Pub Date : 2023-10-18DOI: 10.1007/s00029-023-00887-2
Ko Aoki
Abstract We discuss two known sheaf-cosheaf duality theorems: Curry’s for the face posets of finite regular CW complexes and Lurie’s for compact Hausdorff spaces, i.e., covariant Verdier duality. We provide a uniform formulation for them and prove their generalizations. Our version of the former works over the sphere spectrum and for more general finite posets, which we characterize in terms of the Gorenstein* condition. Our version of the latter says that the stabilization of a proper separated $$infty $$ ∞ -topos is rigid in the sense of Gaitsgory. As an application, for stratified topological spaces, we clarify the relation between these two duality equivalences.
{"title":"Posets for which Verdier duality holds","authors":"Ko Aoki","doi":"10.1007/s00029-023-00887-2","DOIUrl":"https://doi.org/10.1007/s00029-023-00887-2","url":null,"abstract":"Abstract We discuss two known sheaf-cosheaf duality theorems: Curry’s for the face posets of finite regular CW complexes and Lurie’s for compact Hausdorff spaces, i.e., covariant Verdier duality. We provide a uniform formulation for them and prove their generalizations. Our version of the former works over the sphere spectrum and for more general finite posets, which we characterize in terms of the Gorenstein* condition. Our version of the latter says that the stabilization of a proper separated $$infty $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>∞</mml:mi> </mml:math> -topos is rigid in the sense of Gaitsgory. As an application, for stratified topological spaces, we clarify the relation between these two duality equivalences.","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135825405","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 : 2023-10-13DOI: 10.1007/s00029-023-00872-9
Manuel Saorín, Jan Šťovíček
Abstract We study when the heart of a t -structure in a triangulated category $$mathcal {D}$$ D with coproducts is AB5 or a Grothendieck category. If $$mathcal {D}$$ D satisfies Brown representability, a t -structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective t -cogenerating object. If $$mathcal {D}$$ D is standard well generated, such a heart is automatically a Grothendieck category. For compactly generated t -structures (in any ambient triangulated category with coproducts), we prove that the heart is a locally finitely presented Grothendieck category. We use functor categories and the proofs rely on two main ingredients. Firstly, we express the heart of any t -structure in any triangulated category as a Serre quotient of the category of finitely presented additive functors for suitable choices of subcategories of the aisle or the co-aisle that we, respectively, call t -generating or t -cogenerating subcategories. Secondly, we study coproduct-preserving homological functors from $$mathcal {D}$$ D to complete AB5 abelian categories with injective cogenerators and classify them, up to a so-called computational equivalence, in terms of pure-injective objects in $$mathcal {D}$$ D . This allows us to show that any standard well generated triangulated category $$mathcal {D}$$ D possesses a universal such coproduct-preserving homological functor, to develop a purity theory and to prove that pure-injective objects always cogenerate t -structures in such triangulated categories.
{"title":"t-Structures with Grothendieck hearts via functor categories","authors":"Manuel Saorín, Jan Šťovíček","doi":"10.1007/s00029-023-00872-9","DOIUrl":"https://doi.org/10.1007/s00029-023-00872-9","url":null,"abstract":"Abstract We study when the heart of a t -structure in a triangulated category $$mathcal {D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>D</mml:mi> </mml:math> with coproducts is AB5 or a Grothendieck category. If $$mathcal {D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>D</mml:mi> </mml:math> satisfies Brown representability, a t -structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective t -cogenerating object. If $$mathcal {D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>D</mml:mi> </mml:math> is standard well generated, such a heart is automatically a Grothendieck category. For compactly generated t -structures (in any ambient triangulated category with coproducts), we prove that the heart is a locally finitely presented Grothendieck category. We use functor categories and the proofs rely on two main ingredients. Firstly, we express the heart of any t -structure in any triangulated category as a Serre quotient of the category of finitely presented additive functors for suitable choices of subcategories of the aisle or the co-aisle that we, respectively, call t -generating or t -cogenerating subcategories. Secondly, we study coproduct-preserving homological functors from $$mathcal {D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>D</mml:mi> </mml:math> to complete AB5 abelian categories with injective cogenerators and classify them, up to a so-called computational equivalence, in terms of pure-injective objects in $$mathcal {D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>D</mml:mi> </mml:math> . This allows us to show that any standard well generated triangulated category $$mathcal {D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>D</mml:mi> </mml:math> possesses a universal such coproduct-preserving homological functor, to develop a purity theory and to prove that pure-injective objects always cogenerate t -structures in such triangulated categories.","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135804552","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 : 2023-10-12DOI: 10.1007/s00029-023-00879-2
Anton Freund, Michael Rathjen
Abstract We introduce ordinal collapsing principles that are inspired by proof theory but have a set theoretic flavor. These principles are shown to be equivalent to iterated $$Pi ^1_1$$ Π11 -comprehension and the existence of admissible sets, over weak base theories. Our work extends a previous result on the non-iterated case, which had been conjectured in Montalbán’s “Open questions in reverse mathematics" (Bull Symb Log 17(3):431–454, 2011). This previous result has already been applied to the reverse mathematics of combinatorial and set theoretic principles. The present paper is a significant contribution to a general approach that connects these fields.
{"title":"Well ordering principles for iterated $$Pi ^1_1$$-comprehension","authors":"Anton Freund, Michael Rathjen","doi":"10.1007/s00029-023-00879-2","DOIUrl":"https://doi.org/10.1007/s00029-023-00879-2","url":null,"abstract":"Abstract We introduce ordinal collapsing principles that are inspired by proof theory but have a set theoretic flavor. These principles are shown to be equivalent to iterated $$Pi ^1_1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> -comprehension and the existence of admissible sets, over weak base theories. Our work extends a previous result on the non-iterated case, which had been conjectured in Montalbán’s “Open questions in reverse mathematics\" (Bull Symb Log 17(3):431–454, 2011). This previous result has already been applied to the reverse mathematics of combinatorial and set theoretic principles. The present paper is a significant contribution to a general approach that connects these fields.","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135969646","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 : 2023-10-10DOI: 10.1007/s00029-023-00880-9
Aaron Bertram, Thomas Goller, Drew Johnson
{"title":"Le Potier’s strange duality, quot schemes, and multiple point formulas for del Pezzo surfaces","authors":"Aaron Bertram, Thomas Goller, Drew Johnson","doi":"10.1007/s00029-023-00880-9","DOIUrl":"https://doi.org/10.1007/s00029-023-00880-9","url":null,"abstract":"","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254852","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 : 2023-10-09DOI: 10.1007/s00029-023-00882-7
Davide Bricalli, Filippo Francesco Favale, Gian Pietro Pirola
Abstract Gordan and Noether proved in their fundamental theorem that an hypersurface $$X=V(F)subseteq {{mathbb {P}}}^n$$ X=V(F)⊆Pn with $$nle 3$$ n≤3 is a cone if and only if F has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that the statement is false if $$nge 4$$ n≥4 , by giving some counterexamples. Since their proof, several others have been proposed in the literature. In this paper we give a new one by using a different perspective which involves the study of standard Artinian Gorenstein $${{mathbb {K}}}$$ K -algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra $$R={{mathbb {K}}}[x_0,dots ,x_4]/J$$ R=K[x0,⋯,x4]/J with J generated by a regular sequence of quadrics has the strong Lefschetz property. In particular, this holds for Jacobian rings associated to smooth cubic threefolds.
Gordan和Noether在其基本定理中证明了$$nle 3$$ n≤3的超曲面$$X=V(F)subseteq {{mathbb {P}}}^n$$ X = V (F)≥P n是锥当且仅当F具有消失的hessian(即hessian矩阵的行列式)。他们还通过给出一些反例，证明了如果$$nge 4$$ n≥4，该陈述是错误的。在他们的证明之后，文献中又提出了其他几个证明。本文从一个不同的角度给出了一个新的代数，它涉及到对标准的Artinian Gorenstein $${{mathbb {K}}}$$ K -代数和Lefschetz性质的研究。作为我们的设置的进一步应用，我们证明了一个标准的Artinian Gorenstein代数$$R={{mathbb {K}}}[x_0,dots ,x_4]/J$$ R = K [x 0，⋯，x 4] / J，其中由正则二次序列生成的J具有强Lefschetz性质。特别地，这适用于与光滑三次折叠相关的雅可比环。
{"title":"A theorem of Gordan and Noether via Gorenstein rings","authors":"Davide Bricalli, Filippo Francesco Favale, Gian Pietro Pirola","doi":"10.1007/s00029-023-00882-7","DOIUrl":"https://doi.org/10.1007/s00029-023-00882-7","url":null,"abstract":"Abstract Gordan and Noether proved in their fundamental theorem that an hypersurface $$X=V(F)subseteq {{mathbb {P}}}^n$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>V</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> with $$nle 3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> is a cone if and only if F has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that the statement is false if $$nge 4$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , by giving some counterexamples. Since their proof, several others have been proposed in the literature. In this paper we give a new one by using a different perspective which involves the study of standard Artinian Gorenstein $${{mathbb {K}}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>K</mml:mi> </mml:math> -algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra $$R={{mathbb {K}}}[x_0,dots ,x_4]/J$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>=</mml:mo> <mml:mi>K</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mo>]</mml:mo> <mml:mo>/</mml:mo> <mml:mi>J</mml:mi> </mml:mrow> </mml:math> with J generated by a regular sequence of quadrics has the strong Lefschetz property. In particular, this holds for Jacobian rings associated to smooth cubic threefolds.","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135092806","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 : 2023-09-27DOI: 10.1007/s00029-023-00871-w
Fabian Ziltener
Abstract Let G be a compact and connected Lie group. The Hamiltonian G -model functor maps the category of symplectic representations of closed subgroups of G to the category of exact Hamiltonian G -actions. Based on previous joint work with Y. Karshon, the restriction of this functor to the momentum proper subcategory on either side induces a bijection between the sets of isomorphism classes. This classifies all momentum proper exact Hamiltonian G -actions (of arbitrary complexity). As an extreme case, we obtain a version of the Eliashberg cotangent bundle conjecture for transitive smooth actions. As another extreme case, the momentum proper Hamiltonian G -actions on contractible manifolds are exactly the symplectic G -representations, up to isomorphism.
{"title":"Classification of momentum proper exact Hamiltonian group actions and the equivariant Eliashberg cotangent bundle conjecture","authors":"Fabian Ziltener","doi":"10.1007/s00029-023-00871-w","DOIUrl":"https://doi.org/10.1007/s00029-023-00871-w","url":null,"abstract":"Abstract Let G be a compact and connected Lie group. The Hamiltonian G -model functor maps the category of symplectic representations of closed subgroups of G to the category of exact Hamiltonian G -actions. Based on previous joint work with Y. Karshon, the restriction of this functor to the momentum proper subcategory on either side induces a bijection between the sets of isomorphism classes. This classifies all momentum proper exact Hamiltonian G -actions (of arbitrary complexity). As an extreme case, we obtain a version of the Eliashberg cotangent bundle conjecture for transitive smooth actions. As another extreme case, the momentum proper Hamiltonian G -actions on contractible manifolds are exactly the symplectic G -representations, up to isomorphism.","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135476208","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 : 2023-09-21DOI: 10.1007/s00029-023-00869-4
Ivan Cheltsov, Arman Sarikyan
Abstract We classify finite subgroups $$Gsubset {textrm{PGL}}_4({mathbb {C}})$$ G⊂PGL4(C) such that $${mathbb {P}}^3$$ P3 is not G -birational to conic bundles and del Pezzo fibrations, and explicitly describe all G -Mori fibre spaces that are G -birational to $${mathbb {P}}^3$$ P3 for these subgroups.
{"title":"Equivariant pliability of the projective space","authors":"Ivan Cheltsov, Arman Sarikyan","doi":"10.1007/s00029-023-00869-4","DOIUrl":"https://doi.org/10.1007/s00029-023-00869-4","url":null,"abstract":"Abstract We classify finite subgroups $$Gsubset {textrm{PGL}}_4({mathbb {C}})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>⊂</mml:mo> <mml:msub> <mml:mtext>PGL</mml:mtext> <mml:mn>4</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> such that $${mathbb {P}}^3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> is not G -birational to conic bundles and del Pezzo fibrations, and explicitly describe all G -Mori fibre spaces that are G -birational to $${mathbb {P}}^3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> for these subgroups.","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136235809","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 : 2023-09-20DOI: 10.1007/s00029-023-00875-6
Francois Petit, Pierre Schapira
Given a topological space X, a thickening kernel is a monoidal presheaf on $$({{mathbb {R}}}_{ge 0},+)$$ with values in the monoidal category of derived kernels on X. A bi-thickening kernel is defined on $$({{mathbb {R}}},+)$$ . To such a thickening kernel, one naturally associates an interleaving distance on the derived category of sheaves on X. We prove that a thickening kernel exists and is unique as soon as it is defined on an interval containing 0, allowing us to construct (bi-)thickenings in two different situations. First, when X is a “good” metric space, starting with small usual thickenings of the diagonal. The associated interleaving distance satisfies the stability property and Lipschitz kernels give rise to Lipschitz maps. Second, by using (Guillermou et al. in Duke Math J 161:201–245, 2012), when X is a manifold and one is given a non-positive Hamiltonian isotopy on the cotangent bundle. In case X is a complete Riemannian manifold having a strictly positive convexity radius, we prove that it is a good metric space and that the two bi-thickening kernels of the diagonal, one associated with the distance, the other with the geodesic flow, coincide.
给定一个拓扑空间X，一个增厚核是一个在$$({{mathbb {R}}}_{ge 0},+)$$上的一元预层，其值在X上的派生核的一元范畴内。一个双增厚核在$$({{mathbb {R}}},+)$$上定义。对于这样的增厚核，人们很自然地将x上的束的派生范畴上的交错距离联系起来。我们证明了增厚核存在，并且一旦在包含0的区间上定义，它就是唯一的，从而允许我们在两种不同的情况下构造(双-)增厚。首先，当X是一个“好的”度量空间时，从对角线的通常增厚开始。相关的交错距离满足稳定性性质，利普希茨核产生利普希茨映射。其次，通过使用(Guillermou et al. in Duke Math J 161:201 - 245,2012)，当X是流形并且在共切束上给定非正哈密顿同位素时。如果X是具有严格正凸半径的完全黎曼流形，我们证明它是一个很好的度量空间，并且对角线的两个双增厚核，一个与距离有关，另一个与测地线流有关，重合。
{"title":"Thickening of the diagonal and interleaving distance","authors":"Francois Petit, Pierre Schapira","doi":"10.1007/s00029-023-00875-6","DOIUrl":"https://doi.org/10.1007/s00029-023-00875-6","url":null,"abstract":"Given a topological space X, a thickening kernel is a monoidal presheaf on $$({{mathbb {R}}}_{ge 0},+)$$ with values in the monoidal category of derived kernels on X. A bi-thickening kernel is defined on $$({{mathbb {R}}},+)$$ . To such a thickening kernel, one naturally associates an interleaving distance on the derived category of sheaves on X. We prove that a thickening kernel exists and is unique as soon as it is defined on an interval containing 0, allowing us to construct (bi-)thickenings in two different situations. First, when X is a “good” metric space, starting with small usual thickenings of the diagonal. The associated interleaving distance satisfies the stability property and Lipschitz kernels give rise to Lipschitz maps. Second, by using (Guillermou et al. in Duke Math J 161:201–245, 2012), when X is a manifold and one is given a non-positive Hamiltonian isotopy on the cotangent bundle. In case X is a complete Riemannian manifold having a strictly positive convexity radius, we prove that it is a good metric space and that the two bi-thickening kernels of the diagonal, one associated with the distance, the other with the geodesic flow, coincide.","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136263999","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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