Journal of Mathematical Analysis and Applications最新文献

{"title":"The existence and Hadamard well-posedness of cooperative equilibria for discontinuous socially structured games","authors":"Xin-Yi Chi,&nbsp;Qi-Qing Song","doi":"10.1016/j.jmaa.2026.130430","DOIUrl":"10.1016/j.jmaa.2026.130430","url":null,"abstract":"<div><div>In a socially structured game with characteristic function forms, a certain coalition can organize its members as some kinds of internal organizations, and different internal organizations determine different social strengths and attainable payoffs of their members. This study introduces socially structured games with organization utilities, and proposes <span><math><mi>E</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-core, <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>-core, and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-core of such games. The sufficient conditions for the existence of <span><math><mi>E</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-core and <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>-core are given for discontinuous games, and the Hadamard well-posedness of the <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>-core of such games is proven. By introducing a collectively feasible condition and a coalitionally <em>C</em>-secure condition for discontinuous socially structured games, the existence of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-core is also established.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130430"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Norms of partial sums operators for a basis with respect to a filter","authors":"V. Kadets ,&nbsp;M. Manskova","doi":"10.1016/j.jmaa.2026.130481","DOIUrl":"10.1016/j.jmaa.2026.130481","url":null,"abstract":"<div><div>Basis of a Banach space with respect to a filter <span><math><mi>F</mi></math></span> on <span><math><mi>N</mi></math></span> (<span><math><mi>F</mi></math></span>-basis for short) is a generalization of basis, where the ordinary convergence of series is substituted by convergence of partial sums with respect to the filter <span><math><mi>F</mi></math></span>. We study the behavior of the norms of partial sums operators for an <span><math><mi>F</mi></math></span>-basis, depending on the filter and on the space. One of the central results is:</div><div>The following properties of a sequence <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub><mo>⊂</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> are equivalent:<ul><li><span>(i)</span><span><div><span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub><msubsup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><mo>∞</mo></math></span>.</div></span></li><li><span>(ii)</span><span><div>There are a free filter <span><math><mi>F</mi></math></span> on <span><math><mi>N</mi></math></span>, an infinite-dimensional Banach space <em>X</em> and an <span><math><mi>F</mi></math></span>-basis <span><math><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> of <em>X</em> such that the norms of the partial sums operators with respect to <span><math><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> are equal to the corresponding <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>.</div></span></li></ul></div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130481"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146173898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The convergence rate of the viscosity vanishing limit for a Keller-Segel-fluid system of consumption type","authors":"Min Li","doi":"10.1016/j.jmaa.2026.130399","DOIUrl":"10.1016/j.jmaa.2026.130399","url":null,"abstract":"<div><div>In this paper, we investigate the viscosity vanishing limit of the Cauchy problem of the Keller-Segel-Navier-Stokes system with consumption type<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>⋅</mo><mi>∇</mi><msub><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msub></mtd><mtd><mo>=</mo><mi>ϵ</mi><mi>Δ</mi><msub><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>−</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mi>∇</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>⋅</mo><mi>∇</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>ϵ</mi></mrow></msub></mtd><mtd><mo>=</mo><mi>Δ</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>⋅</mo><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>+</mo><mi>∇</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>ϵ</mi></mrow></msub></mtd><mtd><mo>=</mo><mi>ϵ</mi><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mi>∇</mi><mi>ϕ</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>∇</mi><mo>⋅</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>ϵ</mi></mrow></msub></mtd><mtd><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></math></span></span></span> in the two-dimensional and three-dimensional setting as <span><math><mi>ϵ</mi><mo>→</mo><mn>0</mn></math></span>. Precisely, we reveal that the solutions of the Keller-Segel-Navier-Stokes system with viscosity will converge to the corresponding solution of the partially inviscid Keller-Segel-Euler system, and establish an algebraic convergence rate of the viscosity vanishing limit for general initial data by developing a series of subtle coupled functional evolution estimates.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130399"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the existence for vector solutions of Schrödinger system with symmetry in tetrahedral group","authors":"Ohsang Kwon","doi":"10.1016/j.jmaa.2026.130415","DOIUrl":"10.1016/j.jmaa.2026.130415","url":null,"abstract":"<div><div>We consider the following elliptic system which is motivated from Schrödinger equations<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>U</mi><mo>+</mo><mi>U</mi><mo>=</mo><msup><mrow><mi>U</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>α</mi><mi>U</mi><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>V</mi><mo>+</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mi>μ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mi>V</mi><mo>=</mo><msup><mrow><mi>V</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>α</mi><msup><mrow><mi>U</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>V</mi></mtd></mtr></mtable></mrow></mrow><mtext>in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>α</mi><mo>∈</mo><mi>R</mi></math></span> is the coupling constant, <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> is sufficiently small constant, and <span><math><mi>μ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>→</mo><mi>R</mi></math></span> is a potential function. We construct a solution in which one component exhibits a single peak at the origin, while the other component concentrates at the vertices of a regular tetrahedron. Existence of such a solution is obtained for all sufficiently small <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span>, highlighting that an arbitrarily small perturbation in the potential term of the second equation can induce a striking symmetry-breaking phenomenon. Our construction is achieved via a perturbative variational reduction method, which balances a central peak in <em>U</em> against multiple peripheral peaks in <em>V</em> in a highly symmetric configuration. These results extend the recent progress on segregated multi-peak solutions in coupled Schrödinger systems to a new three-dimensional symmetry pattern.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130415"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Impulsive delay systems with general end-times","authors":"Giovanni Fusco ,&nbsp;Monica Motta ,&nbsp;Richard Vinter","doi":"10.1016/j.jmaa.2025.130362","DOIUrl":"10.1016/j.jmaa.2025.130362","url":null,"abstract":"<div><div>In this paper we study impulsive control systems with state time delays in the dynamics and scalar inputs, and derive related necessary optimality conditions. We adopt a definition of state trajectories, motivated by applications in the aerospace field, in which impulse controls are interpreted as idealizations of high-intensity control actions of short duration. In previous work, the authors have shown that, for such systems, the limiting state trajectories due to different distribution sense approximations of a given measure control by strict sense, i.e. absolutely continuous, controls may not be the same. (This is a surprising phenomenon since, for delay-free impulse control systems with scalar input, the limiting state trajectory is unique.) Under the hypothesis that the end-time is an integer multiple of the delay time (the ‘commensurate end-time’ assumption), it has also been shown however that, if measure controls are supplemented by ‘attached controls’ then, together, they unambiguously define a state trajectory. For controls in this broader sense, we can prove closure properties of reachable sets for state trajectories, provide conditions for existence of minimizers for impulsive optimal control problems with time delay and derive necessary conditions of optimality. The key idea is to use re-parameterization techniques to reformulate the impulse control system as a classical control system; the desired properties of the impulse control system are then demonstrated by establishing the analogous properties of the reformulated control system. In applications we can expect that the end-time and the time delay will be independently specified. The commensurate end-time hypothesis is therefore highly artificial. The purpose of this paper is to show that closure properties of reachable sets are retained and necessary conditions of optimality can be derived, even when the commensurate end-time hypothesis is discarded. These goals are achieved by means of a new kind of parameterization, which reduces the impulse control system with time delay to a coupled family of control systems, known as ‘multiprocess’ systems.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130362"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Population dynamics with resource-dependent dispersal in a predator-prey model","authors":"Meijia Fu,&nbsp;Ping Liu","doi":"10.1016/j.jmaa.2025.130351","DOIUrl":"10.1016/j.jmaa.2025.130351","url":null,"abstract":"<div><div>This paper primarily focuses on investigating a predator-prey system that incorporates a Holling type II functional response and resource-dependent diffusion, examining how the predator diffusion rate and this resource-dependent diffusion mechanism affect the system's dynamical behavior. By employing topological degree theory, we establish sufficient conditions governing whether positive steady states exist or not. Furthermore, local bifurcation analysis demonstrates the emergence of non-constant steady state solutions. By constructing a Lyapunov function, we demonstrate the global stability of the positive steady state solution. Notably, our results reveal that resource-dependent diffusion significantly enhances species coexistence, particularly under large predator dispersal rates. All theoretical findings are verified by numerical simulations.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130351"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Insensitizing control of nonlinear coupled parabolic systems with a nonlocal spatial term","authors":"Landry Djomegne ,&nbsp;Cyrille Kenne","doi":"10.1016/j.jmaa.2025.130359","DOIUrl":"10.1016/j.jmaa.2025.130359","url":null,"abstract":"<div><div>This work addresses the problem of insensitizing controls for a nonlinear coupled parabolic system with a nonlocal integral term. The goal is to identify controls that make a prescribed functional of the system states, known as the sentinel, insensitive to small perturbations in the initial data. The main novelty of our contribution is the explicit treatment of the nonlocal spatial interaction, which introduces significant analytical challenges. To overcome these, we use recent techniques to establish a new Carleman estimate adapted to the nonlocal operator. This estimate allows us to apply classical techniques to prove null controllability of the linearized extended system, which in turn leads to the desired insensitizing property for the original nonlinear model.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130359"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The uncertainty principles for the multidimensional Fourier–Bessel transform","authors":"Hui Xu ,&nbsp;Xingyu Zhao ,&nbsp;Longben Wei","doi":"10.1016/j.jmaa.2026.130414","DOIUrl":"10.1016/j.jmaa.2026.130414","url":null,"abstract":"<div><div>The aim of this paper is to establish two uncertainty principles for the multidimensional Fourier–Bessel transform. The first one is an extension of the Amrein–Berthier theorem, stating that a nonzero function <em>f</em> and its multidimensional Fourier–Bessel transform <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> cannot both have supports of finite measure. The second one is the Logvinenko–Sereda theorem, which provides a quantitative version of the uncertainty principle by showing that a function whose multidimensional Fourier–Bessel transform has bounded support can be controlled by its restriction to any relatively dense subset. Both results extend the corresponding one-dimensional uncertainty principles for the Fourier–Bessel transform to the multidimensional setting.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130414"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal control of nonlinear Fokker-Planck equations with reflecting boundary conditions","authors":"Cyrille Kenne","doi":"10.1016/j.jmaa.2026.130431","DOIUrl":"10.1016/j.jmaa.2026.130431","url":null,"abstract":"<div><div>Fokker-Planck equations are widely used to describe the evolution of probability density functions in stochastic systems, with applications ranging from physics and biology to finance and engineering. While these equations admit analytical solutions in certain linear or low-dimensional settings, their analysis becomes significantly more challenging for nonlinear systems, especially when combined with reflecting boundary conditions and external control inputs. Such settings are highly relevant in physical contexts, including confined diffusions and controlled transport phenomena. In this work, we study the optimal control of nonlinear Fokker-Planck equations subject to reflecting boundaries and pointwise control constraints. We rigorously establish the existence, uniqueness, and boundedness of solutions. Additionally, we prove regularity properties of the control-to-state mapping and derive first-order necessary optimality conditions in the form of variational inequalities coupled with an adjoint system. Our analysis employs the Schauder fixed-point theorem, De Giorgi's iteration, and functional analysis arguments, adapted to address the nonlinear structure of the problem. These results contribute to the theoretical understanding of controlled stochastic processes in nonlinear physical systems.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130431"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Increasingly global convergence of Hermite series","authors":"Vít Musil ,&nbsp;S. Spektor","doi":"10.1016/j.jmaa.2025.130360","DOIUrl":"10.1016/j.jmaa.2025.130360","url":null,"abstract":"<div><div>We study the convergence of Hermite series of measurable functions on the real line. We characterize the norm convergence of truncated partial Hermite sums in rearrangement invariant spaces provided that the truncations increase sufficiently slowly. Moreover, we provide necessary and sufficient conditions for convergence in the Orlicz modular.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130360"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145852433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}