is difficult to solve completely even if (n=2,3), in which the certain type of the above equation is also interesting and significant. In this paper, we first to consider the Bi-Fermat type quadratic partial differential equation
{"title":"Meromorphic solutions of Bi-Fermat type partial differential and difference equations","authors":"Yingchun Gao, Kai Liu","doi":"10.1007/s13324-024-00989-w","DOIUrl":"10.1007/s13324-024-00989-w","url":null,"abstract":"<div><p>Fermat type functional equation with four terms </p><div><div><span>$$begin{aligned} f(z)^{n}+g(z)^{n}+h(z)^{n}+k(z)^{n}=1 end{aligned}$$</span></div></div><p>is difficult to solve completely even if <span>(n=2,3)</span>, in which the certain type of the above equation is also interesting and significant. In this paper, we first to consider the Bi-Fermat type quadratic partial differential equation </p><div><div><span>$$begin{aligned} f(z_{1},z_{2})^{2}+left( frac{partial f(z_{1},z_{2})}{partial z_{1}}right) ^{2}+g(z_{1},z_{2})^{2}+left( frac{partial g(z_{1},z_{2})}{partial z_{1}}right) ^{2}=1 end{aligned}$$</span></div></div><p>in <span>(mathbb {C}^{2})</span>. In addition, we consider the Bi-Fermat type cubic difference equation </p><div><div><span>$$begin{aligned} f(z)^{3}+g(z)^{3}+f(z+c)^{3}+g(z+c)^{3}=1 end{aligned}$$</span></div></div><p>in <span>(mathbb {C})</span> and obtain partial meromorphic solutions on the above equation.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 6","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142645697","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}
Pub Date : 2024-11-16DOI: 10.1016/j.na.2024.113699
Philipp Reiser
We give new examples of manifolds that appear as cross sections of tangent cones of non-collapsed Ricci limit spaces. It was shown by Colding–Naber that the homeomorphism types of the tangent cones of a fixed point of such a space do not need to be unique. In fact, they constructed an example in dimension 5 where two different homeomorphism types appear at the same point. In this note, we extend this result and construct limit spaces in all dimensions at least 5 where any finite collection of manifolds that admit core metrics, a type of metric introduced by Perelman and Burdick to study Riemannian metrics of positive Ricci curvature on connected sums, can appear as cross sections of tangent cones of the same point.
{"title":"Examples of tangent cones of non-collapsed Ricci limit spaces","authors":"Philipp Reiser","doi":"10.1016/j.na.2024.113699","DOIUrl":"10.1016/j.na.2024.113699","url":null,"abstract":"<div><div>We give new examples of manifolds that appear as cross sections of tangent cones of non-collapsed Ricci limit spaces. It was shown by Colding–Naber that the homeomorphism types of the tangent cones of a fixed point of such a space do not need to be unique. In fact, they constructed an example in dimension 5 where two different homeomorphism types appear at the same point. In this note, we extend this result and construct limit spaces in all dimensions at least 5 where any finite collection of manifolds that admit <em>core metrics</em>, a type of metric introduced by Perelman and Burdick to study Riemannian metrics of positive Ricci curvature on connected sums, can appear as cross sections of tangent cones of the same point.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"252 ","pages":"Article 113699"},"PeriodicalIF":1.3,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142656316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-16DOI: 10.1016/j.cnsns.2024.108446
Conggang Liang, Dongyang Shi
The main aim of this paper is to propose a 2-step backward differential formula (BDF2) fully discrete scheme with the bilinear Q11 finite element method (FEM) for the nonlinear reaction–diffusion equation. By use of the combination technique of the element’s interpolation and Ritz projection, and through the interpolation post-processing approach, the superclose and global superconvergence estimates with order O(h2+τ2) in H1-norm are deduced rigorously. Furthermore, with the help of the asymptotic error expansion of the Q11 element, a new suitable fully discrete scheme is developed, and the extrapolation result of order O(h3+τ2) in H1-norm is derived, which is one order higher than that of the above traditional superconvergence estimate with respect to h. Here h is the mesh size and τ is the time step. Finally, some numerical results are provided to verify the theoretical analysis. It seems that the extrapolation of the fully discrete finite element scheme has never been seen in the previous studies.
{"title":"Superconvergence analysis and extrapolation of a BDF2 fully discrete scheme for nonlinear reaction–diffusion equations","authors":"Conggang Liang, Dongyang Shi","doi":"10.1016/j.cnsns.2024.108446","DOIUrl":"https://doi.org/10.1016/j.cnsns.2024.108446","url":null,"abstract":"The main aim of this paper is to propose a 2-step backward differential formula (BDF2) fully discrete scheme with the bilinear <mml:math altimg=\"si1.svg\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub></mml:math> finite element method (FEM) for the nonlinear reaction–diffusion equation. By use of the combination technique of the element’s interpolation and Ritz projection, and through the interpolation post-processing approach, the superclose and global superconvergence estimates with order <mml:math altimg=\"si2.svg\" display=\"inline\"><mml:mrow><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>τ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> in <mml:math altimg=\"si3.svg\" display=\"inline\"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>-norm are deduced rigorously. Furthermore, with the help of the asymptotic error expansion of the <mml:math altimg=\"si1.svg\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub></mml:math> element, a new suitable fully discrete scheme is developed, and the extrapolation result of order <mml:math altimg=\"si5.svg\" display=\"inline\"><mml:mrow><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>τ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> in <mml:math altimg=\"si3.svg\" display=\"inline\"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>-norm is derived, which is one order higher than that of the above traditional superconvergence estimate with respect to <mml:math altimg=\"si7.svg\" display=\"inline\"><mml:mi>h</mml:mi></mml:math>. Here <mml:math altimg=\"si7.svg\" display=\"inline\"><mml:mi>h</mml:mi></mml:math> is the mesh size and <mml:math altimg=\"si9.svg\" display=\"inline\"><mml:mi>τ</mml:mi></mml:math> is the time step. Finally, some numerical results are provided to verify the theoretical analysis. It seems that the extrapolation of the fully discrete finite element scheme has never been seen in the previous studies.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"71 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679156","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}
We consider the patch problem for the ‐(surface quasi‐geostrophic) SQG system with the values and being the 2D Euler and the SQG equations respectively. It is well‐known that the Euler patches are globally wellposed in non‐endpoint Hölder spaces, as well as in , spaces. In stark contrast to the Euler case, we prove that for , the ‐SQG patch problem is strongly illposed in every Hölder space with . Moreover, in a suitable range of regularity, the same strong illposedness holds for every Sobolev space unless .
{"title":"The α$alpha$‐SQG patch problem is illposed in C2,β$C^{2,beta }$ and W2,p$W^{2,p}$","authors":"Alexander Kiselev, Xiaoyutao Luo","doi":"10.1002/cpa.22236","DOIUrl":"https://doi.org/10.1002/cpa.22236","url":null,"abstract":"We consider the patch problem for the ‐(surface quasi‐geostrophic) SQG system with the values and being the 2D Euler and the SQG equations respectively. It is well‐known that the Euler patches are globally wellposed in non‐endpoint Hölder spaces, as well as in , spaces. In stark contrast to the Euler case, we prove that for , the ‐SQG patch problem is strongly illposed in <jats:italic>every</jats:italic> Hölder space with . Moreover, in a suitable range of regularity, the same strong illposedness holds for <jats:italic>every</jats:italic> Sobolev space unless .","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"197 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142645950","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}
Pub Date : 2024-11-16DOI: 10.1016/j.dam.2024.11.002
Menglu He, Zemin Jin
An edge-colored graph is called rainbow if no two edges of have the same color. For a graph and a subgraph , the anti-Ramsey number is the maximum number of colors in an edge-coloring of such that contains no rainbow copy of . Recently, the anti-Ramsey problem for disjoint union of graphs received much attention. In particular, several researchers focused on the problem for graphs consisting of small components. In this paper, we continue the work in this direction. We refine the bound and obtain the precise value of for all . Additionally, we determine the value of for any integers and .
如果一个边着色的图 G 没有两条边的颜色相同，则称其为彩虹图。对于图 G 和子图 H⊆G，反拉姆齐数 AR(G,H) 是指在 G 的边染色中，G 不包含 H 的彩虹副本的最大颜色数。特别是，一些研究人员重点研究了由小分量组成的图的反拉姆齐问题。在本文中，我们将继续这一方向的研究。我们完善了边界，并得到了所有 n≥2t+3 时 AR(Kn,P3∪tP2) 的精确值。此外，我们还确定了任意整数 t≥1 和 n≥2t+7 时的 AR(Kn,2P3∪tP2) 值。
{"title":"Rainbow short linear forests in edge-colored complete graph","authors":"Menglu He, Zemin Jin","doi":"10.1016/j.dam.2024.11.002","DOIUrl":"10.1016/j.dam.2024.11.002","url":null,"abstract":"<div><div>An edge-colored graph <span><math><mi>G</mi></math></span> is called rainbow if no two edges of <span><math><mi>G</mi></math></span> have the same color. For a graph <span><math><mi>G</mi></math></span> and a subgraph <span><math><mrow><mi>H</mi><mo>⊆</mo><mi>G</mi></mrow></math></span>, the anti-Ramsey number <span><math><mrow><mi>A</mi><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is the maximum number of colors in an edge-coloring of <span><math><mi>G</mi></math></span> such that <span><math><mi>G</mi></math></span> contains no rainbow copy of <span><math><mi>H</mi></math></span>. Recently, the anti-Ramsey problem for disjoint union of graphs received much attention. In particular, several researchers focused on the problem for graphs consisting of small components. In this paper, we continue the work in this direction. We refine the bound and obtain the precise value of <span><math><mrow><mi>A</mi><mi>R</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>∪</mo><mi>t</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>3</mn></mrow></math></span>. Additionally, we determine the value of <span><math><mrow><mi>A</mi><mi>R</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mn>2</mn><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>∪</mo><mi>t</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> for any integers <span><math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>7</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"361 ","pages":"Pages 523-536"},"PeriodicalIF":1.0,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142650685","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}
Pub Date : 2024-11-16DOI: 10.1016/j.dam.2024.11.013
Shaohan Xu, Kexiang Xu
<div><div>Let <span><math><mi>H</mi></math></span> be a graph with vertex set <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow></math></span>. The generalized join graph <span><math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> is obtained from <span><math><mi>H</mi></math></span> by replacing each vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> with a graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and joining each vertex in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> with each vertex in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> provided <span><math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>. If every <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is an independent set of <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> vertices, then we write <span><math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> as <span><math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>, which is called the blow-up of <span><math><mi>H</mi></math></span>. In this paper we introduce the local complement transformation in electrical networks and obtain an electrically equivalent graph of <span><math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>. As their applications, we obtain formulae for resistance distances of some vertex-weighted graphs and give a unified technique to compute resistance distances in <span><math><mrow><mi>H</mi>
设 H 是顶点集 V(H)={v1,v2,...,vk} 的图。将每个顶点 vi 替换为一个图 Gi，并将 Gi 中的每个顶点与 Gj 中的每个顶点连接（条件是 vivj∈E(H)），就得到广义连接图 H[G1,G2,...,Gk]。如果每个 Gi 都是 ni 个顶点的独立集合，那么我们就把 H[G1,G2,...,Gk] 写成 H[n1,n2,...,nk]，这就是所谓的 H 放大图。作为其应用，我们得到了一些顶点加权图的电阻距离公式，并给出了一种统一的技术来计算 H[G1,G2,...Gk] 中的电阻距离，当 1≤i≤k 的每个 Gi 都是由匹配顶点和孤立顶点组成的图时，从而得到了当 H 是一些给定图时 H[G1,G2,...Gk] 的电阻距离的封闭公式。
{"title":"Resistance distances in generalized join graphs","authors":"Shaohan Xu, Kexiang Xu","doi":"10.1016/j.dam.2024.11.013","DOIUrl":"10.1016/j.dam.2024.11.013","url":null,"abstract":"<div><div>Let <span><math><mi>H</mi></math></span> be a graph with vertex set <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow></math></span>. The generalized join graph <span><math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> is obtained from <span><math><mi>H</mi></math></span> by replacing each vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> with a graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and joining each vertex in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> with each vertex in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> provided <span><math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>. If every <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is an independent set of <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> vertices, then we write <span><math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> as <span><math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>, which is called the blow-up of <span><math><mi>H</mi></math></span>. In this paper we introduce the local complement transformation in electrical networks and obtain an electrically equivalent graph of <span><math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>. As their applications, we obtain formulae for resistance distances of some vertex-weighted graphs and give a unified technique to compute resistance distances in <span><math><mrow><mi>H</mi>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 18-33"},"PeriodicalIF":1.0,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653071","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}
Pub Date : 2024-11-16DOI: 10.1016/j.chaos.2024.115737
Rami Ahmad El-Nabulsi
Nonlinear partial differential equations admitting traveling wave solutions play an important role in the description and analysis of real-life physical processes and nonlinear phenomena. In this study, we prove that the excitable reaction-diffusion-convection system introduced by Kopell and Howard can exhibit, in fractal dimensions, a large variety of spatial patterns. We have considered two independent models: a local reaction-diffusion-convection model characterized by variable coefficients that are subject to particular power laws and a nonlocal reaction-diffusion model characterized by symmetric kernels and a variable diffusion coefficient. Each model is characterized by a number of motivating properties and features. In the 1st model, the amplitude is governed by a 2nd-order differential equation, whereas in the 2nd-model, the amplitude is governed by a 4th-order differential equation, which is, under some conditions, comparable to the Swift-Hohenberg equation with variable coefficients that arise in the study of pattern formation, which belongs to the family of extended Fisher-Kolmogorov stationary equations used to study pattern-forming systems in biological and chemical systems. We report the emergence of superstructures that are suppressed for fractal dimensions much less than unity. These superstructures include superspiral waves characterized by a circular symmetry detected in various oscillatory media and the emergence of reflection of waves that take place in non-uniform reaction-diffusion systems, besides the emergence of micro-spiral waves that emerge at the cellular level. A transition from spiral waves to perfectly rotating waves is observed, besides a transition from Mexican hat shaped solutions to upside-down Mexican hat shaped solutions. The domain size has a very strong impact on the rotational frequency of spiral and circular waves. These new phenomena associated with configuration patterns through a reaction-diffusion-convection system with different scales and characterized by variable coefficients can be applied for modeling a wide class of reaction-diffusion-convection problems. Supplementary properties have been obtained and discussed accordingly.
{"title":"Transition from circular to spiral waves and from Mexican hat to upside-down Mexican hat-solutions: The cases of local and nonlocal λ−ω reaction-diffusion-convection fractal systems with variable coefficients","authors":"Rami Ahmad El-Nabulsi","doi":"10.1016/j.chaos.2024.115737","DOIUrl":"10.1016/j.chaos.2024.115737","url":null,"abstract":"<div><div>Nonlinear partial differential equations admitting traveling wave solutions play an important role in the description and analysis of real-life physical processes and nonlinear phenomena. In this study, we prove that the excitable <span><math><mi>λ</mi><mo>−</mo><mi>ω</mi></math></span>reaction-diffusion-convection system introduced by Kopell and Howard can exhibit, in fractal dimensions, a large variety of spatial patterns. We have considered two independent models: a local reaction-diffusion-convection model characterized by variable coefficients that are subject to particular power laws and a nonlocal reaction-diffusion model characterized by symmetric kernels and a variable diffusion coefficient. Each model is characterized by a number of motivating properties and features. In the 1st model, the amplitude is governed by a 2nd-order differential equation, whereas in the 2nd-model, the amplitude is governed by a 4th-order differential equation, which is, under some conditions, comparable to the Swift-Hohenberg equation with variable coefficients that arise in the study of pattern formation, which belongs to the family of extended Fisher-Kolmogorov stationary equations used to study pattern-forming systems in biological and chemical systems. We report the emergence of superstructures that are suppressed for fractal dimensions much less than unity. These superstructures include superspiral waves characterized by a circular symmetry detected in various oscillatory media and the emergence of reflection of waves that take place in non-uniform reaction-diffusion systems, besides the emergence of micro-spiral waves that emerge at the cellular level. A transition from spiral waves to perfectly rotating waves is observed, besides a transition from Mexican hat shaped solutions to upside-down Mexican hat shaped solutions. The domain size has a very strong impact on the rotational frequency of spiral and circular waves. These new phenomena associated with configuration patterns through a reaction-diffusion-convection system with different scales and characterized by variable coefficients can be applied for modeling a wide class of reaction-diffusion-convection problems. Supplementary properties have been obtained and discussed accordingly.</div></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"189 ","pages":"Article 115737"},"PeriodicalIF":5.3,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142650720","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}
Pub Date : 2024-11-16DOI: 10.1016/j.dam.2024.11.010
Cui-Fang Sun, Zhi Cheng
<div><div>For any positive integer <span><math><mi>m</mi></math></span>, let <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be the set of residue classes modulo <span><math><mi>m</mi></math></span>. For <span><math><mrow><mi>A</mi><mo>⊆</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> and <span><math><mrow><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span>, let the representation function <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span> denote the number of solutions of the equation <span><math><mrow><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>=</mo><mover><mrow><mi>a</mi></mrow><mo>¯</mo></mover><mo>+</mo><mover><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mo>¯</mo></mover></mrow></math></span> with unordered pairs <span><math><mrow><mrow><mo>(</mo><mover><mrow><mi>a</mi></mrow><mo>¯</mo></mover><mo>,</mo><mover><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mo>¯</mo></mover><mo>)</mo></mrow><mo>∈</mo><mi>A</mi><mo>×</mo><mi>A</mi></mrow></math></span>. Let <span><math><mrow><mi>m</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>α</mi></mrow></msup><mi>M</mi></mrow></math></span>, where <span><math><mi>α</mi></math></span> is a nonnegative integer and <span><math><mi>M</mi></math></span> is a positive odd integer. In this paper, we prove that if <span><math><mrow><mi>M</mi><mo>=</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mn>2</mn><mo>∤</mo><mi>α</mi></mrow></math></span>, then there exist two distinct sets <span><math><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊆</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> with <span><math><mrow><mi>A</mi><mo>∪</mo><mi>B</mi><mo>=</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>∖</mo><mrow><mo>{</mo><mover><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>¯</mo></mover><mo>}</mo></mrow><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>=</mo><mrow><mo>{</mo><mover><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>¯</mo></mover><mo>}</mo></mrow></mrow></math></span> such that <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>B</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span>. We also prove that if <span><math><mrow><mi>M</mi><mo>≥</mo><mn>3</mn></mrow></math>
{"title":"Partitions of Zm with identical representation functions","authors":"Cui-Fang Sun, Zhi Cheng","doi":"10.1016/j.dam.2024.11.010","DOIUrl":"10.1016/j.dam.2024.11.010","url":null,"abstract":"<div><div>For any positive integer <span><math><mi>m</mi></math></span>, let <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be the set of residue classes modulo <span><math><mi>m</mi></math></span>. For <span><math><mrow><mi>A</mi><mo>⊆</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> and <span><math><mrow><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span>, let the representation function <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span> denote the number of solutions of the equation <span><math><mrow><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>=</mo><mover><mrow><mi>a</mi></mrow><mo>¯</mo></mover><mo>+</mo><mover><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mo>¯</mo></mover></mrow></math></span> with unordered pairs <span><math><mrow><mrow><mo>(</mo><mover><mrow><mi>a</mi></mrow><mo>¯</mo></mover><mo>,</mo><mover><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mo>¯</mo></mover><mo>)</mo></mrow><mo>∈</mo><mi>A</mi><mo>×</mo><mi>A</mi></mrow></math></span>. Let <span><math><mrow><mi>m</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>α</mi></mrow></msup><mi>M</mi></mrow></math></span>, where <span><math><mi>α</mi></math></span> is a nonnegative integer and <span><math><mi>M</mi></math></span> is a positive odd integer. In this paper, we prove that if <span><math><mrow><mi>M</mi><mo>=</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mn>2</mn><mo>∤</mo><mi>α</mi></mrow></math></span>, then there exist two distinct sets <span><math><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊆</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> with <span><math><mrow><mi>A</mi><mo>∪</mo><mi>B</mi><mo>=</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>∖</mo><mrow><mo>{</mo><mover><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>¯</mo></mover><mo>}</mo></mrow><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>=</mo><mrow><mo>{</mo><mover><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>¯</mo></mover><mo>}</mo></mrow></mrow></math></span> such that <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>B</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span>. We also prove that if <span><math><mrow><mi>M</mi><mo>≥</mo><mn>3</mn></mrow></math>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 1-10"},"PeriodicalIF":1.0,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653073","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}
This paper deals with the derivation of the mean‐field limit for multi‐agent systems on a large class of sparse graphs. More specifically, the case of non‐exchangeable multi‐agent systems consisting of non‐identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect the structure of the connectivities in the network and has critical effects on the collective dynamics. In this article some of the main restrictive hypothesis in the previous literature on the connectivities between the agents (dense graphs) and the cooperation between them (symmetric interactions) are removed.
{"title":"Mean‐field limit of non‐exchangeable systems","authors":"Pierre‐Emmanuel Jabin, David Poyato, Juan Soler","doi":"10.1002/cpa.22235","DOIUrl":"https://doi.org/10.1002/cpa.22235","url":null,"abstract":"This paper deals with the derivation of the mean‐field limit for multi‐agent systems on a large class of sparse graphs. More specifically, the case of non‐exchangeable multi‐agent systems consisting of non‐identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect the structure of the connectivities in the network and has critical effects on the collective dynamics. In this article some of the main restrictive hypothesis in the previous literature on the connectivities between the agents (dense graphs) and the cooperation between them (symmetric interactions) are removed.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"248 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142645951","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}
Pub Date : 2024-11-16DOI: 10.1016/j.nonrwa.2024.104260
Duong Xuan Vinh , Mai Duc Thanh , Nguyen Huu Hiep
Undercompressive phase transitions violating Lax shock inequalities in a model of fluid flows in a nozzle with discontinuous cross-section area are studied. The Riemann problem involving phase transitions is considered. Depending on the choice of admissibility criteria suitable for a specific application, one can obtain a Riemann solver, which may involve nonclassical shock wave. The resonance phenomenon is also observed as multiple shocks waves of the same speed can apparently appear in a single solution. The Riemann problem may admit a unique solution in some region, but may have up to three distinct solutions in other regions.
{"title":"Undercompressive phase transitions for the model of fluid flows in a nozzle with discontinuous cross-sectional area","authors":"Duong Xuan Vinh , Mai Duc Thanh , Nguyen Huu Hiep","doi":"10.1016/j.nonrwa.2024.104260","DOIUrl":"10.1016/j.nonrwa.2024.104260","url":null,"abstract":"<div><div>Undercompressive phase transitions violating Lax shock inequalities in a model of fluid flows in a nozzle with discontinuous cross-section area are studied. The Riemann problem involving phase transitions is considered. Depending on the choice of admissibility criteria suitable for a specific application, one can obtain a Riemann solver, which may involve nonclassical shock wave. The resonance phenomenon is also observed as multiple shocks waves of the same speed can apparently appear in a single solution. The Riemann problem may admit a unique solution in some region, but may have up to three distinct solutions in other regions.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"82 ","pages":"Article 104260"},"PeriodicalIF":1.8,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659195","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}
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