Pub Date : 2024-05-13DOI: 10.1016/j.exmath.2024.125574
Fan Wang , Dachun Yang , Wen Yuan
Assume that is a space of homogeneous type introduced by R. R. Coifman and G. Weiss. In this article, the authors establish a geometric characterization of Ahlfors regular spaces via the dyadic cubes constructed by T. Hytönen and A. Kairema. As applications, the authors show that Lipschitz spaces defined via the quasi-metric under consideration and Lipschitz spaces defined via the measure under consideration coincide with equivalent norms if and only if is an Ahlfors regular space. Moreover, the authors also prove that Lipschitz spaces defined via the quasi-metric under consideration and Campanato spaces defined via balls coincide with equivalent norms if and only if is an Ahlfors regular space.
假设 (X,d,μ) 是由 R. R. Coifman 和 G. Weiss 引入的均质型空间。在本文中，作者通过海托宁（T. Hytönen ）和凯尔玛（A. Kairema）构建的二元立方体建立了阿赫弗斯正则空间的几何特征。作为应用，作者证明了当且仅当 X 是一个 Ahlfors 正则空间时，通过所考虑的准度量定义的 Lipschitz 空间和通过所考虑的度量定义的 Lipschitz 空间以等效规范重合。此外，作者还证明，如果且仅如果 X 是一个 Ahlfors 正则空间，通过所考虑的准度量定义的 Lipschitz 空间和通过球定义的 Campanato 空间与等效规范重合。
{"title":"Geometric characterization of Ahlfors regular spaces in terms of dyadic cubes related to wavelets with its applications to equivalences of Lipschitz spaces","authors":"Fan Wang , Dachun Yang , Wen Yuan","doi":"10.1016/j.exmath.2024.125574","DOIUrl":"10.1016/j.exmath.2024.125574","url":null,"abstract":"<div><p>Assume that <span><math><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></math></span> is a space of homogeneous type introduced by R. R. Coifman and G. Weiss. In this article, the authors establish a geometric characterization of Ahlfors regular spaces via the dyadic cubes constructed by T. Hytönen and A. Kairema. As applications, the authors show that Lipschitz spaces defined via the quasi-metric under consideration and Lipschitz spaces defined via the measure under consideration coincide with equivalent norms if and only if <span><math><mi>X</mi></math></span> is an Ahlfors regular space. Moreover, the authors also prove that Lipschitz spaces defined via the quasi-metric under consideration and Campanato spaces defined via balls coincide with equivalent norms if and only if <span><math><mi>X</mi></math></span> is an Ahlfors regular space.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141040800","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}
Pub Date : 2024-05-01DOI: 10.1016/j.exmath.2024.125583
P. Jorgensen, Myung-Sin Song, James Tian
{"title":"Kernel-algorithms in frame-approximations","authors":"P. Jorgensen, Myung-Sin Song, James Tian","doi":"10.1016/j.exmath.2024.125583","DOIUrl":"https://doi.org/10.1016/j.exmath.2024.125583","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141134319","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}
Pub Date : 2024-05-01DOI: 10.1016/j.exmath.2024.125573
Trond Digernes
{"title":"A short review of finite approximations and unconventional physics","authors":"Trond Digernes","doi":"10.1016/j.exmath.2024.125573","DOIUrl":"https://doi.org/10.1016/j.exmath.2024.125573","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-01","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-04DOI: 10.1016/j.exmath.2024.125571
Hanka Řada , Štěpán Starosta , Vítězslav Kala
We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend and use it to prove that a number of vectors have an eventually periodic expansion in the Algebraic Jacobi–Perron algorithm. Further, we give criteria for vectors to have purely periodic expansions; in particular, the vector cannot be totally positive.
{"title":"Periodicity of general multidimensional continued fractions using repetend matrix form","authors":"Hanka Řada , Štěpán Starosta , Vítězslav Kala","doi":"10.1016/j.exmath.2024.125571","DOIUrl":"https://doi.org/10.1016/j.exmath.2024.125571","url":null,"abstract":"<div><p>We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend and use it to prove that a number of vectors have an eventually periodic expansion in the Algebraic Jacobi–Perron algorithm. Further, we give criteria for vectors to have purely periodic expansions; in particular, the vector cannot be totally positive.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140554317","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}
Pub Date : 2024-04-01DOI: 10.1016/j.exmath.2024.125572
Glenn D. Appleby
{"title":"A variation on von Neumann Entropy and a result of Varadarajan","authors":"Glenn D. Appleby","doi":"10.1016/j.exmath.2024.125572","DOIUrl":"https://doi.org/10.1016/j.exmath.2024.125572","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-01","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}
Pub Date : 2024-03-20DOI: 10.1016/j.exmath.2024.125548
J. Blackman , S. Kristensen , M.J. Northey
In this paper, we investigate the base- expansions of putative counterexamples to the -adic Littlewood conjecture of de Mathan and Teulié. We show that if a counterexample exists, then so does a counterexample whose base- expansion is uniformly recurrent. Furthermore, we show that if the base- expansion of is a morphic word where contains a subword of the form with , then satisfies the -adic Littlewood conjecture. In the special case when , we show that the conjecture holds for all pure morphic words.
在本文中，我们研究了 de Mathan 和 Teulié 的 p-adic Littlewood 猜想的推定反例的基 p 展开。我们证明，如果一个反例存在，那么一个其基p展开是均匀递归的反例也存在。此外，我们还证明，如果 x 的基 p 扩展是一个形态词 τ(φω(a))，其中 φω(a) 包含一个形式为 uXuXu 的子词，且 limn→∞|φn(u)|=∞, 那么 x 满足 p-adic Littlewood 猜想。在 p=2 的特殊情况下，我们证明该猜想对所有纯形声字都成立。
{"title":"A note on the base-p expansions of putative counterexamples to the p-adic Littlewood conjecture","authors":"J. Blackman , S. Kristensen , M.J. Northey","doi":"10.1016/j.exmath.2024.125548","DOIUrl":"10.1016/j.exmath.2024.125548","url":null,"abstract":"<div><p>In this paper, we investigate the base-<span><math><mi>p</mi></math></span> expansions of putative counterexamples to the <span><math><mi>p</mi></math></span>-adic Littlewood conjecture of de Mathan and Teulié. We show that if a counterexample exists, then so does a counterexample whose base-<span><math><mi>p</mi></math></span> expansion is uniformly recurrent. Furthermore, we show that if the base-<span><math><mi>p</mi></math></span> expansion of <span><math><mi>x</mi></math></span> is a morphic word <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><msup><mrow><mi>φ</mi></mrow><mrow><mi>ω</mi></mrow></msup><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> where <span><math><mrow><msup><mrow><mi>φ</mi></mrow><mrow><mi>ω</mi></mrow></msup><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow></math></span> contains a subword of the form <span><math><mrow><mi>u</mi><mi>X</mi><mi>u</mi><mi>X</mi><mi>u</mi></mrow></math></span> with <span><math><mrow><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><mrow><mo>|</mo><msup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>=</mo><mi>∞</mi></mrow></math></span>, then <span><math><mi>x</mi></math></span> satisfies the <span><math><mi>p</mi></math></span>-adic Littlewood conjecture. In the special case when <span><math><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></math></span>, we show that the conjecture holds for all pure morphic words.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S072308692400015X/pdfft?md5=9882f79608644e821115bc0ed83923d6&pid=1-s2.0-S072308692400015X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140281675","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}
Pub Date : 2024-03-18DOI: 10.1016/j.exmath.2024.125563
Florent P. Baudier , Christian Rosendal
We describe several ordinal indices that are capable of detecting, according to various metric notions of faithfulness, the embeddability between pairs of Polish spaces. These embeddability ranks are of theoretical interest but seem difficult to estimate in practice. Embeddability ranks, which are easier to estimate in practice, are embeddability ranks generated by Schauder bases. These embeddability ranks are inspired by the nonlinear indices à la Bourgain. In particular, we resolve a problem raised by F. Baudier, G. Lancien, P. Motakis, and Th. Schlumprecht in Coarse and Lipschitz universality, Fund. Math. 254 (2021), no. 2, 181–214, regarding the necessity of additional set-theoretic axioms regarding the main coarse universality result there.
{"title":"Abstract embeddability ranks","authors":"Florent P. Baudier , Christian Rosendal","doi":"10.1016/j.exmath.2024.125563","DOIUrl":"10.1016/j.exmath.2024.125563","url":null,"abstract":"<div><p>We describe several ordinal indices that are capable of detecting, according to various metric notions of faithfulness, the embeddability between pairs of Polish spaces. These embeddability ranks are of theoretical interest but seem difficult to estimate in practice. Embeddability ranks, which are easier to estimate in practice, are embeddability ranks generated by Schauder bases. These embeddability ranks are inspired by the nonlinear indices à la Bourgain. In particular, we resolve a problem raised by F. Baudier, G. Lancien, P. Motakis, and Th. Schlumprecht in Coarse and Lipschitz universality, Fund. Math. 254 (2021), no. 2, 181–214, regarding the necessity of additional set-theoretic axioms regarding the main coarse universality result there.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140182011","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}
Pub Date : 2024-03-18DOI: 10.1016/j.exmath.2024.125570
M. Ram Murty , Jagannath Sahoo , Akshaa Vatwani
The Wiener–Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series . 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.
{"title":"A simple proof of the Wiener–Ikehara Tauberian Theorem","authors":"M. Ram Murty , Jagannath Sahoo , 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":null,"pages":null},"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}
Pub Date : 2024-03-01DOI: 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":null,"pages":null},"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}
Pub Date : 2024-02-28DOI: 10.1016/j.exmath.2024.125547
Giorgio Saracco
We review some geometric criteria and prove a refined version, that yield existence of capillary surfaces in tubes in a gravity free environment, in the case of physical interest, that is, for bounded, open, and simply connected . These criteria rely on suitable weak one-sided bounds on the curvature of the boundary of the cross-section .
{"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":null,"pages":null},"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}
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