Pub Date : 2024-12-11DOI: 10.1016/j.geomphys.2024.105399
Chengwei Wang , Mengyao Chen , Jipeng Cheng
The determinant formulas for the vacuum expectation values are obtained through the application of Toda Darboux transformations. Initially, it is noted that the 2–Toda hierarchy can be regarded as the 2–component bosonization of the fermionic KP hierarchy. Subsequently, two fundamental Toda Darboux transformation operators, namely and , are constructed from the changes in the Toda (adjoint) wave functions, by employing the 2–component boson–fermion correspondence. On this basis, the aforementioned vacuum expectation values can be realized as the multi–step Toda Darboux transformations. Therefore, the corresponding determinant formulas are derived from the determinant representations of these Toda Darboux transformations. Ultimately, by adopting similar methodologies, we also present the determinant formulas for , which are associated with the KP tau functions.
{"title":"Toda Darboux transformations and vacuum expectation values","authors":"Chengwei Wang , Mengyao Chen , Jipeng Cheng","doi":"10.1016/j.geomphys.2024.105399","DOIUrl":"10.1016/j.geomphys.2024.105399","url":null,"abstract":"<div><div>The determinant formulas for the vacuum expectation values <span><math><mo>〈</mo><mi>s</mi><mo>+</mo><mi>k</mi><mo>+</mo><mi>n</mi><mo>−</mo><mi>m</mi><mo>,</mo><mo>−</mo><mi>s</mi><mo>|</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></msup><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>⋯</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><msub><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⋯</mo><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>g</mi><mo>|</mo><mi>k</mi><mo>〉</mo></math></span> are obtained through the application of Toda Darboux transformations. Initially, it is noted that the 2–Toda hierarchy can be regarded as the 2–component bosonization of the fermionic KP hierarchy. Subsequently, two fundamental Toda Darboux transformation operators, namely <span><math><msub><mrow><mi>T</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><mi>Λ</mi><mo>(</mo><mi>q</mi><mo>)</mo><mo>⋅</mo><mi>Δ</mi><mo>⋅</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>r</mi></math></span>, are constructed from the changes in the Toda (adjoint) wave functions, by employing the 2–component boson–fermion correspondence. On this basis, the aforementioned vacuum expectation values can be realized as the multi–step Toda Darboux transformations. Therefore, the corresponding determinant formulas are derived from the determinant representations of these Toda Darboux transformations. Ultimately, by adopting similar methodologies, we also present the determinant formulas for <span><math><mo>〈</mo><mi>n</mi><mo>−</mo><mi>m</mi><mo>|</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>H</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></msup><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>⋯</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><msub><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⋯</mo><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>g</mi><mo>|</mo><mi>k</mi><mo>〉</mo></math></span>, which are associated with the KP tau functions.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105399"},"PeriodicalIF":1.6,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096602","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 : 2024-12-09DOI: 10.1016/j.geomphys.2024.105400
Leonid Gorodetsky , Nikita Markarian
The main result of the paper is a description of conormal Lie algebras of Feigin–Odesskii Poisson structures. In order to obtain it, we introduce a new variant of a definition of a Feigin–Odesskii Poisson structure: we define it using a differential on the second page of a certain spectral sequence. In the general case, this spectral sequence computes morphisms and higher between filtered objects in an Abelian category. Moreover, we use our definition to give another proof of the description of symplectic leaves of Feigin–Odesskii Poisson structures.
{"title":"On conormal Lie algebras of Feigin–Odesskii Poisson structures","authors":"Leonid Gorodetsky , Nikita Markarian","doi":"10.1016/j.geomphys.2024.105400","DOIUrl":"10.1016/j.geomphys.2024.105400","url":null,"abstract":"<div><div>The main result of the paper is a description of conormal Lie algebras of Feigin–Odesskii Poisson structures. In order to obtain it, we introduce a new variant of a definition of a Feigin–Odesskii Poisson structure: we define it using a differential on the second page of a certain spectral sequence. In the general case, this spectral sequence computes morphisms and higher <span><math><msup><mrow><mi>Ext</mi></mrow><mrow><mo>′</mo></mrow></msup><mspace></mspace><mi>s</mi></math></span> between filtered objects in an Abelian category. Moreover, we use our definition to give another proof of the description of symplectic leaves of Feigin–Odesskii Poisson structures.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105400"},"PeriodicalIF":1.6,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095978","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 : 2024-12-06DOI: 10.1016/j.geomphys.2024.105398
Josef F. Dorfmeister , Hui Ma
We extend the techniques introduced in [10] for contractible Riemann surfaces to construct minimal Lagrangian immersions from arbitrary Riemann surfaces into the complex projective plane via the loop group method. Based on the potentials of translationally equivariant minimal Lagrangian surfaces, we introduce perturbed equivariant minimal Lagrangian surfaces in and construct a class of minimal Lagrangian cylinders. Furthermore, we show that these minimal Lagrangian cylinders approximate Delaunay cylinders with respect to some weighted Wiener norm of the twisted loop group .
{"title":"Minimal Lagrangian surfaces in CP2 via the loop group method part II: The general case","authors":"Josef F. Dorfmeister , Hui Ma","doi":"10.1016/j.geomphys.2024.105398","DOIUrl":"10.1016/j.geomphys.2024.105398","url":null,"abstract":"<div><div>We extend the techniques introduced in <span><span>[10]</span></span> for contractible Riemann surfaces to construct minimal Lagrangian immersions from arbitrary Riemann surfaces into the complex projective plane <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> via the loop group method. Based on the potentials of translationally equivariant minimal Lagrangian surfaces, we introduce perturbed equivariant minimal Lagrangian surfaces in <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and construct a class of minimal Lagrangian cylinders. Furthermore, we show that these minimal Lagrangian cylinders approximate Delaunay cylinders with respect to some weighted Wiener norm of the twisted loop group <span><math><mi>Λ</mi><mi>S</mi><mi>U</mi><msub><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mi>σ</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105398"},"PeriodicalIF":1.6,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096017","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 : 2024-12-06DOI: 10.1016/j.geomphys.2024.105401
Huiyang Xu , Cece Li , Cheng Xing
Lagrangian H-umbilical submanifolds in complex space forms, as the “simplest” Lagrangian submanifolds next to the geodesic ones, were introduced and determined by B.-Y. Chen. Many interesting examples belong to this class, such as the Whitney spheres, isotropic non-minimal immersions, and special Calabi product immersions. In this paper, such submanifolds are proved to be of a conformally flat, quasi-Einstein metric and the pseudo-parallel cubic form. As the main results, we find a geometric characterization of those submanifolds as not being of constant sectional curvature. Meanwhile, for Lagrangian submanifolds in complex space forms with pseudo-parallel cubic form, we completely determine the three dimensional case, and all dimensions for the conformally flat case.
{"title":"Lagrangian H-umbilical submanifolds in complex space forms and pseudo-parallel cubic form","authors":"Huiyang Xu , Cece Li , Cheng Xing","doi":"10.1016/j.geomphys.2024.105401","DOIUrl":"10.1016/j.geomphys.2024.105401","url":null,"abstract":"<div><div>Lagrangian <em>H</em>-umbilical submanifolds in complex space forms, as the “simplest” Lagrangian submanifolds next to the geodesic ones, were introduced and determined by B.-Y. Chen. Many interesting examples belong to this class, such as the Whitney spheres, isotropic non-minimal immersions, and special Calabi product immersions. In this paper, such submanifolds are proved to be of a conformally flat, quasi-Einstein metric and the pseudo-parallel cubic form. As the main results, we find a geometric characterization of those submanifolds as not being of constant sectional curvature. Meanwhile, for Lagrangian submanifolds in complex space forms with pseudo-parallel cubic form, we completely determine the three dimensional case, and all dimensions for the conformally flat case.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105401"},"PeriodicalIF":1.6,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096600","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 : 2024-12-06DOI: 10.1016/j.geomphys.2024.105391
Xavier Blot , Danilo Lewański , Paolo Rossi , Sergei Shadrin
We propose a new system of conjectural relations in the tautological ring of the moduli space of curves involving stable rooted trees with level structure decorated by Hodge and Ω-classes and prove these conjectures in different cases.
{"title":"Stable tree expressions with Omega-classes and double ramification cycles","authors":"Xavier Blot , Danilo Lewański , Paolo Rossi , Sergei Shadrin","doi":"10.1016/j.geomphys.2024.105391","DOIUrl":"10.1016/j.geomphys.2024.105391","url":null,"abstract":"<div><div>We propose a new system of conjectural relations in the tautological ring of the moduli space of curves involving stable rooted trees with level structure decorated by Hodge and Ω-classes and prove these conjectures in different cases.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105391"},"PeriodicalIF":1.6,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096601","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 : 2024-12-05DOI: 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}
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