Journal of Algebra最新文献

{"title":"A generalization of the Arad–Ward theorem on Hall subgroups","authors":"N. Yang ,&nbsp;A.A. Buturlakin","doi":"10.1016/j.jalgebra.2025.05.001","DOIUrl":"10.1016/j.jalgebra.2025.05.001","url":null,"abstract":"<div><div>For a set of primes <em>π</em>, denote by <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span> the class of finite groups containing a Hall <em>π</em>-subgroup. We establish that <span><math><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>∩</mo><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> is contained in <span><math><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span>. As a corollary, we prove that if <em>π</em> is a set of primes, <em>l</em> is an integer such that <span><math><mn>2</mn><mo>⩽</mo><mi>l</mi><mo>&lt;</mo><mo>|</mo><mi>π</mi><mo>|</mo></math></span> and <em>G</em> is a finite group that contains a Hall <em>ρ</em>-subgroup for every subset <em>ρ</em> of <em>π</em> of size <em>l</em>, then <em>G</em> contains a solvable Hall <em>π</em>-subgroup.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"679 ","pages":"Pages 28-36"},"PeriodicalIF":0.8,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144084155","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}

{"title":"Hopf–Galois structures on parallel extensions","authors":"Andrew Darlington","doi":"10.1016/j.jalgebra.2025.04.025","DOIUrl":"10.1016/j.jalgebra.2025.04.025","url":null,"abstract":"<div><div>Let <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> be a finite separable extension of fields of degree <em>n</em>, and let <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> be its Galois closure. Greither and Pareigis showed how to find all Hopf–Galois structures on <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span>. We will call a subextension <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>/</mo><mi>K</mi></math></span> of <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> <em>parallel</em> to <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> if <span><math><mo>[</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>:</mo><mi>K</mi><mo>]</mo><mo>=</mo><mi>n</mi></math></span>.</div><div>In this paper, we investigate the relationship between the Hopf–Galois structures on an extension <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> and those on the related parallel extensions. We give an example of a transitive subgroup corresponding to an extension admitting a Hopf–Galois structure but that has a parallel extension admitting no Hopf–Galois structures. We show that once one has such a situation, it can be extended into an infinite family of transitive subgroups admitting this phenomenon. We also investigate this fully in the case of extensions of degree <em>pq</em> with <span><math><mi>p</mi><mo>,</mo><mi>q</mi></math></span> distinct odd primes, and show that there is no example of such an extension admitting the phenomenon.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"679 ","pages":"Pages 1-27"},"PeriodicalIF":0.8,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069639","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}

{"title":"Corrigendum to “The Gruenberg-Kegel graph of finite solvable rational groups” [J. Algebra 642 (2024) 470–479]","authors":"Sara C. Debón ,&nbsp;Diego García-Lucas ,&nbsp;Ángel del Río","doi":"10.1016/j.jalgebra.2025.03.051","DOIUrl":"10.1016/j.jalgebra.2025.03.051","url":null,"abstract":"","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 618-629"},"PeriodicalIF":0.8,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143918221","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}

{"title":"On N-graded vertex algebras associated with Gorenstein algebras","authors":"Alex Keene,&nbsp;Christian Soltermann,&nbsp;Gaywalee Yamskulna","doi":"10.1016/j.jalgebra.2025.04.032","DOIUrl":"10.1016/j.jalgebra.2025.04.032","url":null,"abstract":"<div><div>This paper investigates the algebraic structure of indecomposable <span><math><mi>N</mi></math></span>-graded vertex algebras <span><math><mi>V</mi><mo>=</mo><msubsup><mrow><mo>⨁</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>V</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, emphasizing the intricate interactions between the commutative associative algebra <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, the Leibniz algebra <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and how non-degenerate bilinear forms on <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> influence their overall structure. We establish foundational properties for indecomposability and locality in <span><math><mi>N</mi></math></span>-graded vertex algebras, with our main result demonstrating the equivalence of locality, indecomposability, and specific structural conditions on semiconformal-vertex algebras. The study of symmetric invariant bilinear forms of semiconformal-vertex algebra is investigated. We also examine the structural characteristics of <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, demonstrating conditions under which certain <span><math><mi>N</mi></math></span>-graded vertex algebras cannot be quasi vertex operator algebras, semiconformal-vertex algebras, or vertex operator algebras, and explore <span><math><mi>N</mi></math></span>-graded vertex algebras <span><math><mi>V</mi><mo>=</mo><msubsup><mrow><mo>⨁</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>V</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> associated with Gorenstein algebras. Our analysis includes examining the socle, Poincaré duality properties, and invariant bilinear forms of <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and their influence on <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, providing conditions for embedding rank-one Heisenberg vertex operator algebras within <em>V</em>. Supporting examples and detailed theoretical insights further illustrate these algebraic structures.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"678 ","pages":"Pages 729-768"},"PeriodicalIF":0.8,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143934666","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}

{"title":"Homotopy equivalences and Grothendieck duality over rings with finite Gorenstein weak global dimension","authors":"Junpeng Wang ,&nbsp;Sergio Estrada","doi":"10.1016/j.jalgebra.2025.04.033","DOIUrl":"10.1016/j.jalgebra.2025.04.033","url":null,"abstract":"<div><div>Let <em>R</em> be a ring with Gwgldim<span><math><mo>(</mo><mi>R</mi><mo>)</mo><mo>&lt;</mo><mo>∞</mo></math></span>. We obtain a triangle-equivalence <span><math><mtext>K</mtext><mo>(</mo><mi>R</mi><mtext>-</mtext><mtext>GProj</mtext><mo>)</mo><mo>≃</mo><mtext>K</mtext><mo>(</mo><mi>R</mi><mtext>-</mtext><mtext>GInj</mtext><mo>)</mo></math></span> which restricts to a triangle-equivalence <span><math><mtext>K</mtext><mo>(</mo><mi>R</mi><mtext>-</mtext><mtext>Proj</mtext><mo>)</mo></math></span> <span><math><mo>≃</mo><mtext>K</mtext><mo>(</mo><mi>R</mi><mtext>-</mtext><mtext>Inj</mtext><mo>)</mo></math></span>. This class of rings includes, among others, (left) Gorenstein rings, Ding–Chen rings and the more general Gorenstein <em>n</em>-coherent rings (<span><math><mi>n</mi><mo>∈</mo><mi>N</mi><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo><mo>,</mo><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>). As application, we establish some triangle-equivalences of Grothendieck duality over Ding–Chen rings and Gorenstein <em>n</em>-coherent rings.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"678 ","pages":"Pages 769-808"},"PeriodicalIF":0.8,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935669","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}

{"title":"On the uniqueness of maximal solvable extensions of nilpotent Leibniz superalgebras","authors":"B.A. Omirov ,&nbsp;G.O. Solijanova","doi":"10.1016/j.jalgebra.2025.04.035","DOIUrl":"10.1016/j.jalgebra.2025.04.035","url":null,"abstract":"<div><div>In the present paper under certain conditions the description of the maximal solvable extension of complex finite-dimensional nilpotent Leibniz superalgebras is obtained. Specifically, we establish that under the condition ensuring the fulfillment of Lie's theorem for a maximal solvable extension of a special kind of nilpotent Leibniz superalgebra (which is consistent and <em>d</em>-locally diagonalizable), it is decomposed into a semidirect sum of a nilpotent Leibniz superalgebra and a maximal torus on it. In other words, under certain conditions the direct sum of the nilpotent superalgebra and its torus (as a vector spaces), admits a solvable Leibniz superalgebra structure. In addition, for the left-side action of a maximal torus on nilpotent Leibniz superalgebra, which does not admit <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> as a direct summand and is diagonalizable, we prove the uniqueness of the maximal extension. Along with the answer to Šnobl's conjecture for Lie algebras this result covers several already known results for Lie (super)algebras and Leibniz algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 798-832"},"PeriodicalIF":0.8,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935173","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}

{"title":"On structural connections between sandpile monoids and weighted Leavitt path algebras","authors":"Roozbeh Hazrat ,&nbsp;Tran Giang Nam","doi":"10.1016/j.jalgebra.2025.04.034","DOIUrl":"10.1016/j.jalgebra.2025.04.034","url":null,"abstract":"<div><div>In this article, we establish the relations between a sandpile graph, its sandpile monoid and the weighted Leavitt path algebra associated with it. Namely, we show that the lattice of all idempotents of the sandpile monoid <span><math><mtext>SP</mtext><mo>(</mo><mi>E</mi><mo>)</mo></math></span> of a sandpile graph <em>E</em> is both isomorphic to the lattice of all nonempty saturated hereditary subsets of <em>E</em>, the lattice of all order-ideals of <span><math><mtext>SP</mtext><mo>(</mo><mi>E</mi><mo>)</mo></math></span> and the lattice of all ideals of the weighted Leavitt path algebra <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>E</mi><mo>,</mo><mi>ω</mi><mo>)</mo></math></span> generated by vertices. Also, we describe the sandpile group of a sandpile graph <em>E</em> via archimedean classes of <span><math><mtext>SP</mtext><mo>(</mo><mi>E</mi><mo>)</mo></math></span>, and prove that all maximal subgroups of <span><math><mtext>SP</mtext><mo>(</mo><mi>E</mi><mo>)</mo></math></span> are exactly the Grothendieck groups of these archimedean classes. Finally, we give the structure of the Leavitt path algebra <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>E</mi><mo>)</mo></math></span> of a sandpile graph <em>E</em> via a finite chain of graded ideals being invariant under every graded automorphism of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>E</mi><mo>)</mo></math></span>, and completely describe the structure of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>E</mi><mo>)</mo></math></span> such that the lattice of all idempotents of <span><math><mtext>SP</mtext><mo>(</mo><mi>E</mi><mo>)</mo></math></span> is a chain. Consequently, we completely describe the structure of the weighted Leavitt path algebra of a sandpile graph <em>E</em> such that <span><math><mtext>SP</mtext><mo>(</mo><mi>E</mi><mo>)</mo></math></span> has exactly two idempotents.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"678 ","pages":"Pages 543-569"},"PeriodicalIF":0.8,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143917760","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}

{"title":"Invariant theory of the queer color group","authors":"Tinu Dhali,&nbsp;Santosha Pattanayak,&nbsp;Preena Samuel","doi":"10.1016/j.jalgebra.2025.04.031","DOIUrl":"10.1016/j.jalgebra.2025.04.031","url":null,"abstract":"<div><div>In this paper, we introduce the notion of the queer color group, analogous to that of the queer supergroup over the infinite Grassmann algebra. We obtain a Schur-Weyl duality theorem for this group and thereby construct an explicit spanning set of invariants of the associated symmetric algebra of the mixed tensor space of a <em>G</em>-graded vector space where <em>G</em> is a finite abelian group. As a consequence, we obtain a generating set for the polynomial invariants, under the simultaneous action of the queer color group on color analogues of several copies of matrices. We also introduce the notion of concomitants for the queer color group analogous to that of Procesi in <span><span>[16]</span></span> and obtain a generating set for the algebra of concomitants. These results generalize those of Berele in <span><span>[3]</span></span> to the color setting.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"678 ","pages":"Pages 706-728"},"PeriodicalIF":0.8,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143934935","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}

{"title":"On the minimal parabolic induction","authors":"Xinyu Li","doi":"10.1016/j.jalgebra.2025.04.026","DOIUrl":"10.1016/j.jalgebra.2025.04.026","url":null,"abstract":"<div><div>Motivated by Beilinson–Bernstein's proof of the Jantzen conjectures <span><span>[4]</span></span>, we define the minimal parabolic induction functor for Kac–Moody algebras, and establish some basic properties.</div><div>As applications of the formal theory, we examine first extension groups between simple highest weight modules in the category of weight modules, and analyze the annihilators of some simple highest weight modules.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 585-617"},"PeriodicalIF":0.8,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143903505","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}

{"title":"Varieties of MV-monoids and positive MV-algebras","authors":"Marco Abbadini ,&nbsp;Paolo Aglianò ,&nbsp;Stefano Fioravanti","doi":"10.1016/j.jalgebra.2025.04.027","DOIUrl":"10.1016/j.jalgebra.2025.04.027","url":null,"abstract":"<div><div>MV-monoids are algebras <span><math><mo>〈</mo><mi>A</mi><mo>,</mo><mo>∨</mo><mo>,</mo><mo>∧</mo><mo>,</mo><mo>⊕</mo><mo>,</mo><mo>⊙</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>〉</mo></math></span> where <span><math><mo>〈</mo><mi>A</mi><mo>,</mo><mo>∨</mo><mo>,</mo><mo>∧</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>〉</mo></math></span> is a bounded distributive lattice, both <span><math><mo>〈</mo><mi>A</mi><mo>,</mo><mo>⊕</mo><mo>,</mo><mn>0</mn><mo>〉</mo></math></span> and <span><math><mo>〈</mo><mi>A</mi><mo>,</mo><mo>⊙</mo><mo>,</mo><mn>1</mn><mo>〉</mo></math></span> are commutative monoids, and some further connecting axioms are satisfied. Every MV-algebra in the signature <span><math><mo>{</mo><mo>⊕</mo><mo>,</mo><mo>¬</mo><mo>,</mo><mn>0</mn><mo>}</mo></math></span> is term equivalent to an algebra that has an MV-monoid as a reduct, by defining, as standard, <span><math><mn>1</mn><mo>≔</mo><mo>¬</mo><mn>0</mn></math></span>, <span><math><mi>x</mi><mo>⊙</mo><mi>y</mi><mo>≔</mo><mo>¬</mo><mo>(</mo><mo>¬</mo><mi>x</mi><mo>⊕</mo><mo>¬</mo><mi>y</mi><mo>)</mo></math></span>, <span><math><mi>x</mi><mo>∨</mo><mi>y</mi><mo>≔</mo><mo>(</mo><mi>x</mi><mo>⊙</mo><mo>¬</mo><mi>y</mi><mo>)</mo><mo>⊕</mo><mi>y</mi></math></span> and <span><math><mi>x</mi><mo>∧</mo><mi>y</mi><mo>≔</mo><mo>¬</mo><mo>(</mo><mo>¬</mo><mi>x</mi><mo>∨</mo><mo>¬</mo><mi>y</mi><mo>)</mo></math></span>. Particular examples of MV-monoids are positive MV-algebras, i.e., the <span><math><mo>{</mo><mo>∨</mo><mo>,</mo><mo>∧</mo><mo>,</mo><mo>⊕</mo><mo>,</mo><mo>⊙</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-subreducts of MV-algebras. Positive MV-algebras form a peculiar quasivariety in the sense that, albeit having a logical motivation (being the quasivariety of subreducts of MV-algebras), it is not the equivalent quasivariety semantics of any logic.</div><div>In this paper, we study the lattices of subvarieties of MV-monoids and of positive MV-algebras. In particular, we characterize and axiomatize all almost minimal varieties of MV-monoids, we characterize the finite subdirectly irreducible positive MV-algebras, and we characterize and axiomatize all varieties of positive MV-algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"677 ","pages":"Pages 690-744"},"PeriodicalIF":0.8,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922297","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}