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Proceedings of the 17th International Workshop on Real and Complex Singularities 第 17 届实数与复杂奇异性国际研讨会论文集
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-08-23 DOI: 10.1007/s40687-024-00465-8
Raimundo Nonato Araújo dos Santos, Alex Carlucci Rezende, Toru Ohmoto, Kentaro Saji
{"title":"Proceedings of the 17th International Workshop on Real and Complex Singularities","authors":"Raimundo Nonato Araújo dos Santos, Alex Carlucci Rezende, Toru Ohmoto, Kentaro Saji","doi":"10.1007/s40687-024-00465-8","DOIUrl":"https://doi.org/10.1007/s40687-024-00465-8","url":null,"abstract":"","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"293 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Splitting hypergeometric functions over roots of unity 在统一根上分割超几何函数
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-08-22 DOI: 10.1007/s40687-024-00468-5
Dermot McCarthy, Mohit Tripathi
{"title":"Splitting hypergeometric functions over roots of unity","authors":"Dermot McCarthy, Mohit Tripathi","doi":"10.1007/s40687-024-00468-5","DOIUrl":"https://doi.org/10.1007/s40687-024-00468-5","url":null,"abstract":"<p>We examine hypergeometric functions in the finite field, <i>p</i>-adic and classical settings. In each setting, we prove a formula which splits the hypergeometric function into a sum of lower order functions whose arguments differ by roots of unity. We provide multiple applications of these results, including new reduction and summation formulas for finite field hypergeometric functions, along with classical analogues; evaluations of special values of these functions which apply in both the finite field and <i>p</i>-adic settings; and new relations to Fourier coefficients of modular forms.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"6 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Evaluations and relations for finite trigonometric sums 有限三角和的求值和关系
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-08-19 DOI: 10.1007/s40687-024-00469-4
Bruce C. Berndt, Sun Kim, Alexandru Zaharescu
{"title":"Evaluations and relations for finite trigonometric sums","authors":"Bruce C. Berndt, Sun Kim, Alexandru Zaharescu","doi":"10.1007/s40687-024-00469-4","DOIUrl":"https://doi.org/10.1007/s40687-024-00469-4","url":null,"abstract":"<p>Several methods are used to evaluate finite trigonometric sums. In most cases, either the sum had not previously been evaluated, or it had been evaluated, but only by analytic means, e.g., by complex analysis or modular transformation formulas. We establish both reciprocity and three sum relations for trigonometric sums. Motivated by certain sums that we have evaluated, we add coprime conditions to the summands and thereby define analogues of Ramanujan sums, which we in turn evaluate. One of these analogues leads to a criterion for the Riemann Hypothesis, analogous to the Franel–Landau criterion.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"22 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Tropical adic spaces I: the continuous spectrum of a topological semiring 热带自旋空间 I：拓扑配线的连续谱
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-08-13 DOI: 10.1007/s40687-024-00467-6
Netanel Friedenberg, Kalina Mincheva
{"title":"Tropical adic spaces I: the continuous spectrum of a topological semiring","authors":"Netanel Friedenberg, Kalina Mincheva","doi":"10.1007/s40687-024-00467-6","DOIUrl":"https://doi.org/10.1007/s40687-024-00467-6","url":null,"abstract":"<p>Toward building tropical analogues of adic spaces, we study certain spaces of prime congruences as a topological semiring replacement for the space of continuous valuations on a topological ring. This requires building the theory of topological idempotent semirings, and we consider semirings of convergent power series as a primary example. We consider the semiring of convergent power series as a topological space by defining a metric on it. We check that, in tropical toric cases, the proposed objects carry meaningful geometric information. In particular, we show that the dimension behaves as expected. We give an explicit characterization of the points in terms of classical polyhedral geometry in a follow-up paper.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"31 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Algebraic aspects of holomorphic quantum modular forms 全形量子模态的代数方面
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-08-05 DOI: 10.1007/s40687-024-00464-9
Ni An, Stavros Garoufalidis, Shana Yunsheng Li
{"title":"Algebraic aspects of holomorphic quantum modular forms","authors":"Ni An, Stavros Garoufalidis, Shana Yunsheng Li","doi":"10.1007/s40687-024-00464-9","DOIUrl":"https://doi.org/10.1007/s40687-024-00464-9","url":null,"abstract":"<p>Matrix-valued holomorphic quantum modular forms are intricate objects associated to 3-manifolds (in particular to knot complements) that arise in successive refinements of the volume conjecture of knots and involve three holomorphic, asymptotic and arithmetic realizations. It is expected that the algebraic properties of these objects can be deduced from the algebraic properties of descendant state integrals, and we illustrate this for the case of the <span>((-2,3,7))</span>-pretzel knot.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141931373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Natural model reduction for kinetic equations 动力学方程的自然模型还原
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-08-03 DOI: 10.1007/s40687-024-00466-7
Zeyu Jin, Ruo Li
{"title":"Natural model reduction for kinetic equations","authors":"Zeyu Jin, Ruo Li","doi":"10.1007/s40687-024-00466-7","DOIUrl":"https://doi.org/10.1007/s40687-024-00466-7","url":null,"abstract":"<p>A promising approach to investigating high-dimensional problems is to identify their intrinsically low-dimensional features, which can be achieved through recently developed techniques for effective low-dimensional representation of functions such as machine learning. Based on available finite-dimensional approximate solution manifolds, this paper proposes a novel model reduction framework for kinetic equations. The method employs projections onto tangent bundles of approximate manifolds, naturally resulting in first-order hyperbolic systems. Under certain conditions on the approximate manifolds, the reduced models preserve several crucial properties, including hyperbolicity, conservation laws, entropy dissipation, finite propagation speed, and linear stability. For the first time, this paper rigorously discusses the relation between the H-theorem of kinetic equations and the linear stability conditions of reduced systems, determining the choice of Riemannian metrics involved in the model reduction. The framework is widely applicable for the model reduction of many models in kinetic theory.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"14 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141931374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
The growth rate of multicolor Ramsey numbers of 3-graphs 3 图形的多色拉姆齐数增长率
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-07-16 DOI: 10.1007/s40687-024-00463-w
Domagoj Bradač, Jacob Fox, Benny Sudakov
{"title":"The growth rate of multicolor Ramsey numbers of 3-graphs","authors":"Domagoj Bradač, Jacob Fox, Benny Sudakov","doi":"10.1007/s40687-024-00463-w","DOIUrl":"https://doi.org/10.1007/s40687-024-00463-w","url":null,"abstract":"<p>The <i>q</i>-color Ramsey number of a <i>k</i>-uniform hypergraph <i>G</i>, denoted <i>r</i>(<i>G</i>; <i>q</i>), is the minimum integer <i>N</i> such that any coloring of the edges of the complete <i>k</i>-uniform hypergraph on <i>N</i> vertices contains a monochromatic copy of <i>G</i>. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior of <i>r</i>(<i>G</i>; <i>q</i>) for fixed <i>G</i> and <i>q</i> tending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height of <i>r</i>(<i>G</i>; <i>q</i>) as a function of <i>q</i>. More precisely, given a hypergraph <i>G</i>, we determine when <i>r</i>(<i>G</i>; <i>q</i>) behaves polynomially, exponentially or double exponentially in <i>q</i>. This answers a question of Axenovich, Gyárfás, Liu and Mubayi.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"45 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Identities on Zagier’s rank two examples for Nahm’s problem 纳姆问题中扎吉尔排名的两个例子的特征
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-07-02 DOI: 10.1007/s40687-024-00460-z
Liuquan Wang
{"title":"Identities on Zagier’s rank two examples for Nahm’s problem","authors":"Liuquan Wang","doi":"10.1007/s40687-024-00460-z","DOIUrl":"https://doi.org/10.1007/s40687-024-00460-z","url":null,"abstract":"<p>Let <span>(rge 1)</span> be a positive integer, <i>A</i> a real positive definite symmetric <span>(rtimes r)</span> matrix, <i>B</i> a vector of length <i>r</i>, and <i>C</i> a scalar. Nahm’s problem is to describe all such <i>A</i>, <i>B</i> and <i>C</i> with rational entries for which a specific <i>r</i>-fold <i>q</i>-hypergeometric series (denoted by <span>(f_{A,B,C}(q))</span>) involving the parameters <i>A</i>, <i>B</i>, <i>C</i> is modular. When the rank <span>(r=2)</span>, Zagier provided eleven sets of examples of (<i>A</i>, <i>B</i>, <i>C</i>) for which <span>(f_{A,B,C}(q))</span> is likely to be modular. We present a number of Rogers–Ramanujan type identities involving double sums, which give modular representations for Zagier’s rank two examples. Together with several known cases in the literature, we verified ten of Zagier’s examples and give conjectural identities for the remaining example.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"29 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Invariants of vanishing Brauer classes 布劳尔消失类的不变式
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-06-28 DOI: 10.1007/s40687-024-00459-6
Federica Galluzzi, Bert van Geemen
{"title":"Invariants of vanishing Brauer classes","authors":"Federica Galluzzi, Bert van Geemen","doi":"10.1007/s40687-024-00459-6","DOIUrl":"https://doi.org/10.1007/s40687-024-00459-6","url":null,"abstract":"<p>A specialization of a <i>K</i>3 surface with Picard rank one to a <i>K</i>3 with rank two defines a vanishing class of order two in the Brauer group of the general <i>K</i>3 surface. We give the <i>B</i>-field invariants of this class. We apply this to the <i>K</i>3 double plane defined by a cubic fourfold with a plane. The specialization of such a cubic fourfold whose group of codimension two cycles has rank two to one which has rank three induces such a specialization of the double planes. We determine the Picard lattice of the specialized double plane as well as the vanishing Brauer class and its relation to the natural ‘Clifford’ Brauer class. This provides more insight in the specializations. It allows us to explicitly determine the <i>K</i>3 surfaces associated with infinitely many of the conjecturally rational cubic fourfolds obtained as such specializations.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"45 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Bruce–Roberts numbers and quasihomogeneous functions on analytic varieties 解析变体上的布鲁斯-罗伯茨数和准均质函数
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-06-26 DOI: 10.1007/s40687-024-00458-7
C. Bivià-Ausina, K. Kourliouros, M. A. S. Ruas
{"title":"Bruce–Roberts numbers and quasihomogeneous functions on analytic varieties","authors":"C. Bivià-Ausina, K. Kourliouros, M. A. S. Ruas","doi":"10.1007/s40687-024-00458-7","DOIUrl":"https://doi.org/10.1007/s40687-024-00458-7","url":null,"abstract":"<p>Given a germ of an analytic variety <i>X</i> and a germ of a holomorphic function <i>f</i> with a stratified isolated singularity with respect to the logarithmic stratification of <i>X</i>, we show that under certain conditions on the singularity type of the pair (<i>f</i>, <i>X</i>), the following relative analog of the well-known K. Saito’s theorem holds true: equality of the relative Milnor and Tjurina numbers of <i>f</i> with respect to <i>X</i> (also known as Bruce–Roberts numbers) is equivalent to the relative quasihomogeneity of the pair (<i>f</i>, <i>X</i>), i.e. to the existence of a coordinate system such that both <i>f</i> and <i>X</i> are quasihomogeneous with respect to the same positive rational weights.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"24 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
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