This paper attributes the sudden emergence of mathematical probability and statistics in the second half of the seventeenth century to Calvin's Reformed theology. Calvin accommodated Epicurean chance with Stoic determinism and synthesised emph{phronesis/prudentia}, founded personal experience and employed to deal with emph{tyche/fortuna}, and emph{episteme/scientia}, universal knowledge. This meant that matters of chance, which had previously been considered too particular for mathematical treatment, became part of emph{episteme/scientia}. Clear evidence of the significance of Calvin in mathematics is in the facts that Huygens considered using the word 'hope' to describe mathematical expectation and French mathematics still uses emph{esp'erance} for mathematical expectation. Calvin asserted that Hope represented a universal, objective and indubitable idea making it characteristic of mathematics. The argument is built on a review of how the ideas of Hope, Faith and Prudence have evolved in European thought that highlights Calvin's innovations. The conclusion identifies contemporary issues in the application of mathematics in society that are illuminated in light of Calvin's doctrine.
{"title":"Faith Believes, Hope Expects: The Impact of Calvin's Theology on the Mathematics of Chance","authors":"Timothy C. Johnson","doi":"arxiv-2407.13312","DOIUrl":"https://doi.org/arxiv-2407.13312","url":null,"abstract":"This paper attributes the sudden emergence of mathematical probability and\u0000statistics in the second half of the seventeenth century to Calvin's Reformed\u0000theology. Calvin accommodated Epicurean chance with Stoic determinism and\u0000synthesised emph{phronesis/prudentia}, founded personal experience and\u0000employed to deal with emph{tyche/fortuna}, and emph{episteme/scientia},\u0000universal knowledge. This meant that matters of chance, which had previously\u0000been considered too particular for mathematical treatment, became part of\u0000emph{episteme/scientia}. Clear evidence of the significance of Calvin in\u0000mathematics is in the facts that Huygens considered using the word 'hope' to\u0000describe mathematical expectation and French mathematics still uses\u0000emph{esp'erance} for mathematical expectation. Calvin asserted that Hope\u0000represented a universal, objective and indubitable idea making it\u0000characteristic of mathematics. The argument is built on a review of how the\u0000ideas of Hope, Faith and Prudence have evolved in European thought that\u0000highlights Calvin's innovations. The conclusion identifies contemporary issues\u0000in the application of mathematics in society that are illuminated in light of\u0000Calvin's doctrine.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Much has been written about the golden ratio $phi=frac{1+sqrt{5}}{2}$ and this strange number appears mysteriously in many mathematical calculations. In this article, we review the appearance of this number in the graph theory. More precisely, we review the relevance of this number in topics such as the number of spanning trees, topological indices, energy, chromatic roots, domination roots and the number of domatic partitions of graphs.
{"title":"Golden ratio in graph theory: A survey","authors":"Saeid Alikhani, Nima Ghanbari","doi":"arxiv-2407.15860","DOIUrl":"https://doi.org/arxiv-2407.15860","url":null,"abstract":"Much has been written about the golden ratio $phi=frac{1+sqrt{5}}{2}$ and\u0000this strange number appears mysteriously in many mathematical calculations. In\u0000this article, we review the appearance of this number in the graph theory. More\u0000precisely, we review the relevance of this number in topics such as the number\u0000of spanning trees, topological indices, energy, chromatic roots, domination\u0000roots and the number of domatic partitions of graphs.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141774626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The monography examines the problem of constructing a group of automorphisms of a graph. A graph automorphism is a mapping of a set of vertices onto itself that preserves adjacency. The set of such automorphisms forms a vertex group of a graph or simply a graph group. The basis for constructing a group of graph automorphisms is the concept of orbit. The construction of an orbit is closely related to the quantitative assessment of a vertex or edge of a graph, called weight. To determine the weight of an element, graph invariants built on the spectrum of edge cuts and the spectrum of edge cycles are used. The weight of the graph elements allows identifying generating cycles and forming orbits. Examples are given of constructing a group of automorphisms for some types of graphs.
{"title":"Algorithmic methods of finite discrete structures. Automorphism of Nonseparable Graphs","authors":"Sergey Kurapov, Maxim Davidovsky","doi":"arxiv-2407.12045","DOIUrl":"https://doi.org/arxiv-2407.12045","url":null,"abstract":"The monography examines the problem of constructing a group of automorphisms\u0000of a graph. A graph automorphism is a mapping of a set of vertices onto itself\u0000that preserves adjacency. The set of such automorphisms forms a vertex group of\u0000a graph or simply a graph group. The basis for constructing a group of graph\u0000automorphisms is the concept of orbit. The construction of an orbit is closely\u0000related to the quantitative assessment of a vertex or edge of a graph, called\u0000weight. To determine the weight of an element, graph invariants built on the\u0000spectrum of edge cuts and the spectrum of edge cycles are used. The weight of\u0000the graph elements allows identifying generating cycles and forming orbits.\u0000Examples are given of constructing a group of automorphisms for some types of\u0000graphs.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1953, Enrico Fermi criticized Dyson's model by quoting Johnny von Neumann: "With four parameters I can fit an elephant, and with five I can make him wiggle his trunk." So far, there have been several attempts to fit an elephant using four parameters, but as the problem has not been well-defined, the current methods do not completely satisfy the requirements. This paper defines the problem and presents an attempt.
1953 年，恩里科-费米（Enrico Fermi）引用约翰尼-冯-诺依曼（Johnny von Neumann）的话批评了戴森的模型："用四个参数我就能拟合一头大象，用五个参数我就能让它扭动躯干"。迄今为止，人们已经多次尝试用四个参数来拟合大象，但由于问题没有得到很好的定义，目前的方法并不能完全满足要求。本文对这一问题进行了定义，并提出了一种尝试。
{"title":"Fitting an Elephant with Four non-Zero Parameters","authors":"Dian Jin, Junze Yuan","doi":"arxiv-2407.07909","DOIUrl":"https://doi.org/arxiv-2407.07909","url":null,"abstract":"In 1953, Enrico Fermi criticized Dyson's model by quoting Johnny von Neumann:\u0000\"With four parameters I can fit an elephant, and with five I can make him\u0000wiggle his trunk.\" So far, there have been several attempts to fit an elephant\u0000using four parameters, but as the problem has not been well-defined, the\u0000current methods do not completely satisfy the requirements. This paper defines\u0000the problem and presents an attempt.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"2011 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bryan Kaiser, Tailin Wu, Maike Sonnewald, Colin Thackray, Skylar Callis
Nobel laureate Philip Anderson and Elihu Abrahams once stated that, "even if machines did contribute to normal science, we see no mechanism by which they could create a Kuhnian revolution and thereby establish a new physical law." In this Perspective, we draw upon insights from the philosophies of science and artificial intelligence (AI) to propose necessary conditions of precisely such a mechanism for generating revolutionary mathematical theories. Recent advancements in AI suggest that satisfying the proposed necessary conditions by machines may be plausible; thus, our proposed necessary conditions also define a moonshot challenge. We also propose a heuristic definition of the intelligibility of mathematical theories to accelerate the development of machine theorists.
{"title":"A Moonshot for AI Oracles in the Sciences","authors":"Bryan Kaiser, Tailin Wu, Maike Sonnewald, Colin Thackray, Skylar Callis","doi":"arxiv-2406.17836","DOIUrl":"https://doi.org/arxiv-2406.17836","url":null,"abstract":"Nobel laureate Philip Anderson and Elihu Abrahams once stated that, \"even if\u0000machines did contribute to normal science, we see no mechanism by which they\u0000could create a Kuhnian revolution and thereby establish a new physical law.\" In\u0000this Perspective, we draw upon insights from the philosophies of science and\u0000artificial intelligence (AI) to propose necessary conditions of precisely such\u0000a mechanism for generating revolutionary mathematical theories. Recent\u0000advancements in AI suggest that satisfying the proposed necessary conditions by\u0000machines may be plausible; thus, our proposed necessary conditions also define\u0000a moonshot challenge. We also propose a heuristic definition of the\u0000intelligibility of mathematical theories to accelerate the development of\u0000machine theorists.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Topology at the undergraduate level is often a theoretical mathematics course, introducing concepts from point-set topology or possibly algebraic topology. However, the last two decades have seen an explosion of growth in applied topology and topological data analysis, which are topics that can be presented in an accessible way to undergraduate students and can encourage exciting projects. For the past several years, the Topology course at Macalester College has included content from point-set and algebraic topology, as well as applied topology, culminating in a project chosen by the students. In the course, students work through a topology scavenger hunt as an activity to introduce the ideas and software behind some of the primary tools in topological data analysis, namely, persistent homology and mapper. This scavenger hunt includes a variety of point clouds of varying dimensions, such as an annulus in 2D, a bouquet of loops in 3D, a sphere in 4D, and a torus in 400D. The students' goal is to analyze each point cloud with a variety of software to infer the topological structure. After completing this activity, students are able to extend the ideas learned in the scavenger hunt to an open-ended capstone project. Examples of past projects include: using persistence to explore the relationship between country development and geography, to analyze congressional voting patterns, and to classify genres of a large corpus of texts by combining with tools from natural language processing and machine learning.
{"title":"A Topology Scavenger Hunt to Introduce Topological Data Analysis","authors":"Lori Ziegelmeier","doi":"arxiv-2406.15580","DOIUrl":"https://doi.org/arxiv-2406.15580","url":null,"abstract":"Topology at the undergraduate level is often a theoretical mathematics\u0000course, introducing concepts from point-set topology or possibly algebraic\u0000topology. However, the last two decades have seen an explosion of growth in\u0000applied topology and topological data analysis, which are topics that can be\u0000presented in an accessible way to undergraduate students and can encourage\u0000exciting projects. For the past several years, the Topology course at\u0000Macalester College has included content from point-set and algebraic topology,\u0000as well as applied topology, culminating in a project chosen by the students.\u0000In the course, students work through a topology scavenger hunt as an activity\u0000to introduce the ideas and software behind some of the primary tools in\u0000topological data analysis, namely, persistent homology and mapper. This\u0000scavenger hunt includes a variety of point clouds of varying dimensions, such\u0000as an annulus in 2D, a bouquet of loops in 3D, a sphere in 4D, and a torus in\u0000400D. The students' goal is to analyze each point cloud with a variety of\u0000software to infer the topological structure. After completing this activity,\u0000students are able to extend the ideas learned in the scavenger hunt to an\u0000open-ended capstone project. Examples of past projects include: using\u0000persistence to explore the relationship between country development and\u0000geography, to analyze congressional voting patterns, and to classify genres of\u0000a large corpus of texts by combining with tools from natural language\u0000processing and machine learning.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"96 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jacob Collard, Valeria de Paiva, Eswaran Subrahmanian
Mathematics is a highly specialized domain with its own unique set of challenges. Despite this, there has been relatively little research on natural language processing for mathematical texts, and there are few mathematical language resources aimed at NLP. In this paper, we aim to provide annotated corpora that can be used to study the language of mathematics in different contexts, ranging from fundamental concepts found in textbooks to advanced research mathematics. We preprocess the corpora with a neural parsing model and some manual intervention to provide part-of-speech tags, lemmas, and dependency trees. In total, we provide 182397 sentences across three corpora. We then aim to test and evaluate several noteworthy natural language processing models using these corpora, to show how well they can adapt to the domain of mathematics and provide useful tools for exploring mathematical language. We evaluate several neural and symbolic models against benchmarks that we extract from the corpus metadata to show that terminology extraction and definition extraction do not easily generalize to mathematics, and that additional work is needed to achieve good performance on these metrics. Finally, we provide a learning assistant that grants access to the content of these corpora in a context-sensitive manner, utilizing text search and entity linking. Though our corpora and benchmarks provide useful metrics for evaluating mathematical language processing, further work is necessary to adapt models to mathematics in order to provide more effective learning assistants and apply NLP methods to different mathematical domains.
{"title":"Mathematical Entities: Corpora and Benchmarks","authors":"Jacob Collard, Valeria de Paiva, Eswaran Subrahmanian","doi":"arxiv-2406.11577","DOIUrl":"https://doi.org/arxiv-2406.11577","url":null,"abstract":"Mathematics is a highly specialized domain with its own unique set of\u0000challenges. Despite this, there has been relatively little research on natural\u0000language processing for mathematical texts, and there are few mathematical\u0000language resources aimed at NLP. In this paper, we aim to provide annotated\u0000corpora that can be used to study the language of mathematics in different\u0000contexts, ranging from fundamental concepts found in textbooks to advanced\u0000research mathematics. We preprocess the corpora with a neural parsing model and\u0000some manual intervention to provide part-of-speech tags, lemmas, and dependency\u0000trees. In total, we provide 182397 sentences across three corpora. We then aim\u0000to test and evaluate several noteworthy natural language processing models\u0000using these corpora, to show how well they can adapt to the domain of\u0000mathematics and provide useful tools for exploring mathematical language. We\u0000evaluate several neural and symbolic models against benchmarks that we extract\u0000from the corpus metadata to show that terminology extraction and definition\u0000extraction do not easily generalize to mathematics, and that additional work is\u0000needed to achieve good performance on these metrics. Finally, we provide a\u0000learning assistant that grants access to the content of these corpora in a\u0000context-sensitive manner, utilizing text search and entity linking. Though our\u0000corpora and benchmarks provide useful metrics for evaluating mathematical\u0000language processing, further work is necessary to adapt models to mathematics\u0000in order to provide more effective learning assistants and apply NLP methods to\u0000different mathematical domains.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"139 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Despite the extensive amount of scholarly work done on Indian mathematics in the last 200 years, the conditions under which it originated and evolved is still not clear. Often, one reads the ancient texts with the present concepts and methods in mind. The fact of absence of script over a long stretch of Indian history in ancient times also gets overlooked in such readings. The purpose of this article is to explore the journey of mathematics by examining what the ancient texts tell us about the nature of mathematics in their times. What one finds from the investigation of arithmetic, geometry and algebra is that while it was concrete and context bound, rooted in solving practical problems in ancient times, Indian mathematics transitioned to context free, abstract stage with the advent of algebra supported by writing.
{"title":"From Concrete to Abstract in Indian Mathematics","authors":"Jaidev Dasgupta","doi":"arxiv-2406.10147","DOIUrl":"https://doi.org/arxiv-2406.10147","url":null,"abstract":"Despite the extensive amount of scholarly work done on Indian mathematics in\u0000the last 200 years, the conditions under which it originated and evolved is\u0000still not clear. Often, one reads the ancient texts with the present concepts\u0000and methods in mind. The fact of absence of script over a long stretch of\u0000Indian history in ancient times also gets overlooked in such readings. The\u0000purpose of this article is to explore the journey of mathematics by examining\u0000what the ancient texts tell us about the nature of mathematics in their times.\u0000What one finds from the investigation of arithmetic, geometry and algebra is\u0000that while it was concrete and context bound, rooted in solving practical\u0000problems in ancient times, Indian mathematics transitioned to context free,\u0000abstract stage with the advent of algebra supported by writing.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"173 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is a colloquium talk in CAU, Kiel delivered on June 7, 2024 on the occasion of Walter Bergweiler's retirement. Walter's work on meromorphic functions consists of two parts: generalizations of Picard's theorem to differential polynomials, and the applications of the rescaling principle known as the Bloch Principle. Since the talk was aimed at the general audience, a brief introduction to Nevanlinna theory is included.
{"title":"The work of Walter Bergweiler in value distribution of meromorphic functions","authors":"Alexandre Eremenko","doi":"arxiv-2406.09992","DOIUrl":"https://doi.org/arxiv-2406.09992","url":null,"abstract":"This is a colloquium talk in CAU, Kiel delivered on June 7, 2024 on the\u0000occasion of Walter Bergweiler's retirement. Walter's work on meromorphic\u0000functions consists of two parts: generalizations of Picard's theorem to\u0000differential polynomials, and the applications of the rescaling principle known\u0000as the Bloch Principle. Since the talk was aimed at the general audience, a\u0000brief introduction to Nevanlinna theory is included.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gottlob Frege ingeniously presented a purely logical definition of the concept of number. However, one can claim that his definition is, in some way, circular, as it relies on the concept of one-to-one relation. The concept of number only makes sense when it presents the property of projection/reflection or binding. When we consider a number as an abstraction of objects, whatever they may be, saying that a number that belongs to the concept F is the same as that which belongs to the concept G means there is a projection/reflection, or binding, between the objects in F and the objects in G. We present a definition based on both equivalent approaches. First, we introduce the definition based on the relations of projection and reflection; then, we present the definition based on the relation of binding.
戈特洛布-弗雷格巧妙地提出了一个关于数概念的纯逻辑定义。然而，我们可以说他的定义在某种程度上是循环论证，因为它依赖于一对一关系的概念。只有当 "数 "的概念具有 "投射/反射 "或 "约束 "的特性时，它才是有意义的。当我们把数字视为对象（无论它们是什么）的抽象时，说属于概念 F 的数字与属于概念 G 的数字相同，就意味着 F 中的对象与 G 中的对象之间存在着投射/反射或绑定。首先，我们介绍基于投影和反射关系的定义；然后，我们介绍基于绑定关系的定义。
{"title":"A non-circular concept of number inspired by Gottlob Frege's definition","authors":"Marco Aurélio Spohn","doi":"arxiv-2406.08715","DOIUrl":"https://doi.org/arxiv-2406.08715","url":null,"abstract":"Gottlob Frege ingeniously presented a purely logical definition of the\u0000concept of number. However, one can claim that his definition is, in some way,\u0000circular, as it relies on the concept of one-to-one relation. The concept of\u0000number only makes sense when it presents the property of projection/reflection\u0000or binding. When we consider a number as an abstraction of objects, whatever\u0000they may be, saying that a number that belongs to the concept F is the same as\u0000that which belongs to the concept G means there is a projection/reflection, or\u0000binding, between the objects in F and the objects in G. We present a definition\u0000based on both equivalent approaches. First, we introduce the definition based\u0000on the relations of projection and reflection; then, we present the definition\u0000based on the relation of binding.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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