{"title":"Extension of an inequality of Ramanujan","authors":"Horst Alzer","doi":"10.1016/j.exmath.2023.02.002","DOIUrl":"10.1016/j.exmath.2023.02.002","url":null,"abstract":"<div><p>We prove that <span><span><span><math><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><mfenced><mfrac><mrow><mi>n</mi><mo>+</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mfenced><mfrac><mrow><msup><mrow><mi>k</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>k</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msup></mrow></mfrac><mo><</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow></math></span></span></span>holds for all integers <span><math><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></math></span> and real numbers <span><math><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></math></span><span>. This extends a result of Ramanujan, who submitted the inequality with </span><span><math><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow></math></span> as a problem to the “Journal of the Indian Mathematical Society”.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 2","pages":"Pages 448-450"},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49478775","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":"Fourfolds of Weil type and the spinor map","authors":"Bert van Geemen","doi":"10.1016/j.exmath.2023.04.006","DOIUrl":"10.1016/j.exmath.2023.04.006","url":null,"abstract":"<div><p>Recent papers by Markman and O’Grady give, besides their main results on the Hodge conjecture and on hyperkähler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six.</p><p>In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of <span><math><mrow><mi>S</mi><mi>O</mi><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mrow></math></span>, is used intensively. For simplicity, we use basic representation theory and we avoid the use of triality.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 2","pages":"Pages 418-447"},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44667689","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 Rochberg garden","authors":"Jesús M.F. Castillo , Raúl Pino","doi":"10.1016/j.exmath.2023.04.002","DOIUrl":"10.1016/j.exmath.2023.04.002","url":null,"abstract":"<div><p><span>In 1996, it was published the seminal work of Rochberg “Higher order estimates in complex interpolation theory” (Rochberg, 1996). Among many other things, the paper contains a new method to construct new Banach spaces<span> having an intriguing behaviour: they are simultaneously interpolation spaces and twisted sums of increasing complexity. The fundamental idea of Rochberg is to consider for each </span></span><span><math><mrow><mi>z</mi><mo>∈</mo><mi>S</mi></mrow></math></span><span><span> the space formed by the arrays of the truncated sequence of the Taylor coefficients of the elements of the Calderón space. What was probably unforeseen is that the Rochberg constructions would lead to a deep theory connecting Interpolation theory, Homology, </span>Operator Theory and the Geometry of Banach spaces. This work aims to synthetically present such connections, an up-to-date account of the theory and a list of significative open problems.</span></p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 2","pages":"Pages 333-397"},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45812008","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":"Relative plus constructions","authors":"Guille Carrión Santiago , Jérôme Scherer","doi":"10.1016/j.exmath.2023.03.001","DOIUrl":"10.1016/j.exmath.2023.03.001","url":null,"abstract":"<div><p>Let <span><math><mi>h</mi></math></span> be a connective homology theory. We construct a functorial relative plus construction as a Bousfield localization functor in the category of maps of spaces. It allows us to associate to a pair <span><math><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></math></span>, consisting of a connected space <span><math><mi>X</mi></math></span> and an <span><math><mi>h</mi></math></span>-perfect normal subgroup <span><math><mi>H</mi></math></span> of the fundamental group <span><math><mrow><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>, an <span><math><mi>h</mi></math></span>-acyclic map <span><math><mrow><mi>X</mi><mo>→</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mi>H</mi></mrow><mrow><mo>+</mo><mi>h</mi></mrow></msubsup></mrow></math></span> inducing the quotient by <span><math><mi>H</mi></math></span> on the fundamental group. We show that this map is terminal among the <span><math><mi>h</mi></math></span>-acyclic maps that kill a subgroup of <span><math><mi>H</mi></math></span>. When <span><math><mi>h</mi></math></span> is an ordinary homology theory with coefficients in a commutative ring with unit <span><math><mi>R</mi></math></span>, this provides a functorial and well-defined counterpart to a construction by cell attachment introduced by Broto, Levi, and Oliver in the spirit of Quillen’s plus construction. We also clarify the necessity to use a strongly <span><math><mi>R</mi></math></span>-perfect group <span><math><mi>H</mi></math></span> in characteristic zero.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 2","pages":"Pages 316-332"},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41568602","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}
Ulrich Bauer , Michael Kerber , Fabian Roll , Alexander Rolle
{"title":"A unified view on the functorial nerve theorem and its variations","authors":"Ulrich Bauer , Michael Kerber , Fabian Roll , Alexander Rolle","doi":"10.1016/j.exmath.2023.04.005","DOIUrl":"10.1016/j.exmath.2023.04.005","url":null,"abstract":"<div><p><span>The nerve theorem is a basic result of algebraic topology that plays a central role in computational and applied aspects of the subject. In topological data analysis, one often needs a nerve theorem that is functorial in an appropriate sense, and furthermore one often needs a nerve theorem for closed covers as well as for open covers. While the techniques for proving such functorial nerve theorems have long been available, there is unfortunately no general-purpose, explicit treatment of this topic in the literature. We address this by proving a variety of functorial nerve theorems. First, we show how one can use elementary techniques to prove nerve theorems for covers by </span>closed convex sets<span><span><span> in Euclidean space, and for covers of a </span>simplicial complex by </span>subcomplexes<span>. Then, we establish a more general, “unified” nerve theorem that subsumes many of the variants, using standard techniques from abstract homotopy theory.</span></span></p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 4","pages":"Article 125503"},"PeriodicalIF":0.7,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48967918","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":"Degeneration locus of Qp-local systems: Conjectures","authors":"A. Cadoret","doi":"10.1016/j.exmath.2023.05.002","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.05.002","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44357718","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}
Ignazio Longhi, Yunzhu Mu , Francesco Maria Saettone
{"title":"Coset topologies on Z and arithmetic applications","authors":"Ignazio Longhi, Yunzhu Mu , Francesco Maria Saettone","doi":"10.1016/j.exmath.2022.10.001","DOIUrl":"https://doi.org/10.1016/j.exmath.2022.10.001","url":null,"abstract":"<div><p>We provide a construction which covers as special cases many of the topologies on integers one can find in the literature. Moreover, our analysis of the Golomb and Kirch topologies inserts them in a family of connected, Hausdorff topologies on <span><math><mi>Z</mi></math></span>, obtained from closed sets of the profinite completion <span><math><mover><mrow><mi>Z</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>. We also discuss various applications to number theory.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 1","pages":"Pages 71-114"},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49834270","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 Stiefel’s parallelizability of 3-manifolds","authors":"Valentina Bais , Daniele Zuddas","doi":"10.1016/j.exmath.2023.01.001","DOIUrl":"10.1016/j.exmath.2023.01.001","url":null,"abstract":"<div><p>We give a new elementary proof of the parallelizability of closed orientable 3-manifolds. We use as the main tool the fact that any such manifold admits a Heegaard splitting.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 1","pages":"Pages 238-243"},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42914149","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":"Groups of prime degree and the Bateman–Horn Conjecture","authors":"Gareth A. Jones , Alexander K. Zvonkin","doi":"10.1016/j.exmath.2022.11.002","DOIUrl":"10.1016/j.exmath.2022.11.002","url":null,"abstract":"<div><p>As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree <span><math><mrow><mrow><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mrow><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> of <span><math><mrow><msub><mrow><mi>PSL</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> is prime. We present heuristic arguments and computational evidence based on the Bateman–Horn Conjecture to support a conjecture that for each prime <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> there are infinitely many primes of this form, even if one restricts to prime values of <span><math><mi>q</mi></math></span>. Similar arguments and results apply to the parameters of the simple groups <span><math><mrow><msub><mrow><mi>PSL</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>PSU</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>PSp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> which arise in the work of Dixon and Zalesskii on linear groups of prime degree.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"41 1","pages":"Pages 1-19"},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43034572","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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