Canadian Journal of Mathematics最新文献

{"title":"Multi-linear forms, graphs, and","authors":"Pablo Bhowmik, Alex Iosevich, Doowon Koh, Thang Pham","doi":"10.4153/s0008414x2300086x","DOIUrl":"https://doi.org/10.4153/s0008414x2300086x","url":null,"abstract":"<p>The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240122134123249-0119:S0008414X2300086X:S0008414X2300086X_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*}T_Kf(x)=int K(x,y) f(y) dmu(y),end{align*} $$</span></span></img></span>where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240122134123249-0119:S0008414X2300086X:S0008414X2300086X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$f: X to {Bbb R}$</span></span></img></span></span>, <span>X</span> a set, finite or infinite, and <span>K</span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240122134123249-0119:S0008414X2300086X:S0008414X2300086X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mu $</span></span></img></span></span> denote a suitable kernel and a measure, respectively. Given a connected ordered graph <span>G</span> on <span>n</span> vertices, consider the multi-linear form <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240122134123249-0119:S0008414X2300086X:S0008414X2300086X_eqnu2.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*}Lambda_G(f_1,f_2, dots, f_n)=int_{x^1, dots, x^n in X} prod_{(i,j) in {mathcal E}(G)} K(x^i,x^j) prod_{l=1}^n f_l(x^l) dmu(x^l),end{align*} $$</span></span></img></span>where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240122134123249-0119:S0008414X2300086X:S0008414X2300086X_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal E}(G)$</span></span></img></span></span> is the edge set of <span>G</span>. Define <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240122134123249-0119:S0008414X2300086X:S0008414X2300086X_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$Lambda _G(p_1, ldots , p_n)$</span></span></img></span></span> as the smallest constant <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240122134123249-0119:S0008414X2300086X:S0008414X2300086X_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$C&gt;0$</span></span></img></span></span> such that the inequality <span><span>(0.1)</span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240122134123249-0119:S0008414X2300086X:S0008414X2300086X_eqn1.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align} Lambda_G(f_1, dots, f_n) leq C prod_{i=1}^n {||f_i||}_{L^{p_i}(X, mu)} end{ali","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139552461","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}

{"title":"Higher dimensional algebraic fiberings for pro-p groups","authors":"Dessislava H. Kochloukova","doi":"10.4153/s0008414x23000895","DOIUrl":"https://doi.org/10.4153/s0008414x23000895","url":null,"abstract":"<p>We prove some conditions for higher-dimensional algebraic fibering of pro-<span>p</span> group extensions, and we establish corollaries about incoherence of pro-<span>p</span> groups. In particular, if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240114040832101-0205:S0008414X23000895:S0008414X23000895_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$1 to K to G to Gamma to 1$</span></span></img></span></span> is a short exact sequence of pro-<span>p</span> groups, such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240114040832101-0205:S0008414X23000895:S0008414X23000895_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$Gamma $</span></span></img></span></span> contains a finitely generated, non-abelian, free pro-<span>p</span> subgroup, <span>K</span> a finitely presented pro-<span>p</span> group with <span>N</span> a normal pro-<span>p</span> subgroup of <span>K</span> such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240114040832101-0205:S0008414X23000895:S0008414X23000895_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$K/ N simeq mathbb {Z}_p$</span></span></img></span></span> and <span>N</span> not finitely generated as a pro-<span>p</span> group, then <span>G</span> is incoherent (in the category of pro-<span>p</span> groups). Furthermore, we show that if <span>K</span> is a finitely generated, free pro-<span>p</span> group with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240114040832101-0205:S0008414X23000895:S0008414X23000895_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$d(K) geq 2$</span></span></img></span></span>, then either <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240114040832101-0205:S0008414X23000895:S0008414X23000895_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm{Aut}_0(K)$</span></span></img></span></span> is incoherent (in the category of pro-<span>p</span> groups) or there is a finitely presented pro-<span>p</span> group, without non-procyclic free pro-<span>p</span> subgroups, that has a metabelian pro-<span>p</span> quotient that is not finitely presented, i.e., a pro-<span>p</span> version of a result of Bieri–Strebel does not hold.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139470251","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}

{"title":"Affine isoperimetric inequalities on flag manifolds","authors":"Susanna Dann, Grigoris Paouris, Peter Pivovarov","doi":"10.4153/s0008414x23000871","DOIUrl":"https://doi.org/10.4153/s0008414x23000871","url":null,"abstract":"<p>Building on work of Furstenberg and Tzkoni, we introduce <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240111032518588-0908:S0008414X23000871:S0008414X23000871_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {r}$</span></span></img></span></span>-flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the Grassmannian can be considered as a special case). We establish affine and linear invariance properties and extend fundamental results to this new setting. In particular, we prove several affine isoperimetric inequalities from convex geometry and their approximate reverse forms. We also introduce functional forms of these quantities and establish corresponding inequalities.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139464941","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}

{"title":"Extension of monotone operators and Lipschitz maps invariant for a group of isometries","authors":"Giulia Cavagnari, Giuseppe Savaré, Giacomo Enrico Sodini","doi":"10.4153/s0008414x23000846","DOIUrl":"https://doi.org/10.4153/s0008414x23000846","url":null,"abstract":"<p>We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240108054719173-0183:S0008414X23000846:S0008414X23000846_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$L^{p}$</span></span></img></span></span>-spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139412943","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}

{"title":"Solvability of Hessian quotient equations in exterior domains","authors":"Limei Dai, Jiguang Bao, Bo Wang","doi":"10.4153/s0008414x23000834","DOIUrl":"https://doi.org/10.4153/s0008414x23000834","url":null,"abstract":"","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"18 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138972481","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}

{"title":"Unified bounds for the independence number of graphs","authors":"Jiang Zhou","doi":"10.4153/s0008414x23000822","DOIUrl":"https://doi.org/10.4153/s0008414x23000822","url":null,"abstract":"","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"17 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139010457","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}

{"title":"Equivariant -correspondences and compact quantum group actions on Pimsner algebras","authors":"Suvrajit Bhattacharjee, Soumalya Joardar","doi":"10.4153/s0008414x23000810","DOIUrl":"https://doi.org/10.4153/s0008414x23000810","url":null,"abstract":"<p>Let <span>G</span> be a compact quantum group. We show that given a <span>G</span>-equivariant <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231223072731647-0534:S0008414X23000810:S0008414X23000810_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$textrm {C}^*$</span></span></img></span></span>-correspondence <span>E</span>, the Pimsner algebra <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231223072731647-0534:S0008414X23000810:S0008414X23000810_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {O}_E$</span></span></img></span></span> can be naturally made into a <span>G</span>-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231223072731647-0534:S0008414X23000810:S0008414X23000810_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$textrm {C}^*$</span></span></img></span></span>-algebra. We also provide sufficient conditions under which it is guaranteed that a <span>G</span>-action on the Pimsner algebra <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231223072731647-0534:S0008414X23000810:S0008414X23000810_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {O}_E$</span></span></img></span></span> arises in this way, in a suitable precise sense. When <span>G</span> is of Kac type, a KMS state on the Pimsner algebra, arising from a quasi-free dynamics, is <span>G</span>-equivariant if and only if the tracial state obtained from restricting it to the coefficient algebra is <span>G</span>-equivariant, under a natural condition. We apply these results to the situation when the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231223072731647-0534:S0008414X23000810:S0008414X23000810_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$textrm {C}^*$</span></span></img></span></span>-correspondence is obtained from a finite, directed graph and draw various conclusions on the quantum automorphism groups of such graphs, both in the sense of Banica and Bichon.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139057100","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}

{"title":"Combinatorial proofs of properties of double-point enhanced grid homology","authors":"Ollie Thakar","doi":"10.4153/s0008414x23000809","DOIUrl":"https://doi.org/10.4153/s0008414x23000809","url":null,"abstract":"<p>We provide a purely combinatorial proof of a skein exact sequence obeyed by double-point enhanced grid homology. We also extend the theory to coefficients over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220175033308-0060:S0008414X23000809:S0008414X23000809_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {Z},$</span></span></img></span></span> and discuss alternatives to the Ozsváth–Szabó <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220175033308-0060:S0008414X23000809:S0008414X23000809_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$tau $</span></span></img></span></span> invariant.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138823569","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}

{"title":"Anisotropic flow, entropy, and -Minkowski problem","authors":"Károly J. Böröczky, Pengfei Guan","doi":"10.4153/s0008414x23000792","DOIUrl":"https://doi.org/10.4153/s0008414x23000792","url":null,"abstract":"<p>We provide a natural simple argument using anistropic flows to prove the existence of weak solutions to Lutwak’s <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214042727186-0503:S0008414X23000792:S0008414X23000792_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$L^p$</span></span></img></span></span>-Minkowski problem on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214042727186-0503:S0008414X23000792:S0008414X23000792_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$S^n$</span></span></img></span></span> which were obtained by other methods.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"83 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138693352","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}

{"title":"Graphical methods and rings of invariants on the symmetric algebra","authors":"Rebecca Bourn, William Q. Erickson, Jeb F. Willenbring","doi":"10.4153/s0008414x23000780","DOIUrl":"https://doi.org/10.4153/s0008414x23000780","url":null,"abstract":"<p>Let <span>G</span> be a complex classical group, and let <span>V</span> be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of <span>G</span>-invariant polynomial functions on the space <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214132544397-0824:S0008414X23000780:S0008414X23000780_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal P^m(V)$</span></span></img></span></span> of degree-<span>m</span> homogeneous polynomial functions on <span>V</span>. In this paper, we replace <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214132544397-0824:S0008414X23000780:S0008414X23000780_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal P^m(V)$</span></span></img></span></span> with the full polynomial algebra <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214132544397-0824:S0008414X23000780:S0008414X23000780_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal P(V)$</span></span></img></span></span>. As a result, the invariant ring is no longer finitely generated. Hence, instead of seeking generators, we aim to write down linear bases for bigraded components. Indeed, when <span>G</span> is of sufficiently high rank, we realize these bases as sets of graphs with prescribed number of vertices and edges. When the rank of <span>G</span> is small, there arise complicated linear dependencies among the graphs, but we remedy this setback via representation theory: in particular, we determine the dimension of an arbitrary component in terms of branching multiplicities from the general linear group to the symmetric group. We thereby obtain an expression for the bigraded Hilbert series of the ring of invariants on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231214132544397-0824:S0008414X23000780:S0008414X23000780_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal P(V)$</span></span></img></span></span>. We conclude with examples using our graphical notation, several of which recover classical results.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"97 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138693353","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}