Pub Date : 2024-08-20DOI: 10.1007/s00605-024-02005-6
Mingzhu Chen, Ilya Gorshkov, Natalia V. Maslova, Nanying Yang
If G is a finite group, then the spectrum (omega (G)) is the set of all element orders of G. The prime spectrum (pi (G)) is the set of all primes belonging to (omega (G)). A simple graph (Gamma (G)) whose vertex set is (pi (G)) and in which two distinct vertices r and s are adjacent if and only if (rs in omega (G)) is called the Gruenberg–Kegel graph or the prime graph of G. In this paper, we prove that if G is a group of even order, then the set of vertices which are non-adjacent to 2 in (Gamma (G)) forms a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg–Kegel graph of a finite group.
如果 G 是一个有限群，那么谱 (omega (G)) 是 G 的所有元素阶的集合。素数谱 (pi (G)) 是属于 (omega (G)) 的所有素数的集合。一个简单的图（(Gamma (G))）的顶点集合是（(pi (G))），并且其中两个不同的顶点 r 和 s 相邻，当且仅当（(rs in omega (G))）是且仅当（(rs in omega (G))），这个图被称为格伦伯格-凯格尔图或 G 的素数图。在本文中，我们证明了如果 G 是偶数阶群，那么在 (Gamma (G)) 中与 2 不相邻的顶点集合构成了小群的联合。此外，我们还确定了强规则图何时与有限群的格伦伯格-凯格尔图同构。
{"title":"On combinatorial properties of Gruenberg–Kegel graphs of finite groups","authors":"Mingzhu Chen, Ilya Gorshkov, Natalia V. Maslova, Nanying Yang","doi":"10.1007/s00605-024-02005-6","DOIUrl":"https://doi.org/10.1007/s00605-024-02005-6","url":null,"abstract":"<p>If <i>G</i> is a finite group, then the spectrum <span>(omega (G))</span> is the set of all element orders of <i>G</i>. The prime spectrum <span>(pi (G))</span> is the set of all primes belonging to <span>(omega (G))</span>. A simple graph <span>(Gamma (G))</span> whose vertex set is <span>(pi (G))</span> and in which two distinct vertices <i>r</i> and <i>s</i> are adjacent if and only if <span>(rs in omega (G))</span> is called the Gruenberg–Kegel graph or the prime graph of <i>G</i>. In this paper, we prove that if <i>G</i> is a group of even order, then the set of vertices which are non-adjacent to 2 in <span>(Gamma (G))</span> forms a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg–Kegel graph of a finite group.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142215254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-16DOI: 10.1007/s00605-024-02000-x
Abhishek Ghosh, Michael Ruzhansky
In this article, we address sparse bounds for a class of spectral multipliers that include oscillating multipliers on stratified Lie groups. Our results can be applied to obtain weighted bounds for general Riesz means and for solutions of dispersive equations.
{"title":"Sparse bounds for oscillating multipliers on stratified groups","authors":"Abhishek Ghosh, Michael Ruzhansky","doi":"10.1007/s00605-024-02000-x","DOIUrl":"https://doi.org/10.1007/s00605-024-02000-x","url":null,"abstract":"<p>In this article, we address sparse bounds for a class of spectral multipliers that include oscillating multipliers on stratified Lie groups. Our results can be applied to obtain weighted bounds for general Riesz means and for solutions of dispersive equations.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142215255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.1007/s00605-024-02004-7
Stefan Gerdjikov, Nikolai Nikolov
The main result of this paper is that for any norm on a complex or real n-dimensional linear space, every extremal basis satisfies inverted triangle inequality with scaling factor (2^n-1). Furthermore, the constant (2^n-1) is tight. We also prove that the norms of any two extremal bases are comparable with a factor of (2^n-1), which, intuitively, means that any two extremal bases are quantitatively equivalent with the stated tolerance.
本文的主要结果是，对于复数或实数 n 维线性空间上的任意规范，每个极值基础都满足具有缩放因子 (2^n-1) 的倒三角不等式。此外，常数 (2^n-1) 是紧密的。我们还证明了任意两个极值基的规范是以(2^n-1)的因子进行比较的，直观地说，这意味着任意两个极值基在数量上是等价的，具有所述的容差。
{"title":"Some sharp inequalities for norms in $$mathbb {R}^n$$ and $$mathbb {C}^n$$","authors":"Stefan Gerdjikov, Nikolai Nikolov","doi":"10.1007/s00605-024-02004-7","DOIUrl":"https://doi.org/10.1007/s00605-024-02004-7","url":null,"abstract":"<p>The main result of this paper is that for any norm on a complex or real <i>n</i>-dimensional linear space, every extremal basis satisfies inverted triangle inequality with scaling factor <span>(2^n-1)</span>. Furthermore, the constant <span>(2^n-1)</span> is tight. We also prove that the norms of any two extremal bases are comparable with a factor of <span>(2^n-1)</span>, which, intuitively, means that any two extremal bases are quantitatively equivalent with the stated tolerance.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"159 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142215256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-13DOI: 10.1007/s00605-024-02002-9
Yanghai Yu, Hui Wang
In this paper, we consider the Cauchy problem to the generalized Fokas–Qiao–Xia–Li/generalized Camassa–Holm-modified Camassa–Holm (gFQXL/gCH-mCH) equation, which includes the Camassa–Holm equation, the generalized Camassa–Holm equation, the Novikov equation, the Fokas–Olver–Rosenau–Qiao/Modified Camassa–Holm equation and the Fokas–Qiao–Xia–Li/Camassa–Holm-modified Camassa–Holm equation. We prove the ill-posedness for the Cauchy problem of the gFQXL/gCH-mCH equation in (B^s_{p,infty }) with (s>max {2+1/p, 5/2}) and (1le ple infty ) in the sense that the solution map to this equation starting from (u_0) is discontinuous at (t = 0) in the metric of (B^s_{p,infty }).
{"title":"Ill-posedness for the gCH-mCH equation in Besov spaces","authors":"Yanghai Yu, Hui Wang","doi":"10.1007/s00605-024-02002-9","DOIUrl":"https://doi.org/10.1007/s00605-024-02002-9","url":null,"abstract":"<p>In this paper, we consider the Cauchy problem to the generalized Fokas–Qiao–Xia–Li/generalized Camassa–Holm-modified Camassa–Holm (gFQXL/gCH-mCH) equation, which includes the Camassa–Holm equation, the generalized Camassa–Holm equation, the Novikov equation, the Fokas–Olver–Rosenau–Qiao/Modified Camassa–Holm equation and the Fokas–Qiao–Xia–Li/Camassa–Holm-modified Camassa–Holm equation. We prove the ill-posedness for the Cauchy problem of the gFQXL/gCH-mCH equation in <span>(B^s_{p,infty })</span> with <span>(s>max {2+1/p, 5/2})</span> and <span>(1le ple infty )</span> in the sense that the solution map to this equation starting from <span>(u_0)</span> is discontinuous at <span>(t = 0)</span> in the metric of <span>(B^s_{p,infty })</span>.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142215257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-10DOI: 10.1007/s00605-024-02001-w
Zhigang Li
{"title":"Stability of pseudo peakons for a new fifth order CH type equation with cubic nonlinearities","authors":"Zhigang Li","doi":"10.1007/s00605-024-02001-w","DOIUrl":"https://doi.org/10.1007/s00605-024-02001-w","url":null,"abstract":"","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"10 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141919682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-10DOI: 10.1007/s00605-024-02001-w
Zhigang Li
{"title":"Stability of pseudo peakons for a new fifth order CH type equation with cubic nonlinearities","authors":"Zhigang Li","doi":"10.1007/s00605-024-02001-w","DOIUrl":"https://doi.org/10.1007/s00605-024-02001-w","url":null,"abstract":"","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"3 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141920165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-10DOI: 10.1007/s00605-024-02003-8
Zicong Yang
{"title":"The difference of weighted composition operators on Fock spaces","authors":"Zicong Yang","doi":"10.1007/s00605-024-02003-8","DOIUrl":"https://doi.org/10.1007/s00605-024-02003-8","url":null,"abstract":"","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"14 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141920285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-16DOI: 10.1007/s00605-024-01997-5
Arup Kumar Maity, P. Ratnakumar
{"title":"Weyl multipliers for $$(L^p, L^q)$$","authors":"Arup Kumar Maity, P. Ratnakumar","doi":"10.1007/s00605-024-01997-5","DOIUrl":"https://doi.org/10.1007/s00605-024-01997-5","url":null,"abstract":"","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"5 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141642079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-10DOI: 10.1007/s00605-024-01999-3
Yongcai Ren
We establish results on the conjugacy class sizes of elements of of prime-power order in a finite group. On the way, we improve results by a number of authors.
我们建立了有限群中素幂级数元素共轭类大小的结果。在此过程中，我们改进了一些作者的结果。
{"title":"Some results on conjugacy class sizes of primary elements of a finite group","authors":"Yongcai Ren","doi":"10.1007/s00605-024-01999-3","DOIUrl":"https://doi.org/10.1007/s00605-024-01999-3","url":null,"abstract":"<p>We establish results on the conjugacy class sizes of elements of of prime-power order in a finite group. On the way, we improve results by a number of authors.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-08DOI: 10.1007/s00605-024-01998-4
Peijin Li, Yaxiang Li, Saminathan Ponnusamy
In this paper, we investigate the solutions of the weighted biharmonic differential equation (Delta big ((1-|z|^2)^{-1}Delta big ) Phi =0) in the unit disk (|z|<1), where (Delta =4frac{partial ^2}{partial zpartial overline{z}}) denotes the Laplacian. The primary aim of the paper is to establish counterparts of several important results in the classical geometric function theory for this class of mappings. The main results include Schwarz type lemma and Landau type theorem. A continuous increasing function (omega :, [0, infty )rightarrow [0, infty )) with (omega (0)=0) and (omega (t)/t) is non-increasing for (t>0) is called a fast majorant if for some (delta _0>0) and (0<delta <delta _0), the inequality
$$begin{aligned} int ^{delta }_{0}frac{omega (t)}{t}dtle Comega (delta ), end{aligned}$$
holds for some positive constant C. Then we obtain (omega )-Lipschitz continuity for the solutions to the weighted biharmonic equation, when (omega ) is a fast majorant.
本文研究了加权双谐微分方程 (Delta big ((1-|z|^2)^{-1}Delta big ) 的解。Phi =0) in the unit disk (|z|<1)，其中 (Delta =4frac{partial ^2}{partial zpartial overline{z}}) 表示拉普拉斯函数。本文的主要目的是为这一类映射建立经典几何函数理论中几个重要结果的对应关系。主要结果包括 Schwarz 型 Lemma 和 Landau 型定理。连续增函数 (omega ：(omega(0)=0)并且(omega(t)/t)对于(t>；如果对于某个 (delta _0>0) 和 (0<delta <delta _0)，不等式 $$begin{aligned} 被称为快速大数。｝int ^{delta }_{0}frac{omega (t)}{t}dtle Comega (delta ), end{aligned}$$对于某个正常数 C 成立。
{"title":"On the solutions to the weighted biharmonic equation in the unit disk","authors":"Peijin Li, Yaxiang Li, Saminathan Ponnusamy","doi":"10.1007/s00605-024-01998-4","DOIUrl":"https://doi.org/10.1007/s00605-024-01998-4","url":null,"abstract":"<p>In this paper, we investigate the solutions of the weighted biharmonic differential equation <span>(Delta big ((1-|z|^2)^{-1}Delta big ) Phi =0)</span> in the unit disk <span>(|z|<1)</span>, where <span>(Delta =4frac{partial ^2}{partial zpartial overline{z}})</span> denotes the Laplacian. The primary aim of the paper is to establish counterparts of several important results in the classical geometric function theory for this class of mappings. The main results include Schwarz type lemma and Landau type theorem. A continuous increasing function <span>(omega :, [0, infty )rightarrow [0, infty ))</span> with <span>(omega (0)=0)</span> and <span>(omega (t)/t)</span> is non-increasing for <span>(t>0)</span> is called a <i>fast majorant</i> if for some <span>(delta _0>0)</span> and <span>(0<delta <delta _0)</span>, the inequality </p><span>$$begin{aligned} int ^{delta }_{0}frac{omega (t)}{t}dtle Comega (delta ), end{aligned}$$</span><p>holds for some positive constant <i>C</i>. Then we obtain <span>(omega )</span>-Lipschitz continuity for the solutions to the weighted biharmonic equation, when <span>(omega )</span> is a fast majorant.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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