数学最新文献

{"title":"On the rigidity of Arnoux-Rauzy words","authors":"V. Berthé ,&nbsp;S. Puzynina","doi":"10.1016/j.aam.2025.102932","DOIUrl":"10.1016/j.aam.2025.102932","url":null,"abstract":"<div><div>An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of the same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words that are generated by iterating a substitution are known to be rigid. In the present paper, we prove that all Arnoux-Rauzy words generated by iterating a substitution are rigid. The proof relies on two main ingredients: first, the fact that the primitive substitutions that fix an Arnoux-Rauzy word share a common power, and secondly, the notion of normal form of an episturmian substitution (i.e., a substitution that fixes an Arnoux-Rauzy word). The main difficulty is then of a combinatorial nature and relies on the normalization process when taking powers of episturmian substitutions: the normal form of a square is not necessarily equal to the square of the normal forms.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102932"},"PeriodicalIF":1.0,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579417","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":"Conformality of minimal transversals of maximal cliques","authors":"Endre Boros ,&nbsp;Vladimir Gurvich ,&nbsp;Martin Milanič ,&nbsp;Dmitry Tikhanovsky ,&nbsp;Yushi Uno","doi":"10.1016/j.disc.2025.114657","DOIUrl":"10.1016/j.disc.2025.114657","url":null,"abstract":"<div><div>Given a hypergraph <span><math><mi>H</mi></math></span>, the dual hypergraph of <span><math><mi>H</mi></math></span> is the hypergraph of all minimal transversals of <span><math><mi>H</mi></math></span>. A hypergraph is conformal if it is the family of maximal cliques of a graph. In a recent work, Boros, Gurvich, Milanič, and Uno (Journal of Graph Theory, 2025) studied conformality of dual hypergraphs and proved several results related to this property, leading in particular to a polynomial-time algorithm for recognizing graphs in which, for any fixed <em>k</em>, all minimal transversals of maximal cliques have size at most <em>k</em>. In this follow-up work, we provide a novel aspect to the study of graph clique transversals, by considering the dual conformality property from the perspective of graphs. More precisely, we study graphs for which the family of minimal transversals of maximal cliques is conformal. Such graphs are called clique dually conformal (CDC for short). It turns out that the class of CDC graphs is a rich generalization of the class of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free graphs. As our main results, we completely characterize CDC graphs within the families of triangle-free graphs and split graphs. Both characterizations lead to polynomial-time recognition algorithms. Generalizing the fact that every <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free graph is CDC, we also show that the class of CDC graphs is closed under substitution, in the strong sense that substituting a graph <em>H</em> for a vertex of a graph <em>G</em> results in a CDC graph if and only if both <em>G</em> and <em>H</em> are CDC.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114657"},"PeriodicalIF":0.7,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579926","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":"Sample path properties and small ball probabilities for stochastic fractional diffusion equations","authors":"Yuhui Guo ,&nbsp;Jian Song ,&nbsp;Ran Wang ,&nbsp;Yimin Xiao","doi":"10.1016/j.jde.2025.113604","DOIUrl":"10.1016/j.jde.2025.113604","url":null,"abstract":"<div><div>We consider the following stochastic space-time fractional diffusion equation with vanishing initial condition:<span><span><span><math><msup><mrow><mo>∂</mo></mrow><mrow><mi>β</mi></mrow></msup><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>−</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mn>0</mn><mo>+</mo></mrow><mrow><mi>γ</mi></mrow></msubsup><mrow><mo>[</mo><mover><mrow><mi>W</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>]</mo></mrow><mo>,</mo><mspace></mspace><mi>t</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>α</mi><mo>&gt;</mo><mn>0</mn></math></span>, <span><math><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mi>γ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, <span><math><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span> is the fractional/power of Laplacian and <span><math><mover><mrow><mi>W</mi></mrow><mrow><mo>˙</mo></mrow></mover></math></span> is a fractional space-time Gaussian noise. We prove the existence and uniqueness of the solution and then focus on various sample path regularity properties of the solution. More specifically, we establish the exact uniform and local moduli of continuity and Chung's laws of the iterated logarithm. The small ball probability is also studied.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"446 ","pages":"Article 113604"},"PeriodicalIF":2.4,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579418","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":"Global well-posedness to the three-dimensional compressible Navier–Stokes equations with anisotropic viscous stress tensor","authors":"Ying Wang ,&nbsp;Zhenhua Guo","doi":"10.1016/j.na.2025.113898","DOIUrl":"10.1016/j.na.2025.113898","url":null,"abstract":"<div><div>This paper addresses the Cauchy problem for the three-dimensional Navier–Stokes equations with anisotropic viscosity tensor. Under the condition that the initial energy is small enough, we establish the global existence and uniqueness of classical solutions and derive some decay rates. Notably, we extend the results for small energy solutions with isotropic viscous stress tensors originally established by Huang et al., (2012) to the anisotropic case.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113898"},"PeriodicalIF":1.3,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580084","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 Tuza’s conjecture in dense graphs","authors":"Luis Chahua,&nbsp;Juan Gutiérrez","doi":"10.1016/j.dam.2025.06.049","DOIUrl":"10.1016/j.dam.2025.06.049","url":null,"abstract":"<div><div>In 1982, Tuza conjectured that the size <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a minimum set of edges that intersects every triangle of a graph <span><math><mi>G</mi></math></span> is at most twice the size <span><math><mrow><mi>ν</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a maximum set of edge-disjoint triangles of <span><math><mi>G</mi></math></span>. This conjecture was proved for several graph classes but it remains open even for split graphs. In this paper, we show Tuza’s conjecture for split graphs with minimum degree at least <span><math><mfrac><mrow><mn>3</mn><mi>n</mi></mrow><mrow><mn>5</mn></mrow></mfrac></math></span>. We also show that <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>&lt;</mo><mfrac><mrow><mn>28</mn></mrow><mrow><mn>15</mn></mrow></mfrac><mi>ν</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for every tripartite graph with minimum degree more than <span><math><mfrac><mrow><mn>33</mn><mi>n</mi></mrow><mrow><mn>56</mn></mrow></mfrac></math></span>, and that <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>ν</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>G</mi></math></span> is a complete 4-partite graph. Moreover, this bound is tight.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 225-233"},"PeriodicalIF":1.0,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580306","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":"Secondary Poincaré section for characterizing dynamical behaviours of nonlinear systems","authors":"Zhengyuan Zhang,&nbsp;Liming Dai","doi":"10.1016/j.chaos.2025.116849","DOIUrl":"10.1016/j.chaos.2025.116849","url":null,"abstract":"<div><div>An innovative secondary Poincaré section method is presented in this research to characterize the behaviours of nonlinear dynamical systems, especially quasiperiodicity and chaos. To circumvent the difficulty of computing the intersection between a discrete point set and a plane, a closed curve is iteratively mapped to approach the Poincaré attractor. In this way, the proposed method effectively defines a secondary Poincaré plot that is more accurate and rigorous than existing methods. It lays the groundwork for dimensionality reduction analysis for nonlinear dynamical systems.</div></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"199 ","pages":"Article 116849"},"PeriodicalIF":5.3,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579375","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":"Principal bundles in the category of Z2n-manifolds","authors":"Andrew James Bruce ,&nbsp;Janusz Grabowski","doi":"10.1016/j.difgeo.2025.102269","DOIUrl":"10.1016/j.difgeo.2025.102269","url":null,"abstract":"<div><div>We introduce and examine the notion of principal <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-bundles, i.e., principal bundles in the category of <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-manifolds. The latter are higher graded extensions of supermanifolds in which a <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-grading replaces <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-grading. These extensions have opened up new areas of research of great interest in both physics and mathematics. In principle, the geometry of <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-manifolds is essentially different than that of supermanifolds, as for n &gt; 1 we have formal variables of even parity, so local smooth functions are power series in formal variables. On the other hand, a full version of differential calculus is still valid. We show in this paper that the fundamental properties of classical principal bundles can be generalised to the setting of this ‘higher graded’ geometry, with properly defined frame bundles of <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-vector bundles as canonical examples. Additionally, we propose a new approach to the concept of a vector bundle in this setting. However, formulating these ideas and proving these results relies on many technical upshots established in earlier papers. A comprehensive introduction to <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-manifolds is therefore included together with basic examples.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"100 ","pages":"Article 102269"},"PeriodicalIF":0.6,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579541","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}

{"title":"The Rees algebra and analytic spread of a divisorial filtration","authors":"Steven Dale Cutkosky","doi":"10.1016/j.aim.2025.110428","DOIUrl":"10.1016/j.aim.2025.110428","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper we investigate some properties of Rees algebras of divisorial filtrations and their analytic spread. A classical theorem of McAdam shows that the analytic spread of an ideal &lt;em&gt;I&lt;/em&gt; in a formally equidimensional local ring is equal to the dimension of the ring if and only if the maximal ideal is an associated prime of &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; for some &lt;em&gt;n&lt;/em&gt;. We show in Theorem 1.5 that McAdam's theorem holds for &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtrations in an equidimensional local ring which is essentially of finite type over an excellent local ring of dimension less than or equal to 3. This generalizes an earlier result for &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtrations in an equicharacteristic zero excellent local domain by the author. This theorem does not hold for more general filtrations.&lt;/div&gt;&lt;div&gt;We consider the question of the asymptotic behavior of the function &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;↦&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for a &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtration &lt;span&gt;&lt;math&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-primary ideals on a &lt;em&gt;d&lt;/em&gt;-dimensional normal excellent local ring. It is known from earlier work of the author that the multiplicity&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;munder&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; can be irrational. We show in Lemma 4.1 that the limsup of the first difference function&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;sup&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is always finite for a &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtration. We then give an example in Section 4 showing that this limsup may not exist as a limit.&lt;/div&gt;&lt;div&gt;In the final section, we","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110428"},"PeriodicalIF":1.5,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144581248","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":"New constructions of permutation polynomials of the form x+γTrqq3(h(x)) over finite fields with even characteristic","authors":"Sha Jiang,&nbsp;Kangquan Li,&nbsp;Longjiang Qu","doi":"10.1016/j.ffa.2025.102694","DOIUrl":"10.1016/j.ffa.2025.102694","url":null,"abstract":"<div><div>Permutation polynomials over finite fields have applications in many areas of mathematics and engineering. Particularly, permutation polynomials of the form <span><math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> have been studied for a long time. In this paper, we further investigate permutation polynomials of the form <span><math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> over finite fields with even characteristic. For one thing, by choosing functions <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> with a low <em>q</em>-degree, we propose four classes of permutation polynomials of the form <span><math><mi>x</mi><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span>. For the other thing, we give seven classes of permutations of the form <span><math><mi>x</mi><mo>+</mo><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>q</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msubsup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> with binomials <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span>. Finally, we also show that the permutation polynomials constructed in this paper are not quasi-multiplicative equivalent to the known permutation polynomials.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102694"},"PeriodicalIF":1.2,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579385","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":"Inverse scattering problem for operators with a finite-dimensional non-local potential","authors":"V.A. Zolotarev","doi":"10.1016/j.jde.2025.113606","DOIUrl":"10.1016/j.jde.2025.113606","url":null,"abstract":"<div><div>Scattering problem for a self-adjoint integro-differential operator, which is the sum of the operator of the second derivative and of a finite-dimensional self-adjoint operator, is studied. Jost solutions are found and it is shown that the scattering function has a multiplicative structure, besides, each of the multipliers is a scattering coefficient for a pair of self-adjoint operators, one of which is a one-dimensional perturbation of the other. Solution of the inverse problem is based upon the solutions to the inverse problem for every multiplier. A technique for finding parameters of the finite-dimensional perturbation via the scattering data is described.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"446 ","pages":"Article 113606"},"PeriodicalIF":2.4,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579420","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}