Pub Date : 2026-01-01Epub Date: 2025-12-08DOI: 10.1016/j.jpaa.2025.108153
Hans Franzen , Gianni Petrella , Rachel Webb
We give an effective characterization of the walls in the variation of geometric invariant theory problem associated to a quiver and a dimension vector.
本文给出了一个几何不变理论问题中与一个颤振和一个维向量相关的壁面变化的有效表征。
{"title":"Finding the walls for quiver moduli","authors":"Hans Franzen , Gianni Petrella , Rachel Webb","doi":"10.1016/j.jpaa.2025.108153","DOIUrl":"10.1016/j.jpaa.2025.108153","url":null,"abstract":"<div><div>We give an effective characterization of the walls in the variation of geometric invariant theory problem associated to a quiver and a dimension vector.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108153"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885249","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 : 2026-01-01Epub Date: 2025-12-08DOI: 10.1016/j.jpaa.2025.108148
Marian Aprodu , Andrea Bruno , Edoardo Sernesi
The present paper is a natural continuation of the previous work [2] where we studied the second syzygy scheme of canonical curves. We find sufficient conditions ensuring that the second syzygy scheme of a genus–g curve of degree at least coincides with the curve. If the property is satisfied, the equality is ensured by a more general fact emphasized in [2]. If fails, then the analysis uses the known case of canonical curves.
{"title":"The second syzygy schemes of curves of large degree","authors":"Marian Aprodu , Andrea Bruno , Edoardo Sernesi","doi":"10.1016/j.jpaa.2025.108148","DOIUrl":"10.1016/j.jpaa.2025.108148","url":null,"abstract":"<div><div>The present paper is a natural continuation of the previous work <span><span>[2]</span></span> where we studied the second syzygy scheme of canonical curves. We find sufficient conditions ensuring that the second syzygy scheme of a genus–<em>g</em> curve of degree at least <span><math><mn>2</mn><mi>g</mi><mo>+</mo><mn>2</mn></math></span> coincides with the curve. If the property <span><math><mo>(</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> is satisfied, the equality is ensured by a more general fact emphasized in <span><span>[2]</span></span>. If <span><math><mo>(</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> fails, then the analysis uses the known case of canonical curves.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108148"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145705835","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 : 2026-01-01Epub Date: 2025-12-08DOI: 10.1016/j.jpaa.2025.108157
Jiayi Chen , Bangming Deng , Shiquan Ruan
This paper deals with the triangle singularity defined by the equation for a weight triple , as well as the category of coherent sheaves over the weighted projective line defined by f. We calculate Hall polynomials associated with extension bundles, line bundles and torsion sheaves over . By using derived equivalence, this provides a unified conceptual method for calculating Hall polynomials for representations of tame quivers obtained by Szántó and Szöllősi (2024) [35].
{"title":"Hall polynomials for weighted projective lines","authors":"Jiayi Chen , Bangming Deng , Shiquan Ruan","doi":"10.1016/j.jpaa.2025.108157","DOIUrl":"10.1016/j.jpaa.2025.108157","url":null,"abstract":"<div><div>This paper deals with the triangle singularity defined by the equation <span><math><mi>f</mi><mo>=</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msubsup></math></span> for a weight triple <span><math><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>, as well as the category of coherent sheaves over the weighted projective line <span><math><mi>X</mi></math></span> defined by <em>f</em>. We calculate Hall polynomials associated with extension bundles, line bundles and torsion sheaves over <span><math><mi>X</mi></math></span>. By using derived equivalence, this provides a unified conceptual method for calculating Hall polynomials for representations of tame quivers obtained by Szántó and Szöllősi (2024) <span><span>[35]</span></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108157"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145739259","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 : 2026-01-01Epub Date: 2025-12-31DOI: 10.1016/j.jpaa.2025.108168
Ahmed Laghribi, Trisha Maiti
Let F be a field of characteristic 2. The m-Pfister number of a quadratic form is the least number of forms similar to m-fold Pfister forms needed to express φ up to Witt equivalence. Our aim in this note is to discuss the case by giving an inductive formula that explicitly bounds the 3-Pfister number of any form in .
{"title":"On the 3-Pfister number in characteristic 2","authors":"Ahmed Laghribi, Trisha Maiti","doi":"10.1016/j.jpaa.2025.108168","DOIUrl":"10.1016/j.jpaa.2025.108168","url":null,"abstract":"<div><div>Let <em>F</em> be a field of characteristic 2. The <em>m</em>-Pfister number of a quadratic form <span><math><mi>φ</mi><mo>∈</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo>(</mo><mi>F</mi><mo>)</mo></math></span> is the least number of forms similar to <em>m</em>-fold Pfister forms needed to express <em>φ</em> up to Witt equivalence. Our aim in this note is to discuss the case <span><math><mi>m</mi><mo>=</mo><mn>3</mn></math></span> by giving an inductive formula that explicitly bounds the 3-Pfister number of any form in <span><math><msubsup><mrow><mi>I</mi></mrow><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mi>F</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108168"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926215","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 : 2026-01-01Epub Date: 2025-12-08DOI: 10.1016/j.jpaa.2025.108151
Barbara Gatti , Gioia Schulte
In the study of algebraic curves with many points over a finite field, a well known general problem is to understand better the properties of -maximal curves whose genera fall in the higher part of the spectrum of the genera of all -maximal curves. This problem is still open for genera smaller than . In this paper we consider the case of where and the curve is the Galois subcover of the Hermitian curve w.r.t. a cyclic automorphism group of order 4. Our contributions concern Frobenius embedding, Weierstrass semigroups and automorphism groups.
{"title":"On a Galois subcover of the Hermitian curve of genus g=18(q−1)2","authors":"Barbara Gatti , Gioia Schulte","doi":"10.1016/j.jpaa.2025.108151","DOIUrl":"10.1016/j.jpaa.2025.108151","url":null,"abstract":"<div><div>In the study of algebraic curves with many points over a finite field, a well known general problem is to understand better the properties of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>-maximal curves whose genera fall in the higher part of the spectrum of the genera of all <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>-maximal curves. This problem is still open for genera smaller than <span><math><mo>⌊</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>q</mi><mo>+</mo><mn>4</mn><mo>)</mo><mo>⌋</mo></math></span>. In this paper we consider the case of <span><math><mi>g</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>8</mn></mrow></mfrac><msup><mrow><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> where <span><math><mi>q</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span> and the curve is the Galois subcover of the Hermitian curve w.r.t. a cyclic automorphism group of order 4. Our contributions concern Frobenius embedding, Weierstrass semigroups and automorphism groups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108151"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145705833","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 : 2026-01-01Epub Date: 2026-01-05DOI: 10.1016/j.jpaa.2025.108167
Shahriyar Roshan Zamir
Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree d singular at a given set of points. After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992–1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this paper we primarily use commutative algebra to lay the groundwork necessary to prove analogous statements in the weighted projective space, a natural generalization of the projective space. We prove the Hilbert function of general simple points in any n-dimensional weighted projective space exhibits the expected behavior. We also introduce an inductive procedure for weighted projective space, similar to that originally due to A. Terracini from 1915, to demonstrate an example of a weighted projective plane where the analogue of the Alexander-Hirschowitz theorem holds without exceptions and prove our example is the only such plane. Furthermore, Terracini's lemma regarding secant varieties is adapted to give an interpolation bound for an infinite family of weighted projective planes.
{"title":"Interpolation in weighted projective spaces","authors":"Shahriyar Roshan Zamir","doi":"10.1016/j.jpaa.2025.108167","DOIUrl":"10.1016/j.jpaa.2025.108167","url":null,"abstract":"<div><div>Over an algebraically closed field, the <em>double point interpolation</em> problem asks for the vector space dimension of the projective hypersurfaces of degree <em>d</em> singular at a given set of points. After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992–1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this paper we primarily use commutative algebra to lay the groundwork necessary to prove analogous statements in the <em>weighted projective space</em>, a natural generalization of the projective space. We prove the Hilbert function of general simple points in any <em>n</em>-dimensional weighted projective space exhibits the expected behavior. We also introduce an inductive procedure for weighted projective space, similar to that originally due to A. Terracini from 1915, to demonstrate an example of a weighted projective plane where the analogue of the Alexander-Hirschowitz theorem holds without exceptions and prove our example is the only such plane. Furthermore, Terracini's lemma regarding secant varieties is adapted to give an interpolation bound for an infinite family of weighted projective planes.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108167"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926216","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}
We introduce a new technique to describe partial reductions and inverse Hamiltonian reductions between affine -algebras along the closure relations of associated nilpotent orbits in the case of , fulfilling all the missing constructions in the literature. We also apply the partial reductions to modules in the Kazhdan–Lusztig category and show compatibility with the usual reductions of Weyl modules.
{"title":"Connecting affine W-algebras: A case study on sl4","authors":"Justine Fasquel , Zachary Fehily , Ethan Fursman , Shigenori Nakatsuka","doi":"10.1016/j.jpaa.2025.108149","DOIUrl":"10.1016/j.jpaa.2025.108149","url":null,"abstract":"<div><div>We introduce a new technique to describe partial reductions and inverse Hamiltonian reductions between affine <span><math><mi>W</mi></math></span>-algebras along the closure relations of associated nilpotent orbits in the case of <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, fulfilling all the missing constructions in the literature. We also apply the partial reductions to modules in the Kazhdan–Lusztig category and show compatibility with the usual reductions of Weyl modules.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108149"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798977","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 : 2026-01-01Epub Date: 2025-12-05DOI: 10.1016/j.jpaa.2025.108155
Adele Maltempo , Carolina Vallejo
Let N be normal subgroup of a finite group G, p be a prime, P be a Sylow p-subgroup of G and θ be a P-invariant irreducible character of N. Suppose that is a p-solvable group. In this note we show that, whenever a finite group A acts on G stabilizing P, there exists an A-equivariant McKay bijection between irreducible characters lying over θ of degree prime to p of G and . This is a consequence of a recent result of D. Rossi. Our approach here is independent from Rossi's and follows the original idea of the proof of the McKay conjecture for p-solvable groups. In particular, we rely on the so-called Okuyama-Wajima argument to deal with characters above Glauberman correspondents. For this purpose, we generalize a classical result of P. X. Gallagher on the number of irreducible characters of G lying over θ.
{"title":"The McKay conjecture with group automorphisms and the Okuyama-Wajima argument","authors":"Adele Maltempo , Carolina Vallejo","doi":"10.1016/j.jpaa.2025.108155","DOIUrl":"10.1016/j.jpaa.2025.108155","url":null,"abstract":"<div><div>Let <em>N</em> be normal subgroup of a finite group <em>G</em>, <em>p</em> be a prime, <em>P</em> be a Sylow <em>p</em>-subgroup of <em>G</em> and <em>θ</em> be a <em>P</em>-invariant irreducible character of <em>N</em>. Suppose that <span><math><mi>G</mi><mo>/</mo><mi>N</mi></math></span> is a <em>p</em>-solvable group. In this note we show that, whenever a finite group <em>A</em> acts on <em>G</em> stabilizing <em>P</em>, there exists an <em>A</em>-equivariant McKay bijection between irreducible characters lying over <em>θ</em> of degree prime to <em>p</em> of <em>G</em> and <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo><mi>N</mi></math></span>. This is a consequence of a recent result of D. Rossi. Our approach here is independent from Rossi's and follows the original idea of the proof of the McKay conjecture for <em>p</em>-solvable groups. In particular, we rely on the so-called Okuyama-Wajima argument to deal with characters above Glauberman correspondents. For this purpose, we generalize a classical result of P. X. Gallagher on the number of irreducible characters of <em>G</em> lying over <em>θ</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108155"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145739258","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 : 2026-01-01Epub Date: 2025-12-08DOI: 10.1016/j.jpaa.2025.108156
Emile Bouaziz
We write down an explicit operator on the chiral de Rham complex of a Calabi-Yau variety X which intertwines the usual module structure with its twist by the spectral flow automorphism of the , producing the expected spectral flow equivariance. Taking the trace of the operators and on cohomology, and using the obvious interaction of spectral flow with characters, we obtain an explicit categorification of ellipticity of the elliptic genus of X, which is well known by other means.
{"title":"Spectral flow equivariance for Calabi-Yau Sigma models","authors":"Emile Bouaziz","doi":"10.1016/j.jpaa.2025.108156","DOIUrl":"10.1016/j.jpaa.2025.108156","url":null,"abstract":"<div><div>We write down an explicit operator on the chiral de Rham complex of a Calabi-Yau variety <em>X</em> which intertwines the usual <span><math><mi>N</mi><mo>=</mo><mn>2</mn></math></span> module structure with its twist by the spectral flow automorphism of the <span><math><mi>N</mi><mo>=</mo><mn>2</mn></math></span>, producing the expected <em>spectral flow equivariance</em>. Taking the trace of the operators <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>J</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> on cohomology, and using the obvious interaction of spectral flow with characters, we obtain an explicit categorification of ellipticity of the elliptic genus of <em>X</em>, which is well known by other means.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108156"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145705832","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 : 2026-01-01Epub Date: 2025-12-08DOI: 10.1016/j.jpaa.2025.108154
Taito Shimoji
Let Γ be a lattice in a simply-connected nilpotent Lie group N whose Lie algebra is p-filiform. We show that Γ is either abelian or 2-step nilpotent if Γ is isomorphic to the fundamental group of a smooth complex algebraic variety. Moreover as an application of our result, we give a required condition of a lattice in a simply-connected nilpotent Lie group of dimension less than or equal to six to be isomorphic to the fundamental group of a smooth complex algebraic variety.
{"title":"Gradings on nilpotent Lie algebras associated with the nilpotent fundamental groups of smooth complex algebraic varieties","authors":"Taito Shimoji","doi":"10.1016/j.jpaa.2025.108154","DOIUrl":"10.1016/j.jpaa.2025.108154","url":null,"abstract":"<div><div>Let Γ be a lattice in a simply-connected nilpotent Lie group <em>N</em> whose Lie algebra <span><math><mi>n</mi></math></span> is <em>p</em>-filiform. We show that Γ is either abelian or 2-step nilpotent if Γ is isomorphic to the fundamental group of a smooth complex algebraic variety. Moreover as an application of our result, we give a required condition of a lattice in a simply-connected nilpotent Lie group of dimension less than or equal to six to be isomorphic to the fundamental group of a smooth complex algebraic variety.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108154"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145705831","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}
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