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Given that we work close to ng = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71, the largest separation is for flux values close to Φext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This tacq △/2π 4c tstab fiatch 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 Time (μs)1-2 4 6a(b) (pink) (green) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='8 (ng + ong, ext Contrast (P3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='6 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='2 green) 0 Wd/2π(GHz6 is related to both the large variation in the ground state energy of the cCPT at these DC bias points, and con- sequently the linear resonance frequency, ω0, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2(a) [16], as well as the variation of the metapotential landscape in the bistable region that limits the extent of switching between the two oscillation amplitude states at a given set of drive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The separation between the ∆0 for two distinct cCPT bias points is also found to be largest at low input powers, Pin (data not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 4(b) shows the variation of the width of the sigmoids, γ, plotted against the theoretically computed value of the Kerr coefficient, K, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 1(b) at different cCPT bias points (ng,Φext), for different Pin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The cCPT DC bias points we are interested in based on the separation of the centers of the sigmoids in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 4(a) correspond to K/2π = −600 to -800 kHz, and we see γ is much smaller for lower Pin at these bias points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This is related both to the reduction in width of the bistable region with de- creasing Pin, as seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (5), and to the reduction in barrier height of the metapotential with increasing Pin [40, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This reduced barrier height enables switching between the two oscillation amplitude states within ex- perimental timescales over a wider range of detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The other consideration in demonstrating single-shot readout is the resolution of the two Gaussians in the in- set of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 5(a) shows the separation between the centers of the two Gaussians for all the cCPT DC bias points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 4(a) for a range of Pin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The small- est Kerr strengths, ∣K∣, and the lowest drive strengths, Pin, give us maximum separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We understand this as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The detuning corresponding to the maximum oscillator response is either negative (spring softening) or positive (spring hardening), based on sgn(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The de- gree of softening of the oscillation response curve (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 1(c)) and hence the reflection coefficient, S11, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 1(d-e) depends on ∣K∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The low oscillation am- plitude response is close to zero at all detunings in the bistable region, with the corresponding S11 close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Meanwhile, the high oscillation amplitude increases from close to zero at the upper bifurcation point to a maxi- mum value at the lower bifurcation point, with a slope inversely proportional to ∣K∣, with similar behavior for S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' At detunings close to the upper bifurcation point around which we see non-zero probability for both high and low oscillation amplitude states within experimental lifetimes, the high amplitude oscillation response and the corresponding Phase(S11) at a given detuning assume a finite non-zero value whose magnitude depends inversely on ∣K∣, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 5(a), such that the difference in Phase(S11) between the high and the zero phase low amplitude state also depends inversely on ∣K∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' To understand the variation with Pin, we compare the blue and the green curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 1(c-e), which are both for the same K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We see that the slope of the amplitude response of both bistable states is nearly independent of Pin, but close to the upper bifurcation point, the corre- sponding S11 yields smaller separation between the high and low oscillation amplitude states for higher Pin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 800 700 600 500 400 300 200 0 50 100 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 128 125 122 119 116 0 2 4 6 8 10 80 85 90 95 100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (a) Measured separation between peaks of Gaussians as in inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(b) vs K for different Pin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (b) Histogram count N (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='62) (N (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71)) of the phase of the reflected signal for optimal charge sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The data is for Ntot = 20, 000 trials each at (ng, Φext) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='62, 0) (blue) and (ng, Φext) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71, 0) (red) respectively, while driving the cCPT with an input tone at ωd/2π = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='8013 GHz with Pin = −128 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The dashed line denotes the threshold phase, φth, used to discriminate the charge state in a single-shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (c) Measurement fidelity as a function of averaging time tacq for the drive conditions in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 5(a) hence suggests that we work at low ∣K∣ and at low Pin in order to observe maximum separation between the reflected phase of the high and the low oscillation am- plitude states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' While we saw that the latter condition also yields S-curves with the smallest widths (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 4(b)), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 4(a) shows that the Φext corresponding to small ∣K∣ values correspond to poor separation between ∆0 for two cCPT ng values, which is contradictory to our goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The pursuit of low ∣K∣ and low Pin suggests operating the cCPT on the cusp of bistability where a large gain in the dispersive readout is expected at a certain detuning [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' However, operation of the device studied here is not possible in that regime as discussed further in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' X（6。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=') 3000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71, N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='62) 2000 1000 0 40 20 20 40 60 O3000 N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71) 2000 1000 0(b) xt)=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='0)7 With these considerations, we obtain a maximum con- trast of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='61% in the S-curves between when the cCPT is biased at (ng,Φext) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='62,0) and at (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71,0), and driven at ωd/2π = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='8013 GHz with Pin = −128 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' For this drive strength, using an averaging time, tacq = 3 µs, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 5(b) shows the obtained histograms with counts N (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='62)(φ) and N (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71)(φ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The separation between the Gaussian peak centers is 36○ for this bias point and Pin, and the width of each Gaussian is 12○ for this tacq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Using a threshold value φth at the center of the two Gaussian peaks as denoted by the dashed line, we assign a charge state to each histogram data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Defining the fidelity F (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='62) = 1 − 1 Ntot ∑180 φ=φth N (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='62)(φ) and F (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71) = 1 − 1 Ntot ∑φth φ=−180 N (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71)(φ), we obtain an average fidelity F = 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='59 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The similarity between the obtained fidelity and the measured maximum contrast which is agnostic to the overlap of the Gaussians caused by the amplifier noise shows that for this tacq, the limiting factor of our measurement is not the signal-to-noise ratio of the amplifier chain, but is the broadening of the S- curves caused by fluctuation-induced switching between the metastable oscillation states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Finally, it is worth not- ing that using the above DC bias and drive parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' (5) along with damping rates κint, κext extracted as described in [16, 39], we find that the intracavity photon number, n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='1 at ng = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='62, and n = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='94 at ng = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71 respectively in the high oscillation amplitude state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' At ng = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71, for the optimal drive tone, the oscillator resides predominantly in the low oscillation amplitude state with an intracavity occupation on the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='2 photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' These are orders of magnitude fewer photons than used by devices such as the rf-SETs [13, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' DISCUSSION The cCPT operating in the Kerr bistable regime is thus sensitive to changes in its electrostatic environment that produce a shift of δng = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='09e and we have demonstrated real-time single-shot high fidelity detection of this charge difference in 3 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This corresponds to a charge sensi- tivity per unit bandwidth = δng √tacq = 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='89 µe/ √ Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The bandwidth of this electrometer is set by κtot/2π ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='5 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This readout is performed with only a few tens of intracavity photons, which is several orders of magnitude smaller than in other state-of-the-art devices [13, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' An application of the cCPT as a charge sensor is to detect the state of a quantum-dot spin qubit using spin-to-charge conversion [9, 10, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The backaction by the charge sen- sor on the system being measured is proportional to the number of intracavity photons [49], making such low cav- ity number operation desirable [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Using techniques such as defining the SET in the Si substrate [50, 51], and extending the cCPT island [5] in order to increase the δng induced on the cCPT island in the event of a spin tunnel- ing out of a quantum dot, we could work at larger Pin for the same Φext and corresponding K, while still achiev- ing sufficient contrast and comparable fidelity for much smaller tacq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' For the more relaxed δng requirement, low power operation with smaller tacq without compromising fidelity would be possible at other Φext corresponding to smaller ∣K∣ and larger phase separation between the high and the low oscillation amplitude states as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 5(a), while still retaining a large contrast value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The major limitation to this mode of operation of the cCPT as a charge sensor are the spontaneous fluctuation- induced transitions between the high and low oscillation amplitude states in the bistable regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The metapoten- tial landscape governing these transitions depends on Pin and K [40, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The ∣K∣/κtot range of the cCPT lies in the interesting ‘mesoscopic’ region where quantum effects begin to become important [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Mapping the metapo- tential and corresponding switching rates between the high and low oscillation amplitude state for such a device could guide understanding of single-photon Kerr devices [52] which have been proposed as single-photon sources [53], to generate ultra-fast pulses [54] and to be used to implement quantum non-demolition measurements [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' For a given metapotential, the intensity of the fluc- tuations present in the system is the other factor that affects the switching rates and hence the width of the S- curves, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Given that thermal activation is unlikely since ̵hωd > kBT, one commonly studied source of fluctuations is the dephasing of the oscillator caused by the modula- tion of the resonant frequency [36], or equivalently, of the detuning of the drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The phase noise of the signal gen- erator is typically 1/f in nature and is quite small in the frequency range relevant for escape dynamics such as ob- served in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The resonant frequency fluctuations for systems such as the cCPT has been studied in de- tail [39], and characterized [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The resonant frequency fluctuations due to charge noise arising from fluctuating two-level systems close to the cCPT island, and mag- netic flux noise arising due to unpaired surface spins are both 1/f in character, and should not be relevant to the switching dynamics either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' However, the frequency mod- ulation due to white photon shot noise [49] induced Kerr shift considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' [39] would explain the increase in the width of the S-curves at larger ∣K∣ as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 4(b), working in tandem with the reduced barrier metapoten- tial barrier at larger ∣K∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' A careful study showing a direct correlation between the switching rates of the cCPT and Pin would confirm this hypothesis, since the frequency independent power spectral density of the photon shot noise depends on the average cavity occupation, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Another avenue to increase the sensitivity of the de- vice would be to reduce the quasiparticle poisoning (QP) in the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We observe substantial switching out of the even to the odd manifold of the CPT (ng → 1 − ng) [56] for ∣ng∣ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='71 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2(a), the cCPT resonance frequency, ω0, is most sensitive to ng close to ∣ng∣ = 1, and employing techniques such as effec- tive shielding from Cooper-pair-breaking, quasiparticle generating radiation [57, 58] could greatly enhance the performance of the cCPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 8 Still using this inherent Kerr nonlinearity, but by driv- ing the cCPT with a Pin ∼ P (c) in close to but before the onset of bistability where dS11 d∆ → ∞ for some ∆, we should be able to realize a large gain in the charge sensitivity [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The presence of gate and flux noise in- duced resonance frequency fluctuations [16, 39] make it hard to operate at the precise ∆ where this enhancement is expected, but using the resonance frequency stabiliz- ing feedback scheme demonstrated in [37] should enable such operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Enhanced cooling of a nanomechanical resonator coupled to a nonlinear cavity operating in this regime has been shown [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The cCPT Hamiltonian also has other nonlinear terms such as those of a degenerate parametric amplifier which can be driven into resonance using an appropriate flux pump at close to 2ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The amplitude of the parametric oscillations induced [59–61] depends on the gate bias of the cCPT [62], and can be used as a charge sensor, similar to the qubit state detector operating on this principle [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Ethan Williams, Hui Wang, Archana Ka- mal, Chandrasekhar Ramanathan, William Braasch and Hory Mohammadpour for helpful discussions and useful feedback on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This work was supported by the NSF under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' DMR-1807785 (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' R) and DMR-1507383 (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' ), and by a Google research award (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Appendix A: Experimental setup Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 6 shows the rf circuitry used in the experiments described in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The input tone from a Keysight N5183B signal generator is mixed with an intermedi- ate frequency tone from a Tektronix arbitrary waveform generator whose amplitude envelope can be ramped, or whose frequency can be chirped for the charge sensing protocol (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This signal passes through various stages of attenuation in the dilution refrigera- tor before driving the cCPT, which is mounted inside a magnetic shield at the mixing chamber stage of the re- frigerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The reflected signal goes through a circulator to a traveling wave parametric amplifier (TWPA) [28] which serves as the first stage amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The signal is then amplified by a Low Noise Factory LNF LNC4 8C high electron mobility transistor (HEMT) and a room temperature low noise field effect transistor (FET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' This signal is then mixed down to an intermediate frequency of 21 MHz, filtered, digitally sampled, and demodulated to extract the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The TWPA has an average gain of 18 dB over the op- erating bandwidth of the cCPT, ensuring that the added noise of the amplifier chain is dominated by the noise added by the TWPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The added noise density referred 300 K 20 dB 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='2 K 20 dB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='8 K cCPT 30 mK 700 mK 20 dB TWPA HEMT FET FET R L I R L I AWG LPF BPF ADC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Microwave circuitry used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' to the input of the amplifier chain is separately measured to be ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='67 photons/Hz (noise temperature of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='28 K), close to the quantum limit of 1 photon/Hz [49] for the phase insensitive TWPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Appendix B: Charge sensing protocol initialization Here, we elaborate upon the initialization section of the charge sensing protocol illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' In order to initialize the oscillator in the high oscillation amplitude state, we start from a detuning in the monos- table region on the positive detuning side in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 1(c-e), and ramp the detuning by framp/2π = −41 MHz in a time tr = 530 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The final detuning is close to a bifurca- tion edge, denoted by the black dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The oscillator is driven with this constant tone for a time tstab = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content='9 µs, during which time fluctuations could cause a transition to the low oscillation amplitude state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' These values for tr and tstab were settled upon after performing QuTiP [64] simulations using a master equation solver for the exact input tone in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 3(a), and seeing the system through a transient evolution period to the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 9 This value of tstab is also close to the nominal value of 5/(κtot/2π) over which transients of oscillating systems are expected to decay, even in the region where switch- ing between high and low oscillation amplitude states is observed, where the oscillator dynamics are considerably slowed [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' The detuning could then be ramped to a slightly larger blue-detuning, flatch, to reduce the proba- bility of a switching event during the measurement time, hence ‘latching’ the oscillator in the oscillation amplitude state attained at the end of the stabilization time [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' However, unlike the systems studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' [23, 43], the low oscillation amplitude state is often not a long- lived state in our system, making the latching a little less likely, and causing the switching statistics to depend on the additional parameters flatch and tacq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' We thus set flatch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Dresselhaus, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Ji, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Han, J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Devoret, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Vion, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Esteve, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 94, 027005 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Esteve, Nature Physics 5, 791 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Dash, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' More, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Arora, and A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' B 100, 245427 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' [31] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Frattini, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Sivak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Lingenfelter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Shankar, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Devoret, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Applied 10, 054020 (2018).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 18, 064033 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Rodriguez, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Casteels, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Applied 14, 054026 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Andersen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Kamal, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Masluk, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Pop, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Blais, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Devoret, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} +page_content=' Nori, Computer Physics Communications 184, 1234 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQfTvcg/content/2301.00110v1.pdf'} diff --git a/LtAzT4oBgHgl3EQfyv51/content/tmp_files/2301.01758v1.pdf.txt b/LtAzT4oBgHgl3EQfyv51/content/tmp_files/2301.01758v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4c7dadcf69e99dd6ffd8f9278e321026e1c593ce --- /dev/null +++ b/LtAzT4oBgHgl3EQfyv51/content/tmp_files/2301.01758v1.pdf.txt @@ -0,0 +1,2254 @@ +PRE-PRINT. SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING. +1 +An Ensemble Mobile-Cloud Computing Method +for Affordable and Accurate Glucometer Readout +Navidreza Asadi, and Maziar Goudarzi, Senior Member, IEEE +Abstract—Despite essential efforts towards advanced wireless medical devices for regular monitoring of blood properties, many such +devices are not available or not affordable for everyone in many countries. Alternatively using ordinary devices, patients ought to log +data into a mobile health-monitoring manually. According to medical specialists, it causes several issues: (1) due to the direct human +intervention, it is prone to errors, and clients reportedly tend to enter unrealistic data; (2) typing values several times a day is bothersome +and causes clients to leave the mobile app. Thus, there is a strong need to use now-ubiquitous smartphones, reducing error by capturing +images from the screen of medical devices and extracting useful information automatically. Nevertheless, there are a few challenges in its +development: (1) data scarcity has led to impractical methods with very low accuracy: to our knowledge, only small datasets are available +in this case; (2) accuracy-availability tradeoff: one can execute a less accurate algorithm on a mobile phone to maintain higher availability, +or alternatively deploy a more accurate and more compute-intensive algorithm on the cloud, however, at the cost of lower availability +in poor/no connectivity situations. We present an ensemble learning algorithm, a mobile-cloud computing service architecture, and a +simple compression technique to achieve higher availability and faster response time while providing higher accuracy by integrating +cloud- and mobile-side predictions. Additionally, we propose an algorithm to generate synthetic training data which facilitates utilizing +deep learning models to improve accuracy. Our proposed method achieves three main objectives: (1) 92.1% and 97.7% accuracy on two +different datasets, improving previous methods by ∼40%, (2) reducing required bandwidth by 45× with ∼1% drop in accuracy, (3) and +providing better availability compared to mobile-only, cloud-only, split computing, and early exit service models. +Index Terms—Mobile Computing, Ensemble Learning, Data Generation, Deep Learning, Smart Health +! +1 +INTRODUCTION +M +ANY mHealth/uHealth medical devices, especially +those affordable in middle/low-income countries, +show the measured quantity on a digital or seven-segment +screen alongside other additional information such as date, +time, diagrams and measurement units. In most commonly +used mHealth services, particularly for diabetics, patients +are required to manually type sensed values into their +mobile app. As illustrated in Fig. 1(a), a client first reads +a value from the medical device, opens the app, navigates +to logging interface, and eventually logs the information +through typing. These steps should be repeated every time +they log a measurement. +1.1 +Motivation +According to the reports [1], [2] as well as our own expe- +rience from our diabetes management application iDia [3], +this procedure has a few drawbacks: (1) Manual logging +multiple times a day, is deterring; it is bothersome and time- +consuming, and based on the feedbacks we have received, +leads users to eventually lose their interest in using the +app. (2) It is prone to human errors. More importantly, +the observations show that patients are tempted to enter +fake information that are more acceptable and closer to the +N. Asadi is now with Computer Engineering Department, Technical Univer- +sity of Munich, Germany (navidreza.asadi@tum.de). He was with Computer +Engineering Department, Sharif University of Technology, Iran while working +on this project. +M. Goudarzi is with the Computer Engineering Department, Sharif Univer- +sity of Technology, Iran (goudarzi@sharif.edu). +Manuscript submitted to IEEE Transactions on Mobile Computing. +normal values. This can have considerable negative effects +on the whole process of prevention, control, and treatment. +Currently, there are two better approaches: (1) Some +medical devices are able to transmit their data to mobile +devices via distant communication technologies (e.g., Blue- +tooth), facilitating the logging procedure (Fig. 1(b)). Never- +theless, they are far more expensive and in some cases less +accurate [4]. More importantly, most of them can only in- +teract with their own software applications and do not per- +mit third-party apps to receive data. In addition, different +versions of transmission technologies cause incompatibility +between different mobile and medical devices. Despite a +potentially bright future, these devices currently hold less +than one percent of the market [5]. This number is even +lower in developing countries. (2) An alternative which +has grown interest in academia, uses image capturing and +computing capabilities of mobile devices (i.e., smartphones); +they are available to almost everybody and are able to +perform light-weight computing tasks. Fig. 1(c) illustrates +this alternative: the mobile phone is used to capture an +image of the medical device, and then image processing +is applied to recognize the sensed values automatically. In +this approach (Fig. 2), each digit is considered as an object +having a region of interest (RoI) and an ordered sequence +of digits (i.e., RoIs) forms a read string. A correct read +means not only all digits are predicted correctly, but also +in the same order as displayed on the device. In this paper +we follow this methodology for its broader applicability +especially in middle/low-income countries. +arXiv:2301.01758v1 [cs.DC] 4 Jan 2023 + +PRE-PRINT. SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING. +2 +Read +Medical Device +12.4 +mmol +Take a Test +Open the App & +Navigate to Log Interface +Manually Type +Observed Value +1 +2 +3 +4 +Mobile Phone +Patient +(a) Manual +Medical Device +12.4 +mmol +Take a Test +Open the App & +Navigate to Log Interface +1 +2 +Transfer Sensed Value via Bluetooth +3 +Mobile Phone +Patient +(b) Wireless Communication +Medical Device +12.4 +mmol +Take a Test +Open the App & +Navigate to Log Interface +1 +2 +Capture an Image from Screen +3 +Patient +Mobile Phone +(c) Image-Based +Fig. 1. Logging methods. Removing human interventions (red arrows) is preferred to avoid false logs. + RoI  + RoI  + RoI   RoI  +. +1 +1 +9 +1 +2 +11.9 +mmol/L +Fig. 2. Image-based method, showing Regions-of-Interest (RoIs) and +the correct labels. The expected correct readout is “11.9 mmol/L”. +1.2 +Challenges +Accuracy. Although automated image-based reading may +seem basic and easy, there are a few points that prove it +a challenging problem in this specific case: (1) the current +state-of-the-art [6], which has largely improved the previous +algorithms, uses only two medical devices, yet achieves +just 51.5% accuracy, meaning it misreads almost half of +the captured images. (2) as we discuss in Sections 2 and 3, +the diversity of medical devices and a variety of structural +and visual differences as well as noisy information on their +screens (e.g., date and time) make it extremely difficult to +read sensed value with business-as-usual image processing +techniques. +Data Scarcity. The accuracy of deep learning algorithms +relies on either big volume of annotated data to be trained +in a supervised manner or a model pre-trained on a related +task. To the best of our knowledge, there exists neither a big +dataset nor a related pre-trained model in our particular task +of imaged-based reading. Generative models (e.g., GANs) +also would not help there because they require similar data +to train, which is not applicable in our case. Additionally, +it might be really difficult (if not impossible) to generate +automatic annotation for our usecase. +Resource Constraints. Current state-of-the-art deep learn- +ing models are compute-intensive and need to be deployed +on specialized cloud infrastructures. Thus, they introduce +new challenges, including availability during poor network +conditions, that is genuinely an expected issue in our target +under-developed countries. Edge Computing [7], referring +to every device near data source with some compute ca- +pacity (e.g., smartphones), is considered as a promising ap- +proach to improve availability and quality of service (QoS) +by reducing delay. Mobile devices are usually resource- +constrained, and therefore, can only run simpler deep neural +network (DNN) models with much lower accuracy. Cloud +Computing, however, is at the opposite side. +1.3 +Main Contributions +We propose practical solutions to address the challenges +above. We present an ensemble mobile-cloud computing ar- +chitecture to get the best of both worlds: higher availability +on the mobile, as well as higher accuracy and enhanced per- +formability (i.e., a measure of level of performance/service- +quality of the system) by integrating cloud and mobile +modules (Fig. 3, described in §3). The followings items are +our main contributions: +(1) A hybrid mobile-cloud service architecture, together +with a compatible ensemble deep learning algorithm. This +enables an accuracy-availability tradeoff based on network +connectivity. We addressed the challenges of combining +predictions of two separate models; e.g., differences and +overlaps in the identified bounding boxes for each data +element. +(2) A simple yet effective compression method. Combined +with our ensemble model, this provides higher accuracy +despite little communicated data. +(3) A high-fidelity data synthesizer algorithm, making utiliz- +ing deep learning models possible. This has basically turned +the challenge into an opportunity; the variety of glucometer +models, data formats, units, fonts, etc. is a challenge to con- +ventional methods, but we used it in our data synthesizer +mechanism to produce enough reasonable data to train high +accuracy models that cover many varieties including those +not seen before. +Our proposed method achieves 92.1% and 97.7% accu- +racy on two real-world datasets, and improves previously +published results by more than 40%. It reduces the required +bandwidth by 45×, and maintains higher availability com- +pared to mobile-only, cloud-only, split computing, and early +exit service models. Our proposal can be easily extended to +other usecases with minor modifications. +The rest of this paper is organized as follows: In Sec- +tion 2, we review related work. In Section 3, we present our +proposed method. Section 4 explains our dataset generation +algorithm. We evaluate our methods and algorithms in +Section 5 and conclude in Section 6. +2 +RELATED WORK +We separate related work into two different parts. The +first one presents algorithms for reading sensed values or +detection and recognition of digits on digital or seven- +segment screens. The second one summarizes the studies +that attempt to deploy part or whole of a deep learning +model on the mobile. + +mmo +14:20 +10-11 +GlucoMen +areoPRE-PRINT. SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING. +3 +Camera Sensor +Edge Side +(1) Capture Image +Edge Computing +(2) Pre-Process Image +(3.1) Call Server-Side Inference API +(5) Ensemble Results +(6) Post-Process +Incorrect +Label? +Predicted Label +Cloud Side +    Web Services + + + + +Storage +(3.2) Run Edge-Side Inference Engine +(4) Run Cloud-Side +Inference Engine +Get Prediction +Yes +(8) Store Image and +Correct Label +Send Image +and Parameters +Send Image and +Correct Label +(7) Call WrongLabel API +Fig. 3. Our mobile-cloud architecture. +2.1 +Image-Based Automated Reading +Most of the proposed methods break the problem into +multiple steps including image enhancement, localization of +RoIs, detection and classification, and eventually ordering. +We review related work within five criteria, as summarized +in Table 1. +Automated Localization. Locating the RoIs is a crucial +step and impacts the final accuracy. Many works try to +simplify the problem while assuming the localization step +is somehow already done, either manually by a client, or by +fixing the device or using special markers [8], [9], [10], [11], +[13], [15]. A few others [6], [12], [16], and our work take a +more holistic approach, applying automated localization as +well. +Accuracy. To our knowledge, no previous work achieves +a reasonable readout accuracy. [6] with 51.5% has by far +the best performance. General OCR engines also are not +helpful; our previous experiments along with the reports +from [15], [16], and [6] confirm the state-of-the-art OCR +engine, Tesseract’s [14] poor performance (<10%) in this +specific task. On the other hand, our method can reach over +90% accuracy on both datasets. +Robustness. Different methods, especially those using con- +ventional algorithms are usually sensitive to different condi- +tions such as skewness, various noises, camera perspective +and angle, exposure, and illumination. Here, we define +robustness as performing consistently in different condi- +tions. That said, only few works [6], [16] try to adress it. +We simulate all these variations in our data synthesizer. +TABLE 1 +Summary of Literature Review on Image-Based Automated Reading. +Work +Localization +Accuracy +Robustness* +Generalization +Response Time +[8] + + + + + +[9] + + + + +? +[10] + + + + +? +[11] + + + + +? +[12] + + + + + +[13] + + + + + +[14] + + + + + +[15] + + + + + +[16] + + + + + +[6] + + + + + +Ours + + + + + +* Robustness to different environmental conditions. +? Information not available. +Addtionally, the ensemble of two different models, mitigates +such errors. +Generalization. There are a variety of medical devices each +having unique characteristics, such as different font styles +(including various types of seven-segments and/or digital +styles), background and foreground colors, screen size and +shape, units, backlit, etc. (Fig. 4). Nevertheless, the previous +studies, except [16], use one or very few specific devices +so that their proposed algorithms highly depend on the +properties of those selected devices; hence, may not be con- +sidered as a general solution. For example, [13] is designed +for a particular medical device with a blue backlit screen, [6] +considers only seven-segment screens, and [15] assumes the +largest contour as the screen, and hardcodes exact location +of RoIs. In contrast, we cover a broad market, and illustrate +this using an additional public dataset from Oxford [6]. +Respone Time. Since larger portion of the previous stud- +ies use conventional and light-weight image processing +methods, they can achieve reasonable response time. For +instance, [13] is implemented on a Samsung Galaxy S i9000 +mobile device, and can process 20 frames per second, or +[12] is deployed on an N95 mobile device, and achieves five +frames per second. The only exception is [15] which uses +a deep convolutional neural network (CNN) for the digits +classification step. It takes 10 seconds, that is unsatisfactory. +Although we leverage CNN-based object detection models, +we choose parameters so that the response is prepared +in less than half a second. Besides, we design a simple +compression technique when using the cloud-side engine +in poor network conditions, and, therefore, reduce the end- +to-end response time. +2.2 +Deep Learning on Edge +In practice, an edge device can be any computing machine +(indluding smartphones) that generates data or is near a +data generation source. Edge devices are usually resource- +constrained. So it is challenging to deploy big deep learning +models on edge. A concise comparison of different methods +is provided in Table 2. The efforts to overcome the limita- +tions can be divided into three major directions. +The first direction is designing light-weight DNN mod- +els or optimizing existing ones. Several successful works +have studied light-weight models including [17], [18], [19], +[20], [21], and [30]. In general, related work in this cate- +gory leverage a combination of using convolution blocks +with lower parameters (e.g., separable convolutions), quan- +tization, pruning, and model distillation. These techniques +TABLE 2 +Related Work on Deep Learning at Edge: A summary. +Method +Works +Accuracy +Availability +No-Cloud* +Bandwidth** +Response Time +Deployment +D1 +[17], [18], [19], +[20], [21] + + + +- + +E +D2 A1 +[22], [23], [24] + + + + + +E,C +D2 A2 +[25], [26], [27] + + + + +? +E,C +D3 +[27], [28], [29] + + + + + +E or C +Ours + + + + + +E,C +* Independence from a central cloud entity. +** Bandwidth usage optimization. +D: Direction +A: Approach +E: Edge +C: Cloud + +388PRE-PRINT. SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING. +4 +(a) +(b) +(c) +(d) +(e) +(f) +(h) +(i) +(j) +Fig. 4. Examples of images captured from medical devices. +Output +Input +(a) Cloud-Only +Output +Input +(b) Mobile-Only +Input +Output +(c) Split Computing +Early Output +Output +Input +(d) Early Exit +Output +Ensemble Algorithm +Post Process  +Lightweight Less-Accurate +Object Detector +Bigger More Accurate Object Detector +Cloud-Side +Output +Input +Edge-Side +Output +Input +(e) Our Architecture +Fig. 5. Deep Learning Serving Architectures. +mainly focus on optimizing response time and memory +footprint, and thereby sacrifice accuracy (Fig. 5(b)). +The second direction, distributes computation across +edge and central cloud, vertically (Fig. 5(c)). Some tech- +niques aim to reduce the required computation and band- +width by dropping insignificant frames of input stream at +the edge, before sending them to the cloud, and depending +on the nature of a task may follow different filtering policies +[22], [23], [24]. Some distribute the inference model across +edge and cloud (Split Computing) [25], [26], [27], and usu- +ally trade accuracy for better response time or bandwidth +usage. These methods rely on central cloud; hence, in the +case of network outages, they become unavailable. +Another recent direction determines one or more exit +points +within +neural +model +(including +pre-processing +steps). Exit points are usually designed so that at least one +of them stay on the edge. Depending on the task and its +requirements, the model can exit early, sacrificing accuracy +to meet delay constraints [27], [28], [29]. Despite their ap- +pealing results, they still are immature and are evaluated on +simple tasks such as classification (Fig. 5(d)). +In comparison, the strength of our proposed mobile- +cloud architecture (Fig. 5(e)) is its ability to take advantage +of both worlds: it improves accuracy, and in general per- +formability as well as availability, thanks to the independent +nature of our ensemble models. +3 +PROPOSED METHOD +Our service captures images from medical devices using +phone’s camera (Fig. 3(1)), then performs a pre-processing +step on mobile (Fig. 3(2)). The mobile device concurrently +sends the prepared image to cloud (Fig. 3(3.1, 4)), while +executing light-weight inference engine locally (Fig. 3(3.2)). +After receiving predictions of both sides, the mobile device +runs our ensemble algorithm (Fig. 3(5)). It also performs +a post-processing step, to correct the initial answer that +is produced by the mobile, if required (Fig. 3(6)). If user +recognizes a misprediction, they can send the image and +its corresponding true reading (Fig. 3(7)) to cloud storage +for future analysis and training iterations (Fig. 3(8)). The +mobile model provides availability, while the cloud model +improves performability (Refer to §5.5). Both models to- +gether improve accuracy. +We reduce the problem into object detection so that digits +of the sensed value (and not other digits illustrating noisy +information such as temperature, time and date) are objects +of interest, together with a post-processing step to reorder +objects and prepare the final response. DNN-based methods +have shown remarkable results, but to get the most out of +them, we need a well-annotated training dataset, which is +not available for our problem. We design a better workflow +and a data generation algorithm to automatically synthesize +thousands of training images with well-aligned annotations. +Our data generation workflow is described in detail in §4. +We use mobile smartphones to capture images from +medical devices and to extract useful information. While in- +ference on the mobile device provides low latency and high +availability for users, it usually suffers from low accuracy +due to resource constraints. On the other hand, inference +on the cloud provides accuracy and performability, but + +D01 +42 +6.59m +arkraukra388ACCU-CHEK +AvivaConnec +12:00 +3/11/15 +106 +AddComment +8:38 +85mg/dL +0Optium +105 +6h +12-10 +APRE-PRINT. SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING. +5 +high latency and unavailability in the case of poor network +connection are its big weaknesses. As depicted in Fig. 3, our +proposed hybrid mobile-cloud serving architecture takes +advantage of both mobile and cloud computing. Addition- +ally, we designed a specialized ensemble model to further +improve our desired performance metrics. +3.1 +Deep Learning Models +We reduce the problem of image-based reading from +screen of medical devices to an object detection and post- +processing task. Each digit and decimal point belonging to +the sensed value occupies a region of interest (RoI) and has +a corresponding class (Fig. 2-1). In post-processing step, we +convert a group of objects to a meaningful string (Fig. 2- +2). For the object detection part, we design and train two +convolutional models (CNNs) based on single-shot detector +(SSD) architecture [31]. +In general, we have powerful resources on cloud, but +limitation at the mobile. Thus, we consider two proportion- +ate backbones for our SSD architectures. One is highly op- +timized for smartphones and has fewer parameters; hence, +lower accuracy. Another one is more accurate and consumes +more resources so that can not be deployed on mobile. +We prefer SSD-based networks for both sides, because they +achieve better end-to-end latency [32]. +3.1.1 +Cloud-Side Model +We employ ResNet-50 [33] as the backbone of our SSD +model on the cloud-side. It contains 16 convolutional blocks +with shortcuts and one fully connected layer. It has more +than 25M parameters, and requires 4 billion multiplication- +accumulation operations (MACs) per sample. We remove +its last few layers including the classification head to use +the remainder as a feature extractor backbone for the SSD +architecture. Since the size of RoIs in input images varies, +we utilize feature pyramid network (FPN) [34]. It leverages +intrinsic pyramid structure of modern CNNs (i.e., ResNet) +to generate multi-scale feature maps. FPN can improve +accuracy of the model, and is much more compute efficient +than feeding an image with different sizes multiple times. +3.1.2 +Mobile-Side Model +Similar to the cloud-side, we stacked up a CNN-based +classification model as a feature extraction backbone for our +mobile-side SSD architecture. However, we employed edge- +friendly architectures both for the backbone and the detec- +tion head. Our mobile-side model is directly inspired by +the current state-of-the-art detection architecture for mobile +devices, the MobileDets [35]. The reason of using depth-wise +separable convolutional layers instead of regular convolu- +tional layers was their fewer parameters and MACs, while +hoping these metrics directly relate to less train and infer- +ence time. However, recent studies [36], [37] have shown +network parameters and MACs may not be good proxies +to model inference throughput and latency as they are not +the only factors. Therefore, MobileDets expands the neural +architecture search space by adding regular convolutions +as well. After multiple experiments on different backbone +architectures, we found the architecture proposed by [35] +for Mobile CPUs is the most appropriate for our mobile- +side model. Since mobile devices are resource-constrained, +we do not leverage FPN. +3.2 +Ensemble Algorithm +As illustrated in Fig. 3, we design and deploy a light-weight +engine at the mobile along with a more compute-intensive +and accurate one on the cloud. We integrate predictions +of our two deep learning models through our ensemble +algorithm (Algorithm 1). Having regions of interest (RoIs), +labels and confidence scores of the objects detected by both +models, our ensemble algorithm first finds corresponding +RoIs among mobile and cloud predictions (line 3). They +must have identical labels (e.g., class of digit ’5’) and their +distance should not be more than a tolerable amount ϵ. +The intuition behind is, the RoI coordinates predicted by +different object detection models may not exactly match +with each other. ϵ controls maximum distance between two +corresponding predictions from two different models. There +may exist more than one object with same labels. So if we +do not check that, it will lead to misrecognition of RoIs. +However, strict comparison is a bad idea because two quite +different deep learning models may have small differences +in RoI refinement after training. Hence, we add a tolerable +distance to make it more suitable. +After finding, when both scores are higher than specified +thresholds, the ensemble algorithm adds them up as the +final confidence score (line 7). Otherwise, picks the high- +est score (line 9). In the case of only one prediction for a +Algorithm 1 Ensemble Learning Algorithm +Input: 1⃝ maximum number of RoIs (N). 2⃝ RoIs matrices (R), +and their corresponding vectors of 3⃝ confidence (ρ) and +4⃝ labels (L). 5⃝ confidence thresholds (T). 6⃝ tolerable +distance (ϵ). +▷ Each RK +l +comprises four member points: left (xmin), +right (xmax), up (ymin), down (ymax). +▷ Superscripts C, M and E stand for Cloud, Mobile (Edge) +and Ensemble, respectively. +Output: A string equivalent to final prediction (e.g., ”10.6”). +1: RE ← ∅, ρE ← ∅, LE ← ∅ +2: for i ← 1 to N do +3: +Find j such that: +-RC +i [xmin] ≈ RM +j [xmin] ± ϵ and +-RC +i [xmax] ≈ RM +j [xmax] ± ϵ and +-LC +i = LM +j +4: +if j ̸= Ø then +5: +LE ← LC +i ∪ LE, RE ← RM +j ∪ RE +6: +if ρC +i ≥ T C +i and ρM +j +≥ T M +i +then +7: +ρE ← (ρM +j + ρC +i ) ∪ ρE +8: +else +9: +ρE ← max(ρM +j , ρC +i ) ∪ ρE +10: +end if +11: +RM ← RM−RM +j , ρM ← ρM−ρM +j , LM ← LM−LM +j +12: +else +13: +LE ← LC +i ∪ LE, ρE ← ρC +i ∪ ρE, RE ← RC +i ∪ RE +14: +end if +15: end for +16: RE ← RM ∪ RE, ρE ← ρM ∪ ρE, LE ← LM ∪ LE +17: I ← {i ∈ N | i ≤ |ρE|, ρE +i ≥ T E} +18: LE ← [LE +i∈I | ∀(i, j) : i < j ⇔ RE +i [xmin] ≤ RE +j [xmin]] +19: return ∥n +i=1LE +i +▷ (∥ concats every element within LE). + +PRE-PRINT. SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING. +6 +particular object on either mobile or cloud, simply adds it +to our final results (lines 13, 16). The second elimination +step happens in (line 17), where all remaining objects are +compared via a higher universal threshold. Eventually, our +algorithm reorders the remaining objects by their placement +within image (line 18), and concatenates a sequence of cor- +responding labels to generate the output string (line 19). +3.3 +Post-Processing Algorithm +To remove redundant objects more, and to improve accu- +racy even further, we add an additional post-processing +algorithm right before the line 18 in Algorithm 1. We apply +a modification of Non-Max Suppression (NMS) [38] to be +compatible with our problem. Here, our objective is to find +and remove those RoIs that significantly overlap with each +other. It means that there are more than one object of interest +in a region while it must not. Leveraging NMS reduces false +positive (FP). Our modified NMS is defined in Algorithm 2. +It executes iteratively. Having RoIs and their corresponding +confidence scores and labels, it first sorts RoIs based on their +confidence in a descending order (line 7). It then removes +those objects that their overlap of the occupied area with +another object is high enough (> Tnms), and their confi- +dence is simultaneously lower (lines 13, 14). The overlap +between two objects is calculated by intersection over union +(IoU) of their corresponding RoIs. Although this works well, +using a single threshold for all labels introduces a problem +in our specific task. The unit label, representing the decimal +point of sensed numbers, considerably overlaps with other +objects, but it must not be removed. This also applies to the +classes of ’1’ and ’7’. To avoid that, we consider the unit +label separately in our calculations within NMS procedure +(lines 2-6). +Algorithm 2 Post-Processing Algorithm +Input: 1⃝ overlap threshold (Tnms). 2⃝ RoIs matrix (R), and its +corresponding vectors of 3⃝ confidence (ρ) and 4⃝ labels +(L). +Output: reduced RoIs matrix, and its corresponding vectors of +confidence and labels. +1: RM ← ∅, ρM ← ∅, LM ← ∅ +2: i ← j ∈ N | j ≥ |L|, Lj =’.’ +▷ (’.’≡ decimal point) +3: if i ̸= Ø then +4: +LM ← Li, ρM ← ρi, RM ← Ri +5: +L ← L − Li, ρ ← ρ − ρi, R ← R − Ri +6: end if +7: (L, ρ, R) ← [(Li, ρi, Ri) | ∀(i, j) : i < j ⇔ ρi ≥ ρj] +8: while R ̸= ∅ do +9: +RM ← R1 ∪ RM, R ← R − R1 +10: +LM ← L1 ∪ LM, L ← L − L1 +11: +ρM ← ρ1 ∪ ρM, ρ ← ρ − ρ1 +12: +for i ← 1 to |R| do +13: +if IoU(Ri, RM +1 ) ≥ Tnms then +14: +R ← R − Ri, L ← L − Li +15: +end if +16: +end for +17: end while +18: return RM ∪ R, LM ∪ L, ρM ∪ ρ +3.4 +Bandwidth Optimization +Our target usecase is in underdeveloped countries, thus +more often than not, some users may be located in areas +or situations that do not have access to the internet or their +connection is poor, e.g., due to limited available bandwidth +or network congestion problems. Therefore, we prefer pro- +viding the highest availability at the cost of less accurate +response, to increase performability, in general. One can +still get the mobile answer in zero-connection situations. +Nevertheless, user will experience accuracy degradation. +For poor network conditions, we present a simple image +compression technique which reduces the bandwidth usage +when sending captured images (Algorithm 3). We down- +scale the image, transform it from RGB to HSV colorspace, +and then only send the value (V ) channel of the image to +the cloud. On the cloud, H and S are filled with predefined +constants and are up-scaled to the original dimension before +performing inference. +Algorithm 3 Simple Lossy Compression +Input: 1⃝ Img 2⃝ output size (Hν, Wν) 3⃝ filling constant (K) +Output: At mobile : Imgcomp. +At cloud : Imgdecomp. +At mobile +1: Imgcomp. ← Resize Img to (3 × Hν × Wν). +2: Transform Imgcomp. from RGB colorspace to HSV . +3: Imgcomp. ← Drop channels H (Hue) and S (Saturation). +4: return Imgcomp. +At cloud +5: Imgdecomp. ← Add channels H and S, and fill pixels with +K. +6: Resize Imgdecomp. to original size of Img. +7: return Imgdecomp. +3.5 +Complexity of Proposed Algorithm +For every detected object we sweep the other objects. Simi- +larly, this is done again in the post-processing algorithm, for +fewer objects. The time complexity of our algorithm, there- +fore, is O(N 2+N ′2) where N is the number of predicted +RoIs and N ′ is the number of objects to post-process. Since +N ′ ϵ) intersection between two different Items. +Consequently, +it +calls +a +transformation +procedure +(line 10) that modifies each generated image together with +its annotation. Our transformation procedure consists of +two major sections: geometrical and visual. In the geomet- +rical part, we apply scaling, cropping, rotation, shearing, +perspective transformation, and translation. In the visual +part, we modify color, contrast, lightness, and sharpness. +We insert noise, and arbitrarily drop some pixels to simulate + +PRE-PRINT. SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING. +8 +light reflections or scratches in display screen. The range and +likelihood of each transformation depend on the generation +set (Set). In general, we designate a broader range and +higher likelihood in the training set rather than validation +set. The intuition behind is, we aim to generalize in the +training time, while in validation, a set of data much more +similar to real-world images is needed. Note that we never +apply our Data Synthesizer on test set. In addition, to prevent +any meaningful leakage, we make sure there is no similarity +in medical devices in test set and training/validation set. +After applying transformations, Value must still be within +Screen RoI (RScreen). Otherwise, we apply transformation +again until the condition is met. A few synthesized samples, +generated using only one image from Images, is depicted +in Fig. 7. +Value in synthesized images, ranges from 0 to 1000 and +follows discrete pseudo iid distribution. Half of the num- +bers are integers. 35% are single decimal (e.g., 12.5), and +remainders are double decimal (e.g., 1.25). The generated +measurement unit must be compatible with its synthetic +Value. Backlit and display background colors are not uni- +formly random, rather our algorithm selects colors that are +more likely to appear in real-world devices with higher +probability, but there is a probability to generate completely +new colors in order to make the models robust against +new devices or outliers. Each item ∈ Items may randomly +appear or disappear on Screen. The Images are split into +train (Imagestrain) and validation (Imagesval) parts, in 90 +and 10 shares, respectively. Each image comprises a different +medical device. +5 +EVALUATION +5.1 +Experimental Setup +Training Dataset. We generated 1M distinct well-annotated +training samples using our Data Synthesizer. The image size +directly affects training throughput, inference latency, size of +detection models, and the space needed to store the dataset. +Therefore, we down-scale each image to 3×3202 and save in +JPEG format. Note that the input sizes of our CNN models +are different. It occupies 45GB disk space and takes ∼11 +hours to generate and store 1M samples. +Test Dataset. We collected 300 images directly captured by +our clients from various glucometer devices without any +modifications except down-sizing to the desired scale. As +Fig. 8 depicts, the class of ’1’ has the most frequency of +occurrence, and the class of ’.’ has the least among others. +Additionally, we evaluate our method on another publicly +available dataset from CameraLab from University of Ox- +ford (referred to as Oxford for simplicity from now on) [6]. +Training. We trained both our mobile and cloud neural +models on a single server. We use an in-house GPU server +(Table 6) to train our models. We designed and trained our +models via TensorFlow framework. We reduce the time and +resources needed for training the models by employing +transfer learning from pre-trained weights on COCO dataset +[39]. It helps our models extract better low-level features. We +trained each model for 500-600k epochs (∼3.5 days), and +applied simple augmentations during training to prevent +overfitting. +Inference Setup. For the proof of concept, we use the same +server as the cloud side. For the mobile side, we use Galaxy +Tab A (2019), a mediocre tablet (Table 5). We leveraged +TensorFlow Lite for Android to convert, optimize, measure +the performance of our mobile model. During the inference, +input images get resized to 3 × 4162 and 3 × 3502 for the +cloud and mobile models, respectively. +5.2 +Accuracy Metric +We report the accuracy of our algorithm as formulated in +(1). It is more illustrative in comparison with the previously +reported formulas, namely Precision, Recall and F1-score for +classification and localization. That said, it is the strictest as +well, and is necessary due to medical nature of the values +being read. +Given a prediction ˆY and its ground-truth Λ. Let: +ˆY := ˆy1 ˆy2 · · · ˆ +ym +Λ := λ1λ2 · · · λn +ˆY is then considered as a correct prediction if: +m = n +and +∀i : ˆyi = λi, i ∈ N, i ≤ n +(1) +A prediction is correct if 1⃝ all ground-truth RoIs are cor- +rectly detected, 2⃝ the labels are correctly predicted, 3⃝ ob- +jects are in the same order as they are in the ground-truth, +and 4⃝ no additional object (e.g., from Items) is detected. +For instance, assuming a ground-truth label 11.9 (Fig. 2), +none of 119, 1.19, 111, 1149, 1109, 11, 19, 1.9, etc. are correct +predictions, which makes it more difficult than Precision or +Recall. +5.3 +Accuracy on Our Test Set +We first evaluate our algorithms against 300 images directly +captured by our clients from various glucometer devices +without any modifications except down-sizing to the de- +sired scale. As Fig. 8 depicts, the class of ‘1’ has the most +frequency of occurrence, and the class of ‘.’ has the least +among others. While cloud- and mobile-only predictions +fluctuate around 89 − 90% (Fig. 9), our ensemble model +improves the accuracy by 7.3 percentage points. The reason +behind is illustrated in Fig. 10. In each confusion matrix, +every column represents objects in a predicted class while +each row represents the objects in its actual class. Making +matrices more informative, we added another row and +column. The last column represents the objects in a true class +that are ignored, and the last row shows the detected objects +that do not belong to any classes. To get a better perspective, +each matrix is scaled to one in rows. Our cloud model per- +forms well on most classes except ‘’.’ and ‘1’. On the other +side, our mobile model performs poorly on ‘0’, ‘4’ and ‘9’, +but predicts ‘.’ and ‘1’ much better than the cloud. The last +matrix confirms that our ensemble algorithm perform better +than the single models. It also reduces false positives and +false negatives. The mispredicted images usually contain +some special objects/symbols on their screen. For example, +the device shown in Fig. 12(f) displays an arrow on the +screen which is pretty similar to ‘7’, and sometimes misleads +the models. + +PRE-PRINT. SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING. +9 +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Fig. 7. A few examples of the images synthesized and transformed by Algorithm 4. (a) is the original image, and (b) is its manually transformed +image. Note that for better understanding, only images generated from one source image are shown. +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 . (dot) +Class (Label) +0 +40 +80 +120 +160 +200 +Frequency +64 +211 +138 +80 +52 +62 +55 +64 +64 +61 +10 +18 +35 +19 +13 +38 +25 +48 +40 +19 +20 +127 +Our test set (sample size: 300 +Oxford test set (sample size: 127) +Fig. 8. Overview of the test sets used for evaluation. Our dataset con- +tains 300 samples, and Oxford’s [6] contains 127. +Method +86 +88 +90 +92 +94 +96 +98 +Accuracy (%) +88.7 +90 +96.7 +97.2 +97.7 +Edge-Only +Cloud-Only +Ensemble +Ensemble+PostProcess +Ensemble+PostProcess+Knowledge bit +Fig. 9. Accuracy on our test set in different settings. +5.3.1 +Post-Processing Impact +Our post-processing algorithm shows its positive effect on +prediction of test samples. It increases accuracy by 0.5 up +to 1 on our test set (Fig. 9). The main reason for fewer +false positive errors is applying post-processing after the +ensemble step (Fig. 10). +5.3.2 +Device-Aware Prediction (Knowledge Injection) +The least accurate prediction in our proposed method be- +longs to the class ‘.’ (decimal point) with 90% accuracy. The +current state-of-the-art [6] only detects objects and simply +adds a decimal point manually. This is because their dataset +contains two devices that both use mmol/L as measurement +unit, meaning that all values are floating point. In contrast, +our dataset is more general and comprises mg/DL as well. +We can improve the accuracy of our algorithm by injecting +one bit prior knowledge about the measurement unit of a +medical device (mg/DL or mmol/L). As shown in Fig. 9, +it increases the accuracy by 0.5. However, considering the +displayed measurement unit on the screen as another object +of interest can be used in future work. +5.4 +Generalization: Accuracy on Oxford’s Dataset +the Oxford’s dataset [6] gives us the opportunity to evaluate +robustness and generalization of our proposed algorithm. +We can also have a direct comparison with the current state- +of-the-art [6]. It consists of two different glucometer devices, +each evenly divided into training and test sets. The two sets +are quite similar (including the image capturing device), +except the values on the screens. Each set contains 127 +samples, and all of them use mmol/L as the measurement +unit on seven-segment displays (Fig. 8). The size of each +image is ∼3×2600×4600 pixels. We first report the accuracy +of our models on the test set without learning the training +set, to measure how robust our method is. We achieved +89.8% accuracy in this configuration. Next, to get a fair +comparison, we fine-tuned our models on the training set. +In both cases, knowledge injection was not employed. All +parameters except T E were fixed between the evaluations +on the two datasets, we used 0.72 for Oxford’s and 0.82 +for ours. To prevent Class Imbalance Problem, the Oxford’s +training set was combined with a small part of our own +training set. After fine-tuning the accuracy improved to +92.1%, while the previous method achieved 51.5% [6]. As +the results suggest, our algorithm outperforms the state- +of-the-art both with (+40.6%) and without observing the +training set (+38.3%). Detailed assessment showed that in +most misread images, devices are far from the camera so +that in some cases it is difficult to read them correctly. +The main reason is lower resolution of our models. Input +resolution significantly impacts accuracy of RoI detection +[32]. In general, reducing size of image by a factor of 4 +decreases accuracy by almost 16% and lowers response time +by ∼27%. The input resolution of our models are <1% of the +images within Oxford’s dataset. Thus, one may be able to +achieve even lower error rates by increasing the input size of +the neural models, but at the cost of longer training and re- +sponse time. The memory and storage needed at the mobile- +side are also the constraints that must be provisioned. +5.5 +Availability and Performability +Our mobile device is able to prepare its prediction in 260ms +and 360ms in GPU-enabled and only-CPU settings, respec- +tively. For the situations that there is no internet connection, +clients can still rely on the results of their smartphones. They +will lose some accuracy, while higher availability can be +achieved. When the connection quality is poor, users can +enable our simple compression algorithm. It can reduce the +bandwidth usage by 45× while resulting <1% degradation +in accuracy, or by 90× and ∼2% accuracy loss. This is +achieved by our Data Synthesizer algorithm. It forces our +deep learning models to learn channel-invariable features. +In addition, the ensemble approach inherently improves +robustness. The response time breakdown in Fig. 11 depicts +that by applying our compression technique, the transmis- +sion delay, which is the major contributor to total delay, +collapses. Hence, users can still get a reasonable response +in <500ms with 512Kbps available bandwidth and 100ms +RTT. +To compare related service architectures (Fig. 5), we as- +sume a SLA in which Performability is described as reaching + +69210.2162 +15Glh +81.4 +01-60-0E2.79604942PRE-PRINT. SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING. +10 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +. +None +Predicted Class +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +. +None +True Class +0.98 +0.02 +0.93 +0.07 +1 +0.98 +0.02 +1 +0.98 +0.02 +1 +0.97 +0.03 +1 +1 +0.1 0.9 +0.6 +0.2 0.2 +(a) Cloud Model +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +. +None +Predicted Class +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +. +None +True Class +0.95 +0.05 +0.96 +0.04 +1 +0.98 +0.02 +0.92 +0.08 +0.97 +0.03 +1 +0.97 +0.03 +1 +0.95 +0.05 +0.7 0.3 +0.39 +0.09 +0.220.090.170.04 +(b) Mobile Model +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +. +None +Predicted Class +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +. +None +True Class +0.98 +0.02 +0.99 +0.01 +1 +0.99 +0.01 +1 +1 +1 +1 +1 +1 +0.9 0.1 +0.25 +0.5 +0.25 +(c) Ensemble Model +Fig. 10. Confusion matrix of different models. Matrices are row normalized. +0 +200 +400 +600 +800 +Response Time (ms) +378 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +378 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +378 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +378 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +100 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +60 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +100 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +60 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +410 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +410 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +41 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +41 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +9 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +9 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +1 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +1 +w/o +BW 0.5Mbps +RTT 100ms +w/o +BW 0.5Mbps +RTT 60ms +w/o +BW 5.0Mbps +RTT 100ms +w/o +BW 5.0Mbps +RTT 60ms +w +w +w +w +Inference Time +RTT +Transmission Time +Fig. 11. Cloud-side response time breakdown with (w) and without (w/o) +applying compression, in two different bandwidth and RTT settings. RTT +is the round-trip time, and can be measured by the command ping. Two +RTT values of 100ms, 60ms assumed for illustration. +Lk accuracy while being able to prepare response within 5s. +Availability can be achieved by responding in <500ms, as +a soft deadline. Here, L1 (  + ) and L2 () stand for >90% +and >85% accuracy, respectively. We evaluate each service +in three different connection qualities: excellent, poor (e.g., +due to congestion or limited bandwidth) and zero (no con- +nectivity). As summarized in Table 4, our model performs +better in all three cases. To the best of our knowledge, no +comparable Split Computing or Early Exit method currently +significantly compresses its intermediate data while losing +little to no accuracy (rows 4 and 6). +6 +CONCLUSION +We presented a mobile-cloud automated image-based glu- +cometer reading system broadly applicable to various med- +ical devices; our method uses the camera, the wireless com- +munication, and the computing capabilities of the mobile +phone to provide a low-cost alternative to expensive devices +TABLE 4 +Overview of Availability (Avail.) and Performability (Perf.) in different +methods and connections (Conn.). +Service +Excellent Conn. +Poor Conn. +Zero Conn. +Avail. +Perf. +Avail. +Perf. +Avail. +Perf. +Mobile-Only + + + + + + + + + +Cloud-Only + + + + ∗ +⇊ + + ∗ + + +Split Comp. + + + + ∗ + + + + +Split Comp.+ + + + + ∗ +⇊ + + ∗ + + +Early Exit + + + + ∗ + + +or + + +or +Early Exit+ + + + + ∗ + + + + ∗ + + +or +Ours + + + + + + + + + + + ++ Assuming intermediate data were compressed by a lossless method. +⇊ Noticeable degradation. +∗ Poses extra cost. +available more in developed countries. Our deep learning- +based ensemble algorithm together with our mobile-cloud +service architecture achieves higher availability and per- +formability compared with mobile-only, cloud-only, split +computing, and early exit rival models. We proposed a data +generation algorithm to address the data scarcity problem, +and synthesized one million well-annotated samples. Note +that the massive varieties in glucometer devices and their +screen outputs, has conventionally posed challenges to ap- +plicability of existing techniques, but we instead took benefit +from them in our data generation techniques to produce +more training samples from existing photos, and thus to +achieve a more robust model. +Our method is capable of proper readout even in im- +perfect conditions such as dark ambience, reflections on +the screen, blurry and out-of-focus photos. Our ensemble +algorithm efficiently combines results obtained from two +separate DNN models. 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SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING. +12 +APPENDIX A +MOBILE DEVICE AND CLOUD SPECIFICATIONS +TABLE 5 +Mobile Device Specification +Samsung Galaxy Tab A 8.0 with S Pen (2019, SM-P205) +CPU +Octa-core big.LITTLE +(2×1.8GHz Cortex-A73 and 6×1.6GHz Cortex-A53) +Chipset +Exynos 7 Octa 7904 (14 nm) – 64bit +GPU +ARM Mali-G71 MP2 +Memory +3GB LPDDR4X +OS +Android 10 +Network +2G/3G/4G(LTE) & Wi-Fi 802.11 +TABLE 6 +Cloud Server Specifications +CPU +Intel Xeon E5-2630-v3 (x86 64) +Frequency: 2.4GHz (1.2-3.2GHz) +Cache: 32K-32K-256K-20480K +#Cores: 24 (double threaded) +GPU +NVidia GeForce GTX 1080 Ti +Total Memory: 11GB +CUDA V9.0.176 , Compute Capacity 6.1 +Memory +32GB (1600MHz) +OS +Gnu/Linux Ubuntu 18.04 +APPENDIX B +RESPONSE TIME +Table 7 shows total response time on the mobile and cloud in +different settings. For mobile inference, Tensorflow bench- +marking tool for Android was used, and for the cloud, we +measured the time it takes from sending a REST request to +the server to get a response. The reported response time at +the cloud side in TABLE 4 is the summation of inference +time, typical transmission time (considering 5Mbps band- +width), and typical round trip time (RTT equals 60ms). +TABLE 7 +Response Time in Different Settings. +Engine +Platform +Response Time (ms) +Mobile +CPU (1 Thread) +494 ± 1 +CPU (4 Threads) +360 ± 25 +GPU (1 CPU Thread) +260 ± 1 +Cloud +CPU +479 ± 20 +APPENDIX C +BANDWIDTH OPTIMIZATION +Table 8 illustrates how our compression method affects +bandwidth and accuracy with respect to different input +resolutions. +TABLE 8 +Bandwidth Usage Reduction Vs. Accuracy (%) +Method +Original RGB +Our Compression +Image Size +Accuracy +BW Reduction +Accuracy +BW Reduction +64×64 +93.0% +30× +95.3% +90× +64×128 +95.7% +15× +96.7% +45× +128×128 +95.7% +7.5× +97% +22.5× +256×256 +97.2% +2.5× +97.2% +5.5× +350×350 +97.2% +1.0× +97.2% +3.0× +APPENDIX D +EXAMPLES OF MISRECOGNITION +We provide samples that our algorithm did not recognize +them correctly. As shown in Fig 12, in some situations, +additional elements on the screen mislead our algorithm. +For example, in Fig. 12(f) and Fig. 12(c) an arrow and battery +information are wrongly detected as ‘7’ and ‘1’, respectively. +This can be corrected by adding similar samples to the +training set in the next iterations. +(a) 53 → 5.3 +(b) 61 → 67 +(c) 5.81 → 5.8 +(d) 99 → 399 +(e) 94 → 9.4 +(f) 1067 → 106 +Fig. 12. Examples of mispredicted images. (prediction → ground-truth) + +performa +60:E +6 +mg/dLmol/ +5.8 +1-2216:00kraVITA +ContourTS +4106-~ \ No newline at end of file diff --git a/LtAzT4oBgHgl3EQfyv51/content/tmp_files/load_file.txt b/LtAzT4oBgHgl3EQfyv51/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..111a7f1307eb27eeb89e6d65f5dd30e21fdd1ae9 --- /dev/null +++ b/LtAzT4oBgHgl3EQfyv51/content/tmp_files/load_file.txt @@ -0,0 +1,1174 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf,len=1173 +page_content='PRE-PRINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1 An Ensemble Mobile-Cloud Computing Method for Affordable and Accurate Glucometer Readout Navidreza Asadi, and Maziar Goudarzi, Senior Member, IEEE Abstract—Despite essential efforts towards advanced wireless medical devices for regular monitoring of blood properties, many such devices are not available or not affordable for everyone in many countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Alternatively using ordinary devices, patients ought to log data into a mobile health-monitoring manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' According to medical specialists, it causes several issues: (1) due to the direct human intervention, it is prone to errors, and clients reportedly tend to enter unrealistic data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' (2) typing values several times a day is bothersome and causes clients to leave the mobile app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Thus, there is a strong need to use now-ubiquitous smartphones, reducing error by capturing images from the screen of medical devices and extracting useful information automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Nevertheless, there are a few challenges in its development: (1) data scarcity has led to impractical methods with very low accuracy: to our knowledge, only small datasets are available in this case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' (2) accuracy-availability tradeoff: one can execute a less accurate algorithm on a mobile phone to maintain higher availability, or alternatively deploy a more accurate and more compute-intensive algorithm on the cloud, however, at the cost of lower availability in poor/no connectivity situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We present an ensemble learning algorithm, a mobile-cloud computing service architecture, and a simple compression technique to achieve higher availability and faster response time while providing higher accuracy by integrating cloud- and mobile-side predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Additionally, we propose an algorithm to generate synthetic training data which facilitates utilizing deep learning models to improve accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Our proposed method achieves three main objectives: (1) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='1% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='7% accuracy on two different datasets, improving previous methods by ∼40%, (2) reducing required bandwidth by 45× with ∼1% drop in accuracy, (3) and providing better availability compared to mobile-only, cloud-only, split computing, and early exit service models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Index Terms—Mobile Computing, Ensemble Learning, Data Generation, Deep Learning, Smart Health !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1 INTRODUCTION M ANY mHealth/uHealth medical devices, especially those affordable in middle/low-income countries, show the measured quantity on a digital or seven-segment screen alongside other additional information such as date, time, diagrams and measurement units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' In most commonly used mHealth services, particularly for diabetics, patients are required to manually type sensed values into their mobile app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1(a), a client first reads a value from the medical device, opens the app, navigates to logging interface, and eventually logs the information through typing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' These steps should be repeated every time they log a measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='1 Motivation According to the reports [1], [2] as well as our own expe- rience from our diabetes management application iDia [3], this procedure has a few drawbacks: (1) Manual logging multiple times a day, is deterring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' it is bothersome and time- consuming, and based on the feedbacks we have received, leads users to eventually lose their interest in using the app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' (2) It is prone to human errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' More importantly, the observations show that patients are tempted to enter fake information that are more acceptable and closer to the N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Asadi is now with Computer Engineering Department, Technical Univer- sity of Munich, Germany (navidreza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='asadi@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' He was with Computer Engineering Department, Sharif University of Technology, Iran while working on this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Goudarzi is with the Computer Engineering Department, Sharif Univer- sity of Technology, Iran (goudarzi@sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Manuscript submitted to IEEE Transactions on Mobile Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' normal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' This can have considerable negative effects on the whole process of prevention, control, and treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Currently, there are two better approaches: (1) Some medical devices are able to transmit their data to mobile devices via distant communication technologies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', Blue- tooth), facilitating the logging procedure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Never- theless, they are far more expensive and in some cases less accurate [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' More importantly, most of them can only in- teract with their own software applications and do not per- mit third-party apps to receive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' In addition, different versions of transmission technologies cause incompatibility between different mobile and medical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Despite a potentially bright future, these devices currently hold less than one percent of the market [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' This number is even lower in developing countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' (2) An alternative which has grown interest in academia, uses image capturing and computing capabilities of mobile devices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', smartphones);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' they are available to almost everybody and are able to perform light-weight computing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1(c) illustrates this alternative: the mobile phone is used to capture an image of the medical device, and then image processing is applied to recognize the sensed values automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' In this approach (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 2), each digit is considered as an object having a region of interest (RoI) and an ordered sequence of digits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', RoIs) forms a read string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' A correct read means not only all digits are predicted correctly, but also in the same order as displayed on the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' In this paper we follow this methodology for its broader applicability especially in middle/low-income countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='01758v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='DC] 4 Jan 2023 PRE-PRINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 2 Read Medical Device 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='4 mmol Take a Test Open the App & Navigate to Log Interface Manually Type Observed Value 1 2 3 4 Mobile Phone Patient (a) Manual Medical Device 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='4 mmol Take a Test Open the App & Navigate to Log Interface 1 2 Transfer Sensed Value via Bluetooth 3 Mobile Phone Patient (b) Wireless Communication Medical Device 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='4 mmol Take a Test Open the App & Navigate to Log Interface 1 2 Capture an Image from Screen 3 Patient Mobile Phone (c) Image-Based Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Logging methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Removing human interventions (red arrows) is preferred to avoid false logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' RoI RoI RoI RoI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1 1 9 1 2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='9 mmol/L Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Image-based method, showing Regions-of-Interest (RoIs) and the correct labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The expected correct readout is “11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='9 mmol/L”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='2 Challenges Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Although automated image-based reading may seem basic and easy, there are a few points that prove it a challenging problem in this specific case: (1) the current state-of-the-art [6], which has largely improved the previous algorithms, uses only two medical devices, yet achieves just 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='5% accuracy, meaning it misreads almost half of the captured images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' (2) as we discuss in Sections 2 and 3, the diversity of medical devices and a variety of structural and visual differences as well as noisy information on their screens (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', date and time) make it extremely difficult to read sensed value with business-as-usual image processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Data Scarcity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The accuracy of deep learning algorithms relies on either big volume of annotated data to be trained in a supervised manner or a model pre-trained on a related task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' To the best of our knowledge, there exists neither a big dataset nor a related pre-trained model in our particular task of imaged-based reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Generative models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', GANs) also would not help there because they require similar data to train, which is not applicable in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Additionally, it might be really difficult (if not impossible) to generate automatic annotation for our usecase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Resource Constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Current state-of-the-art deep learn- ing models are compute-intensive and need to be deployed on specialized cloud infrastructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Thus, they introduce new challenges, including availability during poor network conditions, that is genuinely an expected issue in our target under-developed countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Edge Computing [7], referring to every device near data source with some compute ca- pacity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', smartphones), is considered as a promising ap- proach to improve availability and quality of service (QoS) by reducing delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Mobile devices are usually resource- constrained, and therefore, can only run simpler deep neural network (DNN) models with much lower accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Cloud Computing, however, is at the opposite side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='3 Main Contributions We propose practical solutions to address the challenges above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We present an ensemble mobile-cloud computing ar- chitecture to get the best of both worlds: higher availability on the mobile, as well as higher accuracy and enhanced per- formability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', a measure of level of performance/service- quality of the system) by integrating cloud and mobile modules (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3, described in §3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The followings items are our main contributions: (1) A hybrid mobile-cloud service architecture, together with a compatible ensemble deep learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' This enables an accuracy-availability tradeoff based on network connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We addressed the challenges of combining predictions of two separate models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', differences and overlaps in the identified bounding boxes for each data element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' (2) A simple yet effective compression method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Combined with our ensemble model, this provides higher accuracy despite little communicated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' (3) A high-fidelity data synthesizer algorithm, making utiliz- ing deep learning models possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' This has basically turned the challenge into an opportunity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' the variety of glucometer models, data formats, units, fonts, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' is a challenge to con- ventional methods, but we used it in our data synthesizer mechanism to produce enough reasonable data to train high accuracy models that cover many varieties including those not seen before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Our proposed method achieves 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='1% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='7% accu- racy on two real-world datasets, and improves previously published results by more than 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' It reduces the required bandwidth by 45×, and maintains higher availability com- pared to mobile-only, cloud-only, split computing, and early exit service models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Our proposal can be easily extended to other usecases with minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The rest of this paper is organized as follows: In Sec- tion 2, we review related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' In Section 3, we present our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Section 4 explains our dataset generation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We evaluate our methods and algorithms in Section 5 and conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 2 RELATED WORK We separate related work into two different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The first one presents algorithms for reading sensed values or detection and recognition of digits on digital or seven- segment screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The second one summarizes the studies that attempt to deploy part or whole of a deep learning model on the mobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' mmo 14:20 10-11 GlucoMen areoPRE-PRINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3 Camera Sensor Edge Side (1) Capture Image Edge Computing (2) Pre-Process Image (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='1) Call Server-Side Inference API (5) Ensemble Results (6) Post-Process Incorrect Label?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Predicted Label Cloud Side Web Services Storage (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='2) Run Edge-Side Inference Engine (4) Run Cloud-Side Inference Engine Get Prediction Yes (8) Store Image and Correct Label Send Image and Parameters Send Image and Correct Label (7) Call WrongLabel API Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Our mobile-cloud architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='1 Image-Based Automated Reading Most of the proposed methods break the problem into multiple steps including image enhancement, localization of RoIs, detection and classification, and eventually ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We review related work within five criteria, as summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Automated Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Locating the RoIs is a crucial step and impacts the final accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Many works try to simplify the problem while assuming the localization step is somehow already done, either manually by a client, or by fixing the device or using special markers [8], [9], [10], [11], [13], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' A few others [6], [12], [16], and our work take a more holistic approach, applying automated localization as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' To our knowledge, no previous work achieves a reasonable readout accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' [6] with 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='5% has by far the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' General OCR engines also are not helpful;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' our previous experiments along with the reports from [15], [16], and [6] confirm the state-of-the-art OCR engine, Tesseract’s [14] poor performance (<10%) in this specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' On the other hand, our method can reach over 90% accuracy on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Different methods, especially those using con- ventional algorithms are usually sensitive to different condi- tions such as skewness, various noises, camera perspective and angle, exposure, and illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Here, we define robustness as performing consistently in different condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' That said, only few works [6], [16] try to adress it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We simulate all these variations in our data synthesizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' TABLE 1 Summary of Literature Review on Image-Based Automated Reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Work Localization Accuracy Robustness* Generalization Response Time [8] \x17 \x17 \x17 \x17 \x13 [9] \x17 \x17 \x17 \x17 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' [10] \x17 \x17 \x17 \x17 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' [11] \x17 \x17 \x17 \x17 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' [12] \x13 \x17 \x17 \x17 \x13 [13] \x13 \x17 \x17 \x17 \x13 [14] \x13 \x17 \x17 \x17 \x13 [15] \x17 \x17 \x17 \x17 \x17 [16] \x13 \x17 \x13 \x13 \x13 [6] \x13 \x17 \x13 \x17 \x13 Ours \x13 \x13 \x13 \x13 \x13 Robustness to different environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Information not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Addtionally, the ensemble of two different models, mitigates such errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' There are a variety of medical devices each having unique characteristics, such as different font styles (including various types of seven-segments and/or digital styles), background and foreground colors, screen size and shape, units, backlit, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Nevertheless, the previous studies, except [16], use one or very few specific devices so that their proposed algorithms highly depend on the properties of those selected devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' hence, may not be con- sidered as a general solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' For example, [13] is designed for a particular medical device with a blue backlit screen, [6] considers only seven-segment screens, and [15] assumes the largest contour as the screen, and hardcodes exact location of RoIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' In contrast, we cover a broad market, and illustrate this using an additional public dataset from Oxford [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Respone Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Since larger portion of the previous stud- ies use conventional and light-weight image processing methods, they can achieve reasonable response time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' For instance, [13] is implemented on a Samsung Galaxy S i9000 mobile device, and can process 20 frames per second, or [12] is deployed on an N95 mobile device, and achieves five frames per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The only exception is [15] which uses a deep convolutional neural network (CNN) for the digits classification step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' It takes 10 seconds, that is unsatisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Although we leverage CNN-based object detection models, we choose parameters so that the response is prepared in less than half a second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Besides, we design a simple compression technique when using the cloud-side engine in poor network conditions, and, therefore, reduce the end- to-end response time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='2 Deep Learning on Edge In practice, an edge device can be any computing machine (indluding smartphones) that generates data or is near a data generation source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Edge devices are usually resource- constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' So it is challenging to deploy big deep learning models on edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' A concise comparison of different methods is provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The efforts to overcome the limita- tions can be divided into three major directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The first direction is designing light-weight DNN mod- els or optimizing existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Several successful works have studied light-weight models including [17], [18], [19], [20], [21], and [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' In general, related work in this cate- gory leverage a combination of using convolution blocks with lower parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', separable convolutions), quan- tization, pruning, and model distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' These techniques TABLE 2 Related Work on Deep Learning at Edge: A summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Method Works Accuracy Availability No-Cloud* Bandwidth** Response Time Deployment D1 [17], [18], [19], [20], [21] \x17 \x13 \x13 \x13 E D2 A1 [22], [23], [24] \x17 \x13 \x17 \x13 \x13 E,C D2 A2 [25], [26], [27] \x13 \x17 \x17 \x13 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' E,C D3 [27], [28], [29] \x17 \x13 \x13 \x17 \x13 E or C Ours \x13 \x13 \x13 \x13 \x13 E,C Independence from a central cloud entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ** Bandwidth usage optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' D: Direction A: Approach E: Edge C: Cloud 388PRE-PRINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 4 (a) (b) (c) (d) (e) (f) (h) (i) (j) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Examples of images captured from medical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Output Input (a) Cloud-Only Output Input (b) Mobile-Only Input Output (c) Split Computing Early Output Output Input (d) Early Exit Output Ensemble Algorithm Post Process Lightweight Less-Accurate Object Detector Bigger More Accurate Object Detector Cloud-Side Output Input Edge-Side Output Input (e) Our Architecture Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Deep Learning Serving Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' mainly focus on optimizing response time and memory footprint, and thereby sacrifice accuracy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The second direction, distributes computation across edge and central cloud, vertically (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 5(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Some tech- niques aim to reduce the required computation and band- width by dropping insignificant frames of input stream at the edge, before sending them to the cloud, and depending on the nature of a task may follow different filtering policies [22], [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Some distribute the inference model across edge and cloud (Split Computing) [25], [26], [27], and usu- ally trade accuracy for better response time or bandwidth usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' These methods rely on central cloud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' hence, in the case of network outages, they become unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Another recent direction determines one or more exit points within neural model (including pre-processing steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Exit points are usually designed so that at least one of them stay on the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Depending on the task and its requirements, the model can exit early, sacrificing accuracy to meet delay constraints [27], [28], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Despite their ap- pealing results, they still are immature and are evaluated on simple tasks such as classification (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 5(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' In comparison, the strength of our proposed mobile- cloud architecture (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 5(e)) is its ability to take advantage of both worlds: it improves accuracy, and in general per- formability as well as availability, thanks to the independent nature of our ensemble models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3 PROPOSED METHOD Our service captures images from medical devices using phone’s camera (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3(1)), then performs a pre-processing step on mobile (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The mobile device concurrently sends the prepared image to cloud (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='1, 4)), while executing light-weight inference engine locally (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' After receiving predictions of both sides, the mobile device runs our ensemble algorithm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3(5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' It also performs a post-processing step, to correct the initial answer that is produced by the mobile, if required (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3(6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' If user recognizes a misprediction, they can send the image and its corresponding true reading (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3(7)) to cloud storage for future analysis and training iterations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3(8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The mobile model provides availability, while the cloud model improves performability (Refer to §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Both models to- gether improve accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We reduce the problem into object detection so that digits of the sensed value (and not other digits illustrating noisy information such as temperature, time and date) are objects of interest, together with a post-processing step to reorder objects and prepare the final response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' DNN-based methods have shown remarkable results, but to get the most out of them, we need a well-annotated training dataset, which is not available for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We design a better workflow and a data generation algorithm to automatically synthesize thousands of training images with well-aligned annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Our data generation workflow is described in detail in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We use mobile smartphones to capture images from medical devices and to extract useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' While in- ference on the mobile device provides low latency and high availability for users, it usually suffers from low accuracy due to resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' On the other hand, inference on the cloud provides accuracy and performability, but D01 42 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='59m arkraukra388ACCU-CHEK AvivaConnec 12:00 3/11/15 106 AddComment 8:38 85mg/dL 0Optium 105 6h 12-10 APRE-PRINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 5 high latency and unavailability in the case of poor network connection are its big weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3, our proposed hybrid mobile-cloud serving architecture takes advantage of both mobile and cloud computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Addition- ally, we designed a specialized ensemble model to further improve our desired performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='1 Deep Learning Models We reduce the problem of image-based reading from screen of medical devices to an object detection and post- processing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Each digit and decimal point belonging to the sensed value occupies a region of interest (RoI) and has a corresponding class (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 2-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' In post-processing step, we convert a group of objects to a meaningful string (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 2- 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' For the object detection part, we design and train two convolutional models (CNNs) based on single-shot detector (SSD) architecture [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' In general, we have powerful resources on cloud, but limitation at the mobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Thus, we consider two proportion- ate backbones for our SSD architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' One is highly op- timized for smartphones and has fewer parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' hence, lower accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Another one is more accurate and consumes more resources so that can not be deployed on mobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We prefer SSD-based networks for both sides, because they achieve better end-to-end latency [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='1 Cloud-Side Model We employ ResNet-50 [33] as the backbone of our SSD model on the cloud-side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' It contains 16 convolutional blocks with shortcuts and one fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' It has more than 25M parameters, and requires 4 billion multiplication- accumulation operations (MACs) per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We remove its last few layers including the classification head to use the remainder as a feature extractor backbone for the SSD architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Since the size of RoIs in input images varies, we utilize feature pyramid network (FPN) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' It leverages intrinsic pyramid structure of modern CNNs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', ResNet) to generate multi-scale feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' FPN can improve accuracy of the model, and is much more compute efficient than feeding an image with different sizes multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='2 Mobile-Side Model Similar to the cloud-side, we stacked up a CNN-based classification model as a feature extraction backbone for our mobile-side SSD architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' However, we employed edge- friendly architectures both for the backbone and the detec- tion head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Our mobile-side model is directly inspired by the current state-of-the-art detection architecture for mobile devices, the MobileDets [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The reason of using depth-wise separable convolutional layers instead of regular convolu- tional layers was their fewer parameters and MACs, while hoping these metrics directly relate to less train and infer- ence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' However, recent studies [36], [37] have shown network parameters and MACs may not be good proxies to model inference throughput and latency as they are not the only factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Therefore, MobileDets expands the neural architecture search space by adding regular convolutions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' After multiple experiments on different backbone architectures, we found the architecture proposed by [35] for Mobile CPUs is the most appropriate for our mobile- side model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Since mobile devices are resource-constrained, we do not leverage FPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='2 Ensemble Algorithm As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3, we design and deploy a light-weight engine at the mobile along with a more compute-intensive and accurate one on the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We integrate predictions of our two deep learning models through our ensemble algorithm (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Having regions of interest (RoIs), labels and confidence scores of the objects detected by both models, our ensemble algorithm first finds corresponding RoIs among mobile and cloud predictions (line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' They must have identical labels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', class of digit ’5’) and their distance should not be more than a tolerable amount ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The intuition behind is, the RoI coordinates predicted by different object detection models may not exactly match with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ϵ controls maximum distance between two corresponding predictions from two different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' There may exist more than one object with same labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' So if we do not check that, it will lead to misrecognition of RoIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' However, strict comparison is a bad idea because two quite different deep learning models may have small differences in RoI refinement after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Hence, we add a tolerable distance to make it more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' After finding, when both scores are higher than specified thresholds, the ensemble algorithm adds them up as the final confidence score (line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Otherwise, picks the high- est score (line 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' In the case of only one prediction for a Algorithm 1 Ensemble Learning Algorithm Input: 1⃝ maximum number of RoIs (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 2⃝ RoIs matrices (R), and their corresponding vectors of 3⃝ confidence (ρ) and 4⃝ labels (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 5⃝ confidence thresholds (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 6⃝ tolerable distance (ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ▷ Each RK l comprises four member points: left (xmin), right (xmax), up (ymin), down (ymax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ▷ Superscripts C, M and E stand for Cloud, Mobile (Edge) and Ensemble, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Output: A string equivalent to final prediction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', ”10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='6”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1: RE ← ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρE ← ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' LE ← ∅ 2: for i ← 1 to N do 3: Find j such that: RC i [xmin] ≈ RM j [xmin] ± ϵ and RC i [xmax] ≈ RM j [xmax] ± ϵ and LC i = LM j 4: if j ̸= Ø then 5: LE ← LC i ∪ LE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' RE ← RM j ∪ RE 6: if ρC i ≥ T C i and ρM j ≥ T M i then 7: ρE ← (ρM j + ρC i ) ∪ ρE 8: else 9: ρE ← max(ρM j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρC i ) ∪ ρE 10: end if 11: RM ← RM−RM j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρM ← ρM−ρM j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' LM ← LM−LM j 12: else 13: LE ← LC i ∪ LE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρE ← ρC i ∪ ρE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' RE ← RC i ∪ RE 14: end if 15: end for 16: RE ← RM ∪ RE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρE ← ρM ∪ ρE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' LE ← LM ∪ LE 17: I ← {i ∈ N | i ≤ |ρE|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρE i ≥ T E} 18: LE ← [LE i∈I | ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' j) : i < j ⇔ RE i [xmin] ≤ RE j [xmin]] 19: return ∥n i=1LE i ▷ (∥ concats every element within LE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' PRE-PRINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 6 particular object on either mobile or cloud, simply adds it to our final results (lines 13, 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The second elimination step happens in (line 17), where all remaining objects are compared via a higher universal threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Eventually, our algorithm reorders the remaining objects by their placement within image (line 18), and concatenates a sequence of cor- responding labels to generate the output string (line 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='3 Post-Processing Algorithm To remove redundant objects more, and to improve accu- racy even further, we add an additional post-processing algorithm right before the line 18 in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We apply a modification of Non-Max Suppression (NMS) [38] to be compatible with our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Here, our objective is to find and remove those RoIs that significantly overlap with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' It means that there are more than one object of interest in a region while it must not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Leveraging NMS reduces false positive (FP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Our modified NMS is defined in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' It executes iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Having RoIs and their corresponding confidence scores and labels, it first sorts RoIs based on their confidence in a descending order (line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' It then removes those objects that their overlap of the occupied area with another object is high enough (> Tnms), and their confi- dence is simultaneously lower (lines 13, 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The overlap between two objects is calculated by intersection over union (IoU) of their corresponding RoIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Although this works well, using a single threshold for all labels introduces a problem in our specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The unit label, representing the decimal point of sensed numbers, considerably overlaps with other objects, but it must not be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' This also applies to the classes of ’1’ and ’7’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' To avoid that, we consider the unit label separately in our calculations within NMS procedure (lines 2-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Algorithm 2 Post-Processing Algorithm Input: 1⃝ overlap threshold (Tnms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 2⃝ RoIs matrix (R), and its corresponding vectors of 3⃝ confidence (ρ) and 4⃝ labels (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Output: reduced RoIs matrix, and its corresponding vectors of confidence and labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 1: RM ← ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρM ← ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' LM ← ∅ 2: i ← j ∈ N | j ≥ |L|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Lj =’.’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ▷ (’.’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='≡ decimal point) 3: if i ̸= Ø then 4: LM ← Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρM ← ρi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' RM ← Ri 5: L ← L − Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρ ← ρ − ρi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' R ← R − Ri 6: end if 7: (L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' R) ← [(Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Ri) | ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' j) : i < j ⇔ ρi ≥ ρj] 8: while R ̸= ∅ do 9: RM ← R1 ∪ RM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' R ← R − R1 10: LM ← L1 ∪ LM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' L ← L − L1 11: ρM ← ρ1 ∪ ρM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρ ← ρ − ρ1 12: for i ← 1 to |R| do 13: if IoU(Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' RM 1 ) ≥ Tnms then 14: R ← R − Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' L ← L − Li 15: end if 16: end for 17: end while 18: return RM ∪ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' LM ∪ L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ρM ∪ ρ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='4 Bandwidth Optimization Our target usecase is in underdeveloped countries, thus more often than not, some users may be located in areas or situations that do not have access to the internet or their connection is poor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=', due to limited available bandwidth or network congestion problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Therefore, we prefer pro- viding the highest availability at the cost of less accurate response, to increase performability, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' One can still get the mobile answer in zero-connection situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Nevertheless, user will experience accuracy degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' For poor network conditions, we present a simple image compression technique which reduces the bandwidth usage when sending captured images (Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' We down- scale the image, transform it from RGB to HSV colorspace, and then only send the value (V ) channel of the image to the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' On the cloud, H and S are filled with predefined constants and are up-scaled to the original dimension before performing inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Algorithm 3 Simple Lossy Compression Input: 1⃝ Img 2⃝ output size (Hν, Wν) 3⃝ filling constant (K) Output: At mobile : Imgcomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' At cloud : Imgdecomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' At mobile 1: Imgcomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ← Resize Img to (3 × Hν × Wν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 2: Transform Imgcomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' from RGB colorspace to HSV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3: Imgcomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ← Drop channels H (Hue) and S (Saturation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 4: return Imgcomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' At cloud 5: Imgdecomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' ← Add channels H and S, and fill pixels with K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 6: Resize Imgdecomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' to original size of Img.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 7: return Imgdecomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content='5 Complexity of Proposed Algorithm For every detected object we sweep the other objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Simi- larly, this is done again in the post-processing algorithm, for fewer objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' The time complexity of our algorithm, there- fore, is O(N 2+N ′2) where N is the number of predicted RoIs and N ′ is the number of objects to post-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf'} +page_content=' Since N ′ +τ (fm) +pc=∞ +pc=4 GeV +pc=3 GeV +pc=2.5 GeV +pc=2 GeV +FIG. 3. +(Color online) Time evolution of ⟨uτ⟩ within the +freeze-out hypersurface without momentum cutoff (solid line) +and with pc = 4, 3, 2.5, and 2 GeV (dash-dotted, dash-double- +dotted, dashed, and dotted lines, respectively). +because of the quick convergence of the pressure and its +gradients to the values in the pc → ∞ limit. This implies +that it would be difficult to distinguish the pc scenarios +based on the experimental data of hadronic production. +B. +Photon particle spectra +Transverse momentum spectra of photons at midra- +pidity are estimated in numerical simulations. Figure 4 +shows pT spectra of (a) thermal photons, (b) prompt and +thermal photons, and (c) direct photons estimated as the +sum of prompt, thermal, and high-pT photons. Thermal +photon spectra is visibly reduced above pc once the cut- +off is introduced to the emission rate [Fig. 4 (a)]. The +effect competes with that of higher initial temperatures +which is supposed to increase the spectrum for smaller +pc. When combined with prompt photons, the cutoff ef- +fect becomes relatively small and reduces the spectra only +around 2-3 GeV because prompt photons become impor- +tant above around 3 GeV [Fig. 4 (b)]. Figure 4 (c) shows +direct photon spectra where non-thermal photons from +the high momentum particles are included. It should be +noted that the high pT photons are mimicked by pseudo- +thermal photons with medium-high momenta. While the +spectrum is increased by a few percent for smaller pc be- +cause the initial temperature is higher and the thermal +photon contribution becomes relatively larger, it is found +to be mostly insensitive to the choice of the cutoff mo- +mentum. +C. +Photon differential elliptic flow +Differential elliptic flow of photons vγ +2 (pT ) at midra- +pidity are investigated in this section. +The azimuthal + +5 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +(a) +dNγ +th/2πpTdpTdη +pc=∞ +pc=4 GeV +pc=3 GeV +pc=2.5 GeV +pc=2 GeV +10-4 +10-3 +10-2 +10-1 +100 +101 +(b) +dNγ +th+pr/2πpTdpTdη +10-4 +10-3 +10-2 +10-1 +100 +101 +(b) +dNγ +th+pr/2πpTdpTdη +10-4 +10-3 +10-2 +10-1 +100 +101 +(b) +dNγ +th+pr/2πpTdpTdη +10-4 +10-3 +10-2 +10-1 +100 +101 +(b) +dNγ +th+pr/2πpTdpTdη +10-4 +10-3 +10-2 +10-1 +100 +101 +(b) +dNγ +th+pr/2πpTdpTdη +10-4 +10-3 +10-2 +10-1 +100 +101 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +(c) +dNγ +th+pr+high/2πpTdpTdη +pT (GeV) +10-4 +10-3 +10-2 +10-1 +100 +101 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +(c) +dNγ +th+pr+high/2πpTdpTdη +pT (GeV) +10-4 +10-3 +10-2 +10-1 +100 +101 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +(c) +dNγ +th+pr+high/2πpTdpTdη +pT (GeV) +10-4 +10-3 +10-2 +10-1 +100 +101 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +(c) +dNγ +th+pr+high/2πpTdpTdη +pT (GeV) +10-4 +10-3 +10-2 +10-1 +100 +101 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +(c) +dNγ +th+pr+high/2πpTdpTdη +pT (GeV) +FIG. 4. (Color online) pT spectra of (a) thermal photons (b) +thermal and prompt photons, and (c) thermal, prompt, and +high pT photons without momentum cutoff (solid line) and +with pc = 4, 3, 2.5, and 2 GeV (dash-dotted, dash-double- +dotted, dashed, and dotted lines, respectively). +momentum anisotropy is estimated as +vγ +2 (pT ) = +� +dφ cos[2(φ − Ψ)] +dN γ +dφpT dpT dy +� +dφ +dN γ +dφpT dpT dy +, +(11) +where φ is the azimuthal momentum angle and Ψ is the +event plane angle. +Thermal photons are the source of +anisotropy here because prompt and high pT photons are +assumed to be isotropic. +Figure 5 (a) shows thermal photon v2 for different +pc. +It is visibly enhanced for smaller values of pc be- +cause the photons above pc are late time contributions + 0 + 0.02 + 0.04 + 0.06 + 0.08 + 0.1 + 0.12 + 0.14 + 0.16 + 0.18 +(a) +v2 +γ,th +pc=∞ +pc=4 GeV +pc=3 GeV +pc=2.5 GeV +pc=2 GeV + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0.03 + 0.035 +(b) +v2 +γ,th+pr + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0.03 + 0.035 +(b) +v2 +γ,th+pr + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0.03 + 0.035 +(b) +v2 +γ,th+pr + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0.03 + 0.035 +(b) +v2 +γ,th+pr + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0.03 + 0.035 +(b) +v2 +γ,th+pr + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0.03 + 0.035 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +(c) +v2 +γ,th+pr+high +pT (GeV) + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0.03 + 0.035 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +(c) +v2 +γ,th+pr+high +pT (GeV) + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0.03 + 0.035 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +(c) +v2 +γ,th+pr+high +pT (GeV) + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0.03 + 0.035 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +(c) +v2 +γ,th+pr+high +pT (GeV) + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0.03 + 0.035 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +(c) +v2 +γ,th+pr+high +pT (GeV) +FIG. 5. (Color online) Differential v2 of (a) thermal photons +(b) thermal and prompt photons, and (c) thermal, prompt, +and high pT photons without momentum cutoff (solid line) +and with pc = 4, 3, 2.5, and 2 GeV (dash-dotted, dash-double- +dotted, dashed, and dotted lines, respectively). +where their momentum is Lorentz-boosted by the ra- +dial flow. Note that the number of thermal photons are +small above pc. Once prompt photons are added, v2 is +still enhanced by the momentum cutoff around pT ∼ +2-3 GeV when pc ∼ 2-3 GeV but is suppressed above +those momenta [Fig. 5 (b)]. The contributions of high +pT photons with zero anisotropy further tend to reduce +v2 [Fig. 5 (c)]. Compared with the thermal limit pc → ∞, +the upper bound on the momenta of constituent particles +might decrease the overall momentum anisotropy because +(i) the initial temperature is higher and the number of +early thermal photons with underdeveloped anisotropy + +6 +increases and (ii) the non-thermal contributions above pc +do not have anisotropy. The results suggest that whether +the elliptic flow is enhanced or reduced could depend on +non-thermal photons from the medium-high pT sector. +It is also implied that direct photon v2 might be an ob- +servable sensitive to the momentum dependence of ther- +malization. +It should be noted that the results are shown to demon- +strate the qualitative behavior of the photon observables +caused by the momentum cutoff and cannot be compared +quantitatively to experimental data because of the lack +of event-by-event fluctuations and viscous modifications, +which would affect flow harmonics and, to smaller ex- +tent, pT spectra. Those effects, on the other hand, are +expected to be roughly orthogonal to the the cutoff ef- +fects and can be studied independently. +IV. +CONCLUSION AND OUTLOOK +I have developed the relativistic hydrodynamic model +of heavy-ion collisions where the medium is constituted +by low momentum components because the experimen- +tal data of high-energy heavy-ion collisions suggest that +the system is thermalized only up to pT ∼ 2-4 GeV. The +QCD equation of state has been constructed using the +hadron resonance gas and parton gas models by introduc- +ing the upper limit pc on the momenta of constituents. +It has been found that the effect of the momentum limit +on the hydrostatic pressure is visible only in the QGP +phase and becomes small for larger pc. +Numerical hydrodynamic simulations of heavy-ion col- +lisions have indicated that the effects of the momentum +cutoff on the thermodynamic variables such as the tem- +perature and pressure are mostly limited to early times +before τ ∼ 1 fm/c. The radial flow development is thus +not much affected by the choice of pc. It is also consis- +tent with the observation that the effective temperature +that characterizes the radial expansion in heavy-ion col- +lisions at RHIC and LHC are 0.2-0.3 GeV [100], because +the equations of state at pc = 2-5 GeV are shown to +converge below 0.3 GeV. +Direct photons, on the other hand, have been found +to be sensitive to the cutoff momentum. Thermal pho- +ton radiation is reduced above pT ∼ pc, where the small +number of high-momentum thermal photons with rela- +tively large azimuthal momentum anisotropy are pro- +duced through the Lorentz boost of radial expansion. +Adding prompt photons make the pT spectra mostly in- +sensitive to the cutoff because they dominates the spectra +above pT ∼ 3 GeV. Elliptic flow, on the other hand, is en- +hanced around pT ∼ 2-3 GeV compared with that of the +pc → ∞ case. If one assumes that non-hydrodynamic +high pT components emit photons with no anisotropy, +the pT spectra are slightly enhanced for smaller pc be- +cause of the higher initial temperature. Elliptic flow is +reduced for smaller values of pc in this case because the +early thermal photons have small momentum anisotropy +and the medium-high pT photons have zero anisotropy. +The photon puzzle indicates that direct photon v2 +might need to be larger than that calculated in conven- +tional methods. This might imply that the high pT pho- +tons should not be abundant in the framework of the +low-momentum hydrodynamic model. Direct photon v2 +might be informative of the momentum dependence of +thermalization in heavy-ion collisions as they retain the +information about the early stages of hydrodynamic evo- +lution. +Future prospects include the inclusion of viscosity and +event-by-event fluctuations for qualitative comparisons +with experimental data. Also, non-thermal photons from +medium-high momentum components might be more +quantitatively described by using a weak-coupling pic- +ture such as the partonic and hadronic transport models. +It would also be interesting to consider the time depen- +dence of pc because particles would exchange momentum +elastically and inelastically. The model may be extended +by introducing finite chemical potentials to elucidate the +momentum dependence of thermalization at beam energy +scan energies. +ACKNOWLEDGMENTS +The author is grateful for the valuable discussion with +M. Kitazawa, K. Murase, A. Ohnishi, and H. Suganuma +during a seminar at Yukawa Institute for Theoretical +Physics, Kyoto University. +[1] J. Letessier and J. Rafelski, Hadrons and quark - gluon +plasma, Vol. 18 (Cambridge University Press, 2002). +[2] J. Kapusta, B. Muller, +and J. Rafelski, Quark-Gluon +Plasma: Theoretical Foundations (Elsevier, 2003). +[3] K. Yagi, T. Hatsuda, +and Y. 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C 96, 044902 +(2017), arXiv:1707.08466 [nucl-th]. + diff --git a/NNAyT4oBgHgl3EQfs_m_/content/tmp_files/load_file.txt b/NNAyT4oBgHgl3EQfs_m_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bb073c5ccc382d306583f666b5d5f2484e80cbea --- /dev/null +++ b/NNAyT4oBgHgl3EQfs_m_/content/tmp_files/load_file.txt @@ -0,0 +1,1088 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf,len=1087 +page_content='Hydrodynamic model of heavy-ion collisions with low momentum components Akihiko Monnai1, ∗ 1Department of General Education, Faculty of Engineering, Osaka Institute of Technology, Osaka 535-8585, Japan (Dated: January 3, 2023) Relativistic heavy-ion collisions suggest that low momentum regions of the observed particle spectra are thermal and hydrodynamic, while medium-high momentum regions are non-thermal and perturbative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' In this study, I construct a hydrodynamic model of heavy-ion collisions by cutting off the medium-high momentum contributions and investigate the phenomenological consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Numerical simulations indicate that the temperature of the quark matter can be higher at earlier times owning to the modification of the equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' It is also suggested that direct photon elliptic flow can be sensitive to the momentum dependence of thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' PACS numbers: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='-q, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='Nq, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='Ld I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' INTRODUCTION Heavy-ion collisions at relativistic energies can produce quark-gluon plasma (QGP) [1–4], a deconfined quantum chromodynamic (QCD) matter that is considered to have filled the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' BNL Relativistic Heavy Ion Collider (RHIC) [5–8] and CERN Large Hadron Col- lider (LHC) [9–11] have played pivotal roles in experi- mental exploration of the properties of the QGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The hot and dense medium is quantified as a fluid with ex- tremely small kinetic viscosity by the success of relativis- tic hydrodynamic description of hadronic momentum dis- tributions and their azimuthal momentum anisotropies [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Observed large elliptic flow v2, which can be defined as the second order Fourier coefficient of a trans- verse momentum (pT ) spectrum, is considered as a sig- nature of fluidity because it strongly reflects the spatial anisotropy of the collision system [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Comparisons of hydrodynamic simulations and exper- imental data of pT spectra [16, 17] indicate that low mo- mentum particles below 2-4 GeV are thermalized and participate in the soft medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Medium to high momen- tum regions are, with several exceptions such as the hy- drodynamic models [18–20] based on nonextensive statis- tics [21, 22], explained by perturbative QCD approaches and are not considered to constitute the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The particle spectra are quantitatively well-described by the combined model in a wide transverse momentum range [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' However, most hydrodynamic models assume thermalization of constituent particles at all momenta in the construction of the equation of state and in the calculation of particle production, which can affect the estimation of the observables in collider experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Photons are also abundantly produced in relativistic heavy-ion collisions [26–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Inclusive photons are di- vided into direct photons and decay photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The former are emitted from the interacting QCD system while the latter are produced after the system is decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Since ∗ akihiko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='monnai@oit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='jp the QCD medium is transparent in terms of electromag- netic interactions, direct photons are understood as an observable informative of the space-time evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Direct photon spectra and flow harmonics, on the other hand, are not completely described by hydro- dynamic models [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The situation is known as the photon puzzle, and much efforts have been made to solve the issue [42–86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The discrepancy between the theoret- ical estimations and experimental data of direct photon pT spectra is visible at all momenta while that of direct photon v2 and v3 – the third order Fourier coefficient known as triangular flow – is more apparent at higher momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' One of the difficulties of the puzzle is that the medium-high momentum photons are mainly produced at early times in the time evolution but mechanisms that increase early photon emission reduce v2 and v3 because the flow anisotropy is underdeveloped at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' This implies that a momentum-dependent explanation is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' In this study, I develop a hydrodynamic model as- suming only low momentum components are thermal- ized, which may be referred to as “red” hydrodynamic model for cutting off the contributions of the particles with high momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The modified equation of state is estimated based on hadron resonance gas and parton gas models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' I show in hydrodynamic model simulations that the medium temperature can be higher at early times but the radial flow development is mostly insensitive to the cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Direct photons are affected by the modification of the medium evolution and the thermal photon emission rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Numerical simulations indicate that while the ef- fect on direct photon pT spectra is minimal, differential elliptic flow v2(pT ) can be sensitive to the momentum dependence of thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' II, I for- mulate a relativistic hydrodynamic model with momen- tum cutoffs for the particles participating in the medium to describe momentum-dependent thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The equation of state is constructed accordingly so that ther- modynamic consistency is preserved for macroscopic vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The results of numerical estimations are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Direct photon spectra and elliptic flow are dis- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='00588v1 [nucl-th] 2 Jan 2023 2 cussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Conclusions and outlook are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' I use the natural unit c = ℏ = kB = 1 and the mostly- minus Minkowski metric gµν = diag(+, −, −, −) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' HYDRODYNAMIC MODEL The relativistic hydrodynamic model with low momen- tum components is developed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' I consider the situation where the thermal sector is mostly indepen- dent of the non-thermal sector, as commonly assumed in a standard hydrodynamic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Energy-momentum conservation ∂µT µν = 0 provides the equation of motion in the limit of vanishing densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The modifications of the QCD equation of state and the photon production are discussed in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Equation of state The equation of state is affected by the momentum cutoff for constituent particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' I construct the equation of state based on the hadron resonance gas and parton gas models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The grand-canonical partition function Zi would be expressed as ln Zi = ±V � pc 0 gid3p (2π)3 ln � 1 ± exp � − Ei T �� , (1) in relativistic kinetic theory, which leads to the hydro- static pressure P = 1 V � i T ln Zi = ±T � i � pc 0 gid3p (2π)3 ln � 1 ± exp � − Ei T �� = 1 3 � i � pc 0 gid3p (2π)3 p2 Ei 1 exp(Ei/T) ± 1 + Φc, (2) where the thermodynamic correction is Φc = Tp3 c 6π2 � i ln � 1 ± exp � − Ec i T �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' (3) pc is the cutoff momentum, V is the volume, gi is the degeneracy factor, Ei is the energy, and T is the tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' i denotes particle species;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' u, d, s quarks and gluons are considered in the QGP phase, and hadron res- onances with u, d, s components with the mass below 2 GeV in the hadronic phase [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The hadronic pressure (Phad) and the QGP pressure (PQGP) are connected as follows [20]: P(T) = � � � � � Phad(T) (T < Tc) PQGP(T){1 − exp[−c(T − Tc)]} +Phad(Tc) exp[−c(T − Tc)] (T ≥ Tc) (4) 0 1 2 3 4 5 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='7 P/T4 T (GeV) pc=∞ pc=5 GeV pc=4 GeV pc=3 GeV pc=2 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' (Color online) The equation of state with momentum cutoffs pc = 5, 4, 3, and 2 GeV (dashed, dash-dotted, dash- double-dotted, and dotted lines, respectively) compared to that without a cutoff (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' where the constant factor is c = P ′ had(Tc) PQGP(Tc) − Phad(Tc), (5) which makes P(T) continuous and smooth at the con- necting temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The pressure of the hadron res- onance gas is used up to Tc and then is exponentially damped to the pressure of the parton gas above Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' This prescription has advantages over the conventional hy- perbolic connection when the hadronic and QGP pres- sures have a gap near the crossover region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' It puts more emphasis on the hadron resonance gas model, which is known to be consistent with lattice QCD simulations be- low the crossover temperature, than on the parton gas model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Also, the Cooper-Frye prescription of kinetic freeze-out [88] can be used below Tc without suffering from the loss of entropy density when applied to the hy- drodynamic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The difference between the resulting equation of state in the limit of pc → ∞ and the lat- tice QCD data [89, 90] is within 10% for the temperature range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='13 ≤ T ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='4 TeV when Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='14 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Other state variables such as the energy density e and entropy density s are estimated using thermodynamic relations s = dP/dT and e + P = Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The dimensionless pressure P/T 4 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 1 for several different values of pc with Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='14 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' One can see that the effects of momentum cutoff are more ap- parent at higher temperatures and mostly negligible in the hadronic phase for pc = 2-5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' This implies that the space-time evolution of the medium is affected mostly at early times in nuclear collisions where the temperature is high and the effects on the energy-momentum match- ing of Cooper-Frye prescription at kinetic freeze-out is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' It should be noted that the pressure P without dimensional normalization monotonously increases as a function of the temperature in all cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=', the entropy density s(T) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Photon production Electromagnetic probes are expected to be sensitive to the state of the hot QCD system in early stages of nuclear collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' In this study, direct photons are con- sidered as the target observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' I consider thermal pho- tons and prompt photons as the conventional sources of direct photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Additionally, non-thermal contributions from medium-high momentum components are consid- ered because while they are not part of the medium, they could still emit photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Pre-equilibrium photons and hadron gas photons, which are emitted before and after the medium formation are neglected to focus on the modification of the hydrodynamic stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The thermal photon contributions outside the freeze-out hypersurface are included [64] to partially compensate for the lack of post-equilibrium emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' I assume that the momentum-truncated thermal pho- ton emission rate in the QGP phase is expressed based on the small angle prescription [91, 92] as E dRγ QGP d3p = � f e2 f 4 π2 αEMαs log � 1 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='919 g2 � × h(p)fq(p) � pc 0 d3p′ (2π)3 1 p′ [fg(p′) + fq(p′)], (6) where h(p) = 1 2 � 1 − tanh �p − pc ∆pc �� , (7) is the hyperbolic factor introduced for a smooth momen- tum cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' ∆pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='2pc is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The subscripts q and g denote quarks and gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' ef is the charge for the flavor f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The couplings are set to αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='2 and αEM = 1/137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The total thermal photon emission rate is calculated by connecting the QGP rate with the hadronic rate [93–95] based on the massive Yang-Mills theory [96, 97] truncated with the factor h(p) as E dRγ th d3p = 1 2 � 1 − tanh �T − Tph ∆Tph �� E dRγ had d3p + 1 2 � 1 + tanh �T − Tph ∆Tph �� E dRγ QGP d3p , (8) where Tph = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='17 GeV and ∆Tph = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='1Tph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Thermal photon spectra is estimated by replacing E with p · u for taking account of the Lorentz boost before integrating over the space-time volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' As mentioned earlier, medium-high pT components might also emit non-thermal photons, which in con- ventional frameworks are estimated as thermal photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Owing to the lack of the complete description of this mo- mentum sector, their contribution would be estimated assuming the difference between the thermal rate at pc and that in the limit of pc → ∞ represents the high pT contribution by introducing the effective temperature de- duced from the medium temperature, E dRγ high d3p = E dRγ th d3p (∞) − E dRγ th d3p (pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' (9) Since the momentum sector is weakly-coupled, it is sim- ply assumed to have zero momentum anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The prescription keeps the direct photon spectra unchanged because the lack of thermal photons is compensated by non-thermal photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' This might be improved by in- troducing a transport model designed to describe the medium-high pT components in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Prompt photons are estimated using the parametriza- tion based on the scaling of p + p collision data [93] as E dN γ pr d3p = 6745 √s (pT )5 Ncoll σin pp pb GeV2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' (10) Ncoll is the number of collisions and σin pp is the inelastic cross section of nucleon-nucleon collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' They are also assumed to have no azimuthal momentum anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' NUMERICAL RESULTS The (2+1)-dimensional inviscid hydrodynamic model [56] is used to numerically elucidate the effects of mo- mentum limits on thermal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Event-averaged initial conditions for √sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='76 TeV Pb+Pb colli- sions are constructed using Monte-Carlo Glauber model [98] at the impact parameter of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='6 fm to imitate 0- 20 % centrality class events for demonstrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Full quantitative analyses including event-by-event fluc- tuation will be discussed elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The initial time τhyd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='4 fm/c and σin pp = 65 mb are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The kinetic freeze-out temperature of Tf = 140 MeV is considered to calculate the hadronic yields [88] before estimating the resonance decay effects [99] for normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Thermal photon contributions within the extended hypersurface of T = 110 MeV are taken into account to imitate the effects of hadron gas photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Space-time evolution of the medium The time evolutions of the energy density, pressure, and temperature at the center of the medium without a momentum cutoff and with pc = 4, 3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5, and 2 GeV are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The initial energy density is fixed for all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' One can see that the energy density is mostly insensitive to the momentum cutoff [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 2 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The pres- sure, on the other hand, is larger for smaller cutoffs as the deviation from conformality becomes larger [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 2 (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The initial temperature, likewise, is higher for smaller pc because of the reduction in the effective degrees of free- dom for a given temperature [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 2 (c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The effects of the momentum cutoff on the pressure and temperature are found to be mostly negligible above pc ∼ 4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The modification becomes small as the medium temperature 4 0 20 40 60 80 100 120 (a) e (GeV/fm3) pc=∞ pc=4 GeV pc=3 GeV pc=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 GeV pc=2 GeV 0 10 20 30 40 50 60 (b) P (GeV/fm3) 0 10 20 30 40 50 60 (b) P (GeV/fm3) 0 10 20 30 40 50 60 (b) P (GeV/fm3) 0 10 20 30 40 50 60 (b) P (GeV/fm3) 0 10 20 30 40 50 60 (b) P (GeV/fm3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='6 1 10 (c) T (GeV) τ (fm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='6 1 10 (c) T (GeV) τ (fm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='6 1 10 (c) T (GeV) τ (fm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='6 1 10 (c) T (GeV) τ (fm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='6 1 10 (c) T (GeV) τ (fm) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' (Color online) Time evolution of (a) energy density, (b) pressure, and (c) temperature at the center of the medium without momentum cutoff (solid line) and with pc = 4, 3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5, and 2 GeV (dash-dotted, dash-double-dotted, dashed, and dotted lines, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' decreases and the results converge around τ ∼ 1 fm/c and T ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='3 GeV, which is consistent with the expectations based on the equation of state shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Figure 3 is the time-like flow component uτ averaged over the space volume within the freeze-out hypersurface for different pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The quantity deviates from unity in the presence of the transverse flow because of the normal- ization condition u · u = 1 when the longitudinal flow component uηs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' It increases as a function of time until the peripheral regions start to freeze out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Despite the pressure differences at early times, the flow develop- ment is found to be mostly independent of pc, possibly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='35 1 10 τ (fm) pc=∞ pc=4 GeV pc=3 GeV pc=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 GeV pc=2 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' (Color online) Time evolution of ⟨uτ⟩ within the freeze-out hypersurface without momentum cutoff (solid line) and with pc = 4, 3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5, and 2 GeV (dash-dotted, dash-double- dotted, dashed, and dotted lines, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' because of the quick convergence of the pressure and its gradients to the values in the pc → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' This implies that it would be difficult to distinguish the pc scenarios based on the experimental data of hadronic production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Photon particle spectra Transverse momentum spectra of photons at midra- pidity are estimated in numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Figure 4 shows pT spectra of (a) thermal photons, (b) prompt and thermal photons, and (c) direct photons estimated as the sum of prompt, thermal, and high-pT photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Thermal photon spectra is visibly reduced above pc once the cut- off is introduced to the emission rate [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 4 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The effect competes with that of higher initial temperatures which is supposed to increase the spectrum for smaller pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' When combined with prompt photons, the cutoff ef- fect becomes relatively small and reduces the spectra only around 2-3 GeV because prompt photons become impor- tant above around 3 GeV [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 4 (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Figure 4 (c) shows direct photon spectra where non-thermal photons from the high momentum particles are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' It should be noted that the high pT photons are mimicked by pseudo- thermal photons with medium-high momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' While the spectrum is increased by a few percent for smaller pc be- cause the initial temperature is higher and the thermal photon contribution becomes relatively larger, it is found to be mostly insensitive to the choice of the cutoff mo- mentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Photon differential elliptic flow Differential elliptic flow of photons vγ 2 (pT ) at midra- pidity are investigated in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The azimuthal 5 10-5 10-4 10-3 10-2 10-1 100 (a) dNγ th/2πpTdpTdη pc=∞ pc=4 GeV pc=3 GeV pc=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 GeV pc=2 GeV 10-4 10-3 10-2 10-1 100 101 (b) dNγ th+pr/2πpTdpTdη 10-4 10-3 10-2 10-1 100 101 (b) dNγ th+pr/2πpTdpTdη 10-4 10-3 10-2 10-1 100 101 (b) dNγ th+pr/2πpTdpTdη 10-4 10-3 10-2 10-1 100 101 (b) dNγ th+pr/2πpTdpTdη 10-4 10-3 10-2 10-1 100 101 (b) dNγ th+pr/2πpTdpTdη 10-4 10-3 10-2 10-1 100 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 4 (c) dNγ th+pr+high/2πpTdpTdη pT (GeV) 10-4 10-3 10-2 10-1 100 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 4 (c) dNγ th+pr+high/2πpTdpTdη pT (GeV) 10-4 10-3 10-2 10-1 100 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 4 (c) dNγ th+pr+high/2πpTdpTdη pT (GeV) 10-4 10-3 10-2 10-1 100 101 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 4 (c) dNγ th+pr+high/2πpTdpTdη pT (GeV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' (Color online) pT spectra of (a) thermal photons (b) thermal and prompt photons, and (c) thermal, prompt, and high pT photons without momentum cutoff (solid line) and with pc = 4, 3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5, and 2 GeV (dash-dotted, dash-double- dotted, dashed, and dotted lines, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' momentum anisotropy is estimated as vγ 2 (pT ) = � dφ cos[2(φ − Ψ)] dN γ dφpT dpT dy � dφ dN γ dφpT dpT dy , (11) where φ is the azimuthal momentum angle and Ψ is the event plane angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Thermal photons are the source of anisotropy here because prompt and high pT photons are assumed to be isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Figure 5 (a) shows thermal photon v2 for different pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' It is visibly enhanced for smaller values of pc be- cause the photons above pc are late time contributions 0 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5 4 (c) v2 γ,th+pr+high pT (GeV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' (Color online) Differential v2 of (a) thermal photons (b) thermal and prompt photons, and (c) thermal, prompt, and high pT photons without momentum cutoff (solid line) and with pc = 4, 3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='5, and 2 GeV (dash-dotted, dash-double- dotted, dashed, and dotted lines, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' where their momentum is Lorentz-boosted by the ra- dial flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Note that the number of thermal photons are small above pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Once prompt photons are added, v2 is still enhanced by the momentum cutoff around pT ∼ 2-3 GeV when pc ∼ 2-3 GeV but is suppressed above those momenta [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 5 (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The contributions of high pT photons with zero anisotropy further tend to reduce v2 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' 5 (c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Compared with the thermal limit pc → ∞, the upper bound on the momenta of constituent particles might decrease the overall momentum anisotropy because (i) the initial temperature is higher and the number of early thermal photons with underdeveloped anisotropy 6 increases and (ii) the non-thermal contributions above pc do not have anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The results suggest that whether the elliptic flow is enhanced or reduced could depend on non-thermal photons from the medium-high pT sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' It is also implied that direct photon v2 might be an ob- servable sensitive to the momentum dependence of ther- malization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' It should be noted that the results are shown to demon- strate the qualitative behavior of the photon observables caused by the momentum cutoff and cannot be compared quantitatively to experimental data because of the lack of event-by-event fluctuations and viscous modifications, which would affect flow harmonics and, to smaller ex- tent, pT spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Those effects, on the other hand, are expected to be roughly orthogonal to the the cutoff ef- fects and can be studied independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' CONCLUSION AND OUTLOOK I have developed the relativistic hydrodynamic model of heavy-ion collisions where the medium is constituted by low momentum components because the experimen- tal data of high-energy heavy-ion collisions suggest that the system is thermalized only up to pT ∼ 2-4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The QCD equation of state has been constructed using the hadron resonance gas and parton gas models by introduc- ing the upper limit pc on the momenta of constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' It has been found that the effect of the momentum limit on the hydrostatic pressure is visible only in the QGP phase and becomes small for larger pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Numerical hydrodynamic simulations of heavy-ion col- lisions have indicated that the effects of the momentum cutoff on the thermodynamic variables such as the tem- perature and pressure are mostly limited to early times before τ ∼ 1 fm/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The radial flow development is thus not much affected by the choice of pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' It is also consis- tent with the observation that the effective temperature that characterizes the radial expansion in heavy-ion col- lisions at RHIC and LHC are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='3 GeV [100], because the equations of state at pc = 2-5 GeV are shown to converge below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Direct photons, on the other hand, have been found to be sensitive to the cutoff momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Thermal pho- ton radiation is reduced above pT ∼ pc, where the small number of high-momentum thermal photons with rela- tively large azimuthal momentum anisotropy are pro- duced through the Lorentz boost of radial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Adding prompt photons make the pT spectra mostly in- sensitive to the cutoff because they dominates the spectra above pT ∼ 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Elliptic flow, on the other hand, is en- hanced around pT ∼ 2-3 GeV compared with that of the pc → ∞ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' If one assumes that non-hydrodynamic high pT components emit photons with no anisotropy, the pT spectra are slightly enhanced for smaller pc be- cause of the higher initial temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Elliptic flow is reduced for smaller values of pc in this case because the early thermal photons have small momentum anisotropy and the medium-high pT photons have zero anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The photon puzzle indicates that direct photon v2 might need to be larger than that calculated in conven- tional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' This might imply that the high pT pho- tons should not be abundant in the framework of the low-momentum hydrodynamic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Direct photon v2 might be informative of the momentum dependence of thermalization in heavy-ion collisions as they retain the information about the early stages of hydrodynamic evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Future prospects include the inclusion of viscosity and event-by-event fluctuations for qualitative comparisons with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Also, non-thermal photons from medium-high momentum components might be more quantitatively described by using a weak-coupling pic- ture such as the partonic and hadronic transport models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' It would also be interesting to consider the time depen- dence of pc because particles would exchange momentum elastically and inelastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' The model may be extended by introducing finite chemical potentials to elucidate the momentum dependence of thermalization at beam energy scan energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' ACKNOWLEDGMENTS The author is grateful for the valuable discussion with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Kitazawa, K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Monnai and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Ollitrault, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content=' C 96, 044902 (2017), arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} +page_content='08466 [nucl-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfs_m_/content/2301.00588v1.pdf'} diff --git a/RdAyT4oBgHgl3EQf7_on/content/tmp_files/2301.00847v1.pdf.txt b/RdAyT4oBgHgl3EQf7_on/content/tmp_files/2301.00847v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c958518f1f6ee7f5480598f2c1ccfa9c27138add --- /dev/null +++ b/RdAyT4oBgHgl3EQf7_on/content/tmp_files/2301.00847v1.pdf.txt @@ -0,0 +1,1349 @@ +Popularity Ranking of Database Management +Systems +Aleem Akhtar +Dept. of Computer Science +Virtual University of Pakistan +aleem.akhtar@seecs.edu.pk +Abstract—Databases are considered to be integral part of mod- +ern information systems. Almost every web or mobile application +uses some kind of database. Database management systems are +considered to be a crucial element from both business and techno- +logical standpoint. This paper divides different types of database +management systems into two main categories (relational and +non-relational) and several sub categories. Ranking of various +sub categories for the month of July, 2021 are presented in the +form of popularity score calculated and managed by DB-Engines. +Popularity trend for each category is also presented to look at +the change in popularity since 2013. Complete ranking and trend +of top 20 systems has shown that relational models are still most +popular systems with Oracle and MySQL being two most popular +systems. However, recent trends have shown DBMSs like Time +Series and Document Store getting more and more popular with +their wide use in IOT technology and BigData, respectively. +Index Terms—DBMS, Ranking, NoSQL +I. INTRODUCTION +Databases are considered to be integral part of modern +information systems. Almost every web or mobile application +uses some kind of database. Databases are ubiquitously used in +data centers and to maintain records in educational institutes, +medical and healthcare systems, and all kinds of private and +government institutions. Database and database management +system (DBMS) are two distinct things. Organized collection +of data is called database while DBMS on the other hand is +a software which interacts with the database and acts as an +interface between the user and database. But database term is +usually used to refer to both the DBMS and database itself +[1]. +The DBMSs with their accompanying data communications +systems, has made possible for users from all industries to +develop both batch and online applications in a cost effective +and timely manner. These database systems has served as base +for most of the applications in every government agency and +industry. These systems were driving force behind the sale +of mainframe computers during the 1970s. Historians and +industry analysts consider DBMS a crucial part from both +business and technological standpoint [2]. Some of the reasons +supplied by analysts are: +• An efficient and cost effective method to program com- +plex applications is provided by DBMS without rewriting +the data retrieval and access functions for each applica- +tion. +• A simple and standard way is provided by them to share +data among multiple users and multiple applications. +• Specialized user-oriented languages were created. +• Standard interfaces are provided by DBMSs for data +communication programs so that the development, test- +ing, and maintenance of online transaction processing +applications could be done efficiently in terms of time +and cost. +• Databases are easily managed on various sequential and +random-access devices without the application program- +mer to think about difference between them. +• Portability is provided i.e. enabling users to move appli- +cations from one operating system or platform to other +with an ease. +• Companies marketing these systems became largest inde- +pendent software products companies in the late 1970s. +• A tremendous amount of hardware was sold by IBM and +many independent storage device and terminal compa- +nies. +Rest of the paper is divided into four sections. The first +section provides a brief description on different types of +database systems followed by ranking and trends in the second +section. The third section presents analysis on ranking and +trends. The fourth and last section concludes the paper. +II. TYPES OF DATASE MANAGEMENT SYSTEMS +Database management systems can be generally divided into +two main categories: relational database management systems +or RDMBS that supports relational data model, and in case +it supports other data models are often subsumed as NoSQL +systems. However, there are different subcategories of each +DBMS and a complete hierarchy is given in Figure 1. +As mentioned earlier, DBMSs can be divided into two broad +categories i.e. relational and non-relational. Each category with +subcategory is explained below. +A. Relational DBMSs +Relational database management systems support the rela- +tional or table-oriented data model with a pre-defined database +schema. Each table/relation has a unique name in the database +and fixed number of attributes (columns) with fixed data types. +Each row in the table defines a unique record. Normalization +is used to generate table schemas during the data modeling +process. Relational DBMS allows different types of operations +arXiv:2301.00847v1 [cs.DB] 2 Jan 2023 + +DBMSs +Relational +Traditional +RDBMS +Advanced +OODBMS +Multivalue +Non-Relational +Spatial +Temporal +Time Series +Event Stores +NoSQL +Key- +Value +Wide +Column +Documen +t +Graph +RDF +Native +XML +Content +Stores +Search +Engines +Fig. 1. Types of DBMSs +such as classical set operations (intersection, union, and differ- +ence), Selection, Projection, and Joins. Some other operations +to create, modify, delete table schemas, user management, +and transaction controlling are also performed. These basic +and advanced operations are performed using some kind of +database language, with Structured Query Language (SQL) +being well-established standard [3]. +RDBMS have been most common types of DBMS type +since they were first introduced in the early 1980s. Over the +years, relational databases have been expanded with advanced +non-relational concepts such as non-atomic values (multi- +valued), hierarchies, inheritance, and user-defined data types +which are sometimes referred to as object-oriented database +management systems (OODBMS) [4]. +Most popular examples of traditional RDBMS are: Oracle +[5][6], MySQL [7][8], Microsoft SQL Server [9][10], Post- +greSQL [11][12], and IBM Db2 [13][14]. +1) Object Oriented DBMS: In the 1980s, common use of +object-oriented programming languages motivated the devel- +opment of object-oriented database management systems or +simply object databases. Main objective behind introduction +of OODBMS was to store the objects and their relationships +(inheritance) in the database in a way that relates to their rep- +resentation in the programming language, without converting +or decomposing them [15]. +An OODBMS thus follows an object-oriented data model +with methods, properties, and classes (objects schema). An +object is always managed as a whole. In other words, insertion +or reading of object is done in one atomic operation which in +relational model will take multiple tables to store that object +and complex joins to retrieve it. To perform these operations, +a query language similar to SQL is used for manipulation of +objects in OODBMS. +Most popular examples of OODBMS are InterSystems +Cach´e [16], Db4o [17], InterSystems IRIS [18], ObjectStore +[19], and Actian NoSQL Database [20]. +2) Multivalue DBMS: Multivalue DBMS are similar to +traditional relational systems and store data in tables. However, +major difference between multivalue DBMS and traditional +RDBMS is that they have flexibility of storing multiple (non- +atomic) values to one attribute. These are often called non-first +normal form or NF2 systems as storing non-atomic values +contradicts the condition for first normal form. +Most popular examples of multivalue DBMS are Adabas +[21], UniData,UniVerse [22], jBASE [23], Model 204 [24], +and D3 [25]. +B. Non-Relational DBMS +Non-relational DBMS are further divided into three subcat- +egories. Each of them is briefly explained. +1) NoSQL: NoSQL database systems do not use a relational +data model like RDBMS and generally have no SQL interface. +NoSQL databases have been in existence for many years but +the term NoSQL was first introduced in 2009 when many +new systems were developed in order to cope with the new +requirements for DBMS at that time e.g. scalability, Big Data, +and fault tolerance [26]. Big Data has been at the heart of +development of these types of databases. Apache Science +Foundation has played one of the most important role in Big +Data projects [27]. +NoSQL systems are a heterogeneous group of very different +database systems. Therefore every effort of classification fails +in classifying one or another system. However, the following +categories are well accepted: +• Search Engines +• Content Stores +• Native XML DBMS +• RDF Stores +• Graph DBMS +• Document Stores +• Wide Column Stores +• Key-Value Stores + +Purpose of the paper is to look at the trends of database +systems, therefore explanation of each system is skipped. +2) Spatial DBMS: +Spatial DBMS is different type of +DBMS that can efficiently store, query, and manipulate spatial +data. Objects in geometric space such as polygons and points +are represented by spatial data. Dedicated data types and +spatial indices are provided by spatial DBMSs to optimize +the storing and access of spatial data [28]. +Spatial DBMSs provide features of intersecting or merging +objects, computing distances, and calculating properties of +objects such as areas. Geospatial data are an important subset +of spatial data that deals with the locations on surface of Earth. +Geographic Information Systems (GIS) are able to work with +geospatial data [29]. In some cases, spatial data is combined +with temporal data to form spatio-temporal data that offers +more dimensions to store and manipulate data. +Most popular examples of Spatial DBMS are PostGIS [30], +SpatiaLite [31], and GeoMesa [32]. +3) Temporal DBMS: Temporal DBMS deals with the data +related to timestamps or events. Temporal DBMS are classified +into two categories i.e. Time Series DBMS and Event Stores. +A Time Series DBMS is a database management system that +is optimized for handling time series data: each entry is as- +sociated with a timestamp [33]. For example, time series data +may be produced by smart meters, sensors, or RFIDs in the +IoT. Time Series DBMS are aimed to efficiently gather, save +and query various time series with high transaction volumes. +While time series data can be managed with other categories of +DBMS i.e. key-value stores or relational systems, specialized +systems are often required to handle specific challenges. Most +popular DBMSs in this subcategory are: InfluxDB [34], Kdb+ +[35], and Prometheus [36]. +Event stores are database management systems that imple- +ment the event sourcing concepts. All state changing events +for an object are preserved by these systems with a timestamp. +To inferred current state of an object, all the events from time +0 to current time are replayed. On the other hand, other types +of DBMS lose the history of previous states if not modelled +explicitly. IBM Db2 is most popular system in this category +[37]. +III. RANKING AND TRENDS +A. Method to calculate score +DB-Engines website is used as source for all the ranking +and trends screenshots. DB-Engines uses a custom formula to +calculate a normalized score to rank popular database man- +agement systems [38]. Popularity ranking does not measure +the DBMSs use within IT systems or the total number of +installations on the systems. However, popularity score relate +to broad use of a certain system. Following parameters are +used by the DB-Engines ranking system to calculate popularity +score: +• Number of job offers in which system is mentioned on +Indeed and Simply Hired websites. +• Frequency of the searches in Google Trends +• Number of search results in the Google and Bing search +engines for particular system +• Frequency of the technical discussion on Stack Overflow +and DBA Stack Exchange +• Number of profiles on LinkedIn with the mention of +certain system +• Number of tweets where certain system is mentioned +using Hashtag +B. Ranking and Trends by Type +In this section, we present ranking of top 5 most popular +database management systems by each type for July, 2021. +Trends display popularity of these systems from November +2012 to July, 2021. Logarithmic scale is used to display +popularity score. +1) Relational DBMS: Table I presents ranking of top 5 +relational DBMS for the month of July, 2021 [39]. Oracle +being the most popular system followed by MySQL at second +rank. Score for these two systems is very close to each other +showing how popular these two systems are. +TABLE I +RANKING OF RELATIONAL DBMS +Rank +DBMS +Score +1 +Oracle +1262.66 +2 +MySQL +1228.38 +3 +Microsoft SQL Server +981.95 +4 +PostgreSQL +577.15 +5 +IBM Db2 +165.15 +Figure 2 shows popularity trend of top five relational +database management systems. Oracle, MySQL, and Microsoft +SQL Server have been competing for top spot since 2013. +Popularity of PostgreSQL has been on constant rise and is +getting closer to the top systems. +2) Object Oriented DBMS: Table II presents ranking of top +5 Object Oriented DBMS for the month of July, 2021 [40]. A +very low score of these systems show that they are still not +very popular as other relational systems and it might take a +long time to be globally adopted by industry and academia. +TABLE II +RANKING OF OBJECT ORIENTED DBMS +Rank +DBMS +Score +1 +InterSystems Cach´e +2.86 +2 +Db4o +1.71 +3 +InterSystems IRIS +1.7 +4 +ObjectStore +1.48 +5 +Actian NoSQL Database +1.4 +Figure 3 shows popularity trend of top five Object Oriented +database management systems. It can be clearly see in the +graph that all systems in this category have average popularity +score between 1 and 2, and nothing can be said sure about +which system is more popular than other. +3) Multivalue DBMS: Table III presents ranking of top 5 +Multivalue DBMS for the month of July, 2021 [41]. Similar +to OODBMSs, a very low score of these systems show that + +they are also not very popular as other relational systems and +it can be said that they are still in developmental phases. +TABLE III +RANKING OF MULTIVALUE DBMS +Rank +DBMS +Score +1 +Adabas +4.46 +2 +UniData,UniVerse +3.87 +3 +jBASE +1.82 +4 +Model 204 +1.24 +5 +D3 +1.23 +Figure 4 shows popularity trend of top five Multivalue +database management systems. It can be clearly see in the +graph that all systems in this category have average popularity +score between 1 and 2, and nothing can be said for sure about +which system is more popular than other. +4) Key-Value Stores: Table IV presents ranking of top 5 +Key-Value Store DBMS for the month of July, 2021 [42]. +Redis is most popular system from this category with popu- +larity score being more than double compared to second spot +Amazon DynamoDB. Score for last three systems is very close +to each other but difference between Redis and other systems +is very high at this time. +TABLE IV +RANKING OF KEY-VALUE STORES DBMS +Rank +DBMS +Score +1 +Redis +168.31 +2 +Amazon DynamoDB +75.2 +3 +Microsoft Azure Cosmos DB +36.7 +4 +Memcached +25.34 +5 +etcd +10.1 +Figure 5 shows popularity trend of top five Key-Value Store +database management systems. Even though Redis has been +on the top of popularity chart since 2013 but systems from the +Amazon and Microsoft have been on constant rise while on +the other hand Memcached is slowly decreasing in popularity. +5) Document Store: Table V presents ranking of top 5 +Document Store DBMS for the month of July, 2021 [43]. Mon- +goDB is by far the most popular Document Store DBMS with +the popularity score of nearly 500. Amazon DynamoDB and +Microsoft Azure Cosmos DB which are also Key-Value Store +systems as well come at second and third place respectively. +TABLE V +RANKING OF DOCUMENT STORE DBMS +Rank +DBMS +Score +1 +MongoDB +496.16 +2 +Amazon DynamoDB +75.2 +3 +Microsoft Azure Cosmos DB +36.7 +4 +Couchbase +28.46 +5 +Firebase Realtime Database +17.23 +Figure 6 shows popularity trend of top five Document +Store database management systems. MongoDB has been on +constant rise in popularity since its introduction but other +Document Store DBs have started to gain popularity, Firebase +Realtime Database by Google is one of them. +6) Wide Column Store: Table VI presents ranking of top +5 Wide Column Store DBMS for the month of July, 2021 +[44]. Cassandra being the most popular system followed by +HBase at second rank. Score of Cassandra is higher than next +four systems combined which shows how popular it is for this +category. +TABLE VI +RANKING OF WIDE COLUMN STORE DBMS +Rank +DBMS +Score +1 +Cassandra +114 +2 +HBase +44.07 +3 +Microsoft Azure Cosmos DB +36.7 +4 +Datastax Enterprise +7.52 +5 +Microsoft Azure Table Storage +5.09 +Figure 7 shows popularity trend of top five Wide Column +Store database management systems. Oracle and HBase have +been competing for top spot since 2013. Popularity of Mi- +crosoft Azure Cosmos DB has been on constant rise since its +introduction in 2015 and is getting closer to the top systems. +7) Graph DBMS: Table VII presents ranking of top 5 Graph +DBMS for the month of July, 2021 [45]. Neo4j is most popular +system from this category with popularity score being higher +than next four systems combined. Score for last three systems +is very close to each other but difference between Neo4j and +other systems is very high at this time. +TABLE VII +RANKING OF GRAPH DBMS +Rank +DBMS +Score +1 +Neo4j +57.16 +2 +Microsoft Azure Cosmos DB +36.7 +3 +ArangoDB +4.73 +4 +OrientDB +4.16 +5 +Virtuoso +4.01 +Figure 8 shows popularity trend of top five Graph database +management systems. Even though Neo4j has been on the top +of popularity chart since 2013 but system from the Microsoft +has been on constant rise and getting closer to the most popular +system in this category while on the other hand all other +systems are slowly decreasing in popularity. +8) Search Engines: Table VIII presents ranking of top +5 Search Engine DBMS for the month of July, 2021 [46]. +ElasticSearch is by far the most popular Document Store +DBMS with the popularity score of nearly 156. Splunk and +Solr come at second and third place respectively followed by +MarkLogic and Sphnix at last two spots. +Figure 9 shows popularity trend of top five Search Engine +database management systems. Solr used to be most popular +system in this category but it was overtaken by ElasticSearch +at the start of 2016. Increase in popularity of Splunk made it +second most popular system in January 2018. +9) Ranking of Top 20: Table IX presents complete ranking +of top 20 database management systems for July, 2021 [55]. +It can be seen that Oracle and MySQL are top 2 ranked +systems with the score of above 1200. Microsoft SQL Server + +Fig. 2. Trend of Relational DBMS [47] +Fig. 3. Trend of Object Oriented DBMS [48] +Fig. 4. Trend of Multivalue DBMS [49] +Fig. 5. Trend of Key-Value Stores DBMS [50] +Fig. 6. Trend of Document Store DBMS [51] +Fig. 7. Trend of Wide Column Store DBMS [52] +Fig. 8. Trend of Graph DBMS [53] +Fig. 9. Trend of Search Engines DBMS [54] + +DB-Engines Ranking of Relational DBMS +Oracle + MySQL +Microsoft SQL Server +PostgreSQL +IBM Db2 +SQLite +Microsoft Access +1k +MariaDB +Hive +800 +Microsoft Azure SQL Database +Score (logarithmic s cale) +Teradata +600 + SAP HANA + FileMaker +- SAP Adaptive Server +Google BigQuery +400 + Snowflake + Firebird +Amazon Redshift + Informix + Spark SQL + Vertica +200 + Impala +Netezza +Microsoft Azure Synapse Analytics + dBASE +- Greenplum + July 2021, DB-Engines.com +Presto +100 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +A1/7VDB-Engines Ranking of Object Oriented DBMS +InterSystems Cache +Db4o +InterSystems IRIS +ObjectStore +ActianNoSQLDatabase +Matisse +GemStone/S +s cale) +ObjectBox +Objectivity/DB +Score (logarithmic : +Perst +0.8 +ObjectDB +0.6 +GigaSpaces InsightEdge +Jade +atoti +0.4 +Starcounter +Actian FastObjects +VelocityDB +OrigoDB +0.2 +Siaqodb +WakandaDB +Eloquera +0.1 +@ July 2021, DB-Engines.com +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021DB-Engines Ranking of Multivalue DBMS +Adabas +UniData,UniVerse +s cale +jBASE +Model204 +(logarithmic +D3 +SciDB +Openlnsight +Rasdaman +Score +0.8 +Northgate Reality +OpenQM +0.6 +Milvus +0.4 +0.2 +@ July 2021, DB-Engines.com +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021DB-Engines Ranking of Key-value Stores +Redis +Amazon DynamoDB +200 +Microsoft Azure Cosmos DB +Memcached +etcd +100 +Hazelcast +Ehcache +Aerospike +Riak KV +40 +ArangoDB +Score (logarithmic scale) +Ignite +20 +OrientDB +Oracle NoSQL +RocksDB +10 + ScyllaDB + LevelDB + Oracle Berkeley DB + InterSystems Caché + Infinispan + Oracle Coherence +2 + LMDB +Amazon SimpleDB +Geode +1 +GridGain +InterSystems IRIS +Tarantool +@ July 2021, DB-Engines.com +GT.M +0.4 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +A&/lVDB-Engines Ranking of Document Stores +MongoDB +AmazonDynamoDB +Microsoft Azure Cosmos DB +Couchbase +Firebase Realtime Database +CouchDB +Realm +100 +MarkLogic +Google Cloud Firestore +ArangoDB +Score (logarithmic scale) +Google Cloud Datastore +OrientDB +Virtuoso +10 +Oracle NoSQL +RavenDB +RethinkDB +IBM Cloudant +PouchDB +Apache Drill +CloudKit +1 +Fauna +InterSystems IRIS +Datameer +Mnesia +AllegroGraph +FoundationDB +@ July 2021, DB-Engines.com +Amazon DocumentDB +0.1 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +A&/lVDB-Engines Ranking of Wide Column Stores +100 +Cassandra +40 +HBase +Microsoft Azure Cosmos DB +Score (logarithmic s cale) +Datastax Enterprise +20 +Microsoft Azure Table Storage +Accumulo +10 +Google Cloud Bigtable +ScyllaDB + HPE Ezmeral Data Fabric +- +Elassandra +4 +Amazon Keyspaces + Alibaba Cloud Table Store +Hypertable +2 +Sqrrl +1 +0.4 +@ July 2021, DB-Engines.com +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021DB-Engines Ranking of Graph DBMS +Neo4j +Microsoft Azure Cosmos DB +ArangoDB +OrientDB +Virtuoso + GraphDB +JanusGraph +10 +Amazon Neptune +TigerGraph +Stardog +Score (logarithmic s cale) +Dgraph +Fauna +AllegroGraph + Giraph + Nebula Graph + TypeDB +0.1 +Blazegraph + Graph Engine + InfiniteGraph +AnzoGraph DB +Fluree +FlockDB +0.01 + Memgraph + HyperGraphDB +TerminusDB + TinkerGraph + July 2021, DB-Engines.com +HugeGraph +0.001 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +^1/2VDB-Engines Ranking of Search Engines +Elasticsearch +Splunk +Solr +MarkLogic +100 +Sphinx + Algolia +Microsoft Azure Search +ArangoDB +- Virtuoso +Score (logarithmic s cale) +40 +Amazon CloudSearch +Xapian +CrateDB +20 +Alibaba Cloud Log Service +SearchBlox +Weaviate +Exorbyte +10 +FinchDB +Manticore Search +Google Search Appliance +Indica + searchxml +Endeca +DBSight +Srch2 +@ July 2021, DB-Engines.com +Compass +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021TABLE VIII +RANKING OF SEARCH ENGINES DBMS +Rank +DBMS +Score +1 +Elasticsearch +155.76 +2 +Splunk +90.05 +3 +Solr +51.79 +4 +MarkLogic +9.45 +5 +Sphinx +8 +is ranked third followed by PostgreSQL at fourth. All of these +top 4 are relational systems which shows the importance of +this category. In fact, 12 of the top 20 are relational systems. +3 of them are Search Engines, and one each of Key-Value, +Document, Wide Column, and Graph. Amazon Dynamo DB +is one system which is a multi-model system. Popularity score +difference between MongoDB at 5 and Redis at 6 shows that +top 5 are solid at these positions for foreseeable future. +TABLE IX +RANKING OF TOP 20 DBMS +Rank +DBMS +Score +Type +1 +Oracle +1262.66 +Relational +2 +MySQL +1228.38 +Relational +3 +Microsoft SQL Server +981.95 +Relational +4 +PostgreSQL +577.15 +Relational +5 +MongoDB +496.16 +Document +6 +Redis +168.31 +Key-value +7 +IBM Db2 +165.15 +Relational +8 +Elasticsearch +155.76 +Search engine +9 +SQLite +130.2 +Relational +10 +Cassandra +114 +Wide column +11 +Microsoft Access +113.45 +Relational +12 +MariaDB +97.98 +Relational +13 +Splunk +90.05 +Search engine +14 +Hive +82.68 +Relational +15 +Microsoft Azure SQL Database +75.22 +Relational +16 +Amazon DynamoDB +75.2 +Multi-model +17 +Teradata +68.95 +Relational +18 +Neo4j +57.16 +Graph +19 +SAP HANA +53.81 +Relational +20 +Solr +51.79 +Search engine +Figure 10 presents popularity trend of top 20 database +management systems since 2013. A close competition can +be seen between Oracle, MySQL, and Microsoft SQL Server +from the beginning of 2013. However, SQL Server has been +on slight decrease in popularity and the gap is getting wider. +Popularity of PostgreSQL is on constant rise and getting closer +to SQL Server. MongoDB has become one of the fasted +growing document store system which shows the interest from +the developers community in model different than relational. +IV. ANALYSIS +This section breaks down the popularity of database man- +agement systems by model and provides a brief analysis. +A. Number of systems per category +Figure 11 presents number of systems of per category. +There are total 373 database management systems of which +40% are relational systems followed by 17% of Key-value +Fig. 10. Trend of Top 20 DBMS [56] +stores. Relational DBMS, Document stores, and key-value +stores comprise of total 262 systems (70%) out of 373. +Fig. 11. Number of Systems Per Category [57] +B. Ranking score per category +Figure 12 shows ranking score per category in percentage +for July, 2021. Popularity score of all individual systems for +each category is used to calculate total percentage. As Table +IX showed that Oracle, MySQL, and MS SQL Server makes +highest score for all relational systems therefore this is also +depicted in Figure 12 where relational DBMS have 72.7% +of total popularity. Document stores are other set of systems +which are getting popular with MongoDB being most popular +of them all. +Fig. 12. Ranking Score Per Category [57] +C. Popularity Changes +The following set of charts show the historical trend of the +categories’ popularity. In the ranking of each month the best + +DB +B-Engines Ranking +Oracle +MySQL +Microsoft SQL Server +PostgreSQL +MongoDB +1 k +Redis +IBM Db2 +Elasticsearch +SQLite +Cassandra +Microsoft Access + scale) +MariaDB +100 +Splunk + Score (logarithmic +Hive +Microsoft Azure SQL Database +Amazon DynamoDB +Teradata + Neo4j +SAP HANA +10 +Solr +FileMaker +SAP Adaptive Server +HBase +Google BigQuery +N + Snowflake +Microsoft Azure Cosmos DB +@ July 2021, DB-Engines.com +PostGIS +1 +2013 +2015 +2017 +2018 +1/16V +2014 +2016 +2019 +2020 +2021Wide column stores: 13 / Content stores: 2 +TimeSeriesDBMS:36 +Documentstores:52 +Spatial DBMS: 5 +EventStores:3 +Search engines:21 +GraphDBMS:32 +Key-value stores:63 +Multivalue DBMS: 11 +NativeXMLDBMS:7 +RelationalDBMS:147 +ObjectorientedDBMS:22 +RDF stores:19Wide column stores 2.9%. +Document stores10.1% +Time SeriesDBMS 0.9% +Graph DBMS 1.7% +Spatial DBMS 0.5% +Key-value stores 5.4% +Search engines 4.6% +Multivalue DBMS 0.2% +NativeXMLDBMS0.3% +ObiectorientedDBMS0.2% +RDF stores 0.4% +RelationalDBMS72.7%three systems per category are chosen and the average of their +ranking scores is calculated. In order to allow comparisons, +the initial value is normalized to 100. +1) Complete Popularity Trend: The chart in 13 shows +that Graph DBMS has been on constant rise in popularity +because of their main use in social networking websites such +as Facebook, Twitter, and TikTok. Time Series systems are +another type of DBMSs which have gain popularity in recent +times due to advancements in IOT technology. +Fig. 13. Complete Popularity Trend [57] +2) Trend for last 24 months: This chart 14 shows change +in popularity trend for last 24 months. As mentioned earlier, +Time Series DBMS has seen rise in popularity in recent times +due to their use in Internet of Things. Relational systems on +the other hand are at the same level of popularity as in July +2019. However, DBMS like Native XML and RDF stores are +on decline. +Fig. 14. Trend for last 24 months [57] +V. CONCLUSION AND FUTURE WORK +Databases are considered to be integral part of modern +information systems. Almost every web or mobile application +uses some kind of database. Database management systems +are considered to be a crucial element from both business and +technological standpoint. This paper divided different types +of database management systems into two main categories +i.e. relational and non-relational and several sub categories. +Ranking of various sub categories for the month of July, 2021 +are presented in the form of popularity score calculated and +managed by DB-Engines. Popularity trend for each category is +also presented to look at the change in popularity since 2013. +Complete ranking and trend of top 20 systems has shown that +relational models are still most popular systems comprising of +12 systems in the top 20 with top 4 all being relational systems. +Relational DBMS also make the most number of systems and +highest percentage in popularity. Oracle and MySQL being +two most popular systems. 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Available: https://db-engines.com/en/ +ranking categories + diff --git a/RdAyT4oBgHgl3EQf7_on/content/tmp_files/load_file.txt b/RdAyT4oBgHgl3EQf7_on/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1379a26d585559cdbe562fc9fd0c196bcf4a9338 --- /dev/null +++ b/RdAyT4oBgHgl3EQf7_on/content/tmp_files/load_file.txt @@ -0,0 +1,640 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf,len=639 +page_content='Popularity Ranking of Database Management Systems Aleem Akhtar Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' of Computer Science Virtual University of Pakistan aleem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='akhtar@seecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='pk Abstract—Databases are considered to be integral part of mod- ern information systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Almost every web or mobile application uses some kind of database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Database management systems are considered to be a crucial element from both business and techno- logical standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' This paper divides different types of database management systems into two main categories (relational and non-relational) and several sub categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Ranking of various sub categories for the month of July, 2021 are presented in the form of popularity score calculated and managed by DB-Engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Popularity trend for each category is also presented to look at the change in popularity since 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Complete ranking and trend of top 20 systems has shown that relational models are still most popular systems with Oracle and MySQL being two most popular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' However, recent trends have shown DBMSs like Time Series and Document Store getting more and more popular with their wide use in IOT technology and BigData, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Index Terms—DBMS, Ranking, NoSQL I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' INTRODUCTION Databases are considered to be integral part of modern information systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Almost every web or mobile application uses some kind of database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Databases are ubiquitously used in data centers and to maintain records in educational institutes, medical and healthcare systems, and all kinds of private and government institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Database and database management system (DBMS) are two distinct things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Organized collection of data is called database while DBMS on the other hand is a software which interacts with the database and acts as an interface between the user and database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' But database term is usually used to refer to both the DBMS and database itself [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' The DBMSs with their accompanying data communications systems, has made possible for users from all industries to develop both batch and online applications in a cost effective and timely manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' These database systems has served as base for most of the applications in every government agency and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' These systems were driving force behind the sale of mainframe computers during the 1970s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Historians and industry analysts consider DBMS a crucial part from both business and technological standpoint [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Some of the reasons supplied by analysts are: An efficient and cost effective method to program com- plex applications is provided by DBMS without rewriting the data retrieval and access functions for each applica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' A simple and standard way is provided by them to share data among multiple users and multiple applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Specialized user-oriented languages were created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Standard interfaces are provided by DBMSs for data communication programs so that the development, test- ing, and maintenance of online transaction processing applications could be done efficiently in terms of time and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Databases are easily managed on various sequential and random-access devices without the application program- mer to think about difference between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Portability is provided i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' enabling users to move appli- cations from one operating system or platform to other with an ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Companies marketing these systems became largest inde- pendent software products companies in the late 1970s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' A tremendous amount of hardware was sold by IBM and many independent storage device and terminal compa- nies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Rest of the paper is divided into four sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' The first section provides a brief description on different types of database systems followed by ranking and trends in the second section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' The third section presents analysis on ranking and trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' The fourth and last section concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' TYPES OF DATASE MANAGEMENT SYSTEMS Database management systems can be generally divided into two main categories: relational database management systems or RDMBS that supports relational data model, and in case it supports other data models are often subsumed as NoSQL systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' However, there are different subcategories of each DBMS and a complete hierarchy is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' As mentioned earlier, DBMSs can be divided into two broad categories i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' relational and non-relational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Each category with subcategory is explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Relational DBMSs Relational database management systems support the rela- tional or table-oriented data model with a pre-defined database schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Each table/relation has a unique name in the database and fixed number of attributes (columns) with fixed data types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Each row in the table defines a unique record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Normalization is used to generate table schemas during the data modeling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Relational DBMS allows different types of operations arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='00847v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='DB] 2 Jan 2023 DBMSs Relational Traditional RDBMS Advanced OODBMS Multivalue Non-Relational Spatial Temporal Time Series Event Stores NoSQL Key- Value Wide Column Documen t Graph RDF Native XML Content Stores Search Engines Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Types of DBMSs such as classical set operations (intersection, union, and differ- ence), Selection, Projection, and Joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Some other operations to create, modify, delete table schemas, user management, and transaction controlling are also performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' These basic and advanced operations are performed using some kind of database language, with Structured Query Language (SQL) being well-established standard [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' RDBMS have been most common types of DBMS type since they were first introduced in the early 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Over the years, relational databases have been expanded with advanced non-relational concepts such as non-atomic values (multi- valued), hierarchies, inheritance, and user-defined data types which are sometimes referred to as object-oriented database management systems (OODBMS) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Most popular examples of traditional RDBMS are: Oracle [5][6], MySQL [7][8], Microsoft SQL Server [9][10], Post- greSQL [11][12], and IBM Db2 [13][14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 1) Object Oriented DBMS: In the 1980s, common use of object-oriented programming languages motivated the devel- opment of object-oriented database management systems or simply object databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Main objective behind introduction of OODBMS was to store the objects and their relationships (inheritance) in the database in a way that relates to their rep- resentation in the programming language, without converting or decomposing them [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' An OODBMS thus follows an object-oriented data model with methods, properties, and classes (objects schema).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' An object is always managed as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' In other words, insertion or reading of object is done in one atomic operation which in relational model will take multiple tables to store that object and complex joins to retrieve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' To perform these operations, a query language similar to SQL is used for manipulation of objects in OODBMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Most popular examples of OODBMS are InterSystems Cach´e [16], Db4o [17], InterSystems IRIS [18], ObjectStore [19], and Actian NoSQL Database [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2) Multivalue DBMS: Multivalue DBMS are similar to traditional relational systems and store data in tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' However, major difference between multivalue DBMS and traditional RDBMS is that they have flexibility of storing multiple (non- atomic) values to one attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' These are often called non-first normal form or NF2 systems as storing non-atomic values contradicts the condition for first normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Most popular examples of multivalue DBMS are Adabas [21], UniData,UniVerse [22], jBASE [23], Model 204 [24], and D3 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Non-Relational DBMS Non-relational DBMS are further divided into three subcat- egories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Each of them is briefly explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 1) NoSQL: NoSQL database systems do not use a relational data model like RDBMS and generally have no SQL interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' NoSQL databases have been in existence for many years but the term NoSQL was first introduced in 2009 when many new systems were developed in order to cope with the new requirements for DBMS at that time e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' scalability, Big Data, and fault tolerance [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Big Data has been at the heart of development of these types of databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Apache Science Foundation has played one of the most important role in Big Data projects [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' NoSQL systems are a heterogeneous group of very different database systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Therefore every effort of classification fails in classifying one or another system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' However, the following categories are well accepted: Search Engines Content Stores Native XML DBMS RDF Stores Graph DBMS Document Stores Wide Column Stores Key-Value Stores Purpose of the paper is to look at the trends of database systems, therefore explanation of each system is skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2) Spatial DBMS: Spatial DBMS is different type of DBMS that can efficiently store, query, and manipulate spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Objects in geometric space such as polygons and points are represented by spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Dedicated data types and spatial indices are provided by spatial DBMSs to optimize the storing and access of spatial data [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Spatial DBMSs provide features of intersecting or merging objects, computing distances, and calculating properties of objects such as areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Geospatial data are an important subset of spatial data that deals with the locations on surface of Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Geographic Information Systems (GIS) are able to work with geospatial data [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' In some cases, spatial data is combined with temporal data to form spatio-temporal data that offers more dimensions to store and manipulate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Most popular examples of Spatial DBMS are PostGIS [30], SpatiaLite [31], and GeoMesa [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 3) Temporal DBMS: Temporal DBMS deals with the data related to timestamps or events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Temporal DBMS are classified into two categories i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Time Series DBMS and Event Stores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' A Time Series DBMS is a database management system that is optimized for handling time series data: each entry is as- sociated with a timestamp [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' For example, time series data may be produced by smart meters, sensors, or RFIDs in the IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Time Series DBMS are aimed to efficiently gather, save and query various time series with high transaction volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' While time series data can be managed with other categories of DBMS i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' key-value stores or relational systems, specialized systems are often required to handle specific challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Most popular DBMSs in this subcategory are: InfluxDB [34], Kdb+ [35], and Prometheus [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Event stores are database management systems that imple- ment the event sourcing concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' All state changing events for an object are preserved by these systems with a timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' To inferred current state of an object, all the events from time 0 to current time are replayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' On the other hand, other types of DBMS lose the history of previous states if not modelled explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' IBM Db2 is most popular system in this category [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' RANKING AND TRENDS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Method to calculate score DB-Engines website is used as source for all the ranking and trends screenshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' DB-Engines uses a custom formula to calculate a normalized score to rank popular database man- agement systems [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Popularity ranking does not measure the DBMSs use within IT systems or the total number of installations on the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' However, popularity score relate to broad use of a certain system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Following parameters are used by the DB-Engines ranking system to calculate popularity score: Number of job offers in which system is mentioned on Indeed and Simply Hired websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Frequency of the searches in Google Trends Number of search results in the Google and Bing search engines for particular system Frequency of the technical discussion on Stack Overflow and DBA Stack Exchange Number of profiles on LinkedIn with the mention of certain system Number of tweets where certain system is mentioned using Hashtag B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Ranking and Trends by Type In this section, we present ranking of top 5 most popular database management systems by each type for July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Trends display popularity of these systems from November 2012 to July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Logarithmic scale is used to display popularity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 1) Relational DBMS: Table I presents ranking of top 5 relational DBMS for the month of July, 2021 [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Oracle being the most popular system followed by MySQL at second rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Score for these two systems is very close to each other showing how popular these two systems are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' TABLE I RANKING OF RELATIONAL DBMS Rank DBMS Score 1 Oracle 1262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='66 2 MySQL 1228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='38 3 Microsoft SQL Server 981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='95 4 PostgreSQL 577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='15 5 IBM Db2 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='15 Figure 2 shows popularity trend of top five relational database management systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Oracle, MySQL, and Microsoft SQL Server have been competing for top spot since 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Popularity of PostgreSQL has been on constant rise and is getting closer to the top systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2) Object Oriented DBMS: Table II presents ranking of top 5 Object Oriented DBMS for the month of July, 2021 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' A very low score of these systems show that they are still not very popular as other relational systems and it might take a long time to be globally adopted by industry and academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' TABLE II RANKING OF OBJECT ORIENTED DBMS Rank DBMS Score 1 InterSystems Cach´e 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='86 2 Db4o 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='71 3 InterSystems IRIS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='7 4 ObjectStore 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='48 5 Actian NoSQL Database 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='4 Figure 3 shows popularity trend of top five Object Oriented database management systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' It can be clearly see in the graph that all systems in this category have average popularity score between 1 and 2, and nothing can be said sure about which system is more popular than other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 3) Multivalue DBMS: Table III presents ranking of top 5 Multivalue DBMS for the month of July, 2021 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Similar to OODBMSs, a very low score of these systems show that they are also not very popular as other relational systems and it can be said that they are still in developmental phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' TABLE III RANKING OF MULTIVALUE DBMS Rank DBMS Score 1 Adabas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='46 2 UniData,UniVerse 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='87 3 jBASE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='82 4 Model 204 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='24 5 D3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='23 Figure 4 shows popularity trend of top five Multivalue database management systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' It can be clearly see in the graph that all systems in this category have average popularity score between 1 and 2, and nothing can be said for sure about which system is more popular than other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 4) Key-Value Stores: Table IV presents ranking of top 5 Key-Value Store DBMS for the month of July, 2021 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Redis is most popular system from this category with popu- larity score being more than double compared to second spot Amazon DynamoDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Score for last three systems is very close to each other but difference between Redis and other systems is very high at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' TABLE IV RANKING OF KEY-VALUE STORES DBMS Rank DBMS Score 1 Redis 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='31 2 Amazon DynamoDB 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2 3 Microsoft Azure Cosmos DB 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='7 4 Memcached 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='34 5 etcd 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='1 Figure 5 shows popularity trend of top five Key-Value Store database management systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Even though Redis has been on the top of popularity chart since 2013 but systems from the Amazon and Microsoft have been on constant rise while on the other hand Memcached is slowly decreasing in popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 5) Document Store: Table V presents ranking of top 5 Document Store DBMS for the month of July, 2021 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Mon- goDB is by far the most popular Document Store DBMS with the popularity score of nearly 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Amazon DynamoDB and Microsoft Azure Cosmos DB which are also Key-Value Store systems as well come at second and third place respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' TABLE V RANKING OF DOCUMENT STORE DBMS Rank DBMS Score 1 MongoDB 496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='16 2 Amazon DynamoDB 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2 3 Microsoft Azure Cosmos DB 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='7 4 Couchbase 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='46 5 Firebase Realtime Database 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='23 Figure 6 shows popularity trend of top five Document Store database management systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' MongoDB has been on constant rise in popularity since its introduction but other Document Store DBs have started to gain popularity, Firebase Realtime Database by Google is one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 6) Wide Column Store: Table VI presents ranking of top 5 Wide Column Store DBMS for the month of July, 2021 [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Cassandra being the most popular system followed by HBase at second rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Score of Cassandra is higher than next four systems combined which shows how popular it is for this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' TABLE VI RANKING OF WIDE COLUMN STORE DBMS Rank DBMS Score 1 Cassandra 114 2 HBase 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='07 3 Microsoft Azure Cosmos DB 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='7 4 Datastax Enterprise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='52 5 Microsoft Azure Table Storage 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='09 Figure 7 shows popularity trend of top five Wide Column Store database management systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Oracle and HBase have been competing for top spot since 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Popularity of Mi- crosoft Azure Cosmos DB has been on constant rise since its introduction in 2015 and is getting closer to the top systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 7) Graph DBMS: Table VII presents ranking of top 5 Graph DBMS for the month of July, 2021 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Neo4j is most popular system from this category with popularity score being higher than next four systems combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Score for last three systems is very close to each other but difference between Neo4j and other systems is very high at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' TABLE VII RANKING OF GRAPH DBMS Rank DBMS Score 1 Neo4j 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='16 2 Microsoft Azure Cosmos DB 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='7 3 ArangoDB 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='73 4 OrientDB 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='16 5 Virtuoso 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='01 Figure 8 shows popularity trend of top five Graph database management systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Even though Neo4j has been on the top of popularity chart since 2013 but system from the Microsoft has been on constant rise and getting closer to the most popular system in this category while on the other hand all other systems are slowly decreasing in popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 8) Search Engines: Table VIII presents ranking of top 5 Search Engine DBMS for the month of July, 2021 [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' ElasticSearch is by far the most popular Document Store DBMS with the popularity score of nearly 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Splunk and Solr come at second and third place respectively followed by MarkLogic and Sphnix at last two spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Figure 9 shows popularity trend of top five Search Engine database management systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Solr used to be most popular system in this category but it was overtaken by ElasticSearch at the start of 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Increase in popularity of Splunk made it second most popular system in January 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 9) Ranking of Top 20: Table IX presents complete ranking of top 20 database management systems for July, 2021 [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' It can be seen that Oracle and MySQL are top 2 ranked systems with the score of above 1200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Microsoft SQL Server Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Trend of Relational DBMS [47] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Trend of Object Oriented DBMS [48] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Trend of Multivalue DBMS [49] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Trend of Key-Value Stores DBMS [50] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Trend of Document Store DBMS [51] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Trend of Wide Column Store DBMS [52] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Trend of Graph DBMS [53] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Trend of Search Engines DBMS [54] DB-Engines Ranking of Relational DBMS Oracle MySQL Microsoft SQL Server PostgreSQL IBM Db2 SQLite Microsoft Access 1k MariaDB Hive 800 Microsoft Azure SQL Database Score (logarithmic s cale) Teradata 600 SAP HANA FileMaker SAP Adaptive Server Google BigQuery 400 Snowflake Firebird Amazon Redshift Informix Spark SQL Vertica 200 Impala Netezza Microsoft Azure Synapse Analytics dBASE Greenplum July 2021, DB-Engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com Presto 100 2013 2014 2015 2016 2017 2018 2019 2020 2021 A1/7VDB-Engines Ranking of Object Oriented DBMS InterSystems Cache Db4o InterSystems IRIS ObjectStore ActianNoSQLDatabase Matisse GemStone/S s cale) ObjectBox Objectivity/DB Score (logarithmic : Perst 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='8 ObjectDB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='6 GigaSpaces InsightEdge Jade atoti 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='4 Starcounter Actian FastObjects VelocityDB OrigoDB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2 Siaqodb WakandaDB Eloquera 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='1 @ July 2021, DB-Engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com 2013 2014 2015 2016 2017 2018 2019 2020 2021DB-Engines Ranking of Multivalue DBMS Adabas UniData,UniVerse s cale jBASE Model204 (logarithmic D3 SciDB Openlnsight Rasdaman Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='8 Northgate Reality OpenQM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='6 Milvus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2 @ July 2021, DB-Engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com 2013 2014 2015 2016 2017 2018 2019 2020 2021DB-Engines Ranking of Key-value Stores Redis Amazon DynamoDB 200 Microsoft Azure Cosmos DB Memcached etcd 100 Hazelcast Ehcache Aerospike Riak KV 40 ArangoDB Score (logarithmic scale) Ignite 20 OrientDB Oracle NoSQL RocksDB 10 ScyllaDB LevelDB Oracle Berkeley DB InterSystems Caché Infinispan Oracle Coherence 2 LMDB Amazon SimpleDB Geode 1 GridGain InterSystems IRIS Tarantool @ July 2021, DB-Engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2013 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2014 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2015 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2016 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2017 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2018 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='A&/lVDB-Engines Ranking of Document Stores ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='MongoDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='AmazonDynamoDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Microsoft Azure Cosmos DB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Couchbase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Firebase Realtime Database ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='CouchDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Realm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='MarkLogic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Google Cloud Firestore ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='ArangoDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Score (logarithmic scale) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Google Cloud Datastore ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='OrientDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Virtuoso ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Oracle NoSQL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='RavenDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='RethinkDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='IBM Cloudant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='PouchDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Apache Drill ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='CloudKit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Fauna ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='InterSystems IRIS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Datameer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='Mnesia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='AllegroGraph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='FoundationDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='@ July 2021,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' DB-Engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com Amazon DocumentDB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='1 2013 2014 2015 2016 2017 2018 2019 2020 2021 A&/lVDB-Engines Ranking of Wide Column Stores 100 Cassandra 40 HBase Microsoft Azure Cosmos DB Score (logarithmic s cale) Datastax Enterprise 20 Microsoft Azure Table Storage Accumulo 10 Google Cloud Bigtable ScyllaDB HPE Ezmeral Data Fabric Elassandra 4 Amazon Keyspaces Alibaba Cloud Table Store Hypertable 2 Sqrrl 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='4 @ July 2021, DB-Engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com 2013 2014 2015 2016 2017 2018 2019 2020 2021DB-Engines Ranking of Graph DBMS Neo4j Microsoft Azure Cosmos DB ArangoDB OrientDB Virtuoso GraphDB JanusGraph 10 Amazon Neptune TigerGraph Stardog Score (logarithmic s cale) Dgraph Fauna AllegroGraph Giraph Nebula Graph TypeDB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='1 Blazegraph Graph Engine InfiniteGraph AnzoGraph DB Fluree FlockDB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='01 Memgraph HyperGraphDB TerminusDB TinkerGraph July 2021, DB-Engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com HugeGraph 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='001 2013 2014 2015 2016 2017 2018 2019 2020 2021 ^1/2VDB-Engines Ranking of Search Engines Elasticsearch Splunk Solr MarkLogic 100 Sphinx Algolia Microsoft Azure Search ArangoDB Virtuoso Score (logarithmic s cale) 40 Amazon CloudSearch Xapian CrateDB 20 Alibaba Cloud Log Service SearchBlox Weaviate Exorbyte 10 FinchDB Manticore Search Google Search Appliance Indica searchxml Endeca DBSight Srch2 @ July 2021, DB-Engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com Compass 2013 2014 2015 2016 2017 2018 2019 2020 2021TABLE VIII RANKING OF SEARCH ENGINES DBMS Rank DBMS Score 1 Elasticsearch 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='76 2 Splunk 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='05 3 Solr 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='79 4 MarkLogic 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='45 5 Sphinx 8 is ranked third followed by PostgreSQL at fourth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' All of these top 4 are relational systems which shows the importance of this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' In fact, 12 of the top 20 are relational systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 3 of them are Search Engines, and one each of Key-Value, Document, Wide Column, and Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Amazon Dynamo DB is one system which is a multi-model system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Popularity score difference between MongoDB at 5 and Redis at 6 shows that top 5 are solid at these positions for foreseeable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' TABLE IX RANKING OF TOP 20 DBMS Rank DBMS Score Type 1 Oracle 1262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='66 Relational 2 MySQL 1228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='38 Relational 3 Microsoft SQL Server 981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='95 Relational 4 PostgreSQL 577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='15 Relational 5 MongoDB 496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='16 Document 6 Redis 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='31 Key-value 7 IBM Db2 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='15 Relational 8 Elasticsearch 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='76 Search engine 9 SQLite 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2 Relational 10 Cassandra 114 Wide column 11 Microsoft Access 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='45 Relational 12 MariaDB 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='98 Relational 13 Splunk 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='05 Search engine 14 Hive 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='68 Relational 15 Microsoft Azure SQL Database 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='22 Relational 16 Amazon DynamoDB 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2 Multi-model 17 Teradata 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='95 Relational 18 Neo4j 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='16 Graph 19 SAP HANA 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='81 Relational 20 Solr 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='79 Search engine Figure 10 presents popularity trend of top 20 database management systems since 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' A close competition can be seen between Oracle, MySQL, and Microsoft SQL Server from the beginning of 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' However, SQL Server has been on slight decrease in popularity and the gap is getting wider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Popularity of PostgreSQL is on constant rise and getting closer to SQL Server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' MongoDB has become one of the fasted growing document store system which shows the interest from the developers community in model different than relational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' ANALYSIS This section breaks down the popularity of database man- agement systems by model and provides a brief analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Number of systems per category Figure 11 presents number of systems of per category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' There are total 373 database management systems of which 40% are relational systems followed by 17% of Key-value Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Trend of Top 20 DBMS [56] stores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Relational DBMS, Document stores, and key-value stores comprise of total 262 systems (70%) out of 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Number of Systems Per Category [57] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Ranking score per category Figure 12 shows ranking score per category in percentage for July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Popularity score of all individual systems for each category is used to calculate total percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' As Table IX showed that Oracle, MySQL, and MS SQL Server makes highest score for all relational systems therefore this is also depicted in Figure 12 where relational DBMS have 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='7% of total popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Document stores are other set of systems which are getting popular with MongoDB being most popular of them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Ranking Score Per Category [57] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Popularity Changes The following set of charts show the historical trend of the categories’ popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' In the ranking of each month the best DB B-Engines Ranking Oracle MySQL Microsoft SQL Server PostgreSQL MongoDB 1 k Redis IBM Db2 Elasticsearch SQLite Cassandra Microsoft Access scale) MariaDB 100 Splunk Score (logarithmic Hive Microsoft Azure SQL Database Amazon DynamoDB Teradata Neo4j SAP HANA 10 Solr FileMaker SAP Adaptive Server HBase Google BigQuery N Snowflake Microsoft Azure Cosmos DB @ July 2021, DB-Engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com PostGIS 1 2013 2015 2017 2018 1/16V 2014 2016 2019 2020 2021Wide column stores: 13 / Content stores: 2 TimeSeriesDBMS:36 Documentstores:52 Spatial DBMS: 5 EventStores:3 Search engines:21 GraphDBMS:32 Key-value stores:63 Multivalue DBMS: 11 NativeXMLDBMS:7 RelationalDBMS:147 ObjectorientedDBMS:22 RDF stores:19Wide column stores 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Document stores10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='1% Time SeriesDBMS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='9% Graph DBMS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='7% Spatial DBMS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='5% Key-value stores 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='4% Search engines 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='6% Multivalue DBMS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2% NativeXMLDBMS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='3% ObiectorientedDBMS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='2% RDF stores 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='4% RelationalDBMS72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='7%three systems per category are chosen and the average of their ranking scores is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' In order to allow comparisons, the initial value is normalized to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 1) Complete Popularity Trend: The chart in 13 shows that Graph DBMS has been on constant rise in popularity because of their main use in social networking websites such as Facebook, Twitter, and TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Time Series systems are another type of DBMSs which have gain popularity in recent times due to advancements in IOT technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Complete Popularity Trend [57] 2) Trend for last 24 months: This chart 14 shows change in popularity trend for last 24 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' As mentioned earlier, Time Series DBMS has seen rise in popularity in recent times due to their use in Internet of Things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Relational systems on the other hand are at the same level of popularity as in July 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' However, DBMS like Native XML and RDF stores are on decline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Trend for last 24 months [57] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK Databases are considered to be integral part of modern information systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Almost every web or mobile application uses some kind of database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Database management systems are considered to be a crucial element from both business and technological standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' This paper divided different types of database management systems into two main categories i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' relational and non-relational and several sub categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Ranking of various sub categories for the month of July, 2021 are presented in the form of popularity score calculated and managed by DB-Engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Popularity trend for each category is also presented to look at the change in popularity since 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Complete ranking and trend of top 20 systems has shown that relational models are still most popular systems comprising of 12 systems in the top 20 with top 4 all being relational systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Relational DBMS also make the most number of systems and highest percentage in popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Oracle and MySQL being two most popular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' However, recent trends have shown DBMSs like Time Series and Document Store getting more and more popular with their wide use in IOT technology and BigData, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Future work can include deep analysis of each category with more focus on last 24 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' A comparative analysis for Document Store systems versus Relational systems can present with the results on whether systems like MongoDB and Google Firebase can challenge hugely popular systems like Oracle and MySQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' It will 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/en/ranking/ key-value+store [43] “Db-engines ranking of document stores,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2021, last Accessed: July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Available: https://db-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/en/ranking/ document+store [44] “Db-engines ranking of wide column stores,” Jul.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Available: https: //db-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/en/ranking trend/object+oriented+dbms [49] “Db-engines ranking - trend of multivalue dbms popularity,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2021, last Accessed: July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Available: https://db-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/ en/ranking trend/multivalue+dbms [50] “Db-engines ranking - trend of key-value stores popularity,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2021, last Accessed: July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Available: https://db-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/ en/ranking trend/key-value+store [51] “Db-engines ranking - trend of document stores popularity,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2021, last Accessed: July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Available: https://db-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/ en/ranking trend/document+store [52] “Db-engines ranking - trend of wide column stores popularity,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2021, last Accessed: July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Available: https: //db-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/en/ranking trend/wide+column+store [53] “Db-engines ranking - trend of graph dbms popularity,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2021, last Accessed: July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Available: https://db-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/en/ ranking trend/graph+dbms [54] “Db-engines ranking - trend of search engines popularity,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2021, last Accessed: July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Available: https://db-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/ en/ranking trend/search+engine [55] “Db-engines ranking,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2021, last Accessed: July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Available: https://db-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/en/ranking [56] “Db-engines ranking - trend popularity,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2021, last Accessed: July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Available: https://db-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/en/ranking trend [57] “Dbms popularity broken down by database model,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' 2021, last Accessed: July, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content=' Available: https://db-engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} +page_content='com/en/ ranking categories' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQf7_on/content/2301.00847v1.pdf'} diff --git a/UNE1T4oBgHgl3EQfIQPc/content/tmp_files/2301.02938v1.pdf.txt b/UNE1T4oBgHgl3EQfIQPc/content/tmp_files/2301.02938v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d26d45432163774b22fff14124768acd7e02b2ad --- /dev/null +++ b/UNE1T4oBgHgl3EQfIQPc/content/tmp_files/2301.02938v1.pdf.txt @@ -0,0 +1,1186 @@ +Crafting with a Robot Assistant: Use Social Cues to Inform +Adaptive Handovers in Human-Robot Collaboration +Leimin Tian +leimin.tian@monash.edu +Monash University +Clayton, VIC, Australia +Kerry He +kerry.he@monash.edu +Monash University +Clayton, VIC, Australia +Shiyu Xu +sxuu0041@student.monash.edu +Monash University +Clayton, VIC, Australia +Akansel Cosgun +akan.cosgun@deakin.edu.au +Deakin University +Burwood, VIC, Australia +Dana Kulić +dana.kulic@monash.edu +Monash University +Clayton, VIC, Australia +ABSTRACT +We study human-robot handovers in a naturalistic collaboration +scenario, where a mobile manipulator robot assists a person during +a crafting session by providing and retrieving objects used for +wooden piece assembly (functional activities) and painting +(creative activities). We collect quantitative and qualitative data +from 20 participants in a Wizard-of-Oz study, generating the +Functional And Creative Tasks Human-Robot Collaboration +dataset (the FACT HRC dataset), available to the research +community. This work illustrates how social cues and task context +inform +the +temporal-spatial +coordination +in +human-robot +handovers, and how human-robot collaboration is shaped by and +in turn influences people’s functional and creative activities. +CCS CONCEPTS +• Human-centered computing → Collaborative and social +computing; Collaborative interaction; • Computer systems +organization → Robotic autonomy. +KEYWORDS +HRI, human-robot collaboration, handover, social robots, adaptation +ACM Reference Format: +Leimin Tian, Kerry He, Shiyu Xu, Akansel Cosgun, and Dana Kulić. 2023. +Crafting with a Robot Assistant: Use Social Cues to Inform Adaptive +Handovers in Human-Robot Collaboration. In Proceedings of the 2023 +ACM/IEEE International Conference on Human-Robot Interaction (HRI ’23), +March 13–16, 2023, Stockholm, Sweden. ACM, New York, NY, USA, 9 pages. +https://doi.org/10.1145/3568162.3576998 +1 +INTRODUCTION +Human-Robot Collaboration (HRC) is an active research field that +investigates how robots can work together with people as a +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than the +author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or +republish, to post on servers or to redistribute to lists, requires prior specific permission +and/or a fee. Request permissions from permissions@acm.org. +HRI ’23, March 13–16, 2023, Stockholm, Sweden +© 2023 Copyright held by the owner/author(s). Publication rights licensed to ACM. +ACM ISBN 978-1-4503-9964-7/23/03...$15.00 +https://doi.org/10.1145/3568162.3576998 +Figure 1: Experimental space layout: the participant and +the experimenter sat at the work space and storage space, +respectively. The Fetch robot moves in between to pass +objects in a basket. The operator sat in a neighbouring room. +synergistic team to achieve beneficial outcomes, such as in +education [25], healthcare [23], or manufacturing [15]. A common +HRC task is human-robot handover, which is the physical +exchange of an object between a person and a robot at a mutually +agreed location and time [20]. A successful handover requires +spatial and temporal coordination between the person and the +robot. Further, how the physical exchange unfolds is influenced by +environmental, personal, and task constraints [22]. Current efforts +in developing better handovers in HRC have focused on creating +models that can identify human intentions and adapt the robot’s +motion planning and execution accordingly. However, as identified +by Ortenzi et al. [20], existing studies investigated handovers in +isolation without incorporating them in a naturalistic context with +pre- and/or post-handover HRC tasks. Further, handover +evaluation has focused on the performance and technical aspects, +e.g., handover time and success rate, rather than the social +interaction. This limits the understanding of human perceptions +and behaviours during handovers, as well as the development of +an effective and socially appropriate HRC. +Our goal is to understand how people perceive a robot and +exchange objects with it during task-focused HRC. To build +towards +autonomous +robot +capabilities, +we +conduct +a +Wizard-of-Oz (WoZ) study to investigate how human operators +achieve adaptive, efficient, and socially appropriate HRC. We +design a crafting task in which a person assembles and paints a +wooden birdhouse with the assistance of a mobile manipulator +arXiv:2301.02938v1 [cs.RO] 7 Jan 2023 + +Robot moving space +Operator performing +Experimenter at +Participant at the +Wizard-of-Oz controls +OAK-D camera +the Storage Space +Work SpaceHRI ’23, March 13–16, 2023, Stockholm, Sweden +Tian et al. +robot, as shown in Figure 1. The person and the robot exchange a +set of task-relevant objects with diverse usage purposes (see +Figure 2). We collect quantitative and qualitative data to +understand human experience and behaviour during HRC, as well +as effective and adaptive collaboration strategies. Recordings of +the HRC sessions and the teleoperation framework we developed +are made publicly available. +2 +BACKGROUND +HRC relies on an effective and appropriate social communication +between humans and robots [26, 28]. Expression and perception of +social cues is key to successful HRC, for instance, using hesitation +gestures to negotiate access to shared resources [16]. Similarly, +previous research found that people use body stances and arm +gestures to communicate their intent to when and where a +handover happens [4], both as the giver [21] and as the +receiver [8]. The quality of human-robot handover can be assessed +on various dimensions using both subjective and objective metrics. +For example, Murphy and Schreckenghost [18] evaluated +handover quality as success rate, human/robot idle time, task +completion time, and cost functions relating to movement +trajectories, +while +Hoffman +[10] +measured +human-robot +cooperation and contribution using subjective questionnaires. +Previous studies typically investigate handovers in isolation +where objects are exchanged without a pre- and/or post-handover +task context, for example, to develop better models for predicting +human motions [5, 29], to predict the appropriate location of +object transfers [14, 19], to create more natural and fluent robot +motions [3, 6], or to develop control models adaptive to contextual +factors, such as human activities, preferences, and object +semantics [7, 12, 13]. In these studies, it is unclear how a person’s +perception and behaviours during handovers may change when +they are integrated as part of an HRC task. +An example study that investigated handovers as embedded in a +collaborative task context is Huang et al. [11]. The authors studied +timing coordination in human-human handovers when they +collaborated at a task of unloading a dish rack. Based on +observations of 8 pairs of participants, Huang et al. [11] developed +and evaluated a rule-based model for a robotic arm mounted on a +fixed table, which adapted its movement speed and pause duration +in between robot-to-human handovers to replicate human +decisions during the dish unloading task. This work provided +valuable insights on temporal coordination in handovers. However, +further research is required to understand human-robot handovers +as a social interaction for both temporal and spatial coordination, +as well as for diverse types and goals of the object transfer. +3 +METHODOLOGY +Here we provide an overview to our study design, including human +research ethics, experimental protocols, custom robot teleoperation +framework, and collection of measurements. +3.1 +Human Ethics and Data Collection +Our study protocols were reviewed and approved by the Monash +University human research ethics committee (No.31927). We +recruited 20 participants from the university’s staff and student +population (10 females and 10 males, age 27.00 ± 5.17): fifteen +were from the faculty of engineering, four from the faculty of art +(P7, P8, P10, P15), and one (P13) from the faculty of business and +economics. All participants provided informed consent for taking +part in the study and having their data recorded as part of a public +dataset for research purposes. As a WoZ study, the participants +were not told explicitly before the experiment that the robot was +teleoperated. Instead, the experimenter explained the purpose and +details of the WoZ approach in the debriefing afterwards. +3.2 +Experiment Protocol +In the collaborative task, a person assembles and paints a +birdhouse/feeder +with +the +assistance +of +a +Fetch +mobile +manipulator. +The +participants +kept +their +creation +as +an +appreciation for their time, which is expected to increase their +engagement in the task. A session involves three human roles, as +shown in Figure 1: a Participant sitting at a work space at one side +of the room with a bin and a desk too small to fit all the required +objects at once; an Experimenter sitting at a storage space at the +other side of the room with all objects required during the session +stored in a box; an Operator sitting behind the participant hidden +from their view who controls the robot. The robot moves in +between the work space and storage space and performs object +handovers on both sides. The detailed state machine during each +handover episode is shown in Figure 3. The robot (and operator) +observes the participant using its onboard head-mounted camera, +as well as an RGB-D camera (OAK-D) installed next to the work +desk to the front right of the participant. An instruction sheet +listing all the objects used in a session is displayed on the top right +corner of the work desk for the participants to refer to at any time. +Note that because we are interested in identifying how a robot can +best provide assistance in terms of handover timing and object +transfer point (OTP), we simplified the handover problem: The +robot holds a basket in which objects are transferred, rather than +using its gripper to grasp the various objects directly. +The robot-assisted crafting task consists of three stages: Stage 1 +(Preparation), Stage 2 (Assembly), and Stage 3 (Painting). The +robot and the participant engaged in handovers during each stage, +in which the participant can either be the receiver when the robot +brings them an object from the storage space, or be the giver when +they need the robot to return an object to the storage space. +Figure 2 provides an overview of the experiment session, which +has a total duration of approximately one hour. Stage 1 is aimed at +preparing for the collaborative task and an implicit mutual +learning between the participant and the robot (operator). Stage 2 +is aimed at understanding handovers and HRC in a functional task +context. Stage 3 is aimed at understanding handovers and HRC in +a creative task context. The functional and creative activities have +different pre- and post-handover task context, which facilitates our +goal of investigating contextualised handovers. In functional +activities, participants are expected to focus on assembling the +birdhouse quickly and ensuring its structure is sound. In creative +activities, participants are expected to have different mental (i.e., +design birdhouse appearance to their liking) and behavioural (i.e., +realising this personal design by painting) context, which + +Crafting with a Robot Assistant +HRI ’23, March 13–16, 2023, Stockholm, Sweden +Introduction to the robot and +the experiment +Pre-study questionnaire +Stage 3 evaluative questionnaire and +overall experience questions +Debriefing and post-study interview +Stage 3: Painting (Creative tasks) +Apron +Gloves +Tissues +Stage 1: Preparation +Stage 2: Assembly (Functional tasks) +Glue +Wooden pieces (came separately) +Stage 2 evaluative questionnaire +Gloves +Tissues +Palette +Bundle of paint brushes +Paints (came separately) +Thread +Scissors +Water +Return bag +Stage 1 evaluative questionnaire +Figure 2: Session overview: the one-hour session includes +three stages with various objects used during each stage. +Participants filled in an online questionnaire during the +session. A debriefing and interview is conducted at the end. +influences how and when the handovers should happen, and +participants’ perception and experience during the tasks. +An operator controlled the robot via a teleoperation interface +detailed in Section 3.3. The operator sat in a neighbouring room +(see Figure 1) and took their position after the participant and +the experimenter were seated. As a WoZ study, the participants +were not informed of the operator’s presence before or during the +experiment. This allows us to understand what social cues people +express during HRC and how these social cues can be used to +achieve adaptive handovers applicable to autonomous robots. +3.3 +Teleoperation Framework +We developed a custom teleoperation framework using the Robot +Operating System (ROS).1 One expert Fetch operator was trained +on using the teleoperation framework, and controlled the robot +during all experimental sessions. The operator’s role is to identify +the appropriate timing for initiating and completing each handover +episode, to identify the appropriate OTP, and to evaluate the quality +of a handover episode. Further, after each session, the operator +participated in the interviews to describe their strategies and to +incorporate the participants’ feedback. +The operator controlled the robot using a hand-held controller +and a computer interface. As shown in Figure 4, the operator has +access to a live feed of the OAK-D camera and the robot’s onboard +camera, as well as estimates of the participants’ upper body and +facial keypoints from the OAK-D camera feed using Blazepose [2], +and categorical emotion estimates from both camera feeds using +EmoNet [27]. The participant’s pose, the robot’s pose, and the +robot’s end effector’s (EE) pose are shown in a 3D map view, which +plots the path that the robot follows to travel between the work +space and the storage space. The operator can drive the robot along +a predefined path and rotate the robot’s base. They can also make +fine adjustments to the positions of the path’s endpoints during or +1The teleoperation framework: https://github.com/tianleimin/fetch-teleop +in between episodes. To perform handovers, the operator can extend +the EE towards an OTP (arm stretching), or retract it to a default +tucked position (arm tucking). The handover motion sequence is +based on He et al. [9]. We specified three default OTPs: to the left of +the robot (right-hand-side of a participant), to the right of the robot +(left-hand-side of a participant), and in the centre. The position +and orientation of these OTPs can be fine-tuned by the operator. +Changes in the path endpoints and OTPs can be saved or reset. +After each episode, the operator uses the controller to evaluate the +quality and specify the nature of the participant-robot handover +(human-to-robot, robot-to-human, or bidirectional). During arm +stretching, the robot automatically tilts its head downwards to point +at the EE and display joint attention [17, 30]. During arm tucking, +the robot restores its head to neutral position. +3.4 +Measurements +Data from the teleoperation framework is recorded continuously +using ROS bags, which includes synchronised operator control +signals, video recordings, robot status, estimates of facial and +upper body keypoints from the OAK-D camera feed, and the +emotion estimates from both camera feeds. Further, we collected +participants’ responses to a questionnaire. This includes pre-study +questions on participants’ demographic backgrounds and robot +self-efficacy [24]. After each of the three stages, the participants +evaluated their experience during that stage with 5-point Likert +scale measurements of fluency, trust, and working alliance +between the themselves and the robot [20], their perception of the +robot as the GodSpeed questionnaire [1], their overall enjoyment +and satisfaction, and a text comment box. After all stages are +completed and rated, before the debriefing and interview, +participants also chose whether they thought the robot was +teleoperated or fully autonomous, whether they thought the robot +was adaptive, and gave any additional comments in a text box. +In addition to the quantitative data captured by the +questionnaire and the teleoperation framework recording, we +collected qualitative observation notes and post-session interviews. +The experimenter took observation notes during the session on +participants’ behaviours, operator’s controls and adaptations, and +events of interest. In interviews participants were asked to +describe their impression of the robot and the HRC experience, +their guesses as to how the robot functioned, their reasoning +behind certain events of interest during the session, their design +and creative processes, and future improvements that the robot +can incorporate to become a better assistant. The operator also +joined the interview to describe their observational and adaptive +strategies, and to answer any questions from participants. +4 +THE FACT HRC DATASET +We collected a dataset of WoZ controlled human-robot handovers +and collaboration in Functional And Creative Tasks (FACT). The +FACT HRC dataset includes three segments: +(1) FACT-raw2 (requires a signed EULA to access) contains +identifiable ROS bag recordings described in Section 3.4. +2FACT-raw: https://doi.org/10.26180/21671789.v1 + +eenex7 +7 +Bottom +Left/Right +Front/BackWall +Panel +Roof +6 +Front/BackWall +Left/Right +Left/Right +Roof +WallOTANIA +OVES +L +100GUAHRI ’23, March 13–16, 2023, Stockholm, Sweden +Tian et al. +Stationary +(facing participant) +Storage Space +Move to work space +Stationary +(facing participant) +Moving Space +Work Space +Stretch arm +Object Transfer +(with participant) +Tuck arm +Rotate (towards +experimenter) +Stationary +(facing experimenter) +Move to storage +space +Stationary +(facing experimenter) +Stretch arm +Object Transfer +(with experimenter) +Tuck arm +Rotate (towards +participant) +Episode Start +Episode End +Object transfer with participant +Object transfer with experimenter +Timing adaptation +Object transfer point (OTP) adaptation +Timing adaptation +Figure 3: An episode of handover: the operator performs temporal adaptation when waiting to move the robot to the work +space and when waiting to initiate object transfers with participant. Spatial adaptation also happens during object transfers. +Figure 4: Visualisation of the teleoperation framework: the +operator can see live feeds of the robot’s onboard camera, +the additional RGB-D camera, a 3D map view with the +robot’s pose as well as estimated participant’s poses and +emotions. +(2) FACT-processed3 contains non-identifiable csv data extracted +from the synchronised raw data at intervals of 0.1s (in total +approximately 600K data instances), including the robot’s +status, the operator’s controls, the facial and upper body +keypoints estimated from the OAK-D camera feed, and the +emotion estimates from both camera feeds. In addition, the +de-identified questionnaire responses are provided. +(3) FACT-support contains supporting materials, including the +instruction sheet, operator’s cheat-sheet for teleoperation +framework controls, the questionnaire, CAD file for laser +cutting the birdhouse pieces, implementation of the +teleoperation framework and data processing scripts. +The FACT HRC dataset includes 565 handover episodes, namely: +• 52 Human-to-robot handovers (9.2%): the person giving an +object to the robot by placing it in the empty basket that the +robot holds in its gripper. +3FACT-processed and FACT-support: https://doi.org/10.26180/21671768.v1 +• 320 Robot-to-human handovers (56.6%): the person receiving +an object from the robot by taking it from the basket. +• 193 Bidirectional handovers (34.2%): object exchange +between the person and the robot, either simultaneously or +in two consecutive give and receive motions, during which +the robot’s arm can remain stretched out or be tucked to +wait between the give and receive motions. +5 +OPERATOR’S ADAPTATION STRATEGIES +The operator adapted handovers based on participants’ intent and +preference inferred from gaze and gestures, safety considerations +of the handover, and its efficient integration with the task context. +5.1 +Observation and Expression of Social Cues +The operator reported monitoring the person’s gaze (e.g., eye +contact with the robot), hand gestures (e.g., waving), and upper +body movements (e.g., leaning forward), which aligns with +existing research [4, 20]. The operator did not rely on the +estimated emotion print-outs, as the automatic model is less +accurate than a human’s performance. However, emotion +estimations may benefit an autonomous model as they can +distinguish handover episodes rated as good vs. bad by the +operator, with more positive or neutral emotions predicted in the +episodes rated as good. In addition, the operator monitored pre- +and post-handover context, including object layout on the work +desk, the space where the person is performing the crafting tasks +in, and whether or not one or both of their hands are occupied. +The operator used the position and movement of the robot to +communicate its intention to the human collaborator. When the +robot is waiting to deliver or receive an object to or from a +participant, if the waiting time is estimated to be long, the operator +controlled the robot to wait at the storage space to avoid +pressuring the person to rush through their tasks, as suggested by +some participants in the interviews (Section 6.3); if the wait is +estimated to be short, the operator controlled the robot to wait at +the work space so that it could perform the handover as soon as +the person is ready. Five participants commented on the robot’s +head tilting behaviour during object transfers as an indication of + +[] Select +$ Focus Camera + Measure + 2D Pose Estimate + 2D Nav Goal + Publish Point ++ + Displays +* +Views +* +、游 Globaloptions +Type: ThirdPersonFoll - + V Global status: ok +Zero +PID +Human Pose +> current View +ThirdPersonFol... +、Map +Near Clip +0.01 +O TF +Estimations +Invert Z Axis + Marker +Target Fra... + + LaserScan +End Effector Pose +Distance +3.04145 +Focal shap... 0.05 +iyRobotModel +Emotion +Focal shap... + Path +Yaw +5.73227 +Estimations +Pitch +0.129796 +Add +Duplicate +Remove +Rename +Focal Point +-2.9296; 14.474; ... + Image +Fetch: Neutral +Robot's +OAK-D: Neutral +Camera +hanelevtoa + Image +OAK-D +Camera +Map and Path +Save +Remove +Rename + Time +* +PauseSynchronization: off +→Source:LaserScan +ROs Time:1654147954.58 +V Experimental +Reset +11 fpsCrafting with a Robot Assistant +HRI ’23, March 13–16, 2023, Stockholm, Sweden +the robot’s awareness to itself and its collaborator. The operator’s +observation and control strategies suggest that perception and +expression of social cues is key to facilitates HRC. +5.2 +Adaptation in Handover Timing +The operator focused on choosing the appropriate moment to +move the robot to the work space for initiating the handover +episode (i.e., deciding the wait time at the storage space) and to +stretch out the robot’s arm for initiating the object transfer within +a handover episode (i.e., deciding wait time at the work space), as +shown in Figure 3. A handover was initiated when the participant +displayed a personal “start” gesture (Section 6.1). In addition to +these participant-initiated handovers, the operator performed +robot-initiated handovers in Stage 2 and 3 by moving the robot to +the work space before the participant signalled, when they +anticipated that it will take the person shorter to finish their task +at hand than the time (9s) the robot needs to move from the +storage space to the work space. Robot-initiated handovers were +also performed when the item to be delivered is of immediate need +to the person while they were continuing their task at hand . +The average duration of a handover episode in Stage 1 is +113.6±42.1s, in Stage 2 is 97.1±22.8s, in Stage 3 is 108.1±22.0s, i.e., +the episodes are the shortest in the functional activities of +assembly (S1 vs. S2 𝑝 = 0.03, S2 vs. S3 𝑝 = 0.02, S1 vs. S3, +𝑝 = 0.26). The larger variance in episode duration in Stage 1 is in +line with our discussion in Section 3.2 on it being intended for +implicit mutual learning between the participant and the operator. +In Stage 1 the average pause at the work space (robot waiting at +the participant’s side for the right time to initiate object transfer) +within the handover episodes is 7.5±3.8s, and the average pause at +the storage space (waiting for the right time to move towards the +participant) is 47.9±24.4s; In Stage 2, the average pause at the work +space is 19.0±15.6s, and at the storage space is 26.3±14.7s; In Stage +3, the average pause at the work space is 16.2±13.6s, and at the +storage space is 40.1±27.9s, i.e., the operator controlled the robot +to wait longer at the storage space than at the work space +(𝑝 ≪ 0.01 in S1, 𝑝 = 0.08 in S2, 𝑝 = 0.004 in S3) to avoid +pressuring the participants, in line with our discussion in +Section 5.1. +5.3 +Adaptation in OTP +The operator considered efficiency and safety when deciding on +the OTP. For efficiency, they aimed at stopping the EE +approximately where the person’s dominant or free hand is +positioned, or where they signalled the “start” gesture. For safety, +they factored in object layout on the work desk to avoid collision, +especially with the birdhouse that the person is crafting. When the +two considerations conflicted the operator prioritised safety. OTP +adaptations can happen live in response to a participant’s +behaviours or between object transfers based on past observations. +While the operator can move the robot’s base closer or further +away from the person when it is at the work space, they chose not +to use this function. Instead, OTP was adapted by changing the EE +goal. As described in Section 3.3, there are three default EE goals +(left, centre, right). The operator can make changes to these default +OTPs. However, such fine adjustments were not performed as one +of the three defaults was considered close enough to the operator’s +envisioned OTP. Further, the operator stopped the arm stretching +motion prematurely before the default EE goal was reached (i.e., the +OTP is further away from the person than the default) if the person +reached out their hand(s) before the arm stretching completes. +It takes 12s for the robot to complete an object transfer with the +arm ready for one of the three default OTPs or when it is switching +between central and left/right OTPs. However, switching between +the left and right OTPs takes 30s with bigger motions. To avoid +interrupting the collaboration flow, the operator did not perform +such left-right switches during robot-participant handovers. +Instead, this was done during robot-experimenter handovers or +while the robot was waiting at the storage space. In comparison, +switching between central and left/right OTPs was performed +more frequently in robot-participant handovers to quickly adapt to +the person’s preferences. Out of all 61 OTP switches, 49 were +left-middle switches (80.3%), 7 were right-middle switches (11.5%), +and 5 were left-right switches (8.2%). In the 565 handover episodes, +186 (32.9%) were left handover (right-hand-side of the participant), +367 (65.0%) were central handover, 12 (2.1%) were right handover. +The preference of left OTP over right OTP aligns with our +participant population with only P1 being left-handed. +5.4 +Operator’s vs. Participant’s Perceptions +Out of all handover episodes, the operator rated 397 (70.3%) as +good, 32 (5.7%) as bad, and 136 (24.1%) as neutral. The operator +incorporated the participants’ feedback provided in the debriefing +and interview after each session in their observational and adaptive +strategies in later sessions. Thus, we saw a continual development +of the operator’s expertise and control strategies: With P1 and +P2, the operator rated the sessions as having the most number of +bad handover episodes (5 episodes each). Further, there was one +collision incident with P1 where the robot hit the work desk in +the first episode due to a setup issue, and one collision incident +with P2 in Stage 3 where the robot pushed the assembled birdhouse +when stretching its arm to perform central handover to deliver the +scissors to P2. P1 and P2 addressed the collision incident during +interview, but did not consider the other bad episodes to stand out. +With P5, P7, and P12, the operator rated the sessions as having +3 bad episodes each, while P5 considered there were “one or two +miscommunications”, P7 did not notice the robot’s mistakes or its +changes in OTPs, and P12 commented on “one weird incident” with +the robot’s arm movement appearing unnatural. For remaining +sessions with one or two bad episodes as rated by the operator, +participants did not address these episodes in the interviews. +To further understand the differences in operator’s and +participants’ perceptions, for each stage, we calculated an +operator’s handover quality score 𝑄𝑜 = 0.5 × ( 𝑁𝑔𝑜𝑜𝑑−𝑁𝑏𝑎𝑑 +𝑁𝑡𝑜𝑡𝑎𝑙 ++ 1), +and a participant’s handover quality score as the min-max +normalised sum of their ratings on the enjoyable 𝐸 and +satisfactory 𝑆 level 𝑄𝑝 = 𝐸+𝑆−2 +10−2 (𝑄𝑜,𝑄𝑝 ∈ [0, 1]). Figure 5 shows +the stage-wise mean (𝑄) and standard deviation (𝜎𝑜, 𝜎𝑝) of 𝑄𝑜 and +𝑄𝑝 in each session. Note that the participant’s ratings are +independent from one another, while the operator’s ratings of a +session can be dependant on their experiences in previous sessions. +As shown here, the operator and participants’ perception of + +HRI ’23, March 13–16, 2023, Stockholm, Sweden +Tian et al. +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +Participant ID +Subjective scores +Role: +Operator +Participant +Figure 5: Operator’s vs. participant’s perceptions: the +operator had a handover-centred perspective while the +participants were influenced by the task context. +handover quality do not always align. In 9 sessions where 𝜎𝑜 > 0.1 +(P4, P7-P9, P11-P14, P19), 𝜎𝑝 = 0.07, 𝜎𝑜 = 0.20. This suggests that +compared to the operator who had a handover-centred perspective, +the participants’ perception may be less influenced by the +individual handovers compared to the task context. This is also +found in the interviews as discussed in Sections 6.2 and 6.3. +Comparing 𝑄𝑜 and 𝑄𝑝 in earlier (P1-P10) vs. later (P11-P20) +sessions, the average scores in each of the three stages are +consistently higher in the later half, although the difference is only +statistically significant in Stage 2 of 𝑄𝑜 (𝑝 = 0.03). The increase in +the quality scores is bigger for 𝑄𝑜 than for 𝑄𝑝. This indicates that +the operator gained expertise in handover adaptation, which +improved the participants’ experience in later sessions. +6 +USER’S PERCEPTION AND BEHAVIOURS +We investigate how people collaborate with and perceive the robot +from subjective questionnaires, observation notes, and interview +data. Our analysis showed that participants have both positive and +negative impressions towards the robot and the HRC experience. +Their perceptions are more positive in Stage 2 and 3 compared +to Stage 1, as the operator adapted to their preferences and the +participants became more focused on the functional and creative +activities beyond object handovers. Further, their social cues change +within a session and differ between participants. Moreover, the HRC +directly influenced people’s design and creative processes, even +though only human-robot object handovers were performed. +6.1 +User Behaviours During the HRC +Interestingly, all participants signalled to the robot’s “face” +(onboard camera), even though they were aware that the OAK-D +camera was also used for monitoring their behaviours and was +positioned closer to them. P6, who has worked with robots before, +was the only participant who attempted signalling to both cameras +in the first handover episode and switched to only signalling to the +robot’s onboard camera in all later episodes. P6 commented that he +signalled to the OAK-D camera at the start because it had “better +angle”, but later it felt more natural to signal to the robot’s +onboard camera. Further, P15 commented that at times he felt it +was “weird” or “too formal” to use big gestures to signal the robot +to come, as he considered the robot to be an equal partner working +on a task together with him in a casual way, while waving your +hand at someone’s face can be considered impolite. These +highlight the social interaction aspect of human-robot handovers. +Although they were told that the robot has no speech recognition +or interaction capabilities, five participants still spoke to the robot, +such as saying “thank you” after handovers. This suggests that +speech is a natural interactive modality that can benefit HRC. +When signalling to the robot, participants used social cues +combining one or more modalities, including gaze, movements of +hand(s), and body gestures. The most common social cue to signal +the initiation of handovers, i.e., the “start” gesture, is to gaze up at +the robot while leaning forward and showing a single-handed +wave, either side-to-side or forward-backward, as used by 17 +participants. Several variations of this “start” gesture were +observed, such as using only gaze without hand or body +movements, holding the object to be given to the robot when +waving, or using semantic gestures to demonstrate the object +needed (e.g., showing gestural imitation of putting on gloves to +initiate glove box handover). Twelve participants used more than +one “start” gestures during the session, with 11 switching to +lower-effort “start” gestures in later episodes (except for P13), for +instance, from using both gaze and hand movements to gaze-only. +During the interview, P1, P3, P10, and P13 discussed that they +experimented with different gestures and observed the robot’s +reactions to decide on which gestures were the most effective. P10 +commented that she used bigger gestures at the start of the session +because she was more excited. P14 expressed that she changed to +smaller gestures later in the session as she became more familiar +with how the robot functions and felt that she needed less effort to +communicate with it. After the robot arrived at the work space, +five participants displayed a follow-up “start” gesture to signal the +robot to stretch out its arm and initiate object transfer by reaching +out one hand. After completing object transfers, six participants +displayed an “end” gesture to signal the robot to tuck its arm and +return to the storage space, by showing a single-handed gesture of +“OK” while nodding or an outward “go-away” single-handed wave. +During the sessions, participants also adapted to the robot’s +behaviours. As the robot’s movements are relatively slow, to +increase efficiency, participants performed bidirectional object +exchanges instead of single-directional object handovers when +possible. They also filled the waiting time with other activities, +namely reading the instruction sheet, inspecting their assembly or +painting +progresses, +and +checking +their +phones. +Further, +participants configured the layout of their work desk dynamically +to keep the area that they preferred for object transfers accessible. +Our analysis showed that participants treated the robot as a +social actor during the collaboration and used various social cues, +namely gaze, hand movements, and body gestures, to coordinate +the timings and locations for object handovers. They also adapted +to the robot to achieve more efficient collaboration by changing +their behaviours, such as filling in the waiting time with other tasks +and performing bidirectional handovers when possible. +6.2 +Perception of the Robot +Before the HRC session, participants rated their expectations on +robot operation and application efficacy. No significant differences +were found between participants from the faculty of engineering +and participants from the faculties of art and business. + +Crafting with a Robot Assistant +HRI ’23, March 13–16, 2023, Stockholm, Sweden +* p < 0.05 +** p < 0.01 +* +* +* +* +** +** +** +** +Figure 6: Subjective ratings of robot impression: participants +had more positive impressions of the robot in Stage 2 & 3. +Before debriefing, participants guessed whether the robot was +autonomous or teleoperated, and whether they thought it was +adaptive. Fifteen participants answered the robot was adaptive +during the session, three answered it was not adaptive, and two +were unsure. Nine participants thought the robot was fully +autonomous, nine thought that it was being teleoperated, and two +were unsure. None of the participants identified where the +operator was located until they were introduced during the +debriefing. The main reason participants gave for considering the +robot to be teleoperated or to be unsure about their guesses is the +robot’s ability to adapt to their behaviours and the task context, +which they consider to be too difficult for a fully autonomous +robot. This demonstrates that improving a robot’s adaptive +capabilities is key to achieving natural HRC. +In Figure 6 and Table 1, we report participants’ subjective +impressions of the robot in each stage (sub-item means, value +range [1,5]). As shown here, participants have significantly more +positive impressions of the robot in Stages 2 and 3 compared to +Stage 1 (except for Intelligence). One possible cause of the higher +ratings in Stages 2 and 3 is that the operator learned an effective +adaptation strategy during Stage 1 and performed more fluent +handovers in Stages 2 and 3. However, as discussed in Section 5.4, +another reason may be the participants being more focused on the +functional and creative activities in Stages 2 and 3 rather than +object transfers, i.e., the participant’s perceptions were influenced +more by the collaborative task than by the individual handovers. +In the interviews, participants expressed both positive and +negative impressions towards the robot. Eight participants +commented that the robot was “cool”, “impressive”, “amazing”, +“likeable”, or they were “fascinated”, “excited”, or “satisfied” +interacting with it; Four considered the robot “responsive” or +“adaptive” in its interaction; Three thought the robot was “useful”, +“effective”, or “helpful” for the tasks; Two considered the robot +“reliable” or “trustworthy” in executing actions and performing its +tasks. However, eight participants also mentioned slowness and +experiences of waiting for the robot. Although this sense of +slowness is not necessarily a negative experience for every +participant. As discussed by P15, the robot being slow made it feel +predictable and non-threatening, which led to him considering the +robot “docile” and “cute”. Further, P2 and P18 discussed a sense of +uncertainty or confusion in how to interact with or trigger certain +behaviours in the robot despite having prior exposure to the robot. +Table 1: Participants’ impressions of the robot and the HRC +(mean ± std). Stage 2 & 3 are perceived more positively +overall. +Robot impression +Stage 1 +Stage 2 +Stage 3 +Anthropomorphism +2.78 ± 0.78 +3.06 ± 0.95 +3.12 ± 0.96 +Animacy +2.98 ± 0.70 +3.16 ± 0.71 +3.28 ± 0.81 +Likeability +3.76 ± 0.66 +4.10 ± 0.74 +4.13 ± 0.70 +Intelligence +3.65 ± 0.71 +3.73 ± 0.66 +3.73 ± 0.83 +Perceived safety +3.77 ± 0.68 +3.99 ± 0.62 +4.04 ± 0.70 +HRC impression +Stage 1 +Stage 2 +Stage 3 +Fluency +3.62 ± 0.74 +3.95 ± 0.69 +4.12 ± 0.75 +Trust +3.93 ± 0.91 +4.18 ± 0.78 +4.20 ± 0.85 +Working Alliance +3.97 ± 0.91 +4.09 ± 0.76 +4.14 ± 0.86 +Enjoyable +3.70 ± 0.86 +4.35 ± 0.59 +4.30 ± 0.66 +Satisfactory +3.65 ± 0.75 +4.10 ± 0.85 +4.10 ± 0.85 +Participants also discussed future improvements they would +want the robot to have in order for it to be a better assistant. Eight +participants commented that speech interaction or voice commands +in addition to the current gesture-based approach can help increase +the interactivity and flexibility of the collaboration. Similarly, three +participants suggested that developing a dictionary that associates +gestural commands to specific objects can help minimise confusion +and increase flexibility. Related to this desire for more flexibility, six +participants discussed that it would be good to allow customising +the order and set of objects that the robot delivers. Regarding the +robot’s behaviours, four participants expressed wanting the robot to +be anticipatory instead of being reactive in more handover episodes, +two participants expressed preference for more consistency in its +reaction time and/or behaviours, three participants suggested more +involvement of the robot beyond object exchange, for instance, +working on the assembly directly with the person by holding a +piece while the person is gluing it to the rest of the birdhouse. +As shown in the questionnaires and interview responses, while +having diverse impressions of the robot, overall participants +considered it to be an adaptive and helpful collaborator, especially +in the assembly and painting stages. Improvement that can benefit +the robot’s perceived fluency and contribution to the task includes +increased participation beyond object handovers and multimodal +interaction combining speech and gestures. +6.3 +Perception of the Collaboration +Figure 7 and Table 1 showed participants’ perception of the HRC. +Similar to perception towards the robot, participants have +significantly more positive impressions of the HRC in Stage 2 and +3 compared to Stage 1 in terms of Enjoyable, Satisfactory, and +Fluency. In the interview, four participants expressed that they +found Stage 2 to be the most collaborative out of the three stages +where them and the robot worked as an efficient team, while three +participants considered Stage 3 to be the most collaborative. +Participants discussed both positive and negative perceptions +of the HRC in interviews. Four participants commented that the +experience was “fun”, for instance, P17 said the wooden pieces in +Stage 2 “arrived like presents”; six expressed that the interaction + +HRI ’23, March 13–16, 2023, Stockholm, Sweden +Tian et al. +* p < 0.05 +** p < 0.01 +* +* +** +** +** +** +Figure 7: Subjective ratings of HRC: participants perceived +the collaboration more positively in Stage 2 & 3. +was “smooth”, “efficient”, or “effective”; five considered the session +was “adaptive” with the robot improving over time to better address +their needs. However, five participants also mentioned that when +the robot was waiting for them, especially at the work space, it +created a sense of pressure or a desire to speed up their actions. +Participants can also comment in the free text area of the +questionnaire after each stage. For Stage 1, four participants +commented about their uncertainty in how to signal to the robot to +come and to perform object transfer, and their experience gaining +better understanding of how the robot will behave over time. For +Stage 2, six participants gave comments that highlighted they had +a better collaboration experience compared to Stage 1 as the robot +felt more involved, efficient, and adaptive. For Stage 3, six +participants gave comments discussing their positive impression of +feeling relaxed, as well as negative impressions, namely part of the +interaction being slow, wanting the objects to be delivered in a +different sequence, and the robot handing objects at an undesirable +location in some episodes. At the end of the questionnaire, nine +participants gave additional comments, expressing both positive +impressions of the robot being human-like and adaptive, and +negative impressions of the interaction feeling slow or being +unsure about the robot’s intention at times. +In the interview, twelve participants noticed the robot changing +the OTPs during their sessions, five participants did not notice any +location changes, while all handover episodes were at the central +location in the sessions with P13, P14, and P19. In addition, three +participants noticed the robot changing its arm stretching distance +to be closer or further away from the person. Nine participants +addressed that the robot was proactive during some episodes, while +eleven participants had the impression that the robot was entirely +reactive and only initiated handovers when they signalled to it. Two +participants also discussed that they were adapting to the robot +themselves and became more in tune or familiar with the robot’s +behaviours as the session progressed. +Our analysis shows that overall participants considered the +collaborative experience to be enjoyable, efficient, and adaptive, +especially in the assembly and painting stages, despite not noticing +all the handover timing and OTP adaptations. +6.4 +Functional and Creative Activities +In the interview, participants discussed how the collaboration with +the robot influenced their functional and creative activities during +the crafting session. Regarding the functional tasks of assembling +the birdhouses, P1 commented that the robot helped avoid +cluttered work space. Similarly, P13, P15, and P17 expressed that +the robot bringing the wooden pieces one by one guided them to +work through an unfamiliar task in a structured manner. Further, +P8 referred to her past experience attending furniture design +classes, in which people spent long time waiting to use shared +equipment, and commented that the robot could be helpful in this +scenario by bringing objects to the shared equipment for +processing while the person continues with other tasks. +Regarding the creative activities of designing the appearance +of the birdhouse, all participants conceptualised the design during +the experiment. Since only the five prime colours were offered, six +participants used their phones during the painting stage to search +for colour mixing methods (e.g., P2 searched how to create brown) +or pictures of specific items that they intended to include in the +design (e.g., P9 searched photos of the superb fairywren). Nine +participants stated that the order in which the robot delivered the +paints directly inspired or influenced their design, while eleven +participants commented that as from the instruction sheet they +already knew which colours would be provided, their design was +not influenced by the order in which the colours arrived. +Moreover, six participants expressed that they wished the +session could have been longer as they were deeply invested in the +crafting. Three participants commented that as the human-robot +handovers were integrated as part of the crafting session instead of +being at the centre of attention, they were able to focus on the +creative design process. Specifically, P15 discussed that as he was +able to focus on the creative aspect, he recalled the birdhouse’s +purpose of feeding small birds and changed his design to a more +camouflaged look by avoiding bright colours that may attract +predators. P10 referred to her past experience interacting with +small swarm painting robots, during which the robots made many +mistakes and she was concerned that they would break something +or hurt someone. However, in this session, she found the robot +“smooth” and “helpful”, which allowed her to concentrate on +creating and experimenting with different colour mixes. Further, +P8 commented from her experiences tutoring children’s art classes, +where tidying up the supplies at the end of a class is often time +consuming. She expressed that the robot helped structure the +painting stage to be more organised by delivering and retrieving +items in order, which can potentially reduce the workload of the +tutors in the children’s art class scenario. +As shown in the interviews, participants’ functional and creative +activities were influenced by the robot’s behaviours. The functional +collaboration of object handovers not only benefited the person’s +task efficiency, but also led to design and creative benefits. +7 +CONCLUSIONS +We studied human-robot handovers in a naturalistic HRC context, +where a WoZ-controlled mobile robot assisted users in functional +and creative tasks. Our analysis of the collected FACT HRC dataset +showed that social cues, especially gaze and gestures, are +informative for temporal and spatial adaptation in handovers. +Further, handovers are shaped by the pre- and post-handover task +contexts. Our findings can be generalised to other HRC tasks, such +as a cooking assistance robot, to develop socially aware robot + +Crafting with a Robot Assistant +HRI ’23, March 13–16, 2023, Stockholm, Sweden +behaviours. In the next step, we plan to train an automatic +handover model from the collected dataset, which predicts the +operator’s controls from the social and task context. +ACKNOWLEDGMENTS +This work is funded by the Australian Research Council Future +Fellowship FT200100761. +REFERENCES +[1] Christoph Bartneck, Dana Kulić, Elizabeth Croft, and Susana Zoghbi. 2009. +Measurement instruments for the anthropomorphism, animacy, likeability, +perceived intelligence, and perceived safety of robots. International journal +of social robotics 1, 1 (2009), 71–81. +[2] Valentin Bazarevsky, Ivan Grishchenko, Karthik Raveendran, Tyler Zhu, Fan +Zhang, and Matthias Grundmann. 2020. Blazepose: On-device real-time body +pose tracking. arXiv preprint arXiv:2006.10204 (2020). +[3] Hildo Bijl, Thomas B Schön, Jan-Willem van Wingerden, and Michel Verhaegen. +2017. System identification through online sparse Gaussian process regression +with input noise. 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International Journal +of Social Robotics 7, 5 (2015), 783–798. + diff --git a/UNE1T4oBgHgl3EQfIQPc/content/tmp_files/load_file.txt b/UNE1T4oBgHgl3EQfIQPc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b654403c52506ed1e483f238e7532afacc2519e --- /dev/null +++ b/UNE1T4oBgHgl3EQfIQPc/content/tmp_files/load_file.txt @@ -0,0 +1,709 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf,len=708 +page_content='Crafting with a Robot Assistant: Use Social Cues to Inform Adaptive Handovers in Human-Robot Collaboration Leimin Tian leimin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='tian@monash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='edu Monash University Clayton, VIC, Australia Kerry He kerry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='he@monash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='edu Monash University Clayton, VIC, Australia Shiyu Xu sxuu0041@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='monash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='edu Monash University Clayton, VIC, Australia Akansel Cosgun akan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='cosgun@deakin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='au Deakin University Burwood, VIC, Australia Dana Kulić dana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='kulic@monash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='edu Monash University Clayton, VIC, Australia ABSTRACT We study human-robot handovers in a naturalistic collaboration scenario, where a mobile manipulator robot assists a person during a crafting session by providing and retrieving objects used for wooden piece assembly (functional activities) and painting (creative activities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' We collect quantitative and qualitative data from 20 participants in a Wizard-of-Oz study, generating the Functional And Creative Tasks Human-Robot Collaboration dataset (the FACT HRC dataset), available to the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' This work illustrates how social cues and task context inform the temporal-spatial coordination in human-robot handovers, and how human-robot collaboration is shaped by and in turn influences people’s functional and creative activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' CCS CONCEPTS Human-centered computing → Collaborative and social computing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Collaborative interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' • Computer systems organization → Robotic autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' KEYWORDS HRI, human-robot collaboration, handover, social robots, adaptation ACM Reference Format: Leimin Tian, Kerry He, Shiyu Xu, Akansel Cosgun, and Dana Kulić.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Crafting with a Robot Assistant: Use Social Cues to Inform Adaptive Handovers in Human-Robot Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In Proceedings of the 2023 ACM/IEEE International Conference on Human-Robot Interaction (HRI ’23), March 13–16, 2023, Stockholm, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' ACM, New York, NY, USA, 9 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1145/3568162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3576998 1 INTRODUCTION Human-Robot Collaboration (HRC) is an active research field that investigates how robots can work together with people as a 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' HRI ’23, March 13–16, 2023, Stockholm, Sweden © 2023 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Publication rights licensed to ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' ACM ISBN 978-1-4503-9964-7/23/03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1145/3568162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3576998 Figure 1: Experimental space layout: the participant and the experimenter sat at the work space and storage space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The Fetch robot moves in between to pass objects in a basket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The operator sat in a neighbouring room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' synergistic team to achieve beneficial outcomes, such as in education [25], healthcare [23], or manufacturing [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' A common HRC task is human-robot handover, which is the physical exchange of an object between a person and a robot at a mutually agreed location and time [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' A successful handover requires spatial and temporal coordination between the person and the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, how the physical exchange unfolds is influenced by environmental, personal, and task constraints [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Current efforts in developing better handovers in HRC have focused on creating models that can identify human intentions and adapt the robot’s motion planning and execution accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' However, as identified by Ortenzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' [20], existing studies investigated handovers in isolation without incorporating them in a naturalistic context with pre- and/or post-handover HRC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, handover evaluation has focused on the performance and technical aspects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', handover time and success rate, rather than the social interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' This limits the understanding of human perceptions and behaviours during handovers, as well as the development of an effective and socially appropriate HRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Our goal is to understand how people perceive a robot and exchange objects with it during task-focused HRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' To build towards autonomous robot capabilities, we conduct a Wizard-of-Oz (WoZ) study to investigate how human operators achieve adaptive, efficient, and socially appropriate HRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' We design a crafting task in which a person assembles and paints a wooden birdhouse with the assistance of a mobile manipulator arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='02938v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='RO] 7 Jan 2023 Robot moving space Operator performing Experimenter at Participant at the Wizard-of-Oz controls OAK-D camera the Storage Space Work SpaceHRI ’23, March 13–16, 2023, Stockholm, Sweden Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' robot, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The person and the robot exchange a set of task-relevant objects with diverse usage purposes (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' We collect quantitative and qualitative data to understand human experience and behaviour during HRC, as well as effective and adaptive collaboration strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Recordings of the HRC sessions and the teleoperation framework we developed are made publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 2 BACKGROUND HRC relies on an effective and appropriate social communication between humans and robots [26, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Expression and perception of social cues is key to successful HRC, for instance, using hesitation gestures to negotiate access to shared resources [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Similarly, previous research found that people use body stances and arm gestures to communicate their intent to when and where a handover happens [4], both as the giver [21] and as the receiver [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The quality of human-robot handover can be assessed on various dimensions using both subjective and objective metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' For example, Murphy and Schreckenghost [18] evaluated handover quality as success rate, human/robot idle time, task completion time, and cost functions relating to movement trajectories, while Hoffman [10] measured human-robot cooperation and contribution using subjective questionnaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Previous studies typically investigate handovers in isolation where objects are exchanged without a pre- and/or post-handover task context, for example, to develop better models for predicting human motions [5, 29], to predict the appropriate location of object transfers [14, 19], to create more natural and fluent robot motions [3, 6], or to develop control models adaptive to contextual factors, such as human activities, preferences, and object semantics [7, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In these studies, it is unclear how a person’s perception and behaviours during handovers may change when they are integrated as part of an HRC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' An example study that investigated handovers as embedded in a collaborative task context is Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The authors studied timing coordination in human-human handovers when they collaborated at a task of unloading a dish rack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Based on observations of 8 pairs of participants, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' [11] developed and evaluated a rule-based model for a robotic arm mounted on a fixed table, which adapted its movement speed and pause duration in between robot-to-human handovers to replicate human decisions during the dish unloading task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' This work provided valuable insights on temporal coordination in handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' However, further research is required to understand human-robot handovers as a social interaction for both temporal and spatial coordination, as well as for diverse types and goals of the object transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 3 METHODOLOGY Here we provide an overview to our study design, including human research ethics, experimental protocols, custom robot teleoperation framework, and collection of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1 Human Ethics and Data Collection Our study protocols were reviewed and approved by the Monash University human research ethics committee (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='31927).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' We recruited 20 participants from the university’s staff and student population (10 females and 10 males, age 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='00 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='17): fifteen were from the faculty of engineering, four from the faculty of art (P7, P8, P10, P15), and one (P13) from the faculty of business and economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' All participants provided informed consent for taking part in the study and having their data recorded as part of a public dataset for research purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' As a WoZ study, the participants were not told explicitly before the experiment that the robot was teleoperated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Instead, the experimenter explained the purpose and details of the WoZ approach in the debriefing afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='2 Experiment Protocol In the collaborative task, a person assembles and paints a birdhouse/feeder with the assistance of a Fetch mobile manipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The participants kept their creation as an appreciation for their time, which is expected to increase their engagement in the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' A session involves three human roles, as shown in Figure 1: a Participant sitting at a work space at one side of the room with a bin and a desk too small to fit all the required objects at once;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' an Experimenter sitting at a storage space at the other side of the room with all objects required during the session stored in a box;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' an Operator sitting behind the participant hidden from their view who controls the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The robot moves in between the work space and storage space and performs object handovers on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The detailed state machine during each handover episode is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The robot (and operator) observes the participant using its onboard head-mounted camera, as well as an RGB-D camera (OAK-D) installed next to the work desk to the front right of the participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' An instruction sheet listing all the objects used in a session is displayed on the top right corner of the work desk for the participants to refer to at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Note that because we are interested in identifying how a robot can best provide assistance in terms of handover timing and object transfer point (OTP), we simplified the handover problem: The robot holds a basket in which objects are transferred, rather than using its gripper to grasp the various objects directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The robot-assisted crafting task consists of three stages: Stage 1 (Preparation), Stage 2 (Assembly), and Stage 3 (Painting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The robot and the participant engaged in handovers during each stage, in which the participant can either be the receiver when the robot brings them an object from the storage space, or be the giver when they need the robot to return an object to the storage space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Figure 2 provides an overview of the experiment session, which has a total duration of approximately one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Stage 1 is aimed at preparing for the collaborative task and an implicit mutual learning between the participant and the robot (operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Stage 2 is aimed at understanding handovers and HRC in a functional task context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Stage 3 is aimed at understanding handovers and HRC in a creative task context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The functional and creative activities have different pre- and post-handover task context, which facilitates our goal of investigating contextualised handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In functional activities, participants are expected to focus on assembling the birdhouse quickly and ensuring its structure is sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In creative activities, participants are expected to have different mental (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', design birdhouse appearance to their liking) and behavioural (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' realising this personal design by painting) context,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' which Crafting with a Robot Assistant HRI ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' March 13–16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Sweden ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Introduction to the robot and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='the experiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Pre-study questionnaire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stage 3 evaluative questionnaire and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='overall experience questions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Debriefing and post-study interview ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stage 3: Painting (Creative tasks) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Apron ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Gloves ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Tissues ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stage 1: Preparation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stage 2: Assembly (Functional tasks) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Glue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Wooden pieces (came separately) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stage 2 evaluative questionnaire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Gloves ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Tissues ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Palette ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Bundle of paint brushes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Paints (came separately) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Thread ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Scissors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Water ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Return bag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stage 1 evaluative questionnaire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Figure 2: Session overview: the one-hour session includes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='three stages with various objects used during each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Participants filled in an online questionnaire during the session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' A debriefing and interview is conducted at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' influences how and when the handovers should happen, and participants’ perception and experience during the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' An operator controlled the robot via a teleoperation interface detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The operator sat in a neighbouring room (see Figure 1) and took their position after the participant and the experimenter were seated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' As a WoZ study, the participants were not informed of the operator’s presence before or during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' This allows us to understand what social cues people express during HRC and how these social cues can be used to achieve adaptive handovers applicable to autonomous robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3 Teleoperation Framework We developed a custom teleoperation framework using the Robot Operating System (ROS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1 One expert Fetch operator was trained on using the teleoperation framework, and controlled the robot during all experimental sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The operator’s role is to identify the appropriate timing for initiating and completing each handover episode, to identify the appropriate OTP, and to evaluate the quality of a handover episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, after each session, the operator participated in the interviews to describe their strategies and to incorporate the participants’ feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The operator controlled the robot using a hand-held controller and a computer interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' As shown in Figure 4, the operator has access to a live feed of the OAK-D camera and the robot’s onboard camera, as well as estimates of the participants’ upper body and facial keypoints from the OAK-D camera feed using Blazepose [2], and categorical emotion estimates from both camera feeds using EmoNet [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The participant’s pose, the robot’s pose, and the robot’s end effector’s (EE) pose are shown in a 3D map view, which plots the path that the robot follows to travel between the work space and the storage space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The operator can drive the robot along a predefined path and rotate the robot’s base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' They can also make fine adjustments to the positions of the path’s endpoints during or 1The teleoperation framework: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='com/tianleimin/fetch-teleop in between episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' To perform handovers, the operator can extend the EE towards an OTP (arm stretching), or retract it to a default tucked position (arm tucking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The handover motion sequence is based on He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' We specified three default OTPs: to the left of the robot (right-hand-side of a participant), to the right of the robot (left-hand-side of a participant), and in the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The position and orientation of these OTPs can be fine-tuned by the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Changes in the path endpoints and OTPs can be saved or reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' After each episode, the operator uses the controller to evaluate the quality and specify the nature of the participant-robot handover (human-to-robot, robot-to-human, or bidirectional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' During arm stretching, the robot automatically tilts its head downwards to point at the EE and display joint attention [17, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' During arm tucking, the robot restores its head to neutral position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='4 Measurements Data from the teleoperation framework is recorded continuously using ROS bags, which includes synchronised operator control signals, video recordings, robot status, estimates of facial and upper body keypoints from the OAK-D camera feed, and the emotion estimates from both camera feeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, we collected participants’ responses to a questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' This includes pre-study questions on participants’ demographic backgrounds and robot self-efficacy [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' After each of the three stages, the participants evaluated their experience during that stage with 5-point Likert scale measurements of fluency, trust, and working alliance between the themselves and the robot [20], their perception of the robot as the GodSpeed questionnaire [1], their overall enjoyment and satisfaction, and a text comment box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' After all stages are completed and rated, before the debriefing and interview, participants also chose whether they thought the robot was teleoperated or fully autonomous, whether they thought the robot was adaptive, and gave any additional comments in a text box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In addition to the quantitative data captured by the questionnaire and the teleoperation framework recording, we collected qualitative observation notes and post-session interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The experimenter took observation notes during the session on participants’ behaviours, operator’s controls and adaptations, and events of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In interviews participants were asked to describe their impression of the robot and the HRC experience, their guesses as to how the robot functioned, their reasoning behind certain events of interest during the session, their design and creative processes, and future improvements that the robot can incorporate to become a better assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The operator also joined the interview to describe their observational and adaptive strategies, and to answer any questions from participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 4 THE FACT HRC DATASET We collected a dataset of WoZ controlled human-robot handovers and collaboration in Functional And Creative Tasks (FACT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The FACT HRC dataset includes three segments: (1) FACT-raw2 (requires a signed EULA to access) contains identifiable ROS bag recordings described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 2FACT-raw: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='26180/21671789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='v1 eenex7 7 Bottom Left/Right Front/BackWall Panel Roof 6 Front/BackWall Left/Right Left/Right Roof WallOTANIA OVES L 100GUAHRI ’23, March 13–16, 2023, Stockholm, Sweden Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stationary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='(facing participant) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Storage Space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Move to work space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stationary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='(facing participant) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Moving Space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Work Space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stretch arm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Object Transfer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='(with participant) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Tuck arm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Rotate (towards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='experimenter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stationary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='(facing experimenter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Move to storage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stationary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='(facing experimenter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Stretch arm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Object Transfer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='(with experimenter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Tuck arm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Rotate (towards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='participant) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Episode Start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Episode End ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Object transfer with participant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Object transfer with experimenter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Timing adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Object transfer point (OTP) adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Timing adaptation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='Figure 3: An episode of handover: the operator performs temporal adaptation when waiting to move the robot to the work ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='space and when waiting to initiate object transfers with participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Spatial adaptation also happens during object transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Figure 4: Visualisation of the teleoperation framework: the operator can see live feeds of the robot’s onboard camera, the additional RGB-D camera, a 3D map view with the robot’s pose as well as estimated participant’s poses and emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' (2) FACT-processed3 contains non-identifiable csv data extracted from the synchronised raw data at intervals of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1s (in total approximately 600K data instances), including the robot’s status, the operator’s controls, the facial and upper body keypoints estimated from the OAK-D camera feed, and the emotion estimates from both camera feeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In addition, the de-identified questionnaire responses are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' (3) FACT-support contains supporting materials, including the instruction sheet, operator’s cheat-sheet for teleoperation framework controls, the questionnaire, CAD file for laser cutting the birdhouse pieces, implementation of the teleoperation framework and data processing scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The FACT HRC dataset includes 565 handover episodes, namely: 52 Human-to-robot handovers (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='2%): the person giving an object to the robot by placing it in the empty basket that the robot holds in its gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 3FACT-processed and FACT-support: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='26180/21671768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='v1 320 Robot-to-human handovers (56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='6%): the person receiving an object from the robot by taking it from the basket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 193 Bidirectional handovers (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='2%): object exchange between the person and the robot, either simultaneously or in two consecutive give and receive motions, during which the robot’s arm can remain stretched out or be tucked to wait between the give and receive motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 5 OPERATOR’S ADAPTATION STRATEGIES The operator adapted handovers based on participants’ intent and preference inferred from gaze and gestures, safety considerations of the handover, and its efficient integration with the task context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1 Observation and Expression of Social Cues The operator reported monitoring the person’s gaze (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', eye contact with the robot), hand gestures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', waving), and upper body movements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', leaning forward), which aligns with existing research [4, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The operator did not rely on the estimated emotion print-outs, as the automatic model is less accurate than a human’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' However, emotion estimations may benefit an autonomous model as they can distinguish handover episodes rated as good vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' bad by the operator, with more positive or neutral emotions predicted in the episodes rated as good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In addition, the operator monitored pre- and post-handover context, including object layout on the work desk, the space where the person is performing the crafting tasks in, and whether or not one or both of their hands are occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The operator used the position and movement of the robot to communicate its intention to the human collaborator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' When the robot is waiting to deliver or receive an object to or from a participant, if the waiting time is estimated to be long, the operator controlled the robot to wait at the storage space to avoid pressuring the person to rush through their tasks, as suggested by some participants in the interviews (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' if the wait is estimated to be short, the operator controlled the robot to wait at the work space so that it could perform the handover as soon as the person is ready.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Five participants commented on the robot’s head tilting behaviour during object transfers as an indication of [] Select $ Focus Camera Measure 2D Pose Estimate 2D Nav Goal Publish Point + Displays Views 、游 Globaloptions Type: ThirdPersonFoll - V Global status: ok Zero PID Human Pose > current View ThirdPersonFol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 、Map Near Clip 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='01 O TF Estimations Invert Z Axis Marker Target Fra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' LaserScan End Effector Pose Distance 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='04145 Focal shap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='05 iyRobotModel Emotion Focal shap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Path Yaw 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='73227 Estimations Pitch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='129796 Add Duplicate Remove Rename Focal Point 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='9296;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='474;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=" Image Fetch: Neutral Robot's OAK-D: Neutral Camera hanelevtoa Image OAK-D Camera Map and Path Save Remove Rename Time PauseSynchronization: off →Source:LaserScan ROs Time:1654147954." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='58 V Experimental Reset 11 fpsCrafting with a Robot Assistant HRI ’23, March 13–16, 2023, Stockholm, Sweden the robot’s awareness to itself and its collaborator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The operator’s observation and control strategies suggest that perception and expression of social cues is key to facilitates HRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='2 Adaptation in Handover Timing The operator focused on choosing the appropriate moment to move the robot to the work space for initiating the handover episode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', deciding the wait time at the storage space) and to stretch out the robot’s arm for initiating the object transfer within a handover episode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', deciding wait time at the work space), as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' A handover was initiated when the participant displayed a personal “start” gesture (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In addition to these participant-initiated handovers, the operator performed robot-initiated handovers in Stage 2 and 3 by moving the robot to the work space before the participant signalled, when they anticipated that it will take the person shorter to finish their task at hand than the time (9s) the robot needs to move from the storage space to the work space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Robot-initiated handovers were also performed when the item to be delivered is of immediate need to the person while they were continuing their task at hand .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The average duration of a handover episode in Stage 1 is 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='6±42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1s, in Stage 2 is 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='8s, in Stage 3 is 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='0s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', the episodes are the shortest in the functional activities of assembly (S1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' S2 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='03, S2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' S3 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='02, S1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' S3, 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The larger variance in episode duration in Stage 1 is in line with our discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='2 on it being intended for implicit mutual learning between the participant and the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In Stage 1 the average pause at the work space (robot waiting at the participant’s side for the right time to initiate object transfer) within the handover episodes is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='5±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='8s, and the average pause at the storage space (waiting for the right time to move towards the participant) is 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='9±24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='4s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In Stage 2, the average pause at the work space is 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='0±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='6s, and at the storage space is 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='7s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In Stage 3, the average pause at the work space is 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='2±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='6s, and at the storage space is 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1±27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='9s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', the operator controlled the robot to wait longer at the storage space than at the work space (𝑝 ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='01 in S1, 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='08 in S2, 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='004 in S3) to avoid pressuring the participants, in line with our discussion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3 Adaptation in OTP The operator considered efficiency and safety when deciding on the OTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' For efficiency, they aimed at stopping the EE approximately where the person’s dominant or free hand is positioned, or where they signalled the “start” gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' For safety, they factored in object layout on the work desk to avoid collision, especially with the birdhouse that the person is crafting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' When the two considerations conflicted the operator prioritised safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' OTP adaptations can happen live in response to a participant’s behaviours or between object transfers based on past observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' While the operator can move the robot’s base closer or further away from the person when it is at the work space, they chose not to use this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Instead, OTP was adapted by changing the EE goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' As described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3, there are three default EE goals (left, centre, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The operator can make changes to these default OTPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' However, such fine adjustments were not performed as one of the three defaults was considered close enough to the operator’s envisioned OTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, the operator stopped the arm stretching motion prematurely before the default EE goal was reached (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', the OTP is further away from the person than the default) if the person reached out their hand(s) before the arm stretching completes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' It takes 12s for the robot to complete an object transfer with the arm ready for one of the three default OTPs or when it is switching between central and left/right OTPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' However, switching between the left and right OTPs takes 30s with bigger motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' To avoid interrupting the collaboration flow, the operator did not perform such left-right switches during robot-participant handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Instead, this was done during robot-experimenter handovers or while the robot was waiting at the storage space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In comparison, switching between central and left/right OTPs was performed more frequently in robot-participant handovers to quickly adapt to the person’s preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Out of all 61 OTP switches, 49 were left-middle switches (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3%), 7 were right-middle switches (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='5%), and 5 were left-right switches (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In the 565 handover episodes, 186 (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='9%) were left handover (right-hand-side of the participant), 367 (65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='0%) were central handover, 12 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1%) were right handover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The preference of left OTP over right OTP aligns with our participant population with only P1 being left-handed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='4 Operator’s vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Participant’s Perceptions Out of all handover episodes, the operator rated 397 (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3%) as good, 32 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='7%) as bad, and 136 (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1%) as neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The operator incorporated the participants’ feedback provided in the debriefing and interview after each session in their observational and adaptive strategies in later sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Thus, we saw a continual development of the operator’s expertise and control strategies: With P1 and P2, the operator rated the sessions as having the most number of bad handover episodes (5 episodes each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, there was one collision incident with P1 where the robot hit the work desk in the first episode due to a setup issue, and one collision incident with P2 in Stage 3 where the robot pushed the assembled birdhouse when stretching its arm to perform central handover to deliver the scissors to P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' P1 and P2 addressed the collision incident during interview, but did not consider the other bad episodes to stand out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' With P5, P7, and P12, the operator rated the sessions as having 3 bad episodes each, while P5 considered there were “one or two miscommunications”, P7 did not notice the robot’s mistakes or its changes in OTPs, and P12 commented on “one weird incident” with the robot’s arm movement appearing unnatural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' For remaining sessions with one or two bad episodes as rated by the operator, participants did not address these episodes in the interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' To further understand the differences in operator’s and participants’ perceptions, for each stage, we calculated an operator’s handover quality score 𝑄𝑜 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='5 × ( 𝑁𝑔𝑜𝑜𝑑−𝑁𝑏𝑎𝑑 𝑁𝑡𝑜𝑡𝑎𝑙 + 1), and a participant’s handover quality score as the min-max normalised sum of their ratings on the enjoyable 𝐸 and satisfactory 𝑆 level 𝑄𝑝 = 𝐸+𝑆−2 10−2 (𝑄𝑜,𝑄𝑝 ∈ [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Figure 5 shows the stage-wise mean (𝑄) and standard deviation (𝜎𝑜, 𝜎𝑝) of 𝑄𝑜 and 𝑄𝑝 in each session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Note that the participant’s ratings are independent from one another, while the operator’s ratings of a session can be dependant on their experiences in previous sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' As shown here, the operator and participants’ perception of HRI ’23, March 13–16, 2023, Stockholm, Sweden Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Participant ID Subjective scores Role: Operator Participant Figure 5: Operator’s vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' participant’s perceptions: the operator had a handover-centred perspective while the participants were influenced by the task context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' handover quality do not always align.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In 9 sessions where 𝜎𝑜 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1 (P4, P7-P9, P11-P14, P19), 𝜎𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='07, 𝜎𝑜 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' This suggests that compared to the operator who had a handover-centred perspective, the participants’ perception may be less influenced by the individual handovers compared to the task context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' This is also found in the interviews as discussed in Sections 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Comparing 𝑄𝑜 and 𝑄𝑝 in earlier (P1-P10) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' later (P11-P20) sessions, the average scores in each of the three stages are consistently higher in the later half, although the difference is only statistically significant in Stage 2 of 𝑄𝑜 (𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The increase in the quality scores is bigger for 𝑄𝑜 than for 𝑄𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' This indicates that the operator gained expertise in handover adaptation, which improved the participants’ experience in later sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 6 USER’S PERCEPTION AND BEHAVIOURS We investigate how people collaborate with and perceive the robot from subjective questionnaires, observation notes, and interview data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Our analysis showed that participants have both positive and negative impressions towards the robot and the HRC experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Their perceptions are more positive in Stage 2 and 3 compared to Stage 1, as the operator adapted to their preferences and the participants became more focused on the functional and creative activities beyond object handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, their social cues change within a session and differ between participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Moreover, the HRC directly influenced people’s design and creative processes, even though only human-robot object handovers were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='1 User Behaviours During the HRC Interestingly, all participants signalled to the robot’s “face” (onboard camera), even though they were aware that the OAK-D camera was also used for monitoring their behaviours and was positioned closer to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' P6, who has worked with robots before, was the only participant who attempted signalling to both cameras in the first handover episode and switched to only signalling to the robot’s onboard camera in all later episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' P6 commented that he signalled to the OAK-D camera at the start because it had “better angle”, but later it felt more natural to signal to the robot’s onboard camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, P15 commented that at times he felt it was “weird” or “too formal” to use big gestures to signal the robot to come, as he considered the robot to be an equal partner working on a task together with him in a casual way, while waving your hand at someone’s face can be considered impolite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' These highlight the social interaction aspect of human-robot handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Although they were told that the robot has no speech recognition or interaction capabilities, five participants still spoke to the robot, such as saying “thank you” after handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' This suggests that speech is a natural interactive modality that can benefit HRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' When signalling to the robot, participants used social cues combining one or more modalities, including gaze, movements of hand(s), and body gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The most common social cue to signal the initiation of handovers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', the “start” gesture, is to gaze up at the robot while leaning forward and showing a single-handed wave, either side-to-side or forward-backward, as used by 17 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Several variations of this “start” gesture were observed, such as using only gaze without hand or body movements, holding the object to be given to the robot when waving, or using semantic gestures to demonstrate the object needed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', showing gestural imitation of putting on gloves to initiate glove box handover).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Twelve participants used more than one “start” gestures during the session, with 11 switching to lower-effort “start” gestures in later episodes (except for P13), for instance, from using both gaze and hand movements to gaze-only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' During the interview, P1, P3, P10, and P13 discussed that they experimented with different gestures and observed the robot’s reactions to decide on which gestures were the most effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' P10 commented that she used bigger gestures at the start of the session because she was more excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' P14 expressed that she changed to smaller gestures later in the session as she became more familiar with how the robot functions and felt that she needed less effort to communicate with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' After the robot arrived at the work space, five participants displayed a follow-up “start” gesture to signal the robot to stretch out its arm and initiate object transfer by reaching out one hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' After completing object transfers, six participants displayed an “end” gesture to signal the robot to tuck its arm and return to the storage space, by showing a single-handed gesture of “OK” while nodding or an outward “go-away” single-handed wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' During the sessions, participants also adapted to the robot’s behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' As the robot’s movements are relatively slow, to increase efficiency, participants performed bidirectional object exchanges instead of single-directional object handovers when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' They also filled the waiting time with other activities, namely reading the instruction sheet, inspecting their assembly or painting progresses, and checking their phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, participants configured the layout of their work desk dynamically to keep the area that they preferred for object transfers accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Our analysis showed that participants treated the robot as a social actor during the collaboration and used various social cues, namely gaze, hand movements, and body gestures, to coordinate the timings and locations for object handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' They also adapted to the robot to achieve more efficient collaboration by changing their behaviours, such as filling in the waiting time with other tasks and performing bidirectional handovers when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='2 Perception of the Robot Before the HRC session, participants rated their expectations on robot operation and application efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' No significant differences were found between participants from the faculty of engineering and participants from the faculties of art and business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Crafting with a Robot Assistant HRI ’23, March 13–16, 2023, Stockholm, Sweden p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='05 ** p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='01 ** ** ** ** Figure 6: Subjective ratings of robot impression: participants had more positive impressions of the robot in Stage 2 & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Before debriefing, participants guessed whether the robot was autonomous or teleoperated, and whether they thought it was adaptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Fifteen participants answered the robot was adaptive during the session, three answered it was not adaptive, and two were unsure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Nine participants thought the robot was fully autonomous, nine thought that it was being teleoperated, and two were unsure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' None of the participants identified where the operator was located until they were introduced during the debriefing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The main reason participants gave for considering the robot to be teleoperated or to be unsure about their guesses is the robot’s ability to adapt to their behaviours and the task context, which they consider to be too difficult for a fully autonomous robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' This demonstrates that improving a robot’s adaptive capabilities is key to achieving natural HRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In Figure 6 and Table 1, we report participants’ subjective impressions of the robot in each stage (sub-item means, value range [1,5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' As shown here, participants have significantly more positive impressions of the robot in Stages 2 and 3 compared to Stage 1 (except for Intelligence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' One possible cause of the higher ratings in Stages 2 and 3 is that the operator learned an effective adaptation strategy during Stage 1 and performed more fluent handovers in Stages 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' However, as discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='4, another reason may be the participants being more focused on the functional and creative activities in Stages 2 and 3 rather than object transfers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', the participant’s perceptions were influenced more by the collaborative task than by the individual handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In the interviews, participants expressed both positive and negative impressions towards the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Eight participants commented that the robot was “cool”, “impressive”, “amazing”, “likeable”, or they were “fascinated”, “excited”, or “satisfied” interacting with it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Four considered the robot “responsive” or “adaptive” in its interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Three thought the robot was “useful”, “effective”, or “helpful” for the tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Two considered the robot “reliable” or “trustworthy” in executing actions and performing its tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' However, eight participants also mentioned slowness and experiences of waiting for the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Although this sense of slowness is not necessarily a negative experience for every participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' As discussed by P15, the robot being slow made it feel predictable and non-threatening, which led to him considering the robot “docile” and “cute”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, P2 and P18 discussed a sense of uncertainty or confusion in how to interact with or trigger certain behaviours in the robot despite having prior exposure to the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Table 1: Participants’ impressions of the robot and the HRC (mean ± std).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Stage 2 & 3 are perceived more positively overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Robot impression Stage 1 Stage 2 Stage 3 Anthropomorphism 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='78 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='96 Animacy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='81 Likeability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='70 Intelligence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='66 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='83 Perceived safety 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='62 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='70 HRC impression Stage 1 Stage 2 Stage 3 Fluency 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='69 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='75 Trust 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='78 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='85 Working Alliance 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='86 Enjoyable 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='66 Satisfactory 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='85 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='85 Participants also discussed future improvements they would want the robot to have in order for it to be a better assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Eight participants commented that speech interaction or voice commands in addition to the current gesture-based approach can help increase the interactivity and flexibility of the collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Similarly, three participants suggested that developing a dictionary that associates gestural commands to specific objects can help minimise confusion and increase flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Related to this desire for more flexibility, six participants discussed that it would be good to allow customising the order and set of objects that the robot delivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Regarding the robot’s behaviours, four participants expressed wanting the robot to be anticipatory instead of being reactive in more handover episodes, two participants expressed preference for more consistency in its reaction time and/or behaviours, three participants suggested more involvement of the robot beyond object exchange, for instance, working on the assembly directly with the person by holding a piece while the person is gluing it to the rest of the birdhouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' As shown in the questionnaires and interview responses, while having diverse impressions of the robot, overall participants considered it to be an adaptive and helpful collaborator, especially in the assembly and painting stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Improvement that can benefit the robot’s perceived fluency and contribution to the task includes increased participation beyond object handovers and multimodal interaction combining speech and gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='3 Perception of the Collaboration Figure 7 and Table 1 showed participants’ perception of the HRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Similar to perception towards the robot, participants have significantly more positive impressions of the HRC in Stage 2 and 3 compared to Stage 1 in terms of Enjoyable, Satisfactory, and Fluency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In the interview, four participants expressed that they found Stage 2 to be the most collaborative out of the three stages where them and the robot worked as an efficient team, while three participants considered Stage 3 to be the most collaborative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Participants discussed both positive and negative perceptions of the HRC in interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Four participants commented that the experience was “fun”, for instance, P17 said the wooden pieces in Stage 2 “arrived like presents”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' six expressed that the interaction HRI ’23, March 13–16, 2023, Stockholm, Sweden Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='05 ** p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='01 ** ** ** ** Figure 7: Subjective ratings of HRC: participants perceived the collaboration more positively in Stage 2 & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' was “smooth”, “efficient”, or “effective”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' five considered the session was “adaptive” with the robot improving over time to better address their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' However, five participants also mentioned that when the robot was waiting for them, especially at the work space, it created a sense of pressure or a desire to speed up their actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Participants can also comment in the free text area of the questionnaire after each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' For Stage 1, four participants commented about their uncertainty in how to signal to the robot to come and to perform object transfer, and their experience gaining better understanding of how the robot will behave over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' For Stage 2, six participants gave comments that highlighted they had a better collaboration experience compared to Stage 1 as the robot felt more involved, efficient, and adaptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' For Stage 3, six participants gave comments discussing their positive impression of feeling relaxed, as well as negative impressions, namely part of the interaction being slow, wanting the objects to be delivered in a different sequence, and the robot handing objects at an undesirable location in some episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' At the end of the questionnaire, nine participants gave additional comments, expressing both positive impressions of the robot being human-like and adaptive, and negative impressions of the interaction feeling slow or being unsure about the robot’s intention at times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In the interview, twelve participants noticed the robot changing the OTPs during their sessions, five participants did not notice any location changes, while all handover episodes were at the central location in the sessions with P13, P14, and P19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In addition, three participants noticed the robot changing its arm stretching distance to be closer or further away from the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Nine participants addressed that the robot was proactive during some episodes, while eleven participants had the impression that the robot was entirely reactive and only initiated handovers when they signalled to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Two participants also discussed that they were adapting to the robot themselves and became more in tune or familiar with the robot’s behaviours as the session progressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Our analysis shows that overall participants considered the collaborative experience to be enjoyable, efficient, and adaptive, especially in the assembly and painting stages, despite not noticing all the handover timing and OTP adaptations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='4 Functional and Creative Activities In the interview, participants discussed how the collaboration with the robot influenced their functional and creative activities during the crafting session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Regarding the functional tasks of assembling the birdhouses, P1 commented that the robot helped avoid cluttered work space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Similarly, P13, P15, and P17 expressed that the robot bringing the wooden pieces one by one guided them to work through an unfamiliar task in a structured manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, P8 referred to her past experience attending furniture design classes, in which people spent long time waiting to use shared equipment, and commented that the robot could be helpful in this scenario by bringing objects to the shared equipment for processing while the person continues with other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Regarding the creative activities of designing the appearance of the birdhouse, all participants conceptualised the design during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Since only the five prime colours were offered, six participants used their phones during the painting stage to search for colour mixing methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', P2 searched how to create brown) or pictures of specific items that they intended to include in the design (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=', P9 searched photos of the superb fairywren).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Nine participants stated that the order in which the robot delivered the paints directly inspired or influenced their design, while eleven participants commented that as from the instruction sheet they already knew which colours would be provided, their design was not influenced by the order in which the colours arrived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Moreover, six participants expressed that they wished the session could have been longer as they were deeply invested in the crafting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Three participants commented that as the human-robot handovers were integrated as part of the crafting session instead of being at the centre of attention, they were able to focus on the creative design process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Specifically, P15 discussed that as he was able to focus on the creative aspect, he recalled the birdhouse’s purpose of feeding small birds and changed his design to a more camouflaged look by avoiding bright colours that may attract predators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' P10 referred to her past experience interacting with small swarm painting robots, during which the robots made many mistakes and she was concerned that they would break something or hurt someone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' However, in this session, she found the robot “smooth” and “helpful”, which allowed her to concentrate on creating and experimenting with different colour mixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, P8 commented from her experiences tutoring children’s art classes, where tidying up the supplies at the end of a class is often time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' She expressed that the robot helped structure the painting stage to be more organised by delivering and retrieving items in order, which can potentially reduce the workload of the tutors in the children’s art class scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' As shown in the interviews, participants’ functional and creative activities were influenced by the robot’s behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' The functional collaboration of object handovers not only benefited the person’s task efficiency, but also led to design and creative benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 7 CONCLUSIONS We studied human-robot handovers in a naturalistic HRC context, where a WoZ-controlled mobile robot assisted users in functional and creative tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Our analysis of the collected FACT HRC dataset showed that social cues, especially gaze and gestures, are informative for temporal and spatial adaptation in handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Further, handovers are shaped by the pre- and post-handover task contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' Our findings can be generalised to other HRC tasks, such as a cooking assistance robot, to develop socially aware robot Crafting with a Robot Assistant HRI ’23, March 13–16, 2023, Stockholm, Sweden behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' In the next step, we plan to train an automatic handover model from the collected dataset, which predicts the operator’s controls from the social and task context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work is funded by the Australian Research Council Future Fellowship FT200100761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfIQPc/content/2301.02938v1.pdf'} +page_content=' 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a/VdE_T4oBgHgl3EQfxxyN/content/tmp_files/2301.08314v1.pdf.txt b/VdE_T4oBgHgl3EQfxxyN/content/tmp_files/2301.08314v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ab6923857c546991cabe20334e2aa42dfc109be --- /dev/null +++ b/VdE_T4oBgHgl3EQfxxyN/content/tmp_files/2301.08314v1.pdf.txt @@ -0,0 +1,526 @@ +arXiv:2301.08314v1 [hep-ph] 19 Jan 2023 +Dark Matter Phenomenology in 2HDMS +Gudrid Moortgat-Pick*1,2, Juhi Dutta1, Cheng Li2, Merle Schreiber1,2, Sheikh F. Tabira1, +Julia Ziegler1 +1 II. Inst. of Theo. Phys., University of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany +2Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany +The constituents of dark matter are still an unresolved puzzle. Several Beyond Standard Model +(BSM) Physics offer suitable candidates. In this study here we consider the Two Higgs Doublet model +augmented with a complex scalar singlet (2HDMS) and focus on the dark matter phenomenology of +2HDMS with the complex scalar singlet as the dark matter candidate. The parameter space allowed +from existing experimental constraints from dark matter, flavour physics and collider searches has +been studied. The discovery potential for such a 2HDMS at HL-LHC and at future e+e− colliders has +been worked out. +1. +Introduction +Dark Matter (DM) remains an unsolved puzzle at the interface between particle physics and cos- +mology, only 4-5% of the Universe are composed by ’known’ matter components, but about 25% is +built of dark matter. Since the Standard Model (SM) does not accommodate a suitable DM candi- +date, several Beyond Standard Model (BSM) extensions have been proposed to accommodate DM +candidates ranging from scalar, fermion to vector candidates and with mass scales from below eV up +to TeV particles. We concentrate in this contribution on thermal weakly interacting massive particles +(WIMP) that is expected in the mass range of GeV up to TeV, accessible at future collider experiments +at the LHC and a high-energy e+e− linear collider (ILC, CLIC). +Among popular BSM candidates are models with an extended Higgs sector such as the Two Higgs +Doublet model (2HDM) [1], providing a dark matter candidate within the Inert Doublet model [1]. Al- +ternate models are such multi-Higgs models but extended via real or complex singlet scalars serving +as dark matter candidates. Such extensions involving real scalar singlets have been extensively stud- +ied [2–4] while complex scalar extensions to the 2HDM have also been recently studied in the context +of modified Higgs sectors [5]. Such models have also the potential to explain the matter-antimatter +asymmetry and to accommodate both inflation as well as gravitational waves phenomenology [6,7]. +The parameter space of such extensions of the SM [8] gets strong constraints from direct searches for +DM as well as from precision measurements of the 125 GeV SM-like Higgs boson and in particular +from limits of both its visible as well as invisible branching ratios [9]. +2. +Extended Two Higgs Doublet Model +2.1 +Symmetries +We consider the CP-conserving softly broken Type II Two Higgs Doublet model augmented with +a complex scalar singlet (2HDMS) [5] consistent with flavour changing neutral currents (FCNCs) at +tree-level. It allows for the presence of the mixing term between the two Higgs doublets, Φ1 and Φ2, +i.e., m2 +12, while the explicit Z2 breaking terms are absent. The complex scalar singlet S is stabilised by +a Z′ +2 symmetry such that S is odd under Z′ +2 while the SM fields are even under the new Z′ +2 symmetry. +The fields Φ1 and S are even under Z2 while Φ2 is odd under Z2. +We consider the case where Z′ +2 remains unbroken both explicitly and dynamically, i.e. the scalar +1 + +singlet doesn’t obtain a vacuum expectation value. Therefore, the scalar potential V with a softly +broken Z2- and a conserved Z′ +2 symmetry is V = V2HDM + VS, where, the softly broken Z2-symmetric +2HDM potential is: +V2HDM += m2 +11Φ† +1Φ1 + m2 +22Φ† +2Φ2 − (m2 +12Φ† +1Φ2 + h.c) + λ1 +2 (Φ† +1Φ1)2 + λ2 +2 (Φ† +2Φ2)2 +(1) ++λ3(Φ† +1Φ1)(Φ† +2Φ2) + λ4(Φ† +1Φ2)(Φ† +2Φ1) + [λ5 +2 (Φ† +1Φ2)2 + h.c] +(2) +and the Z′ +2-symmetric singlet potential, VS , is +VS += m2 +S S ∗S + ( +m2′ +S +2 S 2 + h.c) + ( +λ′′ +1 +24S 4 + h.c) + ( +λ′′ +2 +6 (S 2S ∗S ) + h.c) + +λ′′ +3 +4 (S ∗S )2 +(3) ++S ∗S [λ′ +1Φ† +1Φ1 + λ′ +2Φ† +2Φ2] + [S 2(λ′ +4Φ† +1Φ1 + λ′ +5Φ† +2Φ2) + h.c.]. +(4) +The doublet fields have the components Φ1 = (h+ +1 , +1√ +2(v1 + h1 + ia1))T, Φ2 = (h+ +2 , +1√ +2(v2 + h2 + ia2))T, +S = +1√ +2(hs + ias) and tan β = v2 +v1 is the ratio of the up-type and down-type Higgs doublet vevs v1,2 +(with v(= v2 +1 + v2 +2) ≃ 246 GeV. Under the assumption that the complex singlet scalar does not develop +a vev —for this study imposed—, the Higgs sector, after EWSB, remains the same as in 2HDM, i.e, +consisting of two CP-even neutral scalar Higgs particles h, H, a pseudoscalar Higgs A and a pair of +charged Higgs particles H± [1]. All Higgs-dark-matter portal couplings are explicitly given in [10]. +2.2 +Theoretical and Phenomenological Constraints +The Sylvester’s criterion and copositivity [11, 12] has been applied to guarantee boundedness +from below for the Higgs potential, leading to constraints on all coupling parameters λi, i = 1, . . . , 5, +λ′ +j, j = 1, . . . , 5, λ′′ +k , k = 1, . . . , 3. The mass of the lightest CP-even Higgs particle mh = 125 GeV has +been chosen to be in concordance with the measured Higgs state via HiggsSignals [13] and collider +constraints from LEP and LHC have been applied for the heavy Higgs states via HiggsBounds [14]. +The branching ratio BR(h → χχ) < 0, 11 (< 0.19), fulfilling the limits from ATLAS (CMS). Elec- +troweak precision constraints on STU parameters have been taken into account as well as constraints +from flavour physics BR(b → sγ), BR(Bs → µ+µ−), using SPheno [15]. ∆(gµ − 2). Concerning the +dark matter particle, the bounds on the relic density from PLANCK measurements, Ωh2 = 0.119 [16], +as well as constraints from direct detection (XENON-1T [17] ) and indirect detection (FERMI-LAT +[18]) experiments have been applied using micrOMEGAs [19]. +3. +Results +3.1 +Benchmark Points +This model has been implemented using SARAH [20] code implemented into SPheno for the +spectrum generation. In order to calculate collider observables the code chain Madgraph [21] -Pythia +[22] -Delphes [23] -Madanalysis [24] has been used. +As can be seen from Figs.1a) and b) the mass of the dark matter particle χ as well as the coupling +λ′ +2 get strongest constraints from the direct detection search from XENON-1T: in the shown example +where the heavier Higgs particles are about 725 GeV, the mass mχ ∼ 338 GeV. Scanning the available +parameter space allowed to specify different benchmark areas, see Table I. BP1 and BP3 are very +similar, however, they differ significantly in the couplings λ′ +1, λ′ +2 and λ3 leading to different collider +phenomenology. +3.2 +Collider Phenomenology +In this section, we discuss the possible signals of this model at HL-LHC and future e+e− colliders. +As already mentioned, the invisible decay of the heavy Higgs into the dark matter candidate is a +2 + +source of missing energy at colliders. Therefore, the direct production of heavy Higgs bosons and +consequent decay of the Higgs to χ along with visible SM particles can give rise to distinct signatures +for this scenario as opposed to the 2HDM like scenario. We investigate these possibilities and their +prospects in the context of √s = 14 TeV LHC at the targeted integrated luminosity of 3-4 ab−1 and +of future e+e− colliders (ILC,CLIC) up to √s = 3 TeV and integrated luminosities of 5 ab−1. +3.2.1 +Prospects at LHC +The main processes contributing to neutral Higgs production at the LHC are gluon fusion (me- +diated by the top quark loop), vector boson fusion (VBF), associated Higgs production (Vhi), b¯bhi, +t¯thi [1]. For the charged Higgs pair, the possible production channels are H+H− and W±H∓ [1]. At +the LHC Run 3 at √s = 14 TeV, all possible Higgs production processes (including SM and BSM +Higgses) are summarised in Table II. +In the presence of the heavy Higgs H decaying into two dark matter candidates, one gets invisible +momentum in the final state and one can look into the following final states: +a) 1j (ISR)+missing ET [25] +b) 2j+ missing ET [26] +We estimate the significance for the mono-jet and VBF channels using the cuts from an existing +cut-and-count analyses performed in [4] for √s = 14 TeV LHC, further details see [10]. For a) we +achieve a cut efficiency for the signal in BP3 of about ∼ 18% and we obtain a 0.111σ excess at 3ab−1 +using gluon fusion production channel (at leading order (LO)). For b) we get a signal efficiency of +4.5% for BP3 and a signal significance of about ∼ 0.2 σ at 3 ab−1. Therefore, we observe that due to +the small invisible branching ratio and heavy Higgs masses ∼ 820 GeV (and hence small production +cross section) in BP3, the final states will be inaccessible at the upcoming HL-LHC run. +3.2.2 +Prospects at a high-energy e−e+ Linear Collider (ILC, CLIC) +The cleaner environment and lower background along the beam line compared to hadron colliders +make the electron-positron linear colliders an attractive choice for precision studies of new physics. +The International Linear Collider (ILC) [27] , is a proposed e+e− linear collider design with +simultaneously polarized e± beams and several stages of center-of-mass energies, i.e. at the SM-like +Higgs threshold ( √s = 250 GeV), at the top threshold ( √s = 350 GeV) and at about √s = 500 GeV +up to √s = 1 TeV with a maximum target integrated luminosity of L = 500 fb−1. The othe proposed +high-energy e+e− linear collider design is CLIC [28, 29] with an energy upgrade up to √s = 1.5, 3 +TeV and at least a polarized e−beam. An overview of the physics potential at future high-energy linear +colliders is given in [30]. ILC (CLIC ) gain advantage over the LHC in the possibility of exploiting +the polarisation of the beams crucial both at the high energy stages but also already at the first stage +of √s = 250 GeV [31, 32]. Although the invisible decay in BP3 is H → χ¯χ ≃ 4.8% and only a +low production cross section times branching ratio is predicted, we observe that the 2b+ missing ET +channel is observable with a = 3.99σ significance at an integrated luminosity of L = 5 ab−1. +4. +Conclusions and Outlook +We have studied dark matter phenomenology in a Two Higgs Doublet model with a complex +scalar singlet where the scalar singlet doesn’t obtain a vacuum expectation value. Benchmark sce- +narios consistent with all current experimental and cosmological constraints have been worked out. +Particular stringent bounds on the available parameter range for the couplings are set by bounds from +the direct detection. Due to very small rates of the heavy Higgs production, the dark matter candi- +dates will probably not be detectable via monojet or di-jet studies even at the HL-LHC. However, +at a high-energy linear collider with polarized beams and precise initial energy, such a dark matter +scenario is expected to be detectable. In addition, the option of direct dark matter pair production plus +3 + +Fig. 1.: Relic density and direct detection cross-section predicted by the model depending on the +DM mass mχ [10]. The parameter m2 +S has been varied in the range from 100-400000 GeV2 and fixed +tan β = 5. The other parameters are chosen as in benchmark scenario BP1, see Table I. +Parameters +BP1 +BP2 +BP3 +λ1 +0.23 +0.1 +0.23 +λ2 +0.25 +0.26 +0.26 +λ3 +0.39 +0.10 +0.2 +λ4 +-0.17 +-0.10 +-0.14 +λ5 +0.001 +0.10 +0.10 +m2 +12(GeV2) +-1.0×105 +-1.0×105 +-1.0×105 +λ′′ +1 +0.1 +0.1 +0.1 +λ′′ +3 +0.1 +0.1 +0.1 +λ′ +1 +0.042 +0.04 +2.0 +λ′ +2 +0.042 +0.001 +0.01 +λ′ +4 +0.1 +0.1 +0.1 +λ′ +5 +0.1 +0.1 +0.1 +tan β +4.9 +6.5 +6.5 +mh (GeV) +125.09 +125.09 +125.09 +mH(GeV) +724.4 +816.4 +821.7 +mA(GeV) +724.4 +812.6 +817.9 +mH± (GeV) +816.3 +816.3 +822.2 +mχ(GeV) +338.0 +76.7 +323.6 +Ωh2 +0.058 +0.119 +0.05 +σS I +p × 1010 (pb) +0.76 +0.052 +2.9 +σS I +n × 1010 (pb) +0.78 +0.054 +3.1 +Table I.: Relevant parameters of the benchmark +points used for the study [10]. +Processes +Cross section (in fb) at √s = 14 TeV +BP1 +BP2 +BP3 +h (ggF) +29.3×103 +29.3×103 +29.3×103 +H +22.61 +5.238 +6.632 +A +35 +8.58 +10.8 +h jj (VBF) +1.296×103 +1.265×103 +1.25×103 +H jj +1.843 +1.845 +0.56 +Ajj +2.885 +2.88 +40.91 +Wh +1.148×103 +1.133 +1.134 pb +WH +1.195×10−3 +1.11e-03 +1.199×10−3 +WA +4.3×10−4 +5.892e-04 +5.734×10−4 +Zh +880.8 +677.2 +697.9 +ZH +0.93 +0.2783 +0.3408 +ZA +3.999 +1.413 +1.689 +bbh +2534 +2541 +2541 pb +bbH +21.52 +17.92 +17.92 fb +bbA +23.39 +18.9 +19.04fb +t¯th +478.3 +477.1 +477.9 +t¯tH +0.1988 +0.06571 +0.7891 +t¯tA +0.2552 +0.08036 +0.09826 +H+H− +0.06603 +0.03033 +0.03416 +W±H∓ +102.4 +3.453 +4.145 +χ¯χ + 1 j +0.006356 +0.0681 +0.8819 +Table II.: The leading order (LO) cross section (in +fb), further details see [10]. +an ISR-photon is still under studies and offers another promising phenomenology. Further studies on +exploring the mixing angles in the Higgs sector to shed light on the dark matter behaviour is still +ongoing as well. +Acknowledgments +GMP would like to thank the organizers of IDM2022 for the great conference. JD and GMP +acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) +4 + +ProtonCross Section +10-43 +NeutronCrossSection +og +CrossSectionincm +upper limit +10-44 +1-o area +2-o area +10-45 +10-46 +10-47 +101 +102 +103 +104 +mx in GeV0.20 +Modelpredictions +Value measured by Planck +1-g area +0.15 +0.05 +0.00 +100 +200 +300 +400 +500 +600 +mx in GeVunder Germany’s Excellence Strategy EXC 2121 ”Quantum Universe”- 390833306. +References +[1] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rept. 516 (2012), +1-102 [arXiv:1106.0034 [hep-ph]]. +[2] B. Grzadkowski and P. Osland, Phys. Rev. D 82 (2010), 125026 [arXiv:0910.4068 [hep-ph]]. +[3] A. Drozd, B. Grzadkowski, J. F. Gunion and Y. 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Fujii, et al. arXiv:1801.02840. +5 + diff --git a/VdE_T4oBgHgl3EQfxxyN/content/tmp_files/load_file.txt b/VdE_T4oBgHgl3EQfxxyN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f32d17012c397db13e3e2dd71fe926dfee8923a --- /dev/null +++ b/VdE_T4oBgHgl3EQfxxyN/content/tmp_files/load_file.txt @@ -0,0 +1,509 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf,len=508 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='08314v1 [hep-ph] 19 Jan 2023 Dark Matter Phenomenology in 2HDMS Gudrid Moortgat-Pick*1,2, Juhi Dutta1, Cheng Li2, Merle Schreiber1,2, Sheikh F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Tabira1, Julia Ziegler1 1 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' of Theo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=', University of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany 2Deutsches Elektronen-Synchrotron DESY, Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' 85, 22607 Hamburg, Germany The constituents of dark matter are still an unresolved puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Several Beyond Standard Model (BSM) Physics offer suitable candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' In this study here we consider the Two Higgs Doublet model augmented with a complex scalar singlet (2HDMS) and focus on the dark matter phenomenology of 2HDMS with the complex scalar singlet as the dark matter candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' The parameter space allowed from existing experimental constraints from dark matter, flavour physics and collider searches has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' The discovery potential for such a 2HDMS at HL-LHC and at future e+e− colliders has been worked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Introduction Dark Matter (DM) remains an unsolved puzzle at the interface between particle physics and cos- mology, only 4-5% of the Universe are composed by ’known’ matter components, but about 25% is built of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Since the Standard Model (SM) does not accommodate a suitable DM candi- date, several Beyond Standard Model (BSM) extensions have been proposed to accommodate DM candidates ranging from scalar, fermion to vector candidates and with mass scales from below eV up to TeV particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' We concentrate in this contribution on thermal weakly interacting massive particles (WIMP) that is expected in the mass range of GeV up to TeV, accessible at future collider experiments at the LHC and a high-energy e+e− linear collider (ILC, CLIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Among popular BSM candidates are models with an extended Higgs sector such as the Two Higgs Doublet model (2HDM) [1], providing a dark matter candidate within the Inert Doublet model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Al- ternate models are such multi-Higgs models but extended via real or complex singlet scalars serving as dark matter candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Such extensions involving real scalar singlets have been extensively stud- ied [2–4] while complex scalar extensions to the 2HDM have also been recently studied in the context of modified Higgs sectors [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Such models have also the potential to explain the matter-antimatter asymmetry and to accommodate both inflation as well as gravitational waves phenomenology [6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' The parameter space of such extensions of the SM [8] gets strong constraints from direct searches for DM as well as from precision measurements of the 125 GeV SM-like Higgs boson and in particular from limits of both its visible as well as invisible branching ratios [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Extended Two Higgs Doublet Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 Symmetries We consider the CP-conserving softly broken Type II Two Higgs Doublet model augmented with a complex scalar singlet (2HDMS) [5] consistent with flavour changing neutral currents (FCNCs) at tree-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' It allows for the presence of the mixing term between the two Higgs doublets, Φ1 and Φ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=', m2 12, while the explicit Z2 breaking terms are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' The complex scalar singlet S is stabilised by a Z′ 2 symmetry such that S is odd under Z′ 2 while the SM fields are even under the new Z′ 2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' The fields Φ1 and S are even under Z2 while Φ2 is odd under Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' We consider the case where Z′ 2 remains unbroken both explicitly and dynamically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' the scalar 1 singlet doesn’t obtain a vacuum expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Therefore, the scalar potential V with a softly broken Z2- and a conserved Z′ 2 symmetry is V = V2HDM + VS, where, the softly broken Z2-symmetric 2HDM potential is: V2HDM = m2 11Φ† 1Φ1 + m2 22Φ† 2Φ2 − (m2 12Φ† 1Φ2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='c) + λ1 2 (Φ† 1Φ1)2 + λ2 2 (Φ† 2Φ2)2 (1) +λ3(Φ† 1Φ1)(Φ† 2Φ2) + λ4(Φ† 1Φ2)(Φ† 2Φ1) + [λ5 2 (Φ† 1Φ2)2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='c] (2) and the Z′ 2-symmetric singlet potential, VS , is VS = m2 S S ∗S + ( m2′ S 2 S 2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='c) + ( λ′′ 1 24S 4 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='c) + ( λ′′ 2 6 (S 2S ∗S ) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='c) + λ′′ 3 4 (S ∗S )2 (3) +S ∗S [λ′ 1Φ† 1Φ1 + λ′ 2Φ† 2Φ2] + [S 2(λ′ 4Φ† 1Φ1 + λ′ 5Φ† 2Φ2) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' (4) The doublet fields have the components Φ1 = (h+ 1 , 1√ 2(v1 + h1 + ia1))T, Φ2 = (h+ 2 , 1√ 2(v2 + h2 + ia2))T, S = 1√ 2(hs + ias) and tan β = v2 v1 is the ratio of the up-type and down-type Higgs doublet vevs v1,2 (with v(= v2 1 + v2 2) ≃ 246 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Under the assumption that the complex singlet scalar does not develop a vev —for this study imposed—, the Higgs sector, after EWSB, remains the same as in 2HDM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='e, consisting of two CP-even neutral scalar Higgs particles h, H, a pseudoscalar Higgs A and a pair of charged Higgs particles H± [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' All Higgs-dark-matter portal couplings are explicitly given in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='2 Theoretical and Phenomenological Constraints The Sylvester’s criterion and copositivity [11, 12] has been applied to guarantee boundedness from below for the Higgs potential, leading to constraints on all coupling parameters λi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' , 5, λ′ j, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' , 5, λ′′ k , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' , 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' The mass of the lightest CP-even Higgs particle mh = 125 GeV has been chosen to be in concordance with the measured Higgs state via HiggsSignals [13] and collider constraints from LEP and LHC have been applied for the heavy Higgs states via HiggsBounds [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' The branching ratio BR(h → χχ) < 0, 11 (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='19), fulfilling the limits from ATLAS (CMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Elec- troweak precision constraints on STU parameters have been taken into account as well as constraints from flavour physics BR(b → sγ), BR(Bs → µ+µ−), using SPheno [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' ∆(gµ − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Concerning the dark matter particle, the bounds on the relic density from PLANCK measurements, Ωh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='119 [16], as well as constraints from direct detection (XENON-1T [17] ) and indirect detection (FERMI-LAT [18]) experiments have been applied using micrOMEGAs [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 Benchmark Points This model has been implemented using SARAH [20] code implemented into SPheno for the spectrum generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' In order to calculate collider observables the code chain Madgraph [21] -Pythia [22] -Delphes [23] -Madanalysis [24] has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' As can be seen from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1a) and b) the mass of the dark matter particle χ as well as the coupling λ′ 2 get strongest constraints from the direct detection search from XENON-1T: in the shown example where the heavier Higgs particles are about 725 GeV, the mass mχ ∼ 338 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Scanning the available parameter space allowed to specify different benchmark areas, see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' BP1 and BP3 are very similar, however, they differ significantly in the couplings λ′ 1, λ′ 2 and λ3 leading to different collider phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='2 Collider Phenomenology In this section, we discuss the possible signals of this model at HL-LHC and future e+e− colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' As already mentioned, the invisible decay of the heavy Higgs into the dark matter candidate is a 2 source of missing energy at colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Therefore, the direct production of heavy Higgs bosons and consequent decay of the Higgs to χ along with visible SM particles can give rise to distinct signatures for this scenario as opposed to the 2HDM like scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' We investigate these possibilities and their prospects in the context of √s = 14 TeV LHC at the targeted integrated luminosity of 3-4 ab−1 and of future e+e− colliders (ILC,CLIC) up to √s = 3 TeV and integrated luminosities of 5 ab−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 Prospects at LHC The main processes contributing to neutral Higgs production at the LHC are gluon fusion (me- diated by the top quark loop), vector boson fusion (VBF), associated Higgs production (Vhi), b¯bhi, t¯thi [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' For the charged Higgs pair, the possible production channels are H+H− and W±H∓ [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' At the LHC Run 3 at √s = 14 TeV, all possible Higgs production processes (including SM and BSM Higgses) are summarised in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' In the presence of the heavy Higgs H decaying into two dark matter candidates, one gets invisible momentum in the final state and one can look into the following final states: a) 1j (ISR)+missing ET [25] b) 2j+ missing ET [26] We estimate the significance for the mono-jet and VBF channels using the cuts from an existing cut-and-count analyses performed in [4] for √s = 14 TeV LHC, further details see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' For a) we achieve a cut efficiency for the signal in BP3 of about ∼ 18% and we obtain a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='111σ excess at 3ab−1 using gluon fusion production channel (at leading order (LO)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' For b) we get a signal efficiency of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='5% for BP3 and a signal significance of about ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='2 σ at 3 ab−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Therefore, we observe that due to the small invisible branching ratio and heavy Higgs masses ∼ 820 GeV (and hence small production cross section) in BP3, the final states will be inaccessible at the upcoming HL-LHC run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='2 Prospects at a high-energy e−e+ Linear Collider (ILC, CLIC) The cleaner environment and lower background along the beam line compared to hadron colliders make the electron-positron linear colliders an attractive choice for precision studies of new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' The International Linear Collider (ILC) [27] , is a proposed e+e− linear collider design with simultaneously polarized e± beams and several stages of center-of-mass energies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' at the SM-like Higgs threshold ( √s = 250 GeV), at the top threshold ( √s = 350 GeV) and at about √s = 500 GeV up to √s = 1 TeV with a maximum target integrated luminosity of L = 500 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' The othe proposed high-energy e+e− linear collider design is CLIC [28, 29] with an energy upgrade up to √s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='5, 3 TeV and at least a polarized e−beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' An overview of the physics potential at future high-energy linear colliders is given in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' ILC (CLIC ) gain advantage over the LHC in the possibility of exploiting the polarisation of the beams crucial both at the high energy stages but also already at the first stage of √s = 250 GeV [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Although the invisible decay in BP3 is H → χ¯χ ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='8% and only a low production cross section times branching ratio is predicted, we observe that the 2b+ missing ET channel is observable with a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='99σ significance at an integrated luminosity of L = 5 ab−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Conclusions and Outlook We have studied dark matter phenomenology in a Two Higgs Doublet model with a complex scalar singlet where the scalar singlet doesn’t obtain a vacuum expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Benchmark sce- narios consistent with all current experimental and cosmological constraints have been worked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Particular stringent bounds on the available parameter range for the couplings are set by bounds from the direct detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Due to very small rates of the heavy Higgs production, the dark matter candi- dates will probably not be detectable via monojet or di-jet studies even at the HL-LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' However, at a high-energy linear collider with polarized beams and precise initial energy, such a dark matter scenario is expected to be detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' In addition, the option of direct dark matter pair production plus 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=': Relic density and direct detection cross-section predicted by the model depending on the DM mass mχ [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' The parameter m2 S has been varied in the range from 100-400000 GeV2 and fixed tan β = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' The other parameters are chosen as in benchmark scenario BP1, see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Parameters BP1 BP2 BP3 λ1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='23 λ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='26 λ3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='2 λ4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='14 λ5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='10 m2 12(GeV2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='0×105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='0×105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='0×105 λ′′ 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 λ′′ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 λ′ 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='0 λ′ 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='01 λ′ 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 λ′ 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 tan β 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='5 mh (GeV) 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='09 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='09 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='09 mH(GeV) 724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='4 816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='4 821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='7 mA(GeV) 724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='4 812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='6 817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='9 mH± (GeV) 816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='3 816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='3 822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='2 mχ(GeV) 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='7 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='6 Ωh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='05 σS I p × 1010 (pb) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='052 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='9 σS I n × 1010 (pb) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='054 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='1 Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=': Relevant parameters of the benchmark points used for the study [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Processes Cross section (in fb) at √s = 14 TeV BP1 BP2 BP3 h (ggF) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='3×103 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='3×103 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='3×103 H 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='61 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='238 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='632 A 35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='58 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='8 h jj (VBF) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='296×103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='265×103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='25×103 H jj 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='843 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='56 Ajj 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='885 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='88 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='91 Wh 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='148×103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='133 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='134 pb WH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='195×10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='11e-03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='199×10−3 WA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='3×10−4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='892e-04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='734×10−4 Zh 880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='8 677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='2 697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='9 ZH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='2783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='03416 W±H∓ 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='453 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='145 χ¯χ + 1 j 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='006356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='0681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='8819 Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' : The leading order (LO) cross section (in fb), further details see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' an ISR-photon is still under studies and offers another promising phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Further studies on exploring the mixing angles in the Higgs sector to shed light on the dark matter behaviour is still ongoing as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Acknowledgments GMP would like to thank the organizers of IDM2022 for the great conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' JD and GMP acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) 4 ProtonCross Section 10-43 NeutronCrossSection og CrossSectionincm upper limit 10-44 1-o area 2-o area 10-45 10-46 10-47 101 102 103 104 mx in GeV0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='20 Modelpredictions Value measured by Planck 1-g area 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content='00 100 200 300 400 500 600 mx in GeVunder Germany’s Excellence Strategy EXC 2121 ”Quantum Universe”- 390833306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Branco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Ferreira, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE_T4oBgHgl3EQfxxyN/content/2301.08314v1.pdf'} +page_content=' Lavoura, M.' metadata={'source': 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+1,1799 @@ +A Comprehensive Survey to Dataset Distillation +Shiye Lei∗ +Dacheng Tao† +Abstract +Deep learning technology has unprecedentedly developed in the last decade and has become the +primary choice in many application domains. This progress is mainly attributed to a systematic +collaboration that rapidly growing computing resources encourage advanced algorithms to deal +with massive data. However, it gradually becomes challenging to cope with the unlimited growth +of data with limited computing power. To this end, diverse approaches are proposed to improve +data processing efficiency. Dataset distillation, one of the dataset reduction methods, tackles the +problem via synthesising a small typical dataset from giant data and has attracted a lot of attention +from the deep learning community. +Existing dataset distillation methods can be taxonomised +into meta-learning and data match framework according to whether explicitly mimic target data. +Albeit dataset distillation has shown a surprising performance in compressing datasets, it still +possesses several limitations such as distilling high-resolution data. This paper provides a holistic +understanding of dataset distillation from multiple aspects, including distillation frameworks and +algorithms, disentangled dataset distillation, performance comparison, and applications. Finally, +we discuss challenges and promising directions to further promote future studies about dataset +distillation. +1 +Introduction +During the past few decades, deep learning has achieved remarkable success in a wide range of applica- +tions, including computer vision [Krizhevsky et al., 2017, He et al., 2016], neural language processing +[Vaswani et al., 2017], autonomous vehicles [Grigorescu et al., 2020], protein structure prediction +[Jumper et al., 2021], etc. An important reason for this success is the powerful computing resource, +which allows deep neural networks to directly tackle giant datasets and bypass complicated manual +feature extraction, which causes potential loss of data information. For example, the powerful large +language model GPT-3 contains 175 billion parameters and is trained on 45 terabyte of text data with +thousands of GPUs [Brown et al., 2020]. However, massive data are generated every day [Sagiroglu +and Sinanc, 2013], which poses a significant threat to training efficiency and data storage, and deep +learning might reach a bottleneck due to the mismatch between the volume of data and computing +resources [Strubell et al., 2019]. Recently, many methods have been proposed to improve the training +efficiency in deep learning from several perspectives as below: +• Quantisation: this approach sacrifices the data byte during the training process for acceleration +[Gupta et al., 2015, Micikevicius et al., 2018]. +• Model pruning: this approach removes trainable parameters that has few influences on the final +performance [Sun et al., 2017, Lym et al., 2019]. +• Optimisation: this approach designs the training algorithms for fast convergence [Zhang et al., +2016] or less memory cost [Bernstein et al., 2018, Karimireddy et al., 2019]. +• Dataset reduction: this approach generates few representative data to constitute a new training +set. Depending on whether the generated data are natural or synthetic, data compression can +be classified into coreset selection [Mirzasoleiman et al., 2020, Coleman et al., 2020] and dataset +distillation [Wang et al., 2018]. +∗S. Lei is associated with School of Computer Science, Faculty of Engineering, The University of Sydney, Darlington +NSW 2008, Australia. Email: slei5230@uni.sydney.edu.au. +†D. Tao is associated with JD Explore Academy, China and The University of Sydney, Australia. +Email: +dacheng.tao@gmail.com. +1 +arXiv:2301.05603v1 [cs.LG] 13 Jan 2023 + +Figure 1: An illustration of dataset distillation. The models trained on the large original dataset and +small synthetic dataset have comparable performance on the test set. +The above narrative shows that efficient deep learning is an extensive task and outsides the scope of +this paper, and readers can refer to [Sze et al., 2017, Menghani, 2021] for more details. In this survey, +we focus on dataset distillation to improve training efficiency by synthesising training data. To better +appreciate dataset distillation, we briefly introduce other cognate methods in dataset reduction. For +the support vector machine (SVMs), its hyperplane is solely determined by “support vectors”, and +removing all other points in the training dataset does not have an influence on the convergence result +[Cortes and Vapnik, 1995]. Therefore, the selection of “support vectors” is a favourable method to +reduce the training burden for SVMs [Nalepa and Kawulok, 2019]. When the scenario expands to deep +learning, the problem of selecting “support vectors” generalises to the well-known coreset selection: +the algorithms select a few prototype examples from the original dataset as the coreset, and then the +model is solely trained on the small coreset to save training costs whilst avoiding large performance +drops. +Various coreset selection algorithms have been successfully leveraged in many applications +like active learning [Wei et al., 2015], continual learning [Aljundi et al., 2019], neural architecture +search [Shim et al., 2021], robust learning [Mirzasoleiman et al., 2020], etc. However, elements in the +coreset are unmodified and constrained by the original data, which considerably restricts the coreset’s +expressiveness, especially when the coreset budget is limited. +Recently, a novel approach, dataset +distillation (DD) 1 has attracted growing attention from the deep learning community. +Different +from coreset selection, dataset distillation removes the restriction of uneditable elements and carefully +modifies a small number of examples to preserve more information, as shown in Figure 4 for synthetic +examples. By distilling the knowledge of the large original dataset into the small synthetic set, models +trained on the synthetic set can acquire a comparable generalisation performance. A general illustration +for dataset distillation is presented in Figure 1. +Due to the property of extreme high-dimension in the deep learning regime, the data information +is hardly disentangled to specific concepts, and thus distilling numerous high-dimensional data into +a few points is not a trivial task. +Based on the objectives applied to mimic target data, dataset +distillation methods can be grouped into meta-learning framework and data match framework, and +these techniques in each framework can be further classified in a much more detailed manner. In the +meta-learning framework, the distilled data are considered hyperparameters and optimised according +to the validation risk on the target data [Maclaurin et al., 2015, Wang et al., 2018]. As for the data +match framework, it updates distilled data by imitating the influence of target data on model training +from parameter or feature space [Zhao et al., 2021, Cazenavette et al., 2022a, Zhao and Bilen, 2021a]. +Figure 3 presents these different categories of DD algorithms in a tree diagram. +Apart from directly considering the synthetic examples as optimisation objectives, some works +design a proxy model consisting of latent codes and decoders to generate high-informative examples +and resort to learning the latent codes and decoders. For example, Such et al. [2020] employ a network +1Dataset distillation is also referred to as dataset condensation in some literature. +2 + +training +test +training +testFigure 2: The schematic structure of dataset distillation and the relationship between the adjacent +sections. The body of this survey mainly contains fundamentals of dataset distillation, the taxonomy +of distillation schemes, types of disentangled DD, distillation algorithms, performance comparison, +applications, challenges, and future directions. Note that ‘Section’ is abbreviated as ‘Sec.’ in this +figure. +Dataset Distillation +Meta-learning Framework +Back-propagation Through Time +DD [Wang et al., 2018] +LD [Bohdal et al., 2020] +SLDD [Sucholutsky and Schonlau, 2021] +GTN [Such et al., 2020] +Deng and Russakovsky [2022] +Kernel Ridge Regression +KIP [Nguyen et al., 2020, 2021] +FRePo [Zhou et al., 2022] +RFAD [Loo et al., 2022] +Data Match Framework +Gradient Match +DC [Zhao et al., 2021] +DCC [Lee et al., 2022b] +IDC [Kim et al., 2022] +Trajectory Match +MTT [Cazenavette et al., 2022a] +FTD [Du et al., 2022] +TESLA [Cui et al., 2022b] +Haba [Liu et al., 2022c] +Distribution Match +DM [Zhao and Bilen, 2021a] +CAFE [Wang et al., 2022] +GAN [Zhao and Bilen, 2022] +KFS [Lee et al., 2022a] +Figure 3: Tree diagram for different categories of dataset distillation algorithms +to generate highly informative data from noise and optimise the network via meta-gradient. Zhao +and Bilen [2022] optimise the vectors and put them into the generator of a well-trained generative +adversarial network (GAN) to produce the synthetic examples. Moreover, [Kim et al., 2022, Deng and +Russakovsky, 2022, Liu et al., 2022c, Lee et al., 2022a] learn a couple of latent codes and decoders, +and then the synthetic data is generated according to the different combinations of latent codes and +decodes. With this disentanglement of synthetic data, the compression ratio of DD can be further +decreased, and the performance can also be improved due to the intra-class information extracted by +latent codes. +In this paper, we present a comprehensive survey on dataset distillation, and the main objectives +of this survey are to (1) present a clear and systematic overview of dataset distillation; (2) review the +recent progress and state-of-the-art algorithms and discuss various applications in dataset distillation; +(3) give a comprehensive performance comparison w.r.t. different dataset distillation algorithms; and +(4) provides a detailed discussion on the limitation and promising directions of dataset distillation +to help future studies prosper. +Recently, Larasati et al. [2022] publish a short review on dataset +distillation. However, they only provide a general overview of DD and focus on the aspect of appli- +cations. Different from Larasati et al. [2022], our paper gives a systematic, comprehensive survey on +dataset distillation from a wide aspect of distillation schemes, algorithms, applications, limitations, +3 + +Performance Comparison +Gradient Match +Back-propagation +ThroughTime +Problem Setup +Trajectory Match +Kernel Ridge +Regression +Meta-learning +Framework +DistributionMatch +Data Match +Neural +Continual +Framework +Architecture +Learning +Code-basedDD +Search +Disentangled DD +Privacy +Federated +Decoder-basedDD +Protection +Learning +Adversarial +Other +Code-decoder DD +Challenges +Directions +Robustness +ApplicationsFigure 4: Example synthetic images distilled from CIFAR-10/100 and Tiny ImageNet by matching +training trajectory [Cazenavette et al., 2022a]. +and promising directions. +The rest of this paper is organised as follows. We first give some notations of dataset distillation in +Section 2. DD methods under meta-learning framework is described and comprehensively analysed in +Section 3, and a description of the data match framed follows in Section 4. Section 5 discusses different +types of disentangled dataset distillation. +We then report the performance of various distillation +algorithms in Section 6. Applications with dataset distillation are shown in Section 7, and we discuss +challenges and future directions in Section 8. +2 +Problem Setup +Before introducing dataset distillation, we first define some notations used in this paper. For a dataset +T = {(xi, yi)}m +i=1, where xi ∈ Rd, d is the dimension of input data, yi is the label, we assume that +(xi, yi) are both independent and identically distributed (i.i.d.) random variables drawn from the data +generating distribution D. We employ fθ to denote the neural network parameterized by θ, and fθ(x) +is the prediction or output of fθ on the point x. For the training algorithm alg, let alg(T , θ(0)) denote +the learned parameters returned by leveraging alg on the dataset T with the initialised parameter +θ(0). We omit θ(0) and use alg(T ) if there is no ambiguity for the sake of simplicity. Then, the +model f trained on the dataset T can be denoted as falg(T ). Moreover, we also define the loss between +prediction and ground truth as ℓ(fθ(x), y), and the expected risk in terms of θ is defined as +RD(θ) = E(x,y)∼D [ℓ (fθ (x) , y)] . +(1) +Since the data generating distribution D is unknown, evaluating the expected risk RD is not practical. +Therefore, it is a practical way to estimate the expected risk by the empirical risk RT , which is defined +as +RT (θ) = E(x,y)∼T [ℓ (fθ (x) , y)] = 1 +m +m +� +i=1 +ℓ (fθ (xi) , yi) . +(2) +In the deep learning paradigm, gradient descent is the dominant algorithm to train a neural network +by minimising the empirical risk step by step: the network’s parameters are initialised with θ(0), then +the parameter is iteratively updated according to the gradient of empirical loss: +θ(k+1) = θ(k) − ηg(k) +T , +(3) +where η is the learning rate and g(k) +T += ∇θ(k)RT (θ(k)) is the gradient. Because deep learning models +are commonly extremely over-parameterized, i.e., the number of model parameters overwhelms the +number of training examples, the empirical risk readily reaches zero. In this case, the generalisation +error, which measures the difference between the expected risk and the empirical risk, can be solely +equal to the expected risk, which is reflected by test loss or test error in practical pipelines. +Given a target dataset (source training dataset) T = {(xi, yi)}m +i=1, the objective of dataset distil- +lation is to extract the knowledge of T into a small synthetic dataset S = {(sj, yj)}n +j=1, where n ≪ m, +4 + +(a) +(b) +(c) +Figure 5: +(a) Meta-gradient back-propagation in BPTT [Zhou et al., 2022]; (b) Meta-gradient back- +propagation in kernel ridge regression; (b) Meta-gradient back-propagation in gradient match. +and the model trained on the small distilled dataset S can achieve a comparable generalisation per- +formance to the large original dataset T : +E(x,y)∼D +� +ℓ +� +falg(T ) (x) , y +�� +≃ E(x,y)∼D +� +ℓ +� +falg(S) (x) , y +�� +. +(4) +Because the training algorithm alg +� +S, θ(0)� +is both determined by the training set S and the +initialised parameter θ(0), many dataset distillation algorithms will take expectation on S w.r.t. θ(0) +in order to improve the robustness of the distilled dataset S to different parameter initialisation. In +the following narrative, we will omit this expectation w.r.t. initialisation for the sake of simplicity. +3 +Meta-learning Framework +From the meta-learning perspective, the distilled data are treated as hyper-parameters and the objec- +tive of DD is to learn the distilled data for improving the generalisation performance of models. By +assuming RD (alg(S)) ≃ RT (alg(S)), we employ the target dataset T as the validation set in terms +of the model alg(S). To this end, the dataset distillation can be formulated to a bi-level optimisation +problem as below: +S∗ = arg min +S RT (alg(S)) +(outer loop) +(5) +subject to +alg(S) = arg min +θ RS (θ) +(inner loop) +(6) +The inner loop optimises the model parameters based on the synthetic dataset and is often realised +by gradient descent for neural networks or regression for kernel method. During the outer loop, the +synthetic set is updated by minimising the model’s risk in terms of the target dataset. With the nested +loop, the synthetic dataset gradually converges to one of the optima. +3.1 +Back-propagation Through Time Approach +As shown in the above formulation of bi-level optimisation, the objective function of DD can be directly +defined as the meta-loss of +L(S) = RT (alg (S)) . +(7) +Then the distilled dataset is updated by S = S −α∇SL(S) with the step size α. However, as the neural +network is adopted in the distillation process, the training algorithm alg on the model parameter θ is +iterative (Eq. 3) and yields a series of intermediate parameter states of {θ(0), θ(1), · · · , θ(T )} for the +inner loop (Eq. 6). Hence, back-propagation through time (BPTT) is required to recursively compute +the meta-gradient ∇SL(S): +∇SL(S) = ∂L +∂S = +∂L +∂θ(T ) +�k=T +� +k=0 +∂θ(T ) +∂θ(k) · ∂θ(k) +∂S +� +, +(8) +and +∂θ(T ) +∂θ(k) = +T +� +i=k+1 +∂θ(i) +∂θ(i−1) . +(9) +5 + +As shown in Figure 5(a), due to the requirement of unrolling the recursive computation graph, BPTT +is both computing and memory expensive, which also severely affects the final distillation performance. +To alleviate the inefficiency in unrolling the long parameter path of {θ(0), · · · , θ(T )}, Wang et al. +[2018] solely adopt a single-step optimisation w.r.t. the model parameter from θ(0) to θ(1), and the loss +is computed based on θ(1) and the target dataset T . Therefore, the distilled data s and learning rate η +can be efficiently updated via the short-range BPTT. Unlike freezing distilled labels, Sucholutsky and +Schonlau [2021] extend Wang et al. [2018] by learning a soft-label in the synthetic dataset S, i.e., the +label y in the synthetic dataset is also trainable for better information compression. Similarly, [Bohdal +et al., 2020] also extend the standard example distillation to label distillation by solely optimising +the labels of synthetic datasets. Moreover, they provide improvements on the efficiency of long inner +loop optimisation via (1) iteratively updating the model parameters θ and the distilled labels y and +(2) employing ridge regression to update the solution of the last linear layer of networks to avoid +second-order gradient computation. Albeit the BPTT framework has been shown to underperform +other algorithms, Deng and Russakovsky [2022] empirically demonstrate that adding momentum term +and longer unrolled trajectory (200 steps) in the inner loop optimisation can considerably increase the +distillation performance. +3.2 +Kernel Ridge Regression Approach +Albeit multi-step gradient descent can gradually approach the optimal network parameters in terms +of the synthetic dataset during the inner loop, this iterative algorithm makes the meta-gradient back- +propagation highly inefficient, as shown in BPTT. Considering the existence of closed-form solutions +in kernel regression regime, Nguyen et al. [2020] replace the neural network in the inner loop with +a kernel model, which bypasses the recursive back-propagation of meta-gradient. For the regression +model f(x) = w⊤ψ(x), where ψ(·) is a non-linear mapping and the corresponding kernel is K(x, x′) = +⟨ψ(x), ψ(x′)⟩, there exists a closed-form solution for w when the regression model is trained on S with +kernel ridge regression (KRR): +w = ψ(Xs)⊤ (KXsXs + λI)−1 ys, +(10) +where KXsXs = [K(si, sj)]ij ∈ Rn×n is called the kernel matrix or Gram matrix associated to K and +the dataset S, and λ > 0 is a fixed regularization parameter [Petersen et al., 2008]. Therefore, the +mean square error (MSE) of predicting T with the model trained on S is +L(S) = 1 +2 +���yt − KXtXs (KXsXs + λI)−1 ys +��� +2 +, +(11) +where KXtXs = [K(xi, sj)]ij ∈ Rm×n. Then the distilled dataset is updated via the meta-gradient of +the above loss. Due to the closed-form solution in KRR, θ does not require an iterative update and the +backward pass of gradient thus bypasses the recursive computation graph, as shown in Figure 5(b). +In the KRR regime, the synthetic dataset S can be directly updated by back-propagating the +meta-gradient through the kernel function. Albeit this formulation is solid, this algorithm is designed +in the KRR scenario and only employs simple kernels, which causes performance drops when the +distilled dataset is transferred to train neural networks. Jacot et al. [2018] propose the neural tangent +kernel (NTK) theory that proves the equivalence between training infinite-width neural networks and +kernel regression. +With this equivalence, Nguyen et al. [2021] employ the infinite-width networks +as the kernel for dataset distillation, which narrows the gap between the scenarios of KRR and deep +learning. However, every entry in the kernel matrix requires to be calculated separately, and computing +the kernel matrix KXsXt owns the complexity of O(|T ||S|), which is severely inefficient for large- +scale datasets. To tackle this problem, Loo et al. [2022] replace NTK kernel with neural network +Gaussian process (NNGP) kernel that only considers the training dynamic of last-layer classifier for +speed up. +With this replacement, the random feature ψ(x) and ψ(s) can be explicitly computed +via multiple sampling from the Gaussian process, and thus the kernel matrix computation can be +decomposed into random feature calculation and random feature matrix multiplication. Because the +matrix multiplication require negligible amounts of time for small distilled datasets, the complexity of +kernel matrix computation degrades to O(|T | + |S|). A similar efficient method is also proposed by +Zhou et al. [2022], which employs the feature extractor in neural networks as the feature map ψ and +accordingly optimises the distilled data via KRR. +6 + +3.3 +Discussion +From the loss surface perspective [Li et al., 2018], the meta-learning framework of decreasing RT (alg (S)) +can be considered to mimic the local minima of target data with the distilled data. However, the loss +landscape w.r.t. parameters are closely related to the network architecture, while only one type of +small network is used in BPTT approach. There is consequently a moderate performance drop when +the distilled dataset is employed to train other complicated networks. Moreover, long unrolled tra- +jectory and second-order gradient computation are also two key challenges for BPTT approach and +hinder its efficiency. KRR approach compensate for these shortcomings by replacing networks with +the non-parametric kernel model which admits closed-form solution. Albeit KRR is non-parametric +and does not involve neural networks during the distillation process. However, previous research has +shown that the training dynamic of neural networks is equal to the kernel method when the width of +networks becomes infinite [Jacot et al., 2018], which partially guarantees the feasibility of the kernel +regression approach and explains its decent performance when transferred to the neural networks. +4 +Data Match Framework +Albeit direct information extraction is not feasible, the information distillation can be achieved by +implicitly matching the by-products of target data and synthetic data from different aspects. The +objective function of data match can be summarised as follows. +L(S) = +T +� +k=0 +D +� +φ(S, θ(k)), φ(T , θ(k)) +� +, +(12) +where D(·, ·) is a distance function, and φ(·) maps the dataset S or T to other informative spaces, +such as gradient, feature, and parameter spaces. +Compared to the aforementioned meta-learning framework, the data match loss does not only focus +on the final parameter alg(S) but also supervise the intermediate states, as shown in the sum operation +�T +k=0. By this, the distilled data can better imitate the influence of target data on training networks +at different training stages. +4.1 +Gradient Match Approach +To achieve comparable generalisation performance, the intuition is imitate the effect on model param- +eters, i.e., matching the training trajectories introduced by S and T . With a fixed parameter initiali- +sation, the training trajectory of {θ(0), θ(1), · · · , θ(T )} is equal to a series of gradients {g(0), · · · , g(T )}. +Therefore, matching the gradients induced by S and T is a convincing proxy to mimic the influence +on model parameters [Zhao et al., 2021], and the objective function can be formulated as +L(S) = +T −1 +� +k=0 +D +� +g(k) +S , g(k) +T +� +, +(13) +where g(k) +S +and g(k) +T +denote the gradient w.r.t. model parameters generated by S and T in the k-th +training epoch, respectively. It is worth noting that these two gradients are induced on the same +parameter θ(k) to be cohesive with the bi-level optimisation. +Concretely, the gradient match is +class-wise: L(k) = �C +c=1 D +� +∇θ(k)R +� +Sc, θ(k)� +, ∇θ(k)R +� +Tc, θ(k)�� +, where c is the class index and Tc +denotes the examples belong to the c-th class. According to Lee et al. [2022b], the class-wise gradi- +ent match pays much attention to the class-common features and overlooks the class-discriminative +features in the target dataset, and the distilled synthetic dataset S does not possess enough class- +discriminative information, especially when the target dataset is fine-grained, i.e., class-common +features are the dominant. +Based on this finding, they propose a improved objective function of +L(k) = D +��C +c=1 ∇θ(k)R +� +Sc, θ(k)� +, �C +c=1 ∇θ(k)R +� +Tc, θ(k)�� +to better capture constractive signals be- +tween different classes. A similar approach is proposed by Jiang et al. [2022], which considers both +intra-class and inter-class gradient match. To alleviate easy overfitting on the small dataset S, Kim +et al. [2022] propose to do inner loop optimisation on the target dataset T instead of S. +7 + +(a) Trajectory match +(b) Distribution match +Figure 6: +(a) An illustration of trajectory match [Cazenavette et al., 2022a]; (b) Distribution match +approach can more comprehensively cover the data distribution in feature space compared to gradient +match. +In spite of data augmentation brings a large performance increase, conducting augmentation on +distilled datasets has no improvement on the final test accuracy, because the synthetic images have +different characteristics compared to natural images and also are not optimised under the supervision +of various transformations. To leverage data augmentation on synthetic datasets. Zhao and Bilen +[2021b] design data siamese augmentation (DSA) that homologously augments the distilled data and +the target data during the distillation process. In DSA, the augmented form of distilled data has a +consistent correspondence w.r.t. the augmented form of the target data, which permits the knowledge +transfer from the transformation of target images to the corresponding transformation of synthetic +images. Consequently, the augmented synthetic images also possess meaningful characteristics of the +natural images. Due to its superior compatibility, DSA has been widely equipped in many data match +methods. +4.2 +Trajectory Match Approach +Unlike circuitously matching the gradients, Cazenavette et al. [2022a] directly matches the long-range +training trajectory between the target dataset and the synthetic dataset. Concretely, they collect the +expert training trajectory w.r.t. the target dataset T into the buffer in advance, and then ingredients +in the buffer are randomly selected to initialise the networks for training S. +After collecting the +trajectory of S, the synthetic dataset is updated by matching their trajectory, as shown in Figure 6(a). +The objective loss of matching trajectory is defined as +L = ∥θ(k+N) − θ(k+M) +T +∥2 +2 +∥θ(k) +T +− θ(k+M) +T +∥2 +2 +, +(14) +where θT denote the target parameter by training the model on T and is stored in the buffer, and +θk+N are the parameter by training the model on S for N epochs with the initialisation of θ(k) +T . The +denominator in the loss function is for normalisation. +Albeit trajectory match received empirical success, Du et al. [2022] propose that there exists an +accumulated trajectory error when matching the trajectory due to the segmented alignment from θ(k) +T +to θ(k+M) +T +, and they alleviate this by adding random noise when initialise the distilled network to +improve robustness w.r.t. the accumulated trajectory error. +Compared to matching gradients, while trajectory match side-steps second-order gradient compu- +tation, it, unfortunately, requires unrolling N SGD updates during the meta-gradient backpropagation +as the existence of θ(k+N). The unrolled gradient computation significantly increases the memory +burden and impedes scalability. By disentangling the meta-gradient w.r.t. synthetic examples into +two passes, Cui et al. [2022b] greatly reduce the memory required by trajectory match. Motivated +by knowledge distillation [Gou et al., 2021], they also propose to assign soft-label to synthetic exam- +ples with pre-trained models in the buffer, and the soft-label helps learn intra-class information and +consequently improves distillation performance. +8 + +二 +g(k +0(k+2) +(I+)0 +θ(k+N) +q(k+1) +o(k+2) +(N++)0 +0(t+M)(a) Non-disentangled dataset distillation +(b) Disentangled dataset distillation +Figure 7: The schematic diagrams of non-disentangled dataset distillation and disentangled dataset +distillation. +4.3 +Distribution Match Approach +Albeit the parameter-wise match shows a satisfying performance, Zhao and Bilen [2021a] visualise +the distilled data in 2-dimension and reveal that there is a large distribution discrepancy between +the distilled data and the target data. In other words, the distilled dataset can not comprehensively +cover the data distribution in feature space, as shown in Figure 6(b). Based on this discovery, they +propose to match the synthetic and target data from the distribution perspective for dataset distillation. +Concretely, they employ the pre-trained feature extractor ψv with the parameter v to achieve the +mapping from input space to feature space. The synthetic data is optimised according to the objective +function +L(S) = +C +� +c=1 +∥ψv(Sc) − ψv(Tc)∥2, +(15) +where c denotes the class index. Though this distribution match drops bi-level optimisation for boost- +ing, it empirically underperforms the above gradient and trajectory match approaches. Wang et al. +[2022] improve the distribution alignment from several aspects: (1) using multiple-layer features other +than only the last-layer inputs for matching; (2) proposing the discrimination loss to enlarge the class +distinction of synthetic data; and (3) recalling the bi-level optimisation that updates S with different +model parameters for better generalisation. Besides, Lee et al. [2022a] analyse that the subsampling +in distribution match is biased, and they propose to use full batch training to mitigate this problem. +4.4 +Discussion +The gradient match approach can be considered as a short-range parameter match, while its back- +propagation requires second-order gradient computation. Albeit the trajectory match approach make- +ups for this imperfection, the long-range trajectory introduces recursive computation graph during +meta-gradient back-propagation. +Different from matching in parameter space, distribution match +approach employ the feature space as the match proxy and also bypass second-order gradient com- +putation. +Albeit distribution match has advantages in scalability w.r.t. +high-dimensional data, it +empirically underperforms trajectory match, which might be attributed to the mismatch between the +comprehensive distribution and decent distillation performance. In detail, distribution match achieves +DD by mimicking features of target data, so that the distilled data are evenly distributed in the feature +space. However, not all features are equally important and distribution match might waste budget on +imitating less informative features, thereby undermining the distillation performance. +5 +Disentangled Dataset Distillation +In representation learning, albeit the image data are in the extremely high-dimensional space, they +may lie on a low-dimensional manifold and rely on a few numbers of features, and one can recover the +source image from the low-dimensional features with specific decoders [Bengio et al., 2013]. For this +reason, it is plausible to implicitly learn synthetic datasets by optimising their disentangled features +and corresponding decoders, which is termed as disentangled dataset distillation, as shown in Figure 7. +9 + +Dataset distillation +algorithmDataset distillation +algorithm +tFor in harmony with decoders, we recall the notion of feature as code in terms of the dataset. Through +generating synthetic datasets with the combination of codes and decoders, the compression ratio can +be further decreased, and information redundancy in distilled images is also reduced. According to +the learnability of code and decoder, we classify disentangled dataset distillation into three categories +of code-based DD, decoder-based DD, and code-decoder DD, respectively. +Code-based DD aims to learn a series of low-dimensional codes for generating high-informative +images through a specific generator. In Zhao and Bilen [2022], they learn the vectors that are put +into the GAN generator for producing informative images. Concretely, they inverse real examples +with a GAN generator and collect corresponding latent codes. Then, the latent vectors are further +optimised with the distribution match algorithm. By this, the optimised latent vectors can induce +more informative synthetic examples with the pre-trained GAN generator and consequently help train +neural networks. Besides using the GAN, Kim et al. [2022] employ a deterministic multi-formation +function as the decoder to create the synthetic data from fewer condensed data, and the condensed +data is optimised in an end-to-end fashion by gradient descent. +Different from code-based DD, decoder-based DD turn to learning a decoder that is employed to +produce high-informative data. In Such et al. [2020], they propose the generative teaching network +that employs a trainable network to generate synthetic images from random noise based on given +labels, and the meta-gradients are back-propagated to update the generator other than the synthetic +data. +Intuitively, code-decoder DD combines code-based and decoder-based DD and allows to train both +codes and decoders. Deng and Russakovsky [2022] generate synthetic images via the matrix multipli- +cation between codes and decodes, which they call memory and addressing function. Then these two +elements are optimised with the meta-loss scheme. Liu et al. [2022c] achieve the code-decoder DD by +feeding codes into networks to generate synthetic data, and they employ an adversarial contrastive +constraint to help networks learn different knowledge for better compression. A similar code-decoder +factorisation method is also presented in Lee et al. [2022a], where they adopt an improved distribution +match to optimise the latent codes and decoders. +Through implicitly learning the latent codes and decodes, disentangled DD possesses severe ad- +vantages like more compact representation and shared representation across classes and consequently +improves the dataset distillation performance. It is worth noting that this code-decoder factorisation +is compatible with the aforementioned distillation schemes. Therefore, the exploration of synthetic +data generation and distillation schemes can promote dataset distillation in parallel. +6 +Performance Comparison +To demonstrate the effectiveness of dataset distillation, we collect and summarize the classification +performance of some characteristic dataset distillation approaches on the following image datasets: +MNIST [LeCun et al., 1998], SVHN [Netzer et al., 2011], CIFAR-10/100 [Krizhevsky and Hinton, +2009], and Tiny ImageNet [Le and Yang, 2015]. Details of these datasets are presented as follows. +Recently, Cui et al. [2022a] publish a benchmark in terms of dataset distillation. However, it only +contains five DD methods, and we provide a comprehensive comparison of over 15 existing dataset +distillation methods in this survey. +MNIST is a black-and-white dataset and consists of 60, 000 training images and 10, 000 test images +from 10 different classes, and each example has the shape of 28 × 28. SVHN is a colourful dataset +and consists of 73, 257 digits and 26, 032 digits for training and testing, respectively, and examples in +SVHN are 32 × 32 RGB images. CIFAR-10/100 are composed of 50, 000 training images and 10, 000 +test images from 10 and 100 different classes, respectively. The RGB images in CIFAR-10/100 have +the shape of 32 × 32. Tiny ImageNet consists of 100, 000 and 10, 000 64 × 64 RGB images from 200 +different classes for training and testing, respectively. +An important factor to affect the test accuracy w.r.t. the distilled dataset is the distillation budget, +which constrains the size of the distilled dataset by the notion of Images allocated Per Class (IPC). +Usually, the distilled dataset is set to have the same number of classes as the target dataset. Therefore, +for a target dataset with 100 classes, setting the distillation budget to IPC = 10 suggests there are +totally 10 × 100 = 1, 000 images in the distilled dataset. For a comprehensive comparison, we also +present the test accuracy w.r.t. random select, coreset approach, and the original target dataset. For +the coreset approach, the Herding algorithm [Rebuffi et al., 2017] is employed in this survey due to +10 + +Table 1: Performance comparison of different dataset distillation methods on MNIST. +Methods +Distillation schemes +Accuracy +IPC=1 +IPC=10 +IPC=50 +Random +- +64.9 ± 3.5 +95.1 ± 0.9 +97.9 ± 0.2 +Herding +- +89.2 ± 1.6 +93.7 ± 0.3 +94.8 ± 0.2 +DD (Wang et al. [2018]) +BPTT +- +79.5 ± 8.1 +- +LD (Bohdal et al. [2020]) +BPTT +60.9 ± 3.2 +87.3 ± 0.7 +93.3 ± 0.3 +DC (Zhao et al. [2021]) +Gradient match +91.7 ± 0.5 +97.4 ± 0.2 +98.8 ± 0.2 +DSA (Zhao and Bilen [2021b]) +Gradient match +88.7 ± 0.6 +97.8 ± 0.1 +99.2 ± 0.1 +DCC (Lee et al. [2022b]) +Gradient match +- +- +- +MTT (Cazenavette et al. [2022a]) +Trajectory match +91.4 ± 0.9 +97.3 ± 0.1 +98.5 ± 0.1 +FTD (Du et al. [2022]) +Trajectory match +- +- +- +DM (Zhao and Bilen [2021a]) +Distribution match +89.2 ± 1.6 +93.7 ± 0.3 +94.8 ± 0.2 +CAFE (Wang et al. [2022]) +Distribution match +90.8 ± 0.5 +97.5 ± 0.1 +98.9 ± 0.2 +KIP (Nguyen et al. [2020, 2021]) +Kernel regression +90.1 ± 0.1 +97.5 ± 0.0 +98.3 ± 0.1 +FRePo (Zhou et al. [2022]) +Kernel regression +93.0 ± 0.4 +98.6 ± 0.1 +99.2 ± 0.0 +RFAD (Loo et al. [2022]) +Kernel regression +94.4 ± 1.5 +98.5 ± 0.1 +98.8 ± 0.1 +IDC (Kim et al. [2022])∗ +Gradient match +- +- +- +Deng and Russakovsky [2022]∗ +BPTT +89.2 ± 1.6 +93.7 ± 0.3 +94.8 ± 0.2 +Haba (Liu et al. [2022c])∗ +Trajectory match +- +- +- +KFS (Lee et al. [2022a])∗ +Distribution match +- +- +- +Whole dataset +- +99.6 ± 0.0 +its superior performance. For most of the methods, ConvNet [Gidaris and Komodakis, 2018] is the +default architecture to obtain the test accuracy if there is no specific annotation. +The empirical comparison results are presented in Table 1-6 associated with different datasets, +respectively. As shown in these tables, many methods are not tested on some datasets due to the +little meaning for easy datasets or scalability problems for large datasets. We preserve these blanks in +tables for a more fair and comprehensive comparison. To make the comparison clear, we use the bold +number for the best of two methods in each category. Besides, we note the disentangled DD methods +with ∗. Based on the performance comparison in the tables, we have several observations as follows: +• Dataset distillation can be realised on many datasets with various sizes. +• The performance of dataset distillation is significantly ahead of random pick and coreset selection. +• Disentangled dataset distillation by optimising the latent codes and decoders can largely improve +the test accuracy. +• KFS [Lee et al., 2022a] has the best performance among different datasets. +7 +Application +Due to the superior performance in compressing massive datasets, dataset distillation has been widely +employed in many application domains that limit training efficiency and storage, including continual +learning and neural architecture search. Furthermore, due to the correspondence between examples and +gradients, dataset distillation can also benefit privacy preservation, federated learning, and adversarial +robustness. In this section, we briefly review these applications w.r.t dataset distillation. +7.1 +Continual Learning +During the training process, when there is a shift in the training data distribution, the model will +suddenly lose the predicted ability on the previous data distribution. This phenomenon is referred +to as catastrophic forgetting and is common in deep learning. To overcome the problem, continual +learning is developed to incrementally learn new tasks whilst preserving performance on old tasks. +11 + +Table 2: Performance comparison of different dataset distillation methods on FashionMNIST. +Methods +Distillation schemes +Accuracy +IPC=1 +IPC=10 +IPC=50 +Random +- +51.4 ± 3.8 +73.8 ± 0.7 +82.5 ± 0.7 +Herding +- +67.0 ± 1.9 +71.1 ± 0.7 +71.9 ± 0.8 +DD (Wang et al. [2018]) +BPTT +- +- +- +LD (Bohdal et al. [2020]) +BPTT +- +- +- +DC (Zhao et al. [2021]) +Gradient match +70.6 ± 0.6 +82.3 ± 0.4 +83.6 ± 0.4 +DSA (Zhao and Bilen [2021b]) +Gradient match +70.6 ± 0.6 +86.6 ± 0.3 +88.7 ± 0.2 +DCC (Lee et al. [2022b]) +Gradient match +- +- +- +MTT (Cazenavette et al. [2022a]) +Trajectory match +75.1 ± 0.9 +87.2 ± 0.3 +88.3 ± 0.1 +FTD (Du et al. [2022]) +Trajectory match +- +- +- +DM (Zhao and Bilen [2021a]) +Distribution match +- +- +- +CAFE (Wang et al. [2022]) +Distribution match +73.7 ± 0.7 +83.0 ± 0.3 +88.2 ± 0.3 +KIP (Nguyen et al. [2020, 2021]) +Kernel regression +73.5 ± 0.5 +86.8 ± 0.1 +88.0 ± 0.1 +FRePo (Zhou et al. [2022]) +Kernel regression +75.6 ± 0.3 +86.2 ± 0.2 +89.6 ± 0.1 +RFAD (Loo et al. [2022]) +Kernel regression +78.6 ± 1.3 +87.0 ± 0.5 +88.8 ± 0.4 +IDC (Kim et al. [2022])∗ +Gradient match +- +- +- +Deng and Russakovsky [2022]∗ +BPTT +88.5 ± 0.1 +90.0 ± 0.7 +91.2 ± 0.3 +Haba (Liu et al. [2022c])∗ +Trajectory match +- +- +- +KFS (Lee et al. [2022a])∗ +Distribution match +- +- +- +Whole dataset +- +93.5 ± 0.1 +Table 3: Performance comparison of different dataset distillation methods on SVHN. +Methods +Distillation schemes +Accuracy +IPC=1 +IPC=10 +IPC=50 +Random +- +14.6 ± 1.6 +35.1 ± 4.1 +70.9 ± 0.9 +Herding +- +20.9 ± 1.3 +50.5 ± 3.3 +72.6 ± 0.8 +DD (Wang et al. [2018]) +BPTT +- +- +- +LD (Bohdal et al. [2020]) +BPTT +- +- +- +DC (Zhao et al. [2021]) +Gradient match +31.2 ± 1.4 +76.1 ± 0.6 +82.3 ± 0.3 +DSA (Zhao and Bilen [2021b]) +Gradient match +27.5 ± 1.4 +79.2 ± 0.5 +84.4 ± 0.4 +DCC (Lee et al. [2022b]) +Gradient match +47.5 ± 2.6 +80.5 ± 0.6 +87.2 ± 0.3 +MTT (Cazenavette et al. [2022a]) +Trajectory match +- +- +- +FTD (Du et al. [2022]) +Trajectory match +- +- +- +DM (Zhao and Bilen [2021a]) +Distribution match +- +- +- +CAFE (Wang et al. [2022]) +Distribution match +42.9 ± 3.0 +77.9 ± 0.6 +82.3 ± 0.4 +KIP (Nguyen et al. [2020, 2021]) +Kernel regression +57.3 ± 0.1 +75.0 ± 0.1 +80.5 ± 0.1 +FRePo (Zhou et al. [2022]) +Kernel regression +- +- +- +RFAD (Loo et al. [2022]) +Kernel regression +52.2 ± 2.2 +74.9 ± 0.4 +80.9 ± 0.3 +IDC (Kim et al. [2022])∗ +Gradient match +68.1 ± 0.1 +87.3 ± 0.2 +90.2 ± 0.1 +Deng and Russakovsky [2022]∗ +BPTT +87.3 ± 0.1 +89.1 ± 0.2 +89.5 ± 0.2 +Haba (Liu et al. [2022c])∗ +Trajectory match +69.8 ± 1.3 +83.2 ± 0.4 +88.3 ± 0.1 +KFS (Lee et al. [2022a])∗ +Distribution match +82.9 ± 0.4 +91.4 ± 0.2 +92.2 ± 0.1 +Whole dataset +- +95.4 ± 0.1 +12 + +Table 4: Performance comparison of different dataset distillation methods on CIFAR-10. +Methods +Distillation schemes +Accuracy +IPC=1 +IPC=10 +IPC=50 +Random +- +14.4 ± 2.0 +26.0 ± 1.2 +43.4 ± 1.0 +Herding +- +21.5 ± 1.2 +31.6 ± 0.7 +40.4 ± 0.6 +DD (Wang et al. [2018]) +BPTT +- +36.8 ± 1.2 +- +LD (Bohdal et al. [2020]) +BPTT +25.7 ± 0.7 +38.3 ± 0.4 +42.5 ± 0.4 +DC (Zhao et al. [2021]) +Gradient match +28.3 ± 0.5 +44.9 ± 0.5 +53.9 ± 0.5 +DSA (Zhao and Bilen [2021b]) +Gradient match +28.8 ± 0.7 +52.1 ± 0.5 +60.6 ± 0.5 +DCC (Lee et al. [2022b]) +Gradient match +47.5 ± 2.6 +80.5 ± 0.6 +87.2 ± 0.3 +MTT (Cazenavette et al. [2022a]) +Trajectory match +46.3 ± 0.8 +65.3 ± 0.7 +71.6 ± 0.2 +FTD (Du et al. [2022]) +Trajectory match +46.8 ± 0.3 +66.6 ± 0.3 +73.8 ± 0.2 +TESLA (Cui et al. [2022b]) +Trajectory match +48.5 ± 0.8 +66.4 ± 0.8 +72.6 ± 0.7 +DM (Zhao and Bilen [2021a]) +Distribution match +26.0 ± 0.8 +48.9 ± 0.6 +63.0 ± 0.4 +CAFE (Wang et al. [2022]) +Distribution match +31.6 ± 0.8 +50.9 ± 0.5 +62.3 ± 0.4 +KIP (Nguyen et al. [2020, 2021]) +Kernel regression +49.9 ± 0.2 +62.7 ± 0.3 +68.6 ± 0.2 +FRePo (Zhou et al. [2022]) +Kernel regression +46.8 ± 0.7 +65.5 ± 0.4 +71.7 ± 0.2 +RFAD (Loo et al. [2022]) +Kernel regression +53.6 ± 1.2 +66.3 ± 0.5 +71.1 ± 0.4 +IDC (Kim et al. [2022])∗ +Gradient match +50.0 ± 0.4 +67.5 ± 0.5 +74.5 ± 0.1 +Deng and Russakovsky [2022]∗ +BPTT +66.4 ± 0.4 +71.2 ± 0.4 +73.6 ± 0.4 +Haba (Liu et al. [2022c])∗ +Trajectory match +48.3 ± 0.8 +69.9 ± 0.4 +74.0 ± 0.2 +KFS (Lee et al. [2022a])∗ +Distribution match +59.8 ± 0.5 +72.0 ± 0.3 +75.0 ± 0.2 +Whole dataset +- +84.8 ± 0.1 +Table 5: Performance comparison of different dataset distillation methods on CIFAR-100. +Methods +Distillation schemes +Accuracy +IPC=1 +IPC=10 +IPC=50 +Random +- +4.2 ± 0.3 +14.6 ± 0.5 +30.0 ± 0.4 +Herding +- +8.4 ± 0.3 +17.3 ± 0.3 +33.7 ± 0.5 +DD (Wang et al. [2018]) +BPTT +- +- +- +LD (Bohdal et al. [2020]) +BPTT +11.5 ± 0.4 +- +- +DC (Zhao et al. [2021]) +Gradient match +12.8 ± 0.3 +26.6 ± 0.3 +32.1 ± 0.3 +DSA (Zhao and Bilen [2021b]) +Gradient match +13.9 ± 0.4 +32.3 ± 0.3 +42.8 ± 0.4 +DCC (Lee et al. [2022b]) +Gradient match +14.6 ± 0.3 +33.5 ± 0.3 +39.3 ± 0.4 +MTT (Cazenavette et al. [2022a]) +Trajectory match +24.3 ± 0.3 +40.1 ± 0.4 +47.7 ± 0.2 +FTD (Du et al. [2022]) +Trajectory match +25.2 ± 0.2 +43.4 ± 0.3 +50.7 ± 0.3 +TESLA (Cui et al. [2022b]) +Trajectory match +24.8 ± 0.4 +41.7 ± 0.3 +47.9 ± 0.3 +DM (Zhao and Bilen [2021a]) +Distribution match +11.4 ± 0.3 +29.7 ± 0.3 +43.6 ± 0.4 +CAFE (Wang et al. [2022]) +Distribution match +14.0 ± 0.3 +31.5 ± 0.2 +42.9 ± 0.2 +KIP (Nguyen et al. [2020, 2021]) +Kernel regression +15.7 ± 0.2 +28.3 ± 0.1 +- +FRePo (Zhou et al. [2022]) +Kernel regression +28.7 ± 0.1 +42.5 ± 0.2 +44.3 ± 0.2 +RFAD (Loo et al. [2022]) +Kernel regression +26.3 ± 1.1 +33.0 ± 0.3 +- +IDC (Kim et al. [2022])∗ +Gradient match +- +44.8 ± 0.2 +- +Deng and Russakovsky [2022]∗ +BPTT +34.0 ± 0.4 +42.9 ± 0.7 +- +Haba (Liu et al. [2022c])∗ +Trajectory match +33.4 ± 0.4 +40.2 ± 0.2 +47.0 ± 0.2 +KFS (Lee et al. [2022a])∗ +Distribution match +40.0 ± 0.5 +50.6 ± 0.2 +- +Whole dataset +- +56.2 ± 0.3 +13 + +Table 6: Performance comparison of different dataset distillation methods on Tiny ImageNet. +Methods +Distillation schemes +Accuracy +IPC=1 +IPC=10 +IPC=50 +Random +- +1.4 ± 0.1 +5.0 ± 0.2 +15.0 ± 0.4 +Herding +- +2.8 ± 0.2 +6.3 ± 0.2 +16.7 ± 0.3 +DD (Wang et al. [2018]) +BPTT +- +- +- +LD (Bohdal et al. [2020]) +BPTT +- +- +- +DC (Zhao et al. [2021]) +Gradient match +- +- +- +DSA (Zhao and Bilen [2021b]) +Gradient match +- +- +- +DCC (Lee et al. [2022b]) +Gradient match +- +- +- +MTT (Cazenavette et al. [2022a]) +Trajectory match +8.8 ± 0.3 +23.2 ± 0.2 +28.0 ± 0.3 +FTD (Du et al. [2022]) +Trajectory match +10.4 ± 0.3 +24.5 ± 0.2 +- +TESLA (Cui et al. [2022b]) +Trajectory match +7.7 ± 0.2 +18.8 ± 1.3 +27.9 ± 1.2 +DM (Zhao and Bilen [2021a]) +Distribution match +3.9 ± 0.2 +12.9 ± 0.4 +24.1 ± 0.3 +CAFE (Wang et al. [2022]) +Distribution match +- +- +- +KIP (Nguyen et al. [2020, 2021]) +Kernel regression +- +- +- +FRePo (Zhou et al. [2022]) +Kernel regression +15.4 ± 0.3 +25.4 ± 0.2 +- +RFAD (Loo et al. [2022]) +Kernel regression +- +- +- +IDC (Kim et al. [2022])∗ +Gradient match +- +- +- +Deng and Russakovsky [2022]∗ +BPTT +16.0 ± 0.7 +- +- +Haba (Liu et al. [2022c])∗ +Trajectory match +- +- +- +KFS (Lee et al. [2022a])∗ +Distribution match +22.7 ± 0.3 +27.8 ± 0.2 +- +Whole dataset +- +37.6 ± 0.4 +[Rebuffi et al., 2017, Castro et al., 2018]. A common method used in continual learning is the replay- +based strategy, which allows a limited memory to store a few training examples for rehearsal in the +following training. Therefore, the key to replay-based strategy is to select high-informative training +examples to store. Benefiting from extracting essence of datasets, dataset distillation technique is +employed to compress data for the memory with limited storage [Zhao et al., 2021, Carta et al., +2022]. Because the incoming data has a changing distribution, the frequency is high for updating +the elements in memory, which leads to strict requirements for the efficiency of dataset distillation +algorithms. To conveniently embed dataset distillation into replay-based strategy, Wiewel and Yang +[2021], Sangermano et al. [2022] decompose the process of synthetic data generation by linear or non- +linear combination, and thus fewer parameters are optimised during the dataset distillation. Besides +enhancing replay-based methods, dataset distillation can also learn a sequence of stable datasets, and +the network trained on these stable datasets will not suffer from catastrophic forgetting [Masarczyk +and Tautkute, 2020]. +7.2 +Neural Architecture Search +For a given dataset, the technique of neural architecture search (NAS) aims to find an optimal archi- +tecture from thousands of network candidates for better generalisation. The process of NAS usually +includes training the network candidates on a small proxy of the original dataset to save training +time, and the generalisation ranking can be estimated according to these trained network candidates. +Therefore, it is important to design the proxy dataset such that models trained on it can reflect the +model’s true performance in terms of the original data, while the size of the proxy dataset requires +control for the sake of efficiency. To construct proxy datasets, conventional methods are developed, +including random selection or greedy search, without altering the original data. Such et al. [2020] first +propose to optimise a highly informative dataset as the proxy for network candidate selection. More +following works consider NAS as a task for testing their proposed dataset distillation algorithms [Zhao +et al., 2021, Zhao and Bilen, 2021b,a, Du et al., 2022, Cui et al., 2022a]. Through simulation on the +synthetic dataset, a fairly accurate generalisation ranking can be collected for selecting the optimal +architecture whilst training time is considerably reduced. +14 + +7.3 +Privacy Protection +As the over-parameterized neural networks easily memorize all the training data, there exists the risk +for privacy leakage via inferring the well-trained networks [Shokri et al., 2017, Long et al., 2018, Salem +et al., 2018, Hu et al., 2022a]. With the synthetic dataset, the network avoids explicitly training on +the original dataset and consequently helps protect the data privacy [Zhou et al., 2022]. Moreover, +Chen et al. [2022a] employ dataset distillation to generate private training sets by adding differential +privacy noise to the gradients of synthetic data. Theoretically, Dong et al. [2022], Anonymous [2023] +build the connection between dataset distillation and differential privacy, and prove the superiority of +dataset distillation in privacy preservation compared to conventional private data generation methods. +7.4 +Federated Learning +Federated learning has received increasing attention in training neural networks in the past few years +due to its advantages in distributed training and private data protection [Yang et al., 2019]. The +federated learning framework consists of multiple clients and one central server, and each client pos- +sesses exclusive data for training the corresponding local model [McMahan et al., 2017]. In one round +of federated learning, clients transmit the induced gradients or model parameters to the server after +training with their exclusive data. Then the central server aggregates received gradients or parameters +to update the model parameters and broadcast new parameters to clients for the next round. In the +federated learning scenario, the data distributed in different clients are often non-i.i.d, which causes +a biased minimum w.r.t. local model and significantly hinders the convergence speed. Consequently, +the data heterogeneity remarkably imposes a communication burden between the server and clients. +Goetz and Tewari [2020] propose to distil a small set of synthetic data from original exclusive data by +matching gradients, and then the synthetic data instead of a large number of gradients are transmitted +to the server for model updating. Other distillation schemes such as meta-loss [Hu et al., 2022b, Zhou +et al., 2020], distribution match [Xiong et al., 2022], and kernel regression [Song et al., 2022], are also +employed to compress the exclusive data for alleviating communication cost at each transition round. +However, Liu et al. [2022b] discover that synthetic data generated by dataset distillation are still het- +erogeneous. To tackle the problem, they propose two strategies of dynamic weight assignment and +meta knowledge sharing during the distillation process, which significantly accelerate the convergence +speed of federated learning. Apart from compressing the local data, Pi et al. [2022] also distil data via +trajectory match in the server, which allows the synthetic data to possess global information. Then +the distilled data can be used to fine-tune the server model for convergence speed up. +7.5 +Adversarial Robustness +Deep learning networks with standard training is brittle to adversarial attacks, and adding impercep- +tible perturbation to the input can completely fool the network [Goodfellow et al., 2014]. Adversarial +training has been widely used to tackle this problem via continuously feeding adversarial examples +during the training process [Madry et al., 2017]. However, the generation of adversarial examples +requires multi-step gradient ascent, which considerably undermines the training efficiency. Recently, +Tsilivis et al. [2022], Wu et al. [2022] employ dataset distillation to extract the information of adver- +sarial examples and generate robust datasets. Then standard network training on the distilled robust +dataset is enough to achieve satisfied robustness to adversarial perturbation, which incredibly saves +computing resources compared to the expensive adversarial training. +7.6 +Other Applications +Apart from the aforementioned applications, we also summarise other applications in terms of dataset +distillation as follows. +As the synthetic dataset owns a small size, it has been applied for explainability [Loo et al., 2022]. +Concretely, due to the small size of synthetic data, it is easy to measure how the synthetic examples +influence the prediction of test examples. +If the test and training images both rely on the same +synthetic images, the training image will greatly influence the prediction of the test image. In other +words, the synthetic set becomes a bridge to connect the train and test examples. +15 + +Using the characteristic of capturing the essence of datasets, [Cazenavette et al., 2022b, Chen +et al., 2022b] utilise dataset distillation for visual design. +Cazenavette et al. [2022b] generate the +representative textures by randomly cropping the synthetic images during the distillation process. In +addition to extracting texture, Chen et al. [2022b] impose the synthetic images to model the outfit +compatibility through dataset distillation. +Besides the continuous image domain, dataset distillation has also been extended to discrete data +such as graphs [Jin et al., 2022b,a, Liu et al., 2022a] and recommender systems [Sachdeva et al., 2022]. +Jin et al. [2022b] first formulate this dataset distillation w.r.t. graph data, and successfully distil a +small, condensed graph via gradient match scheme. Due to the computational inefficiency in distilling +pairwise relation for graph nodes, Jin et al. [2022a] turn to learn a probabilistic graph model, which +allows a differentiable optimisation w.r.t. the discrete graph. They also employ one-step strategy +for further distillation speedup. Moreover, Liu et al. [2022c] accelerate the graph distillation through +distribution match from the perspective of receptive fields. For the recommender system, Sachdeva +et al. [2022] distil the massive dataset to a continuous prior for tackling the discrete data problem. +Due to its small size and abstract visual information, the distilled data can also be applied in +medicine, especially in medical image sharing [Li et al., 2023]. Empirical studies on gastric X-ray +images have shown the advantages of DD in medical image transition and anonymisation of patient +information [Li et al., 2020, 2022]. Through dataset distillation, hospitals can share their valuable +medical data at lower cost and risk to build powerful computer-aided diagnosis systems. +8 +Challenges and Directions +As its superiority in compressing datasets and training speedup, dataset distillation has promising +prospects in a wide range of areas. In the following words, we will discuss existing challenges in terms +of dataset distillation. Furthermore, we investigate the plausible directions and provide insights on +dataset distillation to promote future studies. +8.1 +Challenges +For dataset distillation, there are two main ingredients: (1) measure the difference between the knowl- +edge of synthetic data and real (original) data; and (2) update the synthetic data to narrow the +knowledge difference. Based on the ingredients, we discuss the challenges of dataset distillation from +the perspectives of (1) the definition of the difference between knowledge, (2) the form of synthetic +datasets, (3) the evaluation of synthetic datasets, and (4) the theory behind dataset distillation. +Dataset distillation methods aim to transfer the knowledge of a target dataset into a small synthetic +dataset to achieve comparable generalisation on the original data distribution. Albeit the dataset’s +knowledge is an abstract notion, many DD approaches indirectly measure the difference between their +knowledge via various proxy losses. However, the existing definition w.r.t. the knowledge difference +is not consistent with the original comparable generalisation. For example, the meta-loss and kernel +regression loss measure the knowledge difference with the error in terms of the specific target dataset. +Through this, one can only get a similar training error other than test error w.r.t. the target dataset. +As for match schemes, they intuitively make the knowledge concrete with a series of parameters or +features, which are matched for knowledge transmission. Therefore, a more consistent definition of +knowledge difference should be investigated to improve the dataset distillation performance. +Albeit most dataset distillation methods directly update the synthetic images, implicit optimising +codes and decoders that cooperatively generate synthetic data outperform the direct optimisation by +a large margin, as discussed in Section 6. From this, it can be seen that the pre-defined form of the +code and decoder has a remarkable influence on the distillation performance. Many disentanglement +methods on synthetic data are based on intuition and lack rigorous analysis, and only some vague +reasons are proposed, such as decreasing redundant information across different classes and learn- +ing more compact representation. Hence, it is necessary to explore the influence of the pre-defined +disentanglement on the dataset distillation for learning a more informative distilled dataset. +In practical dataset distillation, the optimisation of synthetic data is closely related to the specific +network architecture (or kernels in KRR approach), and there is naturally a performance drop when +the learned synthetic dataset is applied to other unseen architectures. Therefore, it is one-sided to +compare different DD methods only on one or a few architectures. To compare different DD methods +16 + +in a more comprehensive scenario, it is essential to set a reasonable baseline consisting of multiple +handpicked network architectures. +In spite of various DD approaches have been emerging, there are few works that discuss the theory +behind dataset distillation. Nevertheless, developing the theory is extremely necessary and will con- +siderably promote the dataset distillation by directly improving the performance and also suggesting +the correct direction of development. For example, the theory can help derive a better definition of +the dataset knowledge and consequently increase the performance. Besides, the theoretical correlation +between the distillation budget (the size of synthetic datasets) and performance can provide an upper +bound on the test error, which can offer a holistic understanding and avoid researchers blindly improv- +ing the DD performance. Hence, a solid theory is indispensable to take the development of DD to the +next stage. +8.2 +Future Directions +Albeit the existing DD methods have shown impressive results in compressing massive datasets, there +are still many areas that are worth to explore to promote the dataset distillation. In this section, we +will discuss some promising directions that shed light on future studies. +Fine-tune with synthetic data. With the rapid development of foundational models [Bom- +masani et al., 2021], the mainstream deep learning pipeline gradually converts from scratch learning to +pre-trained learning or transfer learning. For the foundational models of BERT [Devlin et al., 2018], +GPT-3 [Brown et al., 2020], and etc., they are trained with billions of data to learn an appropriate +latent representation. With only fine-tuning on specific datasets, the foundational models can achieve +extraordinary performance on various downstream tasks and outperform models learned from scratch +by a large margin. To this end, conducting dataset distillation in the fine-tune regime will be a promis- +ing direction to cooperate with foundational models. With the synthetic data, the fine-tine process +will be faster whilst the downstream performance is preserved. Albeit possessing a similar goal, dis- +tilling datasets based on pre-trained models might be more challenging compared to initialised models +because the pre-trained models are harder to collect for repeated distillation to enhance synthetic +datasets’ robustness. Apart from condensing datasets in the fine-tune stage, it is also an interesting +topic to distil datasets in the vanilla training stage so that models trained on synthetic datasets can +be easier or faster to fine-tune by specific datasets. +Initialisation augmentation. Benefiting from excellent efficiency and compatibility, data aug- +mentation becomes an essential strategy to enhance the model’s generalisation w.r.t. data distribution +[Shorten and Khoshgoftaar, 2019]. From basic cropping and flipping [Zagoruyko and Komodakis, 2016] +to the mixup method [Zhang et al., 2018], data augmentation efficiently constructs variants of training +images with prior rules to improve the model’s robustness to patterns. In dataset distillation, the syn- +thetic datasets are also sensitive to the parameter initialisation and network architecture adopted in +the distillation process [Wang et al., 2018, Zhao et al., 2021]. Existing works alleviate this problem by +trivial repeated distillation w.r.t. different initialisations and then taking an expectation [Zhou et al., +2022]. However, the random selection of initialisation is not an elaborate strategy and has limited +improvement. [Zhang et al., 2022] shows that employing the early-stage models trained with a few +epochs as the initialisation can achieve better distillation performance, and they further propose weight +perturbation methods to efficiently generate early-stage models for repeated distillation. Therefore, +it is important for many dataset distillation algorithms to investigate how to design the initialisation +augmentation so that synthetic datasets achieve better generalisation to different initialisations and +architectures. +Pre-processing of target datasets. +To better extract knowledge from training data, pre- +processing, such as normalisation, is a common approach that reshapes the structure of data by +well-designed mappings. In existing dataset distillation methods, only a few works discuss the influ- +ence of pre-processing on final distillation performance [Nguyen et al., 2020]. In Nguyen et al. [2020, +2021], Cazenavette et al. 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URL https://openreview.net/forum?id=2clwrA2tfik. +23 + diff --git a/WdE5T4oBgHgl3EQfcg8E/content/tmp_files/load_file.txt b/WdE5T4oBgHgl3EQfcg8E/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..21514dabed674d56fd9169cec40c4fd0eeda9185 --- /dev/null +++ b/WdE5T4oBgHgl3EQfcg8E/content/tmp_files/load_file.txt @@ -0,0 +1,1556 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf,len=1555 +page_content='A Comprehensive Survey to Dataset Distillation Shiye Lei∗ Dacheng Tao† Abstract Deep learning technology has unprecedentedly developed in the last decade and has become the primary choice in many application domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' This progress is mainly attributed to a systematic collaboration that rapidly growing computing resources encourage advanced algorithms to deal with massive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, it gradually becomes challenging to cope with the unlimited growth of data with limited computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To this end, diverse approaches are proposed to improve data processing efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Dataset distillation, one of the dataset reduction methods, tackles the problem via synthesising a small typical dataset from giant data and has attracted a lot of attention from the deep learning community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Existing dataset distillation methods can be taxonomised into meta-learning and data match framework according to whether explicitly mimic target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Albeit dataset distillation has shown a surprising performance in compressing datasets, it still possesses several limitations such as distilling high-resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' This paper provides a holistic understanding of dataset distillation from multiple aspects, including distillation frameworks and algorithms, disentangled dataset distillation, performance comparison, and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Finally, we discuss challenges and promising directions to further promote future studies about dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 1 Introduction During the past few decades, deep learning has achieved remarkable success in a wide range of applica- tions, including computer vision [Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2017, He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2016], neural language processing [Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2017], autonomous vehicles [Grigorescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020], protein structure prediction [Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2021], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' An important reason for this success is the powerful computing resource, which allows deep neural networks to directly tackle giant datasets and bypass complicated manual feature extraction, which causes potential loss of data information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For example, the powerful large language model GPT-3 contains 175 billion parameters and is trained on 45 terabyte of text data with thousands of GPUs [Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, massive data are generated every day [Sagiroglu and Sinanc, 2013], which poses a significant threat to training efficiency and data storage, and deep learning might reach a bottleneck due to the mismatch between the volume of data and computing resources [Strubell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Recently, many methods have been proposed to improve the training efficiency in deep learning from several perspectives as below: Quantisation: this approach sacrifices the data byte during the training process for acceleration [Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2015, Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Model pruning: this approach removes trainable parameters that has few influences on the final performance [Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2017, Lym et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Optimisation: this approach designs the training algorithms for fast convergence [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2016] or less memory cost [Bernstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018, Karimireddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Dataset reduction: this approach generates few representative data to constitute a new training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Depending on whether the generated data are natural or synthetic, data compression can be classified into coreset selection [Mirzasoleiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020, Coleman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020] and dataset distillation [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' ∗S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Lei is associated with School of Computer Science, Faculty of Engineering, The University of Sydney, Darlington NSW 2008, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Email: slei5230@uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='sydney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' †D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Tao is associated with JD Explore Academy, China and The University of Sydney, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Email: dacheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='tao@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='05603v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='LG] 13 Jan 2023 Figure 1: An illustration of dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The models trained on the large original dataset and small synthetic dataset have comparable performance on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The above narrative shows that efficient deep learning is an extensive task and outsides the scope of this paper, and readers can refer to [Sze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2017, Menghani, 2021] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In this survey, we focus on dataset distillation to improve training efficiency by synthesising training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To better appreciate dataset distillation, we briefly introduce other cognate methods in dataset reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For the support vector machine (SVMs), its hyperplane is solely determined by “support vectors”, and removing all other points in the training dataset does not have an influence on the convergence result [Cortes and Vapnik, 1995].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, the selection of “support vectors” is a favourable method to reduce the training burden for SVMs [Nalepa and Kawulok, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' When the scenario expands to deep learning, the problem of selecting “support vectors” generalises to the well-known coreset selection: the algorithms select a few prototype examples from the original dataset as the coreset, and then the model is solely trained on the small coreset to save training costs whilst avoiding large performance drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Various coreset selection algorithms have been successfully leveraged in many applications like active learning [Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2015], continual learning [Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2019], neural architecture search [Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2021], robust learning [Mirzasoleiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, elements in the coreset are unmodified and constrained by the original data, which considerably restricts the coreset’s expressiveness, especially when the coreset budget is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Recently, a novel approach, dataset distillation (DD) 1 has attracted growing attention from the deep learning community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Different from coreset selection, dataset distillation removes the restriction of uneditable elements and carefully modifies a small number of examples to preserve more information, as shown in Figure 4 for synthetic examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' By distilling the knowledge of the large original dataset into the small synthetic set, models trained on the synthetic set can acquire a comparable generalisation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' A general illustration for dataset distillation is presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Due to the property of extreme high-dimension in the deep learning regime, the data information is hardly disentangled to specific concepts, and thus distilling numerous high-dimensional data into a few points is not a trivial task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Based on the objectives applied to mimic target data, dataset distillation methods can be grouped into meta-learning framework and data match framework, and these techniques in each framework can be further classified in a much more detailed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In the meta-learning framework, the distilled data are considered hyperparameters and optimised according to the validation risk on the target data [Maclaurin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2015, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' As for the data match framework, it updates distilled data by imitating the influence of target data on model training from parameter or feature space [Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2021, Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022a, Zhao and Bilen, 2021a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Figure 3 presents these different categories of DD algorithms in a tree diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Apart from directly considering the synthetic examples as optimisation objectives, some works design a proxy model consisting of latent codes and decoders to generate high-informative examples and resort to learning the latent codes and decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For example, Such et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020] employ a network 1Dataset distillation is also referred to as dataset condensation in some literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 2 training test training testFigure 2: The schematic structure of dataset distillation and the relationship between the adjacent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The body of this survey mainly contains fundamentals of dataset distillation, the taxonomy of distillation schemes, types of disentangled DD, distillation algorithms, performance comparison, applications, challenges, and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Note that ‘Section’ is abbreviated as ‘Sec.’ in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Dataset Distillation Meta-learning Framework Back-propagation Through Time DD [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018] LD [Bohdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020] SLDD [Sucholutsky and Schonlau, 2021] GTN [Such et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020] Deng and Russakovsky [2022] Kernel Ridge Regression KIP [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020, 2021] FRePo [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022] RFAD [Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022] Data Match Framework Gradient Match DC [Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2021] DCC [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022b] IDC [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022] Trajectory Match MTT [Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022a] FTD [Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022] TESLA [Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022b] Haba [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022c] Distribution Match DM [Zhao and Bilen, 2021a] CAFE [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022] GAN [Zhao and Bilen, 2022] KFS [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022a] Figure 3: Tree diagram for different categories of dataset distillation algorithms to generate highly informative data from noise and optimise the network via meta-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Zhao and Bilen [2022] optimise the vectors and put them into the generator of a well-trained generative adversarial network (GAN) to produce the synthetic examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Moreover, [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022, Deng and Russakovsky, 2022, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022c, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022a] learn a couple of latent codes and decoders, and then the synthetic data is generated according to the different combinations of latent codes and decodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' With this disentanglement of synthetic data, the compression ratio of DD can be further decreased, and the performance can also be improved due to the intra-class information extracted by latent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In this paper, we present a comprehensive survey on dataset distillation, and the main objectives of this survey are to (1) present a clear and systematic overview of dataset distillation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' (2) review the recent progress and state-of-the-art algorithms and discuss various applications in dataset distillation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' (3) give a comprehensive performance comparison w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' different dataset distillation algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' and (4) provides a detailed discussion on the limitation and promising directions of dataset distillation to help future studies prosper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Recently, Larasati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022] publish a short review on dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, they only provide a general overview of DD and focus on the aspect of appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Different from Larasati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' our paper gives a systematic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' comprehensive survey on dataset distillation from a wide aspect of distillation schemes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' algorithms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' limitations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Performance Comparison ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Gradient Match ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Back-propagation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='ThroughTime ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Problem Setup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Trajectory Match ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Kernel Ridge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Regression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Meta-learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='DistributionMatch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Data Match ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Neural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Continual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Architecture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Code-basedDD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Search ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Disentangled DD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Privacy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Federated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Decoder-basedDD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Protection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Adversarial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Other ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Code-decoder DD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Challenges ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Directions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='Robustness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='ApplicationsFigure 4: Example synthetic images distilled from CIFAR-10/100 and Tiny ImageNet by matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='training trajectory [Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' and promising directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The rest of this paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' We first give some notations of dataset distillation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' DD methods under meta-learning framework is described and comprehensively analysed in Section 3, and a description of the data match framed follows in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Section 5 discusses different types of disentangled dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' We then report the performance of various distillation algorithms in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Applications with dataset distillation are shown in Section 7, and we discuss challenges and future directions in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 2 Problem Setup Before introducing dataset distillation, we first define some notations used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For a dataset T = {(xi, yi)}m i=1, where xi ∈ Rd, d is the dimension of input data, yi is the label, we assume that (xi, yi) are both independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=') random variables drawn from the data generating distribution D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' We employ fθ to denote the neural network parameterized by θ, and fθ(x) is the prediction or output of fθ on the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For the training algorithm alg, let alg(T , θ(0)) denote the learned parameters returned by leveraging alg on the dataset T with the initialised parameter θ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' We omit θ(0) and use alg(T ) if there is no ambiguity for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Then, the model f trained on the dataset T can be denoted as falg(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Moreover, we also define the loss between prediction and ground truth as ℓ(fθ(x), y), and the expected risk in terms of θ is defined as RD(θ) = E(x,y)∼D [ℓ (fθ (x) , y)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' (1) Since the data generating distribution D is unknown, evaluating the expected risk RD is not practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, it is a practical way to estimate the expected risk by the empirical risk RT , which is defined as RT (θ) = E(x,y)∼T [ℓ (fθ (x) , y)] = 1 m m � i=1 ℓ (fθ (xi) , yi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' (2) In the deep learning paradigm, gradient descent is the dominant algorithm to train a neural network by minimising the empirical risk step by step: the network’s parameters are initialised with θ(0), then the parameter is iteratively updated according to the gradient of empirical loss: θ(k+1) = θ(k) − ηg(k) T , (3) where η is the learning rate and g(k) T = ∇θ(k)RT (θ(k)) is the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Because deep learning models are commonly extremely over-parameterized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', the number of model parameters overwhelms the number of training examples, the empirical risk readily reaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In this case, the generalisation error, which measures the difference between the expected risk and the empirical risk, can be solely equal to the expected risk, which is reflected by test loss or test error in practical pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Given a target dataset (source training dataset) T = {(xi, yi)}m i=1, the objective of dataset distil- lation is to extract the knowledge of T into a small synthetic dataset S = {(sj, yj)}n j=1, where n ≪ m, 4 (a) (b) (c) Figure 5: (a) Meta-gradient back-propagation in BPTT [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' (b) Meta-gradient back- propagation in kernel ridge regression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' (b) Meta-gradient back-propagation in gradient match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' and the model trained on the small distilled dataset S can achieve a comparable generalisation per- formance to the large original dataset T : E(x,y)∼D � ℓ � falg(T ) (x) , y �� ≃ E(x,y)∼D � ℓ � falg(S) (x) , y �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' (4) Because the training algorithm alg � S, θ(0)� is both determined by the training set S and the initialised parameter θ(0), many dataset distillation algorithms will take expectation on S w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' θ(0) in order to improve the robustness of the distilled dataset S to different parameter initialisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In the following narrative, we will omit this expectation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' initialisation for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 3 Meta-learning Framework From the meta-learning perspective, the distilled data are treated as hyper-parameters and the objec- tive of DD is to learn the distilled data for improving the generalisation performance of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' By assuming RD (alg(S)) ≃ RT (alg(S)), we employ the target dataset T as the validation set in terms of the model alg(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To this end, the dataset distillation can be formulated to a bi-level optimisation problem as below: S∗ = arg min S RT (alg(S)) (outer loop) (5) subject to alg(S) = arg min θ RS (θ) (inner loop) (6) The inner loop optimises the model parameters based on the synthetic dataset and is often realised by gradient descent for neural networks or regression for kernel method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' During the outer loop, the synthetic set is updated by minimising the model’s risk in terms of the target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' With the nested loop, the synthetic dataset gradually converges to one of the optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 Back-propagation Through Time Approach As shown in the above formulation of bi-level optimisation, the objective function of DD can be directly defined as the meta-loss of L(S) = RT (alg (S)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' (7) Then the distilled dataset is updated by S = S −α∇SL(S) with the step size α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, as the neural network is adopted in the distillation process, the training algorithm alg on the model parameter θ is iterative (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 3) and yields a series of intermediate parameter states of {θ(0), θ(1), · · · , θ(T )} for the inner loop (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Hence, back-propagation through time (BPTT) is required to recursively compute the meta-gradient ∇SL(S): ∇SL(S) = ∂L ∂S = ∂L ∂θ(T ) �k=T � k=0 ∂θ(T ) ∂θ(k) · ∂θ(k) ∂S � , (8) and ∂θ(T ) ∂θ(k) = T � i=k+1 ∂θ(i) ∂θ(i−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' (9) 5 As shown in Figure 5(a), due to the requirement of unrolling the recursive computation graph, BPTT is both computing and memory expensive, which also severely affects the final distillation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To alleviate the inefficiency in unrolling the long parameter path of {θ(0), · · · , θ(T )}, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2018] solely adopt a single-step optimisation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' the model parameter from θ(0) to θ(1), and the loss is computed based on θ(1) and the target dataset T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, the distilled data s and learning rate η can be efficiently updated via the short-range BPTT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Unlike freezing distilled labels, Sucholutsky and Schonlau [2021] extend Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2018] by learning a soft-label in the synthetic dataset S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', the label y in the synthetic dataset is also trainable for better information compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Similarly, [Bohdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020] also extend the standard example distillation to label distillation by solely optimising the labels of synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Moreover, they provide improvements on the efficiency of long inner loop optimisation via (1) iteratively updating the model parameters θ and the distilled labels y and (2) employing ridge regression to update the solution of the last linear layer of networks to avoid second-order gradient computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Albeit the BPTT framework has been shown to underperform other algorithms, Deng and Russakovsky [2022] empirically demonstrate that adding momentum term and longer unrolled trajectory (200 steps) in the inner loop optimisation can considerably increase the distillation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 Kernel Ridge Regression Approach Albeit multi-step gradient descent can gradually approach the optimal network parameters in terms of the synthetic dataset during the inner loop, this iterative algorithm makes the meta-gradient back- propagation highly inefficient, as shown in BPTT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Considering the existence of closed-form solutions in kernel regression regime, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020] replace the neural network in the inner loop with a kernel model, which bypasses the recursive back-propagation of meta-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For the regression model f(x) = w⊤ψ(x), where ψ(·) is a non-linear mapping and the corresponding kernel is K(x, x′) = ⟨ψ(x), ψ(x′)⟩, there exists a closed-form solution for w when the regression model is trained on S with kernel ridge regression (KRR): w = ψ(Xs)⊤ (KXsXs + λI)−1 ys, (10) where KXsXs = [K(si, sj)]ij ∈ Rn×n is called the kernel matrix or Gram matrix associated to K and the dataset S, and λ > 0 is a fixed regularization parameter [Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, the mean square error (MSE) of predicting T with the model trained on S is L(S) = 1 2 ���yt − KXtXs (KXsXs + λI)−1 ys ��� 2 , (11) where KXtXs = [K(xi, sj)]ij ∈ Rm×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Then the distilled dataset is updated via the meta-gradient of the above loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Due to the closed-form solution in KRR, θ does not require an iterative update and the backward pass of gradient thus bypasses the recursive computation graph, as shown in Figure 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In the KRR regime, the synthetic dataset S can be directly updated by back-propagating the meta-gradient through the kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Albeit this formulation is solid, this algorithm is designed in the KRR scenario and only employs simple kernels, which causes performance drops when the distilled dataset is transferred to train neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Jacot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2018] propose the neural tangent kernel (NTK) theory that proves the equivalence between training infinite-width neural networks and kernel regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' With this equivalence, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2021] employ the infinite-width networks as the kernel for dataset distillation, which narrows the gap between the scenarios of KRR and deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, every entry in the kernel matrix requires to be calculated separately, and computing the kernel matrix KXsXt owns the complexity of O(|T ||S|), which is severely inefficient for large- scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To tackle this problem, Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022] replace NTK kernel with neural network Gaussian process (NNGP) kernel that only considers the training dynamic of last-layer classifier for speed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' With this replacement, the random feature ψ(x) and ψ(s) can be explicitly computed via multiple sampling from the Gaussian process, and thus the kernel matrix computation can be decomposed into random feature calculation and random feature matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Because the matrix multiplication require negligible amounts of time for small distilled datasets, the complexity of kernel matrix computation degrades to O(|T | + |S|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' A similar efficient method is also proposed by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022], which employs the feature extractor in neural networks as the feature map ψ and accordingly optimises the distilled data via KRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 Discussion From the loss surface perspective [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018], the meta-learning framework of decreasing RT (alg (S)) can be considered to mimic the local minima of target data with the distilled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, the loss landscape w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' parameters are closely related to the network architecture, while only one type of small network is used in BPTT approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' There is consequently a moderate performance drop when the distilled dataset is employed to train other complicated networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Moreover, long unrolled tra- jectory and second-order gradient computation are also two key challenges for BPTT approach and hinder its efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' KRR approach compensate for these shortcomings by replacing networks with the non-parametric kernel model which admits closed-form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Albeit KRR is non-parametric and does not involve neural networks during the distillation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, previous research has shown that the training dynamic of neural networks is equal to the kernel method when the width of networks becomes infinite [Jacot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018], which partially guarantees the feasibility of the kernel regression approach and explains its decent performance when transferred to the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 4 Data Match Framework Albeit direct information extraction is not feasible, the information distillation can be achieved by implicitly matching the by-products of target data and synthetic data from different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The objective function of data match can be summarised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' L(S) = T � k=0 D � φ(S, θ(k)), φ(T , θ(k)) � , (12) where D(·, ·) is a distance function, and φ(·) maps the dataset S or T to other informative spaces, such as gradient, feature, and parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Compared to the aforementioned meta-learning framework, the data match loss does not only focus on the final parameter alg(S) but also supervise the intermediate states, as shown in the sum operation �T k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' By this, the distilled data can better imitate the influence of target data on training networks at different training stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 Gradient Match Approach To achieve comparable generalisation performance, the intuition is imitate the effect on model param- eters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', matching the training trajectories introduced by S and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' With a fixed parameter initiali- sation, the training trajectory of {θ(0), θ(1), · · · , θ(T )} is equal to a series of gradients {g(0), · · · , g(T )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, matching the gradients induced by S and T is a convincing proxy to mimic the influence on model parameters [Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2021], and the objective function can be formulated as L(S) = T −1 � k=0 D � g(k) S , g(k) T � , (13) where g(k) S and g(k) T denote the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' model parameters generated by S and T in the k-th training epoch, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' It is worth noting that these two gradients are induced on the same parameter θ(k) to be cohesive with the bi-level optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Concretely, the gradient match is class-wise: L(k) = �C c=1 D � ∇θ(k)R � Sc, θ(k)� , ∇θ(k)R � Tc, θ(k)�� , where c is the class index and Tc denotes the examples belong to the c-th class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' According to Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b], the class-wise gradi- ent match pays much attention to the class-common features and overlooks the class-discriminative features in the target dataset, and the distilled synthetic dataset S does not possess enough class- discriminative information, especially when the target dataset is fine-grained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', class-common features are the dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Based on this finding, they propose a improved objective function of L(k) = D ��C c=1 ∇θ(k)R � Sc, θ(k)� , �C c=1 ∇θ(k)R � Tc, θ(k)�� to better capture constractive signals be- tween different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' A similar approach is proposed by Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022], which considers both intra-class and inter-class gradient match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To alleviate easy overfitting on the small dataset S, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022] propose to do inner loop optimisation on the target dataset T instead of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 7 (a) Trajectory match (b) Distribution match Figure 6: (a) An illustration of trajectory match [Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022a];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' (b) Distribution match approach can more comprehensively cover the data distribution in feature space compared to gradient match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In spite of data augmentation brings a large performance increase, conducting augmentation on distilled datasets has no improvement on the final test accuracy, because the synthetic images have different characteristics compared to natural images and also are not optimised under the supervision of various transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To leverage data augmentation on synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Zhao and Bilen [2021b] design data siamese augmentation (DSA) that homologously augments the distilled data and the target data during the distillation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In DSA, the augmented form of distilled data has a consistent correspondence w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' the augmented form of the target data, which permits the knowledge transfer from the transformation of target images to the corresponding transformation of synthetic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Consequently, the augmented synthetic images also possess meaningful characteristics of the natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Due to its superior compatibility, DSA has been widely equipped in many data match methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 Trajectory Match Approach Unlike circuitously matching the gradients, Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a] directly matches the long-range training trajectory between the target dataset and the synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Concretely, they collect the expert training trajectory w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' the target dataset T into the buffer in advance, and then ingredients in the buffer are randomly selected to initialise the networks for training S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' After collecting the trajectory of S, the synthetic dataset is updated by matching their trajectory, as shown in Figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The objective loss of matching trajectory is defined as L = ∥θ(k+N) − θ(k+M) T ∥2 2 ∥θ(k) T − θ(k+M) T ∥2 2 , (14) where θT denote the target parameter by training the model on T and is stored in the buffer, and θk+N are the parameter by training the model on S for N epochs with the initialisation of θ(k) T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The denominator in the loss function is for normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Albeit trajectory match received empirical success, Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022] propose that there exists an accumulated trajectory error when matching the trajectory due to the segmented alignment from θ(k) T to θ(k+M) T , and they alleviate this by adding random noise when initialise the distilled network to improve robustness w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' the accumulated trajectory error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Compared to matching gradients, while trajectory match side-steps second-order gradient compu- tation, it, unfortunately, requires unrolling N SGD updates during the meta-gradient backpropagation as the existence of θ(k+N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The unrolled gradient computation significantly increases the memory burden and impedes scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' By disentangling the meta-gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' synthetic examples into two passes, Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b] greatly reduce the memory required by trajectory match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Motivated by knowledge distillation [Gou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2021], they also propose to assign soft-label to synthetic exam- ples with pre-trained models in the buffer, and the soft-label helps learn intra-class information and consequently improves distillation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 8 二 g(k 0(k+2) (I+)0 θ(k+N) q(k+1) o(k+2) (N++)0 0(t+M)(a) Non-disentangled dataset distillation (b) Disentangled dataset distillation Figure 7: The schematic diagrams of non-disentangled dataset distillation and disentangled dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 Distribution Match Approach Albeit the parameter-wise match shows a satisfying performance, Zhao and Bilen [2021a] visualise the distilled data in 2-dimension and reveal that there is a large distribution discrepancy between the distilled data and the target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In other words, the distilled dataset can not comprehensively cover the data distribution in feature space, as shown in Figure 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Based on this discovery, they propose to match the synthetic and target data from the distribution perspective for dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Concretely, they employ the pre-trained feature extractor ψv with the parameter v to achieve the mapping from input space to feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The synthetic data is optimised according to the objective function L(S) = C � c=1 ∥ψv(Sc) − ψv(Tc)∥2, (15) where c denotes the class index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Though this distribution match drops bi-level optimisation for boost- ing, it empirically underperforms the above gradient and trajectory match approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022] improve the distribution alignment from several aspects: (1) using multiple-layer features other than only the last-layer inputs for matching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' (2) proposing the discrimination loss to enlarge the class distinction of synthetic data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' and (3) recalling the bi-level optimisation that updates S with different model parameters for better generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Besides, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a] analyse that the subsampling in distribution match is biased, and they propose to use full batch training to mitigate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 Discussion The gradient match approach can be considered as a short-range parameter match, while its back- propagation requires second-order gradient computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Albeit the trajectory match approach make- ups for this imperfection, the long-range trajectory introduces recursive computation graph during meta-gradient back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Different from matching in parameter space, distribution match approach employ the feature space as the match proxy and also bypass second-order gradient com- putation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Albeit distribution match has advantages in scalability w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' high-dimensional data, it empirically underperforms trajectory match, which might be attributed to the mismatch between the comprehensive distribution and decent distillation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In detail, distribution match achieves DD by mimicking features of target data, so that the distilled data are evenly distributed in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, not all features are equally important and distribution match might waste budget on imitating less informative features, thereby undermining the distillation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 5 Disentangled Dataset Distillation In representation learning, albeit the image data are in the extremely high-dimensional space, they may lie on a low-dimensional manifold and rely on a few numbers of features, and one can recover the source image from the low-dimensional features with specific decoders [Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For this reason, it is plausible to implicitly learn synthetic datasets by optimising their disentangled features and corresponding decoders, which is termed as disentangled dataset distillation, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 9 Dataset distillation algorithmDataset distillation algorithm tFor in harmony with decoders, we recall the notion of feature as code in terms of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Through generating synthetic datasets with the combination of codes and decoders, the compression ratio can be further decreased, and information redundancy in distilled images is also reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' According to the learnability of code and decoder, we classify disentangled dataset distillation into three categories of code-based DD, decoder-based DD, and code-decoder DD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Code-based DD aims to learn a series of low-dimensional codes for generating high-informative images through a specific generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In Zhao and Bilen [2022], they learn the vectors that are put into the GAN generator for producing informative images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Concretely, they inverse real examples with a GAN generator and collect corresponding latent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Then, the latent vectors are further optimised with the distribution match algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' By this, the optimised latent vectors can induce more informative synthetic examples with the pre-trained GAN generator and consequently help train neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Besides using the GAN, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022] employ a deterministic multi-formation function as the decoder to create the synthetic data from fewer condensed data, and the condensed data is optimised in an end-to-end fashion by gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Different from code-based DD, decoder-based DD turn to learning a decoder that is employed to produce high-informative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In Such et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020], they propose the generative teaching network that employs a trainable network to generate synthetic images from random noise based on given labels, and the meta-gradients are back-propagated to update the generator other than the synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Intuitively, code-decoder DD combines code-based and decoder-based DD and allows to train both codes and decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Deng and Russakovsky [2022] generate synthetic images via the matrix multipli- cation between codes and decodes, which they call memory and addressing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Then these two elements are optimised with the meta-loss scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022c] achieve the code-decoder DD by feeding codes into networks to generate synthetic data, and they employ an adversarial contrastive constraint to help networks learn different knowledge for better compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' A similar code-decoder factorisation method is also presented in Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a], where they adopt an improved distribution match to optimise the latent codes and decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Through implicitly learning the latent codes and decodes, disentangled DD possesses severe ad- vantages like more compact representation and shared representation across classes and consequently improves the dataset distillation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' It is worth noting that this code-decoder factorisation is compatible with the aforementioned distillation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, the exploration of synthetic data generation and distillation schemes can promote dataset distillation in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 6 Performance Comparison To demonstrate the effectiveness of dataset distillation, we collect and summarize the classification performance of some characteristic dataset distillation approaches on the following image datasets: MNIST [LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 1998], SVHN [Netzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2011], CIFAR-10/100 [Krizhevsky and Hinton, 2009], and Tiny ImageNet [Le and Yang, 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Details of these datasets are presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Recently, Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a] publish a benchmark in terms of dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, it only contains five DD methods, and we provide a comprehensive comparison of over 15 existing dataset distillation methods in this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' MNIST is a black-and-white dataset and consists of 60, 000 training images and 10, 000 test images from 10 different classes, and each example has the shape of 28 × 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' SVHN is a colourful dataset and consists of 73, 257 digits and 26, 032 digits for training and testing, respectively, and examples in SVHN are 32 × 32 RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' CIFAR-10/100 are composed of 50, 000 training images and 10, 000 test images from 10 and 100 different classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The RGB images in CIFAR-10/100 have the shape of 32 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Tiny ImageNet consists of 100, 000 and 10, 000 64 × 64 RGB images from 200 different classes for training and testing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' An important factor to affect the test accuracy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' the distilled dataset is the distillation budget, which constrains the size of the distilled dataset by the notion of Images allocated Per Class (IPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Usually, the distilled dataset is set to have the same number of classes as the target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, for a target dataset with 100 classes, setting the distillation budget to IPC = 10 suggests there are totally 10 × 100 = 1, 000 images in the distilled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For a comprehensive comparison, we also present the test accuracy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' random select, coreset approach, and the original target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For the coreset approach, the Herding algorithm [Rebuffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2017] is employed in this survey due to 10 Table 1: Performance comparison of different dataset distillation methods on MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Methods Distillation schemes Accuracy IPC=1 IPC=10 IPC=50 Random 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 Herding 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 DD (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2018]) BPTT 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 LD (Bohdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020]) BPTT 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 DC (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2021]) Gradient match 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 DSA (Zhao and Bilen [2021b]) Gradient match 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 DCC (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b]) Gradient match MTT (Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a]) Trajectory match 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 FTD (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Trajectory match DM (Zhao and Bilen [2021a]) Distribution match 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 CAFE (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Distribution match 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 KIP (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020, 2021]) Kernel regression 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 FRePo (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 RFAD (Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 IDC (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022])∗ Gradient match Deng and Russakovsky [2022]∗ BPTT 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 Haba (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022c])∗ Trajectory match KFS (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a])∗ Distribution match Whole dataset 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 its superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For most of the methods, ConvNet [Gidaris and Komodakis, 2018] is the default architecture to obtain the test accuracy if there is no specific annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The empirical comparison results are presented in Table 1-6 associated with different datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' As shown in these tables, many methods are not tested on some datasets due to the little meaning for easy datasets or scalability problems for large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' We preserve these blanks in tables for a more fair and comprehensive comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To make the comparison clear, we use the bold number for the best of two methods in each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Besides, we note the disentangled DD methods with ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Based on the performance comparison in the tables, we have several observations as follows: Dataset distillation can be realised on many datasets with various sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The performance of dataset distillation is significantly ahead of random pick and coreset selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Disentangled dataset distillation by optimising the latent codes and decoders can largely improve the test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' KFS [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022a] has the best performance among different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 7 Application Due to the superior performance in compressing massive datasets, dataset distillation has been widely employed in many application domains that limit training efficiency and storage, including continual learning and neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Furthermore, due to the correspondence between examples and gradients, dataset distillation can also benefit privacy preservation, federated learning, and adversarial robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In this section, we briefly review these applications w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 Continual Learning During the training process, when there is a shift in the training data distribution, the model will suddenly lose the predicted ability on the previous data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' This phenomenon is referred to as catastrophic forgetting and is common in deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To overcome the problem, continual learning is developed to incrementally learn new tasks whilst preserving performance on old tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 11 Table 2: Performance comparison of different dataset distillation methods on FashionMNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Methods Distillation schemes Accuracy IPC=1 IPC=10 IPC=50 Random 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 Herding 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 DD (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2018]) BPTT LD (Bohdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020]) BPTT DC (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2021]) Gradient match 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 DSA (Zhao and Bilen [2021b]) Gradient match 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 DCC (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b]) Gradient match MTT (Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a]) Trajectory match 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 FTD (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Trajectory match DM (Zhao and Bilen [2021a]) Distribution match CAFE (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Distribution match 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 KIP (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020, 2021]) Kernel regression 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 FRePo (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 RFAD (Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 IDC (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022])∗ Gradient match Deng and Russakovsky [2022]∗ BPTT 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 Haba (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022c])∗ Trajectory match KFS (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a])∗ Distribution match Whole dataset 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 Table 3: Performance comparison of different dataset distillation methods on SVHN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Methods Distillation schemes Accuracy IPC=1 IPC=10 IPC=50 Random 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 Herding 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 DD (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2018]) BPTT LD (Bohdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020]) BPTT DC (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2021]) Gradient match 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 DSA (Zhao and Bilen [2021b]) Gradient match 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 DCC (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b]) Gradient match 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 MTT (Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a]) Trajectory match FTD (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Trajectory match DM (Zhao and Bilen [2021a]) Distribution match CAFE (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Distribution match 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 KIP (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020, 2021]) Kernel regression 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 FRePo (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression RFAD (Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 IDC (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022])∗ Gradient match 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 Deng and Russakovsky [2022]∗ BPTT 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 Haba (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022c])∗ Trajectory match 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 KFS (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a])∗ Distribution match 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 Whole dataset 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 12 Table 4: Performance comparison of different dataset distillation methods on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Methods Distillation schemes Accuracy IPC=1 IPC=10 IPC=50 Random 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 Herding 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 DD (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2018]) BPTT 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 LD (Bohdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020]) BPTT 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 DC (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2021]) Gradient match 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 DSA (Zhao and Bilen [2021b]) Gradient match 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 DCC (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b]) Gradient match 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 MTT (Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a]) Trajectory match 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 FTD (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Trajectory match 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 TESLA (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b]) Trajectory match 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 DM (Zhao and Bilen [2021a]) Distribution match 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 CAFE (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Distribution match 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 KIP (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020, 2021]) Kernel regression 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 FRePo (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 RFAD (Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 IDC (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022])∗ Gradient match 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 Deng and Russakovsky [2022]∗ BPTT 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 Haba (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022c])∗ Trajectory match 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 KFS (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a])∗ Distribution match 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 Whole dataset 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 Table 5: Performance comparison of different dataset distillation methods on CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Methods Distillation schemes Accuracy IPC=1 IPC=10 IPC=50 Random 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 Herding 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 DD (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2018]) BPTT LD (Bohdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020]) BPTT 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 DC (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2021]) Gradient match 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 DSA (Zhao and Bilen [2021b]) Gradient match 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 DCC (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b]) Gradient match 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 MTT (Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a]) Trajectory match 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 FTD (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Trajectory match 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 TESLA (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b]) Trajectory match 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 DM (Zhao and Bilen [2021a]) Distribution match 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 CAFE (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Distribution match 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 KIP (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020, 2021]) Kernel regression 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 FRePo (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 RFAD (Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 IDC (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022])∗ Gradient match 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 Deng and Russakovsky [2022]∗ BPTT 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 Haba (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022c])∗ Trajectory match 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 KFS (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a])∗ Distribution match 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 Whole dataset 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 13 Table 6: Performance comparison of different dataset distillation methods on Tiny ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Methods Distillation schemes Accuracy IPC=1 IPC=10 IPC=50 Random 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 Herding 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 DD (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2018]) BPTT LD (Bohdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020]) BPTT DC (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2021]) Gradient match DSA (Zhao and Bilen [2021b]) Gradient match DCC (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b]) Gradient match MTT (Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a]) Trajectory match 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 FTD (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Trajectory match 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 TESLA (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b]) Trajectory match 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 DM (Zhao and Bilen [2021a]) Distribution match 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 CAFE (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Distribution match KIP (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020, 2021]) Kernel regression FRePo (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 RFAD (Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022]) Kernel regression IDC (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022])∗ Gradient match Deng and Russakovsky [2022]∗ BPTT 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 Haba (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022c])∗ Trajectory match KFS (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a])∗ Distribution match 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 Whole dataset 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 [Rebuffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2017, Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' A common method used in continual learning is the replay- based strategy, which allows a limited memory to store a few training examples for rehearsal in the following training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, the key to replay-based strategy is to select high-informative training examples to store.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Benefiting from extracting essence of datasets, dataset distillation technique is employed to compress data for the memory with limited storage [Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2021, Carta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Because the incoming data has a changing distribution, the frequency is high for updating the elements in memory, which leads to strict requirements for the efficiency of dataset distillation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To conveniently embed dataset distillation into replay-based strategy, Wiewel and Yang [2021], Sangermano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022] decompose the process of synthetic data generation by linear or non- linear combination, and thus fewer parameters are optimised during the dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Besides enhancing replay-based methods, dataset distillation can also learn a sequence of stable datasets, and the network trained on these stable datasets will not suffer from catastrophic forgetting [Masarczyk and Tautkute, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 Neural Architecture Search For a given dataset, the technique of neural architecture search (NAS) aims to find an optimal archi- tecture from thousands of network candidates for better generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The process of NAS usually includes training the network candidates on a small proxy of the original dataset to save training time, and the generalisation ranking can be estimated according to these trained network candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, it is important to design the proxy dataset such that models trained on it can reflect the model’s true performance in terms of the original data, while the size of the proxy dataset requires control for the sake of efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To construct proxy datasets, conventional methods are developed, including random selection or greedy search, without altering the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Such et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020] first propose to optimise a highly informative dataset as the proxy for network candidate selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' More following works consider NAS as a task for testing their proposed dataset distillation algorithms [Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2021, Zhao and Bilen, 2021b,a, Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022, Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Through simulation on the synthetic dataset, a fairly accurate generalisation ranking can be collected for selecting the optimal architecture whilst training time is considerably reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 14 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='3 Privacy Protection As the over-parameterized neural networks easily memorize all the training data, there exists the risk for privacy leakage via inferring the well-trained networks [Shokri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2017, Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018, Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018, Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' With the synthetic dataset, the network avoids explicitly training on the original dataset and consequently helps protect the data privacy [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Moreover, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a] employ dataset distillation to generate private training sets by adding differential privacy noise to the gradients of synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Theoretically, Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022], Anonymous [2023] build the connection between dataset distillation and differential privacy, and prove the superiority of dataset distillation in privacy preservation compared to conventional private data generation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='4 Federated Learning Federated learning has received increasing attention in training neural networks in the past few years due to its advantages in distributed training and private data protection [Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' The federated learning framework consists of multiple clients and one central server, and each client pos- sesses exclusive data for training the corresponding local model [McMahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In one round of federated learning, clients transmit the induced gradients or model parameters to the server after training with their exclusive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Then the central server aggregates received gradients or parameters to update the model parameters and broadcast new parameters to clients for the next round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In the federated learning scenario, the data distributed in different clients are often non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='d, which causes a biased minimum w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' local model and significantly hinders the convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Consequently, the data heterogeneity remarkably imposes a communication burden between the server and clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Goetz and Tewari [2020] propose to distil a small set of synthetic data from original exclusive data by matching gradients, and then the synthetic data instead of a large number of gradients are transmitted to the server for model updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Other distillation schemes such as meta-loss [Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022b, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020], distribution match [Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022], and kernel regression [Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022], are also employed to compress the exclusive data for alleviating communication cost at each transition round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b] discover that synthetic data generated by dataset distillation are still het- erogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To tackle the problem, they propose two strategies of dynamic weight assignment and meta knowledge sharing during the distillation process, which significantly accelerate the convergence speed of federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Apart from compressing the local data, Pi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022] also distil data via trajectory match in the server, which allows the synthetic data to possess global information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Then the distilled data can be used to fine-tune the server model for convergence speed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='5 Adversarial Robustness Deep learning networks with standard training is brittle to adversarial attacks, and adding impercep- tible perturbation to the input can completely fool the network [Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Adversarial training has been widely used to tackle this problem via continuously feeding adversarial examples during the training process [Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, the generation of adversarial examples requires multi-step gradient ascent, which considerably undermines the training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Recently, Tsilivis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022], Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022] employ dataset distillation to extract the information of adver- sarial examples and generate robust datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Then standard network training on the distilled robust dataset is enough to achieve satisfied robustness to adversarial perturbation, which incredibly saves computing resources compared to the expensive adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='6 Other Applications Apart from the aforementioned applications, we also summarise other applications in terms of dataset distillation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' As the synthetic dataset owns a small size, it has been applied for explainability [Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Concretely, due to the small size of synthetic data, it is easy to measure how the synthetic examples influence the prediction of test examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' If the test and training images both rely on the same synthetic images, the training image will greatly influence the prediction of the test image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In other words, the synthetic set becomes a bridge to connect the train and test examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 15 Using the characteristic of capturing the essence of datasets, [Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022b, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022b] utilise dataset distillation for visual design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b] generate the representative textures by randomly cropping the synthetic images during the distillation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In addition to extracting texture, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b] impose the synthetic images to model the outfit compatibility through dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Besides the continuous image domain, dataset distillation has also been extended to discrete data such as graphs [Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022b,a, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022a] and recommender systems [Sachdeva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022b] first formulate this dataset distillation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' graph data, and successfully distil a small, condensed graph via gradient match scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Due to the computational inefficiency in distilling pairwise relation for graph nodes, Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a] turn to learn a probabilistic graph model, which allows a differentiable optimisation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' the discrete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' They also employ one-step strategy for further distillation speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Moreover, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022c] accelerate the graph distillation through distribution match from the perspective of receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For the recommender system, Sachdeva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022] distil the massive dataset to a continuous prior for tackling the discrete data problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Due to its small size and abstract visual information, the distilled data can also be applied in medicine, especially in medical image sharing [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Empirical studies on gastric X-ray images have shown the advantages of DD in medical image transition and anonymisation of patient information [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Through dataset distillation, hospitals can share their valuable medical data at lower cost and risk to build powerful computer-aided diagnosis systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 8 Challenges and Directions As its superiority in compressing datasets and training speedup, dataset distillation has promising prospects in a wide range of areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In the following words, we will discuss existing challenges in terms of dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Furthermore, we investigate the plausible directions and provide insights on dataset distillation to promote future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='1 Challenges For dataset distillation, there are two main ingredients: (1) measure the difference between the knowl- edge of synthetic data and real (original) data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' and (2) update the synthetic data to narrow the knowledge difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Based on the ingredients, we discuss the challenges of dataset distillation from the perspectives of (1) the definition of the difference between knowledge, (2) the form of synthetic datasets, (3) the evaluation of synthetic datasets, and (4) the theory behind dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Dataset distillation methods aim to transfer the knowledge of a target dataset into a small synthetic dataset to achieve comparable generalisation on the original data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Albeit the dataset’s knowledge is an abstract notion, many DD approaches indirectly measure the difference between their knowledge via various proxy losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, the existing definition w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' the knowledge difference is not consistent with the original comparable generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For example, the meta-loss and kernel regression loss measure the knowledge difference with the error in terms of the specific target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Through this, one can only get a similar training error other than test error w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' the target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' As for match schemes, they intuitively make the knowledge concrete with a series of parameters or features, which are matched for knowledge transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, a more consistent definition of knowledge difference should be investigated to improve the dataset distillation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Albeit most dataset distillation methods directly update the synthetic images, implicit optimising codes and decoders that cooperatively generate synthetic data outperform the direct optimisation by a large margin, as discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' From this, it can be seen that the pre-defined form of the code and decoder has a remarkable influence on the distillation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Many disentanglement methods on synthetic data are based on intuition and lack rigorous analysis, and only some vague reasons are proposed, such as decreasing redundant information across different classes and learn- ing more compact representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Hence, it is necessary to explore the influence of the pre-defined disentanglement on the dataset distillation for learning a more informative distilled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In practical dataset distillation, the optimisation of synthetic data is closely related to the specific network architecture (or kernels in KRR approach), and there is naturally a performance drop when the learned synthetic dataset is applied to other unseen architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, it is one-sided to compare different DD methods only on one or a few architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To compare different DD methods 16 in a more comprehensive scenario, it is essential to set a reasonable baseline consisting of multiple handpicked network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In spite of various DD approaches have been emerging, there are few works that discuss the theory behind dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Nevertheless, developing the theory is extremely necessary and will con- siderably promote the dataset distillation by directly improving the performance and also suggesting the correct direction of development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For example, the theory can help derive a better definition of the dataset knowledge and consequently increase the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Besides, the theoretical correlation between the distillation budget (the size of synthetic datasets) and performance can provide an upper bound on the test error, which can offer a holistic understanding and avoid researchers blindly improv- ing the DD performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Hence, a solid theory is indispensable to take the development of DD to the next stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='2 Future Directions Albeit the existing DD methods have shown impressive results in compressing massive datasets, there are still many areas that are worth to explore to promote the dataset distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In this section, we will discuss some promising directions that shed light on future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Fine-tune with synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' With the rapid development of foundational models [Bom- masani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2021], the mainstream deep learning pipeline gradually converts from scratch learning to pre-trained learning or transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For the foundational models of BERT [Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018], GPT-3 [Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020], and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', they are trained with billions of data to learn an appropriate latent representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' With only fine-tuning on specific datasets, the foundational models can achieve extraordinary performance on various downstream tasks and outperform models learned from scratch by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To this end, conducting dataset distillation in the fine-tune regime will be a promis- ing direction to cooperate with foundational models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' With the synthetic data, the fine-tine process will be faster whilst the downstream performance is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Albeit possessing a similar goal, dis- tilling datasets based on pre-trained models might be more challenging compared to initialised models because the pre-trained models are harder to collect for repeated distillation to enhance synthetic datasets’ robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Apart from condensing datasets in the fine-tune stage, it is also an interesting topic to distil datasets in the vanilla training stage so that models trained on synthetic datasets can be easier or faster to fine-tune by specific datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Initialisation augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Benefiting from excellent efficiency and compatibility, data aug- mentation becomes an essential strategy to enhance the model’s generalisation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' data distribution [Shorten and Khoshgoftaar, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' From basic cropping and flipping [Zagoruyko and Komodakis, 2016] to the mixup method [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018], data augmentation efficiently constructs variants of training images with prior rules to improve the model’s robustness to patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In dataset distillation, the syn- thetic datasets are also sensitive to the parameter initialisation and network architecture adopted in the distillation process [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Existing works alleviate this problem by trivial repeated distillation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' different initialisations and then taking an expectation [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' However, the random selection of initialisation is not an elaborate strategy and has limited improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2022] shows that employing the early-stage models trained with a few epochs as the initialisation can achieve better distillation performance, and they further propose weight perturbation methods to efficiently generate early-stage models for repeated distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Therefore, it is important for many dataset distillation algorithms to investigate how to design the initialisation augmentation so that synthetic datasets achieve better generalisation to different initialisations and architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Pre-processing of target datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' To better extract knowledge from training data, pre- processing, such as normalisation, is a common approach that reshapes the structure of data by well-designed mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In existing dataset distillation methods, only a few works discuss the influ- ence of pre-processing on final distillation performance [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2020, 2021], Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' [2022a], they pre-process target datasets with ZCA whitening [Kessy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=', 2018] before distillation and receive satisfying empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Through elaborate pre-processing, more explicit information can emerge, and the target datasets might be easier to be distilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Besides, this pre-processing is compatible with existing dataset distillation algorithms and thus can be deployed without extra effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' For this reason, it is useful to investigate the pre-processing of target datasets for distillation acceleration and also performance improvement.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Dataset condensation with distribution matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' CoRR, abs/2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='04181, 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='org/abs/2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='04181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Bo Zhao and Hakan Bilen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Dataset condensation with differentiable siamese augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In Inter- national Conference on Machine Learning, pages 12674–12685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' PMLR, 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Bo Zhao and Hakan Bilen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Synthesizing informative training samples with gan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='07513, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Bo Zhao, Konda Reddy Mopuri, and Hakan Bilen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Dataset condensation with gradient matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In International Conference on Learning Representations, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' URL https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='net/ forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='id=mSAKhLYLSsl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Yanlin Zhou, George Pu, Xiyao Ma, Xiaolin Li, and Dapeng Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Distilled one-shot federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' arXiv preprint arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='07999, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Yongchao Zhou, Ehsan Nezhadarya, and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Dataset distillation using neural feature regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' In Alice H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors, Advances in Neural Information Processing Systems, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' URL https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='net/forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content='id=2clwrA2tfik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} +page_content=' 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfcg8E/content/2301.05603v1.pdf'} diff --git a/WtE1T4oBgHgl3EQfvgVo/content/tmp_files/2301.03400v1.pdf.txt b/WtE1T4oBgHgl3EQfvgVo/content/tmp_files/2301.03400v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..437d868a7bfc88033bcf611977cef383d97731fc --- /dev/null +++ b/WtE1T4oBgHgl3EQfvgVo/content/tmp_files/2301.03400v1.pdf.txt @@ -0,0 +1,776 @@ +GPDs in asymmetric frames +Shohini Bhattacharya,𝑎,∗ Krzysztof Cichy,𝑏 Martha Constantinou,𝑐 Jack Dodson,𝑐 +Xiang Gao,𝑑 Andreas Metz,𝑐 Swagato Mukherjee,𝑎 Aurora Scapellato,𝑐 Fernanda +Steffens𝑒 and Yong Zhao𝑑 +𝑎Brookhaven National Laboratory, +Upton, New York 11973, USA +𝑏Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 2 +61-614 Poznań, Poland +𝑐Temple University, +Philadelphia, PA 19122 - 1801, USA +𝑑Argonne National Laboratory, +Lemont, IL 60439, USA +𝑒Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn +Nussallee 14-16, 53115 Bonn +E-mail: sbhattach@bnl.gov +It is often taken for granted that Generalized Parton Distributions (GPDs) are defined in the +"symmetric" frame, where the transferred momentum is symmetrically distributed between the +incoming/outgoing hadrons. However, such frames pose computational challenges for the lattice +QCD practitioners. In these proceedings, we lay the foundation for lattice QCD calculations of +GPDs in "asymmetric" frames, where the transferred momentum is not symmetrically distributed +between the incoming/outgoing hadrons. The novelty of our work relies on the parameterization of +the matrix elements in terms of Lorentz-invariant amplitudes, which not only helps in establishing +relations between the said frames but also helps in isolating higher-twist contaminations. As an +example, we focus on the unpolarized GPDs for spin-1/2 particles. +The 39th International Symposium on Lattice Field Theory (Lattice2022), +8-13 August, 2022 +Bonn, Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.03400v1 [hep-lat] 9 Jan 2023 + +GPDs in asymmetric frames +Shohini Bhattacharya +Figure 1: Graphical representation of the two frames employed in this work. Left plot: Symmetric frame. +Right plot: Asymmetric frame. +1. +Introduction +Generalized Parton Distributions (GPDs) are the 3D generalizations of the collinear Parton +Distribution Functions (PDFs) [1, 2]. There are several motivations to study GPDs: +• For 𝜉 = 0 the Fourier transforms of the GPDs are related to the impact-parameter dis- +tributions which provide information about the three-dimensional distribution of partons — +(one-dimensional) longitudinal momentum distribution; (two-dimensional) transverse spatial +distribution, see for example Ref. [3]. +• Twist-2 GPDs are related to the total angular momentum of partons [1]. +• One should look for other ways to access GPDs because of the challenges involved in their +extraction through the processes of Deep Virtual Compton Scattering (DVCS) [2] and meson +production [4]. +Challenges are caused by the sensitivity of differential cross-sections to +only 𝑥-integrals of GPDs, and not GPDs themselves [1, 2]. Therefore, it is desirable to +extract the 𝑥-dependence of the GPDs from first principles within Lattice QCD. However, +for a very long time this was not possible because of time-dependence of these quantities. +As a result, all of the lattice calculations were limited to the calculations of lowest Mellin +moments of the GPDs, see Ref. [5]. In 2013, there was a path-breaking proposal by X. Ji to +calculate instead auxiliary quantities called "quasi-GPDs" [6–8]. This approach relies on the +extraction of matrix elements for boosted hadrons involving spatially-separated fields. Ever +since this proposal, enormous progress has taken place, see some reviews [9–11]. In fact, +Ref. [12] provides the first-ever lattice-QCD results of the unpolarized and helicity GPDs +of the nucleon from the quasi-distribution approach. Lattice QCD calculations have the +potential to not only provide insight into the experimentally-inaccessible features of GPDs, +but also help in extracting the "full" GPDs from the existing experimental data. +2. +Formalisms to calculate GPDs in asymmetric frames +2.1 Frames: Symmetric and asymmetric +The most widely used frame of reference to calculate GPDs is the symmetric frame. For +this frame, the momentum transfer is symmetrically distributed between the incoming (𝑝𝑖) and the +outgoing hadrons (𝑝 𝑓 ) (see left plot of Fig. 1). However, one can also think of a frame where the +2 + +-z/2 +z/2 +s +△s +Ps +Ps+ +2. +GPDs +2 +t=2≥/2 +z/2 +Pa +GPDs +t= aGPDs in asymmetric frames +Shohini Bhattacharya +momentum transfer is not equally shared between the incoming and outgoing hadrons, but is rather +exclusively applied to the incoming hadron (see, right plot of Fig. 1). Such a frame is known as an +asymmetric frame. +Lattice calculations of GPDs has primarily been confined to symmetric frames. However, +such frames pose serious computational challenges because they require separate calculation for +each values of the momentum transfer (Δ), resulting in increased computational costs. So the +question that we strive to address in this work is: Can we lay a formalism to systematically perform +lattice calculations of GPDs in asymmetric frames (which is expected to be computationally less +expensive)? In this work, we argue that there are two approaches to solving this question. In the +first approach, we will show that it is possible to relate the two frames via an appropriate Lorentz +transformation. In the second approach, we will propose a Lorentz-covariant decomposition of +the lattice matrix elements in terms of Lorentz-invariant (frame-independent) amplitudes. These +amplitudes will then be used to make connections between the two frames. As a byproduct, we will +show that this approach helps in identifying higher-twist contaminations which may be present in +quasi-GPDs at finite values of momentum. +2.2 Lorentz transformation approach +In this section, we explain the Lorentz transformation (LT) approach. First, it is straight- +forward to realize that a LT along the 𝑧-direction is not optimal for lattice calculations because this +requires a spatial operator distance (say 𝑧 = (0, 0⊥, 𝑧3 ≠ 0)) to pick up a temporal component (that +is 𝑧 +LT +−−→ (𝑧0 ≠ 0, 0⊥, 𝑧3)). However, a LT applied to any direction transverse to the 𝑧-axis does +not change the spatial nature of operator distances. This transformation is called as the "transverse +boost". We explain this by considering a transverse boost in the 𝑥-direction and for the simplest case +of zero skewness. The logic can be generalized for any general transverse boost and for arbitrary +values of skewness. +We begin by relating the incoming state in the two frames, 𝑝𝑠 +𝑖 = (𝐸𝑠 +𝑖 , −Δ1,𝑠/2, 0, 𝑃3) and +𝑝𝑎 +𝑖 = (𝐸 𝑎 +𝑖 , −Δ1,𝑎, 0, 𝑃3). LT provides 𝑝𝑠 = ΛLT 𝑝𝑎, +������ +� +𝐸𝑠 +𝑖 +𝑝1,𝑠 +𝑖 +𝑝2,𝑠 +𝑖 +𝑝3,𝑠 +𝑖 +������ +� += +������ +� +𝛾 +−𝛾𝛽 +0 +0 +−𝛾𝛽 +𝛾 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +������ +� +× +������ +� +𝐸𝑎 +𝑖 +−Δ1,𝑎 +0 +𝑃3 +������ +� +. +(1) +This gives, +𝐸𝑠 +𝑖 = 𝛾(𝐸 𝑎 +𝑖 + 𝛽Δ1,𝑎) , +(2) +and, +𝑝1,𝑠 +𝑖 += −𝛾(𝛽𝐸𝑎 +𝑖 + Δ1,𝑎) +→ +Δ1,𝑠 = 2𝛾(𝛽𝐸𝑎 +𝑖 + Δ1,𝑎) . +(3) +Similarly, the outgoing state in the two frames, 𝑝𝑠 +𝑓 = (𝐸𝑠 +𝑓 , Δ1,𝑠/2, 0, 𝑃3) and 𝑝𝑎 +𝑓 = (𝐸 𝑎 +𝑓 , 0, 0, 𝑃3) +can also be related. (Keep in mind that the energies of the incoming and outgoing states are different +in the asymmetric frame.) We then find, +𝐸𝑠 +𝑖 = 𝛾𝐸𝑎 +𝑓 , +(4) +3 + +GPDs in asymmetric frames +Shohini Bhattacharya +and, +𝑝1,𝑠 +𝑓 += −𝛾𝛽𝐸𝑎 +𝑓 +→ +Δ1,𝑠 = −2𝛾𝛽𝐸𝑎 +𝑓 . +(5) +From Eqs. (2) and (4), we find, +𝛽 = − +� 𝐸 𝑎 +𝑖 − 𝐸 𝑎 +𝑓 +Δ1,𝑎 +� +. +(6) +From Eqs. (3) and (5), we find, +𝛽 = − +Δ1,𝑎 +𝐸𝑎 +𝑖 + 𝐸 𝑎 +𝑓 +. +(7) +Then, Eqs. (6) and (7) imply, +Δ1,𝑎 = +√︃ +(𝐸 𝑎 +𝑖 )2 − (𝐸𝑎 +𝑓 )2 . +(8) +Hence, 𝛽 can be written as, +𝛽 = − +� +�𝐸 𝑎 +𝑖 − 𝐸 𝑎 +𝑓 +𝐸 𝑎 +𝑖 + 𝐸𝑎 +𝑓 +< 0 . +(9) +This implies Δ0,𝑎 < 0, and +𝛾 = +1 +√︁ +1 − 𝛽2 = +� +�𝐸 𝑎 +𝑖 + 𝐸 𝑎 +𝑓 +2𝐸 𝑎 +𝑓 +. +(10) +Therefore, by using the expressions for (𝛽, 𝛾), we can write down uniquely the symmetric frame +variables (𝐸𝑠 +𝑖 , Δ1,𝑠) in terms of the asymmetric frame variables (𝐸 𝑎 +𝑖 , 𝐸𝑎 +𝑓 , Δ1,𝑎): The energy should +be, +𝐸𝑠 +𝑖 = 𝛾𝐸𝑎 +𝑓 = +√︄ +𝐸𝑎 +𝑓 (𝐸 𝑎 +𝑖 + 𝐸 𝑎 +𝑓 ) +2 +, +(11) +and the transverse-momentum transfer, +Δ1,𝑠 = −2𝛾𝛽𝐸𝑎 +𝑓 , +or, +Δ1,𝑠 = 2 +√︄ +𝐸 𝑎 +𝑓 (𝐸 𝑎 +𝑖 − 𝐸 𝑎 +𝑓 ) +2 += 2 +� +� +𝐸𝑎 +𝑓 +2(𝐸 𝑎 +𝑖 + 𝐸 𝑎 +𝑓 ) Δ1,𝑎 . +(12) +We repeat that the above method can be generalized for �Δ⊥ = (Δ1, Δ2) and for arbitrary values of +skewness. +Now that we have sketched the idea of how to relate the kinematical variables between the two +frames, we proceed to understand how the matrix elements defining quasi-GPDs transform between +4 + +GPDs in asymmetric frames +Shohini Bhattacharya +the two frames. For this purpose, we focus on spin-0 particles such as the pion. (The method can +be generalized for spin-1/2 particles.) The (unpolarized) pion GPD is defined as, +𝐹𝜇(𝑧, 𝑃, Δ) = ⟨𝑝 𝑓 | ¯𝑞(− 𝑧 +2)𝛾𝜇 W(− 𝑧 +2, 𝑧 +2)𝑞( 𝑧 +2)|𝑝𝑖⟩ . +(13) +Here, W is a straight Wilson line required to make the correlator gauge invariant. Historically, +(unpolarized) quasi-GPDs have been defined through matrix elements of the operator 𝛾0, see for +instance Refs. [12, 13]. By applying the transverse boost Eq. (1), we find that the matrix element +⟨..𝛾0..⟩ in the symmetric frame can be expressed in terms of matrix elements of different operators +⟨..(𝛾0 + 𝛾1)..⟩ in the asymmetric frame, +⟨𝑝 𝑓 | ¯𝑞(− 𝑧 +2)𝛾0 W(− 𝑧3 +2 , 𝑧3 +2 ) 𝑞( 𝑧 +2)|𝑝𝑖⟩𝑠 = 𝛾⟨𝑝 𝑓 | ¯𝑞(− 𝑧 +2)𝛾0 W(− 𝑧3 +2 , 𝑧3 +2 ) 𝑞( 𝑧 +2)|𝑝𝑖⟩𝑎 +− 𝛾𝛽⟨𝑝 𝑓 | ¯𝑞(− 𝑧 +2)𝛾1 W(− 𝑧3 +2 , 𝑧3 +2 ) 𝑞( 𝑧 +2)|𝑝𝑖⟩𝑎. +(14) +This equation simply reflects how the 0th component of a 4-vector changes under the Lorentz +transformation Eq. (1). Therefore, this implies that a transverse boost that fixes (𝛽, 𝛾) (Eqs. (9) +and (10)) allows for an exact calculation of quasi-GPDs in the symmetric frame through matrix +elements of the asymmetric frame. However, Eq. (14) also shows that a quasi-GPD defined through +the operator 𝛾0 is not Lorentz invariant. In the limit of a large momentum, we recover, +lim +𝑃3→∞⟨..𝛾0..⟩𝑠 ≈ ⟨..𝛾0..⟩𝑎 + O +� 1 +𝑃3 +� +⟨..𝛾1..⟩𝑎 → ⟨..𝛾0..⟩𝑎 , +(15) +which means that the contribution from the matrix element ⟨..𝛾1..⟩ maybe viewed as a power +correction at finite values of momentum 𝑃3. +2.3 Amplitude approach: Spin-1/2 particles +In this section, we explain the amplitude approach through the example of spin-1/2 particles, +such as the proton. (We refer to Ref. [14] for details on spin-0 particles.) As a first step, we +build a Lorentz-covariant decomposition of the vector matrix element in terms of the available +vectors (𝑃𝜇, 𝑧𝜇, Δ𝜇). By considering constraints from parity, we find that the general structure of +the vector matrix element involves eight linearly-independent Dirac structures multiplied by eight +Lorentz-invariant (frame-independent) amplitudes, +𝐹𝜇(𝑧, 𝑃, Δ) = ¯𝑢(𝑝 𝑓 , 𝜆′) +� 𝑃𝜇 +𝑚 𝐴1 + 𝑚𝑧𝜇𝐴2 + Δ𝜇 +𝑚 𝐴3 + 𝑖𝑚𝜎𝜇𝑧 𝐴4 + 𝑖𝜎𝜇Δ +𝑚 +𝐴5 ++ 𝑃𝜇𝑖𝜎𝑧Δ +𝑚 +𝐴6 + 𝑚𝑧𝜇𝑖𝜎𝑧Δ𝐴7 + Δ𝜇𝑖𝜎𝑧Δ +𝑚 +𝐴8 +� +𝑢(𝑝𝑖, 𝜆) . +(16) +Here 𝜎𝜇𝜈 ≡ +𝑖 +2 (𝛾𝜇𝛾𝜈 − 𝛾𝜈𝛾𝜇), 𝜎𝜇𝑧 ≡ 𝜎𝜇𝜌𝑧𝜌, 𝜎𝜇Δ ≡ 𝜎𝜇𝜌Δ𝜌, 𝜎𝑧Δ ≡ 𝜎𝜌𝜏𝑧𝜌Δ𝜏, 𝑧 ≡ (𝑧0 = +0, 𝑧⊥ = 0⊥, 𝑧3 ≠ 0). (For a derivation of Eq. (16), we refer to Ref. [15]. See also Ref. [16] where +the vector matrix element has been parameterized in the momentum space for a straight Wilson +line.) For brevity, we use the compact notation 𝐴𝑖 ≡ 𝐴𝑖(𝑧 · 𝑃, 𝑧 · Δ, Δ2, 𝑧2), with 𝐴𝑖’s being the +Lorentz-invariant amplitudes whose arguments are functions of Lorentz scalars1. +1In the literature, the amplitudes have also been called generalized Ioffe time distributions (ITDs) [13]. +5 + +GPDs in asymmetric frames +Shohini Bhattacharya +For spin-1/2 particles, the vector matrix element can be parameterized in terms of two light- +cone GPDs 𝐻 and 𝐸 [17], +𝐹+(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) = ¯𝑢𝑠/𝑎(𝑝𝑠/𝑎 +𝑓 +, 𝜆′) +� +𝛾+𝐻(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) ++ 𝑖𝜎+𝜇Δ𝑠/𝑎 +𝜇 +2𝑚 +𝐸(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) +� +𝑢𝑠/𝑎(𝑝𝑠/𝑎 +𝑖 +, 𝜆) . +(17) +By using 𝜇 = + in Eq. (16), followed by a subsequent change of basis, it is possible to map the 𝐴𝑖’s +onto the 𝐻 and 𝐸 GPDs in Eq. (17). The results are, +𝐻(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) = 𝐴1 + Δ+,𝑠/𝑎 +𝑃+,𝑠/𝑎 𝐴3 , +(18) +𝐸(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) = −𝐴1 − Δ+,𝑠/𝑎 +𝑃+,𝑠/𝑎 𝐴3 + 2𝐴5 + 2𝑃+,𝑠/𝑎𝑧−𝐴6 + 2Δ+,𝑠/𝑎𝑧−𝐴8 . +(19) +Keep in mind that the arguments of the 𝐴𝑖’s for light-cone GPDs have no dependence on 𝑧2. Also, +𝑧𝜇 = (0, 𝑧−, 0⊥) and Δ+/𝑃+ = 𝑧 · Δ/𝑧 · 𝑃, etc. Thus, it is possible to write the above expressions in +a Lorentz invariant way as, +𝐻(𝑧 · 𝑃𝑠/𝑎, 𝑧 · Δ𝑠/𝑎, (Δ𝑠/𝑎)2) = 𝐴1 + Δ𝑠/𝑎 · 𝑧 +𝑃𝑠/𝑎 · 𝑧 𝐴3 , +(20) +𝐸(𝑧 · 𝑃𝑠/𝑎, 𝑧 · Δ𝑠/𝑎, (Δ𝑠/𝑎)2) = −𝐴1 − Δ𝑠/𝑎 · 𝑧 +𝑃𝑠/𝑎 · 𝑧 𝐴3 + 2𝐴5 + 2𝑃𝑠/𝑎 · 𝑧𝐴6 + 2Δ𝑠/𝑎 · 𝑧𝐴8 . +(21) +This means the light-cone GPDs are frame-independent as long as the Lorentz scalars (𝑧 · 𝑃𝑠/𝑎, 𝑧 · +Δ𝑠/𝑎, (Δ𝑠/𝑎)2) are the same in the two frames. +Next, we turn to the quasi-GPDs H and E, which historically have been defined in terms of +matrix elements of 𝛾0 operator as [18, 19], +𝐹0(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) = ⟨𝑝𝑠/𝑎 +𝑓 +, 𝜆′| ¯𝑞(− 𝑧 +2)𝛾0𝑞( 𝑧 +2)|𝑝𝑠/𝑎 +𝑖 +, 𝜆⟩ += ¯𝑢𝑠/𝑎(𝑝𝑠/𝑎 +𝑓 +, 𝜆′) +� +𝛾0H 𝑠/𝑎 +0 +(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) ++ 𝑖𝜎0𝜇Δ𝑠/𝑎 +𝜇 +2𝑚 +E𝑠/𝑎 +0 +(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) +� +𝑢𝑠/𝑎(𝑝𝑠/𝑎 +𝑖 +, 𝜆) . +(22) +If we use 𝜇 = 0 in Eq. (16), then after performing a change of basis it is possible to map the 𝐴𝑖’s +onto the quasi-GPDs in Eq. (22). The relations in the symmetric frame read, +H 𝑠 +0 (𝑧, 𝑃𝑠, Δ𝑠) = 𝐴1 + Δ0,𝑠 +𝑃0,𝑠 𝐴3 − 𝑚2Δ0,𝑠𝑧3 +2𝑃0,𝑠𝑃3,𝑠 𝐴4 + +� (Δ0,𝑠)2𝑧3 +2𝑃3,𝑠 +− Δ0,𝑠Δ3,𝑠𝑧3𝑃0,𝑠 +2(𝑃3,𝑠)2 +− 𝑧3(Δ𝑠 +⊥)2 +2𝑃3,𝑠 +� +𝐴6 ++ +� (Δ0,𝑠)3𝑧3 +2𝑃0,𝑠𝑃3,𝑠 − (Δ0,𝑠)2Δ3,𝑠𝑧3 +2(𝑃3,𝑠)2 +− Δ0,𝑠𝑧3(Δ𝑠 +⊥)2 +2𝑃0,𝑠𝑃3,𝑠 +� +𝐴8 , +(23) +E𝑠 +0(𝑧, 𝑃𝑠, Δ𝑠) = −𝐴1 − Δ0,𝑠 +𝑃0,𝑠 𝐴3 + 𝑚2Δ0,𝑠𝑧3 +2𝑃0,𝑠𝑃3,𝑠 𝐴4 + 2𝐴5 + +� +− (Δ0,𝑠)2𝑧3 +2𝑃3,𝑠 ++ 𝑃0,𝑠Δ0,𝑠Δ3,𝑠𝑧3 +2(𝑃3,𝑠)2 ++ 𝑧3(Δ𝑠 +⊥)2 +2𝑃3,𝑠 +− 2𝑧3(𝑃0,𝑠)2 +𝑃3,𝑠 +� +𝐴6 + +� +− (Δ0,𝑠)3𝑧3 +2𝑃0,𝑠𝑃3,𝑠 + (Δ0,𝑠)2Δ3,𝑠𝑧3 +2(𝑃3,𝑠)2 ++ Δ0,𝑠𝑧3(Δ𝑠 +⊥)2 +2𝑃0,𝑠𝑃3,𝑠 +− 2𝑧3𝑃0,𝑠Δ0,𝑠 +𝑃3,𝑠 +� +𝐴8 . +(24) +6 + +GPDs in asymmetric frames +Shohini Bhattacharya +On the other hand, the relations in the asymmetric frame read, +H 𝑎 +0 (𝑧, 𝑃𝑎, Δ𝑎) = 𝐴1 + Δ0,𝑎 +𝑃0,𝑎 𝐴3 − +� 𝑚2Δ0,𝑎𝑧3 +2𝑃0,𝑎𝑃3,𝑎 − +1 +(1 + Δ3,𝑎 +2𝑃3,𝑎 ) +𝑚2Δ0,𝑎Δ3,𝑎𝑧3 +4𝑃0,𝑎(𝑃3,𝑎)2 +� +𝐴4 ++ +� (Δ0,𝑎)2𝑧3 +2𝑃3,𝑎 +− +1 +(1 + Δ3,𝑎 +2𝑃3,𝑎 ) +(Δ0,𝑎)2Δ3,𝑎𝑧3 +4(𝑃3,𝑎)2 +− +1 +(1 + Δ3,𝑎 +2𝑃3,𝑎 ) +𝑃0,𝑎Δ0,𝑎Δ3,𝑎𝑧3 +2(𝑃3,𝑎)2 +− 𝑧3(Δ𝑎 +⊥)2 +2𝑃3,𝑎 +� +𝐴6 ++ +� (Δ0,𝑎)3𝑧3 +2𝑃0,𝑎𝑃3,𝑎 − +1 +(1 + Δ3,𝑎 +2𝑃3,𝑎 ) +(Δ0,𝑎)3Δ3,𝑎𝑧3 +4𝑃0,𝑎(𝑃3,𝑎)2 − +1 +(1 + Δ3,𝑎 +2𝑃3,𝑎 ) +(Δ0,𝑎)2Δ3,𝑎𝑧3 +2(𝑃3,𝑎)2 +− 𝑧3(Δ𝑎 +⊥)2Δ0,𝑎 +2𝑃0,𝑎𝑃3,𝑎 +� +𝐴8 , +(25) +E𝑎 +0 (𝑧, 𝑃𝑎, Δ𝑎) = −𝐴1 − Δ0,𝑎 +𝑃0,𝑎 𝐴3 − +� +− 𝑚2Δ0,𝑎𝑧3 +2𝑃0,𝑎𝑃3,𝑎 − +1 +(1 + Δ3,𝑎 +2𝑃3,𝑎 ) +�𝑚2𝑧3 +𝑃3,𝑎 − 𝑚2Δ0,𝑎Δ3,𝑎𝑧3 +4𝑃0,𝑎(𝑃3,𝑎)2 +�� +𝐴4 + 2𝐴5 ++ +� +− (Δ0,𝑎)2𝑧3 +2𝑃3,𝑎 +− +1 +(1 + Δ3,𝑎 +2𝑃3,𝑎 ) +� 𝑃0,𝑎Δ0,𝑎𝑧3 +𝑃3,𝑎 +− (Δ0,𝑎)2Δ3,𝑎𝑧3 +4(𝑃3,𝑎)2 +� +− +1 +(1 + Δ3,𝑎 +2𝑃3,𝑎 ) +�2𝑧3(𝑃0,𝑎)2 +𝑃3,𝑎 +− 𝑃0,𝑎Δ0,𝑎Δ3,𝑎𝑧3 +2(𝑃3,𝑎)2 +� ++ 𝑧3(Δ𝑎 +⊥)2 +2𝑃3,𝑎 +� +𝐴6 + +� +− (Δ0,𝑎)3𝑧3 +2𝑃0,𝑎𝑃3,𝑎 − +1 +(1 + Δ3,𝑎 +2𝑃3,𝑎 ) +� (Δ0,𝑎)2𝑧3 +𝑃3,𝑎 +− (Δ0,𝑎)3Δ3,𝑎𝑧3 +4𝑃 +0,𝑎(𝑃3,𝑎)2 +� +− +1 +(1 + Δ3,𝑎 +2𝑃3,𝑎 ) +�2𝑧3𝑃0,𝑎Δ0,𝑎 +𝑃3,𝑎 +− (Δ0,𝑎)2Δ3,𝑎𝑧3 +2(𝑃3,𝑎)2 +� ++ 𝑧3(Δ𝑎 +⊥)2Δ0,𝑎 +2𝑃0,𝑎𝑃3,𝑎 +� +𝐴8 . +(26) +However, one can think of other definitions of quasi-GPDs. For this purpose, we recall the +position-space matching relation between, for instance, light-cone GPD 𝐻 and quasi-GPD H [13]: +H �𝑧 · 𝑃, −2𝜉(𝑧 · 𝑃), Δ2, 𝑧2, 𝜇2� = +∫ 1 +−1 +𝑑𝑢 ¯𝐶 (𝑢, 𝑧 · 𝑃, 𝜉, 𝑧2, 𝜇2) 𝐻�𝑢(𝑧 · 𝑃), −2𝑢𝜉(𝑧 · 𝑃), Δ2, 𝜇2� . +(27) +Here, ¯𝐶 is the pertubatively-calculable matching coefficient [13] and 𝜇 is the renormalization scale +in the MS scheme. At leading order in 𝛼𝑠, the above formula indicates that H collapses to 𝐻 in the +light-cone limit 𝑧2 → 0, +lim +𝑧2→0 H (𝑧 · 𝑃, 𝑧 · Δ, Δ2, 𝑧2) = 𝐻(𝑧 · 𝑃, 𝑧 · Δ, Δ2, 0) + O(𝛼𝑠) . +(28) +Therefore, a natural way to define the quasi-GPDs H and E is through a Lorentz-invariant gener- +alization of the light-cone definitions in Eqs. (20) and (21) to 𝑧2 ≠ 0, i.e., +H (𝑧 · 𝑃𝑠/𝑎, 𝑧 · Δ𝑠/𝑎, (Δ𝑠/𝑎)2, 𝑧2) = 𝐴1 + Δ𝑠/𝑎 · 𝑧 +𝑃𝑠/𝑎 · 𝑧 𝐴3 , +(29) +E(𝑧 · 𝑃𝑠/𝑎, 𝑧 · Δ𝑠/𝑎, (Δ𝑠/𝑎)2, 𝑧2) = −𝐴1 − Δ𝑠/𝑎 · 𝑧 +𝑃𝑠/𝑎 · 𝑧 𝐴3 + 2𝐴5 + 2𝑃𝑠/𝑎 · 𝑧𝐴6 + 2Δ𝑠/𝑎 · 𝑧𝐴8 , +(30) +where now the arguments of the 𝐴𝑖’s have a non-zero dependence on 𝑧2. We expect the definitions in +Eqs. (29-(30) to have two advantages: First, these definitions may converge faster to the light-cone +7 + +GPDs in asymmetric frames +Shohini Bhattacharya +GPDs because of the similarities in their functional forms with their (respective) light-cone GPDs. +(Such a statement is inspired from Ref. [20], where similar arguments were made for the quasi- +PDFs. See also the next paragraph for explicit explanations.) Second, these definitions differ from +their light-cone GPDs by frame-independent power corrections; contrast with historic definitions +which are frame-dependent. +We now discuss in detail the various definitions of quasi-GPDs: We notice that for finite values +of the momentum, the historic definitions of quasi-GPDs (H 𝑠/𝑎 +0 +(𝐴𝑖; 𝑧) , E𝑠/𝑎 +0 +(𝐴𝑖; 𝑧)) in Eqs. (23)- +(26) involve additional amplitudes that are not present in the light-cone GPDs, Eqs. (20)-(21). +This is not the case for the Lorentz-invariant definitions of quasi-GPDs (H (𝐴𝑖; 𝑧) , E(𝐴𝑖; 𝑧)) in +Eqs. (29)-(30). (Note that this is different from the (unpolarized) quasi-PDF case where arguments +were made in favor of 𝛾0 (against 𝛾3) because of the absence of such additional amplitudes +relative to the (unpolarized) light-cone PDF case [20].) Therefore, the additional amplitudes in +(H 𝑠/𝑎 +0 +(𝐴𝑖; 𝑧) , E𝑠/𝑎 +0 +(𝐴𝑖; 𝑧)) may be viewed as contaminations from explicit power corrections, which +one would have to suppress by going to larger and larger values of momentum. Hence, we believe +that (H (𝐴𝑖; 𝑧) , E(𝐴𝑖; 𝑧)) may converge relatively faster to their (respective) light-cone GPDs, +simply because of the absence of such additional amplitudes. (Of course, (H (𝐴𝑖; 𝑧) , E(𝐴𝑖; 𝑧)) +also have power corrections, but they are implicit within the amplitudes themselves. Our argument +above is for the power corrections that are explicit.) Our reasoning is perhaps too simple and for sure +needs further substantiation. In fact, it may be that the actual convergence of the various definitions +of quasi-GPDs is determined by the underlying dynamics. Note that the Lorentz non-invariance of +the historical definitions of quasi-GPDs implies that the basis vectors (𝛾0, 𝑖𝜎0Δ𝑠/𝑎) do not form a +complete set for spatially-separated bi-local operators for finite values of momentum. Therefore, we +can argue that the Lorentz-invariant definitions are in fact just a redefinition of quasi-GPDs in terms +of a suitable linear combination of operators (which turns out to be 𝛾⊥) that make them functions +of Lorentz scalars [14]. +In Ref. [14] and [21], we compare numerically the different definitions of quasi-GPDs for +𝜉 = 0 to get an idea about the relative size of power corrections. +Finally, we remark on the +matching coefficient for the different definitions of quasi-GPDs: It is known that the GPD matching +coefficient for the operator 𝛾0 reduces to that for the corresponding PDF when 𝜉 = 0, even if +𝑡 ≠ 0 [13]. The PDF matching coefficient for 𝛾0 is for the amplitude 𝐴1, which is also the only +contributing amplitude to the LI definition of the GPD when 𝜉 = 0. Therefore, the matching +coefficients for the 𝛾0 and the LI definitions of the GPDs are equal. We will elaborate this point +more, including the general case of 𝜉 ≠ 0, in a forthcoming publication. +3. +Summary +In these proceedings, we have laid down the theoretical tools to perform lattice QCD calcu- +lations of GPDs in asymmetric frames. We have highlighted two approaches to performing such +calculations: +• Lorentz transformation (LT) approach (Sec. 2.2): We have shown that there exists a LT called +the "transverse boost" (transverse with respect to the Wilson Line) that allows one to uniquely +relate the kinematical variables as well as the matrix elements in the two frames. +8 + +GPDs in asymmetric frames +Shohini Bhattacharya +• Amplitude approach (Sec. 2.3): We have proposed a Lorentz-covariant decomposition of +the vector matrix element in terms of Lorentz-invariant/frame-independent amplitudes. The +amplitudes can be used as tools to relate the two frames. This approach also shows that at +finite values of the boost momentum the historic definitions of quasi-GPDs (defined through +𝛾0) have additional amplitudes that are not present in the light-cone limit. This motivates us +to come up with alternative definitions of quasi-GPDs that may potentially converge faster. +One such candidate can be the case where one chooses the same functional form as the +light-cone GPDs subjected to include 𝑧2 ≠ 0. +Naively, because of the similarity in the +functional forms (or because of the absence of additional amplitudes), one may expect such +a definition of quasi-GPD to converge faster to the light-cone GPD. Such a definition is also +frame-independent, contrary to the historic definitions. +Acknowledgements +This material is based upon work supported by the U.S. Department of Energy, Office of Science, +Office of Nuclear Physics through Contract No. DE-SC0012704, No. DE-AC02-06CH11357 and +within the framework of Scientific Discovery through Advance Computing (SciDAC) award Fun- +damental Nuclear Physics at the Exascale and Beyond (S. B. and S. M.). K. C. is supported by the +National Science Centre (Poland) grants SONATA BIS no. 2016/22/E/ST2/00013 and OPUS no. +2021/43/B/ST2/00497. M. C., J. D. and A. S. acknowledge financial support by the U.S. Depart- +ment of Energy, Office of Nuclear Physics, Early Career Award under Grant No. DE-SC0020405. +J. D. also received support by the U.S. Department of Energy, Office of Science, Office of Nuclear +Physics, within the framework of the TMD Topical Collaboration. The work of A. M. has been +supported by the National Science Foundation under grant number PHY-2110472, and also by the +U.S. Department of Energy, Office of Science, Office of Nuclear Physics, within the framework of +the TMD Topical Collaboration. F. S. was funded by by the NSFC and the Deutsche Forschungsge- +meinschaft (DFG, German Research Foundation) through the funds provided to the Sino-German +Collaborative Research Center TRR110 “Symmetries and the Emergence of Structure in QCD” +(NSFC Grant No. +12070131001, DFG Project-ID 196253076 - TRR 110). +YZ was partially +supported by an LDRD initiative at Argonne National Laboratory under Project No. 2020-0020. +Computations for this work were carried out in part on facilities of the USQCD Collaboration, +which are funded by the Office of Science of the U.S. Department of Energy. This research used +resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User +Facility supported under Contract DE-AC05-00OR22725. This research was supported in part by +PLGrid Infrastructure (Prometheus supercomputer at AGH Cyfronet in Cracow). Computations +were also partially performed at the Poznan Supercomputing and Networking Center (Eagle su- +percomputer), the Interdisciplinary Centre for Mathematical and Computational Modelling of the +Warsaw University (Okeanos supercomputer), and at the Academic Computer Centre in Gdańsk +(Tryton supercomputer). The gauge configurations have been generated by the Extended Twisted +Mass Collaboration on the KNL (A2) Partition of Marconi at CINECA, through the Prace project +Pra13_3304 “SIMPHYS". Inversions were performed using the DD-𝛼AMG solver [22] with twisted +mass support [23]. +9 + +GPDs in asymmetric frames +Shohini Bhattacharya +References +[1] X. D. Ji, Phys. Rev. Lett. 78, 610-613 (1997) [arXiv:hep-ph/9603249 [hep-ph]]. +[2] A. V. Radyushkin, Phys. Lett. B 380, 417-425 (1996) [arXiv:hep-ph/9604317 [hep-ph]]. +[3] M. Burkardt, Phys. Rev. D 62, 071503 (2000) [erratum: Phys. Rev. D 66, 119903 (2002)] +[arXiv:hep-ph/0005108 [hep-ph]]. +[4] J. C. Collins, L. Frankfurt and M. 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Comput. 36, +A1581-A1608 (2014) [arXiv:1303.1377 [hep-lat]]. +[23] C. Alexandrou, S. Bacchio, J. Finkenrath, A. Frommer, K. Kahl and M. Rottmann, Phys. Rev. +D 94, 114509 (2016) [arXiv:1610.02370 [hep-lat]]. +10 + diff --git a/WtE1T4oBgHgl3EQfvgVo/content/tmp_files/load_file.txt b/WtE1T4oBgHgl3EQfvgVo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0910c7257e65a4c37ff9a0d312c96e9e6977645e --- /dev/null +++ b/WtE1T4oBgHgl3EQfvgVo/content/tmp_files/load_file.txt @@ -0,0 +1,565 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf,len=564 +page_content='GPDs in asymmetric frames Shohini Bhattacharya,𝑎,∗ Krzysztof Cichy,𝑏 Martha Constantinou,𝑐 Jack Dodson,𝑐 Xiang Gao,𝑑 Andreas Metz,𝑐 Swagato Mukherjee,𝑎 Aurora Scapellato,𝑐 Fernanda Steffens𝑒 and Yong Zhao𝑑 𝑎Brookhaven National Laboratory, Upton, New York 11973, USA 𝑏Adam Mickiewicz University, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Uniwersytetu Poznańskiego 2 61-614 Poznań, Poland 𝑐Temple University, Philadelphia, PA 19122 - 1801, USA 𝑑Argonne National Laboratory, Lemont, IL 60439, USA 𝑒Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn Nussallee 14-16, 53115 Bonn E-mail: sbhattach@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='gov It is often taken for granted that Generalized Parton Distributions (GPDs) are defined in the "symmetric" frame, where the transferred momentum is symmetrically distributed between the incoming/outgoing hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' However, such frames pose computational challenges for the lattice QCD practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' In these proceedings, we lay the foundation for lattice QCD calculations of GPDs in "asymmetric" frames, where the transferred momentum is not symmetrically distributed between the incoming/outgoing hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' The novelty of our work relies on the parameterization of the matrix elements in terms of Lorentz-invariant amplitudes, which not only helps in establishing relations between the said frames but also helps in isolating higher-twist contaminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' As an example, we focus on the unpolarized GPDs for spin-1/2 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' The 39th International Symposium on Lattice Field Theory (Lattice2022), 8-13 August, 2022 Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='03400v1 [hep-lat] 9 Jan 2023 GPDs in asymmetric frames Shohini Bhattacharya Figure 1: Graphical representation of the two frames employed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Left plot: Symmetric frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Right plot: Asymmetric frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Introduction Generalized Parton Distributions (GPDs) are the 3D generalizations of the collinear Parton Distribution Functions (PDFs) [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' There are several motivations to study GPDs: For 𝜉 = 0 the Fourier transforms of the GPDs are related to the impact-parameter dis- tributions which provide information about the three-dimensional distribution of partons — (one-dimensional) longitudinal momentum distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (two-dimensional) transverse spatial distribution, see for example Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Twist-2 GPDs are related to the total angular momentum of partons [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' One should look for other ways to access GPDs because of the challenges involved in their extraction through the processes of Deep Virtual Compton Scattering (DVCS) [2] and meson production [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Challenges are caused by the sensitivity of differential cross-sections to only 𝑥-integrals of GPDs, and not GPDs themselves [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Therefore, it is desirable to extract the 𝑥-dependence of the GPDs from first principles within Lattice QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' However, for a very long time this was not possible because of time-dependence of these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' As a result, all of the lattice calculations were limited to the calculations of lowest Mellin moments of the GPDs, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' In 2013, there was a path-breaking proposal by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Ji to calculate instead auxiliary quantities called "quasi-GPDs" [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' This approach relies on the extraction of matrix elements for boosted hadrons involving spatially-separated fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Ever since this proposal, enormous progress has taken place, see some reviews [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' In fact, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' [12] provides the first-ever lattice-QCD results of the unpolarized and helicity GPDs of the nucleon from the quasi-distribution approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Lattice QCD calculations have the potential to not only provide insight into the experimentally-inaccessible features of GPDs, but also help in extracting the "full" GPDs from the existing experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Formalisms to calculate GPDs in asymmetric frames 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='1 Frames: Symmetric and asymmetric The most widely used frame of reference to calculate GPDs is the symmetric frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' For this frame, the momentum transfer is symmetrically distributed between the incoming (𝑝𝑖) and the outgoing hadrons (𝑝 𝑓 ) (see left plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' However, one can also think of a frame where the 2 z/2 z/2 s △s Ps Ps+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' GPDs 2 t=2≥/2 z/2 Pa GPDs t= aGPDs in asymmetric frames Shohini Bhattacharya momentum transfer is not equally shared between the incoming and outgoing hadrons, but is rather exclusively applied to the incoming hadron (see, right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Such a frame is known as an asymmetric frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Lattice calculations of GPDs has primarily been confined to symmetric frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' However, such frames pose serious computational challenges because they require separate calculation for each values of the momentum transfer (Δ), resulting in increased computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' So the question that we strive to address in this work is: Can we lay a formalism to systematically perform lattice calculations of GPDs in asymmetric frames (which is expected to be computationally less expensive)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' In this work, we argue that there are two approaches to solving this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' In the first approach, we will show that it is possible to relate the two frames via an appropriate Lorentz transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' In the second approach, we will propose a Lorentz-covariant decomposition of the lattice matrix elements in terms of Lorentz-invariant (frame-independent) amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' These amplitudes will then be used to make connections between the two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' As a byproduct, we will show that this approach helps in identifying higher-twist contaminations which may be present in quasi-GPDs at finite values of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='2 Lorentz transformation approach In this section, we explain the Lorentz transformation (LT) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' First, it is straight- forward to realize that a LT along the 𝑧-direction is not optimal for lattice calculations because this requires a spatial operator distance (say 𝑧 = (0, 0⊥, 𝑧3 ≠ 0)) to pick up a temporal component (that is 𝑧 LT −−→ (𝑧0 ≠ 0, 0⊥, 𝑧3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' However, a LT applied to any direction transverse to the 𝑧-axis does not change the spatial nature of operator distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' This transformation is called as the "transverse boost".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' We explain this by considering a transverse boost in the 𝑥-direction and for the simplest case of zero skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' The logic can be generalized for any general transverse boost and for arbitrary values of skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' We begin by relating the incoming state in the two frames, 𝑝𝑠 𝑖 = (𝐸𝑠 𝑖 , −Δ1,𝑠/2, 0, 𝑃3) and 𝑝𝑎 𝑖 = (𝐸 𝑎 𝑖 , −Δ1,𝑎, 0, 𝑃3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' LT provides 𝑝𝑠 = ΛLT 𝑝𝑎, ������ � 𝐸𝑠 𝑖 𝑝1,𝑠 𝑖 𝑝2,𝑠 𝑖 𝑝3,𝑠 𝑖 ������ � = ������ � 𝛾 −𝛾𝛽 0 0 −𝛾𝛽 𝛾 0 0 0 0 1 0 0 0 0 1 ������ � × ������ � 𝐸𝑎 𝑖 −Δ1,𝑎 0 𝑃3 ������ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (1) This gives, 𝐸𝑠 𝑖 = 𝛾(𝐸 𝑎 𝑖 + 𝛽Δ1,𝑎) , (2) and, 𝑝1,𝑠 𝑖 = −𝛾(𝛽𝐸𝑎 𝑖 + Δ1,𝑎) → Δ1,𝑠 = 2𝛾(𝛽𝐸𝑎 𝑖 + Δ1,𝑎) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (3) Similarly, the outgoing state in the two frames, 𝑝𝑠 𝑓 = (𝐸𝑠 𝑓 , Δ1,𝑠/2, 0, 𝑃3) and 𝑝𝑎 𝑓 = (𝐸 𝑎 𝑓 , 0, 0, 𝑃3) can also be related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (Keep in mind that the energies of the incoming and outgoing states are different in the asymmetric frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=') We then find, 𝐸𝑠 𝑖 = 𝛾𝐸𝑎 𝑓 , (4) 3 GPDs in asymmetric frames Shohini Bhattacharya and, 𝑝1,𝑠 𝑓 = −𝛾𝛽𝐸𝑎 𝑓 → Δ1,𝑠 = −2𝛾𝛽𝐸𝑎 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (5) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (2) and (4), we find, 𝛽 = − � 𝐸 𝑎 𝑖 − 𝐸 𝑎 𝑓 Δ1,𝑎 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (6) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (3) and (5), we find, 𝛽 = − Δ1,𝑎 𝐸𝑎 𝑖 + 𝐸 𝑎 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (7) Then, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (6) and (7) imply, Δ1,𝑎 = √︃ (𝐸 𝑎 𝑖 )2 − (𝐸𝑎 𝑓 )2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (8) Hence, 𝛽 can be written as, 𝛽 = − � �𝐸 𝑎 𝑖 − 𝐸 𝑎 𝑓 𝐸 𝑎 𝑖 + 𝐸𝑎 𝑓 < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (9) This implies Δ0,𝑎 < 0, and 𝛾 = 1 √︁ 1 − 𝛽2 = � �𝐸 𝑎 𝑖 + 𝐸 𝑎 𝑓 2𝐸 𝑎 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (10) Therefore, by using the expressions for (𝛽, 𝛾), we can write down uniquely the symmetric frame variables (𝐸𝑠 𝑖 , Δ1,𝑠) in terms of the asymmetric frame variables (𝐸 𝑎 𝑖 , 𝐸𝑎 𝑓 , Δ1,𝑎): The energy should be, 𝐸𝑠 𝑖 = 𝛾𝐸𝑎 𝑓 = √︄ 𝐸𝑎 𝑓 (𝐸 𝑎 𝑖 + 𝐸 𝑎 𝑓 ) 2 , (11) and the transverse-momentum transfer, Δ1,𝑠 = −2𝛾𝛽𝐸𝑎 𝑓 , or, Δ1,𝑠 = 2 √︄ 𝐸 𝑎 𝑓 (𝐸 𝑎 𝑖 − 𝐸 𝑎 𝑓 ) 2 = 2 � � 𝐸𝑎 𝑓 2(𝐸 𝑎 𝑖 + 𝐸 𝑎 𝑓 ) Δ1,𝑎 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (12) We repeat that the above method can be generalized for �Δ⊥ = (Δ1, Δ2) and for arbitrary values of skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Now that we have sketched the idea of how to relate the kinematical variables between the two frames, we proceed to understand how the matrix elements defining quasi-GPDs transform between 4 GPDs in asymmetric frames Shohini Bhattacharya the two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' For this purpose, we focus on spin-0 particles such as the pion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (The method can be generalized for spin-1/2 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=') The (unpolarized) pion GPD is defined as, 𝐹𝜇(𝑧, 𝑃, Δ) = ⟨𝑝 𝑓 | ¯𝑞(− 𝑧 2)𝛾𝜇 W(− 𝑧 2, 𝑧 2)𝑞( 𝑧 2)|𝑝𝑖⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (13) Here, W is a straight Wilson line required to make the correlator gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Historically, (unpolarized) quasi-GPDs have been defined through matrix elements of the operator 𝛾0, see for instance Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' By applying the transverse boost Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (1), we find that the matrix element ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.𝛾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.⟩ in the symmetric frame can be expressed in terms of matrix elements of different operators ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.(𝛾0 + 𝛾1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.⟩ in the asymmetric frame, ⟨𝑝 𝑓 | ¯𝑞(− 𝑧 2)𝛾0 W(− 𝑧3 2 , 𝑧3 2 ) 𝑞( 𝑧 2)|𝑝𝑖⟩𝑠 = 𝛾⟨𝑝 𝑓 | ¯𝑞(− 𝑧 2)𝛾0 W(− 𝑧3 2 , 𝑧3 2 ) 𝑞( 𝑧 2)|𝑝𝑖⟩𝑎 − 𝛾𝛽⟨𝑝 𝑓 | ¯𝑞(− 𝑧 2)𝛾1 W(− 𝑧3 2 , 𝑧3 2 ) 𝑞( 𝑧 2)|𝑝𝑖⟩𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (14) This equation simply reflects how the 0th component of a 4-vector changes under the Lorentz transformation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Therefore, this implies that a transverse boost that fixes (𝛽, 𝛾) (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (9) and (10)) allows for an exact calculation of quasi-GPDs in the symmetric frame through matrix elements of the asymmetric frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' However, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (14) also shows that a quasi-GPD defined through the operator 𝛾0 is not Lorentz invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' In the limit of a large momentum, we recover, lim 𝑃3→∞⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.𝛾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.⟩𝑠 ≈ ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.𝛾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.⟩𝑎 + O � 1 𝑃3 � ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.𝛾1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.⟩𝑎 → ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.𝛾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.⟩𝑎 , (15) which means that the contribution from the matrix element ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.𝛾1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='.⟩ maybe viewed as a power correction at finite values of momentum 𝑃3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='3 Amplitude approach: Spin-1/2 particles In this section, we explain the amplitude approach through the example of spin-1/2 particles, such as the proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (We refer to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' [14] for details on spin-0 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=') As a first step, we build a Lorentz-covariant decomposition of the vector matrix element in terms of the available vectors (𝑃𝜇, 𝑧𝜇, Δ𝜇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' By considering constraints from parity, we find that the general structure of the vector matrix element involves eight linearly-independent Dirac structures multiplied by eight Lorentz-invariant (frame-independent) amplitudes, 𝐹𝜇(𝑧, 𝑃, Δ) = ¯𝑢(𝑝 𝑓 , 𝜆′) � 𝑃𝜇 𝑚 𝐴1 + 𝑚𝑧𝜇𝐴2 + Δ𝜇 𝑚 𝐴3 + 𝑖𝑚𝜎𝜇𝑧 𝐴4 + 𝑖𝜎𝜇Δ 𝑚 𝐴5 + 𝑃𝜇𝑖𝜎𝑧Δ 𝑚 𝐴6 + 𝑚𝑧𝜇𝑖𝜎𝑧Δ𝐴7 + Δ𝜇𝑖𝜎𝑧Δ 𝑚 𝐴8 � 𝑢(𝑝𝑖, 𝜆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (16) Here 𝜎𝜇𝜈 ≡ 𝑖 2 (𝛾𝜇𝛾𝜈 − 𝛾𝜈𝛾𝜇), 𝜎𝜇𝑧 ≡ 𝜎𝜇𝜌𝑧𝜌, 𝜎𝜇Δ ≡ 𝜎𝜇𝜌Δ𝜌, 𝜎𝑧Δ ≡ 𝜎𝜌𝜏𝑧𝜌Δ𝜏, 𝑧 ≡ (𝑧0 = 0, 𝑧⊥ = 0⊥, 𝑧3 ≠ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (For a derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (16), we refer to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' See also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' [16] where the vector matrix element has been parameterized in the momentum space for a straight Wilson line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=') For brevity, we use the compact notation 𝐴𝑖 ≡ 𝐴𝑖(𝑧 · 𝑃, 𝑧 · Δ, Δ2, 𝑧2), with 𝐴𝑖’s being the Lorentz-invariant amplitudes whose arguments are functions of Lorentz scalars1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 1In the literature, the amplitudes have also been called generalized Ioffe time distributions (ITDs) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 5 GPDs in asymmetric frames Shohini Bhattacharya For spin-1/2 particles, the vector matrix element can be parameterized in terms of two light- cone GPDs 𝐻 and 𝐸 [17], 𝐹+(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) = ¯𝑢𝑠/𝑎(𝑝𝑠/𝑎 𝑓 , 𝜆′) � 𝛾+𝐻(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) + 𝑖𝜎+𝜇Δ𝑠/𝑎 𝜇 2𝑚 𝐸(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) � 𝑢𝑠/𝑎(𝑝𝑠/𝑎 𝑖 , 𝜆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (17) By using 𝜇 = + in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (16), followed by a subsequent change of basis, it is possible to map the 𝐴𝑖’s onto the 𝐻 and 𝐸 GPDs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' The results are, 𝐻(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) = 𝐴1 + Δ+,𝑠/𝑎 𝑃+,𝑠/𝑎 𝐴3 , (18) 𝐸(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) = −𝐴1 − Δ+,𝑠/𝑎 𝑃+,𝑠/𝑎 𝐴3 + 2𝐴5 + 2𝑃+,𝑠/𝑎𝑧−𝐴6 + 2Δ+,𝑠/𝑎𝑧−𝐴8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (19) Keep in mind that the arguments of the 𝐴𝑖’s for light-cone GPDs have no dependence on 𝑧2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Also, 𝑧𝜇 = (0, 𝑧−, 0⊥) and Δ+/𝑃+ = 𝑧 · Δ/𝑧 · 𝑃, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Thus, it is possible to write the above expressions in a Lorentz invariant way as, 𝐻(𝑧 · 𝑃𝑠/𝑎, 𝑧 · Δ𝑠/𝑎, (Δ𝑠/𝑎)2) = 𝐴1 + Δ𝑠/𝑎 · 𝑧 𝑃𝑠/𝑎 · 𝑧 𝐴3 , (20) 𝐸(𝑧 · 𝑃𝑠/𝑎, 𝑧 · Δ𝑠/𝑎, (Δ𝑠/𝑎)2) = −𝐴1 − Δ𝑠/𝑎 · 𝑧 𝑃𝑠/𝑎 · 𝑧 𝐴3 + 2𝐴5 + 2𝑃𝑠/𝑎 · 𝑧𝐴6 + 2Δ𝑠/𝑎 · 𝑧𝐴8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (21) This means the light-cone GPDs are frame-independent as long as the Lorentz scalars (𝑧 · 𝑃𝑠/𝑎, 𝑧 · Δ𝑠/𝑎, (Δ𝑠/𝑎)2) are the same in the two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Next, we turn to the quasi-GPDs H and E, which historically have been defined in terms of matrix elements of 𝛾0 operator as [18, 19], 𝐹0(𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) = ⟨𝑝𝑠/𝑎 𝑓 , 𝜆′| ¯𝑞(− 𝑧 2)𝛾0𝑞( 𝑧 2)|𝑝𝑠/𝑎 𝑖 , 𝜆⟩ = ¯𝑢𝑠/𝑎(𝑝𝑠/𝑎 𝑓 , 𝜆′) � 𝛾0H 𝑠/𝑎 0 (𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) + 𝑖𝜎0𝜇Δ𝑠/𝑎 𝜇 2𝑚 E𝑠/𝑎 0 (𝑧, 𝑃𝑠/𝑎, Δ𝑠/𝑎) � 𝑢𝑠/𝑎(𝑝𝑠/𝑎 𝑖 , 𝜆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (22) If we use 𝜇 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (16), then after performing a change of basis it is possible to map the 𝐴𝑖’s onto the quasi-GPDs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' The relations in the symmetric frame read,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' H 𝑠 0 (𝑧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑃𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Δ𝑠) = 𝐴1 + Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 𝐴3 − 𝑚2Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑧3 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 𝐴4 + � (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠)2𝑧3 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 − Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑧3𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 2(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠)2 − 𝑧3(Δ𝑠 ⊥)2 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 � 𝐴6 + � (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠)3𝑧3 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 − (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠)2Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑧3 2(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠)2 − Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑧3(Δ𝑠 ⊥)2 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 � 𝐴8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (23) E𝑠 0(𝑧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑃𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Δ𝑠) = −𝐴1 − Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 𝐴3 + 𝑚2Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑧3 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 𝐴4 + 2𝐴5 + � − (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠)2𝑧3 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 + 𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑧3 2(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠)2 + 𝑧3(Δ𝑠 ⊥)2 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 − 2𝑧3(𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠)2 𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 � 𝐴6 + � − (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠)3𝑧3 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 + (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠)2Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑧3 2(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠)2 + Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑧3(Δ𝑠 ⊥)2 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 − 2𝑧3𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑠 � 𝐴8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (24) 6 GPDs in asymmetric frames Shohini Bhattacharya On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' the relations in the asymmetric frame read,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' H 𝑎 0 (𝑧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑃𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Δ𝑎) = 𝐴1 + Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 𝐴3 − � 𝑚2Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 − 1 (1 + Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 ) 𝑚2Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 4𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2 � 𝐴4 + � (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2𝑧3 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 − 1 (1 + Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 ) (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 4(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2 − 1 (1 + Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 ) 𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 2(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2 − 𝑧3(Δ𝑎 ⊥)2 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 � 𝐴6 + � (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)3𝑧3 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 − 1 (1 + Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 ) (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)3Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 4𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2 − 1 (1 + Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 ) (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 2(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2 − 𝑧3(Δ𝑎 ⊥)2Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 � 𝐴8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (25) E𝑎 0 (𝑧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑃𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Δ𝑎) = −𝐴1 − Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 𝐴3 − � − 𝑚2Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 − 1 (1 + Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 ) �𝑚2𝑧3 𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 − 𝑚2Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 4𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2 �� 𝐴4 + 2𝐴5 + � − (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2𝑧3 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 − 1 (1 + Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 ) � 𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 − (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 4(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2 � − 1 (1 + Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 ) �2𝑧3(𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2 𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 − 𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 2(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2 � + 𝑧3(Δ𝑎 ⊥)2 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 � 𝐴6 + � − (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)3𝑧3 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 − 1 (1 + Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 ) � (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2𝑧3 𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 − (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)3Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 4𝑃 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2 � − 1 (1 + Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 ) �2𝑧3𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 − (Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2Δ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑧3 2(𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎)2 � + 𝑧3(Δ𝑎 ⊥)2Δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 2𝑃0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎𝑃3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='𝑎 � 𝐴8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (26) However, one can think of other definitions of quasi-GPDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' For this purpose, we recall the position-space matching relation between, for instance, light-cone GPD 𝐻 and quasi-GPD H [13]: H �𝑧 · 𝑃, −2𝜉(𝑧 · 𝑃), Δ2, 𝑧2, 𝜇2� = ∫ 1 −1 𝑑𝑢 ¯𝐶 (𝑢, 𝑧 · 𝑃, 𝜉, 𝑧2, 𝜇2) 𝐻�𝑢(𝑧 · 𝑃), −2𝑢𝜉(𝑧 · 𝑃), Δ2, 𝜇2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (27) Here, ¯𝐶 is the pertubatively-calculable matching coefficient [13] and 𝜇 is the renormalization scale in the MS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' At leading order in 𝛼𝑠, the above formula indicates that H collapses to 𝐻 in the light-cone limit 𝑧2 → 0, lim 𝑧2→0 H (𝑧 · 𝑃, 𝑧 · Δ, Δ2, 𝑧2) = 𝐻(𝑧 · 𝑃, 𝑧 · Δ, Δ2, 0) + O(𝛼𝑠) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (28) Therefore, a natural way to define the quasi-GPDs H and E is through a Lorentz-invariant gener- alization of the light-cone definitions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (20) and (21) to 𝑧2 ≠ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=', H (𝑧 · 𝑃𝑠/𝑎, 𝑧 · Δ𝑠/𝑎, (Δ𝑠/𝑎)2, 𝑧2) = 𝐴1 + Δ𝑠/𝑎 · 𝑧 𝑃𝑠/𝑎 · 𝑧 𝐴3 , (29) E(𝑧 · 𝑃𝑠/𝑎, 𝑧 · Δ𝑠/𝑎, (Δ𝑠/𝑎)2, 𝑧2) = −𝐴1 − Δ𝑠/𝑎 · 𝑧 𝑃𝑠/𝑎 · 𝑧 𝐴3 + 2𝐴5 + 2𝑃𝑠/𝑎 · 𝑧𝐴6 + 2Δ𝑠/𝑎 · 𝑧𝐴8 , (30) where now the arguments of the 𝐴𝑖’s have a non-zero dependence on 𝑧2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' We expect the definitions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (29-(30) to have two advantages: First, these definitions may converge faster to the light-cone 7 GPDs in asymmetric frames Shohini Bhattacharya GPDs because of the similarities in their functional forms with their (respective) light-cone GPDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (Such a statement is inspired from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' [20], where similar arguments were made for the quasi- PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' See also the next paragraph for explicit explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=') Second, these definitions differ from their light-cone GPDs by frame-independent power corrections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' contrast with historic definitions which are frame-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' We now discuss in detail the various definitions of quasi-GPDs: We notice that for finite values of the momentum, the historic definitions of quasi-GPDs (H 𝑠/𝑎 0 (𝐴𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑧) , E𝑠/𝑎 0 (𝐴𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑧)) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (23)- (26) involve additional amplitudes that are not present in the light-cone GPDs, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (20)-(21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' This is not the case for the Lorentz-invariant definitions of quasi-GPDs (H (𝐴𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑧) , E(𝐴𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑧)) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (29)-(30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (Note that this is different from the (unpolarized) quasi-PDF case where arguments were made in favor of 𝛾0 (against 𝛾3) because of the absence of such additional amplitudes relative to the (unpolarized) light-cone PDF case [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=') Therefore, the additional amplitudes in (H 𝑠/𝑎 0 (𝐴𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑧) , E𝑠/𝑎 0 (𝐴𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑧)) may be viewed as contaminations from explicit power corrections, which one would have to suppress by going to larger and larger values of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Hence, we believe that (H (𝐴𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑧) , E(𝐴𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑧)) may converge relatively faster to their (respective) light-cone GPDs, simply because of the absence of such additional amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' (Of course, (H (𝐴𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑧) , E(𝐴𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 𝑧)) also have power corrections, but they are implicit within the amplitudes themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Our argument above is for the power corrections that are explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=') Our reasoning is perhaps too simple and for sure needs further substantiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' In fact, it may be that the actual convergence of the various definitions of quasi-GPDs is determined by the underlying dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Note that the Lorentz non-invariance of the historical definitions of quasi-GPDs implies that the basis vectors (𝛾0, 𝑖𝜎0Δ𝑠/𝑎) do not form a complete set for spatially-separated bi-local operators for finite values of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Therefore, we can argue that the Lorentz-invariant definitions are in fact just a redefinition of quasi-GPDs in terms of a suitable linear combination of operators (which turns out to be 𝛾⊥) that make them functions of Lorentz scalars [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' [14] and [21], we compare numerically the different definitions of quasi-GPDs for 𝜉 = 0 to get an idea about the relative size of power corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Finally, we remark on the matching coefficient for the different definitions of quasi-GPDs: It is known that the GPD matching coefficient for the operator 𝛾0 reduces to that for the corresponding PDF when 𝜉 = 0, even if 𝑡 ≠ 0 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' The PDF matching coefficient for 𝛾0 is for the amplitude 𝐴1, which is also the only contributing amplitude to the LI definition of the GPD when 𝜉 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Therefore, the matching coefficients for the 𝛾0 and the LI definitions of the GPDs are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' We will elaborate this point more, including the general case of 𝜉 ≠ 0, in a forthcoming publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Summary In these proceedings, we have laid down the theoretical tools to perform lattice QCD calcu- lations of GPDs in asymmetric frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' We have highlighted two approaches to performing such calculations: Lorentz transformation (LT) approach (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='2): We have shown that there exists a LT called the "transverse boost" (transverse with respect to the Wilson Line) that allows one to uniquely relate the kinematical variables as well as the matrix elements in the two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 8 GPDs in asymmetric frames Shohini Bhattacharya Amplitude approach (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='3): We have proposed a Lorentz-covariant decomposition of the vector matrix element in terms of Lorentz-invariant/frame-independent amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' The amplitudes can be used as tools to relate the two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' This approach also shows that at finite values of the boost momentum the historic definitions of quasi-GPDs (defined through 𝛾0) have additional amplitudes that are not present in the light-cone limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' This motivates us to come up with alternative definitions of quasi-GPDs that may potentially converge faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' One such candidate can be the case where one chooses the same functional form as the light-cone GPDs subjected to include 𝑧2 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Naively, because of the similarity in the functional forms (or because of the absence of additional amplitudes), one may expect such a definition of quasi-GPD to converge faster to the light-cone GPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Such a definition is also frame-independent, contrary to the historic definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Acknowledgements This material is based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Nuclear Physics through Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' DE-SC0012704, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' DE-AC02-06CH11357 and within the framework of Scientific Discovery through Advance Computing (SciDAC) award Fun- damental Nuclear Physics at the Exascale and Beyond (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' is supported by the National Science Centre (Poland) grants SONATA BIS no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 2016/22/E/ST2/00013 and OPUS no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 2021/43/B/ST2/00497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' acknowledge financial support by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Depart- ment of Energy, Office of Nuclear Physics, Early Career Award under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' DE-SC0020405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' also received support by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Nuclear Physics, within the framework of the TMD Topical Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' The work of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' has been supported by the National Science Foundation under grant number PHY-2110472, and also by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Nuclear Physics, within the framework of the TMD Topical Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' was funded by by the NSFC and the Deutsche Forschungsge- meinschaft (DFG, German Research Foundation) through the funds provided to the Sino-German Collaborative Research Center TRR110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 12070131001, DFG Project-ID 196253076 - TRR 110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' YZ was partially supported by an LDRD initiative at Argonne National Laboratory under Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 2020-0020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Computations for this work were carried out in part on facilities of the USQCD Collaboration, which are funded by the Office of Science of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' Department of Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' This research used resources of the Oak Ridge Leadership Computing Facility, which is a 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} +page_content=' 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE1T4oBgHgl3EQfvgVo/content/2301.03400v1.pdf'} diff --git a/XtA0T4oBgHgl3EQfFf84/content/tmp_files/2301.02032v1.pdf.txt b/XtA0T4oBgHgl3EQfFf84/content/tmp_files/2301.02032v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9d529114e2b8c5e6156381b5a2a88ac89f22f69 --- /dev/null +++ b/XtA0T4oBgHgl3EQfFf84/content/tmp_files/2301.02032v1.pdf.txt @@ -0,0 +1,1732 @@ +On the fractional transversely isotropic functionally graded nature of soft +biological tissues +G. Sachina, S. Natarajana, O. Barrera∗,b,c +aDepartment of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, 600036, India. +bSchool of Engineering, Computing and Mathematics, Oxford Brookes University, Headington, Oxford OX3 0BP, United +Kingdom. +cDepartment of Engineering Science, University of Oxford, Parks Road, OX1 3PJ, Oxford, United Kingdom. +Abstract +This paper focuses on the origin of the poroelastic anisotropic behaviour of the meniscal tissue and its +spatially varying properties. We present confined compression creep test results on samples extracted from +three parts of the tissue (Central body, Anterior horn and Posterior horn) in three orientations (Circum- +ferential, Radial and Vertical). We show that a poroelastic model in which the fluid flow evolution is ruled +by non-integer order operators (fractional Darcy’s law) provides accurate agreement with the experimental +creep data. The model is validated against two additional sets of experimental data: stress relaxation and +fluid loss during the consolidation process measured as weight reduction. Results show that the meniscus +can be considered as a transversely isotropic poroelastic material. This behaviour is due to the fluid flow +rate being about three times higher in the circumferential direction than in the radial and vertical directions +in the body region of the meniscus. In the anterior horn, the elastic properties are transversely isotropic, +with the aggregate modulus higher in the radial direction than in the circumferential and vertical directions. +The 3D fractional poroelastic model is implemented in finite element software and quantities such as flux of +interstitial fluid during the consolidation process, a non-trivial experimental measure, are determined. +Key words: +Meniscus, Fractional poroelasticity, Confined compression tests, Anomalous diffusion, Finite +element simulation +1. Introduction +Biological soft tissues, such as the meniscal tissue (Figure 1), exhibit a hierarchical porous solid matrix +and an interstitial fluid flowing into the pores [1–5]. The overall mechanical behaviour depends not only +on the solid matrix deformation but also on the movement of the fluid in and out of the pores during +the deformation. Furthermore, the internal architecture of the structure constituting the solid matrix and +collagen determines the anisotropic behaviour of both the elastic and the transport properties [6]. The +poroelastic theory is employed to model the response of biphasic tissue and better understand the anisotropic +nature of the meniscus on the functioning of the knee. +∗Corresponding author +Email addresses: me21d011@smail.iitm.ac.in (G. Sachin), snatarajan@smail.iitm.ac.in (S. Natarajan), +olga.barrera@eng.ox.ac.uk, +44 7585041147. (O. Barrera) +arXiv:2301.02032v1 [math.NA] 5 Jan 2023 + +Femur +Articular cartilage +of the femur +Lateral +Meniscus +Medial +Meniscus +Medial articular +cartilage of tibia +Lateral articular +cartilage of tibia +Tibia +(a) +CONFINED COMPRESSION +Anterior horn +Posterior horn +Central body +REGION +Vertical +DIRECTION +Channels are preferentially +aligned in the circumferential +direction +0.04mm +6.12 mm +0.84 mm +Collagen fibres and channels +with random alignment +Collagen fibres and channels +with random alignment +Collagen fibres and channels +with 30� alignment. +(b) +(c) +Multi-photon microscopy (fresh tissue) +Micro CT scans- 600nm/ pixel (freeze dried) +(a +) +(b +) +10µm +pores +collagen +elastin +3mm +(d) +Figure 1: (a) Lateral and Medial menisci in the knee joint, (b) Regions and the directions in the meniscus, (c) 3D porous +microstructure of meniscus cylindrical sample (d)Micro CT scan and Multi-photon microscopy of meniscus tissue. +The menisci situated between the femoral and tibial cartilage of the knee joint (Figure 1a) have three main +regions, viz., anterior horn, central body and posterior horn (Figure 1b). They perform several functions +like load bearing, lubrication, energy absorption and stability [7, 8]. In light of our recent experimental, +theoretical and numerical studies [1, 2, 9–11], we understand that each region has different functions. In +[1, 2], the unique structure of the body region that fulfils load-bearing and energy absorption capabilities +is discussed. This is achieved through a sandwich-like structure with two thin, stiff outside layers which +take the load-bearing role and a thick, softer internal layer which acts as an energy absorption element, an +effective natural damper [2, 11]. The architectural arrangement of the solid collagen-based matrix appears +to be different and gradually changing in the outside and the internal layers. The outside layers show a +dense distribution of collagen fibres with random orientation and low mean diameter. The elastic modulus +is one order of magnitude higher than the internal layer. Whereas the internal layer of the meniscus relies +on a hierarchical anisotropic network of collagen channels mainly aligned along the circumferential direction +in which fluid flows during the physiological deformation process (Figure 1). In [5], we have shown that the +permeability changes from the anterior/posterior horns to the body region, with the body region being more +permeable. Here, we show that the permeability tensor in this region is also transversely isotropic and that +the permeability in the circumferential direction (the preferential direction of the collagen channel) is higher +than the other two directions (Figure 1). Fluid flowing inside these channels determines the time-dependent +behaviour of the tissue. As the tissue deforms under the action of loads, these channels change morphology, +which results in change in permeability as a function of channels dimension, porosity, and tortuosity. Thus, +2 + +BFemur +0.04mm +CollagenFibersandChannels +withrandomalignment +6.12mm +CollagenFibersandchannels +with30°algnment +CollagenFibersandChannels +0.84mm +withrandomalignment +Tibiaz=-153.7μm +um +75making the permeability tensor not only depending on the pressure but also varying spatially and temporally +because of the local variation of applied pressure gradients. Recent finite element studies of the human knee +have shown that anterior and posterior horns might have a stability function [10, 12] as they deform to allow +the meniscus to be extruded so that the body region has a higher contact area with the cartilage and can +then carry most of the load. +Appropriate modelling of the fluid flow behaviour in the different portions of the meniscal tissue when +subject to a range of loading conditions is essential to gain insight into the biomechanical function of the +tissue and design appropriate artificial tissue substitutes. Current barriers to the clinical and functional +outcomes of the to-date devices [13] are that the transport and structure-properties relationship of the +native tissues is still not well understood. In particular, the relationship between the “micro/nano”-scopic +composition and structure of the meniscus and its coarse-scale anisotropic behaviour structure is still an +open question [14]. +Experimental tests, such as confined compression tests (relaxation and/or creep) are currently used to +characterize the material parameters such as elastic modulus at equilibrium and permeability. The main +models used to identify these two parameters are based on biphasic theory. In [15], the authors examined, +through a confined compression test, the effect of fluid in cartilage is bearing up to 90% of the applied load. +They used a linear biphasic model to find out the material properties in terms of the aggregate modulus +and permeability. In [16], authors performed a creep indentation test on different regions of the meniscus of +different animal species. The different material properties such as the aggregate modulus, Poisson’s ratio, +shear modulus and permeability were estimated by using the biphasic model to fit the experimental data. +Our recent experimental findings on human meniscal samples show that the biphasic model was not +sufficient to reproduce the observed experimental behaviour [5]. To address this, [17] proposed a poroelastic +model wherein the pore pressure diffusion equation is derived by adopting a modified version of Darcy’s law +involving fractional derivatives. It follows that the hydraulic permeability becomes “anomalous” and the +rate of fluid flow is governed by the order of the derivative. It is thus clear that the anomalous hydraulic +permeability, the order of derivative tensors and their anisotropic nature, which govern the transport of fluid +within the complex porous structure of the meniscus, are key properties of the meniscal tissue. Though a +finite element implementation of the fractional poroelastic model is reported in [17], however, the model is +restricted to estimating only the pore pressure field. +The purpose of this paper is to present for the first time creep data of confined compression tests +performed in different regions of the meniscal tissue (anterior-horn, central-body and posterior-horn) and +in different directions (radial, circumferential and vertical). +We adopt the fractional poroelastic model +presented in [5, 17] to fit the experimental data to recover the aggregate modulus, anomalous permeability +and order of derivative and their dependence on the directions (Aggregate modulus, anomalous permeability +and order of derivative tensors). For the first time, we validate the model in terms of Stress, weight loss and +displacement fields by using the parameters recovered from fitting the creep curve to match two additional +sets of data for the same samples, i.e., relaxation and weight loss data. The fractional poroelastic model is +implemented in commercial software Abaqus using UMATHT and UMAT. The numerical simulation results +show a good agreement with analytical solutions regarding both displacements and pore pressure fields +which were lacking in the literature. This enabled us to run consolidation tests numerically and calculate +fields such as the flux of interstitial fluid during the consolidation process, which is a difficult experimental +measure to obtain. Additionally, we perform statistical analysis of the parameters using ANOVA method +to compare the variation of the material parameters in different directions. +2. Material and Methods +2.1. Confined consolidation creep tests +Menisci were harvested from patients undergoing total knee arthroplasty under ethically approved pro- +tocol as reported in [5]. Samples labelled as “degraded” by the gross investigation of the surgeon were +discarded. Menisci were thawed in Phosphate-buffered solution (PBS) for thirty minutes to recover their +hydration. A total of 29 meniscal test samples were harvested. The samples were cylindrical in shape with +3 + +a 3mm diameter and 3-4 mm in height. Samples were extracted from the central body, anterior horn and +posterior horn regions along each principal direction, i.e. circumferential, radial and vertical (Figure 1). +z +F +At 𝑧 = ℎ, +𝑢 = 0, 𝑝 = 0 +Flux out +h +At 𝑧 = 0, +𝑃 = 𝑃�, �� +�� = 0 +Insulated +boundaries +(a) +0 +100 +200 +300 +400 +-0.6 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +(b) +0 +100 +200 +300 +400 +0 +0.2 +0.4 +0.6 +0.8 +(c) +Figure 2: (a) Schematic representation of confined compression test, (b) applied load for confined compression creep test and +(c) measured displacement. +The confined compression test set-up consisted of a chamber with transparent and insulated walls and +a porous plate at the bottom(Figure 2a). The sample to be tested is confined inside the chamber. A load +of 0.5 N (corresponding to a stress of about 0.07 MPa which is a physiological value for menisci [18]) at the +top of the chamber is applied. It is applied through a piston with a stage velocity of 0.3 % height of the +sample till the load 0f 0.5 N is reached and then the load is kept constant for 400s as shown in Figure 2. The +applied force and the obtained displacement that is measured at the top surface are shown in Figures 2b-2c, +respectively. +2.2. Fractional consolidation - governing equations & finite element implementation +Governing equations. Consider a poroelastic material occupying Ω ⊂ Rd, bounded by ∂Ω ⊂ Rd−1, where +d = 1, 2, 3. The boundary accommodates the following decomposition: ∂Ω = ∂ΩN ∪∂ΩD and ∂ΩN ∩∂ΩD = +∅, where Neumann and Dirichlet boundary conditions are applied on ∂ΩN and ∂ΩD, respectively. In the +4 + +absence of body force and inertia, the governing differential for a poroelastic material is given by: +∇ · σ = 0 +(1a) +∂ζ +∂t + ∇ · Jp = 0 +(1b) +supplemented with the following boundary conditions: u = ˆu, and p = ˆp on ∂ΩD and σ·n = F, and Jp·n = +0 on ∂ΩN. In Equation (1), ζ is the variation of the fluid content, p is the pore pressure, Jp is the fluid flux +and σ is the stress. The relation between the stresses, displacements and pressure; and between the flux +and the pressure is given by: +σ = 2Gϵ + λTr(ϵ)I − αpI +(2a) +Jp = −λβ 0Dβ +t ∇p +(2b) +where λ = (3K − 2G) +3 +is Lam´e’s constant, G, K are the shear and the bulk modulus, respectively, α is the +Biot coefficient and 0Dβ +t is the fractional derivative with respect to time of order β. It is important to note +that in case of β = 0, λβ is with dimension of [L]4[F]−1[T]−1. In the case of β ̸= 0, λβ has dimension of +[L]4[F]−1[T]β−1. The pore pressure diffusion equation (c.f. Equation (2b)), written in terms of dilatational +strain, ϵd, is given by: +∂p +∂t = Ku − K +α2 +λβ 0Dβ +t ∇2p − Ku − K +α +∂ϵd +∂t +(3) +where Ku is the undrained bulk modulus. +During confined compression tests, a cylinder of meniscal tissue is inserted inside a cylindrical chamber +with a porous base, see Figure 2a for geometry and boundary conditions. Upon applying pressure at the +top of the sample, the fluid contained inside the sample will flow out of the base. This can be modelled as a +one-dimensional version of Equation (2) as the only non-zero components of strain will be ϵzz. Equation (2) +will be modified to [17]: +� +K + 4G +3 +� ∂ϵzz +∂z − α∂p +∂z = 0 +(4a) +∂p +∂t = Ku − K +α2 +λβ 0Dβ +t +∂2p +∂z2 − Ku − K +α +∂ϵ +∂t +(4b) +and the corresponding boundary conditions are, ∀ 0 ≤ z ≤ h, p(z, t = 0) = γPA; P(z = 0, t) = PA, p(z = +h, t) = 0, ∂p +∂z +��� +z=0 = 0 and u(z = h, t) = 0. where h is the initial height of the sample. A constant compressive +stress, σzz(z = 0, t) = −PA in the z− direction is applied to the cylinder at z = 0. The initial pore pressure +is derived assuming that the material is under undrained conditions at the initial time of loading, given by: +p(z, t = 0) = PA +3(Ku − K) +α (4G + 3Ku) +(5) +It is shown that the analytical solution of the pore pressure is given by [17]: +p(z, t) = PAγ +∞ +� +n=1,3 +E1−β,1 +� +−n2π2¯λt1−β +4h2 +� +cn cos nπz +2h +(6) +where E1−β,1 is the Mittag-Leffler function and +γ = +3 (Ku − K) +α (4G + 3Ku); +cn = 4 +nπ +� +− 1 +� n−1 +2 +¯λ = λβ +(4G + 3K) (Ku − K) +α2 (4G + 3Ku) +(7) +5 + +The analytical solution for the displacement is given by: +u(z, t) = +3PA +3K + 4G +� +(h − z) + γα +∞ +� +n=1,3 +E1−β,1 +� +−n2π2¯λt1−β +4h2 +� +8h +(nπ)2 +� +(−1) +n − 1 +2 +sin nπz +2h − 1 +�� +(8) +The detailed derivation is given in Appendix A. +The expression for the flux is found by substituting the solution for the pore pressure (c.f. Equation (6)) in +the flux definition (c.f. Equation (2b)) and solving the fractional derivative as follows: +Jp = λβPAγt−β +∞ +� +n=1,3 +E1−β,1−β +� +−n2π2¯λt1−β +4h2 +� +2 +h sin +�nπz +2h +� +(9) +The weight of the sample over time W(t) is related to the initial weight of the sample W0 and the fluid flux +Jp as follows: +W(t) = W0 − +t +� +0 +Jp · ws A dt +(10) +where ws is the specific weight of the fluid, and A is the cross-section over which the fluid flows out. +Substituting Equation (9) in Equation (10) and evaluating the integral analytically, we obtain the following +relation: +W(t) = W0 − PAAwsγ 2 +h +∞ +� +n=1,3 +� +λβt1−βE1−β,2−β +� +−π2¯λt1−β +4h2 +�� +(11) +For detailed derivation, refer to Appendix B. +Finite Element implementation and verification. Biot’s model with fractional Darcy’s law is implemented +in Abaqus using UMATHT and UMAT subroutines using the method proposed by Barrera [17].In [17] only +the pore pressure field is discussed, here for the first time we show that both the pore pressure and the +displacement fields agree with the analytical solution. Using the similarity between the governing equations +of thermoelasticity and poroelasticity [17], UMATHT of Abaqus, which is generally meant for thermoelas- +ticity, is used for the case of poroelasticity. In Abaqus, Temperature can be treated as Pore pressure, and +the heat flux can be treated as heat flux. UMAT is used for transferring the stress data to the UMATHT +for modelling the coupling behaviour. UVARM can be used for transferring stress values. However, due to +the order of implementation in Abaqus, which is elastic model-UMATHT-UVARM, the stress values from +UVARM reach the UMATHT in the next iteration, which introduces errors in the computation. To avoid +this error, UMAT is used, which allows for sharing stress values in the same increment. UMAT is written +for a linear elastic model. The coefficient coupling the strain and pore pressure αT = α/3K is given as the +thermal expansion coefficient. Abaqus model is validated by comparing with theoretical solutions of a con- +Table 1: Material parameters +Bulk modulus, K +1.67×105Pa +Shear modulus, G +7.69× 104 Pa (E = 0.2×106Pa, ν = 0.3) +Skempton coefficient, B +0.88 +Biot coefficient, α +0.65 +Permeability, λβ +8.33×10−8m4/Ns1−β +Fractional Order, β +0, 0.1, 0.2, 0.4 +fined compression test with material properties given in Table 1 [17]. For this, a cylindrical computational +model with height h = 3mm and diameter d = 3mm is considered. The domain is discretized with 8-noded +trilinear hexahedral elements with 4 degrees of freedom per node (C3D8T elements). The displacements in +6 + +all three directions are restrained on the bottom face and the displacements on the side faces are restrained +laterally. The pore pressure is zero at the bottom and an instantaneous load, PA = 0.07 MPa is applied on +the top surface. Based on a systematic study, a total of 28,110 8-noded hexahedral elements, were found to +be adequate to model the poroelastic response. The displacements and the pore pressure obtained from the +numerical simulation are compared against the theoretical results. Figures 3a-3b shows the comparison of +the pore pressure and displacement as a function for depth for different time steps for β = 0 between theo- +retical solution and numerical computation. The influence of β on the pore pressure and the displacement +is shown in Figures 3c-3d. It is seen from Figure 3 that there is a very good agreement with the numerical +result and theoretical solution and that the fractional order has a strong influence on the pore pressure and +the displacement. Large jumps in the pore pressure and displacement in the initial time in Figures 3c, 3d +are due to the numerical approximation of step function with ramp in the first time increment. +(a) +(b) +(c) +(d) +Figure 3: Comparison of numerical results with theoretical results: (a-b) variation of pore pressure and displacement with +depth at different time steps and β = 0 and (c-d) influence of β on the pore pressure and displacement as a function of time +for a point on the top surface. +7 + +3. Results +3.1. Material parameters: anomalous permeability, order of derivative and aggregate modulus +The proposed fractional poroelastic model is used to characterize the response of the meniscus. To this, +we use the expression in Equation (8) for fitting the experimental data. The meniscus is assumed to be +incompressible material, due to which B = 1, α = 1, and Ku → ∞ reducing Equation (8) to: +u(z, t) = PA +M +� +(h − z) + +∞ +� +n=1,3 +E1−β,1 +� +−n2π2λβMt1−β +4h2 +� +8h +(nπ)2 +� +(−1) +n − 1 +2 +sin nπz +2h − 1 +�� +(12) +where, M = (3K + 4G)/3, is the aggregate modulus. Equation (12) is used to find the material properties +of meniscus tissue by fitting with the creep displacement vs time data. +It is noticed that this model +requires only three material properties, viz., the aggregate modulus (M), the fractional order (β) and the +permeability (λβ). On using β = 0, this theory converts into the classical Biot’s theory. The experiments +were displacement controlled, i.e., a constant piston velocity was used until the load reached the required +value. So, ramp loading is neglected for fitting and approximated as step loading. Displacements measured +in the experiments were fitted with the analytical solution Equation (12) at z = 0, i.e., +u = PA +M +� +h − +∞ +� +n=1,3 +E1−β,1 +�−n2π2λβMt1−β +4h2 +� 8h +(nπ)2 +� +(13) +Fitting is done in MATLAB using the inbuilt function ‘fminsearch’ with material properties as variables and +the RMS error between the analytical and the experimental data as the function to be minimized. Fitting +plots for four samples are shown in Figure 5. Table 2 shows the material properties of the meniscus obtained +from the experimental results for samples in both body and anterior horn for three different directions, viz., +circumferential, vertical and radial. The last two characters of the sample name indicate the location and +orientation information, for example, BC represents Body Circular. The other notations employed are A: +Anterior horn, P: Posterior horn, R: Radial, and V: vertical directions. +8 + +(a) +(b) +(c) +Figure 4: Comparison between Biot’s theory with classical and fractional Darcy’s law to fit creep displacement experiment +data. +To understand the fractional order’s influence, we compared the classical Biot’s poroelastic model and +the proposed one (c.f. Section 2.2). The material properties obtained for one sample TK11BC through a +fitting for classical Biot’s model1 are M = 3.56×10−1 MPa & λ = 6.95×10−13 m4/Ns and whilst that with +the fractional framework are: M = 1.27×10−1 MPa, β = 0.73 & λβ = 2.95×10−12 m4/Ns1−β. The RMS +error using Classical Biot’s theory is 6.53×10−5, and the RMS error while using Biot’s theory with fractional +Darcy’s law is 1.42×10−5. Figure 4 shows a comparison of the displacement, flux out and pore pressure as +a function of time for classical Biot’s theory and fractional model with experiments. It is opined that the +classical Biot’s model does not fit the experimental results well and reemphasizes a need for a fractional +poroelastic framework. +1Refer Appendix C for material properties corresponding to the classical Biot’s theory for all the samples. +9 + +0 +100 +200 +300 +400 +0 +0.2 +0.4 +0.6 +0.8 +1 +(a) TK11 Body Circumferential +0 +100 +200 +300 +400 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(b) TK11 Body Radial +0 +100 +200 +300 +400 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +(c) TK11 Body Vertical +0 +100 +200 +300 +400 +0 +0.2 +0.4 +0.6 +0.8 +1 +(d) TK16 Anteriorhorn Radial +Figure 5: Displacement confined creep data fitting with fractional poroelasticity and modelling using FEM, (a) Sample taken +from the body region of TK11 sample in the circumferential direction, (b) Sample taken from the body region of TK11 sample +in the radial direction, (c) Sample taken from the body region of TK11 sample in the vertical direction, (d) Sample taken from +the anterior horn region of TK16 sample in the radial direction. +Further, to check for anisotropy in the meniscus, the ANOVA test is performed on the body region, +properties with circular, radial and vertical directions as categories and properties are compared in different +directions using MATLAB’s default function ‘anova1’. For a few samples, properties were abnormal com- +pared to the remaining samples of the same category. It could be due to the sample, experiment or fitting +issue. These samples were detected as outliers by the Matlab ’anova1’ function and were not considered for +comparison. The mean and the standard deviation of the groups of the samples without considering the +outliers are given in Table 3 and the results for the ANOVA test are shown in Figure 6. In Figure 6a, the +aggregate modulus is compared. The red line shows the sample’s median, and the black dashed line limit +gives the sample’s range. If the notched blue box of different samples does not overlap, it is concluded that +the true medians are different with 95% confidence. If the probability against the null hypothesis (p-value) +is less than 0.05, it is concluded that the mean of the categories is different with 95% confidence. +For +aggregate modulus, the obtained p-value is 0.329, indicating that the aggregate modulus is not different in +different directions and could possibly be treated as isotropic. The obtained p-value for fractional order β +and permeability λβ are 0.001 and 0.0038, respectively. From Figures 6b-6c, it is inferred that the fractional +10 + +Table 2: Meniscus material parameters obtained from fitting with creep data +S No. +Sample +h +M×105 +β +λβ×10−12 +RMS error +(mm) +(Pa) +(m2/Pa.s1−β) +(×10−5) +1 +TK11BC +3.7 +1.27 +0.73 +2.95 +1.42 +2 +TK11BR +3.3 +0.71 +0.59 +1.48 +1.61 +3 +TK11BV +4.1 +0.83 +0.53 +1.56 +1.74 +4 +TK16BC1 +3.2 +0.76 +0.7 +2.13 +1.39 +5 +TK16BR2 +2.9 +0.82 +0.59 +0.89 +1.30 +6 +TK16BV +3.2 +0.62 +0.59 +0.85 +1.48 +7 +TK16AC +3.2 +0.60 +0.63 +0.97 +1.97 +8 +TK16AR1 +3.33 +0.85 +0.58 +1.01 +2.02 +9 +TK16AV +3.33 +0.57 +0.5 +0.37 +1.63 +10 +TK16BC2 +3.33 +0.90 +0.76 +4.28 +1.53 +11 +TK16BR1 +2.9 +0.55 +0.64 +0.91 +1.83 +12 +TK16BV2 +3.13 +0.61 +0.63 +0.93 +1.56 +13 +TK16PV +2.67 +1.41 +0.75 +1.99 +1.40 +14 +TK17BC +3.2 +0.10 +0.73 +1.03 +2.58 +15 +TK17BR +2.2 +0.41 +0.72 +1.36 +1.55 +16 +TK17BV +2.3 +0.86 +0.54 +0.90 +1.31 +17 +TK17AC +2.73 +1.87 +0.79 +4.96 +1.49 +18 +TK17AR +2.7 +1.94 +0.76 +3.05 +1.44 +19 +TK18BC +4.6 +1.34 +0.74 +141.03 +3.81 +20 +TK18BR +2.9 +0.66 +0.72 +1.90 +1.78 +21 +TK18BV +2.8 +0.68 +0.61 +1.07 +1.63 +22 +TK22BR +3.17 +2.54 +0.49 +85.2 +9.90 +23 +TK22AC +3.67 +0.65 +0.62 +1.40 +2.68 +24 +TK22AR +3.33 +1.60 +0.41 +2.24 +4.69 +25 +TK22AV +2.93 +0.76 +0.51 +0.35 +1.42 +26 +TK36BC +3.37 +0.74 +0.73 +4.40 +2.11 +27 +TK37BC +4.6 +0.90 +0.74 +6.63 +3.53 +28 +TK37BR +3.5 +0.67 +0.59 +0.96 +1.93 +29 +TK37BV +3.2 +0.49 +0.53 +0.55 +1.33 +11 + +order and permeability are relatively higher in circular directions when compared to the other two directions, +viz., vertical and radial. Further, it can be considered similar in vertical and radial directions. +Table 3: Meniscus parameters variation across sample +M×105 +β +λβ×10−12 +(Pa) +(m2/Pa.s1−β) +Part +Mean +SD +Mean +SD +Mean +SD +Body Cir +0.75 +0.49 +0.73 +0.00 +3.75 +2.36 +Body Rad +0.64 +0.14 +0.64 +0.06 +1.25 +0.40 +Body Ver +0.65 +0.14 +0.58 +0.04 +0.86 +0.19 +Anthorn Cir +1.04 +0.72 +0.68 +0.10 +2.44 +2.19 +Anthorn Rad +1.47 +0.56 +0.58 +0.17 +2.10 +1.03 +Anthorn Ver +0.66 +0.14 +0.51 +0.01 +0.36 +0.02 +BC +BR +BV +2 +4 +6 +8 +10 +12 +14 +Aggregate Modulus M(Pa) +104 +(a) +BC +BR +BV +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +Fractional power +(b) +BC +BR +BV +1 +2 +3 +4 +5 +6 +Permeability + (m4/Ns1- ) +10-12 +(c) +Figure 6: Material properties comparison in different orientations in the body region(BC - Body circumferential, BR - Body +Radial, BV - Body Vertical). (a) Aggregate Modulus (M), (b) Fractional power β, (c) Permeability λβ +To study the anisotropic behaviour of the meniscus, the confined creep test is performed as described +in Section 2.1 with material properties taken from the average values of the body region in circumferential, +radial and vertical directions. For the numerical study, the height of the sample is taken as 3mm and pressure +is increased to 0.07 MPa from zero and held constant. Figure 7 shows the pore pressure, displacement and +the flux out of the sample from the bottom (computed using Equation (9)) as a function of time for the +three different regions. +12 + +(a) +(b) +(c) +Figure 7: Flux, displacement and pore pressure for different directions (Circumferential, Radial and Vertical) +3.2. Numerical modelling of fractional consolidation +Numerical modelling of confined creep experiments of the meniscus is done in Abaqus using the param- +eters obtained from fitting the data given in Table 2. Poisson’s ratio and Young’s modulus are the elastic +parameters required for Abaqus. As the confined compression tests depend only on the aggregate modulus, +Poisson’s ratio is arbitrarily assumed to be 0.3, and Young’s modulus is calculated using the aggregate +modulus and Poisson’s ratio. Owing to symmetry, an axisymmetric model is considered for the numerical +simulation. Figure 8 shows an axisymmetric model employed for this study. The domain is discretized with +4-noded bilinear quadrilateral elements (CAX4T) and a structured mesh is used. Based on a systematic +mesh convergence study, a mesh size of 0.05mm was found to be adequate to model the behaviour. Axis- +symmetric boundary conditions are enforced on the left of the computational domain. The displacements +are restrained on the right and bottom faces. Zero pore pressure condition is applied at the bottom for +free fluid flow and pressure PA = 0.07MPa is applied at the top surface as a step load. For the numerical +simulation, a fixed time increment of ∆t = 0.1 s is used, and the simulation is carried over for 400 s. +A comparison between the numerical, theoretical and experimental results is shown in Figure 5 and it +is inferred that a good agreement is seen between the numerical, theoretical and experimental results. For +the subsequent studies, the numerical models are used for complex loading in further sections. +13 + +Z +T +R +X +Y +Z +Figure 8: Axisymmetric model employed for the confined creep numerical simulation +3.3. Fractional poroelastic model validation +To further validate the proposed poroelastic model, a confined compression creep test with and without +initial ramp loading and confined compression stress relaxation is done numercally and compared with +experiments. We also compute the weight loss from the confined compression creep test. +Confined compression creep test with initial ramp loading. In the earlier study, the initial ramp present in +the experiments was assumed as a step load. However, to assess the impact of this assumption on the model +fitting parameters, we ran the FE model with the initial ramp (similar to that in the experiments) for the +creep test and the results from the numerical simulation are compared with experiments. The creep test +with ramp loading within Abaqus is modelled by providing the amplitude of the load in the load module +with a ramp for a specific time period similar to experiments and then kept constant for the rest of the +simulation. Figure 9 shows the comparison of numerical results with the experiments for a few samples, +in particular for samples TK11BC, TK11BR, TK11BV and TK16AR. The results of creep with ramp in +Figure 9 show slight deviation at the end of ramp loading caused due to this assumption. But the deviation +is small and can be neglected. Therefore we conclude that the material properties extraction process as +explained above is correct. +14 + +0 +100 +200 +300 +400 +0 +0.2 +0.4 +0.6 +0.8 +1 +(a) TK11 Body Circumferential +0 +100 +200 +300 +400 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +(b) TK11 Body Radial +0 +100 +200 +300 +400 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +(c) TK11 Body Vertical +0 +100 +200 +300 +400 +500 +0 +0.2 +0.4 +0.6 +0.8 +1 +(d) TK16 Anteriorhorn Radial +Figure 9: Validation by a creep with ramp test (a) Sample taken from the body region of TK11 sample in the circumferential +direction, (b) Sample taken from the body region of TK11 sample in the radial direction, (c) Sample taken from the body +region of TK11 sample in the vertical direction, (d)Sample taken from the anterior horn region of TK16 sample in the radial +direction. +Weight loss from confined compression creep test. Weight loss for confined creep compression problem pre- +sented in Section 2.1 and compared with the experiments carried out by Bulle et al.[5]. Specific weight of +water at 37◦C is used(ws = 997Kg/m3). Fluid flow is collected at the bottom of the cylindrical sample with +the area of the circular cross-section A with diameter d = 3mm. Material properties are taken from the +Table 2 which were obtained from fitting the fractional poroelastic model with displacement data from the +confined creep test. Weight loss with time for some samples is shown in the Figure 10. Theoretical and +experimental results for a few samples match as shown in the Figure 10. Moreover, they do not match well +for some samples, as shown in Figure 1. This could be due to the measurement error as it was carried out +manually. The weight of the sample is measured after every 75 seconds of creep by taking it out. +15 + +0 +100 +200 +300 +400 +15 +20 +25 +(a) TK16 Body Circumfrential +0 +100 +200 +300 +400 +14 +15 +16 +17 +18 +19 +20 +21 +(b) TK16 Body Radial +0 +100 +200 +300 +400 +18 +19 +20 +21 +22 +23 +24 +25 +26 +(c) TK16 Body Vertical +0 +100 +200 +300 +400 +500 +15 +16 +17 +18 +19 +20 +(d) TK17 Anteriorhorn Radial +Figure 10: Comparison of theoretical Weight loss from the sample using parameters obtained from fitting with the experiments +(a) Sample taken from the body region of TK16 sample in the circumferential direction, (b) Sample taken from the body region +of TK16 sample in the radial direction, (c) Sample taken from the body region of TK16 sample in the vertical direction, (d) +Sample taken from the Anterior horn region of TK17 sample in the radial direction. +Confined compression stress relaxation. The fractional poroelastic model is validated using the confined +compression stress relaxation tests performed by Bulle et al. [5]. Tests were performed on the cylindrical +sample of height h, confined on the sides, and the fluid can flow freely from the bottom. Tests consist of +preconditioning the sample with 10%h compression at the ramp velocity 0.3%h/s. Then, five relaxation +steps with 2%h compression at a ramp velocity of 0.3%h/s s were performed. In all the cases, the force +applied is measured with time. The finite element modelling of this confined compression stress relaxation +test is performed in Abaqus similar to the creep test as mentioned in Section 3.2. +Material properties +obtained from fitting the creep data were used. In this test, the displacement is given as input, and the +force is measured as output. Figure 11 shows a comparison of the FEM results with the experiments. The +comparisons of the model with the experiments are reasonably good. +16 + +0 +20 +40 +60 +80 +100 +-40 +-30 +-20 +-10 +0 +(a) TK11 Body Circumferential +0 +20 +40 +60 +80 +100 +-20 +-15 +-10 +-5 +0 +5 +(b) TK11 Body Vertical +0 +10 +20 +30 +40 +50 +60 +-14 +-12 +-10 +-8 +-6 +-4 +-2 +0 +(c) TK11 Body Radial +Figure 11: Validation by comparing stresses from Abaqus with stress relaxation test from experiments (a) Sample taken from +the body region of TK11 sample in the circumferential direction, (b) Sample taken from the body region of TK11 sample in +the vertical direction, (c) Sample taken from the body region of TK11 sample in the radial direction. +4. Discussion +We show that the fractional poroelastic model is more appropriate to model the time-dependent behaviour +of soft tissue presenting a hierarchically arranged porous architecture. The model has been verified and +validated. +Our parameter fittings show that the RMS error = 1.42×10−5 for the fractional poroelastic +model is considerably lower than the one given by the classical model that had an RMS error = 6.53×10−5, +see Figure 4. The theoretical displacement curve starts from 0, whereas the experiment shows an initial +displacement. +This is attributed to the assumption of undrained conditions at the initial time and the +assumption of incompressible material. We obtain a value of fractional order β in the range of β = 0.51–0.73 +as shown in Table 3. From the numerical study, it can be inferred that the transport phenomenon during +the confined compression is ruled by a fractional version of Darcy’s law. At the beginning of the confined +compression test, the solid is fully saturated, therefore the pore pressure reaches a maximum value (see +Figure 3c). This is also predicted by Terzaghi’s consolidation theory, in line with the fact that the fluid +pressure entirely carries the load. During the consolidation process, the fluid flows out of the sample at a +rate depending on the anomalous permeability and on the order of the fractional derivative (see Figure 3). +17 + +The higher the value of β the faster the diffusion process. As the fluid flows out of the sample, the pore +pressure decreases and the solid starts deforming (see Figure 3d). Also, higher values of β imply a faster +solid deformation process. We notice that the aggregate modulus M obtained from fitting mentioned in +Table 2 is of the same order as the literature [19–22]. The anomalous permeability λβ is in the order of +10−12/10−13. From the statistical analysis, it can be concluded that the meniscus in the body region is +elastically isotropic, and pore pressure diffusion is transversely isotropic with symmetry in the vertical and +radial directions. Higher hydraulic permeability in the circular direction can be attributed to the fact that +the fibres are oriented in a circular direction in the body region, see Figure 1b. To model the anisotropy in +the anomalous transport phenomena, the following anisotropic form of fractional Darcy’s law is proposed: +Jp = +� +� +λrad +0 +0 +0 +λcir +0 +0 +0 +λver +� +� +� +�� +� +Λ +� +� +0Dβrad +t +0 +0 +0 +0Dβcir +t +0 +0 +0 +0Dβver +t +� +� +� +�� +� +0Dβ +t +� +∇p +� +(14) +where, Λ is the permeability tensor and 0Dβ +t is the fractional derivative operator. We notice that for the +body region, the anomalous permeability tensor Λ is transversely isotropic with λcir considerably higher +than λver and λrad. Similarly β tensor is transversely isotropic with βcir considerably higher than the βrad +and βver. Consequently, from the Figure 7c, it is observed that the fluid flow rate through the sample’s base +during the consolidation experiment in the circumferential direction is higher than the other two directions. +Pore pressure in the circumferential direction reaches a steady state faster than the other two directions, as +shown in the Figure 7a because of the higher permeability in the circumferential direction. Based on the +mean values and the standard deviation in the anterior horn region, it is opined that the aggregate modulus +in the radial direction is higher than the other two directions, viz., vertical and circular directions. It is +inferred that the permeability and the order of derivative are higher in the radial and vertical directions +than in the circumferential directions. The elastic tensor results are transversely isotropic, with the modulus +in the radial direction being higher, and both Λ and β are transversely isotropic with circumferential being +lower. Different properties in different directions show that the elastic, anomalous permeability and the +order of the derivative tensors are transversely isotropic. Furthermore, properties vary with the anterior +horn and body region, showing that it is not homogeneous. +Vertical +Anterior horn +Central body +• +Transversely isotropic permeability tensor (higher +permeability in the circumferential direction) +• +Isotropic elastic tensor +• +Transversely isotropic permeability tensor (permeability +in radial and vertical higher than in the circumferential +direction) +• +Transversely isotropic elastic tensor (higher aggregate +modulus in the radial direction) +Figure 12: Directional and regional dependence of material properties in the meniscus. +In summary (see Figure 12), we show that the body region which has a load-bearing function exhibits a +18 + +transversely isotropic behaviour due to the rate of fluid flow being about three times higher (faster diffusion) +in the circumferential direction which is consistent with the preferential direction of collagen channels. It +explains the role of fluid pressure in sustaining the load. Furthermore, we show that both the elastic and +the permeability tensors are transversely isotropic in the anterior horn. The aggregate modulus is higher +in the radial direction compared to the circumferential and vertical directions. It explains the role of the +anterior horn to be compliant in the circumferential and radial directions to accommodate the kinematics +of the tissue. +5. Conclusions +The meniscus has a porous, hierarchical, multi-oriented fiber and bundle structure filled with fluid +that provides optimized load support and lubrication properties. +To understand its function, confined +compression creep tests were performed on different regions of meniscus tissue and in different orientations. +Biot’s theory with fractional Darcy’s law is used to find the material properties. +It is observed that, +with the classical Darcy’s law, fitting gives an RMS error of 6.53×10−5, while the fractional Darcy’s law +gives an RMS error of 1.42×10−5. +It is shown that Biot’s theory with fractional Darcy’s law is better +suited for modelling the poroelastic behaviour of the meniscus. Three material parameters, viz., aggregate +modulus(M), fractional order(β) and the permeability(λβ) required for the fractional Biot’s theory were +found by fitting the theory with the confined compression creep experiment results for all the samples. +Material properties in different directions of the body region are compared using ANOVA test methods. +It is observed that the aggregate modulus (Mcir = 75.4 ±48.9 KPa, Mrad = 63.7±14.1 KPa and Mver = +65.3±13.7 KPa) is isotropic and the permeability (λβcir = 3.75×10−12± 2.36×10−12m2/Pa.s1−β, λβrad = +1.25×10−12± 4.04×10−13m2/Pa.s1−β and λβver = 8.61×10−13± 1.94×10−13m2/Pa.s1−β) and the fractional +order (βcir = 0.73±0.00, βrad = 0.64±0.06 and βver = 0.58± 0.04) are transversely isotropic with properties +in the circumferential direction is greater than the vertical and radial directions. +Biot’s theory with fractional Darcy’s law is implemented numerically using the FEM in Abaqus software +using UMATHT and UMAT subroutines. This fractional poroelastic theory is validated by using the material +properties obtained from fitting confined creep tests to model stress relaxation, creep with ramp, and weight +loss tests. The results are then compared with experimental results. The comparison shows that Biot’s theory +with fractional Darcy’s law is capable of better characterising the poroelastic behaviour of the meniscus. +Flux out of the meniscus, which is difficult to measure experimentally, can easily be computed efficiently +using the current model. In future works, Biot’s theory with fractional Darcy’s law will be extended to +include anisotropy and use it to understand the working of the meniscus in the knee joint. +6. Acknowledgements +O.B would like to acknowledge the European Union’s Horizon 2020 -EU.1.3.2. - Nurturing excellence +by means of cross-border and cross-sector mobility under the Marie Sk�lodowska-Curie individual fellowship +MSCA-IF-2017, MetaBioMec, Grant agreement ID: 796405. The authors thank the Rizzoli Orthopaedic +Institute and, in particular, G. Marchiori and M. Berni for their invaluable insights on the experiments. +References +[1] G. Agustoni, F. P. Bonomo, S. P. A. Bordas, O. Barrera, High resolution micro-computed tomography reveals a network +of collagen channels in the body region of the knee meniscus, Annals of Biomedical Engineering (2021). +[2] J. Maritz, G. Agustoni, K. Dragnevski, S. P. A. Bordas, O. Barrera, The functionally grading elastic and viscoelastic +properties of the body region of the knee meniscus, Annals of Biomedical Engineering (2021). +[3] F. P. Bonomo, J. J. Gregory, O. Barrera, A procedure for slicing and characterizing soft heterogeneous and irregular-shaped +tissue, Materials Today: Proceedings (2020). +[4] V. Vetri, K. Dragnevsk, M. Tkaczyk, M. Zingales, G. Marchiori, N. Lopomo, S. Zaffagnini, A. Bondi, J. Kennedy, +D. Murray, O. Barrera, Advanced microscopy analysis of the micro-nanoscale architecture of human menisci., Scientific +Reports (2019). +19 + +[5] R. Bulle, G. Alotta, G. Marchiori, M. Berni, N. F. Lopomo, S. Zaffagnini, S. P. A. Bordas, O. Barrera, The human +meniscus behaves as a functionally graded fractional porous medium under confined compression conditions, Applied +sciences 11 (20) (2021). +[6] G. A. Ateshian, J. A. Weiss, Anisotropic hydraulic permeability under finite deformation, Journal of Biomechanical +Engineering 132 (11) (2010). +[7] H. Kurosawa, T. Fukubayashi, H. Nakajima, Load-bearing mode of the knee joint: physical behavior of the knee joint +with or without menisci., Clinical Orthopaedics and Related Research 149 (1980) 283–290. +[8] N. Shrive, J. O’Connor, J. Goodfellow, Load-bearing in the knee joint., Clinical Orthopaedics and Related Research 131 +(1978) 279–287. +[9] J. Waghorne, E. Elmukashfi, S. L. Giudice, V. Manuri, G. Pitarresi, O. Barrera, On the structure-function relationships of +the meniscal tissue. a data driven approach to map soft tissues performance, To be submitted to Nature Material (2022). +[10] R. Readioff, R. Seil, C. Mouton, L. Marks, O. Barrera, An optimised patient-specific finite element model to study +the influence of the intra-articular parameters on the contact mechanics in the knee, To be submitted to Journal of +Biomechanical Engineering (2022). +[11] O. Barrera, et al., On the characteristics of natural hydraulic dampers: an image driven based approach to study the fluid +flow behavior inside the human meniscal tissue., To be submitted to PNAS (2022). +[12] E. Elmukashfi, G. Marchiori, M. Berni, G. Cassiolas, N. F. Lopomo, H. Rappel, M. Girolami, O. Barrera, Model selection +and sensitivity analysis in the biomechanics of soft tissues: A case study on the human knee meniscus, Advances in Applied +Mechanics, Elsevier, 2022. +[13] A. van Kampen, The knee joint in sports medicine, International Orthopaedics 37 (2) (2013) 177–179. +[14] A. J. S. Fox, A. Bedi, S. A. Rodeo, The basic science of human knee menisci: structure, composition, and function, Sports +Health 4 (4) (2012) 340–351. +[15] M. A. Soltz, G. A. Ateshian, Experimental verification and theoretical prediction of cartilage interstitial fluid pressurization +at an impermeable contact interface in confined compression, Journal of Biomechanics 31 (1998) 927–934. +[16] M. A. Sweigart, C. F. Zhu, D. M. Burt, P. D. Deholl, C. M. Agrawal, T. O. Clanton, K.A.Athanasiou, Intraspecies and +interspecies comparison of the compressive properties of the medial meniscus, Annals of Biomedical Engineering 32 (11) +(2004) 1569–1579. +[17] O. Barrera, A unified modelling and simulation for coupled anomalous transport in porous media and its finite element +implementation, Computational Mechanics 68 (2021) 1267–1282. +[18] A. M. Seitz, F. Galbusera, C. Krais, A. Ignatius, L. D¨urselen, Stress-relaxation response of human menisci under confined +compression conditions, Journal of the Mechanical Behavior of Biomedical Materials 26 (2013) 68 – 80. +[19] M. A. Sweigart, C. F. Zhu, D. M. Burt, P. D. Deholl, C. M. Agrawal, T. O. Clanton, K. A. Athanasiou, Intraspecies and +interspecies comparison of the compressive properties of the medial meniscus, Annals of Biomedical Engineering 32 (11) +(2004) 1569–1579. +[20] J. T. Moyer, R. Priest, T. Buman, A. C. Abraham, T. L. H. Donahue, Indentation properties and glycosaminoglycan +content of human menisci in the deep zone, Acta Biomaterialia 9 (5) (2013) 6624–6629. +[21] A. M. Seitz, F. Galbusera, C. Krais, A. Ignatius, L. Durselen, Stress-relaxation response of human menisci under confined +compression conditions, Journal of the Mechanical Behavior of Biomedical Materials 26 (2013) 68–80. +[22] H. N. Chia, M. L. Hull, Compressive moduli of the human medial meniscus in the axial and radial directions at equilibrium +and at a physiological strain rate, Journal of Orthopaedic Research 26 (7) (2008) 951–956. +[23] I. Podlubny, Fractional Differential Equations, Academic Press, 1999. +A. +Using Equation (4a), the displacement field for the problem described in Section 2.1 can be solved as +follows: +ϵzz = ∂uz +∂z = +3 +3K + 4G +� +− PA + αp +� +(15) +Substituting Equation (6) in Equation (15) and integrating +u(z, t) = +3PA +3K + 4G +� +− z + γα +∞ +� +n=1,3 +E1−β,1 +� +−n2π2¯λt1−β +4h2 +� +bn sin nπz +2h +� ++ d +(16) +where, bn = +8h +(nπ)2 (−1) +n−1 +2 . Applying the boundary condition given by Section 2.2, we get: +u(h, t) = +3PA +3K + 4G +� +− h + γα +∞ +� +n=1,3 +E1−β,1 +� +−n2π2¯λt1−β +4h2 +� +8h +(nπ)2 (−1) +n−1 +2 +sin nπ +2 +� ++ d = 0 +(17) +20 + +Since, +� +− 1 +� n−1 +2 +sin nπ +2 = 1 +for +n = 1, 3, · · · +(18) +Therefore, the constant d is +d = − +3PA +3K + 4G +� +− h + γα +∞ +� +n=1,3 +E1−β,1 +� +−n2π2¯λt1−β +4h2 +� +8h +(nπ)2 +� +(19) +Substituting Equation (19) in Equation (16), Displacement field equation is obtained +u(z, t) = +3PA +3K + 4G +� +(h − z) + γα +∞ +� +n=1,3 +E1−β,1 +� +−n2π2¯λt1−β +4h2 +� +8h +(nπ)2 +� +(−1) +n−1 +2 +sin nπz +2h − 1 +�� +(20) +B. +Weight loss from the sample over time W(t) for the problem described in Section 2.1 can be solved using +the fluid flux as: +W(t) = W0 − +� t +0 +Jp · wsAdt +(21) +where Jp is the fluid flux given by Equation (2b), ws is the specific weight, A is the cross-sectional area +over which the flux is calculated and W0 being the initial weight of the sample. Differentiating Equation (6) +with respect to z, we get: +∂p +∂z = −PAγ +∞ +� +n=1,3 +E1−β,1 +� +− n2π2¯λt1−β +4h2 +� 2 +h(−1) +n−1 +2 +sin +�nπz +2h +� +(22) +Fluid can flow only at the bottom, as shown in Figure 2a. +Hence, the weight loss is solved at z = h. +Equation (22) at z = h is: +∂p +∂z = −PAγ +∞ +� +n=1,3 +E1−β,1 +� +− n2π2¯λt1−β +4h2 +� 2 +h +(23) +Using Equation (23) and Equation (2b) in weight loss Equation (21), we get: +W(t) = W0 − λβwsA 0Dβ−1 +t +� +PAγ +∞ +� +n=1,3 +E1−β,1 +� +− n2π2¯λt1−β +4h2 +� 2 +h +� +(24) +Using the identity of the fractional derivative from [23], which is: +0Dα +t tβ−1Eµ,β(λtµ) = tβ−1−αEµ,β−α(λtµ) +(25) +Equation (24) can be found as: +W(t) = W0 − λβwsAPAγt1−β +∞ +� +n=1,3 +E1−β,2−β +� +− n2π2¯λt1−β +4h2 +� 2 +h +(26) +21 + +C. +S No. +Sample +M ×105(Pa) +λ × 10−13(m2/Pa.s) +RMS ×10−5 +1 +TK11BC +3.56 +6.95 +6.53 +2 +TK11BR +2.13 +3.91 +7.40 +3 +TK11BV +2.30 +4.44 +8.10 +4 +TK16BC1 +2.48 +5.32 +7.04 +5 +TK16BR2 +2.55 +2.45 +5.53 +6 +TK16BV +2.50 +3.00 +6.37 +7 +TK16AC1 +2.55 +2.80 +7.64 +8 +TK16AR1 +2.64 +2.42 +7.16 +9 +TK16AV +2.69 +1.79 +5.83 +10 +TK16BC2 +2.39 +8.87 +8.15 +11 +TK16BR1 +2.31 +2.68 +7.12 +12 +TK16BV2 +2.56 +2.74 +6.96 +13 +TK16PV +3.97 +12.1 +4.66 +14 +TK17BC +1.46 +11.8 +13.4 +15 +TK17BR +1.67 +4.17 +6.93 +16 +TK17BV +1.80 +1.99 +5.36 +17 +TK17AC +3.49 +208 +4.58 +18 +TK17AR +4.01 +23.3 +4.49 +19 +TK18BC +1.44 +1.33e4 +10.5 +20 +TK18BR +2.37 +5.27 +7.54 +21 +TK18BV +2.17 +2.89 +6.45 +22 +TK22BR +2.55 +4.66e3 +1.24 +23 +TK22AC +2.41 +3.69 +9.01 +24 +TK22AR +1.95 +4.74 +10.0 +25 +TK22AV +2.97 +1.31 +4.83 +26 +TK36BC +1.98 +9.33 +10.2 +27 +TK37BC +2.52 +16.5 +12.4 +28 +TK37BR +2.67 +3.23 +7.42 +29 +TK37BV +2.29 +2.66 +6.25 +Table 1: Material properties obtained from fitting using classical Biot’s theory +22 + +D. +0 +100 +200 +300 +400 +27 +28 +29 +30 +31 +32 +33 +(a) TK11 Body Circ +0 +100 +200 +300 +400 +500 +16 +18 +20 +22 +24 +26 +(b) TK11 Body Radial +0 +100 +200 +300 +400 +500 +18 +20 +22 +24 +26 +28 +30 +(c) TK11 Body Vertical +0 +100 +200 +300 +400 +500 +19 +20 +21 +22 +23 +24 +25 +26 +27 +(d) TK16 Anterior horn Radial +Figure 1: Comparison of theoretical weight loss from the sample using parameters obtained from fitting with the experiments +(a) Sample taken from the body region of TK11 sample in the circumferential direction, (b) Sample taken from the body region +of TK11 sample in the radial direction, (c) Sample taken from the body region of TK11 sample in the vertical direction, (d) +Sample taken from the Anthorn region of TK16 sample in the radial direction, +23 + diff --git a/XtA0T4oBgHgl3EQfFf84/content/tmp_files/load_file.txt b/XtA0T4oBgHgl3EQfFf84/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d0c9686c19a0b0da8a749a9077bb749717c4f8f --- /dev/null +++ b/XtA0T4oBgHgl3EQfFf84/content/tmp_files/load_file.txt @@ -0,0 +1,1032 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf,len=1031 +page_content='On the fractional transversely isotropic functionally graded nature of soft biological tissues G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Sachina, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Natarajana, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Barrera∗,b,c aDepartment of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, 600036, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' bSchool of Engineering, Computing and Mathematics, Oxford Brookes University, Headington, Oxford OX3 0BP, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' cDepartment of Engineering Science, University of Oxford, Parks Road, OX1 3PJ, Oxford, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Abstract This paper focuses on the origin of the poroelastic anisotropic behaviour of the meniscal tissue and its spatially varying properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' We present confined compression creep test results on samples extracted from three parts of the tissue (Central body, Anterior horn and Posterior horn) in three orientations (Circum- ferential, Radial and Vertical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' We show that a poroelastic model in which the fluid flow evolution is ruled by non-integer order operators (fractional Darcy’s law) provides accurate agreement with the experimental creep data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The model is validated against two additional sets of experimental data: stress relaxation and fluid loss during the consolidation process measured as weight reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Results show that the meniscus can be considered as a transversely isotropic poroelastic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' This behaviour is due to the fluid flow rate being about three times higher in the circumferential direction than in the radial and vertical directions in the body region of the meniscus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In the anterior horn, the elastic properties are transversely isotropic, with the aggregate modulus higher in the radial direction than in the circumferential and vertical directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The 3D fractional poroelastic model is implemented in finite element software and quantities such as flux of interstitial fluid during the consolidation process, a non-trivial experimental measure, are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Key words: Meniscus, Fractional poroelasticity, Confined compression tests, Anomalous diffusion, Finite element simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Introduction Biological soft tissues, such as the meniscal tissue (Figure 1), exhibit a hierarchical porous solid matrix and an interstitial fluid flowing into the pores [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The overall mechanical behaviour depends not only on the solid matrix deformation but also on the movement of the fluid in and out of the pores during the deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Furthermore, the internal architecture of the structure constituting the solid matrix and collagen determines the anisotropic behaviour of both the elastic and the transport properties [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The poroelastic theory is employed to model the response of biphasic tissue and better understand the anisotropic nature of the meniscus on the functioning of the knee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' ∗Corresponding author Email addresses: me21d011@smail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='iitm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='in (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Sachin), snatarajan@smail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='iitm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Natarajan), olga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='barrera@eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='uk, +44 7585041147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Barrera) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='02032v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='NA] 5 Jan 2023 Femur Articular cartilage of the femur Lateral Meniscus Medial Meniscus Medial articular cartilage of tibia Lateral articular cartilage of tibia Tibia (a) CONFINED COMPRESSION Anterior horn Posterior horn Central body REGION Vertical DIRECTION Channels are preferentially aligned in the circumferential direction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='04mm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='12 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='84 mm Collagen fibres and channels with random alignment Collagen fibres and channels with random alignment Collagen fibres and channels with 30� alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (b) (c) Multi-photon microscopy (fresh tissue) Micro CT scans- 600nm/ pixel (freeze dried) (a ) (b ) 10µm pores collagen elastin 3mm (d) Figure 1: (a) Lateral and Medial menisci in the knee joint, (b) Regions and the directions in the meniscus, (c) 3D porous microstructure of meniscus cylindrical sample (d)Micro CT scan and Multi-photon microscopy of meniscus tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The menisci situated between the femoral and tibial cartilage of the knee joint (Figure 1a) have three main regions, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=', anterior horn, central body and posterior horn (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' They perform several functions like load bearing, lubrication, energy absorption and stability [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In light of our recent experimental, theoretical and numerical studies [1, 2, 9–11], we understand that each region has different functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In [1, 2], the unique structure of the body region that fulfils load-bearing and energy absorption capabilities is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' This is achieved through a sandwich-like structure with two thin, stiff outside layers which take the load-bearing role and a thick, softer internal layer which acts as an energy absorption element, an effective natural damper [2, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The architectural arrangement of the solid collagen-based matrix appears to be different and gradually changing in the outside and the internal layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The outside layers show a dense distribution of collagen fibres with random orientation and low mean diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The elastic modulus is one order of magnitude higher than the internal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Whereas the internal layer of the meniscus relies on a hierarchical anisotropic network of collagen channels mainly aligned along the circumferential direction in which fluid flows during the physiological deformation process (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In [5], we have shown that the permeability changes from the anterior/posterior horns to the body region, with the body region being more permeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Here, we show that the permeability tensor in this region is also transversely isotropic and that the permeability in the circumferential direction (the preferential direction of the collagen channel) is higher than the other two directions (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Fluid flowing inside these channels determines the time-dependent behaviour of the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' As the tissue deforms under the action of loads, these channels change morphology, which results in change in permeability as a function of channels dimension, porosity, and tortuosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Thus, 2 BFemur 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='04mm CollagenFibersandChannels withrandomalignment 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='12mm CollagenFibersandchannels with30°algnment CollagenFibersandChannels 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='84mm withrandomalignment Tibiaz=-153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='7μm um 75making the permeability tensor not only depending on the pressure but also varying spatially and temporally because of the local variation of applied pressure gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Recent finite element studies of the human knee have shown that anterior and posterior horns might have a stability function [10, 12] as they deform to allow the meniscus to be extruded so that the body region has a higher contact area with the cartilage and can then carry most of the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Appropriate modelling of the fluid flow behaviour in the different portions of the meniscal tissue when subject to a range of loading conditions is essential to gain insight into the biomechanical function of the tissue and design appropriate artificial tissue substitutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Current barriers to the clinical and functional outcomes of the to-date devices [13] are that the transport and structure-properties relationship of the native tissues is still not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In particular, the relationship between the “micro/nano”-scopic composition and structure of the meniscus and its coarse-scale anisotropic behaviour structure is still an open question [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Experimental tests, such as confined compression tests (relaxation and/or creep) are currently used to characterize the material parameters such as elastic modulus at equilibrium and permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The main models used to identify these two parameters are based on biphasic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In [15], the authors examined, through a confined compression test, the effect of fluid in cartilage is bearing up to 90% of the applied load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' They used a linear biphasic model to find out the material properties in terms of the aggregate modulus and permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In [16], authors performed a creep indentation test on different regions of the meniscus of different animal species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The different material properties such as the aggregate modulus, Poisson’s ratio, shear modulus and permeability were estimated by using the biphasic model to fit the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Our recent experimental findings on human meniscal samples show that the biphasic model was not sufficient to reproduce the observed experimental behaviour [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' To address this, [17] proposed a poroelastic model wherein the pore pressure diffusion equation is derived by adopting a modified version of Darcy’s law involving fractional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It follows that the hydraulic permeability becomes “anomalous” and the rate of fluid flow is governed by the order of the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It is thus clear that the anomalous hydraulic permeability, the order of derivative tensors and their anisotropic nature, which govern the transport of fluid within the complex porous structure of the meniscus, are key properties of the meniscal tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Though a finite element implementation of the fractional poroelastic model is reported in [17], however, the model is restricted to estimating only the pore pressure field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The purpose of this paper is to present for the first time creep data of confined compression tests performed in different regions of the meniscal tissue (anterior-horn, central-body and posterior-horn) and in different directions (radial, circumferential and vertical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' We adopt the fractional poroelastic model presented in [5, 17] to fit the experimental data to recover the aggregate modulus, anomalous permeability and order of derivative and their dependence on the directions (Aggregate modulus, anomalous permeability and order of derivative tensors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' For the first time, we validate the model in terms of Stress, weight loss and displacement fields by using the parameters recovered from fitting the creep curve to match two additional sets of data for the same samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=', relaxation and weight loss data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The fractional poroelastic model is implemented in commercial software Abaqus using UMATHT and UMAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The numerical simulation results show a good agreement with analytical solutions regarding both displacements and pore pressure fields which were lacking in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' This enabled us to run consolidation tests numerically and calculate fields such as the flux of interstitial fluid during the consolidation process, which is a difficult experimental measure to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Additionally, we perform statistical analysis of the parameters using ANOVA method to compare the variation of the material parameters in different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Material and Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Confined consolidation creep tests Menisci were harvested from patients undergoing total knee arthroplasty under ethically approved pro- tocol as reported in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Samples labelled as “degraded” by the gross investigation of the surgeon were discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Menisci were thawed in Phosphate-buffered solution (PBS) for thirty minutes to recover their hydration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' A total of 29 meniscal test samples were harvested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The samples were cylindrical in shape with 3 a 3mm diameter and 3-4 mm in height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Samples were extracted from the central body, anterior horn and posterior horn regions along each principal direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' circumferential, radial and vertical (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' z F At 𝑧 = ℎ, 𝑢 = 0, 𝑝 = 0 Flux out h At 𝑧 = 0, 𝑃 = 𝑃�, �� �� = 0 Insulated boundaries (a) 0 100 200 300 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 0 (b) 0 100 200 300 400 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='8 (c) Figure 2: (a) Schematic representation of confined compression test, (b) applied load for confined compression creep test and (c) measured displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The confined compression test set-up consisted of a chamber with transparent and insulated walls and a porous plate at the bottom(Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The sample to be tested is confined inside the chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' A load of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='5 N (corresponding to a stress of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='07 MPa which is a physiological value for menisci [18]) at the top of the chamber is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It is applied through a piston with a stage velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3 % height of the sample till the load 0f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='5 N is reached and then the load is kept constant for 400s as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The applied force and the obtained displacement that is measured at the top surface are shown in Figures 2b-2c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Fractional consolidation - governing equations & finite element implementation Governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Consider a poroelastic material occupying Ω ⊂ Rd, bounded by ∂Ω ⊂ Rd−1, where d = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The boundary accommodates the following decomposition: ∂Ω = ∂ΩN ∪∂ΩD and ∂ΩN ∩∂ΩD = ∅, where Neumann and Dirichlet boundary conditions are applied on ∂ΩN and ∂ΩD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In the 4 absence of body force and inertia, the governing differential for a poroelastic material is given by: ∇ · σ = 0 (1a) ∂ζ ∂t + ∇ · Jp = 0 (1b) supplemented with the following boundary conditions: u = ˆu, and p = ˆp on ∂ΩD and σ·n = F, and Jp·n = 0 on ∂ΩN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In Equation (1), ζ is the variation of the fluid content, p is the pore pressure, Jp is the fluid flux and σ is the stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The relation between the stresses, displacements and pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' and between the flux and the pressure is given by: σ = 2Gϵ + λTr(ϵ)I − αpI (2a) Jp = −λβ 0Dβ t ∇p (2b) where λ = (3K − 2G) 3 is Lam´e’s constant, G, K are the shear and the bulk modulus, respectively, α is the Biot coefficient and 0Dβ t is the fractional derivative with respect to time of order β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It is important to note that in case of β = 0, λβ is with dimension of [L]4[F]−1[T]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In the case of β ̸= 0, λβ has dimension of [L]4[F]−1[T]β−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The pore pressure diffusion equation (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Equation (2b)), written in terms of dilatational strain, ϵd, is given by: ∂p ∂t = Ku − K α2 λβ 0Dβ t ∇2p − Ku − K α ∂ϵd ∂t (3) where Ku is the undrained bulk modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' During confined compression tests, a cylinder of meniscal tissue is inserted inside a cylindrical chamber with a porous base, see Figure 2a for geometry and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Upon applying pressure at the top of the sample, the fluid contained inside the sample will flow out of the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' This can be modelled as a one-dimensional version of Equation (2) as the only non-zero components of strain will be ϵzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Equation (2) will be modified to [17]: � K + 4G 3 � ∂ϵzz ∂z − α∂p ∂z = 0 (4a) ∂p ∂t = Ku − K α2 λβ 0Dβ t ∂2p ∂z2 − Ku − K α ∂ϵ ∂t (4b) and the corresponding boundary conditions are, ∀ 0 ≤ z ≤ h, p(z, t = 0) = γPA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' P(z = 0, t) = PA, p(z = h, t) = 0, ∂p ∂z ��� z=0 = 0 and u(z = h, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' where h is the initial height of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' A constant compressive stress, σzz(z = 0, t) = −PA in the z− direction is applied to the cylinder at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The initial pore pressure is derived assuming that the material is under undrained conditions at the initial time of loading, given by: p(z, t = 0) = PA 3(Ku − K) α (4G + 3Ku) (5) It is shown that the analytical solution of the pore pressure is given by [17]: p(z, t) = PAγ ∞ � n=1,3 E1−β,1 � −n2π2¯λt1−β 4h2 � cn cos nπz 2h (6) where E1−β,1 is the Mittag-Leffler function and γ = 3 (Ku − K) α (4G + 3Ku);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' cn = 4 nπ � − 1 � n−1 2 ¯λ = λβ (4G + 3K) (Ku − K) α2 (4G + 3Ku) (7) 5 The analytical solution for the displacement is given by: u(z, t) = 3PA 3K + 4G � (h − z) + γα ∞ � n=1,3 E1−β,1 � −n2π2¯λt1−β 4h2 � 8h (nπ)2 � (−1) n − 1 2 sin nπz 2h − 1 �� (8) The detailed derivation is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The expression for the flux is found by substituting the solution for the pore pressure (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Equation (6)) in the flux definition (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Equation (2b)) and solving the fractional derivative as follows: Jp = λβPAγt−β ∞ � n=1,3 E1−β,1−β � −n2π2¯λt1−β 4h2 � 2 h sin �nπz 2h � (9) The weight of the sample over time W(t) is related to the initial weight of the sample W0 and the fluid flux Jp as follows: W(t) = W0 − t � 0 Jp · ws A dt (10) where ws is the specific weight of the fluid, and A is the cross-section over which the fluid flows out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Substituting Equation (9) in Equation (10) and evaluating the integral analytically, we obtain the following relation: W(t) = W0 − PAAwsγ 2 h ∞ � n=1,3 � λβt1−βE1−β,2−β � −π2¯λt1−β 4h2 �� (11) For detailed derivation, refer to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Finite Element implementation and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Biot’s model with fractional Darcy’s law is implemented in Abaqus using UMATHT and UMAT subroutines using the method proposed by Barrera [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='In [17] only the pore pressure field is discussed, here for the first time we show that both the pore pressure and the displacement fields agree with the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Using the similarity between the governing equations of thermoelasticity and poroelasticity [17], UMATHT of Abaqus, which is generally meant for thermoelas- ticity, is used for the case of poroelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In Abaqus, Temperature can be treated as Pore pressure, and the heat flux can be treated as heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' UMAT is used for transferring the stress data to the UMATHT for modelling the coupling behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' UVARM can be used for transferring stress values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' However, due to the order of implementation in Abaqus, which is elastic model-UMATHT-UVARM, the stress values from UVARM reach the UMATHT in the next iteration, which introduces errors in the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' To avoid this error, UMAT is used, which allows for sharing stress values in the same increment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' UMAT is written for a linear elastic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The coefficient coupling the strain and pore pressure αT = α/3K is given as the thermal expansion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Abaqus model is validated by comparing with theoretical solutions of a con- Table 1: Material parameters Bulk modulus, K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='67×105Pa Shear modulus, G 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='69× 104 Pa (E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2×106Pa, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3) Skempton coefficient, B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='88 Biot coefficient, α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='65 Permeability, λβ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='33×10−8m4/Ns1−β Fractional Order, β 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 fined compression test with material properties given in Table 1 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' For this, a cylindrical computational model with height h = 3mm and diameter d = 3mm is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The domain is discretized with 8-noded trilinear hexahedral elements with 4 degrees of freedom per node (C3D8T elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The displacements in 6 all three directions are restrained on the bottom face and the displacements on the side faces are restrained laterally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The pore pressure is zero at the bottom and an instantaneous load, PA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='07 MPa is applied on the top surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Based on a systematic study, a total of 28,110 8-noded hexahedral elements, were found to be adequate to model the poroelastic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The displacements and the pore pressure obtained from the numerical simulation are compared against the theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Figures 3a-3b shows the comparison of the pore pressure and displacement as a function for depth for different time steps for β = 0 between theo- retical solution and numerical computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The influence of β on the pore pressure and the displacement is shown in Figures 3c-3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It is seen from Figure 3 that there is a very good agreement with the numerical result and theoretical solution and that the fractional order has a strong influence on the pore pressure and the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Large jumps in the pore pressure and displacement in the initial time in Figures 3c, 3d are due to the numerical approximation of step function with ramp in the first time increment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 3: Comparison of numerical results with theoretical results: (a-b) variation of pore pressure and displacement with depth at different time steps and β = 0 and (c-d) influence of β on the pore pressure and displacement as a function of time for a point on the top surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Material parameters: anomalous permeability, order of derivative and aggregate modulus The proposed fractional poroelastic model is used to characterize the response of the meniscus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' To this, we use the expression in Equation (8) for fitting the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The meniscus is assumed to be incompressible material, due to which B = 1, α = 1, and Ku → ∞ reducing Equation (8) to: u(z, t) = PA M � (h − z) + ∞ � n=1,3 E1−β,1 � −n2π2λβMt1−β 4h2 � 8h (nπ)2 � (−1) n − 1 2 sin nπz 2h − 1 �� (12) where, M = (3K + 4G)/3, is the aggregate modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Equation (12) is used to find the material properties of meniscus tissue by fitting with the creep displacement vs time data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It is noticed that this model requires only three material properties, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=', the aggregate modulus (M), the fractional order (β) and the permeability (λβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' On using β = 0, this theory converts into the classical Biot’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The experiments were displacement controlled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=', a constant piston velocity was used until the load reached the required value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' So, ramp loading is neglected for fitting and approximated as step loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Displacements measured in the experiments were fitted with the analytical solution Equation (12) at z = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=', u = PA M � h − ∞ � n=1,3 E1−β,1 �−n2π2λβMt1−β 4h2 � 8h (nπ)2 � (13) Fitting is done in MATLAB using the inbuilt function ‘fminsearch’ with material properties as variables and the RMS error between the analytical and the experimental data as the function to be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Fitting plots for four samples are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Table 2 shows the material properties of the meniscus obtained from the experimental results for samples in both body and anterior horn for three different directions, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=', circumferential, vertical and radial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The last two characters of the sample name indicate the location and orientation information, for example, BC represents Body Circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The other notations employed are A: Anterior horn, P: Posterior horn, R: Radial, and V: vertical directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 8 (a) (b) (c) Figure 4: Comparison between Biot’s theory with classical and fractional Darcy’s law to fit creep displacement experiment data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' To understand the fractional order’s influence, we compared the classical Biot’s poroelastic model and the proposed one (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The material properties obtained for one sample TK11BC through a fitting for classical Biot’s model1 are M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='56×10−1 MPa & λ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='95×10−13 m4/Ns and whilst that with the fractional framework are: M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='27×10−1 MPa, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='73 & λβ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='95×10−12 m4/Ns1−β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The RMS error using Classical Biot’s theory is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='53×10−5, and the RMS error while using Biot’s theory with fractional Darcy’s law is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='42×10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Figure 4 shows a comparison of the displacement, flux out and pore pressure as a function of time for classical Biot’s theory and fractional model with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It is opined that the classical Biot’s model does not fit the experimental results well and reemphasizes a need for a fractional poroelastic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 1Refer Appendix C for material properties corresponding to the classical Biot’s theory for all the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 9 0 100 200 300 400 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='8 1 (a) TK11 Body Circumferential 0 100 200 300 400 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 (b) TK11 Body Radial 0 100 200 300 400 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 (c) TK11 Body Vertical 0 100 200 300 400 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='8 1 (d) TK16 Anteriorhorn Radial Figure 5: Displacement confined creep data fitting with fractional poroelasticity and modelling using FEM, (a) Sample taken from the body region of TK11 sample in the circumferential direction, (b) Sample taken from the body region of TK11 sample in the radial direction, (c) Sample taken from the body region of TK11 sample in the vertical direction, (d) Sample taken from the anterior horn region of TK16 sample in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Further, to check for anisotropy in the meniscus, the ANOVA test is performed on the body region, properties with circular, radial and vertical directions as categories and properties are compared in different directions using MATLAB’s default function ‘anova1’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' For a few samples, properties were abnormal com- pared to the remaining samples of the same category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It could be due to the sample, experiment or fitting issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' These samples were detected as outliers by the Matlab ’anova1’ function and were not considered for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The mean and the standard deviation of the groups of the samples without considering the outliers are given in Table 3 and the results for the ANOVA test are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In Figure 6a, the aggregate modulus is compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The red line shows the sample’s median, and the black dashed line limit gives the sample’s range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' If the notched blue box of different samples does not overlap, it is concluded that the true medians are different with 95% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' If the probability against the null hypothesis (p-value) is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='05, it is concluded that the mean of the categories is different with 95% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' For aggregate modulus, the obtained p-value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='329, indicating that the aggregate modulus is not different in different directions and could possibly be treated as isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The obtained p-value for fractional order β and permeability λβ are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='001 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='0038, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' From Figures 6b-6c, it is inferred that the fractional 10 Table 2: Meniscus material parameters obtained from fitting with creep data S No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Sample h M×105 β λβ×10−12 RMS error (mm) (Pa) (m2/Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='s1−β) (×10−5) 1 TK11BC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='42 2 TK11BR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='61 3 TK11BV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='74 4 TK16BC1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='39 5 TK16BR2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='30 6 TK16BV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='48 7 TK16AC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='97 8 TK16AR1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='02 9 TK16AV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='5 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='33 11 order and permeability are relatively higher in circular directions when compared to the other two directions, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=', vertical and radial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Further, it can be considered similar in vertical and radial directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Table 3: Meniscus parameters variation across sample M×105 β λβ×10−12 (Pa) (m2/Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='s1−β) Part Mean SD Mean SD Mean SD Body Cir 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='36 Body Rad 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='40 Body Ver 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='19 Anthorn Cir 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='19 Anthorn Rad 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='03 Anthorn Ver 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='02 BC BR BV 2 4 6 8 10 12 14 Aggregate Modulus M(Pa) 104 (a) BC BR BV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='75 Fractional power (b) BC BR BV 1 2 3 4 5 6 Permeability (m4/Ns1- ) 10-12 (c) Figure 6: Material properties comparison in different orientations in the body region(BC - Body circumferential, BR - Body Radial, BV - Body Vertical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (a) Aggregate Modulus (M), (b) Fractional power β, (c) Permeability λβ To study the anisotropic behaviour of the meniscus, the confined creep test is performed as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 with material properties taken from the average values of the body region in circumferential, radial and vertical directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' For the numerical study, the height of the sample is taken as 3mm and pressure is increased to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='07 MPa from zero and held constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Figure 7 shows the pore pressure, displacement and the flux out of the sample from the bottom (computed using Equation (9)) as a function of time for the three different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 12 (a) (b) (c) Figure 7: Flux, displacement and pore pressure for different directions (Circumferential, Radial and Vertical) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Numerical modelling of fractional consolidation Numerical modelling of confined creep experiments of the meniscus is done in Abaqus using the param- eters obtained from fitting the data given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Poisson’s ratio and Young’s modulus are the elastic parameters required for Abaqus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' As the confined compression tests depend only on the aggregate modulus, Poisson’s ratio is arbitrarily assumed to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3, and Young’s modulus is calculated using the aggregate modulus and Poisson’s ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Owing to symmetry, an axisymmetric model is considered for the numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Figure 8 shows an axisymmetric model employed for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The domain is discretized with 4-noded bilinear quadrilateral elements (CAX4T) and a structured mesh is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Based on a systematic mesh convergence study, a mesh size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='05mm was found to be adequate to model the behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Axis- symmetric boundary conditions are enforced on the left of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The displacements are restrained on the right and bottom faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Zero pore pressure condition is applied at the bottom for free fluid flow and pressure PA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='07MPa is applied at the top surface as a step load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' For the numerical simulation, a fixed time increment of ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 s is used, and the simulation is carried over for 400 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' A comparison between the numerical, theoretical and experimental results is shown in Figure 5 and it is inferred that a good agreement is seen between the numerical, theoretical and experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' For the subsequent studies, the numerical models are used for complex loading in further sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 13 Z T R X Y Z Figure 8: Axisymmetric model employed for the confined creep numerical simulation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Fractional poroelastic model validation To further validate the proposed poroelastic model, a confined compression creep test with and without initial ramp loading and confined compression stress relaxation is done numercally and compared with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' We also compute the weight loss from the confined compression creep test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Confined compression creep test with initial ramp loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In the earlier study, the initial ramp present in the experiments was assumed as a step load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' However, to assess the impact of this assumption on the model fitting parameters, we ran the FE model with the initial ramp (similar to that in the experiments) for the creep test and the results from the numerical simulation are compared with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The creep test with ramp loading within Abaqus is modelled by providing the amplitude of the load in the load module with a ramp for a specific time period similar to experiments and then kept constant for the rest of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Figure 9 shows the comparison of numerical results with the experiments for a few samples, in particular for samples TK11BC, TK11BR, TK11BV and TK16AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The results of creep with ramp in Figure 9 show slight deviation at the end of ramp loading caused due to this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' But the deviation is small and can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Therefore we conclude that the material properties extraction process as explained above is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 14 0 100 200 300 400 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='8 1 (a) TK11 Body Circumferential 0 100 200 300 400 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 (b) TK11 Body Radial 0 100 200 300 400 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 (c) TK11 Body Vertical 0 100 200 300 400 500 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='8 1 (d) TK16 Anteriorhorn Radial Figure 9: Validation by a creep with ramp test (a) Sample taken from the body region of TK11 sample in the circumferential direction, (b) Sample taken from the body region of TK11 sample in the radial direction, (c) Sample taken from the body region of TK11 sample in the vertical direction, (d)Sample taken from the anterior horn region of TK16 sample in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Weight loss from confined compression creep test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Weight loss for confined creep compression problem pre- sented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 and compared with the experiments carried out by Bulle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Specific weight of water at 37◦C is used(ws = 997Kg/m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Fluid flow is collected at the bottom of the cylindrical sample with the area of the circular cross-section A with diameter d = 3mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Material properties are taken from the Table 2 which were obtained from fitting the fractional poroelastic model with displacement data from the confined creep test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Weight loss with time for some samples is shown in the Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Theoretical and experimental results for a few samples match as shown in the Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Moreover, they do not match well for some samples, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' This could be due to the measurement error as it was carried out manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The weight of the sample is measured after every 75 seconds of creep by taking it out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='(a) TK16 Body Circumfrential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='(b) TK16 Body Radial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='(c) TK16 Body Vertical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='(d) TK17 Anteriorhorn Radial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='Figure 10: Comparison of theoretical Weight loss from the sample using parameters obtained from fitting with the experiments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='(a) Sample taken from the body region of TK16 sample in the circumferential direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (b) Sample taken from the body region of TK16 sample in the radial direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (c) Sample taken from the body region of TK16 sample in the vertical direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (d) Sample taken from the Anterior horn region of TK17 sample in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Confined compression stress relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The fractional poroelastic model is validated using the confined compression stress relaxation tests performed by Bulle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Tests were performed on the cylindrical sample of height h, confined on the sides, and the fluid can flow freely from the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Tests consist of preconditioning the sample with 10%h compression at the ramp velocity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3%h/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Then, five relaxation steps with 2%h compression at a ramp velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3%h/s s were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In all the cases, the force applied is measured with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The finite element modelling of this confined compression stress relaxation test is performed in Abaqus similar to the creep test as mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Material properties obtained from fitting the creep data were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In this test, the displacement is given as input, and the force is measured as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Figure 11 shows a comparison of the FEM results with the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The comparisons of the model with the experiments are reasonably good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 16 0 20 40 60 80 100 40 30 20 10 0 (a) TK11 Body Circumferential 0 20 40 60 80 100 20 15 10 5 0 5 (b) TK11 Body Vertical 0 10 20 30 40 50 60 14 12 10 8 6 4 2 0 (c) TK11 Body Radial Figure 11: Validation by comparing stresses from Abaqus with stress relaxation test from experiments (a) Sample taken from the body region of TK11 sample in the circumferential direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (b) Sample taken from the body region of TK11 sample in the vertical direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (c) Sample taken from the body region of TK11 sample in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Discussion We show that the fractional poroelastic model is more appropriate to model the time-dependent behaviour of soft tissue presenting a hierarchically arranged porous architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The model has been verified and validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Our parameter fittings show that the RMS error = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='42×10−5 for the fractional poroelastic model is considerably lower than the one given by the classical model that had an RMS error = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='53×10−5, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The theoretical displacement curve starts from 0, whereas the experiment shows an initial displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' This is attributed to the assumption of undrained conditions at the initial time and the assumption of incompressible material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' We obtain a value of fractional order β in the range of β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='51–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='73 as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' From the numerical study, it can be inferred that the transport phenomenon during the confined compression is ruled by a fractional version of Darcy’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' At the beginning of the confined compression test, the solid is fully saturated, therefore the pore pressure reaches a maximum value (see Figure 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' This is also predicted by Terzaghi’s consolidation theory, in line with the fact that the fluid pressure entirely carries the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' During the consolidation process, the fluid flows out of the sample at a rate depending on the anomalous permeability and on the order of the fractional derivative (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 17 The higher the value of β the faster the diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' As the fluid flows out of the sample, the pore pressure decreases and the solid starts deforming (see Figure 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Also, higher values of β imply a faster solid deformation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' We notice that the aggregate modulus M obtained from fitting mentioned in Table 2 is of the same order as the literature [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The anomalous permeability λβ is in the order of 10−12/10−13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' From the statistical analysis, it can be concluded that the meniscus in the body region is elastically isotropic, and pore pressure diffusion is transversely isotropic with symmetry in the vertical and radial directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Higher hydraulic permeability in the circular direction can be attributed to the fact that the fibres are oriented in a circular direction in the body region, see Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' To model the anisotropy in the anomalous transport phenomena, the following anisotropic form of fractional Darcy’s law is proposed: Jp = � � λrad 0 0 0 λcir 0 0 0 λver � � � �� � Λ � � 0Dβrad t 0 0 0 0Dβcir t 0 0 0 0Dβver t � � � �� � 0Dβ t � ∇p � (14) where, Λ is the permeability tensor and 0Dβ t is the fractional derivative operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' We notice that for the body region, the anomalous permeability tensor Λ is transversely isotropic with λcir considerably higher than λver and λrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Similarly β tensor is transversely isotropic with βcir considerably higher than the βrad and βver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Consequently, from the Figure 7c, it is observed that the fluid flow rate through the sample’s base during the consolidation experiment in the circumferential direction is higher than the other two directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Pore pressure in the circumferential direction reaches a steady state faster than the other two directions, as shown in the Figure 7a because of the higher permeability in the circumferential direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Based on the mean values and the standard deviation in the anterior horn region, it is opined that the aggregate modulus in the radial direction is higher than the other two directions, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=', vertical and circular directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It is inferred that the permeability and the order of derivative are higher in the radial and vertical directions than in the circumferential directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The elastic tensor results are transversely isotropic, with the modulus in the radial direction being higher, and both Λ and β are transversely isotropic with circumferential being lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Different properties in different directions show that the elastic, anomalous permeability and the order of the derivative tensors are transversely isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Furthermore, properties vary with the anterior horn and body region, showing that it is not homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Vertical Anterior horn Central body Transversely isotropic permeability tensor (higher permeability in the circumferential direction) Isotropic elastic tensor Transversely isotropic permeability tensor (permeability in radial and vertical higher than in the circumferential direction) Transversely isotropic elastic tensor (higher aggregate modulus in the radial direction) Figure 12: Directional and regional dependence of material properties in the meniscus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In summary (see Figure 12), we show that the body region which has a load-bearing function exhibits a 18 transversely isotropic behaviour due to the rate of fluid flow being about three times higher (faster diffusion) in the circumferential direction which is consistent with the preferential direction of collagen channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It explains the role of fluid pressure in sustaining the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Furthermore, we show that both the elastic and the permeability tensors are transversely isotropic in the anterior horn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The aggregate modulus is higher in the radial direction compared to the circumferential and vertical directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It explains the role of the anterior horn to be compliant in the circumferential and radial directions to accommodate the kinematics of the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Conclusions The meniscus has a porous, hierarchical, multi-oriented fiber and bundle structure filled with fluid that provides optimized load support and lubrication properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' To understand its function, confined compression creep tests were performed on different regions of meniscus tissue and in different orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Biot’s theory with fractional Darcy’s law is used to find the material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It is observed that, with the classical Darcy’s law, fitting gives an RMS error of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='53×10−5, while the fractional Darcy’s law gives an RMS error of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='42×10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It is shown that Biot’s theory with fractional Darcy’s law is better suited for modelling the poroelastic behaviour of the meniscus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Three material parameters, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=', aggregate modulus(M), fractional order(β) and the permeability(λβ) required for the fractional Biot’s theory were found by fitting the theory with the confined compression creep experiment results for all the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Material properties in different directions of the body region are compared using ANOVA test methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' It is observed that the aggregate modulus (Mcir = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 ±48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='9 KPa, Mrad = 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='7±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 KPa and Mver = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='7 KPa) is isotropic and the permeability (λβcir = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='75×10−12± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='36×10−12m2/Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='s1−β, λβrad = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='25×10−12± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='04×10−13m2/Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='s1−β and λβver = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='61×10−13± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='94×10−13m2/Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='s1−β) and the fractional order (βcir = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='00, βrad = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='06 and βver = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='58± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='04) are transversely isotropic with properties in the circumferential direction is greater than the vertical and radial directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Biot’s theory with fractional Darcy’s law is implemented numerically using the FEM in Abaqus software using UMATHT and UMAT subroutines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' This fractional poroelastic theory is validated by using the material properties obtained from fitting confined creep tests to model stress relaxation, creep with ramp, and weight loss tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The results are then compared with experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The comparison shows that Biot’s theory with fractional Darcy’s law is capable of better characterising the poroelastic behaviour of the meniscus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Flux out of the meniscus, which is difficult to measure experimentally, can easily be computed efficiently using the current model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' In future works, Biot’s theory with fractional Darcy’s law will be extended to include anisotropy and use it to understand the working of the meniscus in the knee joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Acknowledgements O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='B would like to acknowledge the European Union’s Horizon 2020 -EU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' - Nurturing excellence by means of cross-border and cross-sector mobility under the Marie Sk�lodowska-Curie individual fellowship MSCA-IF-2017, MetaBioMec, Grant agreement ID: 796405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' The authors thank the Rizzoli Orthopaedic Institute and, in particular, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Marchiori and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Berni for their invaluable insights on the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' References [1] G.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Agrawal, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Clanton, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Athanasiou, Intraspecies and interspecies comparison of the compressive properties of the medial meniscus, Annals of Biomedical Engineering 32 (11) (2004) 1569–1579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Moyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Priest, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Buman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Abraham, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Donahue, Indentation properties and glycosaminoglycan content of human menisci in the deep zone, Acta Biomaterialia 9 (5) (2013) 6624–6629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Seitz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Galbusera, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Krais, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Ignatius, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Durselen, Stress-relaxation response of human menisci under confined compression conditions, Journal of the Mechanical Behavior of Biomedical Materials 26 (2013) 68–80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Chia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Hull, Compressive moduli of the human medial meniscus in the axial and radial directions at equilibrium and at a physiological strain rate, Journal of Orthopaedic Research 26 (7) (2008) 951–956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' [23] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Podlubny, Fractional Differential Equations, Academic Press, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Using Equation (4a), the displacement field for the problem described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 can be solved as follows: ϵzz = ∂uz ∂z = 3 3K + 4G � − PA + αp � (15) Substituting Equation (6) in Equation (15) and integrating u(z, t) = 3PA 3K + 4G � − z + γα ∞ � n=1,3 E1−β,1 � −n2π2¯λt1−β 4h2 � bn sin nπz 2h � + d (16) where, bn = 8h (nπ)2 (−1) n−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Applying the boundary condition given by Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' we get: u(h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' t) = 3PA 3K + 4G � − h + γα ∞ � n=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3 E1−β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 � −n2π2¯λt1−β 4h2 � 8h (nπ)2 (−1) n−1 2 sin nπ 2 � + d = 0 (17) 20 Since,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' � − 1 � n−1 2 sin nπ 2 = 1 for n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' · · · (18) Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' the constant d is d = − 3PA 3K + 4G � − h + γα ∞ � n=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3 E1−β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 � −n2π2¯λt1−β 4h2 � 8h (nπ)2 � (19) Substituting Equation (19) in Equation (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Displacement field equation is obtained u(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' t) = 3PA 3K + 4G � (h − z) + γα ∞ � n=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3 E1−β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 � −n2π2¯λt1−β 4h2 � 8h (nπ)2 � (−1) n−1 2 sin nπz 2h − 1 �� (20) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Weight loss from the sample over time W(t) for the problem described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 can be solved using the fluid flux as: W(t) = W0 − � t 0 Jp · wsAdt (21) where Jp is the fluid flux given by Equation (2b), ws is the specific weight, A is the cross-sectional area over which the flux is calculated and W0 being the initial weight of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Differentiating Equation (6) with respect to z, we get: ∂p ∂z = −PAγ ∞ � n=1,3 E1−β,1 � − n2π2¯λt1−β 4h2 � 2 h(−1) n−1 2 sin �nπz 2h � (22) Fluid can flow only at the bottom, as shown in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Hence, the weight loss is solved at z = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Equation (22) at z = h is: ∂p ∂z = −PAγ ∞ � n=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3 E1−β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 � − n2π2¯λt1−β 4h2 � 2 h (23) Using Equation (23) and Equation (2b) in weight loss Equation (21),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' we get: W(t) = W0 − λβwsA 0Dβ−1 t � PAγ ∞ � n=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3 E1−β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 � − n2π2¯λt1−β 4h2 � 2 h � (24) Using the identity of the fractional derivative from [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' which is: 0Dα t tβ−1Eµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='β(λtµ) = tβ−1−αEµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='β−α(λtµ) (25) Equation (24) can be found as: W(t) = W0 − λβwsAPAγt1−β ∞ � n=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3 E1−β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2−β � − n2π2¯λt1−β 4h2 � 2 h (26) 21 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' S No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' Sample M ×105(Pa) λ × 10−13(m2/Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='s) RMS ×10−5 1 TK11BC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='56 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='95 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='53 2 TK11BR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='91 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='40 3 TK11BV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='44 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='10 4 TK16BC1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='32 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='04 5 TK16BR2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='45 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='53 6 TK16BV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='37 7 TK16AC1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='80 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='64 8 TK16AR1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='42 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='16 9 TK16AV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='79 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='83 10 TK16BC2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='39 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='87 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='15 11 TK16BR1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='68 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='12 12 TK16BV2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='74 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='96 13 TK16PV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='97 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='66 14 TK17BC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='46 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 15 TK17BR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='67 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='93 16 TK17BV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='99 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='36 17 TK17AC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='49 208 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='58 18 TK17AR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='01 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='49 19 TK18BC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='33e4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='5 20 TK18BR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='27 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='54 21 TK18BV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='89 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='45 22 TK22BR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='66e3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='24 23 TK22AC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='41 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='69 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='01 24 TK22AR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='95 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='74 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='0 25 TK22AV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='83 26 TK36BC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='98 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='33 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='2 27 TK37BC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='52 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='4 28 TK37BR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='42 29 TK37BV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='66 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='25 Table 1: Material properties obtained from fitting using classical Biot’s theory 22 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='(a) TK11 Body Circ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='(b) TK11 Body Radial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='(c) TK11 Body Vertical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='(d) TK16 Anterior horn Radial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='Figure 1: Comparison of theoretical weight loss from the sample using parameters obtained from fitting with the experiments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content='(a) Sample taken from the body region of TK11 sample in the circumferential direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (b) Sample taken from the body region of TK11 sample in the radial direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (c) Sample taken from the body region of TK11 sample in the vertical direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' (d) Sample taken from the Anthorn region of TK16 sample in the radial direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} +page_content=' 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtA0T4oBgHgl3EQfFf84/content/2301.02032v1.pdf'} diff --git a/Y9E5T4oBgHgl3EQfdw8X/content/tmp_files/2301.05613v1.pdf.txt b/Y9E5T4oBgHgl3EQfdw8X/content/tmp_files/2301.05613v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0ca7f147b5d4d642fd3fda941204425b8817f243 --- /dev/null +++ b/Y9E5T4oBgHgl3EQfdw8X/content/tmp_files/2301.05613v1.pdf.txt @@ -0,0 +1,562 @@ +arXiv:2301.05613v1 [math.GR] 13 Jan 2023 +Elementary equivalence +of stable linear groups over fields of characteristic 2 +E. I. Bunina∗, A. V. Mikhalev, I. O. Solovyev† +January 16, 2023 +In this paper, we prove a criterion of elementary equivalence of stable linear groups over +fields of characteristic two. +1 +Introduction, History and Definitions +1.1 +Elementary equivalence. +Two structures of the same signature are called elementary equivalent if they satisfy the +same first order sentences in their signature. Any two finite structures with the same signature +are elementarily equivalent if and only if they are isomorphic. Any two isomorphic structures +are elementarily equivalent, but the opposite is not always true. +For example, the field of +complex numbers C and field of algebraic numbers Q are elementarily equivalent, but they +cannot be isomorphic due to difference in their cardinalities. +Tarski and Maltsev pioneered the theory of describing groups and rings from an elementarily +equivalence standpoint. Several complete results were obtained: for example, two algebraically +closed fields are elementary equivalent if and only if they have the same characteristics; two +Abellian groups are elementary equivalent if and only if they have the same special “character- +istic numbers” (Szmielew, [26]); similar results with invariants were obtained for Boolean rings +(Ershov–Tarski, [17]). +An outstanding result was the answer for the old problem raised by A. Tarski around 1945: +for free groups the elementary theory doesn’t distinguish these groups (see the series of works +of Kharlampovich–Myasnikov and Z. Sela, e. g. [20], [23]). The similar situation takes place for +the torsion free hyperbolic groups (see Sela, [24]). +∗Bar Ilan University (Israel), Department of mathematics +†M.V. Lomonosov Moscow State University, Faculty of mechanics and mathematics +1 + +1.2 +Maltsev-type theorems for linear groups. +It is also interesting to study connections between logic properties of some basic structures +and logic properties of structures derived from these basic structures. +First results of such type was obtained by Maltsev in 1961 in [21]. He proved that the +groups Gn(K1) and Gm(K2) (where G = GL , SL , PGL , PSL , n, m ⩾ 3, K1, K2 are fields of +characteristics 0) are elementarily equivalent if and only if m = n and the fields K1 and K2 are +elementarily equivalent. +This type of correspondence are often called Maltsev translation. It means that logic prop- +erties are completely translated from basic structures to derived structures and vice-versa. +In 1961–1971 Keisler ([19]) and Shelah ([25]) proved the next important Isomorphism the- +orem: +Theorem 1. Two models U1 and U2 are elementarily equivalent if and only if there exists an +ultrafilter F such that their ultrapowers coincide: +� +F +U1 ∼= +� +F +U2. +This theorem allowed Beidar and Mikhalev in 1992 (see [2]) to generalize Maltsev theorem +for the case when K1 and K2 are skewfields and prime associative rings. This approach for the +groups GL n was generalized in [3] by the following result: +Theorem 2. Let R1 and R2 be associative rings with 1 (1/2) with finite number of central +idempotents and m, n ⩾ 4 (m, n ⩾ 3). Then GL m(R1) ≡ GL n(R2) if and only if there exist +central idempotents e ∈ R and f ∈ S such that eMm(R) ≡ fMn(S) and (1 − e)Mm(R) ≡ +(1 − f)Mn(S)op. +Continuation of investigations in this field were the papers of Bunina 1998–2010. Similar +to Maltsev’s results were obtained not only for classical linear groups GL , PGL , SL , PSL , but +also for unitary linear groups over fields, skewfields, and rings with involutions (see [6], [7]), +for Chevalley groups over fields ([8]), over local rings (see [9]) and arbitrary commutative rings +(see [10]), and also for other different derivative structures. +In some cases elementary equivalence of derivative structures (even a little similar to linear +groups) is equivalent not to elementary but to more strong equivalence of initial structures. +Often second-order equivalence or some of its limitations can appear. +For example, in 2000, V. Tolstykh [28] stated the connection between second-order prop- +erties of skewfields and first-order properties of automorphism groups of infinite-dimensional +linear spaces over them. In 2003, E. I. Bunina and A. V. Mikhalev (see [11]) stated the connec- +tion between second-order properties of associative rings and elementary properties of categories +of modules, endomorphism rings, automorphism groups, and projective spaces of infinite rank +over these rings. Similar results were obtained also for endomorphism rings and automorphism +groups of Abelian p-groups: Bunina, Mikhalev and Roizner (see [?]) proved that two endomor- +phism rings of Abelian p-groups or two automorphism groups of Abelian p-groups for p ⩾ 3 are +elementarily equivalent if and only if initial Abelian groups are equivalent in the full second +order logic (in one exceptional case in its limitation). +2 + +1.3 +Stable Linear Groups +Let R be an associative ring with unit. The following definitions correspond to [18]. +Definition 1. Denote by Mat ∞(R) the ring of matrices with countable number of rows and +columns such that out of the main diagonal there are only a finite number of nonzero elements, +and also there exists a number n such that for any i ⩾ n the elements rii = a, a ∈ R. +It is clear that Mat ∞(R) is a ring. +Let A ∈ GL n(R). We identify A with an element from GL n(R) by the following rule: A is +written in the left upper corner, starting from the position (n, n) the diagonal contains 1, and +all other places contain zeros. +We preserve the notation GL n(R) for the obtained subgroups Mat ∞(R). It is clear that +GL n(R) are subgroups of the groups of invertible elements of the ring Mat ∞(R), and also it is +clear that for m ≥ n we have GL n(R) ⊆ GL m(R). +Definition 2. Let us set +GL (R) = +� +n⩾1 +GL n(R) +(GL n(R) ⊆ Mat ∞(R)). +It is a subgroup of the group of invertible elements of the ring Mat ∞(R). Let us call it the +stable linear group. +For stable linear groups their automorphisms were described: Atkarkaya described auto- +morphisms of stable linear groups E(R) and GL(R) over commutative local rings R with 1/2 +(see [1]). +In our previous paper (see [13]) it was proved that despite an “infinite” dimension of the +stable group, from elementary equivalence of two arbitrary rings with unit it follows elementary +equivalence of stable linear groups over them, i.e., we do not need higher-order logic. In the same +paper we proved that from elementary equivalence of stable linear groups over commutative +local rings with 1/2 it follows elementary equivalence of the corresponding rings. +In the given work we extend this previous result to the stable linear groups over fields of +characteristic 2. +2 +Proof of the main theorem +Consider the group GL (R), where R is a field of characteristic 2. Let us denote its unit +by E. For more simple formulas we will write A ∼ B instead of ∃U A = UBU−1 and call A to +be conjugate to B. +Our first goal is to define elementarily a subgroup isomorphic to the group GL 2(R). We +will use matrices diag [1, 1, T], where +T ∼ diag +��1 +1 +1 +0 +� +, . . . +� +The first part of the proof depends on existence of third roots of unity in the field. +3 + +2.1 +Fields containing third roots of unity +Let K be a field of characteristic 2 containing third roots of unity 1, ξ, ξ2. +The following lemma is proved, for example, in [5]; it follows from classical linear algebra +results. +Lemma 1. For any number of mutually commuting third-order elements of the group GL n(K), +there exists a basis in which all of them have diagonal form with third roots of unity on the +diagonal. +It is clear that for a finite set of commuting matrices the same condition holds also in the +stable linear group GL (K). +Let A be a set of matrices. Denote by |||A||| the number of distinct conjugacy classes of A. +Lemma 2. Let us consider a set D = [d1, d2, d3, . . . , dk] of elements of K. If D includes [ξ, ξ] +or [ξ, ξ2], then |||{AB : A ∼ diag [D], B ∼ diag [D], AB = BA}||| > 2. +Proof. To prove this lemma it is sufficient to consider only one pair of elements di, since other +elements can be just fixed at the first positions. Also it is sufficient to consider only permutation +of diagonal elements. +First consider the case when D contains [ξ, ξ]. We assume that these elements are corre- +sponded to dk−1, dk. The first k −2 elements are fixed, under multiplication they give the same +set of eigenvalues. We can obtain the following matrices: +diag [1, 1, ξ, ξ] · diag [ξ, ξ, 1, 1] = diag [ξ, ξ, ξ, ξ], +diag [1, 1, ξ, ξ] · diag [ξ, 1, ξ, 1] = diag [ξ, 1, ξ2, ξ], +diag [1, 1, ξ, ξ] · diag [1, 1, ξ, ξ] = diag [1, 1, ξ2, ξ2]. +We omitted for simplicity of reading elements d1, d2, d3, . . . , dk−2 and final units of stable ma- +trices. All these matrices have different sets of eigenvalues, so they are pairwise non-conjugate. +If D contains [ξ, ξ2], then +diag [1, 1, ξ, ξ2] · diag [1, 1, ξ, ξ2] = diag [1, 1, ξ2, ξ], +diag [1, 1, ξ, ξ2] · diag [ξ, ξ2, 1, 1] = diag [ξ, ξ2, ξ, ξ2], +diag [1, 1, ξ, ξ2] · diag [1, ξ2, ξ, 1] = diag [1, ξ2, ξ2, ξ2], +the same situation. +As a result of these two lemmas we obtain the following lemma: +Lemma 3. Let A3 = E. If |||{BC : B ∼ A, C ∼ A, CB = BC}||| = 2, then A ∼ diag [ξ] (or +A ∼ diag [ξ2]). +This lemma gives a method how to elementarily define a matrix conjugated to one of diag [ξ] +and diag [ξ2]. +ϕ(A) := ∃X1∃X2∃Y1∃Y2∀Z1∀Z2 (A3 = E) ∧ ¬(A = E)∧ +∧ (X1 ∼ X2 ∼ Y1 ∼ Y2 ∼ A) ∧ (X1X2 = X2X1) ∧ (Y1Y2 = Y2Y1)∧ +∧ +�� +(Z1 ∼ A) ∧ (Z2 ∼ A) ∧ (Z1Z2 = Z2Z1) +� +→ +� +(Z1Z2 ∼ Y1Y2) ∨ (Z1Z2 ∼ X1X2) +�� +(1) +4 + +Suppose this formula holds for A. Assume that A ∼ diag [ξ]. It is possible to obtain according +to a replacement of notations ξ2 → ξ′, ξ → ξ′2. Then A2 ∼ diag [ξ2]. +Consider the formula +ψ(B) := ∃X1∃X2(X1 ∼ A) ∧ (X2 ∼ A2) ∧ (X1X2 = X2X1) ∧ (X1X2 = B) ∧ ¬ϕ(B). +If this formula holds for B, then B ∼ diag [ξ, ξ2]. +Let X1 ∼ X2 ∼ A; X1X2 = X2X1. We can suppose that we chose a basis in such that +X1 = diag [ξ, 1, . . .], X2 = diag [1, ξ, . . .]. +Let us consider the formula +θ(C) := (CX1 = X1C) ∧ (CX2 = X2C) ∧ (C ∼ B) ∧ +� +i +¬ϕ(BXi) ∧ +� +i +¬ϕ(BX2 +i ). +If this formula holds for C = (cij), then since c commutes with Xi, i = 1, 2, we have +c1j = ci1 = 0, i, j > 1; c2j = ci2 = 0, i, j > 2. C has order 3, so c3 +11 = 1, c3 +22 = 1. Suppose +that c11 ̸= 1. Then either c11 = ξ, or c11 = ξ2. Suppose (without loss of generality) that +c11 = ξ. Then, formula ϕ holds for CX1, it contradicts to definition of θ. Therefore c11 = 1, +and similarly c22 = 1. Then C = diag [1, 1, T], where T ∼ diag [ξ, ξ2]. +Lemma 4. diag [ξ, ξ2] ∼ +�1 +1 +1 +0 +� +. +Proof. To prove this lemma we need to find the eigenvalues of the matrix. Its characteristic +polynomial is P(λ) = λ2 + λ + 1. Substituting ξ to the characteristic polynomial, we obtain +P(ξ) = ξ2 + ξ + 1. Note that P(ξ) = ξP(ξ), hence +P(ξ) + ξP(ξ) = 0 ⇒ P(ξ)(1 + ξ) = 0 ⇒ P(ξ) = 0, +Similarly for ξ2. Therefore ξ (and ξ2) are roots of the characteristic polynomial. +2.2 +Fields without third roots of unity +Let K be a field of characteristic 2 without non-trivial roots of the third power of unity. +Lemma 5. For any finite number of mutually commuting elements of the order 3 in GL n(K), +there exists a basis in which all of them have block-diagonal form with E11 + E12 + E21, E22 + +E12 + E21 and E11 + E22 on their diagonals. +This lemma is proved in [5] for the elements from GL n, but it is obviously correct for the +group GL too. +Denote T = E11 + E12 + E21 and Dk = diag [T, T, . . . , T, T +� +�� +� +k +, 1, . . .]. +Lemma 6. If k > 1, then |||{AB : A ∼ Dk, B ∼ Dk, AB = BA, A, B ∈ GL (K)}||| > 3. +5 + +Proof. It is sufficient to consider only two blocks T, which exist in the block-diagonal form +of A. By permutations of these two blocks we can obtain +diag [E, E, T, T] · diag [T, T, E, E] = diag [T, T, T, T], +diag [E, E, T, T] · diag [E, E, T 2, T 2] = diag [E, E, E, E], +diag [E, E, T, T] · diag [E, E, T, T] = diag [E, E, T 2, T 2], +diag [E, E, T 2, T] · diag [E, T, T 2, E] = diag [E, T, T, T], +where E is a unity matrix of order 2 × 2. As it follows from Lemma 4, T 2 ∼ T. +This lemma gives us a formula that holds for matrix A if and only if A ∼ D1: +ϕ′(A) := ∃X1∃X2∃Y1∃Y2∀Z1∀Z2 (A3 = E) ∧ ¬(A = E) ∧ +∧ (X1 ∼ X2 ∼ Y1 ∼ Y2 ∼ A) ∧ (X1X2 = X2X1) ∧ (Y1Y2 = Y2Y1) ∧ +∧ +�� +(Z1 ∼ A) ∧ (Z2 ∼ A) ∧ (Z1Z2 = Z2Z1) +� +→ +→ +� +(Z1Z2 ∼ X1X2) ∨ (Z1Z2 ∼ Y1Y2) ∨ (Z1Z2 ∼ E) +�� +. +(2) +Lemma 7. Let A = (aij) commute with +Gk = diag [1, 1, . . . , 1 +� +�� +� +k−1 +, T, 1, 1 . . .], where k ⩾ 1. +Then +1. If j + 1 < k or k + 1 < j, then +� aj,k +aj,k+1 +aj+1,k +aj+1,k+1 +� += +�0 +0 +0 +0 +� +2. If j + 1 < k or k + 1 < j, then +� ak,j +ak,j+1 +ak+1,j +ak+1,j+1 +� += +�0 +0 +0 +0 +� +3. +� ak,k +ak,k+1 +ak+1,k +ak+1,k+1 +� += +�a +b +b +a + b +� +, +for some a, b. +Proof. (1) Assume that A commutes with Gk and B = (bij) = AGk = GkA. Consider +� +bj,k +bj,k+1 +bj+1,k +bj+1,k+1 +� +. +6 + +When A is multiplied by Gk from the left: +� bj,k +bj,k+1 +bj+1,k +bj+1,k+1 +� += +� aj,k +aj,k+1 +aj+1,k +aj+1,k+1 +� +. +When A multiplied by Gk from the right: +� bj,k +bj,k+1 +bj+1,k +bj+1,k+1 +� += +� +aj,k + aj,k+1 +aj,k +aj+1,k + aj+1,k+1 +aj+1,k +� +. +Then: +� +aj,k +aj,k+1 +aj+1,k +aj+1,k+1 +� += +� +aj,k + aj,k+1 +aj,k +aj+1,k + aj+1,k+1 +aj+1,k +� +. +From the equality of left columns of both matrices we obtain aj,k+1 = aj+1,k+1 = 0. But then +from the equality of right columns of both matrices we obtain that all other elements equal to +zero too. The proposition (2) is proved in exactly the same way as proposition (1). +(3) We have +�ak,k + ak+1,k +ak,k+1 + ak+1,k+1 +ak,k +ak,k+1 +� += +� +ak,k + ak,k+1 +ak,k +ak+1,k + ak+1,k+1 +ak+1,k +� +. +Define b := ak,k+1 = ak+1,k, a := akk, then ak+1,k+1 = a + b. +Let X be a matrix satisfied the formula ϕ′. Consider the formula +θ′(C) := (CX = XC) ∧ (C ∼ X) ∧ (CX ̸= E) ∧ (CX2 ̸= E). +Let C be a matrix such that θ′(C) is true. Let us fix a basis such that X has a form G1. Then +from Lemma 7 we have c1j = ci1 = 0, c2j = ci2 = 0, i, j > 2. Also we have +� +c11 +c12 +c21 +c22 +� += +� +a +b +b +a + b +� +and +� +a +b +b +a + b +�3 += +� +1 +0 +0 +1 +� +or +�a3 + ab2 + b3 +a2b + ab2 +a2b + ab2 +a3 + a2b + b3 +� += +�1 +0 +0 +1 +� +. +It follows that a2b + ab2 = ab(a + b) = 0. There is no zero divisors in K, therefore there are +three possible cases: +1. b = 0. Then a3 = 1 ⇒ a = 1. +2. a = 0. Then b3 = 1 ⇒ b = 1. +3. a = b. Then a3 + ab2 + b3 = 1 ⇒ a3 + a3 + a3 = 1 ⇒ a3 = 1 ⇒ a = b = 1. +The first case corresponds to the unity matrix. The second and the third cases correspond to +� +c11 +c12 +c21 +c22 +� += T and +� +c11 +c12 +c21 +c22 +� += T 2. Both thess cases are impossible due to the formula θ′, +because such C multiplied by X or X2 gives the unity matrix. +So we peoved that C has the form diag [1, 1, T ′], where T ′ ∼ D1. +7 + +2.3 +Elementary definability of GL 2(K) and the main theorem +Assume the formula θ holds only for matrices diag [1, 1, T], where T ∼ +�1 +1 +1 +0 +� +. Let us +determine GL 2(K) using these matrices. +Let us notice that the formula θ holds for Gk, k > 2. If M commutes with G2k+1, k = +1, 2, 3, . . ., then M = diag [M0, M1, M2, M3, M4, . . . , Mk], where M0 ∈ GL 2(K), and Mi = +�ai +bi +bi +ai + bi +� +, i > 0. +If M commutes also with G2k+2, k = 1, 2, 3, . . ., then M = diag [M0, a1, M′ +1, M′ +2, M′ +3, M′ +4, . . . , M′ +k′], +where M′ +i = +� +a′ +i +b′ +i +b′ +i +a′ +i + b′‘i +� +. Comparing block structures of different representations of M, we +obtain +M = diag [M0, M1, M2, M3, M4, . . . , Mk] = diag [M0, a1, M′ +1, M′ +2, M′ +3, M′ +4, . . . , M′ +k′]. +Therefore bi = b′ +i = 0, ai = a′ +i and a′ +i = ai+1. Hence ai = ai+1 for every i. It follows from the +definition of the stable linear group that there is only a finite number of non-unit elements on +the diagonal. Hence ai = 1. +Therefore, we obtain a formula that defines a subgroup isomorphic to GL 2(K): +γ(M) = ∀C +� +θ(C) → (MC = CM) +� +. +Now we are ready to prove the main theorem of this paper. +Theorem 1. Let K1 and K2 be fields of characteristic 2. If the stable linear groups GL (K1) and +GL (K2) are elementarily equivalent, then the fields K1 and K2 are also elementarily equivalent. +Proof. Let GL (K1) ≡ GL (K2). Since in the previous lemmas we proved that the subgroup +GL 2 is elementarily defined in GL for a field of characteristic 2, this means that GL 2(K1) ≡ +GL 2(K2). By the generalization of Maltsev Theorem proved for example in [4] this implies +K1 ≡ K2. +References +[1] A.S. Atkarskaya. Automorphisms of Stable Linear Groups Over Commutative Local Rings +With 1/2. Journal of Mathematical Sciences, 2014, 197, 455–466. +[2] C. I. Beidar, A. V. Mikhalev, On Mal’cev’s theorem on elementary equivalence of linear +groups. Contemporary mathematics, 1992, 131(1), 29–35. +[3] Bragin V., Bunina E. Elementary equivalence of linear groups over rings with a finite +number of central idempotents and over Boolean rings. Journal of Mathematical Sciences, +201, 2014, 438–445. +[4] E. I. Bunina, A. V. Mikhalev, and A. G. 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Elementary equivalence of categories of modules over rings, +endomorphism rings, and automorphism groups of modules. J. Math. Sci., 2006, 137(6), +5275–5335. +[12] Bunina E.I., Mikhalev A.V., Roizner M.A. The criteria of elementary equivalence of auto- +morphism groups and endomorphism rings of Abelian p-groups. Dokl. Ross. Akad. Nauk, +2014, 457(1), 11–12. +[13] E. I. Bunina, A. V. Mikhalev, I. O. Solovyev, Elementary equivalence of stable linear groups +over local commutative rings with 1/2. Journal of Mathematical Sciences, 2018, 233(5), +646–655. +[14] Chang C., Keisler H. Model Theory. North Holland, 1990. +[15] Golubchik. I.Z. Isomorphisms of the general linear group GL n(R), n ⩾ 4, over an asso- +ciative ring. Contemporary mathematics, 1992, 131, 123–137. +[16] Golubchik I.Z., Mikhalev A.V. Isomorphisms of the general linear group over associative +ring. Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., 1983, 3, 61–72 (in Russian). +[17] Goncharov S. Countable Boolean allgebras and decidability. Novosibirsk: Science books, +1996 (in Russian). +[18] Hahn A.J., O’Meara O.T. The Classical Groups and K-Theory. Springer-Verlag, Berlin, +New York, 1989, 565 pp. +[19] Keisler H.J. Ultraproducts and elementary models. Indagationes Mathematicae, 1961, 23, +477–495. +[20] Kharlampovich O., Myasnikov A. Elementary theory of free non-abelian groups. Journal +of Algebra. 302, 2006, 451–552. +9 + +[21] Maltsev A.I. On elementary properties of linear groups. Problems of Mathematics and +Mechanics, Novosibirsk, 1961 (in Russian). +[22] Maltsev A.I. On isomorphic matrix representations of infinite groups. Rec. Math. [Mat. +Sbornik] 8(50), 1940, 405–422 (Russian. English summary). +[23] Sela Z. Diophantine geometry over groups. VI. The elementary theory of a free group. +Geom. Funct. Anal. 16(3), 2016, 707–730. +[24] Sela Z.. Diophantine geometry over groups and the elementary theory of free and hyperbolic +groups. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, +2002), 87–92, Beijing, 2002. +[25] Shelah S. Every two elementarily equivalent models have isomorphic ultrapowers. Israel J. +Math., 1971, 10, 224–233. +[26] Szmielew W. Elementary properties of Abelian groups. Fundamenta Mathematica, 41, +1955, 203–271. +[27] Stephenson W. Lattice isomorphism between modules. J. London Math. Soc., 1969, 1, +177–188. +[28] Tolstykh V. Elementary equivalence of infinite-dimensional classical groups. Ann. Pure +Appl. Logic, 2000, 105, 103–156. +10 + diff --git a/Y9E5T4oBgHgl3EQfdw8X/content/tmp_files/load_file.txt b/Y9E5T4oBgHgl3EQfdw8X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2891859b9eb38a8c87d5a2ecbb4362f8d832da47 --- /dev/null +++ b/Y9E5T4oBgHgl3EQfdw8X/content/tmp_files/load_file.txt @@ -0,0 +1,419 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf,len=418 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content='05613v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content='GR] 13 Jan 2023 Elementary equivalence of stable linear groups over fields of characteristic 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Bunina∗, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Mikhalev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Solovyev† January 16, 2023 In this paper, we prove a criterion of elementary equivalence of stable linear groups over fields of characteristic two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' 1 Introduction, History and Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content='1 Elementary equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Two structures of the same signature are called elementary equivalent if they satisfy the same first order sentences in their signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Any two finite structures with the same signature are elementarily equivalent if and only if they are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Any two isomorphic structures are elementarily equivalent, but the opposite is not always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' For example, the field of complex numbers C and field of algebraic numbers Q are elementarily equivalent, but they cannot be isomorphic due to difference in their cardinalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Tarski and Maltsev pioneered the theory of describing groups and rings from an elementarily equivalence standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Several complete results were obtained: for example, two algebraically closed fields are elementary equivalent if and only if they have the same characteristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' two Abellian groups are elementary equivalent if and only if they have the same special “character- istic numbers” (Szmielew, [26]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' similar results with invariants were obtained for Boolean rings (Ershov–Tarski, [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' An outstanding result was the answer for the old problem raised by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Tarski around 1945: for free groups the elementary theory doesn’t distinguish these groups (see the series of works of Kharlampovich–Myasnikov and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Sela, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' [20], [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' The similar situation takes place for the torsion free hyperbolic groups (see Sela, [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' ∗Bar Ilan University (Israel), Department of mathematics †M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Lomonosov Moscow State University, Faculty of mechanics and mathematics 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content='2 Maltsev-type theorems for linear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' It is also interesting to study connections between logic properties of some basic structures and logic properties of structures derived from these basic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' First results of such type was obtained by Maltsev in 1961 in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' He proved that the groups Gn(K1) and Gm(K2) (where G = GL , SL , PGL , PSL , n, m ⩾ 3, K1, K2 are fields of characteristics 0) are elementarily equivalent if and only if m = n and the fields K1 and K2 are elementarily equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' This type of correspondence are often called Maltsev translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' It means that logic prop- erties are completely translated from basic structures to derived structures and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' In 1961–1971 Keisler ([19]) and Shelah ([25]) proved the next important Isomorphism the- orem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Two models U1 and U2 are elementarily equivalent if and only if there exists an ultrafilter F such that their ultrapowers coincide: � F U1 ∼= � F U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' This theorem allowed Beidar and Mikhalev in 1992 (see [2]) to generalize Maltsev theorem for the case when K1 and K2 are skewfields and prime associative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' This approach for the groups GL n was generalized in [3] by the following result: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let R1 and R2 be associative rings with 1 (1/2) with finite number of central idempotents and m, n ⩾ 4 (m, n ⩾ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Then GL m(R1) ≡ GL n(R2) if and only if there exist central idempotents e ∈ R and f ∈ S such that eMm(R) ≡ fMn(S) and (1 − e)Mm(R) ≡ (1 − f)Mn(S)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Continuation of investigations in this field were the papers of Bunina 1998–2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Similar to Maltsev’s results were obtained not only for classical linear groups GL , PGL , SL , PSL , but also for unitary linear groups over fields, skewfields, and rings with involutions (see [6], [7]), for Chevalley groups over fields ([8]), over local rings (see [9]) and arbitrary commutative rings (see [10]), and also for other different derivative structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' In some cases elementary equivalence of derivative structures (even a little similar to linear groups) is equivalent not to elementary but to more strong equivalence of initial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Often second-order equivalence or some of its limitations can appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' For example, in 2000, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Tolstykh [28] stated the connection between second-order prop- erties of skewfields and first-order properties of automorphism groups of infinite-dimensional linear spaces over them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' In 2003, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Bunina and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Mikhalev (see [11]) stated the connec- tion between second-order properties of associative rings and elementary properties of categories of modules, endomorphism rings, automorphism groups, and projective spaces of infinite rank over these rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Similar results were obtained also for endomorphism rings and automorphism groups of Abelian p-groups: Bunina, Mikhalev and Roizner (see [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=']) proved that two endomor- phism rings of Abelian p-groups or two automorphism groups of Abelian p-groups for p ⩾ 3 are elementarily equivalent if and only if initial Abelian groups are equivalent in the full second order logic (in one exceptional case in its limitation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content='3 Stable Linear Groups Let R be an associative ring with unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' The following definitions correspond to [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Denote by Mat ∞(R) the ring of matrices with countable number of rows and columns such that out of the main diagonal there are only a finite number of nonzero elements, and also there exists a number n such that for any i ⩾ n the elements rii = a, a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' It is clear that Mat ∞(R) is a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let A ∈ GL n(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' We identify A with an element from GL n(R) by the following rule: A is written in the left upper corner, starting from the position (n, n) the diagonal contains 1, and all other places contain zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' We preserve the notation GL n(R) for the obtained subgroups Mat ∞(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' It is clear that GL n(R) are subgroups of the groups of invertible elements of the ring Mat ∞(R), and also it is clear that for m ≥ n we have GL n(R) ⊆ GL m(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let us set GL (R) = � n⩾1 GL n(R) (GL n(R) ⊆ Mat ∞(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' It is a subgroup of the group of invertible elements of the ring Mat ∞(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let us call it the stable linear group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' For stable linear groups their automorphisms were described: Atkarkaya described auto- morphisms of stable linear groups E(R) and GL(R) over commutative local rings R with 1/2 (see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' In our previous paper (see [13]) it was proved that despite an “infinite” dimension of the stable group, from elementary equivalence of two arbitrary rings with unit it follows elementary equivalence of stable linear groups over them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=', we do not need higher-order logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' In the same paper we proved that from elementary equivalence of stable linear groups over commutative local rings with 1/2 it follows elementary equivalence of the corresponding rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' In the given work we extend this previous result to the stable linear groups over fields of characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' 2 Proof of the main theorem Consider the group GL (R), where R is a field of characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let us denote its unit by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' For more simple formulas we will write A ∼ B instead of ∃U A = UBU−1 and call A to be conjugate to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Our first goal is to define elementarily a subgroup isomorphic to the group GL 2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' We will use matrices diag [1, 1, T], where T ∼ diag ��1 1 1 0 � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' � The first part of the proof depends on existence of third roots of unity in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content='1 Fields containing third roots of unity Let K be a field of characteristic 2 containing third roots of unity 1, ξ, ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' The following lemma is proved, for example, in [5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' it follows from classical linear algebra results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' For any number of mutually commuting third-order elements of the group GL n(K), there exists a basis in which all of them have diagonal form with third roots of unity on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' It is clear that for a finite set of commuting matrices the same condition holds also in the stable linear group GL (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let A be a set of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Denote by |||A||| the number of distinct conjugacy classes of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let us consider a set D = [d1, d2, d3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' , dk] of elements of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' If D includes [ξ, ξ] or [ξ, ξ2], then |||{AB : A ∼ diag [D], B ∼ diag [D], AB = BA}||| > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' To prove this lemma it is sufficient to consider only one pair of elements di, since other elements can be just fixed at the first positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Also it is sufficient to consider only permutation of diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' First consider the case when D contains [ξ, ξ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' We assume that these elements are corre- sponded to dk−1, dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' The first k −2 elements are fixed, under multiplication they give the same set of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' We can obtain the following matrices: diag [1, 1, ξ, ξ] · diag [ξ, ξ, 1, 1] = diag [ξ, ξ, ξ, ξ], diag [1, 1, ξ, ξ] · diag [ξ, 1, ξ, 1] = diag [ξ, 1, ξ2, ξ], diag [1, 1, ξ, ξ] · diag [1, 1, ξ, ξ] = diag [1, 1, ξ2, ξ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' We omitted for simplicity of reading elements d1, d2, d3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' , dk−2 and final units of stable ma- trices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' All these matrices have different sets of eigenvalues, so they are pairwise non-conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' If D contains [ξ, ξ2], then diag [1, 1, ξ, ξ2] · diag [1, 1, ξ, ξ2] = diag [1, 1, ξ2, ξ], diag [1, 1, ξ, ξ2] · diag [ξ, ξ2, 1, 1] = diag [ξ, ξ2, ξ, ξ2], diag [1, 1, ξ, ξ2] · diag [1, ξ2, ξ, 1] = diag [1, ξ2, ξ2, ξ2], the same situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' As a result of these two lemmas we obtain the following lemma: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let A3 = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' If |||{BC : B ∼ A, C ∼ A, CB = BC}||| = 2, then A ∼ diag [ξ] (or A ∼ diag [ξ2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' This lemma gives a method how to elementarily define a matrix conjugated to one of diag [ξ] and diag [ξ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' ϕ(A) := ∃X1∃X2∃Y1∃Y2∀Z1∀Z2 (A3 = E) ∧ ¬(A = E)∧ ∧ (X1 ∼ X2 ∼ Y1 ∼ Y2 ∼ A) ∧ (X1X2 = X2X1) ∧ (Y1Y2 = Y2Y1)∧ ∧ �� (Z1 ∼ A) ∧ (Z2 ∼ A) ∧ (Z1Z2 = Z2Z1) � → � (Z1Z2 ∼ Y1Y2) ∨ (Z1Z2 ∼ X1X2) �� (1) 4 Suppose this formula holds for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Assume that A ∼ diag [ξ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' It is possible to obtain according to a replacement of notations ξ2 → ξ′, ξ → ξ′2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Then A2 ∼ diag [ξ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Consider the formula ψ(B) := ∃X1∃X2(X1 ∼ A) ∧ (X2 ∼ A2) ∧ (X1X2 = X2X1) ∧ (X1X2 = B) ∧ ¬ϕ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' If this formula holds for B, then B ∼ diag [ξ, ξ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let X1 ∼ X2 ∼ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' X1X2 = X2X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' We can suppose that we chose a basis in such that X1 = diag [ξ, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' ], X2 = diag [1, ξ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let us consider the formula θ(C) := (CX1 = X1C) ∧ (CX2 = X2C) ∧ (C ∼ B) ∧ � i ¬ϕ(BXi) ∧ � i ¬ϕ(BX2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' If this formula holds for C = (cij), then since c commutes with Xi, i = 1, 2, we have c1j = ci1 = 0, i, j > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' c2j = ci2 = 0, i, j > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' C has order 3, so c3 11 = 1, c3 22 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Suppose that c11 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Then either c11 = ξ, or c11 = ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Suppose (without loss of generality) that c11 = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Then, formula ϕ holds for CX1, it contradicts to definition of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Therefore c11 = 1, and similarly c22 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Then C = diag [1, 1, T], where T ∼ diag [ξ, ξ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' diag [ξ, ξ2] ∼ �1 1 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' To prove this lemma we need to find the eigenvalues of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Its characteristic polynomial is P(λ) = λ2 + λ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Substituting ξ to the characteristic polynomial, we obtain P(ξ) = ξ2 + ξ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Note that P(ξ) = ξP(ξ), hence P(ξ) + ξP(ξ) = 0 ⇒ P(ξ)(1 + ξ) = 0 ⇒ P(ξ) = 0, Similarly for ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Therefore ξ (and ξ2) are roots of the characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content='2 Fields without third roots of unity Let K be a field of characteristic 2 without non-trivial roots of the third power of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' For any finite number of mutually commuting elements of the order 3 in GL n(K), there exists a basis in which all of them have block-diagonal form with E11 + E12 + E21, E22 + E12 + E21 and E11 + E22 on their diagonals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' This lemma is proved in [5] for the elements from GL n, but it is obviously correct for the group GL too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Denote T = E11 + E12 + E21 and Dk = diag [T, T, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' , T, T � �� � k , 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' If k > 1, then |||{AB : A ∼ Dk, B ∼ Dk, AB = BA, A, B ∈ GL (K)}||| > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' It is sufficient to consider only two blocks T, which exist in the block-diagonal form of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' By permutations of these two blocks we can obtain diag [E, E, T, T] · diag [T, T, E, E] = diag [T, T, T, T], diag [E, E, T, T] · diag [E, E, T 2, T 2] = diag [E, E, E, E], diag [E, E, T, T] · diag [E, E, T, T] = diag [E, E, T 2, T 2], diag [E, E, T 2, T] · diag [E, T, T 2, E] = diag [E, T, T, T], where E is a unity matrix of order 2 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' As it follows from Lemma 4, T 2 ∼ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' This lemma gives us a formula that holds for matrix A if and only if A ∼ D1: ϕ′(A) := ∃X1∃X2∃Y1∃Y2∀Z1∀Z2 (A3 = E) ∧ ¬(A = E) ∧ ∧ (X1 ∼ X2 ∼ Y1 ∼ Y2 ∼ A) ∧ (X1X2 = X2X1) ∧ (Y1Y2 = Y2Y1) ∧ ∧ �� (Z1 ∼ A) ∧ (Z2 ∼ A) ∧ (Z1Z2 = Z2Z1) � → → � (Z1Z2 ∼ X1X2) ∨ (Z1Z2 ∼ Y1Y2) ∨ (Z1Z2 ∼ E) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' (2) Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let A = (aij) commute with Gk = diag [1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' , 1 � �� � k−1 , T, 1, 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' ], where k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' If j + 1 < k or k + 1 < j, then � aj,k aj,k+1 aj+1,k aj+1,k+1 � = �0 0 0 0 � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' If j + 1 < k or k + 1 < j, then � ak,j ak,j+1 ak+1,j ak+1,j+1 � = �0 0 0 0 � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' � ak,k ak,k+1 ak+1,k ak+1,k+1 � = �a b b a + b � , for some a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' (1) Assume that A commutes with Gk and B = (bij) = AGk = GkA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Consider � bj,k bj,k+1 bj+1,k bj+1,k+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' 6 When A is multiplied by Gk from the left: � bj,k bj,k+1 bj+1,k bj+1,k+1 � = � aj,k aj,k+1 aj+1,k aj+1,k+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' When A multiplied by Gk from the right: � bj,k bj,k+1 bj+1,k bj+1,k+1 � = � aj,k + aj,k+1 aj,k aj+1,k + aj+1,k+1 aj+1,k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Then: � aj,k aj,k+1 aj+1,k aj+1,k+1 � = � aj,k + aj,k+1 aj,k aj+1,k + aj+1,k+1 aj+1,k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' From the equality of left columns of both matrices we obtain aj,k+1 = aj+1,k+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' But then from the equality of right columns of both matrices we obtain that all other elements equal to zero too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' The proposition (2) is proved in exactly the same way as proposition (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' (3) We have �ak,k + ak+1,k ak,k+1 + ak+1,k+1 ak,k ak,k+1 � = � ak,k + ak,k+1 ak,k ak+1,k + ak+1,k+1 ak+1,k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Define b := ak,k+1 = ak+1,k, a := akk, then ak+1,k+1 = a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let X be a matrix satisfied the formula ϕ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Consider the formula θ′(C) := (CX = XC) ∧ (C ∼ X) ∧ (CX ̸= E) ∧ (CX2 ̸= E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let C be a matrix such that θ′(C) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let us fix a basis such that X has a form G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Then from Lemma 7 we have c1j = ci1 = 0, c2j = ci2 = 0, i, j > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Also we have � c11 c12 c21 c22 � = � a b b a + b � and � a b b a + b �3 = � 1 0 0 1 � or �a3 + ab2 + b3 a2b + ab2 a2b + ab2 a3 + a2b + b3 � = �1 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' It follows that a2b + ab2 = ab(a + b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' There is no zero divisors in K, therefore there are three possible cases: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Then a3 = 1 ⇒ a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Then b3 = 1 ⇒ b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Then a3 + ab2 + b3 = 1 ⇒ a3 + a3 + a3 = 1 ⇒ a3 = 1 ⇒ a = b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' The first case corresponds to the unity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' The second and the third cases correspond to � c11 c12 c21 c22 � = T and � c11 c12 c21 c22 � = T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Both thess cases are impossible due to the formula θ′, because such C multiplied by X or X2 gives the unity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' So we peoved that C has the form diag [1, 1, T ′], where T ′ ∼ D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content='3 Elementary definability of GL 2(K) and the main theorem Assume the formula θ holds only for matrices diag [1, 1, T], where T ∼ �1 1 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let us determine GL 2(K) using these matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let us notice that the formula θ holds for Gk, k > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' If M commutes with G2k+1, k = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=', then M = diag [M0, M1, M2, M3, M4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' , Mk], where M0 ∈ GL 2(K), and Mi = �ai bi bi ai + bi � , i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' If M commutes also with G2k+2, k = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=', then M = diag [M0, a1, M′ 1, M′ 2, M′ 3, M′ 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' , M′ k′], where M′ i = � a′ i b′ i b′ i a′ i + b′‘i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Comparing block structures of different representations of M, we obtain M = diag [M0, M1, M2, M3, M4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' , Mk] = diag [M0, a1, M′ 1, M′ 2, M′ 3, M′ 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' , M′ k′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Therefore bi = b′ i = 0, ai = a′ i and a′ i = ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Hence ai = ai+1 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' It follows from the definition of the stable linear group that there is only a finite number of non-unit elements on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Hence ai = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Therefore, we obtain a formula that defines a subgroup isomorphic to GL 2(K): γ(M) = ∀C � θ(C) → (MC = CM) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Now we are ready to prove the main theorem of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let K1 and K2 be fields of characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' If the stable linear groups GL (K1) and GL (K2) are elementarily equivalent, then the fields K1 and K2 are also elementarily equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Let GL (K1) ≡ GL (K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Since in the previous lemmas we proved that the subgroup GL 2 is elementarily defined in GL for a field of characteristic 2, this means that GL 2(K1) ≡ GL 2(K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' By the generalization of Maltsev Theorem proved for example in [4] this implies K1 ≡ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Atkarskaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E5T4oBgHgl3EQfdw8X/content/2301.05613v1.pdf'} +page_content=' Automorphisms of Stable Linear Groups Over Commutative Local Rings With 1/2.' metadata={'source': 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Zhang,1 F. R. Xu,1, 2, ∗ J. G. Li,3, 4 B. S. Hu,1 Z. H. Cheng,1 N. Michel,3, 4 Y. Z. Ma,1 Q. Yuan,1 and Y. H. Zhang3, 4 +1State Key Laboratory of Nuclear Physics and Technology, +School of Physics, Peking University, Beijing 100871, China +2Southern Center for Nuclear-Science Theory, Institute of Modern Physics, +Chinese Academy of Sciences, Lanzhou 730000, China +3Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China +4School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China +We have used the ab initio Gamow shell model to study the mirror asymmetry in A = 16 nuclei. +Starting from a chiral interaction with the two-nucleon force (2NF) at N3LO and three-nucleon force +(3NF) at N2LO, a complex-momentum psd-shell Hamiltonian was constructed by employing the +many-body perturbation theory in the Gamow Hartree-Fock basis which includes self-consistently +bound, resonant and continuum states. In such an elaborated ab initio Gamow shell model with +both 3NF and continuum coupling included, many-body correlations can be treated properly, and +the structures of A = 16 nuclei, 16F, 16N, 16Ne and 16C, have been investigated well. The mirror +partners of 16F and 16N exhibit different level orders in their excitation spectra, which can be well +explained in the present calculations. The mirror asymmetry between the mirror partners 16Ne and +16C was analyzed in detail by insight into their configuration structures. The interplay between +3NF and the continuum coupling is discussed with the ab initio calculations of the weakly bound +and unbound nuclei. +I. +INTRODUCTION +With next-generation Rare Isotope Beam (RIB) facili- +ties, we have the ability to produce most of the rare iso- +topes located at the edge of the nuclear landscape, thus +shedding light on the origin of elements, the fundamental +problems of nuclear structure, and nuclear forces. How- +ever, theoretical descriptions of proton-rich or neutron- +rich nuclei in these regions are challenging in terms of +theoretical methods and computational demands. As nu- +clei approach driplines, the effect of single-particle long- +distance asymptotic behavior and the coupling to the +continuum are vital in understanding these open quan- +tum systems [1]. Indeed, they lead to novel phenomena +in weakly bound and unbound nuclei, such as halo [2– +4], Borromean structures [5], and Thomas-Ehrmann shift +(TES) [6–8]. +Among these open quantum systems, the A = 16 mir- +ror partners are particular and could provide insights into +nuclear force and the evolution of nuclear properties from +the valley of stability to driplines. +For one thing, the +apparent symmetry breaking in spectra of mirror part- +ners 16F and 16N is observed, which leads to level inver- +sion in ground states (g.s.), which can find explanations +from continuum coupling [9]. Furthermore, the energy +shifts in pairs of isobaric analog states in mirror nuclei +are systematically analyzed in Ref. [10]. +It turns out +that the large energy splittings of 16F-16N mirror pair +exceed the normal isospin-symmetry-breaking (ISB) be- +havior, which requires additional ISB effects. Theoret- +ically, as mentioned in Ref. [9], the phenomenological +Gamow shell model (GSM) and coupled-channel GSM +∗ frxu@pku.edu.cn +calculations with a dedicated treatment of nuclear corre- +lations have been performed to analyze the 16F-16N mir- +ror pair [11]. It shows that only coupled-channel GSM +calculations with corrective factors reproduce the TES +observed in the 16F-16N mirror pair. For the mirror part- +ners 16C and 16Ne, larger isospin asymmetries in the con- +figurations of the g.s. and first 2+ state are demonstrated +and considered as a new mechanism of TES [12, 13]. In +addition, 16Ne is an intermediate nucleus of the cascade +2p emission of the recently observed four-proton unbound +nucleus 18Mg. +The study of mirror asymmetry in the +16C-16Ne mirror pair is meaningful to further understand +the nature of higher 2+ excitation energy in 18Mg [14]. +Nuclei at driplines exhibit novel phenomena arising from +the proximity of the continuum, which also provides a +comprehensive and rigorous test of nuclear theory. +In recent decades, significant progress in ab initio cal- +culations [15–20] has been made with the developments +of chiral effective field theory (χEFT) [21, 22], similar- +ity renormalization group (SRG) [23, 24] and many-body +methods [25–29]. Meanwhile, chiral three-nucleon force +(3NF) has been shown to be crucial for nuclear proper- +ties and detailed nuclear structure [30–46]. However, as +the continuum coupling is essential at the proximity of +driplines, weakly bound and unbound nuclei are challeng- +ing theoretical studies with standard approaches, such as +many-body methods using the HO basis. One choice for +that matter is the Berggren basis [47], which can treat +bound, resonant and continuum states on the same foot- +ing. The GSM [48, 49] is a powerful tool, which provides a +full description of the interplay between continuum cou- +pling and inter-nucleon correlations via the use of the +Berggren basis and configuration mixing. Ab initio meth- +ods have also been proposed in the frame of the Berggren +basis, hence with the continuum coupling included, such +as the complex coupled cluster [34, 50], the complex in- +arXiv:2301.01951v1 [nucl-th] 5 Jan 2023 + +2 +medium similarity renormalization group [51], and the +no-core GSM [52–54]. Added to that, the GSM with a +core has been further developed by generating effective +interactions from realistic forces [55–59]. +The A = 16 nuclear systems provide remarkable cases +of interest, which would lead to new insights into nuclear +properties, nuclear forces, and theoretical methods. In +this work, we will thus perform ab initio GSM calcula- +tions of 16F, 16N, 16C and 16Ne with the most widely +used “Magic” force [40, 41]. We will depict g.s. energies, +spectra, as well as other observables of physical interest. +With both 3NF and the continuum coupling considered, +we will probe the ISB effects in the A = 16 mirror part- +ners. Our calculations will show the necessity to combine +the effects between 3NF and continuum coupling in the +nuclei close to driplines. +II. +THE METHOD +We employ the intrinsic Hamiltonian of the A-nucleon +system +H = +A +� +i=1 +� +1 − 1 +A +� p2 +i +2m + +A +� +i φ(xi) +in this case. Note that by definition, α ⃗G satisfies the colouring constraints imposed by the list assignment L. Let +us then show that α ⃗G is indeed an L-dicolouring. Assume, for a contradiction, that there is a monochromatic +directed cycle C in α ⃗G. For every edge-gadget, say corresponding to xixj, the vertices xi,r, xj,r′ must be coloured +differently. Therefore, all vertices of C must be contained in a single vertex-gadget of D. Let xi be the vertex +such that C is in the vertex-gadget of xi. If φ(xi) = 1, then C must be (zi, xi,1, xi,2, xi,3, zi) and all its vertices +should be coloured 2. This is a contradiction since d− +⃗G(xi) ≥ 1 implies, by construction, that at least one of +xi,1, xi,2, xi,3 is coloured 1. If φ(xi) = 2, then C contains at least two vertices in {xi,1, xi,2, xi,3} and two +vertices in {zi,1, zi,2, zi,3}. Then, at least two vertices of {xi,1, xi,2, xi,3} are coloured 2 because vertices zi,j are. +This is a contradiction since d− +⃗G(xi) ≥ 2 implies, by construction, that at least two vertices of xi,1, xi,2, xi,3 are +coloured 1. +♦ +Let us take α1 = α ⃗G1 and α2 = α ⃗G2 to be the dicolourings obtained from the two proper orientations ⃗G1, ⃗G2. +We will now show that there exists a reorienting sequence in G from ⃗G1 to ⃗G2 if and only if there exists a +redicolouring sequence in D from α1 to α2. +7 + +Assume first that there is a reorienting sequence ⃗Γ1, . . . , ⃗Γp from ⃗G1 to ⃗G2, and let us show how to build a +corresponding redicolouring sequence. Consider any step s of the reorienting sequence, say when ⃗Γs is transformed +into ⃗Γs+1 by reversing an arc xixj into xjxi. We will exhibit a path from α⃗Γs to α⃗Γs+1 in D(D, L). Consider +vertices xi,r, xj,r′ in D, corresponding to the edge xixj, and coloured 2 and 1 respectively in α⃗Γs. We first set the +colour of xj,r′ from 1 to 2. Since ai,j is forced to colour 1, the edge-gadget is not monochromatic at this point. +Moreover since step s reorients arc xixj and still yields a proper orientation, d− +⃗Γs(xj) = d− +⃗Γs+1(xj)+1 ≥ φ(xj)+1. +The resulting colouring is an L-dicolouring by Claim 17.1 (ii). We then set the colour of xi,r from 2 to 1, yielding +dicolouring α⃗Γs+1. Concatenating the redicolouring sequences obtained through the process above from steps +s = 1 to s = p yields a redicolouring sequence from α1 to α2. +Conversely, assume that there is a redicolouring sequence γ1, . . . , γp from α1 to α2. Observe that the only +vertices of D that are possibly recoloured in a step of our sequence are those defined as xi,k for i ∈ [n] and k ∈ [3], +since all others are forced. Now, at any step s of the redicolouring, for each edge xixj of G, at most one of the +two corresponding vertices is coloured 1, because aij is forced to colour 1. This allows us to define an orientation +⃗Γs of G as follows. If the vertices xi,r, xj,r′ ∈ V (D), corresponding to the gadget of edge xixj, are not coloured +the same in γs, orientation ⃗Γs sets xixj to be directed from the vertex coloured 2 towards the vertex coloured 1. +Otherwise, both vertices are coloured 2 in γs and we preserve the orientation of the corresponding edge given by +⃗Γs−1. In the first and last dicolourings, α1 and α2, for each edge xixj the corresponding vertices xi,k and xj,k′ +are coloured differently. Thus ⃗Γ1 = ⃗G1 and ⃗Γp = ⃗G2. Therefore, ⃗Γ1, . . . , ⃗Γp is a sequence of orientations of G +from ⃗G1 to ⃗G2 such that ⃗Γs+1 is either obtained by reversing an arc of ⃗Γs (when one of the xi,k is recoloured to +1, and the edge xixj whose edge-gadget contains xi,k was not oriented towards xi), or equal to ⃗Γs otherwise (and +in particular when one of the xi,k is recoloured to 2). Moreover, at each step s, ⃗Γs is a proper orientation of G. +Indeed, if φ(xi) = 1 (resp. φ(xi) = 2), then at least one vertex (resp. two vertices) of {xi,1, xi,2, xi,3} is coloured +1, and so xi has in-degree at least 1 (resp. at least 2) in ⃗Γs. Hence, taking the subsequence of ⃗Γ1, . . . , ⃗Γp that omits +constant steps yields a reorienting sequence from ⃗G1 to ⃗G2. +Since PLANAR-CUBIC-NCL is PSPACE-complete, at this point our reduction already yields the PSPACE- +completeness of 2-LIST DICOLOURING PATH. By construction, forced vertices zj,k and aij have degree at most +3, and all other vertices have degree at most 5. This achieves the proof of the first case of the result. +Now, since the input instances are already planar, to get the PSPACE-completeness of 2-LIST DICOLOURING +PATH on planar digraphs, it suffices to use planar vertex and edge gadgets. In our current reduction, the only gadget +which is not planar is the vertex-gadget corresponding to xi ∈ V (G) such that φ(xi) = 2. We now consider the +same reduction replacing the vertex-gadget for vertices such that φ(xi) = 2 with a planar one. For these vertices, +the planar vertex-gadget is defined on the same set of vertices, i.e. {xi,1, xi,2, xi,3, zi,1, zi,2, zi,3}, but with the arcs +of the directed 3-cycles (zi,1, xi,1, xi,2, zi,1), (zi,3, xi,1, xi,3, zi,3) and (zi,2, xi,2, xi,3, zi,2), as depicted in Figure 3. +This replacement produces a planar digraph in which all forced vertices still have degree at most 3, and all vertices +have maximum in- and out-degree at most 3. This completes the proof. +xi,1 +xi,2 +xi,3 +zi,1 +zi,2 +zi,3 +Figure 3: A planar vertex-gadget when φ(xi) = 2 +8 + +The problem k-COLOURING PATH is known to be PSPACE-complete for every k ≥ 4 in the undirected +case [3]. Leveraging Theorem 17, we prove that this also holds for its dicolouring analogue for k ≥ 2 colours, in +both directed and oriented graphs. +Theorem 5. +(i) For every k ≥ 2, k-DICOLOURING PATH is PSPACE-complete on digraphs with maximum +degree 2k + 1. +(ii) For every k ≥ 2, k-DICOLOURING PATH is PSPACE-complete on oriented graphs. +(iii) For every 2 ≤ k ≤ 4, k-DICOLOURING PATH is PSPACE-complete on planar digraphs with maximum +degree 2k + 2. +(iv) 2-DICOLOURING PATH is PSPACE-complete on planar oriented graphs of degree at most 6. +Proof. (i) We give a reduction from 2-LIST DICOLOURING PATH on instances where forced vertices have de- +gree at most 3 and the graph has maximum degree 5. The problem is PSPACE-complete by Theorem 17. Let +(D, L, α1, α2) be an instance of the problem, we construct an instance (D′, α′ +1, α′ +2) for k-DICOLOURING PATH as +follows. +We build D′ starting with D′ = D. Then, for every vertex v ∈ V (D), we let ←→ +K v +k be a bidirected complete +graph on vertex set {zv +i | i ∈ [k]}. We then add a digon between v and each zv +i such that i /∈ L(v). We define +dicolourings α′ +1 and α′ +2 on D′ by extending dicolourings α1 and α2 as follows. All vertices of D′ that were vertices +of D are coloured the same, and we set zv +i to colour i for all v ∈ V (D) and all i ∈ [k]. Note that all the vertices +of gadget ←→ +K v +k are then frozen in any k-dicolouring, letting us simulate in D′ the list dicolouring constraints on +D. An L-dicolouring path from α1 to α2 in D is then exactly a dicolouring path from α′ +1 to α′ +2 when restricted to +vertices of D, achieving to show equivalence between the instances. +We will now show that the maximal degree of a vertex v in D′ is 2k + 1. If v belongs to some gadget ←→ +K v +k, +then its degree is at most 2(k − 1) + 2 = 2k. Note that when v ∈ V (D), v is of degree 2, 3 or 5 in D, and our +reduction adds exactly 2k − |L(v)| arcs incident to v. If |L(v)| = 2, this yields dD′(v) ≤ 5 + 2k − 4 = 2k + 1. If +|L(v)| = 1, we know by construction that dD(v) ≤ 3, yielding dD′(v) ≤ 3 + 2k − 2 = 2k + 1. This achieves the +proof that D′ has maximum degree at most 2k + 1, concluding (i). +(ii) We give a reduction from k-DICOLOURING PATH to k-DICOLOURING PATH restricted to oriented graphs. +Let (D, α1, α2) be an instance of k-DICOLOURING PATH, we will build an equivalent instance (⃗G, α′ +1, α′ +2) where +⃗G is an oriented graph. Take ⃗H to be an arbitrary oriented graph with dichromatic number exactly k. We construct +⃗G from D by replacing digons of D as follows. For each digon [u, v] of D, create a copy ⃗Huv of ⃗H, then replace +[u, v] by a single arc from u to v, and add the all arcs from v to ⃗Huv and all arcs from ⃗Huv to u. By construction, +⃗G is an oriented graph. +In the following, we let c be a fixed k-dicolouring of ⃗H. We show how to transform any k-dicolouring α of +D to a k-dicolouring α′ of ⃗G, and vice versa. Given a k-dicolouring α for D, we define α′ for ⃗G by colouring +each copy ⃗Huv of ⃗H with c, and keeping the same colours as α on V (D). Any monochromatic directed dcycle +in (⃗G, α′) must contain a vertex of some ⃗Huv, as otherwise it would be a subdigraph of D and would already be +monochromatic in (D, α). Since c is a dicolouring of ⃗H, the cycle must contain both u, v, but then u and v being +coloured the same would yield a monochromatic digon in (D, α), so α′ is indeed a k-dicolouring of ⃗G. Conversely, +given any k-dicolouring α′ of ⃗G, we define α for D as the restriction of α′ on V (D). Similarly, if (D, α) were +to contain a monochromatic directed dcycle, any arc (u, v) of the cycle that is not present in ⃗G may be replaced +with (u, w, v), taking w ∈ ⃗Huv to be a vertex of the same colour as u and v (since ⃗χ( ⃗H) = k). This would yield a +monochromatic directed dcycle in (⃗G, α′), so α must be a k-dicolouring of D. +Now, we define the k-dicolourings α′ +1, α′ +2 on ⃗G obtained from α1, α2 by the transformation above, and let +our output instance be (⃗G, α′ +1, α′ +2). If there is a redicolouring sequence from α1 to α2 in D, we perform the +same recolouring steps in ⃗G starting from α′ +1 and yielding α′ +2. Since we only recolour vertices of V (D), the last +paragraph yields that this sequence is valid. Conversely, if there is a redicolouring sequence from α′ +1 to α′ +2, its +restriction to V (D) (omitting recolourings of vertices in subgraphs ⃗Huv) yields a valid sequence from α1 to α2 in +D. This achieves the proof of the equivalence of the instances and proves (ii). +9 + +(iii) We give a reduction from 2-LIST DICOLOURING PATH where D is planar, forced vertices have degree at most +3 and D has in- and out-degree at most 3. The problem is PSPACE-complete by Theorem 17. Let (D, L, α1, α2) +be an instance of the problem. We make the same reduction as in the proof of (i) by ensuring that ←→ +K v +k is embedded +and coloured in such a way the (at most 3) forbidden colours of v lie on its external face. This allows us to keep a +planar representation of D′ which has maximum degree 2k + 2. This proves (iii). +(iv) As in (iii), we give a reduction from 2-LIST DICOLOURING PATH where D is planar, forced vertices have +degree at most 3 and every vertex has in- and out-degree at most 3. Since we are considering 2-dicolourings, +vertices with a list of size 2 do not require a gadget to simulate forbidden colours. The main difference with case +(iii) is that we cannot use a the bidirected complete graph ⃗Kv +2 to freeze a vertex v with list of size 1. To overcome +this, we use the gadget depicted in Figure 4, where the colour of all vertices is frozen. Therefore, we simply have +to attach such a gadget on each vertex with list size one, that is, those of the form zi,r or ai,j. This can be done +by creating a directed triangle including the vertex and two vertices of the gadget that are of the opposite colour as +depicted in the figure. +a +Figure 4: How to freeze the vertex a in a planar oriented graph with two colours. +3 +Connectivity and diameter of dicolouring graphs +3.1 +Connectivity +We start by showing some easy bounds on the minimal number of colours, with respects to variants of degeneracy, +ensuring that digraphs or oriented graphs are k-mixing. Bonsma and Cereceda [3] and Dyer et al. [9] independently +proved that any (non-oriented) graph G is k-mixing for every k ≥ δ∗(G) + 2. Theorem 6, which we recall, +generalizes this result for digraphs. +Theorem 6. Every digraph D is k-mixing for every k ≥ δ∗ +min(D) + 2. +Proof. The proof is an adaption of the proof of [9] for undirected graphs. We show the result by induction on the +number of vertices of D, the result being obviously true for the digraph with one vertex. Let D be a digraph D +on at least two vertices, k ≥ δ∗ +min(D) + 2 be an integer and α, β be two k-dicolourings of D. Let v be a vertex +such that min{d+(v), d−(v)} ≤ δ∗ +min(D) and let D′ = D − v. By induction, D′ is k-mixing. Let α′, β′ be the +k-dicolourings of D′ induced by α, β (i.e. for every v ∈ V (D′), α′(v) = α(v) and β′(v) = β(v)). Since D′ is +k-mixing, there exists a redicolouring sequence α′ = γ0, . . . , γℓ = β′ from α′ to β′. +Let us prove that we can transform the redicolouring sequence from D′ into a redicolouring sequence for D. +Starting from α, we perform the same steps as in γ0, . . . , γℓ as long as they produce dicolourings of D. If at some +10 + +step i in the sequence it is not possible to recolour vertex vi to ci in D, it must be because v is currently coloured +ci and recolouring vi to ci would create a monochromatic directed cycle containing both. Assume that v has at +most k − 2 out-neighbours (otherwise, v has at most k − 2 in-neighbours and the case is symmetrical). Now, we +can choose c to be a colour different from ci that is also different from that of the out-neighbours of v. Then, we +recolour v to c, allowing us to recolour vi to ci and continue the sequence. +Theorem 7. Every oriented graph ⃗G is k-mixing for all k ≥ +� +δ∗ +avg(⃗G) +� ++ 1. +Proof. We proceed by induction on the number of vertices of ⃗G, the result being obviously true for the oriented +graph with one vertex. Consider now an oriented graph ⃗G on at least two vertices and k ≥ +� +δ∗ +avg(⃗G) +� ++ 1. Let v +be a vertex such that d+(v) + d−(v) ≤ 2δ∗ +avg(⃗G). By directional duality, we may assume d+(v) ≤ d−(v), then by +assumption d+(v) ≤ k − 1. We let ⃗G′ = ⃗G − v. Then, δ∗ +avg(⃗G′) ≤ δ∗ +avg(⃗G), and the induction hypothesis yields +that ⃗G′ is k-mixing. +Let α and β two k-dicolourings of ⃗G, and let α′, β′ be the k-dicolourings of ⃗G′ induced by α, β. Since ⃗G′ +is k-mixing, there exists a sequence α′ = γ0, . . . , γℓ = β′ of k-dicolourings of ⃗G′ such that γi−1 and γi differ +in the colour of exactly one vertex of ⃗G′ for i ∈ [1, ℓ]. We denote this vertex by vi. Now we consider the same +recolouring steps to recolour ⃗G, starting from α. If for some i it is not possible to recolour vi to ci, this must be +because v is currently coloured ci and recolouring vi to ci would create a monochromatic directed cycle. Note that +such a cycle necessarily contains v, because γi is a dicolouring of G′. +• If d+(v) ≤ k − 2, then v can be recoloured to c, a colour different from ci that does not appear on N +[v], +allowing us to recolour vi to ci and proceed with the sequence. +• Otherwise d+(v) = k − 1. Then since k ≥ +� +δ∗ +avg(⃗G) +� ++ 1, we get d−(v) ≤ k − 1, and then d−(v) = k − 1 +(since d+(v) ≤ d−(v)). Since ⃗G is an oriented graph, and recolouring vi to ci would create a monochromatic +directed cycle, then v has at least one neighbour coloured ci. Take w to be one of these neighbours. Then, +if w ∈ N +(v) (resp. w ∈ N −(v)), we can recolour v to a colour that does not appear in N +(v) (resp. +N −(v)), recolour vi to ci and continue the sequence. +Note that δ∗ +max(⃗G) ≥ +� +δ∗ +avg(⃗G) +� +since δ∗ +max(⃗G) is an integer and δ∗ +max(⃗G) ≥ δ∗ +avg(⃗G). Consequently, every +oriented graph ⃗G is k-mixing for all k ≥ δ∗ +max(⃗G) + 1. A natural question is then whether this result can be +extended to δ∗ +out: is every oriented graph ⃗G k-mixing for all k ≥ δ∗ +out(⃗G) + 1? This does not hold for directed +graphs, as witnessed by the bidirected clique, and the following also answers the question in the negative for +oriented graphs. +Proposition 18. For every positive integer k, there exist k-out-degenerate oriented graphs that are not (k + 1)- +mixing. +Proof. Let ⃗B0 be the oriented graph with one vertex. For k ≥ 1, we construct ⃗Bk from ⃗Bk−1 as follows: take +two disjoint copies ⃗B1 +k−1 and ⃗B2 +k−1 of ⃗Bk−1 and one vertex r; add all arcs from r to ⃗B1 +k−1, all arcs from ⃗B1 +k−1 to +⃗B2 +k−1 and all arcs from ⃗B2 +k−1 to r. +Claim 18.1: ⃗Bk is k-out-degenerate and ⃗χ( ⃗Bk) = k + 1. +Proof of claim. We prove the claim by induction on k, the result holding trivially for k = 0. Assume now that +k ≥ 1, and consider ⃗Bk. By induction, there exist two (k − 1)-out-degeneracy orderings σ1 and σ2 on ⃗B1 +k−1 and +⃗B2 +k−1, respectively. In ⃗Bk, each vertex of V ( ⃗B2 +k−1) has exactly one out-neighbour in V \ V ( ⃗B2 +k−1), namely r. +We may then remove the vertices of ⃗B2 +k−1 following σ2, such that at each step the removed vertex has out-degree +at most k. In the remaining graph, all out-neighbours of vertices in V ( ⃗B1 +k−1) belong to V ( ⃗B1 +k−1), allowing us to +11 + +successively remove vertices of out-degree k − 1 by following σ1. Combining these facts, σ2 · σ1 · (r) yields a +k-out-degeneracy ordering of ⃗Bk. Hence ⃗Bk is k-out degenerate. +The graph ⃗Bk is indeed (k+1)-dicolourable since any dicolouring such that the restriction to ⃗B1 +k−1 and ⃗B2 +k−1 is +a k-dicolouring and r is coloured with colour k+1 is a (k+1)-dicolouring of ⃗Bk. Now, assume for a contradiction +that ⃗Bk has a k-dicolouring α. Set c = α(r). By induction, ⃗χ( ⃗Bk−1) = k, so there is a vertex x1 of ⃗B1 +k−1 and +a vertex x2 of ⃗B2 +k−1 such that α(x1) = α(x2) = c. But then (r, x1, x2, r) is a monochromatic directed cycle, a +contradiction. Thus ⃗χ( ⃗Bk) = k + 1. +♦ +Let k be a positive integer. Let ⃗Gk be the oriented graph obtained from the transitive tournament T Tk+1 on +k + 1 vertices, by adding, for each arc xy of T Tk+1, a copy ⃗Bxy +k +of ⃗Bk, all arcs from y to ⃗Bxy +k +and all arcs from +⃗Bxy +k +to x. Let us prove that ⃗Gk is k-out-degenerate and is not (k + 1)-mixing. +Let (t1, . . . , tk+1) be the acyclic ordering of T Tk+1 (such that there is no arc titj with i > j), and let σi,j be +a k-out-degeneracy ordering of ⃗Btitj +k +for all 1 ≤ i < j ≤ k + 1. We build a k-out-degeneracy ordering of ⃗Gk by +combining these orders as follows: (t1) · σ1,2 · σ1,3 · · · · · σ1,k+1 · (t2) · σ2,3 · σ2,4 · · · · · σ2,k+1 · (t3) · · · · · +(tk) · σk,k+1 · (tk+1). Thus ⃗Gk is k-out-degenerate. +We show that every (k + 1)-dicolouring α of ⃗Gk is such that every vertex v ∈ V (T Tk+1) is frozen. Since any +α′ defined from α by a permutation of the colours is also a (k + 1)-dicolouring, the above being true yields that α′ +cannot be reached from α and implies that ⃗Gk is not (k + 1)-mixing. Consider v ∈ V (T Tk+1), and assume there +is a redicolouring sequence starting from α achieving to recolour v. We let β the dicolouring of ⃗Gk right before +v is recoloured in the sequence, and β′ the dicolouring after recolouring v. It suffices to note that β(v) ̸= β(w) +for any w ∈ V (T Tk+1)\v. Indeed, since ⃗χ(Bvw) = k + 1, there exists some z ∈ V (Bvw) coloured β(v), and +by construction β(v) = β(w) would create a monochromatic directed cycle on vertices v, w, z. Then, since β′ is +obtained from β by recolouring v, there exists some w ∈ V (T Tk+1) coloured the same as v in β′, and the same +argument yields a contradiction. +3.2 +Diameter +In this section, we prove Theorems 9, 10 and 11. +Theorem 11. Let ⃗G be a subcubic oriented graph of order n. Then D2(⃗G) is connected and has diameter at most +2n. +Proof. Let α and β be any two 2-dicolourings of ⃗G. Let x = diff(α, β) = |{v ∈ V (⃗G) | α(v) ̸= β(v)}|. By +induction on x ≥ 0, let us show that there exists a path of length at most 2x from α to β in D2(⃗G). This clearly +holds for x = 0 (i.e., α = β). Assume x > 0 and the result holds for every x′ < x. +Let v ∈ V (⃗G) such that 1 = α(v) ̸= β(v) = 2 (v exists up to swapping the colours). If v can be directly +recoloured with colour 2, then we recolour v with colour 2 and reach a new 2-dicolouring α′ such that diff(α′, β) = +x − 1. Then, the result holds by induction. +Therefore, v cannot be directly recoloured with colour 2, i.e., there exists a directed cycle C containing v such +that α(w) = 2 for every w ∈ V (C) \ {v}. Moreover, there must be a vertex u ∈ V (C) \ {v} such that β(u) = 1. +If u is not a neighbour of v, then u has two neighbours (those in C) coloured with 2 in α. Therefore, u can be +recoloured 1 (it does not create any directed cycle of vertices coloured 1 since u has degree at most 3 in G) and +we reach a new 2-dicolouring α′ such that diff(α′, β) = x − 1. Then, the result holds by induction. Hence, u is a +neighbour of v in C. +Again, we may assume that u cannot be directly recoloured 1 (otherwise the result holds by induction) and so +there exists a directed cycle C′ such that α(w) = 1 for every w ∈ V (C′) \ {u}. Since u and v have degree at most +3 in ⃗G, and because of the α-colours of vertices in C and C′, then V (C′) ∩ V (C) = {v, u}. Let h be the vertex of +V (C′) \ {u, v} minimizing its distance to u and v in C′. Again, since ⃗G is subcubic, h cannot be in any directed +cycle C′′ such that all vertices of C′′ but h are coloured 2 by α. Hence, h can be recoloured with 2, then u can be +recoloured 1, then v can be recoloured with 2 and h recoloured with 1, reaching a new 2-dicolouring α′ such that +diff(α′, β) = x − 2, and the result holds by induction. +12 + +Theorem 9. Let D be a digraph on n vertices and k ≥ 2δ∗ +min(D) + 2 be an integer. Then, the diameter of Dk(D) +is at most (δ∗ +min(D) + 1)n. +Proof. The proof is very similar to the proof of Bousquet and Perarnau [5] for undirected graphs. Let α and β be +two k-dicolourings. Let us show by induction on the number of vertices that there exists a redicolouring sequence +from α to β where every vertex is recoloured at most δ∗ +min(D) + 1 times. +If n = 1 the result is obviously true. Let D be a digraph on n + 1 vertices, and let u be a vertex such that +min{d+(u), d−(u)} ≤ δ∗ +min(D) and let D′ = D − u. By directional duality, we may assume d+(u) ≤ δ∗ +min(D). +We denote by α′ and β′ the dicolourings of D′ induced by α and β. By induction and since δ∗ +min(D′) ≤ δ∗ +min(D), +there exists a redicolouring sequence from α′ to β′ such that each vertex is recoloured at most δ∗ +min(D) + 1 +times. Now we consider the same recolouring steps to recolour D, starting from α. If for some step i, it is not +possible to recolour vi to ci, this must be because u is currently coloured ci and recolouring vi to ci would create +a monochromatic directed cycle. Since u has at most δ∗ +min(D) out-neighbours, and since k ≥ 2δ∗ +min(D) + 2, there +are at least δ∗ +min(D) + 2 colours that does not appear in the out-neighbourhood of u. We choose c among these +colours so that c does not appear in the next δ∗ +min(D) + 1 recolourings of N +(u). +Since u has at most δ∗ +min(D) out-neighbours and since each vertex in D′ is recoloured at most δ∗ +min(D) + 1 +times, there are at most δ∗ +min(D)(δ∗ +min(D)+1) recolourings of an out-neighbourof u in this redicolouring sequence. +Hence, in this new redicolouring sequence, u is recoloured at most δ∗ +min(D) times. We finally have to set u to its +colour in β. Doing so u is recoloured at most δ∗ +min(D) + 1 times. This concludes the proof. +The goal of the rest of this part is to prove the following theorem which generalizes a result of Bousquet and +Heinrich [4] to digraphs. +Theorem 10. For a digraph D on n vertices, and k ≥ 3 +2(δ∗ +min(D) + 1), the diameter of Dk(D) is at most Cn2 +where C is independent from k. +Given an undirected graph G = (V, E), a list assignment L is a-feasible if, for some ordering v1, . . . , vn of V , +|L(vi)| ≥ |N(v) ∩ {vi+1, . . . , vn}| + 1 + a for every i ∈ {1, . . . , n}. Bousquet and Heinrich proved the following +result appearing as the first and second items of Theorem 6 in in [4]: +Theorem 19 (Bousquet and Heinrich [4]). Let G be a graph and a ∈ N. Let L be an a-feasible list assignment +and k be the total number of colours. Then C(G, L) has diameter at most: +(i) kn if k ≤ 2a, +(ii) Cn2 if k ≤ 3a (where C a constant independent of k, a). +For a graph G and a list assignment L of G, we say that an L-colouring c avoids a set of colours S if for every +vertex v ∈ V (G), c(v) does not belong to S. In order to prove Theorem 10, we deduce from Theorem 19 the +following lemma: +Lemma 20. Let G = (V, E) be an undirected graph on n vertices, k ∈ N be the total number of colours, L be +a +� k +3 +� +-feasible list assignment and α an L-colouring of G that avoids a set S of +� k +3 +� +colours. Then for any set of +� k +3 +� +colours S′, there is an L-colouring β of G that avoids S′ and such that there is a recolouring sequence from +α to β of length at most 4k+6 +3 +n. +The proof of Lemma 20 comes from the proof of Lemma 8 in [4]. We give it for the sake of completeness. +Proof. Let S′ be any set of +� k +3 +� +colours. +We start with G coloured by α. Let (v1, . . . , vn) be a degeneracy ordering of G. We consider each vertex from +vn to v1. For each vertex, if it possible we recolour it with a colour of S. We denote by η the obtained L-colouring. +This is done in less than n steps. Observe that, for each colour c ∈ S and each vertex vi of G, at least one of the +following holds: +• η(vi) ∈ S, +• vi has a neighbour in {vi+1, . . . , vn} coloured c, or +13 + +• c /∈ L(vi). +Let H be the subgraph of G induced by the vertices whose colour in η is not in S. We define LH by LH(v) = +L(v) \ S for every v ∈ V (H). Using the previous observation, we get that for every vertex vi of H, v has at least +|L(v) ∩ S| neighbours in {vi+1, . . . , vn} \ H. This implies that LH is a +� k +3 +� +-feasible list assignment of H with a +total number of colours bounded by 2k +3 . By Theorem 19 (i), the diameter of C(H, LH) is at most 2k +3 n. Note that +every recolouring of H starting from ηH (the colouring η induced on H) gives a valid recolouring of G starting +from η. +Consider the following preference ordering on the colours: an arbitrary ordering of [k] \ (S ∪ S′), followed +by an ordering of S′ \ S, and finally the colours from S. Let γ be the L-colouring of G obtained by colouring G +greedily from vn to v1 with this preference ordering. Since L is +� k +3 +� +-feasible, and |S| = +� k +3 +� +, no vertex is coloured +with a colour in S in γ. This implies that γH, the colouring γ induced on H, is an LH-colouring of H. Thus +there is a recolouring from ηH to γH of length at most 2k +3 n steps. This gives a recolouring sequence in G. We can +then recolour the vertices of G − H to their target colour in γ in at most n steps. This shows that, in G, there is a +recolouring sequence from α to γ of length at most 2k+3 +3 +n. +Now observe that, for each colour c ∈ R = [k] \ (S ∪ S′) and each vertex vi of G, at least one of the following +must hold: +• γ(vi) ∈ R, +• vi has a neighbour in {vi+1, . . . , vn} coloured c, or +• c /∈ L(vi) +Let Γ be the subgraph of G induced by all vertices coloured with a colour in S′ by γ. Note that Γ is also the +subgraph induced by all vertices coloured with a colour in (S ∪ S′) by γ, because no vertex is coloured in S by γ. +Let LΓ be the list assignment defined by LΓ(v) = L(v) ∩ (S ∪ S′) for all v ∈ V (Γ). By the previous observation, +LΓ is +� k +3 +� +-feasible, and the total number of colours is |S ∪ S′| ≤ 2 +� k +3 +� +. Thus by Theorem 19 (i), C(Γ, LΓ) has +diameter at most 2 +� k +3 +� +n. Let βΓ be an LΓ-colouring of Γ that avoids the colours of S′ (such a colouring exists +because |S′| = +� k +3 +� +and LΓ is +� k +3 +� +-feasible) and γΓ the colouring γ induced on Γ. +There is a recolouring sequence of length at most 2 +� k +3 +� +n from γΓ to βΓ. This extends to a recolouring sequence +in G from γ to β where β does not use any colour of S′. +The total number of steps to reach β from α is at most 2k +3 n + n + 2 +� k +3 +� +n which is bounded by 4k+6 +3 +n. This +shows the result. +We are now able to prove Theorem 10. +Proof of Theorem 10. Let D = (V, A) be a digraph on n vertices and k ≥ 3 +2(δ∗ +min(D) + 1). Let (v1, . . . , vn) be a +min-degeneracy-orderingof D, that is an ordering such that for each i ∈ [n], vi has at most δ∗ +min(D) out-neighbours +or at most δ∗ +min(D) in-neighbours in {vi+1, . . . , vn}. +We define B a subset of A as follows: for each i ∈ {1, . . . , n}, if vi has at most δ∗ +min(D) out-neighbours +in {vi+1, . . . , vn}, we add all arcs of A from vi to {vi+1, . . . , vn} to B. Otherwise, we add all arcs of A from +{vi+1, . . . , vn} to vi to B. Note that both digraphs (V, B) and (V, A \ B) are acyclic. +Let H be (V, B) and G be the underlying graph of H. Note that, since (V, A \ B) is acyclic, each colouring of +G is a dicolouring of D, but a dicolouring of D is not necessarily a colouring of G. +By construction, G has degeneracy at most δ∗ +min(D). Using Theorem 3, we get that Ck(G) has diameter at +most C0n2 for some constant C0 independent from k. +Set δ∗ = δ∗ +min(D) ≥ δ∗(G), Xi = {vi+1, . . . , vn}, and Hi = G − Xi for all i ∈ [n]. +Let α be any dicolouring of D. Let Li be the list assignment of Hi defined by +Li(vj) = [k] \ {α(v)|v ∈ NG(vj) ∩ Xi} for all j ∈ [i]. +14 + +Since k, the total number of colours, is at least 3 +2(δ∗ + 1), for every j ∈ [i] we have: +|Li(vj)| ≥ k − |NG(vj) ∩ Xi| +≥ k +3 + 2 +3 +3 +2(δ∗ + 1) − |NG(vj) ∩ Xi| +≥ |NG(vj) ∩ {vj+1, . . . , vi}| + 1 + k +3 +Hence, since |Li(vj)| is an integer, Li is a +� k +3 +� +-feasible list assignment of Hi. +Remark 13. Let γ be a dicolouring of D and for some i, γ agrees with α on {vi+1, . . . , vn} and γHi is an Li- +colouring of Hi. Then any recolouring sequence starting from γHi on Hi, valid for Li, is a valid redicolouring +sequence in D. Assume this is not the case and at one step, we get to an Li-colouring ζ of Hi but ζD contains a +monochromatic directed cycle C, where ζD(v) = ζ(v) when v belongs to Hi and ζD(v) = γ(v) otherwise. Let +vj be the vertex of C such that j is minimum. This vertex vj has both an in-neighbour vj1 and an out-neighbour +vj2 coloured ζD(vj) such that j1, j2 ≥ j. We know that either vj1vj or vjvj2 belongs to G. Assume by symmetry +that vj1vj belongs to G. Then either j1 ≤ i and then vj1vj is a monochromatic edge in Hi or j1 ≥ i + 1 but then +ζ(vj1) = α(vj1) does not belong to Li(vj). In both cases, we get a contradiction. +Claim 10.1: Let γi be a dicolouring of D which induces an Li-colouring of Hi avoiding at least +� k +3 +� +colours in +Hi. There is a redicolouring sequence of length at most ck +3 n from γi to a dicolouring γi+⌈ k +3⌉ which induces an +Li+⌈ k +3⌉-colouring of Hi+⌈ k +3⌉ avoiding at least +� k +3 +� +colours in Hi+⌈ k +3⌉. +Proof of claim. +Let S be a set of colours of size exactly +� k +3 +� +avoided by γi on Hi. For each vertex vj in +{vi+1, . . . , vi+⌈ k +3⌉}, we choose a colour cj so that each of the following holds: +• cj belongs to Lj(vj), +• cj does not belong to {α(u)|u ∈ NG(vj) ∩ {vj+1, . . . , vn}}, +• for each ℓ ∈ {i + 1, . . . , j − 1}, cℓ is different from cj. +Note that this is possible because Lj(vj) is +� k +3 +� +-feasible. Now let S′ be {ci+1, . . . , ci+⌈ k +3 ⌉}. Observe that |S′| = +� k +3 +� +. By Lemma 20, there is, in Hi, a recolouring sequence of length at most 4k+6 +3 +n, valid for Li, from γi to +some γ′ +i that avoids S′. This recolouring sequence extends to a redicolouring sequence in D by Remark 13. In the +obtained dicolouring, since γ′ +i avoids S′ on Hi, we can recolour successively vj with cj for all i+1 ≤ j ≤ i+ +� k +3 +� +. +This does not create any monochromatic directed cycle by choice of cj. Let ηi be the resulting dicolouring of D. +Now we define ˜Li a list-assignment of Hi as follows: +˜Li(vj) = {1, . . ., k} \ {ηi(v)|v ∈ N(vj) ∩ {vi+1, . . . , vn})) +Using the same arguments as we did for Li, we get that ˜Li is +� k +3 +� +-feasible for Hi. Note that ηi is an ˜Li- +colouring of Hi that avoids S′. Let S′′ be a set of +� k +3 +� +colours disjoint from S′. By Lemma 20, there is, in Hi, a +recolouring sequence (valid for ˜Li) of length at most 4k+6 +3 +n from ηi to some η′ +i that avoids S′′. This recolouring +sequence extends directly to a redicolouring sequence in D. Since S′ is disjoint from S′′, the obtained dicolouring +is an Li+⌈ k +3 ⌉-colouring of Hi+⌈ k +3⌉ that avoids at least +� k +3 +� +colours in Hi+⌈ k +3⌉. Hence we get a redicolouring +sequence from γi to the desired γi+⌈ k +3⌉, in at most 8k+12 +3 +n + +� k +3 +� +steps. This proves Claim 10.1. +♦ +Note that γ⌈ k +3⌉ can be reached from α in less than n steps: for all j ∈ [ +� k +3 +� +], choose a colour cj so that each of +the following holds: +• cj belongs to Lj(vj), +• cj does not belong to {α(u)|u ∈ NG(vj) ∩ {vj+1, . . . , vn}}, +15 + +• for each ℓ ∈ [j − 1], cℓ is different from cj. +Now we can recolour successively v1, . . . , v⌈ k +3⌉ to their corresponding colour in {c1, . . . , c⌈ k +3⌉}. Then applying +Claim 10.1 iteratively at most +� +n +⌈ k +3⌉ +� +≤ +3n +k times, there is a redicolouring sequence of length at most n + +3n +k +� 8k+6 +3 +n + k +3 +� +from α to a dicolouring of D that is also a colouring of G. Note that there exists a constant C1 +such that n + 3n +k +� 8k+6 +3 +n + k +3 +� +≤ C1n2. +Let α and β be two k-dicolourings of D. As proved above, there is a redicolouring sequence of length at most +C1n2 from α (resp. β) to a dicolouring α′ (resp. β′) of D that is also a colouring of G. Since Ck(G) has diameter at +most C0n2, there is a recolouring sequence of G of length at most C0n2 from α′ to β′, which is also a redicolouring +sequence of D (since every colouring of G is a dicolouring of D). The union of those three sequences yields a +redicolouring sequence from α to β of length at most (2C1 + C0)n2. +4 +Density of non 2-mixing oriented graphs +4.1 +Density of non k-mixing undirected graphs +Let k ∈ N. As observed in the introduction, every non k-mixing graph as maximum average degree at least k − 1. +This bound is tight because the complete graph on k vertices is (k − 1)-regular and is not k-mixing. Moreover, it +is shown in [2] that this bound is tight even when we restrict to graphs of arbitrary large girth. The initial proof +uses the probabilistic method. In this section, we give a new constructive proof of this result, based on an explicit +construction of regular bipartite graphs from Lazebnik and Ustimenko in [12]. +Theorem 22 (Bonamy, Bousquet and Perarnau, [2]). For any k, ℓ ∈ N∗, there exists a (k − 1)-regular k-freezable +graph Gk,ℓ with girth at least ℓ. +We first make the following remark that we will use in the proof of Theorem 22: +Remark 23. Let k ∈ N∗, G be a (k − 1)-regular k-freezable graph and c be a frozen k-colouring of G, then all +colour classes of c have the same size. This follows from the fact that, for every vertex v of G, N[v] use all colours +in c. Thus, given two colours i, j of c, there must be a perfect matching in G between the vertices coloured i and +the vertices coloured j. In particular, this implies that the number of vertices coloured i is the same as the number +of vertices coloured j. +Proof of Theorem 22. Let us fix ℓ ∈ N. We prove the statement by induction on k, the result holding trivially for +k = 1. Let k > 1 and assume that there exists a (k − 2)-regular (k − 1)-freezable graph Gk−1,ℓ with girth at least +ℓ. Let c be a frozen k-colouring of Gk−1,ℓ. +We denote by n the number of vertices of Gk−1,ℓ. Consider H a n-regular bipartite graph with girth at least +ℓ (such a graph exists by a construction from Lazebnik and Ustimenko [12]). Since H is bipartite, we can colour +the edges of H with exactly n colours such that two adjacent edges receive different colours. By Remark 23, all +colour classes of c have the same size. Thus there is an ordering (v1, . . . , vn) of V (Gk−1,ℓ) such that for each +i ∈ [n − k + 1], the vertices vi, . . . , vi+k−1 have different colours by c. +We denote by (A, B) the bipartition of H. Let Gk,ℓ be the graph obtained from H as follows. +• For each a ∈ A, replace a by a copy Ga of Gk−1,ℓ, and connect va +i (the vertex corresponding to vi in Ga) to +the edge coloured i that was incident to a. +• For each b ∈ B, replace b by an independent set Ib = {xb +1, . . . , xbn +k } of size n +k (by Remark 23, n +k is an +integer). Connect xb +i to the edges coloured {k(i − 1) + 1, . . . , ki} that were incident to b. +Observe that Gk,ℓ is k-regular: every vertex in a Ga is adjacent to its k − 1 neighbours in Ga and exactly one +neighbour in one of the Ib; every vertex in an Ib has exactly k neighbours by construction. Moreover, Gk,ℓ has +girth at least ℓ. Indeed, assume, for a contradiction, that it contains a cycle C of length at most ℓ − 1. Then C +16 + +cannot contain an edge of H, otherwise, contracting each copy of Ga would transform C into a cycle of length at +most ℓ − 1 in H. Thus C must be contained in some Ga, which is a copy of Gk−1,ℓ, which is impossible since +Gk−1,ℓ has girth at least ℓ. +Let α be the (k + 1)-colouring of Gk,ℓ such that the restriction of α to each Ga corresponds to c, and α(xb +i) = +k + 1 for all b ∈ B and i ∈ [n/k]. Let v be a vertex of G. If v belongs to some Ga, then since c is frozen in +Gk−1,ℓ, NGa[v] contains all colours of {1, . . . , k}. Moreover, by construction, v has a neighbour in some Ib which +is coloured k + 1. If v is in some Ib, then v is coloured k + 1 and by construction it has exactly one neighbour in +each colour class. In both cases, N[v] = {1, . . . , k + 1}. Thus no vertex can be recoloured and so α is a frozen +colouring of Gk,ℓ. +4.2 +Density of non 2-mixing and 2-freezable oriented graphs +Let k ∈ N and D be a digraph that is not k-mixing. As observed in the introduction, Theorem 6 implies that +Mad(D) ≥ 2k − 2. This bound is tight because the bidirected complete digraph on k vertices is (2k − 2)-regular +and is not k-mixing. However, unlike the undirected case, this result does not extend to digraphs with larger digirth, +even for k = 2. While the above inequality states that every non 2-mixing digraph has maximum average degree +at least 2, we conjecture that every non 2-mixing oriented graph has maximum average degree at least 4. +Conjecture 24. Any non 2-mixing oriented graph has maximum average degree at least 4. +Remark 25. If true, this conjecture would be tight since there exist 2-freezable oriented graphs with maximum +average degree 4. Consider for example the oriented graph ⃗Fn obtained from the disjoint union of two disjoints +directed paths (u1, . . . , un) and (v1, . . . , vn) by adding the set of arcs {uivi | i ∈ [n]} ∪ {vi+1ui | i ∈ [n − 1]} ∪ +{v1u2, unv1, vn−1un} (see Figure 5 for an illustration). Let α be the 2-dicolouring of ⃗Fn in which all the ui are +coloured 1 and all the vi are coloured 2. One can easily check that Mad(⃗Fn) = 4 and α is a 2-frozen dicolouring +of ⃗Fn. +... +... +Figure 5: The 2-freezable oriented graph ⃗Fn and a frozen 2-dicolouring. +We now prove two results supporting Conjecture 24. First we prove that Conjecture 24 holds with the stronger +assumption that G is 2-freezable. +Theorem 14. Let ⃗G = (V, A) be an oriented graph. If ⃗G is 2-freezable, then |A| ≥ 2|V |. +Proof. Let ⃗G = (V, A) be a 2-freezable oriented graph, and c a frozen 2-dicolouring of ⃗G. For a vertex v ∈ V , we +say that a vertex u ∈ V is blocking for v (in dicolouring c), if one of the following holds: +• u is an out-neighbour of v, c(u) ̸= c(v), and there exists a directed path (u, ..., x, v) such that (u, ..., x) is +monochromatic, or +• u is an in-neighbour of v, c(u) ̸= c(v), and there exists a directed path (x, ..., u, v) such that (x, ..., u) is +monochromatic. +We shall use a discharging argument. We set the initial charge of every vertex v to be d(v) = d+(v) + d−(v). +Observe that d(v) ≥ 2 otherwise v can be recoloured in c. We then use the following discharging rule. +(R) every vertex receives 1 from each of its blocking neighbours. +Let f(v) be the final charge of every vertex v. Let us show that f(v) ≥ 4 for every v ∈ V . +Let v ∈ V . Let α be its colour and β the other colour. Since c is frozen, v admits at least one out-neighbour +v+ and one in-neighbour v− coloured β that are blocking, and thus sending 1 to v by (R). Let us now examine the +17 + +charge that v sends to others vertices. Let w be a vertex to which v sends charge. v is blocking for w, so c(w) = β. +Moreover if w is an out-neighbour (resp. in-neighbour) of v, then v has an in-neighbour (resp. out-neighbour) +coloured α. We are in one of the following cases. +• If v sends no charge, then f(v) ≥ d(v) + 2 ≥ 4. +• If v sends charge only to some out-neighbours, then it does not send to its in-neighbours. Since v has at least +two in-neighbours (one blocking v and one coloured α), f(v) ≥ d(v) + 2 − (d(v) − 2) ≥ 4. +• If v sends charge only to some in-neighbours, symmetrically to above, f(v) ≥ 4. +• If v sends charge only to both out-neighbours and in-neighbours, then its has both an in-neighbour and an +out-neighbour coloured α to which it sends no charge. Hence f(v) ≥ d(v) + 2 − (d(v) − 2) ≥ 4. +In all cases, we have f(v) ≥ 4. Consequently, 2|A| = � +v∈V d(v) = � +v∈V f(v) ≥ 4|V |. +We can deduce from Theorem 14 the following lower bound on the density of a k-freezable oriented graph. +Corollary 15. Let ⃗G = (V, A) be an oriented graph. If ⃗G is k-freezable, then |A| ≥ k|V | + k(k − 2). +Proof. Suppose for a contradiction that there is a k-freezable oriented graph ⃗G = (V, A) such that |A| < k|V | + +k(k − 2). Without loss of generality, we may take ⃗G having a minimum number of arcs among all such graphs. +Let c be a frozen k-dicolouring of ⃗G. For each i, j ∈ {1, . . ., k}, let ⃗Gi be the subdigraph of ⃗G induced by the +vertices coloured i in c, and let ⃗Gi,j be the subdigraph of ⃗G induced by the vertices coloured i or j in c. We set +ni = |V (⃗Gi)|, mi = |A(⃗Gi)| and mi,j = |A(⃗Gi,j)|. +We first show that, for any i ∈ {1, . . . , k}, mi ≤ ni − 1. Suppose not. Then, since ⃗Gi is acyclic, it admits +an acyclic ordering (x1, . . . , xni). Now consider ⃗G′ = (⃗G \ A(⃗Gi)) ∪ {xjxj+1 | j ∈ [ni − 1]} with the same +colouring c. Clearly |A(⃗G′)| < |A(⃗G)|. Let v be a vertex of ⃗G′. If v ∈ V (⃗Gi), then x is still blocked in (⃗G′, c) +because it is blocked in (⃗G, c). Now, suppose v /∈ V (⃗Gi). For any colour j distinct from i and c(v), it is impossible +to recolour v with j because it was already impossible in ⃗G. Now, in ⃗G, it was impossible to recolour v to i, so +there is a directed path in ⃗Gi whose initial vertex xk is an out-neighbour of v and whose terminal vertex xℓ is an +in-neighbour of v. Since (x1, . . . , xni) is an acyclic ordering of ⃗Gi, we have k ≤ ℓ. Thus (xk, . . . , xℓ) is a directed +path in ⃗G′. Hence v cannot be recoloured to i in (⃗G′, c), meaning it is also blocked in (⃗G′, c). Since all vertices of +(⃗G′, c) are blocked, c is a frozen k-dicolouring of ⃗G′, contradicting the minimality of ⃗G. +We will now prove the result by bounding S = � +1≤i 7 +2. +• Finally assume d(x) ≥ 6. By Claim 16.6, x gives 1 +2 by (R1) to at most d(x) − 2 of its neighbours. Thus +w∗(x) ≥ d(x) − 1 +2(d(x) − 2) − 2 × 1 +4 = d(x) +2 ++ 1 +2 ≥ 7 +2. +This completes the proof of Theorem 16. +5 +Further research +In this first paper, we established the first results on digraph redicolouring. This is obviously just the tip of the +iceberg and many open questions arise. Forthwidth, we detail a few of them. +In Section 2, we prove that k-DICOLOURING PATH is PSPACE-complete for all k ≥ 2. But we did not prove +any complexity result about DIRECTED IS k-MIXING. +Problem 17. What is the complexity of DIRECTED IS k-MIXING? +We believe that this is PSPACE-hard for all k ≥ 2. To settle the complexity of DIRECTED IS k-MIXING, it +could be helpful to settle the complexity of the following particular case of 2-DICOLOURING PATH. (Recall that +the mirror of a 2-dicolouring α of D, is the 2-dicolouring ¯α of D such that α(v) ̸= ¯α(v) for all v ∈ V (D).) +MIRROR-REACHABILITY +Input: A 2-dicolouring α of a digraph D. +Question: Is there a path between α and its mirror ¯α in D2(D)? +Problem 18. What is the complexity of MIRROR-REACHABILITY? +A particular case of non k-mixing digraphs are k-freezable digraphs. It would then be interesting to consider +the complexity of the following problem. +DIRECTED IS k-FREEZABLE +Input: A k-dicolourable digraph D. +Question: Is D k-freezable? +Note that deciding whether a digraph is k-freezable is NP-complete in general since we can reduce easily from +k-dicolourability for all k ≥ 2. Indeed, for a digraph D, let D′ be the digraph obtained from D by adding on +26 + +each vertex v ∈ V (D) a complete bidirected graph Kv +k which contains v and k − 1 new vertices. Trivially, D′ +is k-dicolourable if and only if D is k-dicolourable, and every k-dicolouring of D′ (if one exists) is necessarily +frozen because of the complete bidirected graphs. Therefore D′ is k-freezable if and only if D is k-dicolourable. +A related problem is the one of deciding whether a vertex is frozen in a given k-dicolouring α of a digraph D. +Recall that a vertex v is frozen in α if β(v) = α(v) for any k-dicolouring β in the same connected component of +α in Dk(D). +k-FROZEN VERTEX +Input: A digraph D, a k-dicolouring α of D, and a vertex v of D. +Question: Is v frozen in α? +2-FROZEN VERTEX is PSPACE-complete. This comes from the following result from [10]: given a cubic +graph G, a mapping φ : V (G) −→ {1, 2}, a proper orientation ⃗G of G, and an edge xy of G, deciding whether there +is a reorienting sequence from ⃗G that reverse xy is PSPACE-complete. Hence, the same reduction as the one used +for Theorem 5 also yields PSPACE-completeness of 2-FROZEN VERTEX. +One can then easily derive that k-FROZEN VERTEX is PSPACE-complete for any k ≥ 2. Indeed, consider a +(non-acyclic) digraph D2 and a 2-dicolouring α2 of D2. Let Dk be the digraph obtained the disjoint union of D2 +and bidirected complete graph ←→ +K k−2 on k − 2 vertices y1, . . . , yk−2 and adding a digon between any vertex of +D2 and any vertex of ←→ +K k−2. Let αk be the k-dicolouring of Dk defined by αk(v) = α2(v) for all v ∈ V (D2) and +αk(yi) = i + 2 for every i ∈ [k − 2]. One can easily check that a vertex v in V (D2) is frozen in α2 if and only if +it is frozen in αk. +For each of the above problems which are proved or conjectured to be PSPACE-complete, it is natural to de- +termine the smallest possible classes of digraphs on which the problem remains PSPACE-complete. For example, +in Theorem 5, we prove that it remains PSPACE-complete for oriented graphs in (ii) and planar oriented graphs +with maximum degree at most 6 in (iv). This raises the following questions. +Problem 19. +(a) What is the smallest Dk (resp. D∗ +k) such that k-DICOLOURING PATH remains PSPACE- +complete for oriented graphs (resp. planar oriented graphs) with maximum degree at most Dk (resp. D∗ +k)? +(b) Does k-DICOLOURING PATH remain PSPACE-complete for digraphs with no directed cycles of length less +than g, for any fixed g. +Regarding Problem 19 (a), Theorem 5 and Theorem 11 imply that D2 ∈ {4, 5}. Thus the first question to +address is the following. +Problem 20. What is the complexity of 2-DICOLOURING PATH restricted to digraphs with maximum degree at +most 4? +In the opposite direction, it would be interesting to determine the largest possible classes of digraphs for which +k-DICOLOURING PATH can be solved in polynomial time. For example, Corollary 12 implies that DIRECTED IS +2-MIXING is polynomial-time solvable for subcubic digraphs. This raises the following questions. +Problem 21. (a) What is the largest Mk such that DIRECTED IS k-MIXING is polynomial-time solvable for di- +graphs with maximum degree at most Mk? +(b) What is the largest M ∗ +k such that DIRECTED IS k-FREEZABLE is polynomial-time solvable for digraphs with +maximum degree at most M ∗ +k? +A first step towards tackling Problem 21 (a) would be to determine whether Theorem 11 extends to digraphs +with maximum degree k for every odd k. +Problem 22. Let k = 2ℓ + 1 be an odd positive integer. Is every oriented graph with maximum degree k (ℓ + 1)- +mixing? +Corollary 15 directly implies that a planar oriented graph is not 3-freezable. It is then natural to ask whether a +planar oriented graph is always 3-mixing. +27 + +Problem 23. Is every planar oriented graph 3-mixing? +In Sections 3 and 4, we established several conditions for a digraph to be k-mixing and sometimes some bounds +on the diameter. It is then natural to ask whether the bound in the diameter is tight or not. For example, Theorem 6 +states that every (k − 2)-min-degenerate digraph D is k-mixing and following its proof one can show that the +diameter of Dk(D) is at most 2n with n = |V (D)|. But it is likely to be smaller, perhaps even polynomial in n. +Problem 24. What is the maximum diameter of Dk(D) over all (k − 2)-min-degenerate digraphs D of order n? +Another example is the diameter of D2(⃗G) for a subcubic oriented graph ⃗G. By Theorem 11, it is at most 2n. +But, we do not know if this bound is tight. +Problem 25. What is the maximum diameter of D2(⃗G) over all subcubic oriented graphs ⃗G of order n? +References +[1] Valentin Bartier. Combinatorial and Algorithmic aspects of Reconfiguration. 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Journal of +Computer and System Sciences, 4(2):177–192, 1970. +28 + diff --git a/a9E1T4oBgHgl3EQfxAV3/content/tmp_files/load_file.txt b/a9E1T4oBgHgl3EQfxAV3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f18f3b21d8eece70ea5708bc2fa88c6d22a0f72 --- /dev/null +++ b/a9E1T4oBgHgl3EQfxAV3/content/tmp_files/load_file.txt @@ -0,0 +1,1437 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf,len=1436 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='03417v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='DM] 9 Jan 2023 Digraph redicolouring * N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Bousquet1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Havet2, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Nisse2, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Picasarri-Arrieta2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Reinald2,3 1 LIRIS, CNRS, Universit´e Claude Bernard Lyon 1, Lyon, France nicolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='bousquet@univ-lyon1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='fr 2 CNRS, Universit´e Cˆote d’Azur, I3S, Inria, Sophia-Antipolis, France {frederic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='havet,nicolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='nisse,lucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='picasarri-arrieta}@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='fr 3 LIRMM, CNRS, Universit´e de Montpellier, Montpellier, France amadeus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='reinald@lirmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='fr Abstract In this work, we generalize several results on graph recolouring to digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Given two k-dicolourings of a digraph D, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for k = 2 and for digraphs with maximum degree 5 or oriented planar graphs with maximum degree 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A digraph is said to be k-mixing if there exists a transformation between any pair of k-dicolourings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We show that every digraph D is k-mixing for all k ≥ δ∗ min(D) + 2, generalizing a result due to Dyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We also prove that every oriented graph ⃗G is k-mixing for all k ≥ δ∗ max( ⃗G) + 1 and for all k ≥ δ∗ avg( ⃗G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Here δ∗ min, δ∗ max, and δ∗ avg denote the min-degeneracy, the max-degeneracy, and the average-degeneracy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We pose as a conjecture that, for every digraph D, the dicolouring graph of D on k ≥ δ∗ min(D) + 2 colours has diameter at most O(|V (D)|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This is the analogue of Cereceda’s conjecture for digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We generalize to digraphs two results supporting Cereceda’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We first prove that the dicolouring graph of any digraph D on k ≥ 2δ∗ min(D) + 2 colours has linear diameter, extending a result from Bousquet and Perarnau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We also prove that the analogue of Cereceda’s conjecture is true when k ≥ 3 2(δ∗ min(D) + 1), which generalizes a result from Bousquet and Heinrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Restricted to the special case of oriented graphs, we prove that the dicolouring graph of any subcubic oriented graph on k ≥ 2 colours is connected and has diameter at most 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We conjecture that every non 2-mixing oriented graph has maximum average degree at least 4, and we provide some support for this conjecture by proving it on the special case of 2-freezable oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' More generally, we show that every k-freezable oriented graph on n vertices must contain at least kn + k(k − 2) arcs, and we give a family of k-freezable oriented graphs that reach this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In the general case, we prove as a partial result that every non 2-mixing oriented graph has maximum average degree at least 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Keywords: Digraphs, oriented graphs, graphs recolouring, reconfiguration, dicolouring 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='1 Graph recolouring We denote by [k] the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=', k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Given a graph G = (V, E), a k-colouring of G is a function c : V −→ [k] such that c(x) ̸= c(y) for every edge xy ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The chromatic number χ(G) is the smallest k such that G admits a k-colouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For any k ≥ χ(G), the k-colouring graph of G, denoted by Ck(G), is the graph whose vertices are the k-colourings of G and in which two k-colourings are adjacent if they differ on exactly one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We identify a path between two given colourings in Ck(G) with a sequence of recolourings, that is, an ordered list of pairs composed of a vertex of G and a new colour for this vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If Ck(G) is connected, we say that G is k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Research supported by research grant DIGRAPHS ANR-19-CE48-0013 and by the French government, through the EUR DS4H Invest- ments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-17-EURE-0004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 1 Given a graph G, one may ask for which values of k it is k-mixing, and when it is, how many steps are required at most to get from one colouring to another, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' what is the diameter of the Ck(G)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Determining if a graph is k-mixing has applications in statistical physics, where colourings represent states of the antiferromagnetic Potts model at temperature zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The questions above were first addressed by researchers studying the Glauber dynamics for sampling k-colourings of a given graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This is a Markov chain used to obtain efficient algorithms for approximately counting or almost uniformly sampling k-colourings of a graph, and the connectedness of the k-colouring graph is a necessary condition for such a Markov chain to be rapidly mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In graph theory, the study of recolouring has been rapidly developing in the last fifteen years, since the works of Cereceda, van den Heuvel and Johnson [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We refer the reader to the PhD thesis of Bartier [1] for a complete overview on graph recolouring and to the surveys of van Heuvel [11] and Nishimura [14] for reconfiguration problems in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' As described above, one of the main problems in recolouring is to decide whether a given graph is k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' IS k-MIXING Input: A graph G Question: Is G k-mixing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that IS 2-MIXING is trivial since only edgeless graphs are 2-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' On the other hand, Cereceda [7] proved that IS 3-MIXING is coNP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For higher values of k, the complexity of this problem is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A related problem is that of recognizing when two given k-colourings of a graph G are in the same connected component of Ck(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Formally, we have the following decision problem: k-COLOURING-PATH Input: A graph G along with two k-colourings α and β of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Question: Is there a path between α and β in Ck(G)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 2-COLOURING PATH is trivial since only isolated vertices can be recoloured in a bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Cereceda, van den Heuvel and Johnson [8] proved that 3-COLOURING PATH is polynomial-time solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Moreover the authors proved that the diameter of each component of Ck(G) is O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In contrast, for every k ≥ 4, Bonsma and Cereceda [3] showed a family Gk of graphs such that for every G ∈ Gk of order n, there exists two k-colourings whose distance in Ck(G) is finite and superpolynomial in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' They also proved that k-COLOURING PATH is PSPACE-complete for all k ≥ 4 even restricted to bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' However, the situation is different for degenerate graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The degeneracy of a graph G, denoted by δ∗(G), is the largest minimum degree of any subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Bonsma and Cereceda [3] and Dyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' [9] independently proved the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 1 (Bonsma and Cereceda [3] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Dyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let k ∈ N and G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If k ≥ δ∗(G) + 2, then G is k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Cereceda [6] also conjectured the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Conjecture 2 (Cereceda [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let k ∈ N and G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If k ≥ δ∗(G) + 2, then the diameter of Ck(G) is at most O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Cereceda [6] proved that this is true when k ≥ 2δ∗(G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This was improved recently by Bousquet and Heinrich [4], who showed the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 3 (Bousquet and Heinrich [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let k ∈ N and G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then Ck(G) has diameter at most: Cn2 if k ≥ 3 2(δ∗(G) + 1) (where C is a constant independent from k), Cεn⌈ 1 ε ⌉ if k ≥ (1 + ε)(δ∗(G) + 2) (where Cε is a constant independent from k), (Cn)d+1 for any k ≥ δ∗(G) + 2 (where C is a constant independent from k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Bousquet and Perarnau [5] also proved the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 4 (Bousquet and Perarnau [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let k ∈ N and G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If k ≥ 2δ∗(G) + 2, then the diameter of Ck(G) is at most (δ∗(G) + 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 2 Let k ∈ N and G be a graph that is not k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' It follows from Theorem 1 that G contains a subgraph H with minimum degree at least k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Thus G has maximum average degree at least k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This bound is tight because the complete graph on k vertices is (k − 1)-regular and is not k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Moreover, it is shown in [2] that this bound is tight even when we restrict to graphs of arbitrary large girth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The initial proof uses the probabilistic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In Section 4, we give a new constructive proof of this result, based on an explicit construction of regular bipartite graphs from Lazebnik and Ustimenko [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='2 Digraph redicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let D be a digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A digon is a pair of arcs in opposite directions between the same vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' An oriented graph is a digraph with no digon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The bidirected graph associated to a graph G, denoted by ←→ G , is the digraph obtained from G, by replacing every edge by a digon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A tournament on k vertices is an orientation of the complete graph on k vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The transitive tournament on k vertices is the only acyclic tournament on k vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In 1982, Neumann-Lara [13] introduced the notions of dicolouring and dichromatic number, which generalize the ones of colouring and chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A k-dicolouring of D is a function c : V (D) → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=', k} such that c−1(i) induces an acyclic subdigraph in D for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The dichromatic number of D, denoted by ⃗χ(D), is the smallest k such that D admits a k-dicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' There is a one-to-one correspondence between the k-colourings of a graph G and the k-dicolourings of the associated bidirected graph ←→ G , and in particular χ(G) = ⃗χ(←→ G ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Hence every result on graph colourings can be seen as a result on dicolourings of bidirected graphs, and it is natural to study whether the result can be extended to all digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In this paper, we study digraph redicolouring, which is a generalization of graph recolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For any k ≥ ⃗χ(D), the k-dicolouring graph of D, denoted by Dk(D), is the graph whose vertices are the k-dicolourings of D and in which two k-dicolourings are adjacent if they differ on exactly one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Observe that Ck(G) = Dk(←→ G ) for any bidirected graph ←→ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A redicolouring sequence between two dicolourings is a path between these dicolourings in Dk(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The digraph D is k-mixing if Dk(D) is connected, and k-freezable if Dk(D) contains an isolated vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A vertex v is blocked to its colour in a k-dicolouring α if, for every colour c ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=', k} different from α(v), recolouring v to c in α creates a monochromatic cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' And we say that v is frozen in α if β(v) = α(v) for any k-dicolouring β in the same connected component of α in Dk(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In Section 2, we consider the directed analogues of IS k-MIXING and k-COLOURING PATH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' DIRECTED IS k-MIXING Input: A digraph D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Question: Is D k-mixing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' k-DICOLOURING PATH Input: A digraph D along with two k-dicolourings α and β of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Question: Is there a path between α and β in Dk(D)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that IS k-MIXING and k-DICOLOURING PATH may be seen as the restrictions of DIRECTED IS k- MIXING and k-DICOLOURING PATH to bidirected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Therefore hardness results transfer to those problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' It follows that DIRECTED IS 3-MIXING is coNP-hard and k-DICOLOURING PATH is PSPACE-complete for all k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We strengthen these results is Section 2 by proving that 2-DICOLOURING PATH is PSPACE-complete, and that k-DICOLOURING PATH remains PSPACE-complete when restricted to some digraph classes: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (i) For every k ≥ 2, k-DICOLOURING PATH is PSPACE-complete on digraphs with maximum degree 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (ii) For every k ≥ 2, k-DICOLOURING PATH is PSPACE-complete on oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (iii) For every 2 ≤ k ≤ 4, k-DICOLOURING PATH is PSPACE-complete on planar digraphs with maximum degree 2k + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (iv) 2-DICOLOURING PATH is PSPACE-complete on planar oriented graphs of degree at most 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 3 In Section 3, we consider generalizations of Theorems 1, 3 and 4 to digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' There are several notions of de- generacy for digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The min-degeneracy of D is δ∗ min(D) = max{min{δ+(H), δ−(H)} | H subdigraph of D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' It is the smallest k such that every subdigraph H of D has a vertex v with min{d+ H(v), d− H(v)} ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The out- degeneracy of D is δ∗ out(D) = max{δ+(H) | H subdigraph of D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' It is the smallest k such that every subdigraph H of D has δ+(H) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The max-degeneracy of D, denoted by δ∗ max(D), is the smallest k such that every subdi- graph H of D has a vertex v with max{d+ H(v), d− H(v)} ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The average-degeneracy of D, denoted by δ∗ avg(D), is the smallest k such that every subdigraph H of D has a vertex v with 1 2(d+ H(v) + d− H(v)) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (In that case k may be half-integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=') Note that if D is an oriented graph, then its average-degeneracy is half the degeneracy of its underlying graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By definition, we have δ∗ min(D) ≤ δ∗ out(D) ≤ δ∗ max(D) and δ∗ min(D) ≤ δ∗ avg(D) ≤ δ∗ max(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If ←→ G is a bidirected graph, then all directed versions of degeneracy above are equal to the degeneracy of the associated graph: δ∗(G) = δ∗ min(←→ G ) = δ∗ out(←→ G ) = δ∗ max(←→ G ) = δ∗ avg(←→ G ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We show the following theorem, which extends Theorem 1 for min-degeneracy and thus also for out- , max- and average-degeneracy by the above inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Every digraph D is k-mixing for every k ≥ δ∗ min(D) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In the special case of oriented graphs, we improve the lower bounds on k by 1 in terms of the max-degeneracy and average-degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Every oriented graph ⃗G is k-mixing for all k ≥ � δ∗ avg(⃗G) � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We also show that such an improvement cannot hold for out-degeneracy (and thus min-degeneracy) by exhibit- ing k-out-degenerate oriented graphs that are not (k + 1)-mixing (Proposition 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A natural question is then the generalization of Conjecture 2 to digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Conjecture 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let D be a digraph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If k ≥ δ∗ min(D) + 2, then the diameter of Dk(D) is at most O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' As evidence towards this conjecture, we establish several results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We first prove the analogue of Theorem 4 for digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let D be a digraph on n vertices and k ≥ 2δ∗ min(D) + 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, the diameter of Dk(D) is at most (δ∗ min(D) + 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We also generalize the first point of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For a digraph D on n vertices, and k ≥ 3 2(δ∗ min(D) + 1), the diameter of Dk(D) is at most Cn2 where C is independent from k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We also consider oriented subcubic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A subcubic graph is a graph with maximum degree at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We prove that any orientation of such a graph is 2-mixing, and that its 2-dicolouring graph has linear diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ⃗G be a subcubic oriented graph of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then D2(⃗G) is connected and has diameter at most 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This result is tight because there exists orientations of 4-regular graphs which are 2-freezable (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since any digraph containing a digon is not 2-mixing, Theorem 11 directly implies the following: Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A subcubic digraph D is 2-mixing if and only if D is an oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In Section 4, we turn our focus to the density of non-mixing graphs and digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In a first part, we first consider undirected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The maximum average degree of a graph G, denoted by Mad(G), is defined by Mad(G) = max � 2|E(H)| |V (H)| | H a non-empty subgraph of G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In the undirected case, one can easily deduce from 4 Figure 1: An orientation of a 4-regular oriented graph with a frozen 2-colouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 1 that any non k-mixing graph G contains a subgraph H with minimum degree at least k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This bound is tight because the complete graph on k vertices is (k − 1)-regular and is not k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Using probabilistic arguments, Bonamy, Bousquet and Perarnau [2] showed that this bound is tight even on graphs of arbitrary large girth (Recall that the girth of a graph is the length of its shortest cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We provide a construction witnessing this fact in Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In a second part, we show that Theorem 22 cannot be generalized to digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Given a digraph D, the maximum average degree of D is defined as max{ 2|A(H)| |V (H)| | H a non-empty subdigraph of D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The digirth of a digraph is the length of its shortest directed cycle if the digraph contains cycles, and +∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In particular, oriented graphs are digraphs with digirth at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' It follows from Theorem 6 that every non k-mixing digraph D contains a subdigraph H with minimum out-degree and minimum in-degree at least k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This shows that such a digraph D has maximum average degree at least 2k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This bound is tight because the bidirected complete digraph on k vertices is 2k−2-regular and is not k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' However, unlike the undirected case, this is not the case for digraphs with arbitrary large digirth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In fact, this is not even the case for oriented graphs, which are exactly the digraphs with digirth at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We pose the following conjecture: Conjecture 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Any non 2-mixing oriented graph has maximum average degree at least 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We prove two results providing some support for this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Firstly, using the Discharging Method, we prove the conjecture in the special case of freezable oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ⃗G = (V, A) be an oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If ⃗G is 2-freezable, then |A| ≥ 2|V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' From this result, we derive the following lower bound on the density of k-freezable oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Corollary 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ⃗G = (V, A) be an oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If ⃗G is k-freezable, then |A| ≥ k|V | + k(k − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We give a family of oriented graphs for which this bound is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Secondly, again with the Discharging Method, we show a statement weaker than Conjecture 13 with 7/2 instead of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ⃗G be an oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If ⃗G is not 2-mixing, then Mad(⃗G) ≥ 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Finally, in Section 5, we conclude with some open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 2 Complexity of k-DICOLOURING PATH In this section, we establish some hardness results for k-DICOLOURING PATH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We need some definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Given a graph G = (V, E) together with a mapping φ : V −→ {1, 2}, an orientation ⃗G of G is proper if for any v ∈ V , d− ⃗G(v) ≥ φ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A reorienting sequence from ⃗G1 to ⃗G2 is a sequence of proper orientations ⃗Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , ⃗Γp of G such that ⃗Γ1 = ⃗G1, ⃗Γp = ⃗G2, every ⃗Γi is proper and every ⃗Γi+1 can be obtained from ⃗Γi by reversing exactly 5 one arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The following problem has been shown to be PSPACE-complete in [10] by a reduction from Quantified Boolean Formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' PLANAR-CUBIC-NCL Input: A cubic planar graph G, a mapping φ : V −→ {1, 2}, two proper orientations ⃗G1 and ⃗G2 of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Question: Is there a reorienting sequence from ⃗G1 to ⃗G2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We will derive the hardness results for DICOLOURING PATH from a hardness result on its list colouring version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let D be a digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A list assignment L is a function which associates a list of colours to every vertex v of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' An L-dicolouring of D is a dicolouring α of D such that α(v) ∈ L(v) for all vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A k-list assignment is a list assignment L such that L(v) ⊆ [k] for all vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We denote by D(D, L) the redicolouring graph of the L-dicolourings of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We call vertices v such that |L(v)| = 1 forced vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We will consider the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' k-LIST DICOLOURING PATH Input: A digraph D, a k-list assignment L, and two L-dicolourings α and β of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Question: Is there a path between α and β in D(D, L) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let us start by proving the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 2-LIST DICOLOURING PATH is PSPACE-complete on digraphs D even when: forced vertices have degree at most 3 and, either all the vertices degree at most 5 or, the digraph D is planar and all the vertices have in and out-degree at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' First note that 2-LIST DICOLOURING PATH is indeed in NPSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Given a digraph D and two dicolour- ings α and β of D together with a redicolouring sequence from α to β, we can easily check with a polynomial amount of space that each dicolouring is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, we get that k-LIST DICOLOURING PATH belongs to PSPACE thanks to Savitch’s Theorem [15], which asserts that PSPACE = NPSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We shall now give a polynomial reduction from PLANAR-CUBIC-NCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let G be a planar cubic graph on n vertices x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , xn with a mapping φ : V (G) → {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ⃗G1 and ⃗G2 be two proper orientations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' From (G, φ) we construct the digraph D and the function L as follows (see Figure 2 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For each vertex xi ∈ V (G), we create a vertex-gadget as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We associate three vertices xi,1, xi,2 and xi,3 in VD so that each of these vertices is associated to exactly one edge of G incident to xi, and each edge of G is associated to exactly two vertices of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The function L assigns to each of these vertices the list {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We complete the vertex-gadget in two ways depending if φ(xi) = 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If φ(xi) = 1, then we create a new vertex zi in such a way that (zi, xi,1, xi,2, xi,3, zi) is a directed 4-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We set L(zi) = {2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If φ(xi) = 2, then we create three new vertices zi,1, zi,2, and zi,3 in such a way that (xi,1, zi,1, xi,2, zi,2, xi,3, zi,3, xi,1) is a directed 6-cycle, and we add the arcs xi,1zi,2, xi,2zi,3 and xi,3zi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The function L assigns the list {2} to zi,1, zi,2, and zi,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For each edge xixj ∈ E(G), where i < j, we create a vertex aij and create the directed 3-cycle (aij, xi,r, xj,r′, aij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We set L(ai,j) = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This directed 3-cycle is the edge-gadget of xixj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 6 xi xj xi,1 xi,2 xi,3 zi xj,1 xj,2 xj,3 zj,1 zj,2 zj,3 aij Figure 2: An example of building D from G, where φ(xi) = 1, φ(xj) = 2, and i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that the digraph D has degree at most 5 and that its forced vertices have degree at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For every proper orientation ⃗G of G, we define the dicolouring associated to ⃗G α ⃗G as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Forced vertices are assigned the colour of their list: vertices zi, zi,1, zi,2, zi,3 are coloured 2, and vertices aij are coloured 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For each arc xixj ∈ A(⃗G), we set xi,r to colour 2 and xj,r′ to colour 1, where xi,r and xj,r′ are the vertices of D corresponding to the edge xixj in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Claim 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='1: For every proper orientation ⃗G of G and corresponding digraph D, the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (i) α ⃗G is an L-dicolouring of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (ii) Unless d− ⃗G(xi) = φ(xi) in ⃗G, changing the colour of xi,r from 1 to 2 still yields a valid L- dicolouring of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let us only show the first item, the second follows by similar arguments noting that d− ⃗G(xi) > φ(xi) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that by definition, α ⃗G satisfies the colouring constraints imposed by the list assignment L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let us then show that α ⃗G is indeed an L-dicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Assume, for a contradiction, that there is a monochromatic directed cycle C in α ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For every edge-gadget, say corresponding to xixj, the vertices xi,r, xj,r′ must be coloured differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Therefore, all vertices of C must be contained in a single vertex-gadget of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let xi be the vertex such that C is in the vertex-gadget of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If φ(xi) = 1, then C must be (zi, xi,1, xi,2, xi,3, zi) and all its vertices should be coloured 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This is a contradiction since d− ⃗G(xi) ≥ 1 implies, by construction, that at least one of xi,1, xi,2, xi,3 is coloured 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If φ(xi) = 2, then C contains at least two vertices in {xi,1, xi,2, xi,3} and two vertices in {zi,1, zi,2, zi,3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, at least two vertices of {xi,1, xi,2, xi,3} are coloured 2 because vertices zi,j are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This is a contradiction since d− ⃗G(xi) ≥ 2 implies, by construction, that at least two vertices of xi,1, xi,2, xi,3 are coloured 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' ♦ Let us take α1 = α ⃗G1 and α2 = α ⃗G2 to be the dicolourings obtained from the two proper orientations ⃗G1, ⃗G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We will now show that there exists a reorienting sequence in G from ⃗G1 to ⃗G2 if and only if there exists a redicolouring sequence in D from α1 to α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 7 Assume first that there is a reorienting sequence ⃗Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , ⃗Γp from ⃗G1 to ⃗G2, and let us show how to build a corresponding redicolouring sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Consider any step s of the reorienting sequence, say when ⃗Γs is transformed into ⃗Γs+1 by reversing an arc xixj into xjxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We will exhibit a path from α⃗Γs to α⃗Γs+1 in D(D, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Consider vertices xi,r, xj,r′ in D, corresponding to the edge xixj, and coloured 2 and 1 respectively in α⃗Γs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We first set the colour of xj,r′ from 1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since ai,j is forced to colour 1, the edge-gadget is not monochromatic at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Moreover since step s reorients arc xixj and still yields a proper orientation, d− ⃗Γs(xj) = d− ⃗Γs+1(xj)+1 ≥ φ(xj)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The resulting colouring is an L-dicolouring by Claim 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='1 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We then set the colour of xi,r from 2 to 1, yielding dicolouring α⃗Γs+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Concatenating the redicolouring sequences obtained through the process above from steps s = 1 to s = p yields a redicolouring sequence from α1 to α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Conversely, assume that there is a redicolouring sequence γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , γp from α1 to α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Observe that the only vertices of D that are possibly recoloured in a step of our sequence are those defined as xi,k for i ∈ [n] and k ∈ [3], since all others are forced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now, at any step s of the redicolouring, for each edge xixj of G, at most one of the two corresponding vertices is coloured 1, because aij is forced to colour 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This allows us to define an orientation ⃗Γs of G as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If the vertices xi,r, xj,r′ ∈ V (D), corresponding to the gadget of edge xixj, are not coloured the same in γs, orientation ⃗Γs sets xixj to be directed from the vertex coloured 2 towards the vertex coloured 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Otherwise, both vertices are coloured 2 in γs and we preserve the orientation of the corresponding edge given by ⃗Γs−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In the first and last dicolourings, α1 and α2, for each edge xixj the corresponding vertices xi,k and xj,k′ are coloured differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Thus ⃗Γ1 = ⃗G1 and ⃗Γp = ⃗G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Therefore, ⃗Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , ⃗Γp is a sequence of orientations of G from ⃗G1 to ⃗G2 such that ⃗Γs+1 is either obtained by reversing an arc of ⃗Γs (when one of the xi,k is recoloured to 1, and the edge xixj whose edge-gadget contains xi,k was not oriented towards xi), or equal to ⃗Γs otherwise (and in particular when one of the xi,k is recoloured to 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Moreover, at each step s, ⃗Γs is a proper orientation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Indeed, if φ(xi) = 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' φ(xi) = 2), then at least one vertex (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' two vertices) of {xi,1, xi,2, xi,3} is coloured 1, and so xi has in-degree at least 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' at least 2) in ⃗Γs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Hence, taking the subsequence of ⃗Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , ⃗Γp that omits constant steps yields a reorienting sequence from ⃗G1 to ⃗G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since PLANAR-CUBIC-NCL is PSPACE-complete, at this point our reduction already yields the PSPACE- completeness of 2-LIST DICOLOURING PATH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By construction, forced vertices zj,k and aij have degree at most 3, and all other vertices have degree at most 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This achieves the proof of the first case of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now, since the input instances are already planar, to get the PSPACE-completeness of 2-LIST DICOLOURING PATH on planar digraphs, it suffices to use planar vertex and edge gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In our current reduction, the only gadget which is not planar is the vertex-gadget corresponding to xi ∈ V (G) such that φ(xi) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We now consider the same reduction replacing the vertex-gadget for vertices such that φ(xi) = 2 with a planar one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For these vertices, the planar vertex-gadget is defined on the same set of vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' {xi,1, xi,2, xi,3, zi,1, zi,2, zi,3}, but with the arcs of the directed 3-cycles (zi,1, xi,1, xi,2, zi,1), (zi,3, xi,1, xi,3, zi,3) and (zi,2, xi,2, xi,3, zi,2), as depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This replacement produces a planar digraph in which all forced vertices still have degree at most 3, and all vertices have maximum in- and out-degree at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' xi,1 xi,2 xi,3 zi,1 zi,2 zi,3 Figure 3: A planar vertex-gadget when φ(xi) = 2 8 The problem k-COLOURING PATH is known to be PSPACE-complete for every k ≥ 4 in the undirected case [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Leveraging Theorem 17, we prove that this also holds for its dicolouring analogue for k ≥ 2 colours, in both directed and oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (i) For every k ≥ 2, k-DICOLOURING PATH is PSPACE-complete on digraphs with maximum degree 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (ii) For every k ≥ 2, k-DICOLOURING PATH is PSPACE-complete on oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (iii) For every 2 ≤ k ≤ 4, k-DICOLOURING PATH is PSPACE-complete on planar digraphs with maximum degree 2k + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (iv) 2-DICOLOURING PATH is PSPACE-complete on planar oriented graphs of degree at most 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (i) We give a reduction from 2-LIST DICOLOURING PATH on instances where forced vertices have de- gree at most 3 and the graph has maximum degree 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The problem is PSPACE-complete by Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let (D, L, α1, α2) be an instance of the problem, we construct an instance (D′, α′ 1, α′ 2) for k-DICOLOURING PATH as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We build D′ starting with D′ = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, for every vertex v ∈ V (D), we let ←→ K v k be a bidirected complete graph on vertex set {zv i | i ∈ [k]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We then add a digon between v and each zv i such that i /∈ L(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We define dicolourings α′ 1 and α′ 2 on D′ by extending dicolourings α1 and α2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' All vertices of D′ that were vertices of D are coloured the same, and we set zv i to colour i for all v ∈ V (D) and all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that all the vertices of gadget ←→ K v k are then frozen in any k-dicolouring, letting us simulate in D′ the list dicolouring constraints on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' An L-dicolouring path from α1 to α2 in D is then exactly a dicolouring path from α′ 1 to α′ 2 when restricted to vertices of D, achieving to show equivalence between the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We will now show that the maximal degree of a vertex v in D′ is 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If v belongs to some gadget ←→ K v k, then its degree is at most 2(k − 1) + 2 = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that when v ∈ V (D), v is of degree 2, 3 or 5 in D, and our reduction adds exactly 2k − |L(v)| arcs incident to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If |L(v)| = 2, this yields dD′(v) ≤ 5 + 2k − 4 = 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If |L(v)| = 1, we know by construction that dD(v) ≤ 3, yielding dD′(v) ≤ 3 + 2k − 2 = 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This achieves the proof that D′ has maximum degree at most 2k + 1, concluding (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (ii) We give a reduction from k-DICOLOURING PATH to k-DICOLOURING PATH restricted to oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let (D, α1, α2) be an instance of k-DICOLOURING PATH, we will build an equivalent instance (⃗G, α′ 1, α′ 2) where ⃗G is an oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Take ⃗H to be an arbitrary oriented graph with dichromatic number exactly k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We construct ⃗G from D by replacing digons of D as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For each digon [u, v] of D, create a copy ⃗Huv of ⃗H, then replace [u, v] by a single arc from u to v, and add the all arcs from v to ⃗Huv and all arcs from ⃗Huv to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By construction, ⃗G is an oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In the following, we let c be a fixed k-dicolouring of ⃗H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We show how to transform any k-dicolouring α of D to a k-dicolouring α′ of ⃗G, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Given a k-dicolouring α for D, we define α′ for ⃗G by colouring each copy ⃗Huv of ⃗H with c, and keeping the same colours as α on V (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Any monochromatic directed dcycle in (⃗G, α′) must contain a vertex of some ⃗Huv, as otherwise it would be a subdigraph of D and would already be monochromatic in (D, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since c is a dicolouring of ⃗H, the cycle must contain both u, v, but then u and v being coloured the same would yield a monochromatic digon in (D, α), so α′ is indeed a k-dicolouring of ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Conversely, given any k-dicolouring α′ of ⃗G, we define α for D as the restriction of α′ on V (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Similarly, if (D, α) were to contain a monochromatic directed dcycle, any arc (u, v) of the cycle that is not present in ⃗G may be replaced with (u, w, v), taking w ∈ ⃗Huv to be a vertex of the same colour as u and v (since ⃗χ( ⃗H) = k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This would yield a monochromatic directed dcycle in (⃗G, α′), so α must be a k-dicolouring of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now, we define the k-dicolourings α′ 1, α′ 2 on ⃗G obtained from α1, α2 by the transformation above, and let our output instance be (⃗G, α′ 1, α′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If there is a redicolouring sequence from α1 to α2 in D, we perform the same recolouring steps in ⃗G starting from α′ 1 and yielding α′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since we only recolour vertices of V (D), the last paragraph yields that this sequence is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Conversely, if there is a redicolouring sequence from α′ 1 to α′ 2, its restriction to V (D) (omitting recolourings of vertices in subgraphs ⃗Huv) yields a valid sequence from α1 to α2 in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This achieves the proof of the equivalence of the instances and proves (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 9 (iii) We give a reduction from 2-LIST DICOLOURING PATH where D is planar, forced vertices have degree at most 3 and D has in- and out-degree at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The problem is PSPACE-complete by Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let (D, L, α1, α2) be an instance of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We make the same reduction as in the proof of (i) by ensuring that ←→ K v k is embedded and coloured in such a way the (at most 3) forbidden colours of v lie on its external face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This allows us to keep a planar representation of D′ which has maximum degree 2k + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This proves (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (iv) As in (iii), we give a reduction from 2-LIST DICOLOURING PATH where D is planar, forced vertices have degree at most 3 and every vertex has in- and out-degree at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since we are considering 2-dicolourings, vertices with a list of size 2 do not require a gadget to simulate forbidden colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The main difference with case (iii) is that we cannot use a the bidirected complete graph ⃗Kv 2 to freeze a vertex v with list of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' To overcome this, we use the gadget depicted in Figure 4, where the colour of all vertices is frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Therefore, we simply have to attach such a gadget on each vertex with list size one, that is, those of the form zi,r or ai,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This can be done by creating a directed triangle including the vertex and two vertices of the gadget that are of the opposite colour as depicted in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' a Figure 4: How to freeze the vertex a in a planar oriented graph with two colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 3 Connectivity and diameter of dicolouring graphs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='1 Connectivity We start by showing some easy bounds on the minimal number of colours, with respects to variants of degeneracy, ensuring that digraphs or oriented graphs are k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Bonsma and Cereceda [3] and Dyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' [9] independently proved that any (non-oriented) graph G is k-mixing for every k ≥ δ∗(G) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 6, which we recall, generalizes this result for digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Every digraph D is k-mixing for every k ≥ δ∗ min(D) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The proof is an adaption of the proof of [9] for undirected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We show the result by induction on the number of vertices of D, the result being obviously true for the digraph with one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let D be a digraph D on at least two vertices, k ≥ δ∗ min(D) + 2 be an integer and α, β be two k-dicolourings of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let v be a vertex such that min{d+(v), d−(v)} ≤ δ∗ min(D) and let D′ = D − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By induction, D′ is k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let α′, β′ be the k-dicolourings of D′ induced by α, β (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' for every v ∈ V (D′), α′(v) = α(v) and β′(v) = β(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since D′ is k-mixing, there exists a redicolouring sequence α′ = γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , γℓ = β′ from α′ to β′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let us prove that we can transform the redicolouring sequence from D′ into a redicolouring sequence for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Starting from α, we perform the same steps as in γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , γℓ as long as they produce dicolourings of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If at some 10 step i in the sequence it is not possible to recolour vertex vi to ci in D, it must be because v is currently coloured ci and recolouring vi to ci would create a monochromatic directed cycle containing both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Assume that v has at most k − 2 out-neighbours (otherwise, v has at most k − 2 in-neighbours and the case is symmetrical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now, we can choose c to be a colour different from ci that is also different from that of the out-neighbours of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, we recolour v to c, allowing us to recolour vi to ci and continue the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Every oriented graph ⃗G is k-mixing for all k ≥ � δ∗ avg(⃗G) � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We proceed by induction on the number of vertices of ⃗G, the result being obviously true for the oriented graph with one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Consider now an oriented graph ⃗G on at least two vertices and k ≥ � δ∗ avg(⃗G) � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let v be a vertex such that d+(v) + d−(v) ≤ 2δ∗ avg(⃗G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By directional duality, we may assume d+(v) ≤ d−(v), then by assumption d+(v) ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We let ⃗G′ = ⃗G − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, δ∗ avg(⃗G′) ≤ δ∗ avg(⃗G), and the induction hypothesis yields that ⃗G′ is k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let α and β two k-dicolourings of ⃗G, and let α′, β′ be the k-dicolourings of ⃗G′ induced by α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since ⃗G′ is k-mixing, there exists a sequence α′ = γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , γℓ = β′ of k-dicolourings of ⃗G′ such that γi−1 and γi differ in the colour of exactly one vertex of ⃗G′ for i ∈ [1, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We denote this vertex by vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now we consider the same recolouring steps to recolour ⃗G, starting from α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If for some i it is not possible to recolour vi to ci, this must be because v is currently coloured ci and recolouring vi to ci would create a monochromatic directed cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that such a cycle necessarily contains v, because γi is a dicolouring of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If d+(v) ≤ k − 2, then v can be recoloured to c, a colour different from ci that does not appear on N +[v], allowing us to recolour vi to ci and proceed with the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Otherwise d+(v) = k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then since k ≥ � δ∗ avg(⃗G) � + 1, we get d−(v) ≤ k − 1, and then d−(v) = k − 1 (since d+(v) ≤ d−(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since ⃗G is an oriented graph, and recolouring vi to ci would create a monochromatic directed cycle, then v has at least one neighbour coloured ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Take w to be one of these neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, if w ∈ N +(v) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' w ∈ N −(v)), we can recolour v to a colour that does not appear in N +(v) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' N −(v)), recolour vi to ci and continue the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that δ∗ max(⃗G) ≥ � δ∗ avg(⃗G) � since δ∗ max(⃗G) is an integer and δ∗ max(⃗G) ≥ δ∗ avg(⃗G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Consequently, every oriented graph ⃗G is k-mixing for all k ≥ δ∗ max(⃗G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' A natural question is then whether this result can be extended to δ∗ out: is every oriented graph ⃗G k-mixing for all k ≥ δ∗ out(⃗G) + 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This does not hold for directed graphs, as witnessed by the bidirected clique, and the following also answers the question in the negative for oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proposition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For every positive integer k, there exist k-out-degenerate oriented graphs that are not (k + 1)- mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ⃗B0 be the oriented graph with one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For k ≥ 1, we construct ⃗Bk from ⃗Bk−1 as follows: take two disjoint copies ⃗B1 k−1 and ⃗B2 k−1 of ⃗Bk−1 and one vertex r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' add all arcs from r to ⃗B1 k−1, all arcs from ⃗B1 k−1 to ⃗B2 k−1 and all arcs from ⃗B2 k−1 to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Claim 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='1: ⃗Bk is k-out-degenerate and ⃗χ( ⃗Bk) = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We prove the claim by induction on k, the result holding trivially for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Assume now that k ≥ 1, and consider ⃗Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By induction, there exist two (k − 1)-out-degeneracy orderings σ1 and σ2 on ⃗B1 k−1 and ⃗B2 k−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In ⃗Bk, each vertex of V ( ⃗B2 k−1) has exactly one out-neighbour in V \\ V ( ⃗B2 k−1), namely r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We may then remove the vertices of ⃗B2 k−1 following σ2, such that at each step the removed vertex has out-degree at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In the remaining graph, all out-neighbours of vertices in V ( ⃗B1 k−1) belong to V ( ⃗B1 k−1), allowing us to 11 successively remove vertices of out-degree k − 1 by following σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Combining these facts, σ2 · σ1 · (r) yields a k-out-degeneracy ordering of ⃗Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Hence ⃗Bk is k-out degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The graph ⃗Bk is indeed (k+1)-dicolourable since any dicolouring such that the restriction to ⃗B1 k−1 and ⃗B2 k−1 is a k-dicolouring and r is coloured with colour k+1 is a (k+1)-dicolouring of ⃗Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now, assume for a contradiction that ⃗Bk has a k-dicolouring α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Set c = α(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By induction, ⃗χ( ⃗Bk−1) = k, so there is a vertex x1 of ⃗B1 k−1 and a vertex x2 of ⃗B2 k−1 such that α(x1) = α(x2) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' But then (r, x1, x2, r) is a monochromatic directed cycle, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Thus ⃗χ( ⃗Bk) = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' ♦ Let k be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ⃗Gk be the oriented graph obtained from the transitive tournament T Tk+1 on k + 1 vertices, by adding, for each arc xy of T Tk+1, a copy ⃗Bxy k of ⃗Bk, all arcs from y to ⃗Bxy k and all arcs from ⃗Bxy k to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let us prove that ⃗Gk is k-out-degenerate and is not (k + 1)-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , tk+1) be the acyclic ordering of T Tk+1 (such that there is no arc titj with i > j), and let σi,j be a k-out-degeneracy ordering of ⃗Btitj k for all 1 ≤ i < j ≤ k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We build a k-out-degeneracy ordering of ⃗Gk by combining these orders as follows: (t1) · σ1,2 · σ1,3 · · · · · σ1,k+1 · (t2) · σ2,3 · σ2,4 · · · · · σ2,k+1 · (t3) · · · · · (tk) · σk,k+1 · (tk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Thus ⃗Gk is k-out-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We show that every (k + 1)-dicolouring α of ⃗Gk is such that every vertex v ∈ V (T Tk+1) is frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since any α′ defined from α by a permutation of the colours is also a (k + 1)-dicolouring, the above being true yields that α′ cannot be reached from α and implies that ⃗Gk is not (k + 1)-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Consider v ∈ V (T Tk+1), and assume there is a redicolouring sequence starting from α achieving to recolour v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We let β the dicolouring of ⃗Gk right before v is recoloured in the sequence, and β′ the dicolouring after recolouring v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' It suffices to note that β(v) ̸= β(w) for any w ∈ V (T Tk+1)\\v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Indeed, since ⃗χ(Bvw) = k + 1, there exists some z ∈ V (Bvw) coloured β(v), and by construction β(v) = β(w) would create a monochromatic directed cycle on vertices v, w, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, since β′ is obtained from β by recolouring v, there exists some w ∈ V (T Tk+1) coloured the same as v in β′, and the same argument yields a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='2 Diameter In this section, we prove Theorems 9, 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ⃗G be a subcubic oriented graph of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then D2(⃗G) is connected and has diameter at most 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let α and β be any two 2-dicolourings of ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let x = diff(α, β) = |{v ∈ V (⃗G) | α(v) ̸= β(v)}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By induction on x ≥ 0, let us show that there exists a path of length at most 2x from α to β in D2(⃗G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This clearly holds for x = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=', α = β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Assume x > 0 and the result holds for every x′ < x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let v ∈ V (⃗G) such that 1 = α(v) ̸= β(v) = 2 (v exists up to swapping the colours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If v can be directly recoloured with colour 2, then we recolour v with colour 2 and reach a new 2-dicolouring α′ such that diff(α′, β) = x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, the result holds by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Therefore, v cannot be directly recoloured with colour 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=', there exists a directed cycle C containing v such that α(w) = 2 for every w ∈ V (C) \\ {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Moreover, there must be a vertex u ∈ V (C) \\ {v} such that β(u) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If u is not a neighbour of v, then u has two neighbours (those in C) coloured with 2 in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Therefore, u can be recoloured 1 (it does not create any directed cycle of vertices coloured 1 since u has degree at most 3 in G) and we reach a new 2-dicolouring α′ such that diff(α′, β) = x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, the result holds by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Hence, u is a neighbour of v in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Again, we may assume that u cannot be directly recoloured 1 (otherwise the result holds by induction) and so there exists a directed cycle C′ such that α(w) = 1 for every w ∈ V (C′) \\ {u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since u and v have degree at most 3 in ⃗G, and because of the α-colours of vertices in C and C′, then V (C′) ∩ V (C) = {v, u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let h be the vertex of V (C′) \\ {u, v} minimizing its distance to u and v in C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Again, since ⃗G is subcubic, h cannot be in any directed cycle C′′ such that all vertices of C′′ but h are coloured 2 by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Hence, h can be recoloured with 2, then u can be recoloured 1, then v can be recoloured with 2 and h recoloured with 1, reaching a new 2-dicolouring α′ such that diff(α′, β) = x − 2, and the result holds by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 12 Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let D be a digraph on n vertices and k ≥ 2δ∗ min(D) + 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, the diameter of Dk(D) is at most (δ∗ min(D) + 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The proof is very similar to the proof of Bousquet and Perarnau [5] for undirected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let α and β be two k-dicolourings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let us show by induction on the number of vertices that there exists a redicolouring sequence from α to β where every vertex is recoloured at most δ∗ min(D) + 1 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If n = 1 the result is obviously true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let D be a digraph on n + 1 vertices, and let u be a vertex such that min{d+(u), d−(u)} ≤ δ∗ min(D) and let D′ = D − u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By directional duality, we may assume d+(u) ≤ δ∗ min(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We denote by α′ and β′ the dicolourings of D′ induced by α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By induction and since δ∗ min(D′) ≤ δ∗ min(D), there exists a redicolouring sequence from α′ to β′ such that each vertex is recoloured at most δ∗ min(D) + 1 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now we consider the same recolouring steps to recolour D, starting from α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If for some step i, it is not possible to recolour vi to ci, this must be because u is currently coloured ci and recolouring vi to ci would create a monochromatic directed cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since u has at most δ∗ min(D) out-neighbours, and since k ≥ 2δ∗ min(D) + 2, there are at least δ∗ min(D) + 2 colours that does not appear in the out-neighbourhood of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We choose c among these colours so that c does not appear in the next δ∗ min(D) + 1 recolourings of N +(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since u has at most δ∗ min(D) out-neighbours and since each vertex in D′ is recoloured at most δ∗ min(D) + 1 times, there are at most δ∗ min(D)(δ∗ min(D)+1) recolourings of an out-neighbourof u in this redicolouring sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Hence, in this new redicolouring sequence, u is recoloured at most δ∗ min(D) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We finally have to set u to its colour in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Doing so u is recoloured at most δ∗ min(D) + 1 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The goal of the rest of this part is to prove the following theorem which generalizes a result of Bousquet and Heinrich [4] to digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For a digraph D on n vertices, and k ≥ 3 2(δ∗ min(D) + 1), the diameter of Dk(D) is at most Cn2 where C is independent from k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Given an undirected graph G = (V, E), a list assignment L is a-feasible if, for some ordering v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn of V , |L(vi)| ≥ |N(v) ∩ {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn}| + 1 + a for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Bousquet and Heinrich proved the following result appearing as the first and second items of Theorem 6 in in [4]: Theorem 19 (Bousquet and Heinrich [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let G be a graph and a ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let L be an a-feasible list assignment and k be the total number of colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then C(G, L) has diameter at most: (i) kn if k ≤ 2a, (ii) Cn2 if k ≤ 3a (where C a constant independent of k, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For a graph G and a list assignment L of G, we say that an L-colouring c avoids a set of colours S if for every vertex v ∈ V (G), c(v) does not belong to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In order to prove Theorem 10, we deduce from Theorem 19 the following lemma: Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let G = (V, E) be an undirected graph on n vertices, k ∈ N be the total number of colours, L be a � k 3 � feasible list assignment and α an L-colouring of G that avoids a set S of � k 3 � colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then for any set of � k 3 � colours S′, there is an L-colouring β of G that avoids S′ and such that there is a recolouring sequence from α to β of length at most 4k+6 3 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The proof of Lemma 20 comes from the proof of Lemma 8 in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We give it for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let S′ be any set of � k 3 � colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We start with G coloured by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn) be a degeneracy ordering of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We consider each vertex from vn to v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For each vertex, if it possible we recolour it with a colour of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We denote by η the obtained L-colouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This is done in less than n steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Observe that, for each colour c ∈ S and each vertex vi of G, at least one of the following holds: η(vi) ∈ S, vi has a neighbour in {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn} coloured c, or 13 c /∈ L(vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let H be the subgraph of G induced by the vertices whose colour in η is not in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We define LH by LH(v) = L(v) \\ S for every v ∈ V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Using the previous observation, we get that for every vertex vi of H, v has at least |L(v) ∩ S| neighbours in {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn} \\ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This implies that LH is a � k 3 � feasible list assignment of H with a total number of colours bounded by 2k 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By Theorem 19 (i), the diameter of C(H, LH) is at most 2k 3 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that every recolouring of H starting from ηH (the colouring η induced on H) gives a valid recolouring of G starting from η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Consider the following preference ordering on the colours: an arbitrary ordering of [k] \\ (S ∪ S′), followed by an ordering of S′ \\ S, and finally the colours from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let γ be the L-colouring of G obtained by colouring G greedily from vn to v1 with this preference ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since L is � k 3 � feasible, and |S| = � k 3 � , no vertex is coloured with a colour in S in γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This implies that γH, the colouring γ induced on H, is an LH-colouring of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Thus there is a recolouring from ηH to γH of length at most 2k 3 n steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This gives a recolouring sequence in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We can then recolour the vertices of G − H to their target colour in γ in at most n steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This shows that, in G, there is a recolouring sequence from α to γ of length at most 2k+3 3 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now observe that, for each colour c ∈ R = [k] \\ (S ∪ S′) and each vertex vi of G, at least one of the following must hold: γ(vi) ∈ R, vi has a neighbour in {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn} coloured c, or c /∈ L(vi) Let Γ be the subgraph of G induced by all vertices coloured with a colour in S′ by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that Γ is also the subgraph induced by all vertices coloured with a colour in (S ∪ S′) by γ, because no vertex is coloured in S by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let LΓ be the list assignment defined by LΓ(v) = L(v) ∩ (S ∪ S′) for all v ∈ V (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By the previous observation, LΓ is � k 3 � feasible, and the total number of colours is |S ∪ S′| ≤ 2 � k 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Thus by Theorem 19 (i), C(Γ, LΓ) has diameter at most 2 � k 3 � n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let βΓ be an LΓ-colouring of Γ that avoids the colours of S′ (such a colouring exists because |S′| = � k 3 � and LΓ is � k 3 � feasible) and γΓ the colouring γ induced on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' There is a recolouring sequence of length at most 2 � k 3 � n from γΓ to βΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This extends to a recolouring sequence in G from γ to β where β does not use any colour of S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The total number of steps to reach β from α is at most 2k 3 n + n + 2 � k 3 � n which is bounded by 4k+6 3 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This shows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We are now able to prove Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let D = (V, A) be a digraph on n vertices and k ≥ 3 2(δ∗ min(D) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn) be a min-degeneracy-orderingof D, that is an ordering such that for each i ∈ [n], vi has at most δ∗ min(D) out-neighbours or at most δ∗ min(D) in-neighbours in {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We define B a subset of A as follows: for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , n}, if vi has at most δ∗ min(D) out-neighbours in {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn}, we add all arcs of A from vi to {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn} to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Otherwise, we add all arcs of A from {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn} to vi to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that both digraphs (V, B) and (V, A \\ B) are acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let H be (V, B) and G be the underlying graph of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that, since (V, A \\ B) is acyclic, each colouring of G is a dicolouring of D, but a dicolouring of D is not necessarily a colouring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By construction, G has degeneracy at most δ∗ min(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Using Theorem 3, we get that Ck(G) has diameter at most C0n2 for some constant C0 independent from k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Set δ∗ = δ∗ min(D) ≥ δ∗(G), Xi = {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn}, and Hi = G − Xi for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let α be any dicolouring of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let Li be the list assignment of Hi defined by Li(vj) = [k] \\ {α(v)|v ∈ NG(vj) ∩ Xi} for all j ∈ [i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 14 Since k, the total number of colours, is at least 3 2(δ∗ + 1), for every j ∈ [i] we have: |Li(vj)| ≥ k − |NG(vj) ∩ Xi| ≥ k 3 + 2 3 3 2(δ∗ + 1) − |NG(vj) ∩ Xi| ≥ |NG(vj) ∩ {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vi}| + 1 + k 3 Hence, since |Li(vj)| is an integer, Li is a � k 3 � feasible list assignment of Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let γ be a dicolouring of D and for some i, γ agrees with α on {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn} and γHi is an Li- colouring of Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then any recolouring sequence starting from γHi on Hi, valid for Li, is a valid redicolouring sequence in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Assume this is not the case and at one step, we get to an Li-colouring ζ of Hi but ζD contains a monochromatic directed cycle C, where ζD(v) = ζ(v) when v belongs to Hi and ζD(v) = γ(v) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let vj be the vertex of C such that j is minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This vertex vj has both an in-neighbour vj1 and an out-neighbour vj2 coloured ζD(vj) such that j1, j2 ≥ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We know that either vj1vj or vjvj2 belongs to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Assume by symmetry that vj1vj belongs to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then either j1 ≤ i and then vj1vj is a monochromatic edge in Hi or j1 ≥ i + 1 but then ζ(vj1) = α(vj1) does not belong to Li(vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In both cases, we get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='1: Let γi be a dicolouring of D which induces an Li-colouring of Hi avoiding at least � k 3 � colours in Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' There is a redicolouring sequence of length at most ck 3 n from γi to a dicolouring γi+⌈ k 3⌉ which induces an Li+⌈ k 3⌉-colouring of Hi+⌈ k 3⌉ avoiding at least � k 3 � colours in Hi+⌈ k 3⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let S be a set of colours of size exactly � k 3 � avoided by γi on Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For each vertex vj in {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vi+⌈ k 3⌉}, we choose a colour cj so that each of the following holds: cj belongs to Lj(vj), cj does not belong to {α(u)|u ∈ NG(vj) ∩ {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn}}, for each ℓ ∈ {i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , j − 1}, cℓ is different from cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that this is possible because Lj(vj) is � k 3 � feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now let S′ be {ci+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , ci+⌈ k 3 ⌉}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Observe that |S′| = � k 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By Lemma 20, there is, in Hi, a recolouring sequence of length at most 4k+6 3 n, valid for Li, from γi to some γ′ i that avoids S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This recolouring sequence extends to a redicolouring sequence in D by Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In the obtained dicolouring, since γ′ i avoids S′ on Hi, we can recolour successively vj with cj for all i+1 ≤ j ≤ i+ � k 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This does not create any monochromatic directed cycle by choice of cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ηi be the resulting dicolouring of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now we define ˜Li a list-assignment of Hi as follows: ˜Li(vj) = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=', k} \\ {ηi(v)|v ∈ N(vj) ∩ {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn})) Using the same arguments as we did for Li, we get that ˜Li is � k 3 � feasible for Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that ηi is an ˜Li- colouring of Hi that avoids S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let S′′ be a set of � k 3 � colours disjoint from S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By Lemma 20, there is, in Hi, a recolouring sequence (valid for ˜Li) of length at most 4k+6 3 n from ηi to some η′ i that avoids S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This recolouring sequence extends directly to a redicolouring sequence in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since S′ is disjoint from S′′, the obtained dicolouring is an Li+⌈ k 3 ⌉-colouring of Hi+⌈ k 3⌉ that avoids at least � k 3 � colours in Hi+⌈ k 3⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Hence we get a redicolouring sequence from γi to the desired γi+⌈ k 3⌉, in at most 8k+12 3 n + � k 3 � steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This proves Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' ♦ Note that γ⌈ k 3⌉ can be reached from α in less than n steps: for all j ∈ [ � k 3 � ], choose a colour cj so that each of the following holds: cj belongs to Lj(vj), cj does not belong to {α(u)|u ∈ NG(vj) ∩ {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn}}, 15 for each ℓ ∈ [j − 1], cℓ is different from cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now we can recolour successively v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , v⌈ k 3⌉ to their corresponding colour in {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , c⌈ k 3⌉}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then applying Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='1 iteratively at most � n ⌈ k 3⌉ � ≤ 3n k times, there is a redicolouring sequence of length at most n + 3n k � 8k+6 3 n + k 3 � from α to a dicolouring of D that is also a colouring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Note that there exists a constant C1 such that n + 3n k � 8k+6 3 n + k 3 � ≤ C1n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let α and β be two k-dicolourings of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' As proved above, there is a redicolouring sequence of length at most C1n2 from α (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' β) to a dicolouring α′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' β′) of D that is also a colouring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since Ck(G) has diameter at most C0n2, there is a recolouring sequence of G of length at most C0n2 from α′ to β′, which is also a redicolouring sequence of D (since every colouring of G is a dicolouring of D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The union of those three sequences yields a redicolouring sequence from α to β of length at most (2C1 + C0)n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 4 Density of non 2-mixing oriented graphs 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='1 Density of non k-mixing undirected graphs Let k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' As observed in the introduction, every non k-mixing graph as maximum average degree at least k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This bound is tight because the complete graph on k vertices is (k − 1)-regular and is not k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Moreover, it is shown in [2] that this bound is tight even when we restrict to graphs of arbitrary large girth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' The initial proof uses the probabilistic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In this section, we give a new constructive proof of this result, based on an explicit construction of regular bipartite graphs from Lazebnik and Ustimenko in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 22 (Bonamy, Bousquet and Perarnau, [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For any k, ℓ ∈ N∗, there exists a (k − 1)-regular k-freezable graph Gk,ℓ with girth at least ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We first make the following remark that we will use in the proof of Theorem 22: Remark 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let k ∈ N∗, G be a (k − 1)-regular k-freezable graph and c be a frozen k-colouring of G, then all colour classes of c have the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This follows from the fact that, for every vertex v of G, N[v] use all colours in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Thus, given two colours i, j of c, there must be a perfect matching in G between the vertices coloured i and the vertices coloured j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In particular, this implies that the number of vertices coloured i is the same as the number of vertices coloured j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof of Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let us fix ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We prove the statement by induction on k, the result holding trivially for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let k > 1 and assume that there exists a (k − 2)-regular (k − 1)-freezable graph Gk−1,ℓ with girth at least ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let c be a frozen k-colouring of Gk−1,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We denote by n the number of vertices of Gk−1,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Consider H a n-regular bipartite graph with girth at least ℓ (such a graph exists by a construction from Lazebnik and Ustimenko [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since H is bipartite, we can colour the edges of H with exactly n colours such that two adjacent edges receive different colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' By Remark 23, all colour classes of c have the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Thus there is an ordering (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn) of V (Gk−1,ℓ) such that for each i ∈ [n − k + 1], the vertices vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vi+k−1 have different colours by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We denote by (A, B) the bipartition of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let Gk,ℓ be the graph obtained from H as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For each a ∈ A, replace a by a copy Ga of Gk−1,ℓ, and connect va i (the vertex corresponding to vi in Ga) to the edge coloured i that was incident to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For each b ∈ B, replace b by an independent set Ib = {xb 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , xbn k } of size n k (by Remark 23, n k is an integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Connect xb i to the edges coloured {k(i − 1) + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , ki} that were incident to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Observe that Gk,ℓ is k-regular: every vertex in a Ga is adjacent to its k − 1 neighbours in Ga and exactly one neighbour in one of the Ib;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' every vertex in an Ib has exactly k neighbours by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Moreover, Gk,ℓ has girth at least ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Indeed, assume, for a contradiction, that it contains a cycle C of length at most ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then C 16 cannot contain an edge of H, otherwise, contracting each copy of Ga would transform C into a cycle of length at most ℓ − 1 in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Thus C must be contained in some Ga, which is a copy of Gk−1,ℓ, which is impossible since Gk−1,ℓ has girth at least ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let α be the (k + 1)-colouring of Gk,ℓ such that the restriction of α to each Ga corresponds to c, and α(xb i) = k + 1 for all b ∈ B and i ∈ [n/k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let v be a vertex of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If v belongs to some Ga, then since c is frozen in Gk−1,ℓ, NGa[v] contains all colours of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Moreover, by construction, v has a neighbour in some Ib which is coloured k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If v is in some Ib, then v is coloured k + 1 and by construction it has exactly one neighbour in each colour class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In both cases, N[v] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Thus no vertex can be recoloured and so α is a frozen colouring of Gk,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='2 Density of non 2-mixing and 2-freezable oriented graphs Let k ∈ N and D be a digraph that is not k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' As observed in the introduction, Theorem 6 implies that Mad(D) ≥ 2k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' This bound is tight because the bidirected complete digraph on k vertices is (2k − 2)-regular and is not k-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' However, unlike the undirected case, this result does not extend to digraphs with larger digirth, even for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' While the above inequality states that every non 2-mixing digraph has maximum average degree at least 2, we conjecture that every non 2-mixing oriented graph has maximum average degree at least 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Conjecture 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Any non 2-mixing oriented graph has maximum average degree at least 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Remark 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If true, this conjecture would be tight since there exist 2-freezable oriented graphs with maximum average degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Consider for example the oriented graph ⃗Fn obtained from the disjoint union of two disjoints directed paths (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , un) and (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , vn) by adding the set of arcs {uivi | i ∈ [n]} ∪ {vi+1ui | i ∈ [n − 1]} ∪ {v1u2, unv1, vn−1un} (see Figure 5 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let α be the 2-dicolouring of ⃗Fn in which all the ui are coloured 1 and all the vi are coloured 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' One can easily check that Mad(⃗Fn) = 4 and α is a 2-frozen dicolouring of ⃗Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Figure 5: The 2-freezable oriented graph ⃗Fn and a frozen 2-dicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We now prove two results supporting Conjecture 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' First we prove that Conjecture 24 holds with the stronger assumption that G is 2-freezable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ⃗G = (V, A) be an oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If ⃗G is 2-freezable, then |A| ≥ 2|V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ⃗G = (V, A) be a 2-freezable oriented graph, and c a frozen 2-dicolouring of ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For a vertex v ∈ V , we say that a vertex u ∈ V is blocking for v (in dicolouring c), if one of the following holds: u is an out-neighbour of v, c(u) ̸= c(v), and there exists a directed path (u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=', x, v) such that (u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=', x) is monochromatic, or u is an in-neighbour of v, c(u) ̸= c(v), and there exists a directed path (x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=', u, v) such that (x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=', u) is monochromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We shall use a discharging argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We set the initial charge of every vertex v to be d(v) = d+(v) + d−(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Observe that d(v) ≥ 2 otherwise v can be recoloured in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We then use the following discharging rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' (R) every vertex receives 1 from each of its blocking neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let f(v) be the final charge of every vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let us show that f(v) ≥ 4 for every v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let α be its colour and β the other colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since c is frozen, v admits at least one out-neighbour v+ and one in-neighbour v− coloured β that are blocking, and thus sending 1 to v by (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let us now examine the 17 charge that v sends to others vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let w be a vertex to which v sends charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' v is blocking for w, so c(w) = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Moreover if w is an out-neighbour (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' in-neighbour) of v, then v has an in-neighbour (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' out-neighbour) coloured α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We are in one of the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If v sends no charge, then f(v) ≥ d(v) + 2 ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If v sends charge only to some out-neighbours, then it does not send to its in-neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since v has at least two in-neighbours (one blocking v and one coloured α), f(v) ≥ d(v) + 2 − (d(v) − 2) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If v sends charge only to some in-neighbours, symmetrically to above, f(v) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If v sends charge only to both out-neighbours and in-neighbours, then its has both an in-neighbour and an out-neighbour coloured α to which it sends no charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Hence f(v) ≥ d(v) + 2 − (d(v) − 2) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' In all cases, we have f(v) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Consequently, 2|A| = � v∈V d(v) = � v∈V f(v) ≥ 4|V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We can deduce from Theorem 14 the following lower bound on the density of a k-freezable oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Corollary 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let ⃗G = (V, A) be an oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If ⃗G is k-freezable, then |A| ≥ k|V | + k(k − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Suppose for a contradiction that there is a k-freezable oriented graph ⃗G = (V, A) such that |A| < k|V | + k(k − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Without loss of generality, we may take ⃗G having a minimum number of arcs among all such graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let c be a frozen k-dicolouring of ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For each i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=', k}, let ⃗Gi be the subdigraph of ⃗G induced by the vertices coloured i in c, and let ⃗Gi,j be the subdigraph of ⃗G induced by the vertices coloured i or j in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We set ni = |V (⃗Gi)|, mi = |A(⃗Gi)| and mi,j = |A(⃗Gi,j)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We first show that, for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , k}, mi ≤ ni − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Then, since ⃗Gi is acyclic, it admits an acyclic ordering (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , xni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now consider ⃗G′ = (⃗G \\ A(⃗Gi)) ∪ {xjxj+1 | j ∈ [ni − 1]} with the same colouring c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Clearly |A(⃗G′)| < |A(⃗G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Let v be a vertex of ⃗G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' If v ∈ V (⃗Gi), then x is still blocked in (⃗G′, c) because it is blocked in (⃗G, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now, suppose v /∈ V (⃗Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' For any colour j distinct from i and c(v), it is impossible to recolour v with j because it was already impossible in ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Now, in ⃗G, it was impossible to recolour v to i, so there is a directed path in ⃗Gi whose initial vertex xk is an out-neighbour of v and whose terminal vertex xℓ is an in-neighbour of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , xni) is an acyclic ordering of ⃗Gi, we have k ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Thus (xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' , xℓ) is a directed path in ⃗G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Hence v cannot be recoloured to i in (⃗G′, c), meaning it is also blocked in (⃗G′, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' Since all vertices of (⃗G′, c) are blocked, c is a frozen k-dicolouring of ⃗G′, contradicting the minimality of ⃗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfxAV3/content/2301.03417v1.pdf'} +page_content=' We will now prove the result by bounding S = � 1≤i