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A simple proof of the Wiener–Ikehara Tauberian Theorem 维纳-池原陶伯定理的简单证明
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2024-03-18 DOI: 10.1016/j.exmath.2024.125570
M. Ram Murty , Jagannath Sahoo , Akshaa Vatwani
{"title":"A simple proof of the Wiener–Ikehara Tauberian Theorem","authors":"M. Ram Murty ,&nbsp;Jagannath Sahoo ,&nbsp;Akshaa Vatwani","doi":"10.1016/j.exmath.2024.125570","DOIUrl":"10.1016/j.exmath.2024.125570","url":null,"abstract":"<div><p>The Wiener–Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></msubsup><mfrac><mrow><mi>a</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></mfrac></mrow></math></span>. We provide a simple and elegant proof of the Wiener–Ikehara Tauberian theorem which relies only on basic Fourier analysis and known estimates for the given Dirichlet series. This method also allows us to derive a version of the Wiener–Ikehara theorem with an error term.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 3","pages":"Article 125570"},"PeriodicalIF":0.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140181844","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
Obituary: Bent Fuglede (1925–2023) 讣告本特-福格勒德（1925-2023）
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2024-03-01 DOI: 10.1016/j.exmath.2023.125531
Liming Ge (Managing Editor)
{"title":"Obituary: Bent Fuglede (1925–2023)","authors":"Liming Ge (Managing Editor)","doi":"10.1016/j.exmath.2023.125531","DOIUrl":"10.1016/j.exmath.2023.125531","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 2","pages":"Article 125531"},"PeriodicalIF":0.7,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S072308692300107X/pdfft?md5=aa8f8838bfe5d66a791a9333d3459373&pid=1-s2.0-S072308692300107X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139409818","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}
引用次数: 0
Geometric criteria for the existence of capillary surfaces in tubes 管道中存在毛细管表面的几何标准
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2024-02-28 DOI: 10.1016/j.exmath.2024.125547
Giorgio Saracco
{"title":"Geometric criteria for the existence of capillary surfaces in tubes","authors":"Giorgio Saracco","doi":"10.1016/j.exmath.2024.125547","DOIUrl":"https://doi.org/10.1016/j.exmath.2024.125547","url":null,"abstract":"<div><p>We review some geometric criteria and prove a refined version, that yield existence of capillary surfaces in tubes <span><math><mrow><mi>Ω</mi><mo>×</mo><mi>R</mi></mrow></math></span> in a gravity free environment, in the case of physical interest, that is, for bounded, open, and simply connected <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>. These criteria rely on suitable weak one-sided bounds on the curvature of the boundary of the cross-section <span><math><mi>Ω</mi></math></span>.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 3","pages":"Article 125547"},"PeriodicalIF":0.7,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0723086924000148/pdfft?md5=8d050c0fb8ec6bbd1f9d3483b6085e70&pid=1-s2.0-S0723086924000148-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140031119","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}
引用次数: 0
Three families of matrices 三个矩阵系列
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2024-02-22 DOI: 10.1016/j.exmath.2024.125546
Alexander Pushnitski
{"title":"Three families of matrices","authors":"Alexander Pushnitski","doi":"10.1016/j.exmath.2024.125546","DOIUrl":"10.1016/j.exmath.2024.125546","url":null,"abstract":"<div><p>This paper has an expository nature. We compare the spectral properties (such as boundedness and compactness) of three families of semi-infinite matrices and point out similarities between them. The common feature of these families is that they can be understood as matrices of some linear operations on appropriate Hardy spaces.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 2","pages":"Article 125546"},"PeriodicalIF":0.7,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946462","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
Reverse engineered Diophantine equations 逆向工程 Diophantine 方程
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2024-02-09 DOI: 10.1016/j.exmath.2024.125545
Stevan Gajović
{"title":"Reverse engineered Diophantine equations","authors":"Stevan Gajović","doi":"10.1016/j.exmath.2024.125545","DOIUrl":"10.1016/j.exmath.2024.125545","url":null,"abstract":"<div><p>We answer a question of Samir Siksek, asked at the open problems session of the conference “Rational Points 2022”, which, in a broader sense, can be viewed as a reverse engineering of Diophantine equations. For any finite set <span><math><mi>S</mi></math></span> of perfect integer powers, using Mihăilescu’s theorem, we construct a polynomial <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>∈</mo><mi>Z</mi><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow></mrow></math></span> such that the set <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>S</mi></mrow></msub><mrow><mo>(</mo><mi>Z</mi><mo>)</mo></mrow></mrow></math></span> contains a perfect integer power if and only if it belongs to <span><math><mi>S</mi></math></span>. We first discuss the easier case where we restrict to all powers with the same exponent. In this case, the constructed polynomials are inspired by Runge’s method and Fermat’s Last Theorem. Therefore we can construct a polynomial–exponential Diophantine equation whose solutions are determined in advance.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 2","pages":"Article 125545"},"PeriodicalIF":0.7,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139924066","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
The three harmonic homologies theorem 三次谐波同调定理
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2024-02-02 DOI: 10.1016/j.exmath.2024.125544
Fahimeh Heidari, Bijan Honari
{"title":"The three harmonic homologies theorem","authors":"Fahimeh Heidari,&nbsp;Bijan Honari","doi":"10.1016/j.exmath.2024.125544","DOIUrl":"10.1016/j.exmath.2024.125544","url":null,"abstract":"<div><p>In this paper, we give a complete answer to the question: “Under what conditions the product of three harmonic homologies of the real projective space <span><math><mrow><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> is a harmonic homology again?” Among other things, we prove the three harmonic homologies theorem in <span><math><mrow><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> by which the product of three harmonic homologies with collinear centers is again a harmonic homology if and only if the hyperplanes are polars of the centers with respect to a quadric. It is shown that the three reflections theorem, the three inversions theorem, notably Pascal’s theorem and Miquel’s theorem in Laguerre geometry are special cases of this theorem.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 2","pages":"Article 125544"},"PeriodicalIF":0.7,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139664268","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
On the Hopf problem and a conjecture of Liu–Maxim–Wang 关于霍普夫问题和刘-马西姆-王的一个猜想
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2024-01-17 DOI: 10.1016/j.exmath.2024.125543
Luca F. Di Cerbo , Rita Pardini
{"title":"On the Hopf problem and a conjecture of Liu–Maxim–Wang","authors":"Luca F. Di Cerbo ,&nbsp;Rita Pardini","doi":"10.1016/j.exmath.2024.125543","DOIUrl":"10.1016/j.exmath.2024.125543","url":null,"abstract":"<div><p>We discuss an approach towards the Hopf problem for aspherical smooth projective varieties recently proposed by Liu et al. (2021). In complex dimension two, we point out that this circle of ideas suggests an intriguing conjecture regarding the geography of aspherical surfaces of general type.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 2","pages":"Article 125543"},"PeriodicalIF":0.7,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139510373","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
Generalised solutions to linear and non-linear Schrödinger-type equations with point defect: Colombeau and non-Colombeau regimes 具有点缺陷的线性和非线性薛定谔型方程的广义解：科伦坡和非科伦坡状态
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-12-30 DOI: 10.1016/j.exmath.2023.125533
Nevena Dugandžija , Alessandro Michelangeli , Ivana Vojnović
{"title":"Generalised solutions to linear and non-linear Schrödinger-type equations with point defect: Colombeau and non-Colombeau regimes","authors":"Nevena Dugandžija ,&nbsp;Alessandro Michelangeli ,&nbsp;Ivana Vojnović","doi":"10.1016/j.exmath.2023.125533","DOIUrl":"10.1016/j.exmath.2023.125533","url":null,"abstract":"<div><p><span>For a semi-linear Schrödinger equation of Hartree type in three spatial dimensions, various approximations of singular, point-like perturbations are considered, in the form of potentials of very small range and very large magnitude, obeying different </span>scaling limits<span>. The corresponding nets of approximate solutions represent actual generalised solutions for the singular-perturbed Schrödinger equation. The behaviour of such nets is investigated, comparing the distinct scaling regimes that yield, respectively, the Hartree equation with point interaction Hamiltonian vs the ordinary Hartree equation with the free Laplacian. In the second case, the distinguished regime admitting a generalised solution in the Colombeau algebra is studied, and for such a solution compatibility with the classical Hartree equation is established, in the sense of the Colombeau generalised solution theory.</span></p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 2","pages":"Article 125533"},"PeriodicalIF":0.7,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139070033","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 new theory of atomic Hp spaces with applications to smoothness of functions 原子 Hp 空间的新理论及其在函数平滑性中的应用
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-12-23 DOI: 10.1016/j.exmath.2023.125532
Steven G. Krantz
{"title":"A new theory of atomic Hp spaces with applications to smoothness of functions","authors":"Steven G. Krantz","doi":"10.1016/j.exmath.2023.125532","DOIUrl":"10.1016/j.exmath.2023.125532","url":null,"abstract":"<div><p>We provide a new definition of Hardy space atoms that avoids use of coordinates to formulate the moment condition. Thus the new theory can be used in abstract settings such as spaces of homogeneous type. We give applications of this theory to the definition of and study of smooth functions on spaces of homogeneous type.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 2","pages":"Article 125532"},"PeriodicalIF":0.7,"publicationDate":"2023-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139029324","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
The best constant in a Hilbert-type inequality 希尔伯特不等式中的最佳常数
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-12-02 DOI: 10.1016/j.exmath.2023.125530
Ole Fredrik Brevig
{"title":"The best constant in a Hilbert-type inequality","authors":"Ole Fredrik Brevig","doi":"10.1016/j.exmath.2023.125530","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.125530","url":null,"abstract":"<div><p>We establish that <span><span><span><math><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><munderover><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><mover><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mo>¯</mo></mover><mfrac><mrow><mi>m</mi><mi>n</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mo>max</mo><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mo>≤</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac><munderover><mrow><mo>∑</mo></mrow><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msup><mrow><mrow><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span></span></span>holds for every square-summable sequence of complex numbers <span><math><mrow><mi>a</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>)</mo></mrow></mrow></math></span> and that the constant <span><math><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></math></span> cannot be replaced by any smaller number. Our proof is rooted in a seminal 1911 paper concerning bilinear forms due to Schur, and we include for expositional reasons an elaboration on his approach.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"42 1","pages":"Article 125530"},"PeriodicalIF":0.7,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0723086923001068/pdfft?md5=e51710329d2cc0195e76c5e071736e56&pid=1-s2.0-S0723086923001068-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138501641","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}
引用次数: 1
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