Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107866
Violeta Borges Marques, Wendy Lowen, Arne Mertens
The framework of templicial objects was put forth in [30] in order to develop higher categorical concepts in the presence of enrichment. In particular, quasi-categories in modules constitute a subclass of templicial modules which may be considered as a kind of “weak dg-categories (concentrated in homologically positive degrees)” according to [29]. The main goal of the present paper is to initiate the deformation theory of templicial modules. In particular, we show that quasi-categories in modules are preserved under levelwise flat infinitesimal deformation.
{"title":"Deformations of quasi-categories in modules","authors":"Violeta Borges Marques, Wendy Lowen, Arne Mertens","doi":"10.1016/j.jpaa.2025.107866","DOIUrl":"10.1016/j.jpaa.2025.107866","url":null,"abstract":"<div><div>The framework of templicial objects was put forth in <span><span>[30]</span></span> in order to develop higher categorical concepts in the presence of enrichment. In particular, quasi-categories in modules constitute a subclass of templicial modules which may be considered as a kind of “weak dg-categories (concentrated in homologically positive degrees)” according to <span><span>[29]</span></span>. The main goal of the present paper is to initiate the deformation theory of templicial modules. In particular, we show that quasi-categories in modules are preserved under levelwise flat infinitesimal deformation.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107866"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163055","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 : 2025-02-01DOI: 10.1016/j.jpaa.2025.107876
Silvia Properzi , Arne Van Antwerpen
We introduce two common divisor graphs associated with a finite skew brace, based on its λ- and θ-orbits. We prove that the number of connected components is at most two and the diameter of a connected component is at most four. Furthermore, we investigate their relationship with isoclinism. Similarly to its group theoretic inspiration, the skew braces with a graph with two disconnected vertices are very restricted and are determined. Finally, we classify all finite skew braces with a graph with one vertex, where four infinite families arise.
{"title":"Common divisor graphs for skew braces","authors":"Silvia Properzi , Arne Van Antwerpen","doi":"10.1016/j.jpaa.2025.107876","DOIUrl":"10.1016/j.jpaa.2025.107876","url":null,"abstract":"<div><div>We introduce two common divisor graphs associated with a finite skew brace, based on its <em>λ</em>- and <em>θ</em>-orbits. We prove that the number of connected components is at most two and the diameter of a connected component is at most four. Furthermore, we investigate their relationship with isoclinism. Similarly to its group theoretic inspiration, the skew braces with a graph with two disconnected vertices are very restricted and are determined. Finally, we classify all finite skew braces with a graph with one vertex, where four infinite families arise.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107876"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163914","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 : 2025-02-01DOI: 10.1016/j.jpaa.2025.107904
Sławomir Rams , Matthias Schütt
We prove that there are at most irreducible rational curves of positive low-degree on high-degree models of K3 surfaces with at most Du Val singularities, where is the number of exceptional divisors on the minimal resolution. We also provide several existence results in the above setting (i.e. for rational curves on quasi-polarized K3 surfaces), which imply that for many values of our bound cannot be improved.
{"title":"Low degree rational curves on quasi-polarized K3 surfaces","authors":"Sławomir Rams , Matthias Schütt","doi":"10.1016/j.jpaa.2025.107904","DOIUrl":"10.1016/j.jpaa.2025.107904","url":null,"abstract":"<div><div>We prove that there are at most <span><math><mo>(</mo><mn>24</mn><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> irreducible rational curves of positive low-degree on high-degree models of K3 surfaces with at most Du Val singularities, where <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is the number of exceptional divisors on the minimal resolution. We also provide several existence results in the above setting (i.e. for rational curves on quasi-polarized K3 surfaces), which imply that for many values of <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> our bound cannot be improved.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107904"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394580","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 : 2025-02-01DOI: 10.1016/j.jpaa.2025.107894
Lingling Han, Tao Zheng
In this paper, we investigate the homotopy properties of , the finite topological space consisting of all non-trivial proper subgroups of a finite group G. For some classes of groups G, we give the relations between the contractibility of and the algebraic properties of G, which is inspired by the study of R. E. Stong on and .
{"title":"On the finite spaces of non-trivial proper subgroups of finite groups","authors":"Lingling Han, Tao Zheng","doi":"10.1016/j.jpaa.2025.107894","DOIUrl":"10.1016/j.jpaa.2025.107894","url":null,"abstract":"<div><div>In this paper, we investigate the homotopy properties of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, the finite topological space consisting of all non-trivial proper subgroups of a finite group <em>G</em>. For some classes of groups <em>G</em>, we give the relations between the contractibility of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and the algebraic properties of <em>G</em>, which is inspired by the study of R. E. Stong on <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107894"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162792","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 : 2025-02-01DOI: 10.1016/j.jpaa.2024.107861
V.V. Bavula
The class of Δ-locally nilpotent algebras introduced in the paper is a wide generalization of the algebras of differential operators on commutative algebras. Examples include all the rings of differential operators on commutative algebras in arbitrary characteristic, all subalgebras of that contain the algebra A, the universal enveloping algebras of nilpotent, solvable and semi-simple Lie algebras, the Poisson universal enveloping algebra of an arbitrary Poisson algebra, iterated Ore extensions , certain generalized Weyl algebras, and others.
In [8], simplicity criteria are given for the algebras differential operators on commutative algebras. To find the simplicity criterion was a long standing problem from 60'th. The aim of the paper is to describe the ideal structure of Δ-locally nilpotent algebras and as a corollary to give simplicity criteria for them. These results are generalizations of the results of [8]. Examples are considered.
{"title":"Δ-locally nilpotent algebras, their ideal structure and simplicity criteria","authors":"V.V. Bavula","doi":"10.1016/j.jpaa.2024.107861","DOIUrl":"10.1016/j.jpaa.2024.107861","url":null,"abstract":"<div><div>The class of Δ-locally nilpotent algebras introduced in the paper is a wide generalization of the algebras of differential operators on commutative algebras. Examples include all the rings <span><math><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> of differential operators on commutative algebras in arbitrary characteristic, all subalgebras of <span><math><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> that contain the algebra <em>A</em>, the universal enveloping algebras of nilpotent, solvable and semi-simple Lie algebras, the Poisson universal enveloping algebra of an arbitrary Poisson algebra, iterated Ore extensions <span><math><mi>A</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>, certain generalized Weyl algebras, and others.</div><div>In <span><span>[8]</span></span>, simplicity criteria are given for the algebras differential operators on commutative algebras. To find the simplicity criterion was a long standing problem from 60'th. The aim of the paper is to describe the ideal structure of Δ-locally nilpotent algebras and as a corollary to give simplicity criteria for them. These results are generalizations of the results of <span><span>[8]</span></span>. Examples are considered.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107861"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107875
Artem Lopatin , Alexandr N. Zubkov
Over an algebraically closed field, we described a minimal set of representatives for -orbits on the set of pairs of octonions.
{"title":"Classification of G2-orbits for pairs of octonions","authors":"Artem Lopatin , Alexandr N. Zubkov","doi":"10.1016/j.jpaa.2025.107875","DOIUrl":"10.1016/j.jpaa.2025.107875","url":null,"abstract":"<div><div>Over an algebraically closed field, we described a minimal set of representatives for <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-orbits on the set <span><math><msup><mrow><mi>O</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of pairs of octonions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107875"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163666","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 : 2025-02-01DOI: 10.1016/j.jpaa.2025.107863
Takeo Uramoto
This paper is an extended version of our proceedings paper [49] announced at LICS'16; in order to complement it, this version is written from a different viewpoint including topos-theoretic aspects of [49] that were not discussed there. Technically, this paper introduces and studies the class of semi-galois categories, which extend galois categories and are dual to profinite monoids in the same way as galois categories are dual to profinite groups; the study on this class of categories is aimed at providing an axiomatic reformulation of Eilenberg's theory of varieties of regular languages—a branch in formal language theory that has been developed since the mid 1960s and particularly concerns systematic classification of regular languages, finite monoids and deterministic finite automata. In this paper, detailed proofs of our central results announced at LICS'16 are presented, together with topos-theoretic considerations. The main results include (I) a proof of the duality theorem between profinite monoids and semi-galois categories, extending the duality theorem between profinite groups and galois categories; based on these results on semi-galois categories we then discuss (II) a reinterpretation of Eilenberg's theory from a viewpoint of the duality theorem; in relation with this reinterpretation of the theory, (III) we also give a purely topos-theoretic characterization of classifying topoi of profinite monoids M among general coherent topoi, which is a topos-theoretic application of (I). This characterization states that a topos is equivalent to the classifying topos of some profinite monoid M if and only if is (i) coherent, (ii) noetherian, and (iii) has a surjective coherent point .
{"title":"Semi-galois Categories I: The classical Eilenberg variety theory","authors":"Takeo Uramoto","doi":"10.1016/j.jpaa.2025.107863","DOIUrl":"10.1016/j.jpaa.2025.107863","url":null,"abstract":"<div><div>This paper is an extended version of our proceedings paper <span><span>[49]</span></span> announced at LICS'16; in order to complement it, this version is written from a different viewpoint including topos-theoretic aspects of <span><span>[49]</span></span> that were not discussed there. Technically, this paper introduces and studies the class of <em>semi-galois categories</em>, which extend galois categories and are dual to profinite monoids in the same way as galois categories are dual to profinite groups; the study on this class of categories is aimed at providing an axiomatic reformulation of <em>Eilenberg's theory of varieties of regular languages</em>—a branch in formal language theory that has been developed since the mid 1960s and particularly concerns systematic classification of regular languages, finite monoids and deterministic finite automata. In this paper, detailed proofs of our central results announced at LICS'16 are presented, together with topos-theoretic considerations. The main results include (I) a proof of the duality theorem between profinite monoids and semi-galois categories, extending the duality theorem between profinite groups and galois categories; based on these results on semi-galois categories we then discuss (II) a reinterpretation of Eilenberg's theory from a viewpoint of the duality theorem; in relation with this reinterpretation of the theory, (III) we also give a purely topos-theoretic characterization of classifying topoi <span><math><mi>B</mi><mi>M</mi></math></span> of profinite monoids <em>M</em> among general coherent topoi, which is a topos-theoretic application of (I). This characterization states that a topos <span><math><mi>E</mi></math></span> is equivalent to the classifying topos <span><math><mi>B</mi><mi>M</mi></math></span> of some profinite monoid <em>M</em> if and only if <span><math><mi>E</mi></math></span> is (i) coherent, (ii) noetherian, and (iii) has a surjective coherent point <span><math><mi>p</mi><mo>:</mo><mrow><mi>Sets</mi></mrow><mo>→</mo><mi>E</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107863"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107870
Anargyros Katsabekis, Apostolos Thoma
The toric ideal is splittable if it has a toric splitting; namely, if there exist toric ideals such that and for all . We provide a necessary and sufficient condition for a toric ideal to be splittable in terms of A, and we apply it to prove or disprove that certain classes of toric ideals are splittable.
{"title":"Toric splittings","authors":"Anargyros Katsabekis, Apostolos Thoma","doi":"10.1016/j.jpaa.2025.107870","DOIUrl":"10.1016/j.jpaa.2025.107870","url":null,"abstract":"<div><div>The toric ideal <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> is splittable if it has a toric splitting; namely, if there exist toric ideals <span><math><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> such that <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> and <span><math><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>≠</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mn>2</mn></math></span>. We provide a necessary and sufficient condition for a toric ideal to be splittable in terms of <em>A</em>, and we apply it to prove or disprove that certain classes of toric ideals are splittable.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107870"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163677","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 : 2025-02-01DOI: 10.1016/j.jpaa.2025.107882
Qian Chao , Jiun-Cheng Chen , Hsian-Hua Tseng
We establish some properties of the derived category of torus-equivariant coherent sheaves on a split toric stack bundle. Our main result is a semi-orthogonal decomposition of such a category.
{"title":"On the derived category of a toric stack bundle","authors":"Qian Chao , Jiun-Cheng Chen , Hsian-Hua Tseng","doi":"10.1016/j.jpaa.2025.107882","DOIUrl":"10.1016/j.jpaa.2025.107882","url":null,"abstract":"<div><div>We establish some properties of the derived category of torus-equivariant coherent sheaves on a split toric stack bundle. Our main result is a semi-orthogonal decomposition of such a category.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107882"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107903
Peter Bubenik , José A. Vélez-Marulanda
We define exact weights on a pretriangulated category to be nonnegative functions on objects satisfying a subadditivity condition with respect to distinguished triangles. Such weights induce a metric on objects in the category, which we call a path metric. Our exact weights generalize the rank functions of J. Chuang and A. Lazarev for triangulated categories and are analogous to the exact weights for an exact category given by the first author and J. Scott and D. Stanley. We show that (co)homological functors from a triangulated category to an abelian category with an additive weight induce an exact weight on the triangulated category. We prove that triangle equivalences induce an isometry for the path metrics induced by cohomological functors. In the perfectly generated or compactly generated case, we use Brown representability to express the exact weight on the triangulated category. We give three characterizations of exactness for a weight on a pretriangulated category and show that they are equivalent. We also define Wasserstein distances for triangulated categories. Finally, we apply our work to derived categories of persistence modules and to representations of continuous quivers of type .
{"title":"Exact weights and path metrics for triangulated categories and the derived category of persistence modules","authors":"Peter Bubenik , José A. Vélez-Marulanda","doi":"10.1016/j.jpaa.2025.107903","DOIUrl":"10.1016/j.jpaa.2025.107903","url":null,"abstract":"<div><div>We define exact weights on a pretriangulated category to be nonnegative functions on objects satisfying a subadditivity condition with respect to distinguished triangles. Such weights induce a metric on objects in the category, which we call a path metric. Our exact weights generalize the rank functions of J. Chuang and A. Lazarev for triangulated categories and are analogous to the exact weights for an exact category given by the first author and J. Scott and D. Stanley. We show that (co)homological functors from a triangulated category to an abelian category with an additive weight induce an exact weight on the triangulated category. We prove that triangle equivalences induce an isometry for the path metrics induced by cohomological functors. In the perfectly generated or compactly generated case, we use Brown representability to express the exact weight on the triangulated category. We give three characterizations of exactness for a weight on a pretriangulated category and show that they are equivalent. We also define Wasserstein distances for triangulated categories. Finally, we apply our work to derived categories of persistence modules and to representations of continuous quivers of type <span><math><mi>A</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107903"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394581","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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