Advances in Mathematics最新文献

{"title":"On the formula for characteristic determinants of boundary value problems for n × n Dirac type systems and its applications","authors":"Anton A. Lunyov , Mark M. Malamud","doi":"10.1016/j.aim.2025.110389","DOIUrl":"10.1016/j.aim.2025.110389","url":null,"abstract":"<div><div>The paper is concerned with the spectral properties of the boundary value problems (BVP) associated with the following <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Dirac type equation:<span><span><span><math><mo>−</mo><mi>i</mi><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>−</mo><mi>i</mi><mi>Q</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>y</mi><mo>=</mo><mi>λ</mi><mi>B</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>y</mi><mo>,</mo><mspace></mspace><mi>y</mi><mo>=</mo><mrow><mi>col</mi></mrow><mo>(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>ℓ</mi><mo>]</mo><mo>,</mo></math></span></span></span> on a finite interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>ℓ</mi><mo>]</mo></math></span> subject to the general two-point boundary conditions <span><math><mi>C</mi><mi>y</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>+</mo><mi>D</mi><mi>y</mi><mo>(</mo><mi>ℓ</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> with <span><math><mi>C</mi><mo>,</mo><mi>D</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span>. Here <span><math><mi>Q</mi><mo>=</mo><msubsup><mrow><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>j</mi><mo>,</mo><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is an integrable potential matrix and <span><math><mi>B</mi><mo>=</mo><mrow><mi>diag</mi></mrow><mo>(</mo><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is a diagonal integrable matrix “weight”. If <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi></math></span> and <span><math><mi>B</mi><mo>(</mo><mo>⋅</mo><mo>)</mo><mo>=</mo><mrow><mi>diag</mi></mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span>, this equation turns into <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Dirac equation.</div><div>First, assuming that <span><math><mrow><mi>supp</mi></mrow><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>)</mo><mo>⊂</mo><mrow><mi>supp</mi></mrow><mo>(</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></math></span>, we show that the deviation <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>λ</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>Φ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>λ</mi><mo>)</mo></math>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110389"},"PeriodicalIF":1.5,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144240710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Banach space with L-orthogonal sequences but without L-orthogonal elements","authors":"Antonio Avilés ,&nbsp;Gonzalo Martínez-Cervantes ,&nbsp;Alejandro Poveda ,&nbsp;Luis Sáenz","doi":"10.1016/j.aim.2025.110391","DOIUrl":"10.1016/j.aim.2025.110391","url":null,"abstract":"<div><div>We prove that the existence of Banach spaces with <em>L</em>-orthogonal sequences but without <em>L</em>-orthogonal elements is independent of the standard foundation of Mathematics, ZFC. This provides a definitive answer to <span><span>[1, Question 1.1]</span></span>. Generalizing classical <em>Q</em>-point ultrafilters, we introduce the notion of <em>Q</em>-measures and provide several results generalizing former theorems by Miller <span><span>[22]</span></span> and Bartoszynski <span><span>[2]</span></span> for <em>Q</em>-point ultrafilters.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110391"},"PeriodicalIF":1.5,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144240707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Archimedean period relations for Rankin-Selberg convolutions","authors":"Yubo Jin ,&nbsp;Dongwen Liu ,&nbsp;Binyong Sun","doi":"10.1016/j.aim.2025.110393","DOIUrl":"10.1016/j.aim.2025.110393","url":null,"abstract":"<div><div>We formulate and prove the archimedean period relations for Rankin-Selberg convolutions of <span><math><mrow><mi>GL</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo><mo>×</mo><mrow><mi>GL</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mrow><mi>GL</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo><mo>×</mo><mrow><mi>GL</mi></mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>, for all generic cohomological representations. As a consequence, we prove the non-vanishing of the archimedean modular symbols. This extends the earlier results in <span><span>[14]</span></span> for essentially tempered representations of <span><math><mrow><mi>GL</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo><mo>×</mo><mrow><mi>GL</mi></mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110393"},"PeriodicalIF":1.5,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144240709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Counting rational points in non-isotropic neighborhoods of manifolds","authors":"Rajula Srivastava","doi":"10.1016/j.aim.2025.110394","DOIUrl":"10.1016/j.aim.2025.110394","url":null,"abstract":"<div><div>In this manuscript, we initiate the study of the number of rational points with bounded denominators, contained in a <em>non-isotropic</em> <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><mo>…</mo><mo>×</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> neighborhood of a compact submanifold <span><math><mi>M</mi></math></span> of codimension <em>R</em> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>M</mi></mrow></msup></math></span>. We establish an upper bound for this counting function which holds when <span><math><mi>M</mi></math></span> satisfies a strong curvature condition, first introduced by Schindler-Yamagishi in <span><span>[22]</span></span>. Further, even in the isotropic case when <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mo>…</mo><mo>=</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>=</mo><mi>δ</mi></math></span>, we obtain an asymptotic formula which holds beyond the range of distance to <span><math><mi>M</mi></math></span> established in <span><span>[22]</span></span>. Our result is also a generalization of the work of J.J. Huang <span><span>[9]</span></span> for hypersurfaces.</div><div>As an application, we establish for the first time an upper bound for the Hausdorff dimension of the set of weighted simultaneously well approximable points on a manifold <span><math><mi>M</mi></math></span> satisfying the strong curvature condition, which agrees with the lower bound obtained by Allen-Wang in <span><span>[2]</span></span>. Moreover, for <span><math><mi>R</mi><mo>&gt;</mo><mn>1</mn></math></span>, we obtain a new upper bound for the number of rational points <em>on</em> <span><math><mi>M</mi></math></span>, which goes beyond the bound in an analogue of Serre's dimension growth conjecture for submanifolds of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>M</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110394"},"PeriodicalIF":1.5,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144222701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On formal non-commutative deformations of smooth varieties","authors":"Yujiro Kawamata","doi":"10.1016/j.aim.2025.110392","DOIUrl":"10.1016/j.aim.2025.110392","url":null,"abstract":"<div><div>We will develop a formal non-commutative (NC) deformation theory of smooth algebraic varieties <em>X</em> defined over a field <em>k</em>, and describe a semi-universal deformation where the tangent space <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> and the obstruction space <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> are given by the Hochschild cohomology groups.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110392"},"PeriodicalIF":1.5,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144222700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-zero to zero curvature transition: Operators along hybrid curves with no quadratic (quasi-)resonances","authors":"Alejandra Gaitan ,&nbsp;Victor Lie","doi":"10.1016/j.aim.2025.110356","DOIUrl":"10.1016/j.aim.2025.110356","url":null,"abstract":"<div><div>Building on <span><span>[20]</span></span>, this paper develops a unifying study on the boundedness properties of several representative classes of <em>hybrid</em> operators, <em>i.e.</em> operators that enjoy both zero and non-zero curvature features. Specifically, via the LGC-method, we provide suitable <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> bounds for three classes of operators: (1) Carleson-type operators, (2) Hilbert transform along variable curves, and, taking the center stage, (3) Bilinear Hilbert transform and bilinear maximal operators along curves. All these classes of operators will be studied in the context of hybrid curves with no quadratic resonances.</div><div>The above study is interposed between two naturally derived topics:<ul><li><span>i)</span><span><div>A prologue providing a first rigorous account on how the presence/absence of a higher order modulation invariance property interacts with and determines the nature of the method employed for treating operators with such features.</div></span></li><li><span>ii)</span><span><div>An epilogue revealing how several key ingredients within our present study can blend and inspire a short, intuitive new proof of the smoothing inequality that plays the central role in the analysis of the curved version of the triangular Hilbert transform treated in <span><span>[3]</span></span>.</div></span></li></ul></div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110356"},"PeriodicalIF":1.5,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144222715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Almost sure dimensional properties for the spectrum and the density of states of Sturmian Hamiltonians","authors":"Jie Cao ,&nbsp;Yanhui Qu","doi":"10.1016/j.aim.2025.110387","DOIUrl":"10.1016/j.aim.2025.110387","url":null,"abstract":"<div><div>In this paper, we find a full Lebesgue measure set of frequencies <span><math><mover><mrow><mi>I</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>⊂</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>∖</mo><mi>Q</mi></math></span> such that for any <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>λ</mi><mo>)</mo><mo>∈</mo><mover><mrow><mi>I</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>×</mo><mo>[</mo><mn>24</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, the Hausdorff and box dimensions of the spectrum of the Sturmian Hamiltonian <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>θ</mi></mrow></msub></math></span> coincide and are independent of <em>α</em>. Denote the common value by <span><math><mi>D</mi><mo>(</mo><mi>λ</mi><mo>)</mo></math></span>, we show that <span><math><mi>D</mi><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> satisfies a Bowen's type formula, and is locally Lipschitz. We also obtain the exact asymptotic behavior of <span><math><mi>D</mi><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> as <em>λ</em> tends to ∞. This considerably improves the result of Damanik and Gorodetski (Comm. Math. Phys. 337, 2015). We also show that for any <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>λ</mi><mo>)</mo><mo>∈</mo><mover><mrow><mi>I</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>×</mo><mo>[</mo><mn>24</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, the density of states measure of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>θ</mi></mrow></msub></math></span> is exact-dimensional; its Hausdorff and packing dimensions coincide and are independent of <em>α</em>. Denote the common value by <span><math><mi>d</mi><mo>(</mo><mi>λ</mi><mo>)</mo></math></span>, we show that <span><math><mi>d</mi><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> satisfies a Young's type formula, and is locally Lipschitz. We also obtain the exact asymptotic behavior of <span><math><mi>d</mi><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> as <em>λ</em> tends to ∞. During the course of study, we also answer or partially answer several questions in the same paper of Damanik and Gorodetski.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110387"},"PeriodicalIF":1.5,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144222702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Big Ramsey degrees using parameter spaces","authors":"Jan Hubička","doi":"10.1016/j.aim.2025.110386","DOIUrl":"10.1016/j.aim.2025.110386","url":null,"abstract":"<div><div>We show that the universal homogeneous partial order has finite big Ramsey degrees and discuss several corollaries. Our proof relies on parameter spaces and the Carlson–Simpson theorem rather than on (a strengthening of) the Halpern–Läuchli theorem and the Milliken tree theorem, which are typically used to bound big Ramsey degrees in the existing literature (originating from the work of Laver and Milliken).</div><div>This new technique has many additional applications. We show that the homogeneous universal triangle-free graph has finite big Ramsey degrees, providing a short proof of a recent result by Dobrinen. Moreover, generalizing an indivisibility (vertex partition) result of Nguyen van Thé and Sauer, we give an upper bound on big Ramsey degrees of metric spaces with finitely many distances. This leads to a new combinatorial argument for the oscillation stability of the Urysohn Sphere.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110386"},"PeriodicalIF":1.5,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144222717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Precise estimates of invariant distances on strongly pseudoconvex domains","authors":"Łukasz Kosiński ,&nbsp;Nikolai Nikolov ,&nbsp;Ahmed Yekta Ökten","doi":"10.1016/j.aim.2025.110388","DOIUrl":"10.1016/j.aim.2025.110388","url":null,"abstract":"<div><div>Studying the behavior of real and complex geodesics we provide sharp estimates for the Kobayashi distance, the Lempert function, and the Carathéodory distance on strongly pseudoconvex domain with <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>-smooth boundary. Similar estimates are also provided for the Bergman distance on strongly pseudoconvex domains with <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span>-smooth boundary.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110388"},"PeriodicalIF":1.5,"publicationDate":"2025-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144213268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regularity of solutions to the Dirichlet problem for fast diffusion equations","authors":"Tianling Jin ,&nbsp;Jingang Xiong","doi":"10.1016/j.aim.2025.110390","DOIUrl":"10.1016/j.aim.2025.110390","url":null,"abstract":"<div><div>We prove global Hölder gradient estimates for bounded positive weak solutions of fast diffusion equations in smooth bounded domains with the homogeneous Dirichlet boundary condition, which then lead us to establish their optimal global regularity. This solves a problem raised by Berryman and Holland in 1980.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"478 ","pages":"Article 110390"},"PeriodicalIF":1.5,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144194778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}