Expositiones Mathematicae最新文献

{"title":"An annotated bibliography for comparative prime number theory","authors":"Greg Martin,&nbsp;Pu Justin Scarfy Yang,&nbsp;Aram Bahrini,&nbsp;Prajeet Bajpai,&nbsp;Kübra Benli̇,&nbsp;Jenna Downey,&nbsp;Yuan Yuan Li,&nbsp;Xiaoxuan Liang,&nbsp;Amir Parvardi,&nbsp;Reginald Simpson,&nbsp;Ethan Patrick White,&nbsp;Chi Hoi Yip","doi":"10.1016/j.exmath.2024.125644","DOIUrl":"10.1016/j.exmath.2024.125644","url":null,"abstract":"<div><div>The goal of this annotated bibliography is to record every publication on the topic of comparative prime number theory together with a summary of its results. We use a unified system of notation for the quantities being studied and for the hypotheses under which results are obtained.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125644"},"PeriodicalIF":0.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143508438","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}

{"title":"Quantum spheres as graph C*-algebras: A review","authors":"Francesco D’Andrea","doi":"10.1016/j.exmath.2024.125632","DOIUrl":"10.1016/j.exmath.2024.125632","url":null,"abstract":"<div><div>In this survey, we discuss the description of Vaksman–Soibelman quantum spheres using graph C*-algebras, following the seminal work of Hong and Szymański. We give a slightly different proof of the isomorphism with a graph C*-algebra, borrowing the idea of Mikkelsen and Kaad of using conditional expectations to prove the desired result.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125632"},"PeriodicalIF":0.8,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143137735","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}

{"title":"The Blaschke–Lebesgue theorem revisited","authors":"Ryan Hynd","doi":"10.1016/j.exmath.2024.125617","DOIUrl":"10.1016/j.exmath.2024.125617","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125617"},"PeriodicalIF":0.8,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143137736","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}

{"title":"On the existence of certain Lehmer numbers modulo a prime","authors":"Bidisha Roy","doi":"10.1016/j.exmath.2024.125628","DOIUrl":"10.1016/j.exmath.2024.125628","url":null,"abstract":"<div><div>A <em>Lehmer number modulo an odd prime number</em> <span><math><mi>p</mi></math></span> is a residue class <span><math><mrow><mi>a</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>×</mo></mrow></msubsup></mrow></math></span> whose multiplicative inverse <span><math><mover><mrow><mi>a</mi></mrow><mrow><mo>̄</mo></mrow></mover></math></span> has opposite parity. Lehmer numbers that are also primitive roots are called <em>Lehmer primitive roots</em>. Analogously, in this article we say that a residue class <span><math><mrow><mi>a</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>×</mo></mrow></msubsup></mrow></math></span> is a <em>Lehmer non-primitive root modulo</em> <span><math><mi>p</mi></math></span> if <span><math><mi>a</mi></math></span> is Lehmer number modulo <span><math><mi>p</mi></math></span> which is not a primitive root. We provide explicit estimates for the difference between the number of Lehmer non-primitive roots modulo a prime <span><math><mi>p</mi></math></span> and their “expected number”, which is <span><math><mfrac><mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Similar explicit bounds are also provided for the number of <span><math><mi>k</mi></math></span>-consecutive Lehmer numbers modulo a prime, and <span><math><mi>k</mi></math></span>-consecutive Lehmer primitive roots We also prove that for any prime number <span><math><mrow><mi>p</mi><mo>&gt;</mo><mn>3</mn><mo>.</mo><mn>05</mn><mo>×</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>14</mn></mrow></msup></mrow></math></span>, there exists a Lehmer non-primitive root modulo <span><math><mi>p</mi></math></span>. Moreover, we show that for any positive integer <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span> (respectively, <span><math><mrow><mi>k</mi><mo>≥</mo><mn>5</mn></mrow></math></span>) and for all primes <span><math><mrow><mi>p</mi><mo>≥</mo><mo>exp</mo><mrow><mo>(</mo><mn>1</mn><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> (respectively, <span><math><mrow><mi>p</mi><mo>≥</mo><mo>exp</mo><mrow><mo>(</mo><mn>6</mn><mo>.</mo><mn>8</mn><msup><mrow><mn>7</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>), there exist <span><math><mi>k</mi></math></span> consecutive Lehmer numbers modulo <span><math><mi>p</mi></math></span> (respectively, <span><math><mi>k</mi></math></span> consecutive Lehmer primitive roots modulo <span><math><mi>p</mi></math></span>). For large primes <span><math><mi>p</mi></math></span>, these theorems generalize two results which were proven in a paper by Cohen and Trudgian appeared in the Journal of Number Theory in 2019.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125628"},"PeriodicalIF":0.8,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659468","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}

{"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":"2024-10-30","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}

{"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":"2024-10-22","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}

{"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":"2024-10-22","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}

{"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":"2024-10-17","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}

{"title":"Brownian motion in a vector space over a local field is a scaling limit","authors":"Tyler Pierce ,&nbsp;Rahul Rajkumar ,&nbsp;Andrea Stine ,&nbsp;David Weisbart ,&nbsp;Adam M. Yassine","doi":"10.1016/j.exmath.2024.125607","DOIUrl":"10.1016/j.exmath.2024.125607","url":null,"abstract":"<div><div>For any natural number <span><math><mi>d</mi></math></span>, the Vladimirov–Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a <span><math><mi>d</mi></math></span>-dimensional vector space <span><math><mi>V</mi></math></span> over a local field <span><math><mi>K</mi></math></span>. Just as the Laplace operator on <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> is the infinitesimal generator of Brownian motion with state space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, the Vladimirov–Taibleson operator on <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></math></span> is the infinitesimal generator of real-time Brownian motion with state space <span><math><mi>V</mi></math></span>. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted <span><math><mi>V</mi></math></span> to be the <span><math><mi>p</mi></math></span>-adic numbers.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 6","pages":"Article 125607"},"PeriodicalIF":0.8,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432092","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}

{"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":"2024-09-26","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}