Pub Date : 2025-01-15DOI: 10.1016/j.matpur.2025.103669
Maria Gordina , Wei Qian , Yilin Wang
We use the SLEκ loop measure to construct a natural representation of the Virasoro algebra of central charge . In particular, we introduce a non-degenerate bilinear Hermitian form (and non positive-definite) using the SLE loop measure and show that the representation is indefinite unitary. Our proof relies on the infinitesimal conformal restriction property of the SLE loop measure.
{"title":"Infinitesimal conformal restriction and unitarizing measures for Virasoro algebra","authors":"Maria Gordina , Wei Qian , Yilin Wang","doi":"10.1016/j.matpur.2025.103669","DOIUrl":"10.1016/j.matpur.2025.103669","url":null,"abstract":"<div><div>We use the SLE<sub><em>κ</em></sub> loop measure to construct a natural representation of the Virasoro algebra of central charge <span><math><mi>c</mi><mo>=</mo><mi>c</mi><mo>(</mo><mi>κ</mi><mo>)</mo><mo>≤</mo><mn>1</mn></math></span>. In particular, we introduce a non-degenerate bilinear Hermitian form (and non positive-definite) using the SLE loop measure and show that the representation is indefinite unitary. Our proof relies on the infinitesimal conformal restriction property of the SLE loop measure.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"195 ","pages":"Article 103669"},"PeriodicalIF":2.1,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-15DOI: 10.1016/j.matpur.2025.103657
Jean-Bernard Bru , Nathan Metraud
Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique solution for all times, for instance in the Schatten norm topology. This system presents remarkable ellipticity properties that turn out to be crucial for the study of the infinite-time limit of its solution, which is proven under relatively weak, albeit probably not necessary, hypotheses on the initial data. This system of differential equations is the elliptic counterpart of an hyperbolic flow applied to quantum field theory to diagonalize Hamiltonians that are quadratic in the bosonic field. In a similar way, this elliptic flow, in particular its asymptotics, has application in quantum field theory: it can be used to diagonalize Hamiltonians that are quadratic in the fermionic field while giving new explicit expressions and properties of these pivotal Hamiltonians of quantum field theory and quantum statistical mechanics.
{"title":"Non-linear operator-valued elliptic flows with application to quantum field theory","authors":"Jean-Bernard Bru , Nathan Metraud","doi":"10.1016/j.matpur.2025.103657","DOIUrl":"10.1016/j.matpur.2025.103657","url":null,"abstract":"<div><div>Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique solution for all times, for instance in the Schatten norm topology. This system presents remarkable ellipticity properties that turn out to be crucial for the study of the infinite-time limit of its solution, which is proven under relatively weak, albeit probably not necessary, hypotheses on the initial data. This system of differential equations is the elliptic counterpart of an hyperbolic flow applied to quantum field theory to diagonalize Hamiltonians that are quadratic in the bosonic field. In a similar way, this elliptic flow, in particular its asymptotics, has application in quantum field theory: it can be used to diagonalize Hamiltonians that are quadratic in the fermionic field while giving new explicit expressions and properties of these pivotal Hamiltonians of quantum field theory and quantum statistical mechanics.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"195 ","pages":"Article 103657"},"PeriodicalIF":2.1,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-15DOI: 10.1016/j.matpur.2025.103659
Lorenzo Ferreri , Bozhidar Velichkov
We study a two-phase free boundary problem in which the two-phases satisfy an impenetrability condition. Precisely, we have two ordered positive functions, which are harmonic in their supports, satisfy a Bernoulli condition on the one-phase part of the free boundary and a transmission condition on the collapsed part of the free boundary. For this two-membrane type problem, we prove an epsilon-regularity theorem with sharp modulus of continuity. Precisely, we show that at flat points each of the two boundaries is regular surface and that the remaining singular set has Hausdorff dimension at most , where N is the dimension of the space.
{"title":"A one-sided two phase Bernoulli free boundary problem","authors":"Lorenzo Ferreri , Bozhidar Velichkov","doi":"10.1016/j.matpur.2025.103659","DOIUrl":"10.1016/j.matpur.2025.103659","url":null,"abstract":"<div><div>We study a two-phase free boundary problem in which the two-phases satisfy an impenetrability condition. Precisely, we have two ordered positive functions, which are harmonic in their supports, satisfy a Bernoulli condition on the one-phase part of the free boundary and a transmission condition on the collapsed part of the free boundary. For this two-membrane type problem, we prove an epsilon-regularity theorem with sharp modulus of continuity. Precisely, we show that at flat points each of the two boundaries is <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span> regular surface and that the remaining singular set has Hausdorff dimension at most <span><math><mi>N</mi><mo>−</mo><mn>5</mn></math></span>, where <em>N</em> is the dimension of the space.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"195 ","pages":"Article 103659"},"PeriodicalIF":2.1,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-14DOI: 10.1016/j.matpur.2025.103658
Habib Ammari , Silvio Barandun , Ping Liu
The aim of this paper is fourfold: (i) to obtain explicit formulas for the eigenpairs of perturbed tridiagonal block Toeplitz matrices; (ii) to make use of such formulas in order to provide a mathematical justification of the non-Hermitian skin effect in dimer systems of subwavelength resonators by proving the condensation of the system's bulk eigenmodes at one of the edges of the system; (iii) to show the topological origin of the non-Hermitian skin effect for dimer systems and (iv) to prove localisation of the interface modes between two dimer structures with non-Hermitian gauge potentials of opposite signs based on new estimates of the decay of the entries of the eigenvectors of block matrices with mirrored blocks.
{"title":"Perturbed block Toeplitz matrices and the non-Hermitian skin effect in dimer systems of subwavelength resonators","authors":"Habib Ammari , Silvio Barandun , Ping Liu","doi":"10.1016/j.matpur.2025.103658","DOIUrl":"10.1016/j.matpur.2025.103658","url":null,"abstract":"<div><div>The aim of this paper is fourfold: (i) to obtain explicit formulas for the eigenpairs of perturbed tridiagonal block Toeplitz matrices; (ii) to make use of such formulas in order to provide a mathematical justification of the non-Hermitian skin effect in dimer systems of subwavelength resonators by proving the condensation of the system's bulk eigenmodes at one of the edges of the system; (iii) to show the topological origin of the non-Hermitian skin effect for dimer systems and (iv) to prove localisation of the interface modes between two dimer structures with non-Hermitian gauge potentials of opposite signs based on new estimates of the decay of the entries of the eigenvectors of block matrices with mirrored blocks.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"195 ","pages":"Article 103658"},"PeriodicalIF":2.1,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.matpur.2024.103628
Gung-Min Gie , James P. Kelliher , Anna L. Mazzucato
The 3D incompressible Euler equations in a bounded domain are most often supplemented with impermeable boundary conditions, which constrain the fluid to neither enter nor leave the domain. We establish well-posedness with inflow, outflow of velocity when either the full value of the velocity is specified on inflow, or only the normal component is specified along with the vorticity (and an additional constraint). We derive compatibility conditions to obtain regularity in a Hölder space with prescribed arbitrary index, and allow multiply connected domains. Our results apply as well to impermeable boundaries, establishing higher regularity of solutions in Hölder spaces.
{"title":"The 3D Euler equations with inflow, outflow and vorticity boundary conditions","authors":"Gung-Min Gie , James P. Kelliher , Anna L. Mazzucato","doi":"10.1016/j.matpur.2024.103628","DOIUrl":"10.1016/j.matpur.2024.103628","url":null,"abstract":"<div><div>The 3D incompressible Euler equations in a bounded domain are most often supplemented with impermeable boundary conditions, which constrain the fluid to neither enter nor leave the domain. We establish well-posedness with inflow, outflow of velocity when either the full value of the velocity is specified on inflow, or only the normal component is specified along with the vorticity (and an additional constraint). We derive compatibility conditions to obtain regularity in a Hölder space with prescribed arbitrary index, and allow multiply connected domains. Our results apply as well to impermeable boundaries, establishing higher regularity of solutions in Hölder spaces.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"193 ","pages":"Article 103628"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143129193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-04DOI: 10.1016/j.matpur.2024.103620
Joaquin Moraga , Roberto Svaldi
Given a projective contraction and a log canonical pair such that is nef over a neighborhood of a closed point , one can define an invariant, the complexity of over , comparing the dimension of X and the relative Picard number of with the sum of the coefficients of those components of B intersecting the fiber over z. We prove that, in the hypotheses above, the complexity of the log pair over is non-negative and that when it is zero then is formally isomorphic to a morphism of toric varieties around . In particular, considering the case when π is the identity morphism, we get a geometric characterization of singularities that are formally isomorphic to toric singularities, thus resolving a conjecture due to Shokurov.
{"title":"A geometric characterization of toric singularities","authors":"Joaquin Moraga , Roberto Svaldi","doi":"10.1016/j.matpur.2024.103620","DOIUrl":"10.1016/j.matpur.2024.103620","url":null,"abstract":"<div><div>Given a projective contraction <span><math><mi>π</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Z</mi></math></span> and a log canonical pair <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> such that <span><math><mo>−</mo><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>+</mo><mi>B</mi><mo>)</mo></math></span> is nef over a neighborhood of a closed point <span><math><mi>z</mi><mo>∈</mo><mi>Z</mi></math></span>, one can define an invariant, the complexity of <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> over <span><math><mi>z</mi><mo>∈</mo><mi>Z</mi></math></span>, comparing the dimension of <em>X</em> and the relative Picard number of <span><math><mi>X</mi><mo>/</mo><mi>Z</mi></math></span> with the sum of the coefficients of those components of <em>B</em> intersecting the fiber over <em>z</em>. We prove that, in the hypotheses above, the complexity of the log pair <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> over <span><math><mi>z</mi><mo>∈</mo><mi>Z</mi></math></span> is non-negative and that when it is zero then <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mo>⌊</mo><mi>B</mi><mo>⌋</mo><mo>)</mo><mo>→</mo><mi>Z</mi></math></span> is formally isomorphic to a morphism of toric varieties around <span><math><mi>z</mi><mo>∈</mo><mi>Z</mi></math></span>. In particular, considering the case when <em>π</em> is the identity morphism, we get a geometric characterization of singularities that are formally isomorphic to toric singularities, thus resolving a conjecture due to Shokurov.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"195 ","pages":"Article 103620"},"PeriodicalIF":2.1,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-04DOI: 10.1016/j.matpur.2024.103625
Leon Bungert , Tim Laux , Kerrek Stinson
We connect adversarial training for binary classification to a geometric evolution equation for the decision boundary. Relying on a perspective that recasts adversarial training as a regularization problem, we introduce a modified training scheme that constitutes a minimizing movements scheme for a nonlocal perimeter functional. We prove that the scheme is monotone and consistent as the adversarial budget vanishes and the perimeter localizes, and as a consequence we rigorously show that the scheme approximates a weighted mean curvature flow. This highlights that the efficacy of adversarial training may be due to locally minimizing the length of the decision boundary. In our analysis, we introduce a variety of tools for working with the subdifferential of a supremal-type nonlocal total variation and its regularity properties.
{"title":"A mean curvature flow arising in adversarial training","authors":"Leon Bungert , Tim Laux , Kerrek Stinson","doi":"10.1016/j.matpur.2024.103625","DOIUrl":"10.1016/j.matpur.2024.103625","url":null,"abstract":"<div><div>We connect adversarial training for binary classification to a geometric evolution equation for the decision boundary. Relying on a perspective that recasts adversarial training as a regularization problem, we introduce a modified training scheme that constitutes a minimizing movements scheme for a nonlocal perimeter functional. We prove that the scheme is monotone and consistent as the adversarial budget vanishes and the perimeter localizes, and as a consequence we rigorously show that the scheme approximates a weighted mean curvature flow. This highlights that the efficacy of adversarial training may be due to locally minimizing the length of the decision boundary. In our analysis, we introduce a variety of tools for working with the subdifferential of a supremal-type nonlocal total variation and its regularity properties.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"192 ","pages":"Article 103625"},"PeriodicalIF":2.1,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-04DOI: 10.1016/j.matpur.2024.103630
Charles Bertucci , Jean-Michel Lasry , Pierre-Louis Lions
This paper is the second of a series devoted to the study of the dynamics of the spectrum of large random matrices. We precise and extend some results of the first part. We study general extensions of the partial differential equation arising to characterize the limit spectral measure of the Dyson Brownian motion. We provide a regularizing result for those generalizations. We also show that several results of part I extend to cases in which there is no spectral dominance property. We then provide several modeling extensions of such models as well as several identities for the Dyson Brownian motion.
{"title":"A spectral dominance approach to large random matrices: Part II","authors":"Charles Bertucci , Jean-Michel Lasry , Pierre-Louis Lions","doi":"10.1016/j.matpur.2024.103630","DOIUrl":"10.1016/j.matpur.2024.103630","url":null,"abstract":"<div><div>This paper is the second of a series devoted to the study of the dynamics of the spectrum of large random matrices. We precise and extend some results of the first part. We study general extensions of the partial differential equation arising to characterize the limit spectral measure of the Dyson Brownian motion. We provide a regularizing result for those generalizations. We also show that several results of part I extend to cases in which there is no spectral dominance property. We then provide several modeling extensions of such models as well as several identities for the Dyson Brownian motion.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"192 ","pages":"Article 103630"},"PeriodicalIF":2.1,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142700825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-04DOI: 10.1016/j.matpur.2024.103632
Bruno Colbois , Corentin Léna , Luigi Provenzano , Alessandro Savo
We consider the first eigenvalue of the magnetic Laplacian in a bounded and simply connected planar domain, with uniform magnetic field and Neumann boundary conditions. We investigate the reverse Faber-Krahn inequality conjectured by S. Fournais and B. Helffer, stating that this eigenvalue is maximized by the disk for a given area. Using the method of level lines, we prove the conjecture for small enough values of the magnetic field (those for which the corresponding eigenfunction in the disk is radial).
我们考虑的是有界且简单连接的平面域中的磁拉普拉斯第一特征值，该域具有均匀磁场和诺伊曼边界条件。我们研究了 S. Fournais 和 B. Helffer 提出的反向 Faber-Krahn 不等式猜想，即在给定区域内，该特征值由圆盘最大化。利用水平线方法，我们证明了磁场值足够小（磁盘中相应的特征函数是径向的）时的猜想。
{"title":"A reverse Faber-Krahn inequality for the magnetic Laplacian","authors":"Bruno Colbois , Corentin Léna , Luigi Provenzano , Alessandro Savo","doi":"10.1016/j.matpur.2024.103632","DOIUrl":"10.1016/j.matpur.2024.103632","url":null,"abstract":"<div><div>We consider the first eigenvalue of the magnetic Laplacian in a bounded and simply connected planar domain, with uniform magnetic field and Neumann boundary conditions. We investigate the reverse Faber-Krahn inequality conjectured by S. Fournais and B. Helffer, stating that this eigenvalue is maximized by the disk for a given area. Using the method of level lines, we prove the conjecture for small enough values of the magnetic field (those for which the corresponding eigenfunction in the disk is radial).</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"192 ","pages":"Article 103632"},"PeriodicalIF":2.1,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-01DOI: 10.1016/j.matpur.2024.103622
Shuang Liu , Yuan Lou
We investigate the joint effects of diffusion and advection on principal eigenvalues of some elliptic operators with shear flow. Some monotonicity and asymptotic behaviors of principal eigenvalues, with respect to diffusion rate and flow amplitude, are established. These analyses lead to a classification of topological structures of level sets for principal eigenvalues, as a function of diffusion rate and flow amplitude. Our analytical results provide a unifying viewpoint to understand mixing enhancement and dispersal-induced growth, which are apparently two unrelated phenomena, one in fluid mechanics and the other in population dynamics. In our analysis, some limiting Hamilton-Jacobi equations, blowup argument and limiting generalized principal eigenvalue problems play critical roles.
{"title":"Monotonicity, asymptotics and level sets for principal eigenvalues of some elliptic operators with shear flow","authors":"Shuang Liu , Yuan Lou","doi":"10.1016/j.matpur.2024.103622","DOIUrl":"10.1016/j.matpur.2024.103622","url":null,"abstract":"<div><div>We investigate the joint effects of diffusion and advection on principal eigenvalues of some elliptic operators with shear flow. Some monotonicity and asymptotic behaviors of principal eigenvalues, with respect to diffusion rate and flow amplitude, are established. These analyses lead to a classification of topological structures of level sets for principal eigenvalues, as a function of diffusion rate and flow amplitude. Our analytical results provide a unifying viewpoint to understand mixing enhancement and dispersal-induced growth, which are apparently two unrelated phenomena, one in fluid mechanics and the other in population dynamics. In our analysis, some limiting Hamilton-Jacobi equations, blowup argument and limiting generalized principal eigenvalue problems play critical roles.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"191 ","pages":"Article 103622"},"PeriodicalIF":2.1,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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