Pub Date : 2026-01-29DOI: 10.1016/j.exmath.2026.125754
Ho Yun Jung , Ja Kyung Koo , Dong Hwa Shin
<div><div>Let <span><math><mi>N</mi></math></span> be a positive integer, and let <span><math><mrow><mi>D</mi><mo>≡</mo><mn>0</mn></mrow></math></span> or <span><math><mrow><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mtext>mod</mtext><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span> be a negative integer. We define the sets <span><math><mrow><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><mi>Y</mi><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> as subsets of the Shimura varieties <span><math><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup></mrow></math></span> and <span><math><mrow><mi>Y</mi><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup></mrow></math></span>, respectively, consisting of CM points of discriminant <span><math><mi>D</mi></math></span> that are primitive modulo <span><math><mi>N</mi></math></span>. By using the theory of definite form class groups, we show that the inverse limits <span><span><span><math><mrow><msub><mrow><munder><mrow><mo>lim</mo></mrow><mo>←</mo></munder></mrow><mrow><mi>N</mi></mrow></msub><mspace></mspace><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow><mspace></mspace><mtext>and</mtext><mspace></mspace><msub><mrow><munder><mrow><mo>lim</mo></mrow><mo>←</mo></munder></mrow><mrow><mi>N</mi></mrow></msub><mspace></mspace><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><mi>Y</mi><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span></span></span>naturally inherit group structures isomorphic to <span><math><mrow><mi>Gal</mi><mrow><mo>(</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>ab</mi></mrow></msup><mo>/</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>Gal</mi><mrow><mo>(</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>ab</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>∞</mi></mrow></msup><mo>)</mo></mrow><mo>/</mo><mi>Q</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, respectively, where <span><math><mrow><mi>K</mi><mo>=</mo><mi>Q</mi><mrow><mo>(</mo><msqrt><mrow><mi>D</mi></mrow></msqrt><mo>)</mo></mrow></mrow></math></span> and <span><math><mi>t</mi></math></span> is a transcendental number. These resu
{"title":"Inverse limits of CM points on certain Shimura varieties","authors":"Ho Yun Jung , Ja Kyung Koo , Dong Hwa Shin","doi":"10.1016/j.exmath.2026.125754","DOIUrl":"10.1016/j.exmath.2026.125754","url":null,"abstract":"<div><div>Let <span><math><mi>N</mi></math></span> be a positive integer, and let <span><math><mrow><mi>D</mi><mo>≡</mo><mn>0</mn></mrow></math></span> or <span><math><mrow><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mtext>mod</mtext><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow></math></span> be a negative integer. We define the sets <span><math><mrow><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><mi>Y</mi><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> as subsets of the Shimura varieties <span><math><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup></mrow></math></span> and <span><math><mrow><mi>Y</mi><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup></mrow></math></span>, respectively, consisting of CM points of discriminant <span><math><mi>D</mi></math></span> that are primitive modulo <span><math><mi>N</mi></math></span>. By using the theory of definite form class groups, we show that the inverse limits <span><span><span><math><mrow><msub><mrow><munder><mrow><mo>lim</mo></mrow><mo>←</mo></munder></mrow><mrow><mi>N</mi></mrow></msub><mspace></mspace><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow><mspace></mspace><mtext>and</mtext><mspace></mspace><msub><mrow><munder><mrow><mo>lim</mo></mrow><mo>←</mo></munder></mrow><mrow><mi>N</mi></mrow></msub><mspace></mspace><mi>CM</mi><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mspace></mspace><mi>Y</mi><msup><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span></span></span>naturally inherit group structures isomorphic to <span><math><mrow><mi>Gal</mi><mrow><mo>(</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>ab</mi></mrow></msup><mo>/</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>Gal</mi><mrow><mo>(</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>ab</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>∞</mi></mrow></msup><mo>)</mo></mrow><mo>/</mo><mi>Q</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, respectively, where <span><math><mrow><mi>K</mi><mo>=</mo><mi>Q</mi><mrow><mo>(</mo><msqrt><mrow><mi>D</mi></mrow></msqrt><mo>)</mo></mrow></mrow></math></span> and <span><math><mi>t</mi></math></span> is a transcendental number. These resu","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 2","pages":"Article 125754"},"PeriodicalIF":0.9,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077601","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 : 2025-12-01DOI: 10.1016/j.exmath.2025.125741
Ulrik Enstad , Franz Luef
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.
{"title":"Bessel duality of Gabor systems: A von Neumann algebraic perspective","authors":"Ulrik Enstad , 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}
Pub Date : 2025-12-01DOI: 10.1016/j.exmath.2025.125742
David P. Blecher
We call an operator algebra reversible if 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 symmetric operator algebras: the subalgebras of 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 . 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.
{"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}
Pub Date : 2025-12-01DOI: 10.1016/j.exmath.2025.125739
Cristian Cruz Hernandez , Juan Ignacio Giribet , Ursula Molter
We revisit the notion of positive bases proposed by Chandler Davis, and connect it with the operator-driven constructions that appear in dynamical sampling. Working in finite-dimensional real inner-product spaces, we derive verifiable criteria ensuring that families of the form are positive generators or positive bases. Under spectral hypotheses on 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.
{"title":"Dynamical sampling for unmanned aerial vehicles","authors":"Cristian Cruz Hernandez , Juan Ignacio Giribet , 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}
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.
{"title":"Stability bounds for correcting from erased and irregular samples","authors":"Duncan Koepke , Huston Wilhite , Rebecca Yoshino , 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}
Pub Date : 2025-12-01DOI: 10.1016/j.exmath.2025.125740
Nazar Emirov , Guohui Song , Qiyu Sun
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 ( distance, quadratic, and entropy) confirm the theory and demonstrate scalability and effectiveness.
{"title":"Exponential convergence of a distributed divide-and-conquer algorithm for constrained convex optimization on networks","authors":"Nazar Emirov , Guohui Song , 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}
Pub Date : 2025-11-29DOI: 10.1016/j.exmath.2025.125743
Dorin Ervin Dutkay , Catalin Georgescu , Gabriel Picioroaga
We provide a new characterization of amenability for countable groups, based on frame representations admitting almost invariant vectors. By relaxing the frame inequalities, thereby weakening amenability, we obtain a large class of countable groups which we call framenable. We show that this class has some permanence properties, stands in contrast with property (T), and contains, for example, all free groups , and , all (countable) lattices of , the Baumslag–Solitar groups , the braid groups , and Thompson’s group .
{"title":"Frame vector group representations and amenability properties","authors":"Dorin Ervin Dutkay , Catalin Georgescu , Gabriel Picioroaga","doi":"10.1016/j.exmath.2025.125743","DOIUrl":"10.1016/j.exmath.2025.125743","url":null,"abstract":"<div><div>We provide a new characterization of amenability for countable groups, based on frame representations admitting almost invariant vectors. By relaxing the frame inequalities, thereby weakening amenability, we obtain a large class of countable groups which we call <em>framenable</em>. We show that this class has some permanence properties, stands in contrast with property (T), and contains, for example, all free groups <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mrow><mtext>Aut</mtext><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mtext>Aut</mtext><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>, all (countable) lattices of <span><math><mrow><mi>S</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, the Baumslag–Solitar groups <span><math><mrow><mi>B</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></mrow></math></span>, the braid groups <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and Thompson’s group <span><math><mi>F</mi></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125743"},"PeriodicalIF":0.9,"publicationDate":"2025-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694512","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 : 2025-11-29DOI: 10.1016/j.exmath.2025.125745
Marcin Bownik , Ziemowit Rzeszotnik , Darrin Speegle
Larson’s problem Larson (2007, Problem 3) asks “Must the support of the Fourier transform of a wavelet contain a wavelet set?”. We give an affirmative answer to a non-measurable variant of this question by proving that the Fourier transform of a wavelet must contain a possibly non-measurable wavelet set. We also provide background results on Larson’s problem and propose two new related problems.
{"title":"Are MSF wavelets minimally supported?","authors":"Marcin Bownik , Ziemowit Rzeszotnik , Darrin Speegle","doi":"10.1016/j.exmath.2025.125745","DOIUrl":"10.1016/j.exmath.2025.125745","url":null,"abstract":"<div><div>Larson’s problem Larson (2007, Problem 3) asks “Must the support of the Fourier transform of a wavelet contain a wavelet set?”. We give an affirmative answer to a non-measurable variant of this question by proving that the Fourier transform of a wavelet must contain a possibly non-measurable wavelet set. We also provide background results on Larson’s problem and propose two new related problems.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125745"},"PeriodicalIF":0.9,"publicationDate":"2025-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145645748","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 : 2025-11-25DOI: 10.1016/j.exmath.2025.125738
Daniel Alpay , Chad Berner , Eric S. Weber
The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert -modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert -modules can be generated by the Kaczmarz algorithm and realized as orbits of bounded operators.
Hilbert空间中的Kaczmarz算法是一种从内积数据中稳定恢复向量的经典迭代方法。在本文中，我们将该算法推广到Hilbert C * -模的集合，并建立了它在有限维和平稳情况下的有效性的类比。因此，我们证明了Hilbert空间中的连续族元素可以用Kaczmarz算法一致恢复。此外，我们开发了连续测度族的规范化Cauchy变换，并利用它提供了Hilbert C(X)-模块中的标准帧可以由Kaczmarz算法生成并实现为有界算子的轨道的充分条件。
{"title":"The Kaczmarz Algorithm in Hilbert C∗-modules","authors":"Daniel Alpay , Chad Berner , Eric S. Weber","doi":"10.1016/j.exmath.2025.125738","DOIUrl":"10.1016/j.exmath.2025.125738","url":null,"abstract":"<div><div>The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>-modules can be generated by the Kaczmarz algorithm and realized as orbits of bounded operators.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 2","pages":"Article 125738"},"PeriodicalIF":0.9,"publicationDate":"2025-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145658446","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 : 2025-11-04DOI: 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), Deguang Han, Palle Jorgensen, Keri Kornelson, 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":"2025-11-04","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：