Pub Date : 2024-12-05DOI: 10.1016/j.geomphys.2024.105376
Pietro Corvaja, Umberto Zannier
{"title":"Corrigendum to “Unramified sections of the Legendre scheme and modular forms” [J. Geom. Phys. 166 (2021) 104266]","authors":"Pietro Corvaja, Umberto Zannier","doi":"10.1016/j.geomphys.2024.105376","DOIUrl":"10.1016/j.geomphys.2024.105376","url":null,"abstract":"","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105376"},"PeriodicalIF":1.6,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096595","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}
Pub Date : 2024-12-04DOI: 10.1016/j.geomphys.2024.105390
Shuwen Chen, Fangyang Zheng
A Hermitian-symplectic metric is a Hermitian metric whose Kähler form is given by the -part of a closed 2-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be Kählerian (i.e., admitting a Kähler metric). The conjecture is known to be true in dimension 2 but is still open in dimensions 3 or higher. In this article, we confirm the conjecture for all 2-step solvmanifolds, namely, compact quotients of 2-step solvable Lie groups by discrete subgroups. In the proofs, we adopted a method of using special non-unitary frames, which enabled us to squeeze out some hidden symmetries to make the proof go through. Hopefully the technique could be further applied. The proof of the main theorem also gives an explicit description of all Hermitian-symplectic metrics on any 2-step solvable Lie algebra equipped with a complex structure.
{"title":"Streets-Tian conjecture holds for 2-step solvmanifolds","authors":"Shuwen Chen, Fangyang Zheng","doi":"10.1016/j.geomphys.2024.105390","DOIUrl":"10.1016/j.geomphys.2024.105390","url":null,"abstract":"<div><div>A Hermitian-symplectic metric is a Hermitian metric whose Kähler form is given by the <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-part of a closed 2-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be Kählerian (i.e., admitting a Kähler metric). The conjecture is known to be true in dimension 2 but is still open in dimensions 3 or higher. In this article, we confirm the conjecture for all 2-step solvmanifolds, namely, compact quotients of 2-step solvable Lie groups by discrete subgroups. In the proofs, we adopted a method of using special <em>non-unitary</em> frames, which enabled us to squeeze out some hidden symmetries to make the proof go through. Hopefully the technique could be further applied. The proof of the main theorem also gives an explicit description of all Hermitian-symplectic metrics on any 2-step solvable Lie algebra equipped with a complex structure.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105390"},"PeriodicalIF":1.6,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096015","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}
Pub Date : 2024-11-29DOI: 10.1016/j.geomphys.2024.105389
Edgar Gasperín , Mariem Magdy , Filipe C. Mena
We use conformal geometry methods and the construction of Friedrich's cylinder at spatial infinity to study the propagation of spin-0 fields (solutions to the wave equation) on n-dimensional Minkowski spacetimes in a neighbourhood of spatial and null infinity. We obtain formal solutions written in terms of series expansions close to spatial and null infinity and use them to compute non-trivial asymptotic spin-0 charges. It is shown that if one examines the most general initial data within the class considered in this paper, the expansion is polyhomogeneous and hence of restricted regularity at null infinity. Furthermore, we derive the conditions on the initial data needed to obtain well-defined limits for the asymptotic charges at the critical sets where null infinity and spatial infinity meet. In four dimensions, we find that there are infinitely many well-defined asymptotic charges at the critical sets, while for higher dimensions there is only a finite number of non-trivial asymptotic charges with well-defined limits at the critical sets.
{"title":"Asymptotics of spin-0 fields and conserved charges on n-dimensional Minkowski spaces","authors":"Edgar Gasperín , Mariem Magdy , Filipe C. Mena","doi":"10.1016/j.geomphys.2024.105389","DOIUrl":"10.1016/j.geomphys.2024.105389","url":null,"abstract":"<div><div>We use conformal geometry methods and the construction of Friedrich's cylinder at spatial infinity to study the propagation of spin-0 fields (solutions to the wave equation) on <em>n</em>-dimensional Minkowski spacetimes in a neighbourhood of spatial and null infinity. We obtain formal solutions written in terms of series expansions close to spatial and null infinity and use them to compute non-trivial asymptotic spin-0 charges. It is shown that if one examines the most general initial data within the class considered in this paper, the expansion is polyhomogeneous and hence of restricted regularity at null infinity. Furthermore, we derive the conditions on the initial data needed to obtain well-defined limits for the asymptotic charges at the critical sets where null infinity and spatial infinity meet. In four dimensions, we find that there are infinitely many well-defined asymptotic charges at the critical sets, while for higher dimensions there is only a finite number of non-trivial asymptotic charges with well-defined limits at the critical sets.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105389"},"PeriodicalIF":1.6,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-29DOI: 10.1016/j.geomphys.2024.105388
Hyungjin Huh
We define an area function for Schrödinger equation. The conservation of the area is shown by considering momentum conservation. The similar idea is applied to Dirac equations.
{"title":"Geometric meaning of momentum conservation","authors":"Hyungjin Huh","doi":"10.1016/j.geomphys.2024.105388","DOIUrl":"10.1016/j.geomphys.2024.105388","url":null,"abstract":"<div><div>We define an area function <span><math><mi>A</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> for Schrödinger equation. The conservation of the area <span><math><mi>A</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>A</mi><mo>(</mo><mn>0</mn><mo>)</mo></math></span> is shown by considering momentum conservation. The similar idea is applied to Dirac equations.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105388"},"PeriodicalIF":1.6,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096596","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}
Pub Date : 2024-11-28DOI: 10.1016/j.geomphys.2024.105385
Safa Braiek , Taoufik Chtioui , Sami Mabrouk
Leibniz algebras are non skew-symmetric generalization of Lie algebras. In this paper we introduce the notion of anti-Leibniz algebras as a “non commutative version” of mock-Lie algebras. Low dimensional classification of such algebras is given. Then we investigate the notion of averaging operators and more general embedding tensors to build some new algebraic structures, namely anti-associative dialgebras, anti-associative trialgebras and anti-Leibniz trialgebras.
{"title":"Anti-Leibniz algebras: A non-commutative version of mock-Lie algebras","authors":"Safa Braiek , Taoufik Chtioui , Sami Mabrouk","doi":"10.1016/j.geomphys.2024.105385","DOIUrl":"10.1016/j.geomphys.2024.105385","url":null,"abstract":"<div><div>Leibniz algebras are non skew-symmetric generalization of Lie algebras. In this paper we introduce the notion of anti-Leibniz algebras as a “non commutative version” of mock-Lie algebras. Low dimensional classification of such algebras is given. Then we investigate the notion of averaging operators and more general embedding tensors to build some new algebraic structures, namely anti-associative dialgebras, anti-associative trialgebras and anti-Leibniz trialgebras.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105385"},"PeriodicalIF":1.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096597","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}
Pub Date : 2024-11-28DOI: 10.1016/j.geomphys.2024.105386
Michael B. Law, Isaac M. Lopez, Daniel Santiago
We show that the weighted positive mass theorem of Baldauf–Ozuch and Chu–Zhu is equivalent to the usual positive mass theorem under suitable regularity, and can be regarded as a positive mass theorem for smooth metric measure spaces. A stronger weighted positive mass theorem is established, unifying and generalizing their results. We also study Dirac operators on certain warped product manifolds associated to smooth metric measure spaces. Applications of this include, among others, an alternative proof for a special case of our positive mass theorem, eigenvalue bounds for the Dirac operator on closed spin manifolds, and a new way to understand the weighted Dirac operator using warped products.
{"title":"Positive mass and Dirac operators on weighted manifolds and smooth metric measure spaces","authors":"Michael B. Law, Isaac M. Lopez, Daniel Santiago","doi":"10.1016/j.geomphys.2024.105386","DOIUrl":"10.1016/j.geomphys.2024.105386","url":null,"abstract":"<div><div>We show that the weighted positive mass theorem of Baldauf–Ozuch and Chu–Zhu is equivalent to the usual positive mass theorem under suitable regularity, and can be regarded as a positive mass theorem for smooth metric measure spaces. A stronger weighted positive mass theorem is established, unifying and generalizing their results. We also study Dirac operators on certain warped product manifolds associated to smooth metric measure spaces. Applications of this include, among others, an alternative proof for a special case of our positive mass theorem, eigenvalue bounds for the Dirac operator on closed spin manifolds, and a new way to understand the weighted Dirac operator using warped products.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105386"},"PeriodicalIF":1.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096598","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}
Pub Date : 2024-11-28DOI: 10.1016/j.geomphys.2024.105387
Xianguo Geng , Feiying Yan , Jiao Wei
In this work, the Cauchy problem for the generalized complex short pulse equation with initial conditions in the weighted Sobolev space is studied by using the Riemann-Hilbert method and the -steepest descent method. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem can be expressed as solution of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, the long-time asymptotics and soliton resolution of the generalized complex short pulse equation in the soliton region are obtained by resorting to the -steepest descent method. The results also indicate that the N-soliton solutions of the generalized complex short pulse equation are asymptotically stable.
{"title":"Soliton resolution for the generalized complex short pulse equation with the weighted Sobolev initial data","authors":"Xianguo Geng , Feiying Yan , Jiao Wei","doi":"10.1016/j.geomphys.2024.105387","DOIUrl":"10.1016/j.geomphys.2024.105387","url":null,"abstract":"<div><div>In this work, the Cauchy problem for the generalized complex short pulse equation with initial conditions in the weighted Sobolev space <span><math><mi>H</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is studied by using the Riemann-Hilbert method and the <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span>-steepest descent method. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem can be expressed as solution of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, the long-time asymptotics and soliton resolution of the generalized complex short pulse equation in the soliton region are obtained by resorting to the <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span>-steepest descent method. The results also indicate that the <em>N</em>-soliton solutions of the generalized complex short pulse equation are asymptotically stable.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105387"},"PeriodicalIF":1.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142747797","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}
Pub Date : 2024-11-26DOI: 10.1016/j.geomphys.2024.105384
Simone Cristofori, Michela Zedda
In this paper we compute the third coefficient arising from the TYCZ-expansion of the ε-function associated to a Kähler-Einstein metric and discuss the consequences of its vanishing.
{"title":"On the third coefficient in the TYCZ–expansion of the epsilon function of Kähler–Einstein manifolds","authors":"Simone Cristofori, Michela Zedda","doi":"10.1016/j.geomphys.2024.105384","DOIUrl":"10.1016/j.geomphys.2024.105384","url":null,"abstract":"<div><div>In this paper we compute the third coefficient arising from the TYCZ-expansion of the <em>ε</em>-function associated to a Kähler-Einstein metric and discuss the consequences of its vanishing.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105384"},"PeriodicalIF":1.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-22DOI: 10.1016/j.geomphys.2024.105375
Miguel Manzano, Marc Mars
In this work we provide a definition of the constraint tensor of a null hypersurface data which is completely explicit in the extrinsic geometry of the hypersurface. The definition is fully covariant and applies for any topology of the hypersurface. For data embedded in a spacetime, the constraint tensor coincides with the pull-back of the ambient Ricci tensor. As applications of the results, we find three geometric quantities on any transverse submanifold S of the data with remarkably simple gauge behaviour, and prove that the restriction of the constraint tensor to S takes a very simple form in terms of them. We also obtain an identity that generalizes the standard near horizon equation of isolated horizons to totally geodesic null hypersurfaces with any topology. Finally, we prove that when a null hypersurface has product topology, its extrinsic curvature can be uniquely reconstructed from the constraint tensor plus suitable initial data on a cross-section.
在这项工作中，我们给出了空超曲面数据的约束张量定义，该定义在超曲面的外几何学中是完全明确的。该定义是完全协变的，适用于超曲面的任何拓扑结构。对于嵌入时空的数据，约束张量与环境里奇张量的回拉重合。作为结果的应用，我们在数据的任意横向子满面 S 上发现了三个几何量，它们具有非常简单的轨距行为，并证明了约束张量对 S 的限制采用了非常简单的形式。我们还得到了一个特性，它将孤立地平线的标准近地平线方程推广到具有任意拓扑结构的完全大地空超曲面。最后，我们证明了当空超曲面具有积拓扑结构时，其外曲率可以通过约束张量加上横截面上合适的初始数据唯一地重建。
{"title":"The constraint tensor for null hypersurfaces","authors":"Miguel Manzano, Marc Mars","doi":"10.1016/j.geomphys.2024.105375","DOIUrl":"10.1016/j.geomphys.2024.105375","url":null,"abstract":"<div><div>In this work we provide a definition of the constraint tensor of a null hypersurface data which is completely explicit in the extrinsic geometry of the hypersurface. The definition is fully covariant and applies for any topology of the hypersurface. For data embedded in a spacetime, the constraint tensor coincides with the pull-back of the ambient Ricci tensor. As applications of the results, we find three geometric quantities on any transverse submanifold <em>S</em> of the data with remarkably simple gauge behaviour, and prove that the restriction of the constraint tensor to <em>S</em> takes a very simple form in terms of them. We also obtain an identity that generalizes the standard near horizon equation of isolated horizons to totally geodesic null hypersurfaces with any topology. Finally, we prove that when a null hypersurface has product topology, its extrinsic curvature can be uniquely reconstructed from the constraint tensor plus suitable initial data on a cross-section.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"208 ","pages":"Article 105375"},"PeriodicalIF":1.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142721953","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}
Pub Date : 2024-11-22DOI: 10.1016/j.geomphys.2024.105373
Liqiang Cai , Zhuo Chen , Honglei Lang , Maosong Xiang
Given a vector bundle A over a smooth manifold M such that the square root of the line bundle exists, the Clifford bundle associated to the standard split pseudo-Euclidean vector bundle admits a spinor bundle , whose section space consists of Berezinian half-densities of the graded manifold . Inspired by Kosmann-Schwarzbach's formula of deriving operator of split Courant algebroid (or proto-bialgebroid) structures on , we give an explicit construction of the associate Dirac generating operator introduced by Alekseev and Xu. We prove that the square of the Dirac generating operator is an invariant of the corresponding split Courant algebroid, and also give an explicit expression of this invariant in terms of modular elements.
给定光滑流形 M 上的向量束 A，使得线束 ∧topA⁎⊗∧topT⁎M 的平方根 L 存在，那么与标准分裂伪欧几里得向量束相关联的克利福德束（E=A⊕A⁎、⋅,〉）允许一个旋量束∧-A⊗L，其截面空间由分级流形 A⁎[1] 的贝雷津半密度组成。受科斯曼-施瓦茨巴赫（Kosmann-Schwarzbach）关于 A⊕A⁎上的分裂库朗网格（或原边网格）结构的导出算子公式的启发，我们给出了阿列克谢耶夫（Alekseev）和徐（Xu）引入的关联狄拉克生成算子的明确构造。我们证明了狄拉克生成算子的平方是相应的分裂库朗拟合结构的不变量，并给出了这个不变量在模元方面的明确表达式。
{"title":"Dirac generating operators of split Courant algebroids","authors":"Liqiang Cai , Zhuo Chen , Honglei Lang , Maosong Xiang","doi":"10.1016/j.geomphys.2024.105373","DOIUrl":"10.1016/j.geomphys.2024.105373","url":null,"abstract":"<div><div>Given a vector bundle <em>A</em> over a smooth manifold <em>M</em> such that the square root <span><math><mi>L</mi></math></span> of the line bundle <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mi>top</mi></mrow></msup><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>⊗</mo><msup><mrow><mo>∧</mo></mrow><mrow><mi>top</mi></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>M</mi></math></span> exists, the Clifford bundle associated to the standard split pseudo-Euclidean vector bundle <span><math><mo>(</mo><mi>E</mi><mo>=</mo><mi>A</mi><mo>⊕</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>,</mo><mo>〈</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>〉</mo><mo>)</mo></math></span> admits a spinor bundle <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mo>•</mo></mrow></msup><mi>A</mi><mo>⊗</mo><mi>L</mi></math></span>, whose section space consists of Berezinian half-densities of the graded manifold <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>[</mo><mn>1</mn><mo>]</mo></math></span>. Inspired by Kosmann-Schwarzbach's formula of deriving operator of split Courant algebroid (or proto-bialgebroid) structures on <span><math><mi>A</mi><mo>⊕</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, we give an explicit construction of the associate Dirac generating operator introduced by Alekseev and Xu. We prove that the square of the Dirac generating operator is an invariant of the corresponding split Courant algebroid, and also give an explicit expression of this invariant in terms of modular elements.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"208 ","pages":"Article 105373"},"PeriodicalIF":1.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142721793","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}
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