Pub Date : 2026-04-15Epub Date: 2026-01-05DOI: 10.1016/j.jalgebra.2025.12.017
Yunsong Gan , Weijun Liu , Binzhou Xia
A regular bipartite graph Γ is called semisymmetric if its full automorphism group acts transitively on the edge set but not on the vertex set. For a subgroup G of that stabilizes the biparts of Γ, we say that Γ is G-biprimitive if G acts primitively on each part. In this paper, we first provide a method to construct infinite families of biprimitive semisymmetric graphs admitting almost simple groups. With the aid of this result, a classification of G-biprimitive semisymmetric graphs is obtained for or . In pursuit of this goal, we determine all pairs of almost simple groups of the same order and all pairs of maximal subgroups of or with the same order.
{"title":"On biprimitive semisymmetric graphs","authors":"Yunsong Gan , Weijun Liu , Binzhou Xia","doi":"10.1016/j.jalgebra.2025.12.017","DOIUrl":"10.1016/j.jalgebra.2025.12.017","url":null,"abstract":"<div><div>A regular bipartite graph Γ is called semisymmetric if its full automorphism group <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> acts transitively on the edge set but not on the vertex set. For a subgroup <em>G</em> of <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> that stabilizes the biparts of Γ, we say that Γ is <em>G</em>-biprimitive if <em>G</em> acts primitively on each part. In this paper, we first provide a method to construct infinite families of biprimitive semisymmetric graphs admitting almost simple groups. With the aid of this result, a classification of <em>G</em>-biprimitive semisymmetric graphs is obtained for <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In pursuit of this goal, we determine all pairs of almost simple groups of the same order and all pairs of maximal subgroups of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with the same order.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 422-462"},"PeriodicalIF":0.8,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2025-12-05DOI: 10.1016/j.jalgebra.2025.11.028
Darlayne Addabbo , Christoph A. Keller
It is known from Zhu's results that under modular transformations, correlators of rational -cofinite vertex operator algebras transform like Jacobi forms. We investigate the modular transformation properties of VOA correlators that have zero modes inserted. We derive recursion relations for such correlators and use them to establish modular transformation properties. For holomorphic VOAs we find that correlators with only zero modes transform like quasi-modular forms, and mixed correlators with both zero modes and vertex operators transform like quasi-Jacobi forms. As an application of our results, we introduce algebras of higher weight fields whose zero mode correlators mimic the properties of those of weight 1 fields. We also give a simplified proof of the weight 1 transformation properties originally proven by Miyamoto.
{"title":"Modularity of vertex operator algebra correlators with zero modes","authors":"Darlayne Addabbo , Christoph A. Keller","doi":"10.1016/j.jalgebra.2025.11.028","DOIUrl":"10.1016/j.jalgebra.2025.11.028","url":null,"abstract":"<div><div>It is known from Zhu's results that under modular transformations, correlators of rational <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-cofinite vertex operator algebras transform like Jacobi forms. We investigate the modular transformation properties of VOA correlators that have zero modes inserted. We derive recursion relations for such correlators and use them to establish modular transformation properties. For holomorphic VOAs we find that correlators with only zero modes transform like quasi-modular forms, and mixed correlators with both zero modes and vertex operators transform like quasi-Jacobi forms. As an application of our results, we introduce algebras of higher weight fields whose zero mode correlators mimic the properties of those of weight 1 fields. We also give a simplified proof of the weight 1 transformation properties originally proven by Miyamoto.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 27-69"},"PeriodicalIF":0.8,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145750306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2025-10-30DOI: 10.1016/j.jalgebra.2025.10.027
Ramla Abdellatif , Mabud Ali Sarkar
In this paper, we construct a class of 2-dimensional formal groups over that provide a higher-dimensional analogue of the usual 1-dimensional Lubin-Tate formal groups, then we initiate the study of the extensions generated by their -torsion points. For instance, we prove that the coordinates of the -torsion points of such a formal group generate an abelian extension over a certain unramified extension of , and we study some ramification properties of these abelian extensions. In particular, we prove that the extension generated by the coordinates of the p-torsion points is in general totally ramified.
{"title":"Constructing 2-dimensional Lubin-Tate formal groups over Zp (I)","authors":"Ramla Abdellatif , Mabud Ali Sarkar","doi":"10.1016/j.jalgebra.2025.10.027","DOIUrl":"10.1016/j.jalgebra.2025.10.027","url":null,"abstract":"<div><div>In this paper, we construct a class of 2-dimensional formal groups over <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> that provide a higher-dimensional analogue of the usual 1-dimensional Lubin-Tate formal groups, then we initiate the study of the extensions generated by their <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>-torsion points. For instance, we prove that the coordinates of the <span><math><msup><mrow><mi>p</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>-torsion points of such a formal group generate an abelian extension over a certain unramified extension of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, and we study some ramification properties of these abelian extensions. In particular, we prove that the extension generated by the coordinates of the <em>p</em>-torsion points is in general totally ramified.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 205-237"},"PeriodicalIF":0.8,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2026-01-08DOI: 10.1016/j.jalgebra.2025.12.021
Gerhard Hiss , Rafał Lutowski
We prove a conjecture of Dekimpe, De Rock and Penninckx concerning the existence of eigenvalues one in certain elements of finite groups acting irreducibly on a real vector space of odd dimension. This yields a sufficient condition for a closed flat manifold to be an -manifold.
{"title":"The eigenvalue one property of finite groups, I","authors":"Gerhard Hiss , Rafał Lutowski","doi":"10.1016/j.jalgebra.2025.12.021","DOIUrl":"10.1016/j.jalgebra.2025.12.021","url":null,"abstract":"<div><div>We prove a conjecture of Dekimpe, De Rock and Penninckx concerning the existence of eigenvalues one in certain elements of finite groups acting irreducibly on a real vector space of odd dimension. This yields a sufficient condition for a closed flat manifold to be an <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-manifold.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 592-626"},"PeriodicalIF":0.8,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2026-01-02DOI: 10.1016/j.jalgebra.2025.12.019
Oksana S. Yakimova
Let be an algebraic Lie algebra of index 1, i.e., a generic Q-orbit on has codimension 1. We show that the following conditions are equivalent: is contact; a generic Q-orbit on is not conical; there is a generic stabiliser for the coadjoint action of . In addition, if is contact, then the subalgebra generated by symmetric semi-invariants of is a polynomial ring. We study also affine seaweed Lie algebras of type A and find some contact as well as non-contact examples among them.
{"title":"Contact Lie algebras, generic stabilisers, and affine seaweeds","authors":"Oksana S. Yakimova","doi":"10.1016/j.jalgebra.2025.12.019","DOIUrl":"10.1016/j.jalgebra.2025.12.019","url":null,"abstract":"<div><div>Let <span><math><mi>q</mi><mo>=</mo><mrow><mi>Lie</mi><mspace></mspace></mrow><mi>Q</mi></math></span> be an algebraic Lie algebra of index 1, i.e., a generic <em>Q</em>-orbit on <span><math><msup><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> has codimension 1. We show that the following conditions are equivalent: <span><math><mi>q</mi></math></span> is contact; a generic <em>Q</em>-orbit on <span><math><msup><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is not conical; there is a generic stabiliser for the coadjoint action of <span><math><mi>q</mi></math></span>. In addition, if <span><math><mi>q</mi></math></span> is contact, then the subalgebra <span><math><mi>S</mi><msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mrow><mi>si</mi></mrow></msub><mo>⊂</mo><mi>S</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> generated by symmetric semi-invariants of <span><math><mi>q</mi></math></span> is a polynomial ring. We study also affine seaweed Lie algebras of type <span>A</span> and find some contact as well as non-contact examples among them.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 401-421"},"PeriodicalIF":0.8,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2025-12-08DOI: 10.1016/j.jalgebra.2025.12.006
Victor Guba
A (discrete) group is called amenable if there exists a finitely additive right-invariant probability measure on it. The question of whether Thompson's group F is amenable is a long-standing open problem. We consider the presentation of F in terms of non-spherical semigroup diagrams. There is a natural partition of F into 7 parts in terms of these diagrams. We show that for any finitely additive right-invariant probability measure on F, all but one of these sets have zero measure. This helps to clarify the structure of Følner sets in F, provided the group is amenable.
{"title":"On zero-measured subsets of Thompson's group F","authors":"Victor Guba","doi":"10.1016/j.jalgebra.2025.12.006","DOIUrl":"10.1016/j.jalgebra.2025.12.006","url":null,"abstract":"<div><div>A (discrete) group is called <em>amenable</em> if there exists a finitely additive right-invariant probability measure on it. The question of whether Thompson's group <em>F</em> is amenable is a long-standing open problem. We consider the presentation of <em>F</em> in terms of non-spherical semigroup diagrams. There is a natural partition of <em>F</em> into 7 parts in terms of these diagrams. We show that for any finitely additive right-invariant probability measure on <em>F</em>, all but one of these sets have zero measure. This helps to clarify the structure of Følner sets in <em>F</em>, provided the group is amenable.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 106-122"},"PeriodicalIF":0.8,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145750307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2025-12-31DOI: 10.1016/j.jalgebra.2025.12.016
Sagar Saha, K.V. Krishna
In this work, we provide the first example of an infinite family of branch groups in the class of non-contracting self-similar groups. We show that these groups are super strongly fractal, not regular branch, and of exponential growth. Further, we prove that these groups do not have the congruence subgroup property by explicitly calculating the structure of their rigid kernels. This class of groups is also the first example of branch groups with non-torsion rigid kernels. As a consequence of these results, we also determine the Hausdorff dimension of these groups.
{"title":"A class of non-contracting branch groups with non-torsion rigid kernels","authors":"Sagar Saha, K.V. Krishna","doi":"10.1016/j.jalgebra.2025.12.016","DOIUrl":"10.1016/j.jalgebra.2025.12.016","url":null,"abstract":"<div><div>In this work, we provide the first example of an infinite family of branch groups in the class of non-contracting self-similar groups. We show that these groups are super strongly fractal, not regular branch, and of exponential growth. Further, we prove that these groups do not have the congruence subgroup property by explicitly calculating the structure of their rigid kernels. This class of groups is also the first example of branch groups with non-torsion rigid kernels. As a consequence of these results, we also determine the Hausdorff dimension of these groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 379-400"},"PeriodicalIF":0.8,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2025-12-08DOI: 10.1016/j.jalgebra.2025.11.026
Rebecca Goldin , Martha Precup
We study families of matrix Hessenberg schemes in the affine scheme of complex matrices, each defined over a fixed sheet in the Lie algebra . Abe, Fujita and Zeng show in [3] that such families over the regular sheet are flat, and every regular Hessenberg scheme degenerates to a regular nilpotent Hessenberg scheme. This paper explores whether flat degenerations exist outside of the regular case.
For each matrix Hessenberg scheme, we introduce a one-parameter family of matrix Hessenberg schemes that degenerates it to a specific nilpotent Hessenberg scheme. Our main theorem states that, when the family lies over the minimal sheet in , this degeneration is flat. The proof leverages commutative algebra on the polynomial ring to identify the structure of the family concretely, and we explore several applications. We conjecture that flatness holds for these families over other sheets as well.
{"title":"A flat family of matrix Hessenberg schemes over the minimal sheet","authors":"Rebecca Goldin , Martha Precup","doi":"10.1016/j.jalgebra.2025.11.026","DOIUrl":"10.1016/j.jalgebra.2025.11.026","url":null,"abstract":"<div><div>We study families of matrix Hessenberg schemes in the affine scheme of complex <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices, each defined over a fixed sheet in the Lie algebra <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span>. Abe, Fujita and Zeng show in <span><span>[3]</span></span> that such families over the regular sheet are flat, and every regular Hessenberg scheme degenerates to a regular nilpotent Hessenberg scheme. This paper explores whether flat degenerations exist outside of the regular case.</div><div>For each matrix Hessenberg scheme, we introduce a one-parameter family of matrix Hessenberg schemes that degenerates it to a specific nilpotent Hessenberg scheme. Our main theorem states that, when the family lies over the minimal sheet in <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span>, this degeneration is flat. The proof leverages commutative algebra on the polynomial ring to identify the structure of the family concretely, and we explore several applications. We conjecture that flatness holds for these families over other sheets as well.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 123-172"},"PeriodicalIF":0.8,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2025-12-09DOI: 10.1016/j.jalgebra.2025.12.009
Daniel Rogalski , Robert Won , James J. Zhang
The notion of a homological integral of an infinite-dimensional weak Hopf algebra is introduced. We show that the homological integral is an invertible object in the associated monoidal category. Using integrals, we prove that the Artin–Schelter property and the Van den Bergh condition are equivalent for a noetherian weak Hopf algebra, and that the antipode is automatically invertible in this case. We also prove that any weak Hopf algebra finite over an affine center is a direct sum of Artin–Schelter Gorenstein weak Hopf algebras.
引入了无穷维弱Hopf代数的同调积分的概念。证明了同调积分在相关的一元范畴中是可逆对象。利用积分证明了noether弱Hopf代数的Artin-Schelter性质和Van den Bergh条件是等价的，并且在这种情况下对映对是自动可逆的。我们还证明了在仿射中心上有限的任何弱Hopf代数是Artin-Schelter Gorenstein弱Hopf代数的直接和。
{"title":"Homological integrals for weak Hopf algebras","authors":"Daniel Rogalski , Robert Won , James J. Zhang","doi":"10.1016/j.jalgebra.2025.12.009","DOIUrl":"10.1016/j.jalgebra.2025.12.009","url":null,"abstract":"<div><div>The notion of a homological integral of an infinite-dimensional weak Hopf algebra is introduced. We show that the homological integral is an invertible object in the associated monoidal category. Using integrals, we prove that the Artin–Schelter property and the Van den Bergh condition are equivalent for a noetherian weak Hopf algebra, and that the antipode is automatically invertible in this case. We also prove that any weak Hopf algebra finite over an affine center is a direct sum of Artin–Schelter Gorenstein weak Hopf algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 173-204"},"PeriodicalIF":0.8,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2025-12-12DOI: 10.1016/j.jalgebra.2025.12.012
Xiangqian Guo , Xuewen Liu , Junzhou Qin
In this paper, we study the higher rank polynomial modules for the quantum group , improving Bavula's description of irreducible nonweight modules using the terminology of generalized Weyl algebras, constructing examples of irreducible polynomial modules of arbitrary finite rank, describing irreducible polynomial modules in terms of Smith normal form, and presenting properties of dual polynomial modules. Then we consider the tensor products of irreducible polynomial modules with finite-dimensional simple modules, obtaining a direct sum decomposition formula, similar to classical Clebsch-Gordan formula, under some conditions. These results are generalizations of our previous results for the rank-1 polynomial modules.
{"title":"Higher rank polynomial modules over Uq(sl2)","authors":"Xiangqian Guo , Xuewen Liu , Junzhou Qin","doi":"10.1016/j.jalgebra.2025.12.012","DOIUrl":"10.1016/j.jalgebra.2025.12.012","url":null,"abstract":"<div><div>In this paper, we study the higher rank polynomial modules for the quantum group <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, improving Bavula's description of irreducible nonweight modules using the terminology of generalized Weyl algebras, constructing examples of irreducible polynomial modules of arbitrary finite rank, describing irreducible polynomial modules in terms of Smith normal form, and presenting properties of dual polynomial modules. Then we consider the tensor products of irreducible polynomial modules with finite-dimensional simple modules, obtaining a direct sum decomposition formula, similar to classical Clebsch-Gordan formula, under some conditions. These results are generalizations of our previous results for the rank-1 polynomial modules.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 238-270"},"PeriodicalIF":0.8,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号