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Generalizing Hurwitz’s quaternionic proof of Lagrange’s and Jacobi’s four-square theorems 推广赫维茨对拉格朗日定理和雅可比定理的四元数证明
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2025-12-01 Epub Date: 2025-07-19 DOI: 10.1016/j.exmath.2025.125715
Matěj Doležálek
{"title":"Generalizing Hurwitz’s quaternionic proof of Lagrange’s and Jacobi’s four-square theorems","authors":"Matěj Doležálek","doi":"10.1016/j.exmath.2025.125715","DOIUrl":"10.1016/j.exmath.2025.125715","url":null,"abstract":"<div><div>A proof of Lagrange’s and Jacobi’s four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orders over number fields, we identify two key components: an order with a good factorization theory and the condition that all orbits under the action of the group of elements of norm 1 acting by multiplication intersect the suborder corresponding to the quadratic form to be studied. We use recent results on class numbers of quaternion orders and then find all suborders satisfying the orbit condition. Subsequently, we obtain universality and formulas for the number of representations by the corresponding quadratic forms. We also present a quaternionic proof of Götzky’s four-square theorem.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125715"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144711948","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
Bessel duality of Gabor systems: A von Neumann algebraic perspective Gabor系统的贝塞尔对偶性：一个冯诺依曼代数的观点
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2025-12-01 Epub Date: 2025-11-20 DOI: 10.1016/j.exmath.2025.125741
Ulrik Enstad , Franz Luef
{"title":"Bessel duality of Gabor systems: A von Neumann algebraic perspective","authors":"Ulrik Enstad ,&nbsp;Franz Luef","doi":"10.1016/j.exmath.2025.125741","DOIUrl":"10.1016/j.exmath.2025.125741","url":null,"abstract":"<div><div>Bessel duality of regular Gabor systems states that a Gabor system over a lattice is a Bessel sequence if and only if the corresponding Gabor system over the adjoint lattice is a Bessel sequence. We show that this fundamental result of time–frequency analysis can be deduced from a theorem in the theory of bimodules over von Neumann algebras, namely that under certain conditions, the left and right bounded vectors of such bimodules coincide.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125741"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623107","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
Commutativity of operator algebras 算子代数的交换性
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2025-12-01 Epub Date: 2025-11-20 DOI: 10.1016/j.exmath.2025.125742
David P. Blecher
{"title":"Commutativity of operator algebras","authors":"David P. Blecher","doi":"10.1016/j.exmath.2025.125742","DOIUrl":"10.1016/j.exmath.2025.125742","url":null,"abstract":"<div><div>We call an operator algebra <span><math><mi>A</mi></math></span> <em>reversible</em> if <span><math><mi>A</mi></math></span> with reversed multiplication is also an abstract operator algebra (in the modern operator space sense). This class of operator algebras is intimately related to the <em>symmetric operator algebras</em>: the subalgebras of <span><math><mrow><mi>B</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> on which the transpose map is a complete isometry. In previous work we studied the unital case, where reversibility is equivalent to commutativity. We give many sufficient conditions under which a nonunital reversible or symmetric operator algebra is commutative. We also give many complementary results of independent interest, and solve a few open questions from previous papers. Not every reversible or symmetric operator algebra is commutative, however we show that they all are 3-commutative. That is, order does not matter in the product of three or more elements from <span><math><mi>A</mi></math></span>. The proof of this relies on a technical analysis involving the injective envelope. Indeed nonunital algebras are often enormously more complicated than unital ones in regard to the topics we consider. On the positive side, our considerations raise very many questions even for low dimensional matrix algebras, some of which are of a computational nature and might be suitable for undergraduate research. The canonical anticommutation relations from mathematical physics play a significant role.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125742"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623228","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
Fractional wavelet phase retrieval of bandlimited functions 带限函数的分数小波相位检索
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2025-12-01 Epub Date: 2025-09-30 DOI: 10.1016/j.exmath.2025.125732
Bei Liu, Meng Ning, Rui Li, Liping Wu, Qingyue Zhang
{"title":"Fractional wavelet phase retrieval of bandlimited functions","authors":"Bei Liu,&nbsp;Meng Ning,&nbsp;Rui Li,&nbsp;Liping Wu,&nbsp;Qingyue Zhang","doi":"10.1016/j.exmath.2025.125732","DOIUrl":"10.1016/j.exmath.2025.125732","url":null,"abstract":"<div><div>In this paper, we demonstrate the real-valued bandlimited signals can be recovered (up to global sign) from the magnitude of their fractional wavelet transforms, also called fractional wavelet phase retrieval. We present a new uniqueness result for fractional wavelet phase retrieval. To be precise, we show that any real-valued bandlimited functions can be recovered uniquely (up to global sign) by fractional wavelets which have finitely many vanishing moments. Furthermore, we show that the real-valued bandlimited function can be uniquely recovered up to global sign from the absolute values of the discrete fractional wavelet transform for fixed <span><math><mrow><mi>a</mi><mo>&gt;</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125732"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145262723","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
Stability bounds for correcting from erased and irregular samples 从擦除和不规则样本校正的稳定性界限
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2025-12-01 Epub Date: 2025-11-26 DOI: 10.1016/j.exmath.2025.125744
Duncan Koepke , Huston Wilhite , Rebecca Yoshino , Sam Scholze
{"title":"Stability bounds for correcting from erased and irregular samples","authors":"Duncan Koepke ,&nbsp;Huston Wilhite ,&nbsp;Rebecca Yoshino ,&nbsp;Sam Scholze","doi":"10.1016/j.exmath.2025.125744","DOIUrl":"10.1016/j.exmath.2025.125744","url":null,"abstract":"<div><div>The Shannon–Whittaker Sampling Theorem allows us to reconstruct frequency band-limited signals from uniformly spaced samples in the time domain. The Reduced Direct Inversion algorithm allows us to recover from erased sampled values, as well as certain types of irregular samples. When truncating series, we introduce error into the reconstruction that can be amplified by a reconstruction algorithm if the algorithm is unstable. Utilizing alternate summation kernels with faster decay can provide greater stability than the standard sinc summation kernel. In this paper, we provide stability bounds for these alternate summation kernels for the cases of erasures as well as irregular sampling. Numerical experiments are also provided to demonstrate the stability of the Reduced Direct Inversion algorithm.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125744"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693325","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
Preface to the Special Issue in honour of David RoyalLarson 纪念大卫·罗亚尔·亚拉森的特刊前言
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2025-12-01 Epub Date: 2025-11-04 DOI: 10.1016/j.exmath.2025.125735
Alan Donsig (Guest Editors), Deguang Han, Palle Jorgensen, Keri Kornelson, Rui Liu
{"title":"Preface to the Special Issue in honour of David RoyalLarson","authors":"Alan Donsig (Guest Editors),&nbsp;Deguang Han,&nbsp;Palle Jorgensen,&nbsp;Keri Kornelson,&nbsp;Rui Liu","doi":"10.1016/j.exmath.2025.125735","DOIUrl":"10.1016/j.exmath.2025.125735","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125735"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528226","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
Dynamical sampling for unmanned aerial vehicles 无人机动态采样
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2025-12-01 Epub Date: 2025-11-20 DOI: 10.1016/j.exmath.2025.125739
Cristian Cruz Hernandez , Juan Ignacio Giribet , Ursula Molter
{"title":"Dynamical sampling for unmanned aerial vehicles","authors":"Cristian Cruz Hernandez ,&nbsp;Juan Ignacio Giribet ,&nbsp;Ursula Molter","doi":"10.1016/j.exmath.2025.125739","DOIUrl":"10.1016/j.exmath.2025.125739","url":null,"abstract":"<div><div>We revisit the notion of <em>positive bases</em> proposed by Chandler Davis, and connect it with the operator-driven constructions that appear in <em>dynamical sampling</em>. Working in finite-dimensional real inner-product spaces, we derive verifiable criteria ensuring that families of the form <span><span><span><math><mrow><mo>{</mo><mspace></mspace><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>A</mi><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>;</mo><mspace></mspace><mo>…</mo><mo>;</mo><mspace></mspace><msub><mrow><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>A</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msup><msub><mrow><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msub><mspace></mspace><mo>}</mo></mrow></math></span></span></span>are positive generators or positive bases. Under spectral hypotheses on <span><math><mi>A</mi></math></span> we give necessary and sufficient conditions, including minimality criteria. We also show that the canonical dual of a positive frame is positive under our framework. Beyond their intrinsic interest, positive generating families model some actuation-constrained systems where inputs are required to be nonnegative (e.g., unmanned aerial vehicles, thruster layouts or multirotor design), and iterates of a system operator produce rich families from a few seeds. Our results offer a unified mathematical framework that explains when the orbits of a system operator give rise to positive generation, and provide explicit spectral conditions for this to occur.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125739"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623227","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
Exponential convergence of a distributed divide-and-conquer algorithm for constrained convex optimization on networks 网络约束凸优化的分布式分治算法的指数收敛性
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2025-12-01 Epub Date: 2025-11-20 DOI: 10.1016/j.exmath.2025.125740
Nazar Emirov , Guohui Song , Qiyu Sun
{"title":"Exponential convergence of a distributed divide-and-conquer algorithm for constrained convex optimization on networks","authors":"Nazar Emirov ,&nbsp;Guohui Song ,&nbsp;Qiyu Sun","doi":"10.1016/j.exmath.2025.125740","DOIUrl":"10.1016/j.exmath.2025.125740","url":null,"abstract":"<div><div>We propose a divide-and-conquer (DAC) algorithm for constrained convex optimization over networks, where the global objective is the sum of local objectives attached to individual agents. The algorithm is fully distributed: each iteration solves local subproblems around selected fusion centers and coordinates only with neighboring fusion centers. Under standard assumptions of smoothness, strong convexity, and locality on the objective function, together with polynomial growth conditions on the underlying graph, we establish exponential convergence of the DAC iterations and derive explicit bounds for both exact and inexact local solvers. Numerical experiments on three representative losses (<span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> distance, quadratic, and entropy) confirm the theory and demonstrate scalability and effectiveness.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 6","pages":"Article 125740"},"PeriodicalIF":0.9,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623106","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 comprehensive approach to multifractal analysis 多重分形分析的综合方法
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2025-09-01 Epub Date: 2025-04-11 DOI: 10.1016/j.exmath.2025.125690
Zhiming Li , Bilel Selmi , Haythem Zyoudi
{"title":"A comprehensive approach to multifractal analysis","authors":"Zhiming Li ,&nbsp;Bilel Selmi ,&nbsp;Haythem Zyoudi","doi":"10.1016/j.exmath.2025.125690","DOIUrl":"10.1016/j.exmath.2025.125690","url":null,"abstract":"<div><div>This paper investigates how specific techniques have been broadened to suit more general contexts. Among them, multifractal analysis stands out for its adaptability and depth, presenting a unified framework to approach these generalizations. We examine the relative multifractal formalism within the context of metric spaces in a general way. The primary aim is to introduce a generalized concept of general relative multifractal Hausdorff and packing measures. In particular, we delve into the characteristics of the generalized multifractal Hausdorff and packing measures and analyze their impact on the broader multifractal spectrum functions. The investigation explores the connection between these generalized multifractal measures and the nature of general multifractal dimensions within this framework. Further, we establish an equivalence relation between general relative multifractal Hausdorff and packing measures by utilizing density theorems. Moreover, we study various properties of the generalized relative multifractal Hausdorff measures, packing measures, and pre-measures. Lastly, our work addresses the question of whether a subset in Euclidean space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with infinite positive Hausdorff measure can contain a compact set of positive finite general relative Hausdorff measures.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125690"},"PeriodicalIF":0.8,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143868741","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
Ends and end cohomology 端和端上同调
IF 0.8 4区 数学
Expositiones Mathematicae Pub Date : 2025-09-01 Epub Date: 2025-04-21 DOI: 10.1016/j.exmath.2025.125692
William G. Bass, Jack S. Calcut
{"title":"Ends and end cohomology","authors":"William G. Bass,&nbsp;Jack S. Calcut","doi":"10.1016/j.exmath.2025.125692","DOIUrl":"10.1016/j.exmath.2025.125692","url":null,"abstract":"<div><div>Ends and end cohomology are powerful invariants for the study of noncompact spaces. We present a self-contained exposition of the topological theory of ends and prove novel extensions including the existence of an exhaustion of a proper map. We define reduced end cohomology as the relative end cohomology of a ray-based space. We use those results to prove a version of a theorem of King that computes the reduced end cohomology of an end sum of two manifolds. We include a complete proof of Freudenthal’s fundamental theorem on the number of ends of a topological group, and we use our results on dimension-zero end cohomology to prove—without using transfinite induction—a theorem of Nöbeling on freeness of certain modules of continuous functions.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 5","pages":"Article 125692"},"PeriodicalIF":0.8,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143906828","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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