Pub Date : 2025-10-27DOI: 10.1016/j.geomphys.2025.105689
Yu.B. Chernyakov , G.I. Sharygin
In this paper we discuss the relation between the functions that give first integrals of full symmetric Toda system (an important Hamilton system on the space of traceless real symmetric matrices) and the vector fields on the group of orthogonal matrices: it is known that this system is equivalent to an ordinary differential equation on the orthogonal group, and we extend this observation further to its first integrals. As a by-product we describe a representation of the Lie algebra of -invariant functions on the dual space of Lie algebra (under the canonical Poisson structure) by vector fields on .
{"title":"Full symmetric Toda system and vector fields on the group SOn(R)","authors":"Yu.B. Chernyakov , G.I. Sharygin","doi":"10.1016/j.geomphys.2025.105689","DOIUrl":"10.1016/j.geomphys.2025.105689","url":null,"abstract":"<div><div>In this paper we discuss the relation between the functions that give first integrals of full symmetric Toda system (an important Hamilton system on the space of traceless real symmetric matrices) and the vector fields on the group of orthogonal matrices: it is known that this system is equivalent to an ordinary differential equation on the orthogonal group, and we extend this observation further to its first integrals. As a by-product we describe a representation of the Lie algebra of <span><math><msup><mrow><mi>B</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>-invariant functions on the dual space of Lie algebra <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> (under the canonical Poisson structure) by vector fields on <span><math><mi>S</mi><msub><mrow><mi>O</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105689"},"PeriodicalIF":1.2,"publicationDate":"2025-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417178","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 : 2025-10-24DOI: 10.1016/j.geomphys.2025.105687
Dmitrii Adler , Valery Gritsenko
We study modular differential equations (MDEs) of high orders and find necessary conditions for weak Jacobi forms to satisfy MDEs of order 3 with respect to the heat operator. We investigate all possible MDEs for weak Jacobi forms of weight 0 and index 3. This is the target space for the elliptic genus of the compact complex manifolds of dimension 6 with trivial first Chern class. We prove that the minimal possible order of MDEs of such Jacobi forms is four. Moreover, we find all such forms and show that only three of them might be the elliptic genus of strict Calabi–Yau six-folds. We describe also a discrete set of Jacobi forms satisfying fifth-order MDEs and the divisor of forms satisfying sixth-order MDEs. Then we prove that a Jacobi form of weight 0 and index 3 which does not belong to a smooth cubic in the space of coefficients satisfies a MDE of order 7. We provide such MDEs for the elliptic genus of 6-dimensional holomorphic symplectic varieties of types , , and OG6.
{"title":"Modular differential equations of minimal orders of the elliptic genus of Calabi–Yau varieties","authors":"Dmitrii Adler , Valery Gritsenko","doi":"10.1016/j.geomphys.2025.105687","DOIUrl":"10.1016/j.geomphys.2025.105687","url":null,"abstract":"<div><div>We study modular differential equations (MDEs) of high orders and find necessary conditions for weak Jacobi forms to satisfy MDEs of order 3 with respect to the heat operator. We investigate all possible MDEs for weak Jacobi forms of weight 0 and index 3. This is the target space for the elliptic genus of the compact complex manifolds of dimension 6 with trivial first Chern class. We prove that the minimal possible order of MDEs of such Jacobi forms is four. Moreover, we find all such forms and show that only three of them might be the elliptic genus of strict Calabi–Yau six-folds. We describe also a discrete set of Jacobi forms satisfying fifth-order MDEs and the divisor of forms satisfying sixth-order MDEs. Then we prove that a Jacobi form of weight 0 and index 3 which does not belong to a smooth cubic in the space of coefficients satisfies a MDE of order 7. We provide such MDEs for the elliptic genus of 6-dimensional holomorphic symplectic varieties of types <span><math><msup><mrow><mtext>Hilb</mtext></mrow><mrow><mo>[</mo><mn>3</mn><mo>]</mo></mrow></msup><mo>(</mo><mi>K</mi><mn>3</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mtext>Kum</mtext></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, and OG6.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105687"},"PeriodicalIF":1.2,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417181","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 : 2025-10-24DOI: 10.1016/j.geomphys.2025.105688
Gianni Manno, Filippo Salis
A classical and long-staying problem addressed, among others, by Calabi and Chern, is that to find a complete list of mutually non-isometric Kähler-Einstein manifolds immersed in a finite-dimensional Kähler space form. We address the same problem in the para-Kähler context and, then, we find a list of mutually non-isometric toric para-Kähler manifolds analytically immersed in a finite-dimensional para-Kähler space form.
{"title":"Toric para-Kähler-Einstein manifolds immersed in para-Kähler space forms","authors":"Gianni Manno, Filippo Salis","doi":"10.1016/j.geomphys.2025.105688","DOIUrl":"10.1016/j.geomphys.2025.105688","url":null,"abstract":"<div><div>A classical and long-staying problem addressed, among others, by Calabi and Chern, is that to find a complete list of mutually non-isometric Kähler-Einstein manifolds immersed in a finite-dimensional Kähler space form. We address the same problem in the para-Kähler context and, then, we find a list of mutually non-isometric toric para-Kähler manifolds analytically immersed in a finite-dimensional para-Kähler space form.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105688"},"PeriodicalIF":1.2,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417183","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 : 2025-10-22DOI: 10.1016/j.geomphys.2025.105681
Miguel Manzano , Argam Ohanyan , Roland Steinbauer
The cut-and-paste method is a procedure for constructing null thin shells by matching two regions of the same spacetime across a null hypersurface. Originally proposed by Penrose, it has so far allowed to describe purely gravitational and null-dust shells in constant-curvature backgrounds. In this paper, we extend the cut-and-paste method to null shells with arbitrary gravitational/matter content. To that aim, we first derive a locally Lipschitz continuous form of the metric of the spacetime resulting from the most general matching of two constant-curvature spacetimes with totally geodesic null boundaries, and then obtain the coordinate transformation that turns this metric into the cut-and-paste form with a Dirac-delta term. The paper includes an example of a null shell with non-trivial energy density, energy flux and pressure in Minkowski space.
{"title":"Generalizing the Penrose cut-and-paste method: Null shells with pressure and energy flux","authors":"Miguel Manzano , Argam Ohanyan , Roland Steinbauer","doi":"10.1016/j.geomphys.2025.105681","DOIUrl":"10.1016/j.geomphys.2025.105681","url":null,"abstract":"<div><div>The cut-and-paste method is a procedure for constructing null thin shells by matching two regions of the same spacetime across a null hypersurface. Originally proposed by Penrose, it has so far allowed to describe purely gravitational and null-dust shells in constant-curvature backgrounds. In this paper, we extend the cut-and-paste method to null shells with arbitrary gravitational/matter content. To that aim, we first derive a locally Lipschitz continuous form of the metric of the spacetime resulting from the most general matching of two constant-curvature spacetimes with totally geodesic null boundaries, and then obtain the coordinate transformation that turns this metric into the cut-and-paste form with a Dirac-delta term. The paper includes an example of a null shell with non-trivial energy density, energy flux and pressure in Minkowski space.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105681"},"PeriodicalIF":1.2,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363229","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 : 2025-10-22DOI: 10.1016/j.geomphys.2025.105683
Nanyan Xu, Yunhe Sheng
In this paper, first we show that a central Leibniz 2-algebra naturally gives rise to a solution of the Zamolodchikov Tetrahedron equation. Then we introduce the notion of linear 2-racks and show that a linear 2-rack also gives rise to a solution of the Zamolodchikov Tetrahedron equation. We show that a central Leibniz 2-algebra gives rise to a linear 2-rack if the underlying 2-vector space is splittable. Finally we discuss the relation between linear 2-racks and 2-racks, and show that a linear 2-rack gives rise to a 2-rack structure on the group-like category. A concrete example of strict 2-racks is constructed from an action of a strict 2-group.
{"title":"Leibniz 2-algebras, linear 2-racks and the Zamolodchikov Tetrahedron equation","authors":"Nanyan Xu, Yunhe Sheng","doi":"10.1016/j.geomphys.2025.105683","DOIUrl":"10.1016/j.geomphys.2025.105683","url":null,"abstract":"<div><div>In this paper, first we show that a central Leibniz 2-algebra naturally gives rise to a solution of the Zamolodchikov Tetrahedron equation. Then we introduce the notion of linear 2-racks and show that a linear 2-rack also gives rise to a solution of the Zamolodchikov Tetrahedron equation. We show that a central Leibniz 2-algebra gives rise to a linear 2-rack if the underlying 2-vector space is splittable. Finally we discuss the relation between linear 2-racks and 2-racks, and show that a linear 2-rack gives rise to a 2-rack structure on the group-like category. A concrete example of strict 2-racks is constructed from an action of a strict 2-group.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105683"},"PeriodicalIF":1.2,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417176","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 : 2025-10-22DOI: 10.1016/j.geomphys.2025.105684
Alberto S. Cattaneo, Nima Moshayedi
The recently introduced equivariant BV formalism is extended to the case of manifolds with boundary under appropriate conditions. AKSZ theories are presented as a practical example.
在适当的条件下，将最近引入的等变BV形式推广到具有边界的流形。并以实例介绍了AKSZ理论。
{"title":"Equivariant BV-BFV formalism","authors":"Alberto S. Cattaneo, Nima Moshayedi","doi":"10.1016/j.geomphys.2025.105684","DOIUrl":"10.1016/j.geomphys.2025.105684","url":null,"abstract":"<div><div>The recently introduced equivariant BV formalism is extended to the case of manifolds with boundary under appropriate conditions. AKSZ theories are presented as a practical example.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105684"},"PeriodicalIF":1.2,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417177","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 : 2025-10-22DOI: 10.1016/j.geomphys.2025.105685
Xiaojian Shi, Xiaoqing Yue
The gap-p Virasoro algebra, which is closely related to the Heisenberg-Virasoro algebra and the algebra of derivations over a quantum torus, plays an important role in both mathematics and mathematical physics. In this paper, we first construct the gap-p Virasoro Lie conformal algebra from gap-p Virasoro algebra. Then we concretely determine the conformal derivations and conformal biderivations of this Lie conformal algebra. Furthermore, we investigate finite irreducible conformal modules and characterize nontrivial central extensions of . Based on these results, we finally give a complete classification of extensions of finite irreducible conformal modules over gap-p Virasoro Lie conformal algebra .
{"title":"Gap-p Virasoro Lie conformal algebra and extensions of modules","authors":"Xiaojian Shi, Xiaoqing Yue","doi":"10.1016/j.geomphys.2025.105685","DOIUrl":"10.1016/j.geomphys.2025.105685","url":null,"abstract":"<div><div>The gap-<em>p</em> Virasoro algebra, which is closely related to the Heisenberg-Virasoro algebra and the algebra of derivations over a quantum torus, plays an important role in both mathematics and mathematical physics. In this paper, we first construct the gap-<em>p</em> Virasoro Lie conformal algebra <span><math><msup><mrow><mi>HV</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> from gap-<em>p</em> Virasoro algebra. Then we concretely determine the conformal derivations and conformal biderivations of this Lie conformal algebra. Furthermore, we investigate finite irreducible conformal modules and characterize nontrivial central extensions of <span><math><msup><mrow><mi>HV</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>. Based on these results, we finally give a complete classification of extensions of finite irreducible conformal modules over gap-<em>p</em> Virasoro Lie conformal algebra <span><math><msup><mrow><mi>HV</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105685"},"PeriodicalIF":1.2,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417179","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 : 2025-10-22DOI: 10.1016/j.geomphys.2025.105686
Yaxi Jiang, Chuangchuang Kang, Jiafeng Lü
-associative algebra is a generalization of associative algebra and is one of the four remarkable types of Lie-admissible algebras, along with associative algebra, left-symmetric algebra and right-symmetric algebra. This paper develops bialgebra theory for -associative algebras. We introduce Manin triples and bialgebras for -associative algebras, prove their equivalence using matched pairs of -associative algebras, and define the -associative Yang-Baxter equation and triangular -associative bialgebras. Additionally, we introduce relative Rota-Baxter operators to provide skew-symmetric solutions of the -associative Yang-Baxter equation.
{"title":"Manin triples, bialgebras and Yang-Baxter equation of A3-associative algebras","authors":"Yaxi Jiang, Chuangchuang Kang, Jiafeng Lü","doi":"10.1016/j.geomphys.2025.105686","DOIUrl":"10.1016/j.geomphys.2025.105686","url":null,"abstract":"<div><div><span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-associative algebra is a generalization of associative algebra and is one of the four remarkable types of Lie-admissible algebras, along with associative algebra, left-symmetric algebra and right-symmetric algebra. This paper develops bialgebra theory for <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-associative algebras. We introduce Manin triples and bialgebras for <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-associative algebras, prove their equivalence using matched pairs of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-associative algebras, and define the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-associative Yang-Baxter equation and triangular <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-associative bialgebras. Additionally, we introduce relative Rota-Baxter operators to provide skew-symmetric solutions of the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-associative Yang-Baxter equation.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105686"},"PeriodicalIF":1.2,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417180","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 : 2025-10-21DOI: 10.1016/j.geomphys.2025.105682
Eric Jankowski
We provide an elementary proof that with the exceptions of certain Π-projective spaces, both the Picard group and the Π-Picard set of the isomeric (i.e. type-Q) supergrassmannian are trivial. We extend this technique to show that the Picard group and the Π-Picard set of a supertorus orbit closure within the isomeric supergrassmannian can be easily calculated from its defining polytope by counting the number of simplex factors. Since the presence of nontrivial invertible sheaves and Π-invertible sheaves depends entirely on factors of Π-projective space, we construct them as symmetric powers of the tautological sheaf and its dual.
{"title":"Invertible sheaves and Π-invertible sheaves on the isomeric supergrassmannian and its toric subvarieties","authors":"Eric Jankowski","doi":"10.1016/j.geomphys.2025.105682","DOIUrl":"10.1016/j.geomphys.2025.105682","url":null,"abstract":"<div><div>We provide an elementary proof that with the exceptions of certain Π-projective spaces, both the Picard group and the Π-Picard set of the isomeric (i.e. type-Q) supergrassmannian are trivial. We extend this technique to show that the Picard group and the Π-Picard set of a supertorus orbit closure within the isomeric supergrassmannian can be easily calculated from its defining polytope by counting the number of simplex factors. Since the presence of nontrivial invertible sheaves and Π-invertible sheaves depends entirely on factors of Π-projective space, we construct them as symmetric powers of the tautological sheaf and its dual.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105682"},"PeriodicalIF":1.2,"publicationDate":"2025-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363230","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 : 2025-10-17DOI: 10.1016/j.geomphys.2025.105679
Yu Feng , Bin Xu
In this paper, we construct a stable parabolic Higgs bundle of rank two, which corresponds to the uniformization associated with a conformal hyperbolic metric on a compact Riemann surface with prescribed singularities. This provides an alternative proof of the classical existence theorem for singular hyperbolic metrics, originally established by Heins (1962) [8]. We also introduce a family of stable parabolic Higgs bundles of rank two on , parametrized by a nonempty open subset of a complex vector space. These bundles correspond to singular hyperbolic metrics with the same type of singularity as the original, but are defined on deformed Riemann surfaces of . Thus, we extend partially the final section of Hitchin's celebrated work (Hitchin (1987) [9]) to the context of hyperbolic metrics with singularities.
{"title":"Stable parabolic Higgs bundles of rank two and singular hyperbolic metrics","authors":"Yu Feng , Bin Xu","doi":"10.1016/j.geomphys.2025.105679","DOIUrl":"10.1016/j.geomphys.2025.105679","url":null,"abstract":"<div><div>In this paper, we construct a stable parabolic Higgs bundle of rank two, which corresponds to the uniformization associated with a conformal hyperbolic metric on a compact Riemann surface <span><math><mover><mrow><mi>X</mi></mrow><mo>‾</mo></mover></math></span> with prescribed singularities. This provides an alternative proof of the classical existence theorem for singular hyperbolic metrics, originally established by Heins (1962) <span><span>[8]</span></span>. We also introduce a family of stable parabolic Higgs bundles of rank two on <span><math><mover><mrow><mi>X</mi></mrow><mo>‾</mo></mover></math></span>, parametrized by a nonempty open subset of a complex vector space. These bundles correspond to singular hyperbolic metrics with the same type of singularity as the original, but are defined on deformed Riemann surfaces of <span><math><mover><mrow><mi>X</mi></mrow><mo>‾</mo></mover></math></span>. Thus, we extend partially the final section of Hitchin's celebrated work (Hitchin (1987) <span><span>[9]</span></span>) to the context of hyperbolic metrics with singularities.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105679"},"PeriodicalIF":1.2,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363231","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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