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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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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
The generators of the K-groups of the sphere 球k群的产生器
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-11-10 DOI: 10.1016/j.exmath.2023.125519
Hermann Schulz-Baldes , Tom Stoiber
{"title":"The generators of the K-groups of the sphere","authors":"Hermann Schulz-Baldes ,&nbsp;Tom Stoiber","doi":"10.1016/j.exmath.2023.125519","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.125519","url":null,"abstract":"<div><p><span>This note presents an elementary iterative construction of the generators for the complex </span><span><math><mi>K</mi></math></span>-groups <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mrow><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> of the <span><math><mi>d</mi></math></span><span>-dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to the unit sphere. Connections to solid state physics are briefly elaborated on.</span></p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134655636","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
Unmarked trace spectrum rigidity on strictly convex real projective surfaces 严格凸实射影表面上的无标记迹谱刚性
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-11-10 DOI: 10.1016/j.exmath.2023.125520
Inkang Kim
{"title":"Unmarked trace spectrum rigidity on strictly convex real projective surfaces","authors":"Inkang Kim","doi":"10.1016/j.exmath.2023.125520","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.125520","url":null,"abstract":"<div><p>We prove that for a given unmarked trace spectrum with multiplicity, there are only a finite number of convex real projective surfaces with that spectrum up to remarking.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134832572","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
Homotopy groups of cubical sets 三次集的同伦群
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2023-10-07 DOI: 10.1016/j.exmath.2023.125518
Daniel Carranza , Krzysztof Kapulkin
{"title":"Homotopy groups of cubical sets","authors":"Daniel Carranza ,&nbsp;Krzysztof Kapulkin","doi":"10.1016/j.exmath.2023.125518","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.125518","url":null,"abstract":"<div><p><span><span>We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the </span>geometric realization </span>functor<span><span>. We also provide purely combinatorial proofs of several classical theorems, including: product preservation, commutativity of higher homotopy groups, the long exact sequence of a </span>fibration, and Whitehead’s theorem.</span></p><p>This is a companion paper to our “Cubical setting for discrete homotopy theory, revisited” in which we apply these results to study the homotopy theory of simple graphs.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49863757","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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