{"title":"Homotopy properties of the complex of frames of a unitary space","authors":"Kevin I. Piterman, Volkmar Welker","doi":"10.1112/jlms.12978","DOIUrl":"https://doi.org/10.1112/jlms.12978","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>V</mi>\u0000 <annotation>$V$</annotation>\u0000 </semantics></math> be a finite-dimensional vector space equipped with a nondegenerate Hermitian form over a field <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>${mathbb {K}}$</annotation>\u0000 </semantics></math>. Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${mathcal {G}}(V)$</annotation>\u0000 </semantics></math> be the graph with vertex set the one-dimensional nondegenerate subspaces of <span></span><math>\u0000 <semantics>\u0000 <mi>V</mi>\u0000 <annotation>$V$</annotation>\u0000 </semantics></math> and adjacency relation given by orthogonality. We give a complete description of when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${mathcal {G}}(V)$</annotation>\u0000 </semantics></math> is connected in terms of the dimension of <span></span><math>\u0000 <semantics>\u0000 <mi>V</mi>\u0000 <annotation>$V$</annotation>\u0000 </semantics></math> and the size of the ground field <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>${mathbb {K}}$</annotation>\u0000 </semantics></math>. Furthermore, we prove that if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>dim</mo>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 <mo>></mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation>$dim (V) &gt; 4$</annotation>\u0000 </semantics></math>, then the clique complex <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${mathcal {F}}(V)$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${mathcal {G}}(V)$</annotation>\u0000 </semantics></math> is simply connected. For finite fields <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>${mathbb {K}}$</annotation>\u0000 </semantics></math>, we also compute the eigenvalues of the adjacency matrix of <span></span><math>\u0000 <semantics>\u0000","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142050539","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 finitely generated Engel branch groups","authors":"J. Moritz Petschick","doi":"10.1112/jlms.12980","DOIUrl":"https://doi.org/10.1112/jlms.12980","url":null,"abstract":"<p>We construct finitely generated Engel branch groups, answering a question of Fernández-Alcober, Noce and Tracey on the existence of such objects. In particular, the groups constructed are not nilpotent, yielding the second known class of examples of finitely generated non-nilpotent Engel groups following a construction by Golod from 1969. To do so, we exhibit groups acting on rooted trees with growing valency on which word lengths of elements are contracting very quickly under section maps. Our methods apply in principle to a wider class of iterated identities, of which the Engel words are a special case.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12980","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142050540","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":"Global classical solutions to a multidimensional radiation hydrodynamics model with symmetry and large initial data","authors":"Jing Wei, Minyi Zhang, Changjiang Zhu","doi":"10.1112/jlms.12973","DOIUrl":"https://doi.org/10.1112/jlms.12973","url":null,"abstract":"<p>As a first stage to study the global large solutions of the radiation hydrodynamics model with viscosity and thermal conductivity in the high-dimensional space, we study the problems in high dimensions with some symmetry, such as the spherically or cylindrically symmetric solutions. Specifically, we will study the global classical large solutions to the radiation hydrodynamics model with spherically or cylindrically symmetric initial data. The key point is to obtain the strict positive lower and upper bounds of the density <span></span><math>\u0000 <semantics>\u0000 <mi>ρ</mi>\u0000 <annotation>$rho$</annotation>\u0000 </semantics></math> and the lower bound of the temperature <span></span><math>\u0000 <semantics>\u0000 <mi>θ</mi>\u0000 <annotation>$theta$</annotation>\u0000 </semantics></math>. Compared with the Navier–Stokes equations, these estimates in the present paper are more complicated due to the influence of the radiation. To overcome the difficulties caused by the radiation, we construct a pointwise estimate between the radiative heat flux <span></span><math>\u0000 <semantics>\u0000 <mi>q</mi>\u0000 <annotation>$q$</annotation>\u0000 </semantics></math> and the temperature <span></span><math>\u0000 <semantics>\u0000 <mi>θ</mi>\u0000 <annotation>$theta$</annotation>\u0000 </semantics></math> by studying the boundary value problem of the corresponding ordinary differential equation. And we consider a general heat conductivity: <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>ρ</mi>\u0000 <mo>,</mo>\u0000 <mi>θ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>⩾</mo>\u0000 <mi>C</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>+</mo>\u0000 <msup>\u0000 <mi>θ</mi>\u0000 <mi>β</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$kappa (rho,theta)geqslant C(1+theta ^beta)$</annotation>\u0000 </semantics></math> if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ρ</mi>\u0000 <mo>⩽</mo>\u0000 <msub>\u0000 <mi>ρ</mi>\u0000 <mo>+</mo>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$rho leqslant rho _+$</annotation>\u0000 </semantics></math>; <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>ρ</mi>\u0000 <mo>,</mo>\u0000 <mi>θ</mi>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142045288","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":"Almost sure bounds for a weighted Steinhaus random multiplicative function","authors":"Seth Hardy","doi":"10.1112/jlms.12979","DOIUrl":"https://doi.org/10.1112/jlms.12979","url":null,"abstract":"<p>We obtain almost sure bounds for the weighted sum <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mo>∑</mo>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩽</mo>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mfrac>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <msqrt>\u0000 <mi>n</mi>\u0000 </msqrt>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation>$sum _{n leqslant t} frac{f(n)}{sqrt {n}}$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$f(n)$</annotation>\u0000 </semantics></math> is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated logarithm, giving sharp upper and lower bounds.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12979","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142045289","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":"Construction of multi-bubble blow-up solutions to the \u0000 \u0000 \u0000 L\u0000 2\u0000 \u0000 $L^2$\u0000 -critical half-wave equation","authors":"Daomin Cao, Yiming Su, Deng Zhang","doi":"10.1112/jlms.12974","DOIUrl":"https://doi.org/10.1112/jlms.12974","url":null,"abstract":"<p>This paper concerns the bubbling phenomena for the <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>-critical half-wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow-up solutions concentrating exactly at these singularities. This provides the first examples of multi-bubble solutions for the half-wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in Krieger, Lenzmann and Raphaël [Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129] for the single-bubble case, and explores the localization techniques in Cao, Su and Zhang [Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 4] and Röckner, Su and Zhang [Trans. Amer. Math. Soc., 377 (2024), no. 1, 517–588] for bubbling solutions to non-linear Schrödinger equations (NLS). However, unlike the single-bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control these interactions, as well as non-local effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half-wave operator, and find that there exists a narrow room between the orders <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>t</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>+</mo>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$|t|^{2+}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>t</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mo>−</mo>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$|t|^{3-}$</annotation>\u0000 </semantics></math> for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi-bubble non-local structure.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142021777","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}