Mathematische Annalen最新文献

{"title":"Discrete stochastic maximal regularity.","authors":"Foivos Evangelopoulos-Ntemiris, Mark Veraar","doi":"10.1007/s00208-026-03348-1","DOIUrl":"10.1007/s00208-026-03348-1","url":null,"abstract":"<p><p>In this paper, we investigate discrete regularity estimates for a broad class of temporal numerical schemes for parabolic stochastic evolution equations. We provide a characterization of discrete stochastic maximal <math><msup><mi>ℓ</mi> <mi>p</mi></msup> </math> -regularity in terms of its continuous counterpart, thereby establishing a unified framework that yields numerous new discrete regularity results. Moreover, as a consequence of the continuous-time theory, we establish several important properties of discrete stochastic maximal regularity such as extrapolation in the exponent <i>p</i> and with respect to a power weight. Furthermore, employing the <math><msup><mi>H</mi> <mi>∞</mi></msup> </math> -functional calculus, we derive a powerful discrete maximal estimate in the trace space norm <math> <mrow><msub><mi>D</mi> <mi>A</mi></msub> <mrow><mo>(</mo> <mn>1</mn> <mo>-</mo> <mfrac><mn>1</mn> <mi>p</mi></mfrac> <mo>,</mo> <mi>p</mi> <mo>)</mo></mrow> </mrow> </math> for <math><mrow><mi>p</mi> <mo>∈</mo> <mo>[</mo> <mn>2</mn> <mo>,</mo> <mi>∞</mi> <mo>)</mo></mrow> </math> .</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"394 2","pages":"42"},"PeriodicalIF":1.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12904967/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146202136","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}

{"title":"Groups acting on trees with Tits' independence property (P): With an appendix by Stephan Tornier.","authors":"Colin D Reid, Simon M Smith","doi":"10.1007/s00208-026-03412-w","DOIUrl":"https://doi.org/10.1007/s00208-026-03412-w","url":null,"abstract":"<p><p>Local actions (actions of a vertex stabiliser on the neighbours of that vertex) have become an important approach to group actions on trees since J. Tits' introduction in 1970 of the independence property <math><mrow><mo>(</mo> <mtext>P</mtext> <mo>)</mo></mrow> </math> and especially since a 2000 paper by M. Burger and Sh. Mozes. This 'local-to-global' approach has been critical in the development of the theory of totally disconnected locally compact groups because it allows the construction of additional group actions on trees while keeping control over the action of a vertex stabiliser, in a way that is not practical under the classical Bass-Serre approach. The majority of constructions of nonlinear nondiscrete locally compact simple groups use <math><mrow><mo>(</mo> <mtext>P</mtext> <mo>)</mo></mrow> </math> and its generalisations. In this article we give a full classification and description of all closed group actions on trees with Tits' independence property <math><mrow><mo>(</mo> <mtext>P</mtext> <mo>)</mo></mrow> </math> using a new coherent theory for local actions that applies to all actions on trees. This theory is a 'local action' complement to classical Bass-Serre theory. On the one hand, our theory gives a decomposition of a group acting on a tree into a 'local action diagram' (a decorated graph that encodes all 'local' information), and on the other hand a construction of a group acting on a tree from a given local action diagram. One can read directly from the local action diagram whether the resulting group has certain properties, like geometric density, compact generation and simplicity.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"395 3","pages":"53"},"PeriodicalIF":1.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13152943/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147868663","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}

{"title":"The profinite rigidity of torsion-free lamplighter groups.","authors":"Nikolay Nikolov, Julian Wykowski","doi":"10.1007/s00208-026-03548-9","DOIUrl":"https://doi.org/10.1007/s00208-026-03548-9","url":null,"abstract":"<p><p>We prove that the torsion-free lamplighter group <math><mrow><mi>Γ</mi> <mo>=</mo> <msup><mrow><mi>Z</mi></mrow> <mi>n</mi></msup> <mo>≀</mo> <mi>Z</mi></mrow> </math> of any rank <math><mrow><mi>n</mi> <mo>∈</mo> <mi>N</mi></mrow> </math> is profinitely rigid in the absolute sense: the finite quotients of <math><mi>Γ</mi></math> determine its isomorphism type uniquely among all finitely generated residually finite groups. The proof combines the theory of profinite rigidity for modules over Noetherian domains with an analysis of the algebraic properties of the lower central series of groups with the same profinite completion as <math><mi>Γ</mi></math> .</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"396 1","pages":"17"},"PeriodicalIF":1.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13493475/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148795001","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}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Low-energy <ns0:math><ns0:mi>α</ns0:mi></ns0:math> -harmonic maps into the round sphere.","authors":"Ben Sharp","doi":"10.1007/s00208-026-03566-7","DOIUrl":"10.1007/s00208-026-03566-7","url":null,"abstract":"<p><p>We classify low-energy <math><mi>α</mi></math> -harmonic maps from a closed non-spherical Riemannian surface <math><mi>Σ</mi></math> of constant curvature to the round sphere via their bubble scales and centres. In particular we show that as <math><mrow><mn>1</mn> <mo><</mo> <mi>α</mi> <mo>↓</mo> <mn>1</mn></mrow> </math> and assuming <math><msub><mi>E</mi> <mi>α</mi></msub> </math> is close to <math><mrow><mo>|</mo> <mi>Σ</mi> <mo>|</mo> <mo>+</mo> <mn>4</mn> <mi>π</mi></mrow> </math> then degree-one <math><mi>α</mi></math> -harmonic maps blow a bubble based at a critical point <math><msub><mi>a</mi> <mi>c</mi></msub> </math> of a an explicit function <math><mi>J</mi></math> and at scale <math> <mrow> <msqrt> <mrow><mrow><mo>|</mo> <mi>J</mi></mrow> <mrow><mo>(</mo> <msub><mi>a</mi> <mi>c</mi></msub> <mo>)</mo></mrow> <msup><mrow><mo>|</mo></mrow> <mrow><mo>-</mo> <mn>1</mn></mrow> </msup> <mrow><mo>(</mo> <mi>α</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> </mrow> </msqrt> <mo>.</mo></mrow> </math> Up to a constant, <math><mi>J</mi></math> is the sum of the squares of any <math><msup><mi>L</mi> <mn>2</mn></msup> </math> -orthonormal basis of holomorphic one-forms on the domain.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"396 2","pages":"33"},"PeriodicalIF":1.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13569565/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148955351","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}

{"title":"Cones of Noether-Lefschetz divisors and moduli spaces of hyperkähler manifolds.","authors":"Ignacio Barros, Pietro Beri, Laure Flapan, Brandon Williams","doi":"10.1007/s00208-026-03372-1","DOIUrl":"10.1007/s00208-026-03372-1","url":null,"abstract":"<p><p>We give a general formula for generators of the NL cone on an orthogonal modular variety. This is the cone of effective divisors linearly equivalent to an effective linear combination of irreducible components of Noether-Lefschetz divisors. We apply this to describe, in terms of minimal generators, the NL cone of various moduli spaces of geometric origin such as those of polarized K3 surfaces, cubic fourfolds, and hyperkähler manifolds. Additionally, we establish uniruledness for many moduli spaces of primitively polarized hyperkähler manifolds of <math><mtext>OG6</mtext></math> and <math><msub><mtext>Kum</mtext> <mi>n</mi></msub> </math> -type. Finally, in analogy with the case of K3 surfaces of degree 2, we show that any family of polarized <math><msub><mtext>Kum</mtext> <mn>2</mn></msub> </math> -type hyperkähler manifolds with divisibility 2 and polarization degree 2 over a projective base is isotrivial.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"394 2","pages":"27"},"PeriodicalIF":1.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12904891/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146202156","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}

{"title":"Automorphisms of relatively hyperbolic groups and the Farrell-Jones conjecture.","authors":"Naomi Andrew, Yassine Guerch, Sam Hughes","doi":"10.1007/s00208-026-03431-7","DOIUrl":"https://doi.org/10.1007/s00208-026-03431-7","url":null,"abstract":"<p><p>We prove the fibred Farrell-Jones conjecture (FJC) in <i>A</i>-, <i>K</i>-, and <i>L</i>-theory for a large class of suspensions of relatively hyperbolic groups, as well as for all suspensions of one-ended hyperbolic groups. We deduce two applications: FJC for the automorphism group of a one-ended group hyperbolic relative to virtually polycyclic subgroups;FJC is closed under extensions of FJC groups with kernel in a large class of relatively hyperbolic groups. Along the way we prove a number of results about JSJ decompositions of relatively hyperbolic groups which may be of independent interest.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"395 4","pages":"89"},"PeriodicalIF":1.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13269154/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148272052","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}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">On manifolds with almost non-negative Ricci curvature and integrally-positive <ns0:math><ns0:msup><ns0:mi>k</ns0:mi> <ns0:mrow><ns0:mi>th</ns0:mi></ns0:mrow> </ns0:msup> </ns0:math> -scalar curvature.","authors":"Alessandro Cucinotta, Andrea Mondino","doi":"10.1007/s00208-026-03406-8","DOIUrl":"https://doi.org/10.1007/s00208-026-03406-8","url":null,"abstract":"<p><p>We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest <i>k</i> eigenvalues of the Ricci tensor. If <math><mrow><mo>(</mo> <msup><mi>M</mi> <mi>n</mi></msup> <mo>,</mo> <mi>g</mi> <mo>)</mo></mrow> </math> is a Riemannian manifold satisfying such curvature bounds for <math><mrow><mi>k</mi> <mo>=</mo> <mn>2</mn></mrow> </math> , then we show that <i>M</i> is contained in a neighbourhood of controlled width of an isometrically embedded 1-dimensional sub-manifold. From this, we deduce several metric and topological consequences: <i>M</i> has at most linear volume growth and at most two ends, it has bounded 1-Urysohn width, the first Betti number of <i>M</i> is bounded above by 1, and there is precise information on elements of infinite order in <math> <mrow><msub><mi>π</mi> <mn>1</mn></msub> <mrow><mo>(</mo> <mi>M</mi> <mo>)</mo></mrow> </mrow> </math> . If <math><mrow><mo>(</mo> <msup><mi>M</mi> <mi>n</mi></msup> <mo>,</mo> <mi>g</mi> <mo>)</mo></mrow> </math> is a Riemannian manifold satisfying such bounds for <math><mrow><mi>k</mi> <mo>≥</mo> <mn>2</mn></mrow> </math> , then we show that <i>M</i> has at most <math><mrow><mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> </math> -dimensional behavior at large scales. If <math><mrow><mi>k</mi> <mo>=</mo> <mi>n</mi> <mo>=</mo> <mtext>dim</mtext> <mo>(</mo> <mi>M</mi> <mo>)</mo></mrow> </math> , so that the integral lower bound is on the scalar curvature, assuming in addition that the <math><mrow><mo>(</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo></mrow> </math> -Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to <math><mrow><mi>n</mi> <mo>-</mo> <mn>2</mn></mrow> </math> . From the above results we deduce topological restrictions, such as upper bounds on the first Betti number.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"394 2","pages":"49"},"PeriodicalIF":1.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12907273/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146213694","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}

{"title":"The shifted convolution problem in function fields.","authors":"Alexandra Florea, Matilde Lalín, Amita Malik, Anurag Sahay","doi":"10.1007/s00208-026-03340-9","DOIUrl":"10.1007/s00208-026-03340-9","url":null,"abstract":"<p><p>We study the shifted convolution problem for the divisor function in function fields in the large degree limit, that is, the average value of <math><mrow><mi>d</mi> <mo>(</mo> <mi>f</mi> <mo>)</mo> <mi>d</mi> <mo>(</mo> <mi>f</mi> <mo>+</mo> <mi>h</mi> <mo>)</mo></mrow> </math> where <i>f</i> runs over monic polynomials in <math> <mrow><msub><mi>F</mi> <mi>q</mi></msub> <mrow><mo>[</mo> <mi>T</mi> <mo>]</mo></mrow> </mrow> </math> of a given degree, and <i>h</i> is a given monic polynomial. We prove an asymptotic formula in the range <math><mrow><mo>deg</mo> <mo>(</mo> <mi>h</mi> <mo>)</mo> <mo><</mo> <mo>(</mo> <mn>2</mn> <mo>-</mo> <mi>ϵ</mi> <mo>)</mo> <mo>deg</mo> <mo>(</mo> <mi>f</mi> <mo>)</mo></mrow> </math> . We also consider mixed correlations and self-correlations of <math> <mrow><msub><mi>r</mi> <mi>χ</mi></msub> <mo>=</mo> <mn>1</mn> <mo>⋆</mo> <mi>χ</mi></mrow> </math> , the convolution of 1 with a Dirichlet character mod <math><mi>ℓ</mi></math> , where <math><mi>ℓ</mi></math> is a monic irreducible polynomial, proving asymptotic formulae in various ranges. This includes the case of quadratic characters, which yields results about correlations of norm-counting functions of quadratic extensions of <math> <mrow><msub><mi>F</mi> <mi>q</mi></msub> <mrow><mo>[</mo> <mi>T</mi> <mo>]</mo></mrow> </mrow> </math> . A novel feature of our work is a Voronoi summation formula (equivalently, a functional equation for the Estermann function) in <math> <mrow><msub><mi>F</mi> <mi>q</mi></msub> <mrow><mo>[</mo> <mi>T</mi> <mo>]</mo></mrow> </mrow> </math> which was not previously available.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"395 1","pages":"6"},"PeriodicalIF":1.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12999764/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147499370","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}

{"title":"Positive curvature conditions on contractible manifolds.","authors":"Paul Sweeney","doi":"10.1007/s00208-026-03333-8","DOIUrl":"https://doi.org/10.1007/s00208-026-03333-8","url":null,"abstract":"<p><p>Our goal is to identify curvature conditions that distinguish Euclidean space in the case of open, contractible manifolds and the disk in the case of compact, contractible manifolds with boundary. First, we show that an open manifold that is the interior of a sufficiently connected, compact, contractible 5-manifold with boundary and supports a complete Riemannian metric with uniformly positive scalar curvature is diffeomorphic to Euclidean 5-space. Next, we investigate the analogous question for compact manifolds with boundary: Must a compact, contractible manifold that supports a Riemannian metric with positive scalar curvature and mean convex boundary necessarily be the disk? We present examples demonstrating that this curvature condition alone cannot distinguish the disk; on the other hand, we exhibit stronger curvature conditions that allow us to draw such a conclusion.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"394 3","pages":"53"},"PeriodicalIF":1.4,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12916530/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147271439","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}

{"title":"Abelian varieties of prescribed order over finite fields.","authors":"Raymond van Bommel, Edgar Costa, Wanlin Li, Bjorn Poonen, Alexander Smith","doi":"10.1007/s00208-024-03084-4","DOIUrl":"https://doi.org/10.1007/s00208-024-03084-4","url":null,"abstract":"<p><p>Given a prime power <i>q</i> and <math><mrow><mi>n</mi> <mo>≫</mo> <mn>1</mn></mrow> </math> , we prove that every integer in a large subinterval of the Hasse-Weil interval <math><mrow><mo>[</mo> <msup><mrow><mo>(</mo> <msqrt><mi>q</mi></msqrt> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mrow><mn>2</mn> <mi>n</mi></mrow> </msup> <mo>,</mo> <msup><mrow><mo>(</mo> <msqrt><mi>q</mi></msqrt> <mo>+</mo> <mn>1</mn> <mo>)</mo></mrow> <mrow><mn>2</mn> <mi>n</mi></mrow> </msup> <mo>]</mo></mrow> </math> is <math><mrow><mo>#</mo> <mi>A</mi> <mo>(</mo> <msub><mi>F</mi> <mi>q</mi></msub> <mo>)</mo></mrow> </math> for some ordinary geometrically simple principally polarized abelian variety <i>A</i> of dimension <i>n</i> over <math><msub><mi>F</mi> <mi>q</mi></msub> </math> . As a consequence, we generalize a result of Howe and Kedlaya for <math><msub><mi>F</mi> <mn>2</mn></msub> </math> to show that for each prime power <i>q</i>, every sufficiently large positive integer is realizable, i.e., <math><mrow><mo>#</mo> <mi>A</mi> <mo>(</mo> <msub><mi>F</mi> <mi>q</mi></msub> <mo>)</mo></mrow> </math> for some abelian variety <i>A</i> over <math><msub><mi>F</mi> <mi>q</mi></msub> </math> . Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse-Weil interval. A separate argument determines, for fixed <i>n</i>, the largest subinterval of the Hasse-Weil interval consisting of realizable integers, asymptotically as <math><mrow><mi>q</mi> <mo>→</mo> <mi>∞</mi></mrow> </math> ; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if <math><mrow><mi>q</mi> <mo>≤</mo> <mn>5</mn></mrow> </math> , then every positive integer is realizable, and for arbitrary <i>q</i>, every positive integer <math><mrow><mo>≥</mo> <msup><mi>q</mi> <mrow><mn>3</mn> <msqrt><mi>q</mi></msqrt> <mo>log</mo> <mi>q</mi></mrow> </msup> </mrow> </math> is realizable.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"392 1","pages":"1167-1202"},"PeriodicalIF":1.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11971235/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143795736","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}