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A short review of finite approximations and unconventional physics 有限近似和非常规物理学简评
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2025-03-01 Epub Date: 2024-05-13 DOI: 10.1016/j.exmath.2024.125573
Trond Digernes
{"title":"A short review of finite approximations and unconventional physics","authors":"Trond Digernes","doi":"10.1016/j.exmath.2024.125573","DOIUrl":"10.1016/j.exmath.2024.125573","url":null,"abstract":"<div><div>We review some topics from our collaboration with V.S. Varadarajan.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125573"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141052781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
A variation on von Neumann Entropy and a result of Varadarajan 冯-诺依曼熵的变体和瓦拉达拉詹的一个结果
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2025-03-01 Epub Date: 2024-04-25 DOI: 10.1016/j.exmath.2024.125572
Glenn D. Appleby
{"title":"A variation on von Neumann Entropy and a result of Varadarajan","authors":"Glenn D. Appleby","doi":"10.1016/j.exmath.2024.125572","DOIUrl":"10.1016/j.exmath.2024.125572","url":null,"abstract":"<div><div>Thirty years ago, I had just completed my Ph.D. under Varadarajan when, as part of a subsequent reading course, he and I considered a generalization of von Neumann entropy, here called a <em>matrix entropy</em>, computed by using the classical entropy function on the diagonals of density matrices. I had asked whether the value of von Neumann entropy was the maximum of the matrix entropy on a given unitary equivalence class. Varadarajan soon sketched a proof of this, which is presented here. It serves as a nice way to see classical entropy sitting in the von Neumann entropy context, and a reminder for this short note’s author of a pleasant time spent working with a remarkable scholar and teacher.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125572"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140796018","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
Representations of real numbers by alternating Perron series and their geometry 用交替的Perron级数及其几何表示实数
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2025-02-01 Epub Date: 2024-12-05 DOI: 10.1016/j.exmath.2024.125635
Mykola Moroz
{"title":"Representations of real numbers by alternating Perron series and their geometry","authors":"Mykola Moroz","doi":"10.1016/j.exmath.2024.125635","DOIUrl":"10.1016/j.exmath.2024.125635","url":null,"abstract":"<div><div>We consider the representation of real numbers by alternating Perron series (<span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span>-representation), which is a generalization of representations of real numbers by Ostrogradsky–Sierpiński–Pierce series (Pierce series), alternating Sylvester series (second Ostrogradsky series), alternating Lüroth series, etc. Namely, we prove the basic topological and metric properties of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span>-representation and find the relationship between <span><math><mi>P</mi></math></span>-representation and <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span>-representation in some measure theory problems.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 1","pages":"Article 125635"},"PeriodicalIF":0.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133974","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
Extremal length and duality 极值长度和对偶性
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2025-02-01 Epub Date: 2024-12-07 DOI: 10.1016/j.exmath.2024.125634
Kai Rajala
{"title":"Extremal length and duality","authors":"Kai Rajala","doi":"10.1016/j.exmath.2024.125634","DOIUrl":"10.1016/j.exmath.2024.125634","url":null,"abstract":"<div><div>Classical extremal length (or conformal modulus) is a conformal invariant involving families of paths on the Riemann sphere. In “Extremal length and functional completion”, Fuglede initiated an abstract theory of extremal length which has since been widely applied. Concentrating on duality properties and applications to quasiconformal analysis, we demonstrate the flexibility of the theory and present recent advances in three different settings:</div><div>(1) Extremal length and uniformization of metric surfaces.</div><div>(2) Extremal length of families of surfaces and quasiconformal maps between <span><math><mi>n</mi></math></span>-dimensional spaces.</div><div>(3) Schramm’s transboundary extremal length and conformal maps between multiply connected plane domains.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 1","pages":"Article 125634"},"PeriodicalIF":0.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Conformal tilings, combinatorial curvature, and the type problem 保角拼接，组合曲率，和类型问题
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2025-02-01 Epub Date: 2024-12-02 DOI: 10.1016/j.exmath.2024.125633
Mohith Raju Nagaraju
{"title":"Conformal tilings, combinatorial curvature, and the type problem","authors":"Mohith Raju Nagaraju","doi":"10.1016/j.exmath.2024.125633","DOIUrl":"10.1016/j.exmath.2024.125633","url":null,"abstract":"<div><div>Roughly, a conformal tiling of a Riemann surface is a tiling where each tile is a suitable conformal image of a Euclidean regular polygon. In 1997, Bowers and Stephenson constructed an edge-to-edge conformal tiling of the complex plane using conformally regular pentagons. In contrast, we show that for all <span><math><mrow><mi>n</mi><mo>≥</mo><mn>7</mn></mrow></math></span>, there is no edge-to-edge conformal tiling of the complex plane using conformally regular <span><math><mi>n</mi></math></span>-gons. More generally, we discuss a relationship between the combinatorial curvature at each vertex of the conformal tiling and the universal cover (sphere, plane, or disc) of the underlying Riemann surface. This result follows from the work of Stone (1976) and Oh (2005) through a rich interplay between Riemannian geometry and combinatorial geometry. We provide an exposition of these proofs and some new applications to conformal tilings.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 1","pages":"Article 125633"},"PeriodicalIF":0.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133973","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 note on the standard zero-free region for L-functions 关于 L 函数标准无零区域的说明
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2024-12-01 Epub Date: 2024-10-17 DOI: 10.1016/j.exmath.2024.125624
Sun-Kai Leung
{"title":"A note on the standard zero-free region for L-functions","authors":"Sun-Kai Leung","doi":"10.1016/j.exmath.2024.125624","DOIUrl":"10.1016/j.exmath.2024.125624","url":null,"abstract":"<div><div>In this short note, we establish a standard zero-free region for a general class of <span><math><mi>L</mi></math></span>-functions for which their logarithms have coefficients with nonnegative real parts, including the Rankin–Selberg <span><math><mi>L</mi></math></span>-functions for unitary cuspidal automorphic representations.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125624"},"PeriodicalIF":0.8,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142525776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Strong Gröbner bases and linear algebra in multivariate polynomial rings over Euclidean domains 欧几里得域上多变量多项式环中的强格罗布纳基和线性代数
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2024-12-01 Epub Date: 2024-10-30 DOI: 10.1016/j.exmath.2024.125627
Erhard Aichinger
{"title":"Strong Gröbner bases and linear algebra in multivariate polynomial rings over Euclidean domains","authors":"Erhard Aichinger","doi":"10.1016/j.exmath.2024.125627","DOIUrl":"10.1016/j.exmath.2024.125627","url":null,"abstract":"<div><div>We provide a self-contained introduction to Gröbner bases of submodules of <span><math><mrow><mi>R</mi><msup><mrow><mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span>, where <span><math><mi>R</mi></math></span> is a Euclidean domain, and explain how to use these bases to solve linear systems over <span><math><mrow><mi>R</mi><mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125627"},"PeriodicalIF":0.8,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Some remarks on rational right triangles 关于有理直角三角形的一些评论
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2024-12-01 Epub Date: 2024-10-22 DOI: 10.1016/j.exmath.2024.125623
Jasbir S. Chahal
{"title":"Some remarks on rational right triangles","authors":"Jasbir S. Chahal","doi":"10.1016/j.exmath.2024.125623","DOIUrl":"10.1016/j.exmath.2024.125623","url":null,"abstract":"<div><div>We determine all rational right triangles that tightly enclose the unit circle and the congruent numbers they generate.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125623"},"PeriodicalIF":0.8,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592557","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
Abelian groups acting on the line 作用于直线的阿贝尔群
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2024-12-01 Epub Date: 2024-09-26 DOI: 10.1016/j.exmath.2024.125619
Nancy Guelman, Matilde Martínez
{"title":"Abelian groups acting on the line","authors":"Nancy Guelman,&nbsp;Matilde Martínez","doi":"10.1016/j.exmath.2024.125619","DOIUrl":"10.1016/j.exmath.2024.125619","url":null,"abstract":"<div><div>We study the action of finitely generated Abelian subgroups of <span><math><mrow><mi>H</mi><mi>o</mi><mi>m</mi><mi>e</mi><msub><mrow><mi>o</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>.</div><div>We propose a presentation where the focus is on understanding the set of stabilizers, which yields a dynamical description of the action that is both elementary and self-contained.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125619"},"PeriodicalIF":0.8,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417662","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
Classification of the conjugacy classes of SL˜(2,R) SL˜(2,R) 共轭类的分类
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2024-12-01 Epub Date: 2024-10-22 DOI: 10.1016/j.exmath.2024.125626
Christian Táfula
{"title":"Classification of the conjugacy classes of SL˜(2,R)","authors":"Christian Táfula","doi":"10.1016/j.exmath.2024.125626","DOIUrl":"10.1016/j.exmath.2024.125626","url":null,"abstract":"<div><div>In this note, we classify the conjugacy classes of <span><math><mrow><msub><mrow><mover><mrow><mi>SL</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, the universal covering group of <span><math><mrow><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>. For any non-central element <span><math><mrow><mi>α</mi><mo>∈</mo><msub><mrow><mover><mrow><mi>SL</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, we show that its conjugacy class may be determined by three invariants: (i) <em>Trace</em>: the trace (valued in the set of positive real numbers <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>) of its image <span><math><mover><mrow><mi>α</mi></mrow><mo>¯</mo></mover></math></span> in <span><math><mrow><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>; (ii) <em>Direction type</em>: the sign behavior of the induced self-homeomorphism of <span><math><mi>R</mi></math></span> determined by the lifting <span><math><mrow><msub><mrow><mover><mrow><mi>SL</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>↷</mo><mi>R</mi></mrow></math></span> of the action <span><math><mrow><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>↷</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></math></span>; (iii) <em>The function</em> <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>♯</mi></mrow></msup></math></span>: a conjugacy invariant length function introduced by Mochizuki (2016).</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125626"},"PeriodicalIF":0.8,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659469","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
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