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Frobenius and quasi-Frobenius left Hopf algebroids
IF 0.7 2区 数学
Journal of Pure and Applied Algebra Pub Date : 2025-01-01 DOI: 10.1016/j.jpaa.2024.107844
Sophie Chemla
{"title":"Frobenius and quasi-Frobenius left Hopf algebroids","authors":"Sophie Chemla","doi":"10.1016/j.jpaa.2024.107844","DOIUrl":"10.1016/j.jpaa.2024.107844","url":null,"abstract":"<div><div>We study when left (op)Hopf algebroids in the sense of Takeuchi-Schauenburg give rise to a Frobenius or quasi-Frobenius extension. The case of Hopf algebroids in the sense of Böhm was treated by G. Böhm (<span><span>[4]</span></span>). Contrary to Hopf algebroids, (op)Hopf left algebroids don't necessarily have an antipode but their Hopf-Galois map is invertible. We make use of recent results about left Hopf algebroids (<span><span>[18]</span></span>, <span><span>[35]</span></span>, <span><span>[25]</span></span>). Our results are applied to the restricted enveloping algebra of a restricted Lie-Rinehart algebra.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107844"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094013","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}
引用次数: 0
Colimits in 2-dimensional slices
IF 0.7 2区 数学
Journal of Pure and Applied Algebra Pub Date : 2025-01-01 DOI: 10.1016/j.jpaa.2024.107855
Luca Mesiti
{"title":"Colimits in 2-dimensional slices","authors":"Luca Mesiti","doi":"10.1016/j.jpaa.2024.107855","DOIUrl":"10.1016/j.jpaa.2024.107855","url":null,"abstract":"<div><div>We generalize to dimension 2 the well-known fact that a colimit in a 1-dimensional slice is precisely the map from the colimit of the domains of the diagram that is induced by the universal property. For this, we find the need to reduce weighted 2-colimits to cartesian-marked oplax conical ones, and as a consequence the need to consider lax slices.</div><div>We prove results of preservation, reflection and lifting of 2-colimits for the domain 2-functor from a lax slice. We thus generalize to dimension 2 the whole fruitful calculus of colimits in 1-dimensional slices. We achieve this within the framework of enhanced (or <span><math><mi>F</mi></math></span>-)category theory. The preservation result assumes products in the base 2-category and uses an original general theorem which states that a lax left adjoint preserves appropriate 2-colimits if the adjunction is strict on one side and suitably <span><math><mi>F</mi></math></span>-categorical.</div><div>Finally, we apply the same general theorem of preservation of 2-colimits to the 2-functor of change of base along a split Grothendieck opfibration between lax slices. We prove that this change of base 2-functor is indeed a left adjoint of the kind described above by laxifying the proof that Conduché functors are exponentiable. We conclude extending the result of preservation of 2-colimits for the change of base 2-functor to any finitely complete 2-category with a dense generator.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107855"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099039","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}
引用次数: 0
Determining the Betti numbers of R/(xpe,ype,zpe) for most even degree hypersurfaces in odd characteristic
IF 0.7 2区 数学
Journal of Pure and Applied Algebra Pub Date : 2025-01-01 DOI: 10.1016/j.jpaa.2024.107858
Heath Camphire
{"title":"Determining the Betti numbers of R/(xpe,ype,zpe) for most even degree hypersurfaces in odd characteristic","authors":"Heath Camphire","doi":"10.1016/j.jpaa.2024.107858","DOIUrl":"10.1016/j.jpaa.2024.107858","url":null,"abstract":"<div><div>Let <em>k</em> be a field of odd characteristic <em>p</em>. Fix an even number <span><math><mi>d</mi><mo>&lt;</mo><mi>p</mi><mo>+</mo><mn>1</mn></math></span> and a power <span><math><mi>q</mi><mo>≥</mo><mi>d</mi><mo>+</mo><mn>3</mn></math></span> of <em>p</em>. For most choices of degree <em>d</em> standard graded hypersurfaces <span><math><mi>R</mi><mo>=</mo><mi>k</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>]</mo><mo>/</mo><mo>(</mo><mi>f</mi><mo>)</mo></math></span> with homogeneous maximal ideal <span><math><mi>m</mi></math></span>, we can determine the graded Betti numbers of <span><math><mi>R</mi><mo>/</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>[</mo><mi>q</mi><mo>]</mo></mrow></msup></math></span>. In fact, given two fixed powers <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mi>d</mi><mo>+</mo><mn>3</mn></math></span>, for most choices of <em>R</em> the graded Betti numbers in high homological degree of <span><math><mi>R</mi><mo>/</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>[</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>]</mo></mrow></msup></math></span> and <span><math><mi>R</mi><mo>/</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>[</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>]</mo></mrow></msup></math></span> are the same up to a constant shift. This paper shows this fact by combining our results with the work of Miller, Rahmati, and R.G. on link-<em>q</em>-compressed polynomials in <span><span>[13]</span></span>. We show that link-<em>q</em>-compressed polynomials are indeed fairly common in many polynomial rings.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107858"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099041","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}
引用次数: 0
Automorphisms of quartic surfaces and Cremona transformations
IF 0.7 2区 数学
Journal of Pure and Applied Algebra Pub Date : 2025-01-01 DOI: 10.1016/j.jpaa.2024.107850
Daniela Paiva , Ana Quedo
{"title":"Automorphisms of quartic surfaces and Cremona transformations","authors":"Daniela Paiva ,&nbsp;Ana Quedo","doi":"10.1016/j.jpaa.2024.107850","DOIUrl":"10.1016/j.jpaa.2024.107850","url":null,"abstract":"<div><div>In this paper, we consider the problem of determining which automorphisms of a smooth quartic surface <span><math><mi>S</mi><mo>⊂</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> are induced by a Cremona transformation of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. We provide the first steps towards a complete solution of this problem when <span><math><mi>ρ</mi><mo>(</mo><mi>S</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span>. In particular, we give several examples of quartics whose automorphism groups are generated by involutions, but no non-trivial automorphism is induced by a Cremona transformation of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, giving a negative answer for Oguiso's question of whether every automorphism of finite order of a smooth quartic surface <span><math><mi>S</mi><mo>⊂</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> is induced by a Cremona transformation.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107850"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094020","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}
引用次数: 0
Free automorphism groups of K3 surfaces with Picard number 3
IF 0.7 2区 数学
Journal of Pure and Applied Algebra Pub Date : 2025-01-01 DOI: 10.1016/j.jpaa.2024.107845
Kenji Hashimoto , Kwangwoo Lee
{"title":"Free automorphism groups of K3 surfaces with Picard number 3","authors":"Kenji Hashimoto ,&nbsp;Kwangwoo Lee","doi":"10.1016/j.jpaa.2024.107845","DOIUrl":"10.1016/j.jpaa.2024.107845","url":null,"abstract":"<div><div>It is known that the automorphism group of any projective K3 surface is finitely generated. In this paper, we consider a certain kind of K3 surfaces with Picard number 3 whose automorphism groups are isomorphic to congruence subgroups of the modular group <span><math><mi>P</mi><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. In particular, we show that a free group of arbitrarily large rank appears as the automorphism group of such a K3 surface.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107845"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094022","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}
引用次数: 0
Minimal and cellular free resolutions over polynomial OI-algebras
IF 0.7 2区 数学
Journal of Pure and Applied Algebra Pub Date : 2025-01-01 DOI: 10.1016/j.jpaa.2024.107856
Nathan Fieldsteel , Uwe Nagel
{"title":"Minimal and cellular free resolutions over polynomial OI-algebras","authors":"Nathan Fieldsteel ,&nbsp;Uwe Nagel","doi":"10.1016/j.jpaa.2024.107856","DOIUrl":"10.1016/j.jpaa.2024.107856","url":null,"abstract":"<div><div>Minimal free resolutions of graded modules over a noetherian polynomial ring have been attractive objects of interest for more than a hundred years. We introduce and study two natural extensions in the setting of graded modules over a polynomial OI-algebra, namely <em>minimal</em> and <em>width-wise minimal</em> free resolutions. A minimal free resolution of an OI-module can be characterized by the fact that the free module in every fixed homological degree, say <em>i</em>, has minimal rank among all free resolutions of the module. We show that any finitely generated graded module over a noetherian polynomial OI-algebra admits a graded minimal free resolution and that it is unique. A width-wise minimal free resolution is a free resolution that provides a minimal free resolution of a module in every width. Such a resolution is necessarily minimal. Its existence is not guaranteed. However, we show that certain monomial OI-ideals do admit width-wise minimal free or, more generally, width-wise minimal flat resolutions. These ideals include families of well-known monomial ideals such as Ferrers ideals and squarefree strongly stable ideals. The arguments rely on the theory of cellular resolutions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107856"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094024","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}
引用次数: 0
Prime group graded rings with applications to partial crossed products and Leavitt path algebras
IF 0.7 2区 数学
Journal of Pure and Applied Algebra Pub Date : 2025-01-01 DOI: 10.1016/j.jpaa.2024.107842
Daniel Lännström , Patrik Lundström , Johan Öinert , Stefan Wagner
{"title":"Prime group graded rings with applications to partial crossed products and Leavitt path algebras","authors":"Daniel Lännström ,&nbsp;Patrik Lundström ,&nbsp;Johan Öinert ,&nbsp;Stefan Wagner","doi":"10.1016/j.jpaa.2024.107842","DOIUrl":"10.1016/j.jpaa.2024.107842","url":null,"abstract":"<div><div>We generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a characterization of prime <em>s</em>-unital strongly group graded rings, and, in particular, of infinite matrix rings and of group rings over <em>s</em>-unital rings, thereby generalizing a well-known result by Connell; (ii) characterizations of prime <em>s</em>-unital partial skew group rings and of prime unital partial crossed products; (iii) a generalization of the well-known characterizations of prime Leavitt path algebras, by Larki and by Abrams-Bell-Rangaswamy.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107842"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094010","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}
引用次数: 0
The second homology group of the commutative case of Kontsevich's symplectic derivation Lie algebra
IF 0.7 2区 数学
Journal of Pure and Applied Algebra Pub Date : 2025-01-01 DOI: 10.1016/j.jpaa.2024.107841
Shuichi Harako
{"title":"The second homology group of the commutative case of Kontsevich's symplectic derivation Lie algebra","authors":"Shuichi Harako","doi":"10.1016/j.jpaa.2024.107841","DOIUrl":"10.1016/j.jpaa.2024.107841","url":null,"abstract":"<div><div>In 1993, Kontsevich introduced the symplectic derivation Lie algebras related to various geometric objects including moduli spaces of graphs and of Riemann surfaces, graph homologies, Hamiltonian vector fields, etc. Each of them is a graded algebra, so that its Chevalley-Eilenberg chain complex has another <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math></span>-grading, called weight, than the usual homological degree. We focus on one of the Lie algebras <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>, called the “commutative case”, and its positive weight part <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>⊂</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>. The symplectic invariant homology of <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> is closely related to the commutative graph homology, hence some computational results are obtained from the viewpoint of graph homology theory. On the other hand, the details of the entire homology group <span><math><msub><mrow><mi>H</mi></mrow><mrow><mo>•</mo></mrow></msub><mo>(</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>)</mo></math></span> are not completely known. We determine <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>)</mo></math></span> by decomposing it by weight and using the classical representation theory of the symplectic groups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107841"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094012","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}
引用次数: 0
Group-graded twisted Calabi–Yau algebras
IF 0.7 2区 数学
Journal of Pure and Applied Algebra Pub Date : 2025-01-01 DOI: 10.1016/j.jpaa.2024.107849
Yasmeen S. Baki
{"title":"Group-graded twisted Calabi–Yau algebras","authors":"Yasmeen S. Baki","doi":"10.1016/j.jpaa.2024.107849","DOIUrl":"10.1016/j.jpaa.2024.107849","url":null,"abstract":"<div><div>Historically, the study of graded (twisted or otherwise) Calabi–Yau algebras has meant the study of such algebras under an <span><math><mi>N</mi></math></span>-grading. In this paper, we propose a suitable definition for a twisted <em>G</em>-graded Calabi–Yau algebra, for <em>G</em> an arbitrary abelian group. Building on the work of Reyes and Rogalski, we show that a <em>G</em>-graded algebra is twisted Calabi–Yau if and only if it is <em>G</em>-graded twisted Calabi–Yau. In the second half of the paper, we prove that localizations of twisted Calabi–Yau algebras at elements which form both left and right denominator sets remain twisted Calabi–Yau. As such, we obtain a large class of <span><math><mi>Z</mi></math></span>-graded twisted Calabi–Yau algebras arising as localizations of Artin–Schelter regular algebras. Throughout the paper, we survey a number of concrete examples of <em>G</em>-graded twisted Calabi–Yau algebras, including the Weyl algebras, families of generalized Weyl algebras, and universal enveloping algebras of finite dimensional Lie algebras.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107849"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094016","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}
引用次数: 0
Kernels of linear maps
IF 0.7 2区 数学
Journal of Pure and Applied Algebra Pub Date : 2025-01-01 DOI: 10.1016/j.jpaa.2024.107847
Arno van den Essen , Jan Schoone
{"title":"Kernels of linear maps","authors":"Arno van den Essen ,&nbsp;Jan Schoone","doi":"10.1016/j.jpaa.2024.107847","DOIUrl":"10.1016/j.jpaa.2024.107847","url":null,"abstract":"<div><div>The theorem of Duistermaat and Van der Kallen from 1998 proved the first case of the Mathieu conjecture. Using the theory of Mathieu-Zhao spaces, we can reformulate this theorem as Ker <em>L</em> is a Mathieu-Zhao space where <em>L</em> is the linear map <span><math><mi>L</mi><mo>:</mo><mi>C</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><msubsup><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>]</mo><mo>→</mo><mi>C</mi><mo>,</mo><mspace></mspace><mi>f</mi><mo>↦</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. In this paper, we generalize this result (for <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span>) to all non-trivial linear maps <span><math><mi>L</mi><mo>:</mo><mi>C</mi><mo>[</mo><mi>X</mi><mo>,</mo><msup><mrow><mi>X</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo><mo>→</mo><mi>C</mi></math></span> such that <span><math><mo>{</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>|</mo><mo>|</mo><mi>n</mi><mo>|</mo><mo>≥</mo><mi>N</mi><mo>}</mo><mo>⊂</mo><mi>Ker</mi><mspace></mspace><mi>L</mi></math></span> for some <span><math><mi>N</mi><mo>≥</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107847"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094015","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}
引用次数: 0
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