Journal of the London Mathematical Society-Second Series最新文献

{"title":"CAT(0) and cubulated Shephard groups","authors":"Katherine M. Goldman","doi":"10.1112/jlms.70050","DOIUrl":"https://doi.org/10.1112/jlms.70050","url":null,"abstract":"<p>Shephard groups are common generalizations of Coxeter groups, Artin groups, and graph products of cyclic groups. Their definition is similar to that of a Coxeter group, but generators may have arbitrary order rather than strictly order 2. We extend a well-known result that Coxeter groups are <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>CAT</mi>\u0000 <mo>(</mo>\u0000 <mn>0</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathrm{CAT}(0)$</annotation>\u0000 </semantics></math> to a class of Shephard groups that have ‘enough’ finite parabolic subgroups. We also show that in this setting, if the associated Coxeter group is type (FC), then the Shephard group acts properly and cocompactly on a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>CAT</mi>\u0000 <mo>(</mo>\u0000 <mn>0</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathrm{CAT}(0)$</annotation>\u0000 </semantics></math> cube complex. As part of our proof of the former result, we introduce a new criteria for a complex made of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mn>3</mn>\u0000 </msub>\u0000 <annotation>$A_3$</annotation>\u0000 </semantics></math> simplices to be <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>CAT</mi>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathrm{CAT}(1)$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70050","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861574","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}

{"title":"Multi-indice \u0000 \u0000 B\u0000 $B$\u0000 -series","authors":"Yvain Bruned,&nbsp;Kurusch Ebrahimi-Fard,&nbsp;Yingtong Hou","doi":"10.1112/jlms.70049","DOIUrl":"https://doi.org/10.1112/jlms.70049","url":null,"abstract":"<p>We propose a novel way to study numerical methods for ordinary differential equations in one dimension via the notion of multi-indice. The main idea is to replace rooted trees in Butcher's <span></span><math>\u0000 <semantics>\u0000 <mi>B</mi>\u0000 <annotation>$B$</annotation>\u0000 </semantics></math>-series by multi-indices. The latter were introduced recently in the context of describing solutions of singular stochastic partial differential equations. The combinatorial shift away from rooted trees allows for a compressed description of numerical schemes. Furthermore, such multi-indices <span></span><math>\u0000 <semantics>\u0000 <mi>B</mi>\u0000 <annotation>$B$</annotation>\u0000 </semantics></math>-series uniquely characterize the Taylor expansion of 1-dimensional local and affine equivariant maps.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861491","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":"The structure and density of \u0000 \u0000 k\u0000 $k$\u0000 -product-free sets in the free semigroup and group","authors":"Freddie Illingworth,&nbsp;Lukas Michel,&nbsp;Alex Scott","doi":"10.1112/jlms.70046","DOIUrl":"https://doi.org/10.1112/jlms.70046","url":null,"abstract":"&lt;p&gt;The free semigroup &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {F}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; on a finite alphabet &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is the set of all finite words with letters from &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; equipped with the operation of concatenation. A subset &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;annotation&gt;$S$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {F}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;annotation&gt;$k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-product-free if no element of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;annotation&gt;$S$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; can be obtained by concatenating &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;annotation&gt;$k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; words from &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;annotation&gt;$S$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, and strongly &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;annotation&gt;$k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-product-free if no element of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;annotation&gt;$S$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a (non-trivial) concatenation of at most &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;annotation&gt;$k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; words from &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;annotation&gt;$S$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We prove that a &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;annotation&gt;$k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-product-free subset of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {F}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; has upper Banach density at most &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861259","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}

{"title":"Chow rings of matroids as permutation representations","authors":"Robert Angarone,&nbsp;Anastasia Nathanson,&nbsp;Victor Reiner","doi":"10.1112/jlms.70039","DOIUrl":"https://doi.org/10.1112/jlms.70039","url":null,"abstract":"<p>Given a matroid with a symmetry group, we study the induced group action on the Chow ring of the matroid with respect to symmetric building sets. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincaré duality and the Hard Lefschetz theorem. We lift these to statements about this permutation action, and suggest further conjectures in this vein.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70039","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861228","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}

{"title":"Convergence and nonconvergence in a nonlocal gradient flow","authors":"Sangmin Park,&nbsp;Robert L. Pego","doi":"10.1112/jlms.70047","DOIUrl":"https://doi.org/10.1112/jlms.70047","url":null,"abstract":"<p>We study the asymptotic convergence as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$trightarrow infty$</annotation>\u0000 </semantics></math> of solutions of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>∂</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <mi>u</mi>\u0000 <mo>=</mo>\u0000 <mo>−</mo>\u0000 <mi>f</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>u</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>+</mo>\u0000 <mo>∫</mo>\u0000 <mi>f</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>u</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$partial _t u=-f(u)+int f(u)$</annotation>\u0000 </semantics></math>, a nonlocal differential equation that is formally a gradient flow in a constant-mass subspace of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$L^2$</annotation>\u0000 </semantics></math> arising from simplified models of phase transitions. In case the solution takes finitely many values, we provide a new proof of stabilization that uses a Łojasiewicz-type gradient inequality near a degenerate curve of equilibria. Solutions with infinitely many values in general <i>need not</i> converge to equilibrium, however, which we demonstrate by providing counterexamples for piecewise linear and cubic functions <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math>. Curiously, the exponential <i>rate</i> of convergence in the finite-value case can jump from order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$O(1)$</annotation>\u0000 </semantics></math> to arbitrarily small values upon perturbation of parameters.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70047","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860813","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}

{"title":"Vanishing of Brauer classes on K3 surfaces under reduction","authors":"Davesh Maulik,&nbsp;Salim Tayou","doi":"10.1112/jlms.70051","DOIUrl":"https://doi.org/10.1112/jlms.70051","url":null,"abstract":"<p>Given a Brauer class on a K3 surface defined over a number field, we prove that there exists infinitely many reductions where the Brauer class vanishes, under certain technical hypotheses, answering a question of Frei–Hassett–Várilly-Alvarado.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860814","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 some matrix counting problems","authors":"Ali Mohammadi,&nbsp;Alina Ostafe,&nbsp;Igor E. Shparlinski","doi":"10.1112/jlms.70044","DOIUrl":"https://doi.org/10.1112/jlms.70044","url":null,"abstract":"<p>We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>. In particular, in the integer case, we improve a recent bound of V. Blomer and J. Li (2022).</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860486","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 the Witt group of the punctured spectrum of a regular semilocal ring","authors":"Stefan Gille,&nbsp;Ivan Panin","doi":"10.1112/jlms.70042","DOIUrl":"https://doi.org/10.1112/jlms.70042","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;annotation&gt;$R$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a regular semilocal ring of dimension &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mn&gt;5&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$4q+1geqslant 5$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; which contains &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;annotation&gt;$frac{1}{2}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$lgeqslant 1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; the number of maximal ideals of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;annotation&gt;$R$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; which are assumed to be all of the same height, and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;U&lt;/mi&gt;\u0000 &lt;annotation&gt;$U$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; the punctured spectrum of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;annotation&gt;$R$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, that is, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;Spec&lt;/mo&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{Spec}R$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; without the maximal ideals. We show that the Witt ring &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;W&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;U&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathrm{W}(U)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;U&lt;/mi&gt;\u0000 &lt;annotation&gt;$U$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; has &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;annotation&gt;$l$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; non-trivial generators &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mtext&gt;…&lt;/mtext&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70042","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860155","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}

{"title":"On the regularity number of a finite group and other base-related invariants","authors":"Marina Anagnostopoulou-Merkouri,&nbsp;Timothy C. Burness","doi":"10.1112/jlms.70035","DOIUrl":"https://doi.org/10.1112/jlms.70035","url":null,"abstract":"&lt;p&gt;A &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;annotation&gt;$k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-tuple &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mtext&gt;…&lt;/mtext&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(H_1, ldots, H_k)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of core-free subgroups of a finite group &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is said to be regular if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; has a regular orbit on the Cartesian product &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mi&gt;⋯&lt;/mi&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$G/H_1 times cdots times G/H_k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. The regularity number of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, denoted by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$R(G)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, is the smallest positive integer &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;annotation&gt;$k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with the property that every such &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;annotation&gt;$k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-tuple is regular. In this paper, we develop some general methods for studying the regularity of subgroup tuples in arbitrary finite groups, and we determine the precise regularity number of all almost simple gro","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70035","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860154","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}

{"title":"On the independence number of sparser random Cayley graphs","authors":"Marcelo Campos,&nbsp;Gabriel Dahia,&nbsp;João Pedro Marciano","doi":"10.1112/jlms.70041","DOIUrl":"https://doi.org/10.1112/jlms.70041","url":null,"abstract":"&lt;p&gt;The Cayley sum graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$Gamma _A$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of a set &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;⊆&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$A subseteq mathbb {Z}_n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is defined to have vertex set &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$mathbb {Z}_n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and an edge between two distinct vertices &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;y&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$x, y in mathbb {Z}_n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;y&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$x + y in A$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Green and Morris proved that if the set &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$A$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-random subset of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$mathbb {Z}_n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$p = 1/2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, then the independence number of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$Gamma _A$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is asymptotically equal to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142764082","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}