数学最新文献

{"title":"Paucity of rational points on fibrations with multiple fibres","authors":"Tim Browning, Julian Lyczak, Arne Smeets","doi":"10.2140/ant.2025.19.2049","DOIUrl":"https://doi.org/10.2140/ant.2025.19.2049","url":null,"abstract":"<p>Given a family of varieties over the projective line, we study the density of fibres that are everywhere locally soluble in the case that components of higher multiplicity are allowed. We use log geometry to formulate a new sparsity criterion for the existence of everywhere locally soluble fibres and formulate new conjectures that generalise previous work of Loughran and Smeets. These conjectures involve geometric invariants of the associated multiplicity orbifolds on the base of the fibration in the spirit of Campana. We give evidence for the conjectures by providing an assortment of bounds using Chebotarev’s theorem and sieve methods, with most of the evidence involving upper bounds. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"24 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145002871","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":"Smooth numbers are orthogonal to nilsequences","authors":"Lilian Matthiesen, Mengdi Wang","doi":"10.2140/ant.2025.19.1881","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1881","url":null,"abstract":"&lt;p&gt;The aim of this paper is to study distributional properties of integers without large or small prime factors. Define an integer to be &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;′&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/math&gt;-smooth if all of its prime factors belong to the interval &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;′&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/math&gt;. We identify suitable weights &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;′&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; for the characteristic function of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;′&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/math&gt;-smooth numbers that allow us to establish strong asymptotic results on their distribution in short arithmetic progressions. Building on these equidistribution properties, we show that (a &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/math&gt;-tricked version of) the function &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;′&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;\u0000&lt;mo&gt;−&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt; is orthogonal to nilsequences. Our results apply in the almost optimal range &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt; &lt;/mo&gt;&lt;!--nolimits--&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;\u0000&lt;mo&gt;&lt;&lt;/mo&gt;\u0000&lt;mi&gt;y&lt;/mi&gt;\u0000&lt;mo&gt;≤&lt;/mo&gt;\u0000&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt; of the smoothness parameter &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/math&gt;, where &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;K&lt;/mi&gt;\u0000&lt;mo&gt;≥&lt;/mo&gt; &lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt; is sufficiently large, and to any &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;′&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;\u0000&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt; min&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt; &lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt; &lt;/mo&gt;&lt;!--nolimits--&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;. &lt;/p&gt;&lt;p&gt; As a first application, we e","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"20 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145002872","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":"Heat equation in a periodic domain with special initial data","authors":"Marcus Rosenberg,&nbsp;Jari Taskinen","doi":"10.1016/j.jde.2025.113754","DOIUrl":"10.1016/j.jde.2025.113754","url":null,"abstract":"<div><div>We consider the initial-boundary value problem with the Neumann boundary condition for the classical linear heat equation in unbounded domains <span><math><mi>Ω</mi><mo>⊊</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> which are periodic in all directions of the Cartesian coordinate system. Generalizing the results of a previous paper by the authors, we apply Floquet transform methods to obtain results on the large time decay rates of the solution in the sup-norm. We observe that for a general, integrable initial data, the solution decays at the same rate <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span> as in the case of the Cauchy problem in the entire Euclidean space. We also consider special initial data with vanishing <em>x</em>-integral and obtain a faster decay rate. In the main results of the paper we pose for the initial data certain more detailed conditions, which are related to the lowest eigenvalue and eigenfunction of the model problem coming from the Floquet transform. Faster decay rates are obtained for such initial data.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"451 ","pages":"Article 113754"},"PeriodicalIF":2.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144997656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On determinants of tournaments and the characterization of D5","authors":"Jing Zeng,&nbsp;Lihua You","doi":"10.1016/j.disc.2025.114766","DOIUrl":"10.1016/j.disc.2025.114766","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;em&gt;T&lt;/em&gt; be a tournament with &lt;em&gt;n&lt;/em&gt; vertices &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. The skew-adjacency matrix of &lt;em&gt;T&lt;/em&gt; is the &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; zero-diagonal matrix &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; in which &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; dominates &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. We define the determinant &lt;span&gt;&lt;math&gt;&lt;mi&gt;det&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of &lt;em&gt;T&lt;/em&gt; as the determinant of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. It is well-known that &lt;span&gt;&lt;math&gt;&lt;mi&gt;det&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; if &lt;em&gt;n&lt;/em&gt; is odd and &lt;span&gt;&lt;math&gt;&lt;mi&gt;det&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the square of an odd integer if &lt;em&gt;n&lt;/em&gt; is even. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be the set of tournaments whose all subtournaments have determinant at most &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;em&gt;k&lt;/em&gt; is a positive odd integer. The necessary and sufficient condition for &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; has been characterized in 2023. In this paper, we characterize the set &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, and we obtain some properties of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. Moreover, for any positive odd integer &lt;em&gt;k&lt;/em&gt;, we give a construction of a tournament &lt;em&gt;T&lt;/em&gt; satisfying &lt;span&gt;&lt;math&gt;&lt;mi&gt;det&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;﹨&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; if &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, which implies &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;﹨&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/spa","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114766"},"PeriodicalIF":0.7,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144996761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global-in-time maximal regularity for the Cauchy problem of the heat equation in BMO and applications","authors":"Xuan Thinh Duong ,&nbsp;Ji Li ,&nbsp;Liangchuan Wu ,&nbsp;Lixin Yan","doi":"10.1016/j.jde.2025.113748","DOIUrl":"10.1016/j.jde.2025.113748","url":null,"abstract":"<div><div>In this article, we establish global-in-time maximal regularity for the Cauchy problem of the classical heat equation <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> with <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></span> in a certain BMO setting, which improves the local-in-time result initially proposed by Ogawa and Shimizu in <span><span>[26]</span></span>, <span><span>[27]</span></span>. In further developing our method originally formulated for the heat equation, we obtain analogous global BMO-maximal regularity associated to the Schrödinger operator <span><math><mi>L</mi><mo>=</mo><mo>−</mo><mi>Δ</mi><mo>+</mo><mi>V</mi></math></span>, where the nonnegative potential <em>V</em> belongs to the reverse Hölder class <span><math><msub><mrow><mi>RH</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> for some <span><math><mi>q</mi><mo>&gt;</mo><mi>n</mi><mo>/</mo><mn>2</mn></math></span>. This extension includes several inhomogeneous estimates as ingredients, such as Carleson-type estimates for the external forces.</div><div>Our new methodology is to exploit elaborate heat kernel estimates, along with matched space-time decomposition on the involving integral-type structure of maximal operators, as well as some global techniques such as those from de Simon's work and Schur's lemma. One crucial trick is to utilize the mean oscillation therein to contribute a higher and necessary decay order for global-in-time estimates.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"451 ","pages":"Article 113748"},"PeriodicalIF":2.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144997555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Large-Time Existence and Decay of Entropy Solutions for Unsteady Isentropic Gas Flow in the Quasi-One-Dimensional Nozzle","authors":"Jianjun Chen,&nbsp;Qiquan Fang,&nbsp;Yun-guang Lu,&nbsp;Naoki Tsuge","doi":"10.1111/sapm.70104","DOIUrl":"https://doi.org/10.1111/sapm.70104","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we apply the viscosity–flux approximation method coupled with the maximum principle to obtain the a priori <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>∞</mi>\u0000 </msup>\u0000 <annotation>$L^{infty }$</annotation>\u0000 </semantics></math> estimates for the viscosity approximation solutions of the unsteady isentropic gas flow in the de Laval nozzle. Then by applying the compensated compactness method, we obtain the global existence of entropy solutions. Finally, we study the large-time behavior of solutions and show the decay of the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>γ</mi>\u0000 </msup>\u0000 <annotation>$L^{gamma }$</annotation>\u0000 </semantics></math> norm of density for any adiabatic exponent <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$gamma &gt;1$</annotation>\u0000 </semantics></math>.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 3","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144998865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamical Stability of Transonic Shock Solutions to Euler-Poisson System in an Annulus","authors":"Qifeng Bai,&nbsp;Yuanyuan Xing","doi":"10.1007/s00021-025-00961-z","DOIUrl":"10.1007/s00021-025-00961-z","url":null,"abstract":"<div><p>This paper concerns the Euler-Poisson system in an annulus with finite radius. The dynamical stability of radially symmetric transonic shock solutions to the Euler-Poisson system is transformed into the global well-posedness of a free boundary problem for a second-order quasilinear hyperbolic equation. One of the crucial ingredients of the analysis is to establish an energy estimate for the associated initial boundary value problem. The steady radial transonic shock solutions are proved to be dynamically and exponentially stable with respect to small perturbations of the initial data.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"27 4","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144990456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nakayama's lemma for Z-graded vertex algebras and its applications","authors":"Hao Wang ,&nbsp;Wei Wang","doi":"10.1016/j.aim.2025.110503","DOIUrl":"10.1016/j.aim.2025.110503","url":null,"abstract":"<div><div>This is the first paper in a series of studies on <span><math><mi>Z</mi></math></span>-graded vertex algebras arising from Lie algebras with triangular decomposition (referred to as triangulated Lie algebras for simplicity). In this paper, we establish the Jacobson radical theory for <span><math><mi>Z</mi></math></span>-graded vertex algebras and prove Nakayama's lemma. As an application, we investigate a specific triangulated Lie algebra: <span><math><mi>g</mi><mo>=</mo><mi>C</mi><mi>f</mi><mo>⊕</mo><mi>C</mi><mi>h</mi><mo>⊕</mo><mi>C</mi><mi>e</mi></math></span> with Lie brackets <span><math><mo>[</mo><mi>h</mi><mo>,</mo><mi>e</mi><mo>]</mo><mo>=</mo><mn>2</mn><mi>e</mi><mo>,</mo><mspace></mspace><mo>[</mo><mi>h</mi><mo>,</mo><mi>f</mi><mo>]</mo><mo>=</mo><mo>−</mo><mn>2</mn><mi>f</mi><mo>,</mo><mspace></mspace><mo>[</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>]</mo><mo>=</mo><mn>0</mn></math></span>. Denote the <span><math><mi>Z</mi></math></span>-graded vertex algebra constructed from <span><math><mi>g</mi></math></span> by <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>l</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>. Using Nakayama's lemma, we classify all the irreducible admissible <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>l</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>-modules. Finally, we construct a class of indecomposable admissible <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>l</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>-modules.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110503"},"PeriodicalIF":1.5,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144996258","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":"Nonlinear dynamics of a quasizero energy harvester forced by flow and kinematic excitations","authors":"Jerzy Margielewicz ,&nbsp;Damian Gąska ,&nbsp;Sławomir Bucki ,&nbsp;Daniil Yurchenko ,&nbsp;Grzegorz Litak","doi":"10.1016/j.chaos.2025.117148","DOIUrl":"10.1016/j.chaos.2025.117148","url":null,"abstract":"<div><div>The paper proposes a novel design of a high-efficiency nonlinear energy harvester that benefits from the synergy between a quasi-zero stiffness system and flow-induced excitations. We demonstrate that this innovative approach enables improved adaptability and energy output. The harvester consists of a cantilever beam in a flag-like configuration subjected to both air-flow and kinematic excitations. In the wake galloping scenario, the variable lift force generated behind a fixed bluff body interacts with the tail-like cantilever beam. Using finite element modeling, we identified the lift force across a wide spectrum of air velocities and incorporated it as an excitation in a dimensionless mathematical model. Subsequently, vibrations of the cantilever beam, leading to the generation of electromotive force in an attached piezoelectric element. The system's displacement and voltage output responses were analyzed using nonlinear dynamics tools. An important advantage of this design is its tunability – by adjusting the elastic elements, the potential function can be configured for optimal performance. Key findings also concern the behavior of the system in chaotic and periodic motion zones and their impact on efficiency. We used tools such as the Lyapunov exponent, bifurcation diagrams, and Poincare sections to analyze nonlinear dynamics. We also identified transient chaos and periodic, chaotic, and quasi-periodic solutions to determine their impact on the performance of the system. The study further explores coexisting solutions and evaluates the effectiveness of the system under different types of excitation, both individually and together. The results demonstrate that the harvester maintains high efficiency even at low excitation levels, highlighting its potential for practical applications.</div></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"200 ","pages":"Article 117148"},"PeriodicalIF":5.6,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144996350","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":"Affine Deligne–Lusztig varieties via the double Bruhat graph, I : Semi-infinite orbits","authors":"Felix Schremmer","doi":"10.2140/ant.2025.19.1973","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1973","url":null,"abstract":"<p>We introduce a new language to describe the geometry of affine Deligne–Lusztig varieties in affine flag varieties. This first part of a two-paper series develops the definition and fundamental properties of the double Bruhat graph by studying semi-infinite orbits. This double Bruhat graph was originally introduced by Naito and Watanabe to study periodic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>R</mi></math>-polynomials. We use it to describe the geometry of many affine Deligne–Lusztig varieties, overcoming a previously ubiquitous regularity condition. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145002869","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}