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On Ramanujan’s continued fractions of order twenty-four 关于拉马努金的24阶连分数
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-08-21 DOI: 10.1016/j.exmath.2023.08.003
Shraddha Rajkhowa, Nipen Saikia
{"title":"On Ramanujan’s continued fractions of order twenty-four","authors":"Shraddha Rajkhowa,&nbsp;Nipen Saikia","doi":"10.1016/j.exmath.2023.08.003","DOIUrl":"10.1016/j.exmath.2023.08.003","url":null,"abstract":"<div><p>Two continued fractions <span><math><mrow><mi>U</mi><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span><span> of order twenty-four are obtained from a general continued fraction identity of Ramanujan. Some theta-function and modular identities for </span><span><math><mrow><mi>U</mi><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> are established to prove general theorems for the explicit evaluations of <span><math><mrow><mi>U</mi><mrow><mo>(</mo><mo>±</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mo>±</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span>. From the theta-function identities of <span><math><mrow><mi>U</mi><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span>, three colour partition identities are derived as application to partition theory of integer. Further, <span><math><mn>2</mn></math></span>-, <span><math><mn>4</mn></math></span>- and <span><math><mn>8</mn></math></span>-dissection formulas are established for the continued fractions <span><math><mrow><msup><mrow><mi>U</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mo>≔</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msup><mi>U</mi><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mi>V</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mo>≔</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mi>V</mi><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span>, and their reciprocals.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 4","pages":"Article 125516"},"PeriodicalIF":0.7,"publicationDate":"2023-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42408513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Composition of Bhargava’s cubes over number fields 巴尔伽瓦立方体在数域上的组合
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-08-19 DOI: 10.1016/j.exmath.2023.08.002
Kristýna Zemková
{"title":"Composition of Bhargava’s cubes over number fields","authors":"Kristýna Zemková","doi":"10.1016/j.exmath.2023.08.002","DOIUrl":"10.1016/j.exmath.2023.08.002","url":null,"abstract":"<div><p><span><span>In this paper, the composition of Bhargava’s cubes is generalized to the ring of integers of a number field of narrow class number one, excluding the case of totally imaginary number fields. The exclusion of the latter case arises from the nonexistence of a </span>bijection between (classes of) binary </span>quadratic forms and an ideal class group. This problem, together with a related mistake in another paper of the author, is addressed in the appendix.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 4","pages":"Article 125515"},"PeriodicalIF":0.7,"publicationDate":"2023-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44892435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 Riemann–Roch theorem for the Adams operations Adams运算的Riemann-Roch定理
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-08-01 DOI: 10.1016/j.exmath.2023.07.002
A. Navarro , J. Navarro
{"title":"The Riemann–Roch theorem for the Adams operations","authors":"A. Navarro ,&nbsp;J. Navarro","doi":"10.1016/j.exmath.2023.07.002","DOIUrl":"10.1016/j.exmath.2023.07.002","url":null,"abstract":"<div><p>We prove the classical Riemann–Roch theorems for the Adams operations <span><math><mrow><mspace></mspace><msup><mrow><mi>ψ</mi></mrow><mrow><mi>j</mi></mrow></msup><mspace></mspace></mrow></math></span> on <span><math><mi>K</mi></math></span>-theory: a statement with coefficients on <span><math><mrow><mi>Z</mi><mrow><mo>[</mo><msup><mrow><mi>j</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo></mrow></mrow></math></span><span><span>, that holds for arbitrary projective morphisms, as well as another statement with </span>integral coefficients<span>, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann–Roch formula for one Adams operation and the analogous formula for the Chern character. To do so, we complete the elementary exposition of the work of Panin–Smirnov that was initiated by the first author in a previous paper. Their notion of oriented cohomology<span> theory on algebraic varieties allows to use classical arguments to prove general and neat statements, which imply all the aforementioned results as particular cases.</span></span></span></p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 4","pages":"Article 125513"},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44076407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
On Briançon–Skoda theorem for foliations 关于叶理的BriançOn–Skoda定理
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-07-20 DOI: 10.1016/j.exmath.2023.07.001
Arturo Fernández-Pérez , Evelia R. García Barroso , Nancy Saravia-Molina
{"title":"On Briançon–Skoda theorem for foliations","authors":"Arturo Fernández-Pérez ,&nbsp;Evelia R. García Barroso ,&nbsp;Nancy Saravia-Molina","doi":"10.1016/j.exmath.2023.07.001","DOIUrl":"10.1016/j.exmath.2023.07.001","url":null,"abstract":"<div><p>We generalize Mattei’s result relative to the Briançon–Skoda theorem for foliations to the family of foliations of the second type. We use this generalization to establish relationships between the Milnor and Tjurina numbers of foliations of second type, inspired by the results obtained by Liu for complex hypersurfaces and we determine a lower bound for the global Tjurina number of an algebraic curve.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 4","pages":"Article 125512"},"PeriodicalIF":0.7,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43076398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
On an identity of Sylvester 以西尔维斯特的身份
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-06-25 DOI: 10.1016/j.exmath.2023.06.003
Bogdan Nica
{"title":"On an identity of Sylvester","authors":"Bogdan Nica","doi":"10.1016/j.exmath.2023.06.003","DOIUrl":"10.1016/j.exmath.2023.06.003","url":null,"abstract":"<div><p>We discuss an algebraic identity, due to Sylvester, as well as related algebraic identities and applications.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 4","pages":"Article 125511"},"PeriodicalIF":0.7,"publicationDate":"2023-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44956206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
A matrix theory introduction to seaweed algebras and their index 海藻代数及其索引的矩阵论导论
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-06-19 DOI: 10.1016/j.exmath.2023.06.001
Alex Cameron , Vincent E. Coll Jr. , Nicholas Mayers , Nicholas Russoniello
{"title":"A matrix theory introduction to seaweed algebras and their index","authors":"Alex Cameron ,&nbsp;Vincent E. Coll Jr. ,&nbsp;Nicholas Mayers ,&nbsp;Nicholas Russoniello","doi":"10.1016/j.exmath.2023.06.001","DOIUrl":"10.1016/j.exmath.2023.06.001","url":null,"abstract":"<div><p><span>The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult to compute. However, for the suggestively-named seaweed algebras, the computation of the index can be reduced to a combinatorial formula based on the connected components of a “meander”: a </span>planar graph<span> associated with the algebra. Our index analysis on seaweed algebras requires only basic linear and abstract algebra. Indeed, the main goal of this article is to introduce a broader audience to seaweed algebras with minimal appeal to specialized language and notation from Lie theory. This said, we present several results that do not appear elsewhere and do appeal to more advanced language in the Introduction to provide added context.</span></p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 4","pages":"Article 125509"},"PeriodicalIF":0.7,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46797897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
On real-analytic Levi-flat hypersurfaces and holomorphic Webs 实解析Levi平超曲面与全纯Webs
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-06-13 DOI: 10.1016/j.exmath.2023.06.002
Ayane Adelina da Silva , Arturo Fernández-Pérez
{"title":"On real-analytic Levi-flat hypersurfaces and holomorphic Webs","authors":"Ayane Adelina da Silva ,&nbsp;Arturo Fernández-Pérez","doi":"10.1016/j.exmath.2023.06.002","DOIUrl":"10.1016/j.exmath.2023.06.002","url":null,"abstract":"<div><p><span>We investigate holomorphic webs tangent to real-analytic Levi-flat hypersurfaces on compact complex surfaces. Under certain conditions, we prove that a holomorphic web tangent to a real-analytic Levi-flat hypersurface admits a multiple-valued meromorphic first integral. We also prove that the Levi foliation of a Levi-flat hypersurface induced by an irreducible real-analytic curve in the Grassmannian </span><span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span><span> extends to an algebraic web on the complex projective space.</span></p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 4","pages":"Article 125510"},"PeriodicalIF":0.7,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49053443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
On three general forms of multiple zeta(-star) values 关于多重zeta(-星)值的三种一般形式
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-06-01 DOI: 10.1016/j.exmath.2023.02.003
Kwang-Wu Chen , Minking Eie
{"title":"On three general forms of multiple zeta(-star) values","authors":"Kwang-Wu Chen ,&nbsp;Minking Eie","doi":"10.1016/j.exmath.2023.02.003","DOIUrl":"10.1016/j.exmath.2023.02.003","url":null,"abstract":"<div><p><span>In this paper, we investigate three general forms of multiple zeta(-star) values. We use these values to give three new sum formulas for multiple zeta(-star) values with height </span><span><math><mrow><mo>≤</mo><mn>2</mn></mrow></math></span> and the evaluation of <span><math><mrow><msup><mrow><mi>ζ</mi></mrow><mrow><mo>⋆</mo></mrow></msup><mrow><mo>(</mo><msup><mrow><mrow><mo>{</mo><mn>1</mn><mo>}</mo></mrow></mrow><mrow><mi>m</mi></mrow></msup><mo>,</mo><msup><mrow><mrow><mo>{</mo><mn>2</mn><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>. We also give a new proof of the sum formula of multiple zeta values.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 2","pages":"Pages 299-315"},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46179870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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: 2
On Hasse’s inequality 关于Hasse不等式
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-06-01 DOI: 10.1016/j.exmath.2023.03.002
M. Ram Murty
{"title":"On Hasse’s inequality","authors":"M. Ram Murty","doi":"10.1016/j.exmath.2023.03.002","DOIUrl":"10.1016/j.exmath.2023.03.002","url":null,"abstract":"<div><p>We give an elementary exposition of the little known work of Harold Davenport related to Hasse’s inequality. We formulate a new conjecture suggested by this proof that has implications for the classical Riemann hypothesis.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 2","pages":"Pages 451-460"},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46346638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
A simple algorithm for expanding a power series as a continued fraction 将幂级数展开为连分式的一个简单算法
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-06-01 DOI: 10.1016/j.exmath.2022.12.001
Alan D. Sokal
{"title":"A simple algorithm for expanding a power series as a continued fraction","authors":"Alan D. Sokal","doi":"10.1016/j.exmath.2022.12.001","DOIUrl":"10.1016/j.exmath.2022.12.001","url":null,"abstract":"<div><p>I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980).</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 2","pages":"Pages 245-287"},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43473940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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: 6
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