Pub Date : 2025-06-17DOI: 10.1016/j.exmath.2025.125710
Takumi Asano
Miyanishi conjecture claims that for any variety over an algebraically closed field of characteristic zero, any endomorphism of such a variety which is injective outside a closed subset of codimension at least 2 is bijective. We prove Miyanishi conjecture for any quasi-projective variety which is a dense open subset of a -factorial normal projective variety such that with the ample canonical divisor or the ample anti-canonical divisor. Also, we observe Miyanishi conjecture without the conditions of its canonical divisor by using minimal model program. In particular, we prove Miyanishi conjecture in the case that has canonical singularities and has the canonical model which is obtained by divisorial contractions.
{"title":"On Miyanishi conjecture for quasi-projective varieties","authors":"Takumi Asano","doi":"10.1016/j.exmath.2025.125710","DOIUrl":"10.1016/j.exmath.2025.125710","url":null,"abstract":"<div><div>Miyanishi conjecture claims that for any variety over an algebraically closed field of characteristic zero, any endomorphism of such a variety which is injective outside a closed subset of codimension at least 2 is bijective. We prove Miyanishi conjecture for any quasi-projective variety <span><math><mi>X</mi></math></span> which is a dense open subset of a <span><math><mi>Q</mi></math></span>-factorial normal projective variety <span><math><mover><mrow><mi>X</mi></mrow><mo>¯</mo></mover></math></span> such that <span><math><mrow><mo>codim</mo><mrow><mo>(</mo><mover><mrow><mi>X</mi></mrow><mo>¯</mo></mover><mo>∖</mo><mi>X</mi><mo>)</mo></mrow><mo>≥</mo><mn>2</mn></mrow></math></span> with the ample canonical divisor or the ample anti-canonical divisor. Also, we observe Miyanishi conjecture without the conditions of its canonical divisor by using minimal model program. In particular, we prove Miyanishi conjecture in the case that <span><math><mover><mrow><mi>X</mi></mrow><mo>¯</mo></mover></math></span> has canonical singularities and <span><math><mover><mrow><mi>X</mi></mrow><mo>¯</mo></mover></math></span> has the canonical model which is obtained by divisorial contractions.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125710"},"PeriodicalIF":0.8,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-13DOI: 10.1016/j.exmath.2025.125701
Valentin Gutev
The classical McShane–Whitney extension theorem for Lipschitz functions is refined by showing that for a closed subset of the domain, it remains valid for any interval of the real line. This result is also extended to the setting of locally (pointwise) Lipschitz functions. In contrast to Lipschitz and pointwise Lipschitz extensions, the construction of locally Lipschitz extensions is based on Lipschitz partitions of unity of countable open covers of the domain. Such partitions of unity are a special case of a more general result obtained by Zdeněk Frolík. To avoid the use of Stone’s theorem (paracompactness of metrizable spaces), it is given a simple direct proof of this special case of Frolík’s result. As an application, it is shown that the locally Lipschitz functions are precisely the locally finite sums of sequences of Lipschitz functions. Also, it is obtained a natural locally Lipschitz version of one of Michael’s selection theorems.
{"title":"On real-valued functions of Lipschitz type","authors":"Valentin Gutev","doi":"10.1016/j.exmath.2025.125701","DOIUrl":"10.1016/j.exmath.2025.125701","url":null,"abstract":"<div><div>The classical McShane–Whitney extension theorem for Lipschitz functions is refined by showing that for a closed subset of the domain, it remains valid for any interval of the real line. This result is also extended to the setting of locally (pointwise) Lipschitz functions. In contrast to Lipschitz and pointwise Lipschitz extensions, the construction of locally Lipschitz extensions is based on Lipschitz partitions of unity of countable open covers of the domain. Such partitions of unity are a special case of a more general result obtained by Zdeněk Frolík. To avoid the use of Stone’s theorem (paracompactness of metrizable spaces), it is given a simple direct proof of this special case of Frolík’s result. As an application, it is shown that the locally Lipschitz functions are precisely the locally finite sums of sequences of Lipschitz functions. Also, it is obtained a natural locally Lipschitz version of one of Michael’s selection theorems.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125701"},"PeriodicalIF":0.8,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-21DOI: 10.1016/j.exmath.2025.125700
Beniamin Bogosel
The sensitivity of the areas of Reuleaux polygons and disk polygons is computed with respect to vertex perturbations. Computations are completed for both constrained and Lagrangian formulations and they imply that the only critical Reuleaux polygons for the area functional are the regular ones. As a consequence, new variational proofs for the Blaschke–Lebesgue and Firey–Sallee theorems are found.
{"title":"New variational arguments regarding the Blaschke–Lebesgue theorem","authors":"Beniamin Bogosel","doi":"10.1016/j.exmath.2025.125700","DOIUrl":"10.1016/j.exmath.2025.125700","url":null,"abstract":"<div><div>The sensitivity of the areas of Reuleaux polygons and disk polygons is computed with respect to vertex perturbations. Computations are completed for both constrained and Lagrangian formulations and they imply that the only critical Reuleaux polygons for the area functional are the regular ones. As a consequence, new variational proofs for the Blaschke–Lebesgue and Firey–Sallee theorems are found.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125700"},"PeriodicalIF":0.8,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144115481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-21DOI: 10.1016/j.exmath.2025.125692
William G. Bass, Jack S. Calcut
Ends and end cohomology are powerful invariants for the study of noncompact spaces. We present a self-contained exposition of the topological theory of ends and prove novel extensions including the existence of an exhaustion of a proper map. We define reduced end cohomology as the relative end cohomology of a ray-based space. We use those results to prove a version of a theorem of King that computes the reduced end cohomology of an end sum of two manifolds. We include a complete proof of Freudenthal’s fundamental theorem on the number of ends of a topological group, and we use our results on dimension-zero end cohomology to prove—without using transfinite induction—a theorem of Nöbeling on freeness of certain modules of continuous functions.
{"title":"Ends and end cohomology","authors":"William G. Bass, Jack S. Calcut","doi":"10.1016/j.exmath.2025.125692","DOIUrl":"10.1016/j.exmath.2025.125692","url":null,"abstract":"<div><div>Ends and end cohomology are powerful invariants for the study of noncompact spaces. We present a self-contained exposition of the topological theory of ends and prove novel extensions including the existence of an exhaustion of a proper map. We define reduced end cohomology as the relative end cohomology of a ray-based space. We use those results to prove a version of a theorem of King that computes the reduced end cohomology of an end sum of two manifolds. We include a complete proof of Freudenthal’s fundamental theorem on the number of ends of a topological group, and we use our results on dimension-zero end cohomology to prove—without using transfinite induction—a theorem of Nöbeling on freeness of certain modules of continuous functions.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125692"},"PeriodicalIF":0.8,"publicationDate":"2025-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143906828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-16DOI: 10.1016/j.exmath.2025.125691
Hongki Jung , Bartosz Langowski , Alexander Ortiz , Truong Vu
In this paper, we present an exposition of the work by Jean Bourgain, in which he resolved the well known conjecture posed by Rudin regarding the existence of -sets.
{"title":"Expository article: “Bounded orthogonal systems and the Λ(p)-set problem” by Jean Bourgain","authors":"Hongki Jung , Bartosz Langowski , Alexander Ortiz , Truong Vu","doi":"10.1016/j.exmath.2025.125691","DOIUrl":"10.1016/j.exmath.2025.125691","url":null,"abstract":"<div><div>In this paper, we present an exposition of the work by Jean Bourgain, in which he resolved the well known conjecture posed by Rudin regarding the existence of <span><math><mrow><mi>Λ</mi><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span>-sets.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125691"},"PeriodicalIF":0.8,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143868724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-11DOI: 10.1016/j.exmath.2025.125690
Zhiming Li , Bilel Selmi , Haythem Zyoudi
This paper investigates how specific techniques have been broadened to suit more general contexts. Among them, multifractal analysis stands out for its adaptability and depth, presenting a unified framework to approach these generalizations. We examine the relative multifractal formalism within the context of metric spaces in a general way. The primary aim is to introduce a generalized concept of general relative multifractal Hausdorff and packing measures. In particular, we delve into the characteristics of the generalized multifractal Hausdorff and packing measures and analyze their impact on the broader multifractal spectrum functions. The investigation explores the connection between these generalized multifractal measures and the nature of general multifractal dimensions within this framework. Further, we establish an equivalence relation between general relative multifractal Hausdorff and packing measures by utilizing density theorems. Moreover, we study various properties of the generalized relative multifractal Hausdorff measures, packing measures, and pre-measures. Lastly, our work addresses the question of whether a subset in Euclidean space with infinite positive Hausdorff measure can contain a compact set of positive finite general relative Hausdorff measures.
{"title":"A comprehensive approach to multifractal analysis","authors":"Zhiming Li , Bilel Selmi , Haythem Zyoudi","doi":"10.1016/j.exmath.2025.125690","DOIUrl":"10.1016/j.exmath.2025.125690","url":null,"abstract":"<div><div>This paper investigates how specific techniques have been broadened to suit more general contexts. Among them, multifractal analysis stands out for its adaptability and depth, presenting a unified framework to approach these generalizations. We examine the relative multifractal formalism within the context of metric spaces in a general way. The primary aim is to introduce a generalized concept of general relative multifractal Hausdorff and packing measures. In particular, we delve into the characteristics of the generalized multifractal Hausdorff and packing measures and analyze their impact on the broader multifractal spectrum functions. The investigation explores the connection between these generalized multifractal measures and the nature of general multifractal dimensions within this framework. Further, we establish an equivalence relation between general relative multifractal Hausdorff and packing measures by utilizing density theorems. Moreover, we study various properties of the generalized relative multifractal Hausdorff measures, packing measures, and pre-measures. Lastly, our work addresses the question of whether a subset in Euclidean space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with infinite positive Hausdorff measure can contain a compact set of positive finite general relative Hausdorff measures.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125690"},"PeriodicalIF":0.8,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143868741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-10DOI: 10.1016/j.exmath.2025.125689
Gessica Alecci , Carsten Elsner
From around 2010 onward, Elsner et al. developed and applied a method in which the algebraic independence of quantities over a field is transferred to further quantities by means of a system of polynomials in variables . In this paper, we systematically study and explain this criterion and its variants. Moreover, we present new results concerning its application to periodic non-regular continued fractions, namely continued fractions with real numbers as partial quotients. We show that given a continued fraction of this type, this criterion can be applied to prove that not only are the convergents algebraically independent from each other, but they are also algebraically independent from the continued fraction.
{"title":"On a criterion for algebraic independence applied to continued fractions","authors":"Gessica Alecci , Carsten Elsner","doi":"10.1016/j.exmath.2025.125689","DOIUrl":"10.1016/j.exmath.2025.125689","url":null,"abstract":"<div><div>From around 2010 onward, Elsner et al.<!--> <!--> <!-->developed and applied a method in which the algebraic independence of <span><math><mi>n</mi></math></span> quantities <span><math><mrow><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></mrow></math></span> over a field is transferred to further <span><math><mi>n</mi></math></span> quantities <span><math><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> by means of a system of polynomials in <span><math><mrow><mn>2</mn><mi>n</mi></mrow></math></span> variables <span><math><mrow><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>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>. In this paper, we systematically study and explain this criterion and its variants. Moreover, we present new results concerning its application to periodic non-regular continued fractions, namely continued fractions with real numbers as partial quotients. We show that given a continued fraction of this type, this criterion can be applied to prove that not only are the convergents algebraically independent from each other, but they are also algebraically independent from the continued fraction.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125689"},"PeriodicalIF":0.8,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143868742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-09DOI: 10.1016/j.exmath.2025.125684
Gene Abrams , Roozbeh Hazrat
Leavitt path algebras, which are algebras associated to directed graphs, were first introduced about 20 years ago. They have strong connections to such topics as symbolic dynamics, operator algebras, non-commutative geometry, representation theory, and even chip firing. In this article we invite the reader to sneak a peek at these fascinating algebras and their interplay with several seemingly disparate parts of mathematics.
{"title":"Monoids, dynamics and Leavitt path algebras","authors":"Gene Abrams , Roozbeh Hazrat","doi":"10.1016/j.exmath.2025.125684","DOIUrl":"10.1016/j.exmath.2025.125684","url":null,"abstract":"<div><div>Leavitt path algebras, which are algebras associated to directed graphs, were first introduced about 20 years ago. They have strong connections to such topics as symbolic dynamics, operator algebras, non-commutative geometry, representation theory, and even chip firing. In this article we invite the reader to sneak a peek at these fascinating algebras and their interplay with several seemingly disparate parts of mathematics.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125684"},"PeriodicalIF":0.8,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143936950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-03DOI: 10.1016/j.exmath.2025.125687
Xiaofei Qi , Huimin Chen , Jinchuan Hou
We generalize the concept of homomorphisms between Hilbert C-modules to the concept of Jordan homomorphisms. Let be a Hilbert C-module over a C-algebra and be a map. Under the condition that is commutative and is -linear bounded, we show that is a Jordan homomorphism if and only if is a homomorphism. In addition, we also discuss the relationship between -unitary maps and automorphisms, and give some conditions under which is an automorphism if and only if is a -unitary map.
我们将Hilbert C * -模间同态的概念推广到Jordan同态的概念。设M是C∗代数a上的Hilbert C *模，且φ:M→M是映射。在A是可交换的且φ是有界的条件下，证明φ是约当同态的当且仅当φ是同态。此外，我们还讨论了Φ-unitary映射与自同构的关系，并给出了φ是自同构当且仅当Φ-unitary映射的一些条件。
{"title":"Jordan homomorphisms on Hilbert C∗-modules","authors":"Xiaofei Qi , Huimin Chen , Jinchuan Hou","doi":"10.1016/j.exmath.2025.125687","DOIUrl":"10.1016/j.exmath.2025.125687","url":null,"abstract":"<div><div>We generalize the concept of homomorphisms between Hilbert C<span><math><msup><mrow></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-modules to the concept of Jordan homomorphisms. Let <span><math><mi>M</mi></math></span> be a Hilbert C<span><math><msup><mrow></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-module over a C<span><math><msup><mrow></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-algebra <span><math><mi>A</mi></math></span> and <span><math><mrow><mi>φ</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>M</mi></mrow></math></span> be a map. Under the condition that <span><math><mi>A</mi></math></span> is commutative and <span><math><mi>φ</mi></math></span> is <span><math><mi>ℂ</mi></math></span>-linear bounded, we show that <span><math><mi>φ</mi></math></span> is a Jordan homomorphism if and only if <span><math><mi>φ</mi></math></span> is a homomorphism. In addition, we also discuss the relationship between <span><math><mi>Φ</mi></math></span>-unitary maps and automorphisms, and give some conditions under which <span><math><mi>φ</mi></math></span> is an automorphism if and only if <span><math><mi>φ</mi></math></span> is a <span><math><mi>Φ</mi></math></span>-unitary map.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125687"},"PeriodicalIF":0.8,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-03DOI: 10.1016/j.exmath.2025.125686
Jingsheng Wang, Pengtong Li
In this paper, we extend the redundancy theorem of J. Gabardo and D. Han for Gabor type unitary systems indexed by full-rank lattices in Euclidean spaces to the setting of locally compact abelian (LCA) groups. Let be an LCA group, let be a uniform lattice in , let be an automorphism of , and let be an automorphism of . We show that the redundancy of a Gabor type unitary system indexed by equals the reciprocal of the density of this index set. As an application, we give a new proof of the famous time-frequency density theorem in Gabor analysis.
{"title":"The redundancy of Gabor type unitary systems on locally compact abelian groups","authors":"Jingsheng Wang, Pengtong Li","doi":"10.1016/j.exmath.2025.125686","DOIUrl":"10.1016/j.exmath.2025.125686","url":null,"abstract":"<div><div>In this paper, we extend the redundancy theorem of J. Gabardo and D. Han for Gabor type unitary systems indexed by full-rank lattices in Euclidean spaces to the setting of locally compact abelian (LCA) groups. Let <span><math><mi>G</mi></math></span> be an LCA group, let <span><math><mi>Λ</mi></math></span> be a uniform lattice in <span><math><mi>G</mi></math></span>, let <span><math><mi>α</mi></math></span> be an automorphism of <span><math><mi>G</mi></math></span>, and let <span><math><mi>β</mi></math></span> be an automorphism of <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>̂</mo></mrow></mover></math></span>. We show that the redundancy of a Gabor type unitary system indexed by <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>Λ</mi><mo>)</mo></mrow><mo>×</mo><mi>β</mi><mrow><mo>(</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mo>⊥</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> equals the reciprocal of the density of this index set. As an application, we give a new proof of the famous time-frequency density theorem in Gabor analysis.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125686"},"PeriodicalIF":0.8,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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