Pub Date : 2025-01-23DOI: 10.1016/j.exmath.2025.125656
Mikhail Ignatev , Mikhail Venchakov
Let be the unitriangular group over a finite field. We consider an interesting class of irreducible complex characters of , so-called characters of depth 2. This is a next natural step after characters of maximal and submaximal dimension, whose description is already known. We explicitly describe the support of a character of depth 2 by a system of defining algebraic equations. After that, we calculate the value of such a character on an element from the support. The main technical tool used in the proofs is the Mackey little group method for semidirect products.
{"title":"Characters of the unitriangular group and the Mackey method","authors":"Mikhail Ignatev , Mikhail Venchakov","doi":"10.1016/j.exmath.2025.125656","DOIUrl":"10.1016/j.exmath.2025.125656","url":null,"abstract":"<div><div>Let <span><math><mi>U</mi></math></span> be the unitriangular group over a finite field. We consider an interesting class of irreducible complex characters of <span><math><mi>U</mi></math></span>, so-called characters of depth 2. This is a next natural step after characters of maximal and submaximal dimension, whose description is already known. We explicitly describe the support of a character of depth 2 by a system of defining algebraic equations. After that, we calculate the value of such a character on an element from the support. The main technical tool used in the proofs is the Mackey little group method for semidirect products.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125656"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349100","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}
Pub Date : 2025-01-20DOI: 10.1016/j.exmath.2025.125654
Leonard Todjihounde
Important basics on dualistic structures on Riemannian manifolds are revisited and presented as a fundamental concept connecting information geometry, affine geometry and Hessian geometry. Since several statistical manifolds can be seen as warped product spaces, we conclude this survey by some results on warped products of dualistic structures.
{"title":"Dualistic structures in information geometry","authors":"Leonard Todjihounde","doi":"10.1016/j.exmath.2025.125654","DOIUrl":"10.1016/j.exmath.2025.125654","url":null,"abstract":"<div><div>Important basics on dualistic structures on Riemannian manifolds are revisited and presented as a fundamental concept connecting information geometry, affine geometry and Hessian geometry. Since several statistical manifolds can be seen as warped product spaces, we conclude this survey by some results on warped products of dualistic structures.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125654"},"PeriodicalIF":0.8,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386889","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}
Pub Date : 2025-01-14DOI: 10.1016/j.exmath.2025.125653
Pratulananda Das, Ayan Ghosh
Very recently in Das and Ghosh (2024), characterized subgroups have been investigated for some special kind of non-arithmetic sequences where certain cardinality related questions were answered. As statistically characterized subgroups Dikranjan et al. (2020) have evolved as non-trivial generalization of characterized subgroups, it is natural to ask the same questions for these subgroups which we try to answer here. The entire investigation emphasizes that these statistically characterized subgroups are mostly larger in size, having cardinality , and exhibit behavior that significantly differs from that of classical characterized subgroups. As a consequence, we are able to present solution of an open problem raised in Dikranjan et al. (2020).
{"title":"Statistically characterized subgroups related to some non-arithmetic sequence of integers","authors":"Pratulananda Das, Ayan Ghosh","doi":"10.1016/j.exmath.2025.125653","DOIUrl":"10.1016/j.exmath.2025.125653","url":null,"abstract":"<div><div>Very recently in Das and Ghosh (2024), characterized subgroups have been investigated for some special kind of non-arithmetic sequences where certain cardinality related questions were answered. As statistically characterized subgroups Dikranjan et al. (2020) have evolved as non-trivial generalization of characterized subgroups, it is natural to ask the same questions for these subgroups which we try to answer here. The entire investigation emphasizes that these statistically characterized subgroups are mostly larger in size, having cardinality <span><math><mi>c</mi></math></span>, and exhibit behavior that significantly differs from that of classical characterized subgroups. As a consequence, we are able to present solution of an open problem raised in Dikranjan et al. (2020).</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125653"},"PeriodicalIF":0.8,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349098","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}
Pub Date : 2025-01-10DOI: 10.1016/j.exmath.2024.125645
Bruno Deschamps
In this article, we state several results relating to the arithmetic of a constants extension of a skew fractions field . As an application, we show a non-commutative version of the Leptin–Waterhouse theorem: for any profinite group , there exist a skew field and an algebraic, outer and Galois extension such that .
{"title":"Quelques considérations galoisiennes relatives à l’extension des constantes d’un corps de fractions tordu","authors":"Bruno Deschamps","doi":"10.1016/j.exmath.2024.125645","DOIUrl":"10.1016/j.exmath.2024.125645","url":null,"abstract":"<div><div>In this article, we state several results relating to the arithmetic of a constants extension of a skew fractions field <span><math><mrow><mi>K</mi><mrow><mo>[</mo><mi>t</mi><mo>,</mo><mi>σ</mi><mo>,</mo><mi>δ</mi><mo>]</mo></mrow></mrow></math></span>. As an application, we show a non-commutative version of the Leptin–Waterhouse theorem: for any profinite group <span><math><mi>Γ</mi></math></span>, there exist a skew field <span><math><mi>K</mi></math></span> and an algebraic, outer and Galois extension <span><math><mrow><mi>L</mi><mo>/</mo><mi>K</mi></mrow></math></span> such that <span><math><mrow><mtext>Gal</mtext><mrow><mo>(</mo><mi>L</mi><mo>/</mo><mi>K</mi><mo>)</mo></mrow><mo>≃</mo><mi>Γ</mi></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125645"},"PeriodicalIF":0.8,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349058","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}
Pub Date : 2025-01-01DOI: 10.1016/j.exmath.2024.125644
Greg Martin, Pu Justin Scarfy Yang, Aram Bahrini, Prajeet Bajpai, Kübra Benli̇, Jenna Downey, Yuan Yuan Li, Xiaoxuan Liang, Amir Parvardi, Reginald Simpson, Ethan Patrick White, Chi Hoi Yip
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.
{"title":"An annotated bibliography for comparative prime number theory","authors":"Greg Martin, Pu Justin Scarfy Yang, Aram Bahrini, Prajeet Bajpai, Kübra Benli̇, Jenna Downey, Yuan Yuan Li, Xiaoxuan Liang, Amir Parvardi, Reginald Simpson, Ethan Patrick White, 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}
Pub Date : 2024-12-01DOI: 10.1016/j.exmath.2024.125632
Francesco D’Andrea
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.
{"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}
Pub Date : 2024-11-02DOI: 10.1016/j.exmath.2024.125628
Bidisha Roy
A Lehmer number modulo an odd prime number is a residue class whose multiplicative inverse has opposite parity. Lehmer numbers that are also primitive roots are called Lehmer primitive roots. Analogously, in this article we say that a residue class is a Lehmer non-primitive root modulo if is Lehmer number modulo which is not a primitive root. We provide explicit estimates for the difference between the number of Lehmer non-primitive roots modulo a prime and their “expected number”, which is . Similar explicit bounds are also provided for the number of -consecutive Lehmer numbers modulo a prime, and -consecutive Lehmer primitive roots We also prove that for any prime number , there exists a Lehmer non-primitive root modulo . Moreover, we show that for any positive integer (respectively, ) and for all primes (respectively, ), there exist consecutive Lehmer numbers modulo (respectively, consecutive Lehmer primitive roots modulo ). For large primes , these theorems generalize two results which were proven in a paper by Cohen and Trudgian appeared in the Journal of Number Theory in 2019.
奇素数 p 的雷默数是一个残差类 a∈Fp×，它的乘法逆 ā 具有相反的奇偶性。同时也是初根的雷默数称为雷默初根。与此类似，在本文中，如果一个残差类 a∈Fp× 是雷默数 modulo p，而 a 不是初等根，我们就说 a∈Fp× 是雷默非初等根 modulo p。我们提供了莱默尔非原始根数 modulo a prime p 与其 "期望数"（即 p-1-j(p-1)2）之差的明确估计值。我们还证明，对于任何素数 p>3.05×1014，都存在一个以 p 为模数的雷默非原始根。此外，我们还证明，对于任意正整数 k≥2（分别为 k≥5）和所有素数 p≥exp(122k3)（分别为 p≥exp(6.87k)），存在 k 个连续的雷默数 modulo p（分别为 k 个连续的雷默原始根 modulo p）。对于大素数 p，这些定理概括了科恩和特鲁吉安发表在 2019 年《数论杂志》上的论文中证明的两个结果。
{"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>></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}
Pub Date : 2024-10-30DOI: 10.1016/j.exmath.2024.125627
Erhard Aichinger
We provide a self-contained introduction to Gröbner bases of submodules of , where is a Euclidean domain, and explain how to use these bases to solve linear systems over .
我们对 R[x1,...,xn]k 的子模的格洛布纳基（其中 R 是欧几里得域）进行了完整的介绍，并解释了如何使用这些基来求解 R[x1,...,xn] 上的线性系统。
{"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}
Pub Date : 2024-10-22DOI: 10.1016/j.exmath.2024.125623
Jasbir S. Chahal
We determine all rational right triangles that tightly enclose the unit circle and the congruent numbers they generate.
我们确定了紧包单位圆的所有有理直角三角形及其产生的全等数。
{"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}
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