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A conjecture of Warnaar-Zudilin from deformations of lie superalgebras. 李超代数变形的Warnaar-Zudilin猜想。
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2026-01-01 Epub Date: 2026-04-02 DOI: 10.1007/s40687-026-00613-2
Thomas Creutzig, Niklas Garner
{"title":"A conjecture of Warnaar-Zudilin from deformations of lie superalgebras.","authors":"Thomas Creutzig, Niklas Garner","doi":"10.1007/s40687-026-00613-2","DOIUrl":"https://doi.org/10.1007/s40687-026-00613-2","url":null,"abstract":"<p><p>We prove a collection of <i>q</i>-series identities conjectured by Warnaar and Zudilin and appearing in recent work with H. Kim in the context of superconformal field theory. Our proof utilizes a deformation of the simple affine vertex operator superalgebra <math> <mrow><msub><mi>L</mi> <mi>k</mi></msub> <mrow><mo>(</mo> <msub><mi>osp</mi> <mrow><mn>1</mn> <mo>|</mo> <mn>2</mn> <mi>n</mi></mrow> </msub> <mo>)</mo></mrow> </mrow> </math> into the principal subsuperspace of <math> <mrow><msub><mi>L</mi> <mi>k</mi></msub> <mrow><mo>(</mo> <mi>s</mi> <msub><mi>l</mi> <mrow><mn>1</mn> <mo>|</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> <mo>)</mo></mrow> </mrow> </math> in a manner analogous to earlier work of Feigin-Stoyanovsky. This result fills a gap left by Stoyanovsky, showing that for all positive integers <i>N</i>, <i>k</i> the character of the principal subspace of type <math><msub><mi>A</mi> <mi>N</mi></msub> </math> at level <i>k</i> can be identified with the (super)character of a simple affine vertex operator (super)algebra at the same level.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"13 2","pages":"33"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13046567/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147624547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Applications of dimension interpolation to orthogonal projections. 维度插值在正交投影中的应用。
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2025-01-01 Epub Date: 2025-01-25 DOI: 10.1007/s40687-025-00496-9
Jonathan M Fraser
{"title":"Applications of dimension interpolation to orthogonal projections.","authors":"Jonathan M Fraser","doi":"10.1007/s40687-025-00496-9","DOIUrl":"https://doi.org/10.1007/s40687-025-00496-9","url":null,"abstract":"<p><p>Dimension interpolation is a novel programme of research which attempts to unify the study of fractal dimension by considering various spectra which live in between well-studied notions of dimension such as Hausdorff, box, Assouad and Fourier dimension. These spectra often reveal novel features not witnessed by the individual notions and this information has applications in many directions. In this survey article, we discuss dimension interpolation broadly and then focus on applications to the dimension theory of orthogonal projections. We focus on three distinct applications coming from three different dimension spectra, namely, the Fourier spectrum, the intermediate dimensions, and the Assouad spectrum. The celebrated Marstrand-Mattila projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. This result has inspired much further research on the dimension theory of projections including the consideration of dimensions other than the Hausdorff dimension, and the study of the exceptional set in the Marstrand-Mattila theorem.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"12 1","pages":"10"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11890391/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143598044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal. 热带方案理论中的射影超曲面I: Macaulay理想。
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2025-01-01 Epub Date: 2025-04-25 DOI: 10.1007/s40687-025-00517-7
Alex Fink, Jeffrey Giansiracusa, Noah Giansiracusa, Joshua Mundinger
{"title":"Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal.","authors":"Alex Fink, Jeffrey Giansiracusa, Noah Giansiracusa, Joshua Mundinger","doi":"10.1007/s40687-025-00517-7","DOIUrl":"https://doi.org/10.1007/s40687-025-00517-7","url":null,"abstract":"<p><p>A \"tropical ideal\" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each <math><mrow><mi>n</mi> <mo>≥</mo> <mn>2</mn></mrow> </math> and <math><mrow><mi>d</mi> <mo>≥</mo> <mn>1</mn></mrow> </math> , our construction yields a non-realizable degree <i>d</i> hypersurface scheme in <math> <msup><mrow><mi>P</mi></mrow> <mi>n</mi></msup> </math> . Maclagan-Rincón produced a non-realizable line in <math> <msup><mrow><mi>P</mi></mrow> <mi>n</mi></msup> </math> for each <i>n</i>, and for <math><mrow><mo>(</mo> <mi>d</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo></mrow> </math> the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"12 2","pages":"30"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12031988/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144035312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Approximate incidence geometry in the plane. 平面上的近似入射几何。
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2025-01-01 Epub Date: 2025-09-05 DOI: 10.1007/s40687-025-00552-4
Tuomas Orponen
{"title":"Approximate incidence geometry in the plane.","authors":"Tuomas Orponen","doi":"10.1007/s40687-025-00552-4","DOIUrl":"https://doi.org/10.1007/s40687-025-00552-4","url":null,"abstract":"<p><p>These are lecture notes for a mini-course given in Banff in June 2024. They discuss the problem of bounding the number of <math><mi>δ</mi></math> <i>-incidences</i> <math> <mrow><msub><mi>I</mi> <mi>δ</mi></msub> <mrow><mo>(</mo> <mi>P</mi> <mo>,</mo> <mi>L</mi> <mo>)</mo></mrow> <mo>:</mo> <mo>=</mo> <mrow><mo>{</mo> <mrow><mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>ℓ</mi> <mo>)</mo></mrow> <mo>∈</mo> <mi>P</mi> <mo>×</mo> <mi>L</mi> <mo>:</mo> <mi>p</mi> <mo>∈</mo> <msub><mrow><mo>[</mo> <mi>ℓ</mi> <mo>]</mo></mrow> <mi>δ</mi></msub> <mo>}</mo></mrow> </mrow> </math> under various hypotheses on <math><mrow><mi>P</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </mrow> </math> and <math><mrow><mi>L</mi> <mo>⊂</mo> <mi>A</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo></mrow> </math> . The main focus will be on hypotheses relevant for the <i>Furstenberg set problem</i>.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"12 4","pages":"65"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12413431/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145015229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions. 不受限制分区、划分为不同部分和过度分区中的残馀类偏差。
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2025-01-01 Epub Date: 2025-02-21 DOI: 10.1007/s40687-025-00502-0
Michael J Schlosser, Nian Hong Zhou
{"title":"Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions.","authors":"Michael J Schlosser, Nian Hong Zhou","doi":"10.1007/s40687-025-00502-0","DOIUrl":"https://doi.org/10.1007/s40687-025-00502-0","url":null,"abstract":"<p><p>We prove specific biases in the number of occurrences of parts belonging to two different residue classes <i>a</i> and <i>b</i>, modulo a fixed nonnegative integer <i>m</i>, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size <i>n</i> that belong to these sets of partitions and have a symmetric residue class bias (i.e., for <math><mrow><mn>1</mn> <mo>≤</mo> <mi>a</mi> <mo><</mo> <mi>m</mi> <mo>/</mo> <mn>2</mn></mrow> </math> and <math><mrow><mi>b</mi> <mo>=</mo> <mi>m</mi> <mo>-</mo> <mi>a</mi></mrow> </math> ), as <i>n</i> tends to infinity.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"12 1","pages":"17"},"PeriodicalIF":1.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11845404/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143484297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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
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