Expositiones Mathematicae最新文献

{"title":"Linearization of Lipschitz framings for Banach spaces","authors":"Qiyao Bao ,&nbsp;Deguang Han ,&nbsp;Rui Liu ,&nbsp;Jie Shen","doi":"10.1016/j.exmath.2025.125680","DOIUrl":"10.1016/j.exmath.2025.125680","url":null,"abstract":"<div><div>Nonlinear framings naturally appear in many applications where nonlinear procedures are necessary. This paper examines two basic issues involving the linearization of Lipschitz framings. We first prove that every Lipschitz framing induces a linear framing which shares the same synthesis operator, and consequently every Banach space admitting a Lipschitz framing has the bounded approximation property. Secondly, we examine the projection-valued dilations of Lipschitz operator-valued measures on Banach spaces. We prove that every Lipschitz operator-valued measure can induce an operator-valued measure by linearization, and every <span><math><mrow><mi>Lip</mi><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span>-valued measure has a projection-valued measure dilation by establishing a nonlinear version of minimal dilation theory. As examples, we discuss a concrete construction of the minimal dilation for the special case when the measure space is <span><math><mrow><mo>(</mo><mi>N</mi><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></math></span>, and how nonlinear sampling naturally induces a Lipschitz framing.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125680"},"PeriodicalIF":0.8,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768160","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":"Divisibility of orders of reductions of elliptic curves","authors":"Antigona Pajaziti ,&nbsp;Mohammad Sadek","doi":"10.1016/j.exmath.2025.125679","DOIUrl":"10.1016/j.exmath.2025.125679","url":null,"abstract":"<div><div>Let <span><math><mi>E</mi></math></span> be an elliptic curve defined over <span><math><mi>Q</mi></math></span> and <span><math><msub><mrow><mover><mrow><mi>E</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub></math></span> denote the reduction of <span><math><mi>E</mi></math></span> modulo a prime <span><math><mi>p</mi></math></span> of good reduction for <span><math><mi>E</mi></math></span>. The divisibility of <span><math><mrow><mo>|</mo><msub><mrow><mover><mrow><mi>E</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mo>|</mo></mrow></math></span> by an integer <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span> for a set of primes <span><math><mi>p</mi></math></span> of density 1 is determined by the torsion subgroups of elliptic curves that are <span><math><mi>Q</mi></math></span>-isogenous to <span><math><mi>E</mi></math></span>. In this work, we give explicit families of elliptic curves <span><math><mi>E</mi></math></span> over <span><math><mi>Q</mi></math></span> together with integers <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>E</mi></mrow></msub></math></span> such that the congruence class of <span><math><mrow><mo>|</mo><msub><mrow><mover><mrow><mi>E</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mo>|</mo></mrow></math></span> modulo <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>E</mi></mrow></msub></math></span> can be computed explicitly. In addition, we can estimate the density of primes <span><math><mi>p</mi></math></span> for which each congruence class occurs. These include elliptic curves over <span><math><mi>Q</mi></math></span> whose torsion grows over a quadratic field <span><math><mi>K</mi></math></span> where <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>E</mi></mrow></msub></math></span> is determined by the <span><math><mi>K</mi></math></span>-torsion subgroups in the <span><math><mi>Q</mi></math></span>-isogeny class of <span><math><mi>E</mi></math></span>. We also exhibit elliptic curves over <span><math><mrow><mi>Q</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than 1 are divisible by given small integers.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125679"},"PeriodicalIF":0.8,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143747610","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":"Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms","authors":"Paweł Goldstein ,&nbsp;Zofia Grochulska ,&nbsp;Piotr Hajłasz","doi":"10.1016/j.exmath.2025.125681","DOIUrl":"10.1016/j.exmath.2025.125681","url":null,"abstract":"<div><div>Cerf and Palais independently proved a remarkable result about extending diffeomorphisms defined on smooth balls in a manifold to global diffeomorphisms of the manifold onto itself. We explain Palais’ argument and show how to extend it to the class of homeomorphisms and bi-Lipschitz homeomorphisms. While Palais’ argument is surprising, it is elementary and short. However, its extension to bi-Lipschitz homeomorphisms and homeomorphisms requires deep results: the stable homeomorphism and the annulus theorems.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125681"},"PeriodicalIF":0.8,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777148","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":"An approach to annihilators in the context of vector field Lie algebras","authors":"Charles H. Conley ,&nbsp;William Goode","doi":"10.1016/j.exmath.2024.125600","DOIUrl":"10.1016/j.exmath.2024.125600","url":null,"abstract":"<div><div>We present a general method for describing the annihilators of modules of Lie algebras under certain conditions, which hold for some tensor modules of vector field Lie algebras. As an example, we apply the method to obtain an efficient proof of previously known results on the annihilators of the bounded irreducible modules of <span><math><mrow><mi>Vec</mi><mspace></mspace><mi>R</mi></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125600"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192927","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":"Exceptional Periodicity and Magic Star algebras","authors":"Piero Truini ,&nbsp;Alessio Marrani ,&nbsp;Michael Rios ,&nbsp;Willem de Graaf","doi":"10.1016/j.exmath.2024.125621","DOIUrl":"10.1016/j.exmath.2024.125621","url":null,"abstract":"<div><div>We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call <span><math><mi>H</mi></math></span> algebras) on each vertex of such a projection. We then focus on the Magic Star algebra <span><math><msubsup><mrow><mi>f</mi></mrow><mrow><mn>4</mn></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msubsup></math></span>, which generalizes the non-simply laced exceptional Lie algebra <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the <span><math><mi>H</mi></math></span> algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125621"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621008","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":"Characterising the Haar measure on the p-adic rotation groups via inverse limits of measure spaces","authors":"Paolo Aniello ,&nbsp;Sonia L’Innocente ,&nbsp;Stefano Mancini ,&nbsp;Vincenzo Parisi ,&nbsp;Ilaria Svampa ,&nbsp;Andreas Winter","doi":"10.1016/j.exmath.2024.125592","DOIUrl":"10.1016/j.exmath.2024.125592","url":null,"abstract":"<div><div>We determine the Haar measure on the compact <span><math><mi>p</mi></math></span>-adic special orthogonal groups of rotations <span><math><mrow><mi>SO</mi><msub><mrow><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> in dimension <span><math><mrow><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math></span>, by exploiting the machinery of inverse limits of measure spaces, for every prime <span><math><mrow><mi>p</mi><mo>&gt;</mo><mn>2</mn></mrow></math></span>. We characterise the groups <span><math><mrow><mi>SO</mi><msub><mrow><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each <span><math><mrow><mi>SO</mi><msub><mrow><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span>. Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on <span><math><mrow><mi>SO</mi><msub><mrow><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span>. Our results pave the way towards the study of the irreducible projective unitary representations of the <span><math><mi>p</mi></math></span>-adic rotation groups, with potential applications to the recently proposed <span><math><mi>p</mi></math></span>-adic quantum information theory.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125592"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141880872","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":"Supergravity in the geometric approach and its hidden graded Lie algebra","authors":"L. Andrianopoli ,&nbsp;R. D’Auria","doi":"10.1016/j.exmath.2024.125631","DOIUrl":"10.1016/j.exmath.2024.125631","url":null,"abstract":"<div><div>In this contribution, we present the geometric approach to supergravity. In the first part, we discuss in some detail the peculiarities of the approach and apply the formalism to the case of pure supergravity in four space-time dimensions. In the second part, we extend the discussion to theories in higher dimensions, which include antisymmetric tensors of degree higher than one, focussing on the case of eleven dimensional space–time. Here, we report the formulation first introduced by R. D’Auria and P. Fré in 1981, corresponding to a generalization of a Chevalley–Eilenberg Lie algebra, together with some more recent results, pointing out the relation of the formalism with the mathematical framework of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> algebras.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125631"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621013","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":"V.S. Varadarajan (1937–2019): In memoriam","authors":"Rita Fioresi","doi":"10.1016/j.exmath.2025.125661","DOIUrl":"10.1016/j.exmath.2025.125661","url":null,"abstract":"<div><div>This article is a personal recollection of some aspects of the life and mathematics of Professor V.S. Varadarajan, who passed away on April 25, 2019, in Santa Monica, California, USA.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125661"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621012","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 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}

{"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}