Journal of the London Mathematical Society-Second Series最新文献

{"title":"Matchings: Source, goal, and faithful companion","authors":"András Sebő","doi":"10.1112/jlms.70703","DOIUrl":"https://doi.org/10.1112/jlms.70703","url":null,"abstract":"<p>Matchings were among the earliest motivations for graph theory. They subsequently remained a central goal, inspiring the development of new mathematical tools that went well beyond problems directly concerning matchings.</p><p>These tools proved widely applicable, accompanying the growth of graph theory over the past century. A milestone in this trajectory is <i>The factorisation of linear graphs</i>, which firmly embedded graph theory—through matchings—into the body of classical mathematics, in particular, linear algebra and polynomials. We revisit this article by Tutte, presenting its original content, sketching some aspects of its impact until some recent progress, and tracing one of its subsequent lines of development, finally leading to a new contribution that answers an open challenge and extends known results.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70703","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149145793","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":"Distinguishing Siegel modular forms","authors":"Arvind Kumar,&nbsp;Ariel Weiss","doi":"10.1112/jlms.70726","DOIUrl":"https://doi.org/10.1112/jlms.70726","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;annotation&gt;$f$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$f^{prime }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be genus 2 cuspidal Siegel paramodular newforms. We prove that if their Hecke eigenvalues &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$a_p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;annotation&gt;$a_p^{prime }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; satisfy a non-trivial polynomial relation &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$P(a_p, a_p^{prime }) = 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for a set of primes &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of positive density, then &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;annotation&gt;$f$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a scalar multiple of a quadratic twist of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$f^{prime }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. This result extends the strong multiplicity one theorem, which handles the case &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;y&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;y&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$P(x,y) = x - y$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, to arbitrary polynomial relations. Our proof analyses the image of the product Galois representation attached to the pair &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149145720","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}

{"title":"Quantitative Carleman-type estimates for holomorphic sections over bounded domains","authors":"Xiangsen Qin","doi":"10.1112/jlms.70717","DOIUrl":"https://doi.org/10.1112/jlms.70717","url":null,"abstract":"<p>This paper establishes quantitative Carleman-type inequalities for holomorphic sections of Hermitian vector bundles over bounded domains in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$mathbb {C}^n$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$n geqslant 2$</annotation>\u0000 </semantics></math>. We first prove a Sobolev-type inequality with explicit constants for the Laplace operator, which leads to quantitative Carleman-type estimates for holomorphic functions. These results are then extended to holomorphic sections of Hermitian vector bundles satisfying certain curvature restrictions, yielding quantitative versions where previously only non-quantitative forms were available. The proofs refine existing methods through careful constant tracking and by estimating the radius of the uniform sphere condition of the boundary through the Lipschitz constant of its outward unit normal vector.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149147013","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}

{"title":"On the geometry of WDVV equations and their Hamiltonian formalism in arbitrary dimension","authors":"S. Opanasenko,&nbsp;R. Vitolo","doi":"10.1112/jlms.70720","DOIUrl":"https://doi.org/10.1112/jlms.70720","url":null,"abstract":"<p>It is known that in low dimensions Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations can be rewritten as commuting quasilinear bi-Hamiltonian systems. We extend some of these results to arbitrary dimension <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> and arbitrary scalar product <span></span><math>\u0000 <semantics>\u0000 <mi>η</mi>\u0000 <annotation>$eta$</annotation>\u0000 </semantics></math>. In particular, we show that suitable subsets of WDVV equations can be interpreted as a set of linear line congruences in suitable Plücker embeddings. This form leads to their representation as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$N-2$</annotation>\u0000 </semantics></math> Hamiltonian systems of conservation laws. Moreover, we show that WDVV equations can be reduced to an orthonomic form, which is also passive in low dimensions <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>⩽</mo>\u0000 <mn>5</mn>\u0000 </mrow>\u0000 <annotation>$Nleqslant 5$</annotation>\u0000 </semantics></math>. This also leads to the commutativity of the Hamiltonian systems of conservation laws (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>⩽</mo>\u0000 <mn>5</mn>\u0000 </mrow>\u0000 <annotation>$Nleqslant 5$</annotation>\u0000 </semantics></math>), after which we can find a solution of the WDVV equations from a joint solution of the Hamiltonian systems. Finally, we conjecture that passivity holds in all dimensions.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70720","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149147014","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":"A quadratic Abramovich–Bertram formula","authors":"Erwan Brugallé,&nbsp;Kirsten Wickelgren","doi":"10.1112/jlms.70705","DOIUrl":"https://doi.org/10.1112/jlms.70705","url":null,"abstract":"<p>Quadratic Gromov–Witten invariants allow one to obtain an arithmetically meaningful count of curves satisfying constraints over a field <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> without assuming that <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> is the field of complex or real numbers. This paper studies the behavior of quadratic genus 0 Gromov–Witten invariants during an algebraic analog of surgery on del Pezzo surfaces. For this, we define and study (twisted) binomial coefficients in the Grothendieck–Witt group, building on work of Serre. We obtain a formula expressing the quadratic genus 0 Gromov–Witten invariants of surfaces obtained as a smoothing of a given nodal surface in terms of those of the one having the largest Picard group. We give applications to quadratic Gromov–Witten invariants of rational del Pezzo surfaces of degree at least 7, some cubic surfaces, for point constraints defined over quadratic extensions of <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>, as well as an invariance result under a Dehn twist.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149147000","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}

{"title":"Spectrum of p-adic linear differential equations II: Variation of the spectrum","authors":"Tinhinane A. Azzouz","doi":"10.1112/jlms.70718","DOIUrl":"https://doi.org/10.1112/jlms.70718","url":null,"abstract":"<p>This paper explores the spectrum of <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-adic linear differential equations, extending previous findings to quasi-smooth Berkovich curves. It focuses on the relationship between the spectrum and the radii of convergence by investigating the continuity and variation of the spectrum for these equations. The study reveals that the spectrum can be influenced by the controlling graph of the radii of convergence and is continuous on the skeleton of annuli. Furthermore, the paper demonstrates that approximating the connection allows for an accurate estimation of spectral radii of convergence. Key results include a refined decomposition theorem with respect to the spectrum and its extension to neighborhoods of specific points. The work builds upon earlier research and introduces new approaches for understanding the spectral properties of differential equations defined over quasi-smooth curves.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149147012","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}

{"title":"The Boolean spectrum of a Grothendieck category","authors":"Henning Krause","doi":"10.1112/jlms.70719","DOIUrl":"https://doi.org/10.1112/jlms.70719","url":null,"abstract":"<p>A notion of support for objects in any Grothendieck category is introduced. This is based on the spectral category of a Grothendieck category and uses its Boolean lattice of localising subcategories. The support provides a classification of all subcategories that are closed under arbitrary coproducts, subobjects, and essential extensions. There is also a notion of exact support which classifies certain thick subcategories. As an application, the coproduct decompositions of objects are described in terms of Boolean lattices. Also, for any ring Crawley-Boevey's correspondence between definable subcategories of modules and closed subsets of the Ziegler spectrum is extended.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70719","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149146755","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":"Quasi-isometries of relatively hyperbolic groups with an elementary hierarchy","authors":"Aaron W. Messerla","doi":"10.1112/jlms.70698","DOIUrl":"https://doi.org/10.1112/jlms.70698","url":null,"abstract":"<p>Sela introduced limit groups in his work on the Tarski problem, and showed that each limit group has a cyclic hierarchy. In this paper, a class of relatively hyperbolic groups, equipped with a hierarchy similar to the one for limit groups, is shown to be closed under quasi-isometry. Additionally, these groups share some of the properties of limit groups. In particular, groups quasi-isometric to limit groups are shown to be locally extended residually finite (LERF) and virtually toral relatively hyperbolic.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149146756","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}

{"title":"Optimal eigenvalues on a metric graph with densities","authors":"Kiyan Naderi,&nbsp;Noema Nicolussi","doi":"10.1112/jlms.70714","DOIUrl":"https://doi.org/10.1112/jlms.70714","url":null,"abstract":"<p>We introduce and study Laplacians on a finite metric graph endowed with generalized densities, that is, measures of finite mass. One important motivation is that this setting provides a common framework for several interesting classes of operators: discrete graph Laplacians, Kirchhoff Laplacians, and Dirichlet-to-Neumann operators on graphs. Our main interest lies in spectral optimization with respect to the underlying measure. In contrast to the setting of domains and manifolds, we prove that a minimal <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>-th eigenvalue exists, whereas the corresponding maximization problem has no meaning. We then establish connections between these optimal eigenvalues and the geometry of the metric graph, including a transparent geometric characterization of the first optimal eigenvalue via the resistance metric, and a Weyl law for the higher optimal eigenvalues.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149048423","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}

{"title":"Mixed motives and linear forms in the Catalan constant","authors":"Payman Eskandari,&nbsp;V. Kumar Murty,&nbsp;Yusuke Nemoto","doi":"10.1112/jlms.70708","DOIUrl":"https://doi.org/10.1112/jlms.70708","url":null,"abstract":"<p>We first give a geometric construction of a two-dimensional mixed motive over <span></span><math>\u0000 <semantics>\u0000 <mi>Q</mi>\u0000 <annotation>${mathbb {Q}}$</annotation>\u0000 </semantics></math> with the Catalan constant <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <msup>\u0000 <mn>3</mn>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <msup>\u0000 <mn>5</mn>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <msup>\u0000 <mn>7</mn>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mi>⋯</mi>\u0000 </mrow>\u0000 <annotation>${mathbf {G}}=1-1/3^2+1/5^2-1/7^2+cdots$</annotation>\u0000 </semantics></math> as a period. We then use this motive to obtain a supply of linear forms in 1 and <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>${mathbf {G}}$</annotation>\u0000 </semantics></math>. We also explicitly compute the coefficients of 1 and <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>${mathbf {G}}$</annotation>\u0000 </semantics></math> in these linear forms.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 4","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70708","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149048422","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}