Advances in Mathematics最新文献

{"title":"Mordell-Tornheim zeta functions and functional equations for Herglotz-Zagier type functions","authors":"Atul Dixit,&nbsp;Sumukha Sathyanarayana ,&nbsp;N. Guru Sharan","doi":"10.1016/j.aim.2025.110303","DOIUrl":"10.1016/j.aim.2025.110303","url":null,"abstract":"<div><div>The Mordell-Tornheim zeta function and the Herglotz-Zagier function <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are two important functions in Mathematics. By generalizing a special case of the former, namely <span><math><mi>Θ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>, we show that the theories of these functions are inextricably woven. We obtain a three-term functional equation for <span><math><mi>Θ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> as well as decompose it in terms of the Herglotz-Hurwitz function <span><math><mi>Φ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>. This decomposition can be conceived as a two-term functional equation for <span><math><mi>Φ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>. Through this result, we are not only able to get Zagier's identity relating <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> with <span><math><mi>F</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>x</mi><mo>)</mo></math></span> but also a two-term functional equation for Ishibashi's generalization of <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, namely, <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, which has been sought after for over twenty years. We further generalize <span><math><mi>Θ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> by incorporating two Gauss sums, each associated to a Dirichlet character, and decompose it in terms of an interesting integral which involves the Fekete polynomial as well as the character polylogarithm. This result gives infinite families of functional equations of Herglotz-type integrals out of which only two, due to Choie and Kumar, were known so far. The first one among the two involves the integral <span><math><mi>J</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> whose special values have received a lot of attention, more recently, in the work of Muzzaffar and Williams, and in that of Radchenko and Zagier. Analytic continuation of our generalization of <span><math><mi>Θ</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> is also accomplished which allows us to obtain transformations between certain double series and Herglotz-type integrals or their explicit evaluations.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110303"},"PeriodicalIF":1.5,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143879396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On subspaces of indecomposable Banach spaces","authors":"Piotr Koszmider,&nbsp;Zdeněk Silber","doi":"10.1016/j.aim.2025.110292","DOIUrl":"10.1016/j.aim.2025.110292","url":null,"abstract":"<div><div>We address the following question: what is the class of Banach spaces isomorphic to subspaces of indecomposable Banach spaces? We show that this class includes all Banach spaces of density not bigger than the continuum which do not admit <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> as a quotient (equivalently do not admit a subspace isomorphic to <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>c</mi><mo>)</mo></math></span>). This includes all Asplund spaces and all weakly Lindelöf determined Banach spaces of density not bigger than the continuum. However, we also show that this class includes some Banach spaces admitting <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> as a quotient. This sheds some light on the question asked in the paper [2] of Argyros and Haydon whether all Banach spaces not containing <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> embed in some indecomposable Banach spaces. Our method of constructing indecomposable Banach spaces above a given Banach space is a considerable modification of the method of constructing Banach spaces of continuous functions with few^⁎ operators developed before by the first-named author.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110292"},"PeriodicalIF":1.5,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143879395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Unimodal sequences and mixed false theta functions","authors":"Kevin Allen,&nbsp;Robert Osburn","doi":"10.1016/j.aim.2025.110293","DOIUrl":"10.1016/j.aim.2025.110293","url":null,"abstract":"<div><div>We consider two-parameter generalizations of Hecke-Appell type expansions for the generating functions of unimodal and special unimodal sequences. We then determine their explicit representations which involve mixed false theta functions. These results complement recent striking work of Mortenson and Zwegers on the mixed mock modularity of the generalized <em>U</em>-function due to Hikami and Lovejoy. As an application, we demonstrate how to recover classical partial theta function identities which appear in Ramanujan's lost notebook and in work of Warnaar.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110293"},"PeriodicalIF":1.5,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143873941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two curious strongly invertible L-space knots","authors":"Kenneth L. Baker ,&nbsp;Marc Kegel ,&nbsp;Duncan McCoy","doi":"10.1016/j.aim.2025.110287","DOIUrl":"10.1016/j.aim.2025.110287","url":null,"abstract":"<div><div>We present two examples of strongly invertible L-space knots whose surgeries are never the double branched cover of a Khovanov thin link in the 3-sphere. Consequently, these knots provide counterexamples to a conjectural characterization of strongly invertible L-space knots due to Watson. We also discuss other exceptional properties of these two knots, for example, these two L-space knots have formal semigroups that are actual semigroups.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110287"},"PeriodicalIF":1.5,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the dynamics of a complex continued fraction map which contains the Gauss map as its real number section","authors":"Hiromi Ei ,&nbsp;Hitoshi Nakada ,&nbsp;Rie Natsui","doi":"10.1016/j.aim.2025.110286","DOIUrl":"10.1016/j.aim.2025.110286","url":null,"abstract":"<div><div>We consider the complex continued fraction map <em>T</em> defined by R. Kaneiwa, I. Shiokawa, and J. Tamura (1975) associated with the Eisenstein field <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>3</mn></mrow></msqrt><mo>)</mo></math></span>. A significant aspect of their continued fraction map is that the real number part of this map <em>T</em> is exactly the simple continued fraction map (Gauss map). In this paper we characterize the set of strictly periodic expansions of continued fraction expansions associated to this map in terms of quadratic extensions of <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>3</mn></mrow></msqrt><mo>)</mo></math></span> in connection with the closure of <span><math><mo>{</mo><mo>−</mo><mfrac><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac><mo>:</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></math></span>, where <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>, is the denominator of the <em>n</em>th principal convergent of the continued fraction expansion. Moreover, we show that the closure of <span><math><mo>{</mo><mo>−</mo><mfrac><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac><mo>:</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></math></span> has positive Lebesgue measure on the complex plane <span><math><mi>C</mi></math></span> though it has infinitely many holes. This gives us that the construction of the natural extension of <em>T</em> on a subset of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>∖</mo><mo>{</mo><mtext>diagonal</mtext><mo>}</mo></math></span> is equivalent to the geodesics over the hyperbolic space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Then the invariant measure for the natural extension map is given by the hyperbolic measure. Hence its projection to the complex plane is obviously the invariant measure for <em>T</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110286"},"PeriodicalIF":1.5,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Braid group actions on branched coverings and full exceptional sequences","authors":"Wen Chang ,&nbsp;Fabian Haiden ,&nbsp;Sibylle Schroll","doi":"10.1016/j.aim.2025.110284","DOIUrl":"10.1016/j.aim.2025.110284","url":null,"abstract":"<div><div>We relate full exceptional sequences in Fukaya categories of surfaces or equivalently in derived categories of graded gentle algebras to branched coverings over the disk, building on a previous classification result of the first and third author <span><span>[5]</span></span>. This allows us to apply tools from the theory of branched coverings such as Birman–Hilden theory and Hurwitz systems to study the natural braid group action on exceptional sequences. As an application, counterexamples are given to a conjecture of Bondal–Polishchuk <span><span>[3]</span></span> on the transitivity of the braid group action on full exceptional sequences in a triangulated category.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110284"},"PeriodicalIF":1.5,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Large-scale geometry of Borel graphs of polynomial growth","authors":"Anton Bernshteyn ,&nbsp;Jing Yu","doi":"10.1016/j.aim.2025.110290","DOIUrl":"10.1016/j.aim.2025.110290","url":null,"abstract":"<div><div>We study graphs of polynomial growth from the perspective of asymptotic geometry and descriptive set theory. The starting point of our investigation is a theorem of Krauthgamer and Lee who showed that every connected graph of polynomial growth admits an injective contraction mapping to <span><math><mo>(</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msub><mrow><mo>‖</mo><mo>⋅</mo><mo>‖</mo></mrow><mrow><mo>∞</mo></mrow></msub><mo>)</mo></math></span> for some <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>. We strengthen and generalize this result in a number of ways. In particular, answering a question of Papasoglu, we construct coarse embeddings from graphs of polynomial growth to <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Moreover, we only require <em>n</em> to be linear in the asymptotic polynomial growth rate of the graph; this confirms a conjecture of Levin and Linial, London, and Rabinovich “in the asymptotic sense.” (The exact form of the conjecture was refuted by Krauthgamer and Lee.) All our results are proved for Borel graphs, which allows us to settle a number of problems in descriptive combinatorics. Roughly, we prove that graphs generated by free Borel actions of <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> are universal for the class of Borel graphs of polynomial growth. This provides a general method for extending results about <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>-actions to all Borel graphs of polynomial growth. For example, an immediate consequence of our main result is that all Borel graphs of polynomial growth are hyperfinite, which answers a well-known question in the area. As another illustration, we show that Borel graphs of polynomial growth support a certain combinatorial structure called a toast. An important technical tool in our arguments is the notion of padded decomposition from computer science, which is closely related to the concept of asymptotic dimension due to Gromov. Along the way we find an alternative, probabilistic proof of a theorem of Papasoglu that graphs of asymptotic polynomial growth rate <span><math><mi>ρ</mi><mo>&lt;</mo><mo>∞</mo></math></span> have asymptotic dimension at most <em>ρ</em> and establish the same bound in the Borel setting.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110290"},"PeriodicalIF":1.5,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"2-torsion in instanton Floer homology","authors":"Zhenkun Li ,&nbsp;Fan Ye","doi":"10.1016/j.aim.2025.110289","DOIUrl":"10.1016/j.aim.2025.110289","url":null,"abstract":"<div><div>This paper studies the existence of 2-torsion in instanton Floer homology with <span><math><mi>Z</mi></math></span> coefficients for closed 3-manifolds and singular knots. First, we show that the non-existence of 2-torsion in the framed instanton Floer homology <span><math><msup><mrow><mi>I</mi></mrow><mrow><mo>♯</mo></mrow></msup><mo>(</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mi>K</mi><mo>)</mo><mo>;</mo><mi>Z</mi><mo>)</mo></math></span> of any nonzero integral <em>n</em>-surgery along a knot <em>K</em> in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> would imply that <em>K</em> is fibered. Also, we show that <span><math><msup><mrow><mi>I</mi></mrow><mrow><mo>♯</mo></mrow></msup><mo>(</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mi>K</mi><mo>)</mo><mo>;</mo><mi>Z</mi><mo>)</mo></math></span> for any nontrivial <em>K</em> with <span><math><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>4</mn></math></span> always has 2-torsion. These two results indicate that the existence of 2-torsion is expected to be a generic phenomenon for Dehn surgeries along knots. Second, we show that for genus-one knots with nontrivial Alexander polynomials and for unknotting-number-one knots, the unreduced singular instanton knot homology <span><math><msup><mrow><mi>I</mi></mrow><mrow><mo>♯</mo></mrow></msup><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><mi>K</mi><mo>;</mo><mi>Z</mi><mo>)</mo></math></span> always has 2-torsion. Finally, some crucial lemmas that help us demonstrate the existence of 2-torsion are motivated by analogous results in Heegaard Floer theory, which may be of independent interest. In particular, we show that, for a knot <em>K</em> in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, if there is a nonzero rational number <em>r</em> such that the dual knot <span><math><msub><mrow><mover><mrow><mi>K</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msub></math></span> inside <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mi>K</mi><mo>)</mo></math></span> is Floer simple, then <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mi>K</mi><mo>)</mo></math></span> must be an L-space and <em>K</em> must be an L-space knot.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110289"},"PeriodicalIF":1.5,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143864896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Restricted Hausdorff spectra of q-adic automorphisms","authors":"Jorge Fariña-Asategui","doi":"10.1016/j.aim.2025.110294","DOIUrl":"10.1016/j.aim.2025.110294","url":null,"abstract":"<div><div>Firstly, we completely determine the self-similar Hausdorff spectrum of the group of <em>q</em>-adic automorphisms where <em>q</em> is a prime power, answering a question of Grigorchuk. Indeed, we take a further step and completely determine its Hausdorff spectra restricted to the most important subclasses of self-similar groups, providing examples differing drastically from the previously known ones in the literature. Our proof relies on a new explicit formula for the computation of the Hausdorff dimension of closed self-similar groups and a generalization of iterated permutational wreath products.</div><div>Secondly, we provide for every prime <em>p</em> the first examples of just infinite branch pro-<em>p</em> groups with zero Hausdorff dimension in <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, giving strong evidence against a well-known conjecture of Boston.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110294"},"PeriodicalIF":1.5,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143864897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Frieze patterns and Farey complexes","authors":"Ian Short,&nbsp;Matty van Son,&nbsp;Andrei Zabolotskii","doi":"10.1016/j.aim.2025.110269","DOIUrl":"10.1016/j.aim.2025.110269","url":null,"abstract":"<div><div>Frieze patterns have attracted significant attention recently, motivated by their relationship with cluster algebras. A longstanding open problem has been to provide a combinatorial model for frieze patterns over the ring of integers modulo <em>N</em> akin to Conway and Coxeter's celebrated model for positive integer frieze patterns. Here we solve this problem using the Farey complex of the ring of integers modulo <em>N</em>; in fact, using more general Farey complexes we provide combinatorial models for frieze patterns over any rings whatsoever.</div><div>Our strategy generalises that of the first author and of Morier-Genoud et al. for integers and that of Felikson et al. for Eisenstein integers. We also generalise results of Singerman and Strudwick on diameters of Farey graphs, we recover a theorem of Morier-Genoud on enumerating friezes over finite fields, and we classify those frieze patterns modulo <em>N</em> that lift to frieze patterns over the integers in terms of the topology of the corresponding Farey complexes.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110269"},"PeriodicalIF":1.5,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}