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

{"title":"The quantization of Maxwell theory in the Cauchy radiation gauge: Hodge decomposition and Hadamard states","authors":"Simone Murro,&nbsp;Gabriel Schmid","doi":"10.1112/jlms.70020","DOIUrl":"https://doi.org/10.1112/jlms.70020","url":null,"abstract":"<p>The aim of this paper is to prove the existence of Hadamard states for the Maxwell equations on any globally hyperbolic spacetime. This will be achieved by introducing a new gauge fixing condition, the <i>Cauchy radiation gauge</i>, that will allow to suppress all the unphysical degrees of freedom. The key ingredient for achieving this gauge is a new Hodge decomposition for differential <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>-forms in Sobolev spaces on complete (possibly noncompact) Riemannian manifolds.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70020","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142598999","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":"The scaling limit of random cubic planar graphs","authors":"Benedikt Stufler","doi":"10.1112/jlms.70018","DOIUrl":"https://doi.org/10.1112/jlms.70018","url":null,"abstract":"<p>We study the random cubic planar graph <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$mathsf {C}_n$</annotation>\u0000 </semantics></math> with an even number <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> of vertices. We show that the Brownian map arises as Gromov–Hausdorff–Prokhorov scaling limit of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$mathsf {C}_n$</annotation>\u0000 </semantics></math> as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>∈</mo>\u0000 <mn>2</mn>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$n in 2 mathbb {N}$</annotation>\u0000 </semantics></math> tends to infinity, after rescaling distances by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <msup>\u0000 <mi>n</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$gamma n^{-1/4}$</annotation>\u0000 </semantics></math> for a specific constant <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$gamma &amp;gt;0$</annotation>\u0000 </semantics></math>. This is the first time a model of random graphs that are not embedded into the plane is shown to converge to the Brownian map. Our approach features a new method that allows us to relate distances on random graphs to first-passage percolation distances on their 3-connected core.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70018","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142588052","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":"Bounds in a popular multidimensional nonlinear Roth theorem","authors":"Sarah Peluse,&nbsp;Sean Prendiville,&nbsp;Xuancheng Shao","doi":"10.1112/jlms.70019","DOIUrl":"https://doi.org/10.1112/jlms.70019","url":null,"abstract":"<p>A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form <span></span><math>\u0000 <semantics>\u0000 <mi>x</mi>\u0000 <annotation>$x$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>+</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$x+d$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>+</mo>\u0000 <msup>\u0000 <mi>d</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$x+d^2$</annotation>\u0000 </semantics></math>. We obtain a multidimensional version of this result, which can be regarded as a first step toward effectivising those cases of the multidimensional polynomial Szemerédi theorem involving polynomials with distinct degrees. In addition, we prove an effective “popular” version of this result, showing that every dense set has some non-zero <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math> such that the number of configurations with difference parameter <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math> is almost optimal. Perhaps surprisingly, the quantitative dependence in this result is exponential, compared to the tower-type bounds encountered in the popular linear Roth theorem.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142596376","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":"A factorization of the GJMS operators of special Einstein products and applications","authors":"Jeffrey S. Case,&nbsp;Andrea Malchiodi","doi":"10.1112/jlms.70023","DOIUrl":"https://doi.org/10.1112/jlms.70023","url":null,"abstract":"&lt;p&gt;We show that the GJMS operators of a special Einstein product factor as a composition of second- and fourth-order differential operators. In particular, our formula applies to the Riemannian product &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$H^{ell } times S^{d-ell }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We also show that there is an integer &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$D = D(k,ell)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$d geqslant D$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, then for any special Einstein product &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$N^ell times M^{d-ell }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, the Green's function for the GJMS operator of order &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$2k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is positive. As a result, these products give new examples of closed Riemannian manifolds for which the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$Q_{2k}$&lt;/annotation&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142574012","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 Hilbert expansion for some nonrelativistic kinetic equations","authors":"Yuanjie Lei,&nbsp;Shuangqian Liu,&nbsp;Qinghua Xiao,&nbsp;Huijiang Zhao","doi":"10.1112/jlms.70016","DOIUrl":"https://doi.org/10.1112/jlms.70016","url":null,"abstract":"<p>The Vlasov–Maxwell–Landau (VML) system and the Vlasov–Maxwell–Boltzmann (VMB) system are fundamental models in dilute collisional plasmas. In this paper, we are concerned with the hydrodynamic limits of both the VML and the noncutoff VMB systems in the entire space. Our primary objective is to rigorously prove that, within the framework of Hilbert expansion, the unique classical solution of the VML or noncutoff VMB system converges globally over time to the smooth global solution of the Euler–Maxwell system as the Knudsen number approaches zero. The core of our analysis hinges on deriving novel interplay energy estimates for the solutions of these two systems, concerning both a local Maxwellian and a global Maxwellian, respectively. Our findings address a problem in the hydrodynamic limit for Landau-type equations and noncutoff Boltzmann-type equations with a magnetic field. Furthermore, the approach developed in this paper can be seamlessly extended to assess the validity of the Hilbert expansion for other types of kinetic equations.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142574010","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":"Continuity of extensions of Lipschitz maps and of monotone maps","authors":"Krzysztof J. Ciosmak","doi":"10.1112/jlms.70014","DOIUrl":"https://doi.org/10.1112/jlms.70014","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> be a subset of a Hilbert space. We prove that if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>:</mo>\u0000 <mi>X</mi>\u0000 <mo>→</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>m</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$vcolon Xrightarrow mathbb {R}^m$</annotation>\u0000 </semantics></math> is such that\u0000\u0000 </p><p>Moreover, if either <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>∈</mo>\u0000 <mo>{</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>3</mn>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$min lbrace 1,2,3rbrace$</annotation>\u0000 </semantics></math> or <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> is convex, we prove the converse: We show that a map <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>:</mo>\u0000 <mi>X</mi>\u0000 <mo>→</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>m</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$vcolon Xrightarrow mathbb {R}^m$</annotation>\u0000 </semantics></math> that allows for a 1-Lipschitz, uniform distance preserving extension of any 1-Lipschitz map on a subset of <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> also satisfies the above set of inequalities. We also prove a similar continuity result concerning extensions of monotone maps. Our results hold true also for maps taking values in infinite-dimensional spaces.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142574067","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":"Computability in infinite Galois theory and algorithmically random algebraic fields","authors":"Wesley Calvert,&nbsp;Valentina Harizanov,&nbsp;Alexandra Shlapentokh","doi":"10.1112/jlms.70017","DOIUrl":"https://doi.org/10.1112/jlms.70017","url":null,"abstract":"<p>We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of <span></span><math>\u0000 <semantics>\u0000 <mi>Q</mi>\u0000 <annotation>${mathbb {Q}}$</annotation>\u0000 </semantics></math> that are random according to our definition. We show that there are noncomputable algebraic fields which are not random. We also partially characterize the index set, relative to an oracle, of the set of random algebraic fields computable relative to that oracle.</p><p>In order to carry out this investigation of randomness for fields, we develop computability in the context of the infinite Galois theory (where the relevant Galois groups are uncountable), including definitions of computable and computably enumerable Galois groups and computability of Haar measure on the Galois groups.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142574068","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 topology of determinantal links","authors":"Matthias Zach","doi":"10.1112/jlms.70012","DOIUrl":"https://doi.org/10.1112/jlms.70012","url":null,"abstract":"&lt;p&gt;We study sections &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;D&lt;/mi&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;∩&lt;/mo&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(D_kcap M_{m,n}^s,0)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of the generic determinantal varieties &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mi&gt;φ&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;:&lt;/mo&gt;\u0000 &lt;mo&gt;rank&lt;/mo&gt;\u0000 &lt;mi&gt;φ&lt;/mi&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$M_{m,n}^s = lbrace varphi in mathbb {C}^{mtimes n}: operatorname{rank}varphi &amp;lt;s rbrace$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; by generic hyperplanes &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$D_k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of various codimensions &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;, the polar multiplicities of these sections, and the cohomology of their real and complex links. Such complex links were shown to provide the basic building blocks in a bouquet decomposition for the (determinantal) smoothings of smoothable isolated determinantal singularities. The detailed vanishing topology of such singularities was still not fully understood beyond isolated complete intersections and a ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142574043","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":"Abundance: Asymmetric graph removal lemmas and integer solutions to linear equations","authors":"António Girão,&nbsp;Eoin Hurley,&nbsp;Freddie Illingworth,&nbsp;Lukas Michel","doi":"10.1112/jlms.70015","DOIUrl":"https://doi.org/10.1112/jlms.70015","url":null,"abstract":"&lt;p&gt;We prove that a large family of pairs of graphs satisfy a polynomial dependence in asymmetric graph removal lemmas. In particular, we give an unexpected answer to a question of Gishboliner, Shapira and Wigderson by showing that for every &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$t geqslant 4$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, there are &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$K_t$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-abundant graphs of chromatic number &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;annotation&gt;$t$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Using similar methods, we also extend work of Ruzsa by proving that 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;mo&gt;{&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&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;N&lt;/mi&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {A}subset lbrace 1,dots,N rbrace$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; which avoids solutions with distinct integers to an equation of genus at least two has size &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;O&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msqrt&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msqrt&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {O}(sqrt {N})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. The best previous bound was &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;o&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;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$N^{1 - o(1)}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and the exponent of &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;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$1/2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is best possible in such a result. Finally, we investigate the relationship between polynomial dependencies in asymmetric removal lemmas and","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142574044","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 power of quantum entanglement in multipartite quantum XOR games","authors":"Marius Junge,&nbsp;Carlos Palazuelos","doi":"10.1112/jlms.70009","DOIUrl":"https://doi.org/10.1112/jlms.70009","url":null,"abstract":"<p>We show that, given <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$kgeqslant 3$</annotation>\u0000 </semantics></math>, there exist <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>-player quantum XOR games for which the entangled bias can be arbitrarily larger than the bias of the game when the players are restricted to separable strategies. In particular, quantum entanglement can be a much more powerful resource than local operations and classical communication to play these games. This result shows a strong contrast to the bipartite case, where it was recently proved that, as a consequence of a noncommutative version of Grothendieck theorem, the entangled bias is always upper bounded by a universal constant times the one-way classical communication bias. In this sense, our main result can be understood as a counterexample to an extension of such a noncommutative Grothendieck theorem to multilinear forms.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142525447","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}