Let $mathcal{F}(M)$ be the Lipschitz-free space of a pointed metric space $M$. For every isometric continuous group action of $G$ we have an induced continuous dual action on the weak-star compact unit ball $B_{mathcal{F}(M)^*}$ of the dual space $mathrm{Lip_0} (M)=mathcal{F}(M)^*$. We pose the question when a given abstract continuous action of $G$ on a topological space $X$ can be represented through a $G$-subspace of $B_{mathcal{F}(M)^*}$. One of such natural examples is the so-called metric compactification (of isometric $G$-spaces) for a pointed metric space. As well as the Gromov $G$-compactification of a bounded metric $G$-space. Note that there are sufficiently many representations of compact $G$-spaces on Lipschitz-free spaces.
{"title":"A note about dual representations of group actions on Lipschitz-free spaces","authors":"Michael Megrelishvili","doi":"arxiv-2408.15208","DOIUrl":"https://doi.org/arxiv-2408.15208","url":null,"abstract":"Let $mathcal{F}(M)$ be the Lipschitz-free space of a pointed metric space\u0000$M$. For every isometric continuous group action of $G$ we have an induced\u0000continuous dual action on the weak-star compact unit ball\u0000$B_{mathcal{F}(M)^*}$ of the dual space $mathrm{Lip_0} (M)=mathcal{F}(M)^*$.\u0000We pose the question when a given abstract continuous action of $G$ on a\u0000topological space $X$ can be represented through a $G$-subspace of\u0000$B_{mathcal{F}(M)^*}$. One of such natural examples is the so-called metric\u0000compactification (of isometric $G$-spaces) for a pointed metric space. As well\u0000as the Gromov $G$-compactification of a bounded metric $G$-space. Note that\u0000there are sufficiently many representations of compact $G$-spaces on\u0000Lipschitz-free spaces.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181066","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 recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset with its spectrum, which is a compact T_1 topological space. In this paper, we focus on the case where such finite sets have a graph structure and the relations belong to a given graph category. We relate topological properties of the spectrum to combinatorial properties of the graph categories involved. We then utilise this to exhibit elementary combinatorial constructions of well-known continua as Fra"iss'e limits of finite graphs in categories with relational morphisms.
{"title":"Generic Compacta from Relations between Finite Graphs: Theory Building and Examples","authors":"Adam Bartoš, Tristan Bice, Alessandro Vignati","doi":"arxiv-2408.15228","DOIUrl":"https://doi.org/arxiv-2408.15228","url":null,"abstract":"In recent work, the authors developed a simple method of constructing\u0000topological spaces from certain well-behaved partially ordered sets -- those\u0000coming from sequences of relations between finite sets. This method associates\u0000a given poset with its spectrum, which is a compact T_1 topological space. In this paper, we focus on the case where such finite sets have a graph\u0000structure and the relations belong to a given graph category. We relate\u0000topological properties of the spectrum to combinatorial properties of the graph\u0000categories involved. We then utilise this to exhibit elementary combinatorial\u0000constructions of well-known continua as Fra\"iss'e limits of finite graphs in\u0000categories with relational morphisms.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181050","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}
With a complete residuated lattice $L$ as the truth value table, we extend the definition of sobriety of classical convex spaces to the framework of $L$-convex spaces. We provide a specific construction for the sobrification of an $L$-convex space, demonstrating that the full subcategory of sober $L$-convex spaces is reflective in the category of $L$-convex spaces with convexity-preserving mappings. Additionally, we introduce the concept of Scott $L$-convex structures on $L$-ordered sets. As an application of this type of sobriety, we obtain a characterization for the $L$-join-semilattice completion of an $L$-ordered set: an $L$-ordered set $Q$ is an $L$-join-semilattice completion of an $L$-ordered set $P$ if and only if the Scott $L$-convex space $(Q, sigma^{ast}(Q))$ is a sobrification of the Scott $L$-convex space $(P, sigma^{ast}(P))$.
{"title":"Sober $L$-convex spaces and $L$-join-semilattices","authors":"Guojun Wu, Wei Yao","doi":"arxiv-2408.08520","DOIUrl":"https://doi.org/arxiv-2408.08520","url":null,"abstract":"With a complete residuated lattice $L$ as the truth value table, we extend\u0000the definition of sobriety of classical convex spaces to the framework of\u0000$L$-convex spaces. We provide a specific construction for the sobrification of\u0000an $L$-convex space, demonstrating that the full subcategory of sober\u0000$L$-convex spaces is reflective in the category of $L$-convex spaces with\u0000convexity-preserving mappings. Additionally, we introduce the concept of Scott\u0000$L$-convex structures on $L$-ordered sets. As an application of this type of\u0000sobriety, we obtain a characterization for the $L$-join-semilattice completion\u0000of an $L$-ordered set: an $L$-ordered set $Q$ is an $L$-join-semilattice\u0000completion of an $L$-ordered set $P$ if and only if the Scott $L$-convex space\u0000$(Q, sigma^{ast}(Q))$ is a sobrification of the Scott $L$-convex space $(P,\u0000sigma^{ast}(P))$.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181053","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}
We constructed two dcpo's whose Scott spaces are sober, but the Scott space of their order product is not sober. This answers an open problem on the sobriety of Scott spaces. Meantime, we show that if $M$ and $N$ are special type of sober complete lattices, then the Scott space of their order product $Mtimes N$ is sober.
{"title":"Products of two sober dcpo's need not be sober","authors":"Hualin Miao, Xiaoyong Xi, Xiaodong Jia, Qingguo Li, Dongsheng Zhao","doi":"arxiv-2408.08587","DOIUrl":"https://doi.org/arxiv-2408.08587","url":null,"abstract":"We constructed two dcpo's whose Scott spaces are sober, but the Scott space\u0000of their order product is not sober. This answers an open problem on the\u0000sobriety of Scott spaces. Meantime, we show that if $M$ and $N$ are special\u0000type of sober complete lattices, then the Scott space of their order product\u0000$Mtimes N$ is sober.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181052","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}
Given two finite abstract simplicial complexes A and B, one can define a new simplicial complex on the set of simplicial maps from A to B. After adding two technicalities, we call this complex Homsc(A, B). We prove the following dichotomy: For a fixed finite abstract simplicial complex B, either Homsc(A, B) is always a disjoint union of contractible spaces or every finite CW-complex can be obtained up to a homotopy equivalence as Homsc(A, B) by choosing A in a right way. We furthermore show that the first case is equivalent to the existence of a nontrivial social choice function and that in this case, the space itself is homotopy equivalent to a discrete set. Secondly, we give a generalization to finite relational structures and show that this dichotomy coincides with a complexity theoretic dichotomy for constraint satisfaction problems, namely in the first case, the problem is in P and in the second case NP-complete. This generalizes a result from [SW24] respectively arXiv:2307.03446 [cs.CC]
给定两个有限抽象单纯复数 A 和 B，我们可以在从 A 到 B 的单纯映射集合上定义一个新闻单纯复数。我们将证明以下二分法：对于一个固定的有限抽象单纯复数 B，要么 Homsc(A, B) 总是可收缩空间的不相交联合，要么每个有限 CW 复数都可以通过选择 A 的正确方法得到一个同调等价的 Homsc(A,B)。我们还进一步证明，第一种情况等同于存在一个非琐碎的社会选择函数，在这种情况下，空间本身等同于一个离散集合。其次，我们给出了对有限关系结构的推广，并证明这种二分法与复杂性理论中对约束满足问题的二分法是一致的，即在第一种情况下，问题是在潘德（Pand）中完成的，而在第二种情况下，问题是 NP-完成的。这概括了[SW24]分别来自 arXiv:2307.03446 [cs.CC] 的一个结果。
{"title":"A Dichotomy for Finite Abstract Simplicial Complexes","authors":"Sebastian Meyer","doi":"arxiv-2408.08199","DOIUrl":"https://doi.org/arxiv-2408.08199","url":null,"abstract":"Given two finite abstract simplicial complexes A and B, one can define a new\u0000simplicial complex on the set of simplicial maps from A to B. After adding two\u0000technicalities, we call this complex Homsc(A, B). We prove the following dichotomy: For a fixed finite abstract simplicial\u0000complex B, either Homsc(A, B) is always a disjoint union of contractible spaces\u0000or every finite CW-complex can be obtained up to a homotopy equivalence as\u0000Homsc(A, B) by choosing A in a right way. We furthermore show that the first case is equivalent to the existence of a\u0000nontrivial social choice function and that in this case, the space itself is\u0000homotopy equivalent to a discrete set. Secondly, we give a generalization to finite relational structures and show\u0000that this dichotomy coincides with a complexity theoretic dichotomy for\u0000constraint satisfaction problems, namely in the first case, the problem is in P\u0000and in the second case NP-complete. This generalizes a result from [SW24]\u0000respectively arXiv:2307.03446 [cs.CC]","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181058","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}
Serhii Bardyla, Branislav Novotný, Jaroslav Šupina
In this paper, we study an interplay between local and global properties of spaces of minimal usco maps equipped with the topology of uniform convergence on compact sets. In particular, for each locally compact space $X$ and metric space $Y$, we characterize the space of minimal usco maps from $X$ to $Y$, satisfying one of the following properties: (i) compact, (ii) locally compact, (iii) $sigma$-compact, (iv) locally $sigma$-compact, (v) metrizable, (vi) ccc, (vii) locally ccc, where in the last two items we additionally assumed that $Y$ is separable and non-discrete. Some of the aforementioned results complement ones of v{L}ubica Hol'a and Duv{s}an Hol'y. Also, we obtain analogical characterizations for spaces of minimal cusco maps.
{"title":"Local and global properties of spaces of minimal usco maps","authors":"Serhii Bardyla, Branislav Novotný, Jaroslav Šupina","doi":"arxiv-2408.07409","DOIUrl":"https://doi.org/arxiv-2408.07409","url":null,"abstract":"In this paper, we study an interplay between local and global properties of\u0000spaces of minimal usco maps equipped with the topology of uniform convergence\u0000on compact sets. In particular, for each locally compact space $X$ and metric\u0000space $Y$, we characterize the space of minimal usco maps from $X$ to $Y$,\u0000satisfying one of the following properties: (i) compact, (ii) locally compact,\u0000(iii) $sigma$-compact, (iv) locally $sigma$-compact, (v) metrizable, (vi)\u0000ccc, (vii) locally ccc, where in the last two items we additionally assumed\u0000that $Y$ is separable and non-discrete. Some of the aforementioned results\u0000complement ones of v{L}ubica Hol'a and Duv{s}an Hol'y. Also, we obtain\u0000analogical characterizations for spaces of minimal cusco maps.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181055","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}
Cesar A. Ipanaque Zapata, Felipe A. Torres Estrella
For a Hausdorff space $Y$, a topological space $X$ and a map $g:Xto Y$, we present a connection between the relative sectional number of the first coordinate projection $pi_{2,1}^Y:F(Y,2)to Y$ with respect to $g$, and the coincidence property (CP) for $(X,Y;g)$, where $(X,Y;g)$ has the coincidence property (CP) if, for every map $f:Xto Y$, there is a point $x$ of $X$ such that $f(x)=g(x)$. Explicitly, we demonstrate that $(X,Y;g)$ has the CP if and only if 2 is the minimal cardinality of open covers ${U_i}$ of $X$ such that each $U_i$ admits a local lifting for $g$ with respect to $pi_{2,1}^Y$. This characterisation connects a standard problem in coincidence theory to current research trends in sectional category and topological robotics. Motivated by this connection, we introduce the notion of relative topological complexity of a map.
{"title":"Relative sectional number and the coincidence property","authors":"Cesar A. Ipanaque Zapata, Felipe A. Torres Estrella","doi":"arxiv-2408.07316","DOIUrl":"https://doi.org/arxiv-2408.07316","url":null,"abstract":"For a Hausdorff space $Y$, a topological space $X$ and a map $g:Xto Y$, we\u0000present a connection between the relative sectional number of the first\u0000coordinate projection $pi_{2,1}^Y:F(Y,2)to Y$ with respect to $g$, and the\u0000coincidence property (CP) for $(X,Y;g)$, where $(X,Y;g)$ has the coincidence\u0000property (CP) if, for every map $f:Xto Y$, there is a point $x$ of $X$ such\u0000that $f(x)=g(x)$. Explicitly, we demonstrate that $(X,Y;g)$ has the CP if and\u0000only if 2 is the minimal cardinality of open covers ${U_i}$ of $X$ such that\u0000each $U_i$ admits a local lifting for $g$ with respect to $pi_{2,1}^Y$. This\u0000characterisation connects a standard problem in coincidence theory to current\u0000research trends in sectional category and topological robotics. Motivated by\u0000this connection, we introduce the notion of relative topological complexity of\u0000a map.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180957","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}
We investigate approach spaces generated by probabilistic metric spaces with respect to a continuous t-norm $*$ on the unit interval $[0,1]$. Let $k^*$ be the supremum of the idempotent elements of $*$ in $[0,1)$. It is shown that if $k^*=1$ (resp. $k^*<1$), then an approach space is probabilistic metrizable with respect to $*$ if and only if it is probabilistic metrizable with respect to the minimum (resp. product) t-norm.
{"title":"On the probabilistic metrizability of approach spaces","authors":"Hongliang Lai, Lili Shen, Junche Yu","doi":"arxiv-2408.07548","DOIUrl":"https://doi.org/arxiv-2408.07548","url":null,"abstract":"We investigate approach spaces generated by probabilistic metric spaces with\u0000respect to a continuous t-norm $*$ on the unit interval $[0,1]$. Let $k^*$ be\u0000the supremum of the idempotent elements of $*$ in $[0,1)$. It is shown that if\u0000$k^*=1$ (resp. $k^*<1$), then an approach space is probabilistic metrizable\u0000with respect to $*$ if and only if it is probabilistic metrizable with respect\u0000to the minimum (resp. product) t-norm.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181054","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 this paper we introduce and study the notion of I-convergence of sequences in a metric-like space, where I is an ideal of subsets of the set N of all natural numbers. Further introducing the notion of I*-convergence of sequences in a metric-like space we study its relationship with I-convergence.
本文介绍并研究了类公空间中序列的 I- 收敛概念，其中 I 是所有自然数集合 N 的理想子集。我们进一步引入了类公空间中序列的 I* 收敛概念，并研究了它与 I 收敛的关系。
{"title":"I-convergence of sequences in metric-like spaces","authors":"Prasanta Malik, Saikat Das","doi":"arxiv-2408.13264","DOIUrl":"https://doi.org/arxiv-2408.13264","url":null,"abstract":"In this paper we introduce and study the notion of I-convergence of sequences\u0000in a metric-like space, where I is an ideal of subsets of the set N of all\u0000natural numbers. Further introducing the notion of I*-convergence of sequences\u0000in a metric-like space we study its relationship with I-convergence.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181057","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}
Fahimeh Arabyani Neyshaburi, Ali Akbar Arefijamaal, Ghadir Sadeghi
Projective Hilbert spaces as the underlying spaces of this paper are obtained by identifying two vectors of a Hilbert space $mathcal{H}$ which have the same phase and denoted by $hat{mathcal{H}}$. For a family $Phi$ of vectors of $mathcal{H}$ we introduce a topology $tau_{Phi}$ on $hat{mathcal{H}}$ and provide a topology-based approach for analyzing $hat{mathcal{H}}$. This leads to a new classification of phase retrieval property. We prove that $(hat{mathcal{H}}, tau_{Phi})$ is $sigma$-compact, as well as it is Hausdorff if and only if $Phi$ does phase retrieval. In particular, if $Phi$ is phase retrieval, then we prove that $(hat{mathcal{H}}, tau_{Phi})$ is metrizable and $hat{mathcal{H}}$ is paracompact by a direct limit topology. Also, we make a comparison between $tau_{Phi}$ and some known topologies including the quotient topology, the weak topology and the direct-limit topology. Furthermore, we establish a metric $d_{Phi}$ on $hat{mathcal{H}}$ and show that $d_{Phi}$ is weaker than the Bures-Wasserstein distance on $hat{mathcal{H}}$. As a result, in the finite dimensional case, we prove that $tau_{Phi}$ coincides with the weak topology and $tau_{d_{Phi}}$ on $hat{mathcal{H}}$ if and only if $Phi$ is phase retrieval.
{"title":"Topological structure of projective Hilbert spaces associated with phase retrieval vectors","authors":"Fahimeh Arabyani Neyshaburi, Ali Akbar Arefijamaal, Ghadir Sadeghi","doi":"arxiv-2408.05317","DOIUrl":"https://doi.org/arxiv-2408.05317","url":null,"abstract":"Projective Hilbert spaces as the underlying spaces of this paper are obtained\u0000by identifying two vectors of a Hilbert space $mathcal{H}$ which have the same\u0000phase and denoted by $hat{mathcal{H}}$. For a family $Phi$ of vectors of\u0000$mathcal{H}$ we introduce a topology $tau_{Phi}$ on $hat{mathcal{H}}$ and\u0000provide a topology-based approach for analyzing $hat{mathcal{H}}$. This leads\u0000to a new classification of phase retrieval property. We prove that\u0000$(hat{mathcal{H}}, tau_{Phi})$ is $sigma$-compact, as well as it is\u0000Hausdorff if and only if $Phi$ does phase retrieval. In particular, if $Phi$\u0000is phase retrieval, then we prove that $(hat{mathcal{H}}, tau_{Phi})$ is\u0000metrizable and $hat{mathcal{H}}$ is paracompact by a direct limit topology.\u0000Also, we make a comparison between $tau_{Phi}$ and some known topologies\u0000including the quotient topology, the weak topology and the direct-limit\u0000topology. Furthermore, we establish a metric $d_{Phi}$ on $hat{mathcal{H}}$\u0000and show that $d_{Phi}$ is weaker than the Bures-Wasserstein distance on\u0000$hat{mathcal{H}}$. As a result, in the finite dimensional case, we prove that\u0000$tau_{Phi}$ coincides with the weak topology and $tau_{d_{Phi}}$ on\u0000$hat{mathcal{H}}$ if and only if $Phi$ is phase retrieval.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181056","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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