Bulletin of the London Mathematical Society最新文献

{"title":"Towards quantum hierarchy for the Gromov–Witten theory of elliptic curves","authors":"Paolo Rossi,&nbsp;Sergey Shadrin,&nbsp;Ishan Jaztar Singh","doi":"10.1112/blms.70486","DOIUrl":"https://doi.org/10.1112/blms.70486","url":null,"abstract":"<p>We construct the quantum double ramification (DR) hierarchy associated with the Gromov–Witten theory of elliptic curves. We use results of Oberdieck and Pixton on the intersection numbers of the DR cycle, the Gromov–Witten classes of the elliptic curve, and the Hodge class <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>λ</mi>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$lambda _{g-1}$</annotation>\u0000 </semantics></math>, together with vanishing results for <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>λ</mi>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$lambda _{g-2}$</annotation>\u0000 </semantics></math> to produce a closed, modular expression for the resulting integrable hierarchy. It is the first explicit nontrivial example of a quantum integrable hierarchy from a cohomological field theory containing fermionic fields, which correspond to the odd classes in the cohomology of the elliptic curve.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 9","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70486","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148848998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Uniqueness problem for accretive Schrödinger operators with complex singular coefficients","authors":"Vladimir Mikhailets,&nbsp;Volodymyr Molyboga","doi":"10.1112/blms.70490","DOIUrl":"https://doi.org/10.1112/blms.70490","url":null,"abstract":"&lt;p&gt;This paper studies the uniqueness problem for the one-dimensional Schrödinger operator associated with the formal differential expression &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;mrow&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;mo&gt;=&lt;/mo&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$l[u] =-u^{prime prime }+qu + i[(ru)^{prime }+ru^{prime }] $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in the complex Hilbert space &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$L^{2}(mathbb {R})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. The coefficients of the expression are complex-valued and satisfy\u0000&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;mi&gt;o&lt;/mi&gt;\u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 9","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70490","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148849036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geometric inverse semigroup theory: a note on the Milnor–Schwarz lemma for inverse monoids","authors":"Giorgio Mangioni,&nbsp;Francesco Tesolin","doi":"10.1112/blms.70482","DOIUrl":"https://doi.org/10.1112/blms.70482","url":null,"abstract":"<p>We generalise the Milnor–Schwarz lemma to inverse monoids acting on presheaves of geodesic metric spaces. We provide two proofs of this fact: one only uses elementary techniques, inspired by the arguments for group actions on metric spaces; the other involves a version of the Vietoris–Rips complex, and builds on work of Chung–Martínez–Szakács.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 9","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70482","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148849035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On tested Bousfield–Friedlander localizations","authors":"Niall Taggart","doi":"10.1112/blms.70485","DOIUrl":"https://doi.org/10.1112/blms.70485","url":null,"abstract":"<p>We show that a Bousfield–Friedlander localization of a model category of functors with respect to a set of test morphisms, as introduced by Bandklayder, Bergner, Griffiths, Johnson and Santhanam, can be characterized as a left Bousfield localization at a specific tensoring of the set of homotopy generators. This reformulation immediately strengthens their main result by establishing that such localizations always yield cellular or combinatorial model categories. We apply this framework to functor calculus in two key ways. First, we construct a homogeneous model structure for any calculus arising from tested Bousfield–Friedlander localization. As a consequence, we construct a Quillen adjunction between homogeneous functors in Goodwillie calculus and homogeneous functors in the discrete calculus of Bauer, Johnson and McCarthy. Second, we show that the polynomial model structure of Weiss calculus is a specific instance of this localization framework, suggesting a unifying perspective on various flavours of functor calculus.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 9","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70485","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148848768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Merryfield's inequality for multiparameter martingales","authors":"Guillermo Rey","doi":"10.1112/blms.70479","DOIUrl":"https://doi.org/10.1112/blms.70479","url":null,"abstract":"<p>We extend an inequality of Merryfield, valid in the continuous setting, to discrete multiparameter martingales. As a consequence, we obtain the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msup>\u0000 <annotation>$L^p$</annotation>\u0000 </semantics></math> comparison of the maximal function with the square function:\u0000\u0000 </p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 9","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148848775","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}

{"title":"Convergence and combinatorics of the Reverse algorithm","authors":"Hiroaki Ito,&nbsp;Niels Langeveld,&nbsp;Jörg Thuswaldner","doi":"10.1112/blms.70470","DOIUrl":"https://doi.org/10.1112/blms.70470","url":null,"abstract":"<p>We study the Reverse algorithm, a multidimensional continued fraction algorithm, which is not unimodular. We show that the Reverse algorithm is ergodic and, by proving that its second Lyapunov exponent is negative, that it is a.e. exponentially convergent. In addition to that, we attach substitutions to this algorithm and study the <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math>-adic languages generated by sequences of these substitutions. The negativity of the second Lyapunov exponent implies that almost all of these languages are balanced. By a thorough study of the combinatorics of the substitutions, we are even able to obtain a concrete generic family of balanced languages that is characterized in terms of a simple condition on the underlying sequence of substitutions.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 9","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70470","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148848774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Calderón–Zygmund estimates for generalized double phase equations with matrix weights","authors":"Sun-Sig Byun,&nbsp;Hongsoo Kim","doi":"10.1112/blms.70487","DOIUrl":"https://doi.org/10.1112/blms.70487","url":null,"abstract":"<p>We prove Calderón–Zygmund estimates for generalized double phase equations with Orlicz growth and variable matrix weights. The operator combines a nonuniformly elliptic double phase structure with a degenerate or singular matrix weight satisfying a small log-<span></span><math>\u0000 <semantics>\u0000 <mi>BMO</mi>\u0000 <annotation>$mathrm{BMO}$</annotation>\u0000 </semantics></math> condition. Under appropriate structural assumptions, we show that higher integrability of the weighted datum yields higher integrability of the weighted gradient of weak solutions. Our results extend the existing Calderón–Zygmund theory for double phase problems and weighted elliptic equations to a unified framework capturing the interaction between Orlicz growth and matrix-weighted structures, thereby building upon and unifying previous results.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 9","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148785223","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}

{"title":"The DGA of planar loops when \u0000 \u0000 \u0000 2\u0000 n\u0000 =\u0000 4\u0000 \u0000 $2n=4$","authors":"Guy Boyde","doi":"10.1112/blms.70476","DOIUrl":"https://doi.org/10.1112/blms.70476","url":null,"abstract":"<p>The DGA of planar loops is a pictorial chain complex introduced in a recent paper of the author, Boyd, Randal-Williams and Sroka, where a minimal model for it was given. In the first non-trivial case, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 <mo>=</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation>$2n=4$</annotation>\u0000 </semantics></math>, we give a new model which incorporates two natural ‘reflection’ involutions, as well as a more explicit description of the existing model.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 9","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70476","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148785224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On a Sobolev critical problem for the superposition of a local and nonlocal operator with the “wrong sign”","authors":"Stefano Biagi,&nbsp;Serena Dipierro,&nbsp;Enrico Valdinoci,&nbsp;Eugenio Vecchi","doi":"10.1112/blms.70477","DOIUrl":"https://doi.org/10.1112/blms.70477","url":null,"abstract":"<p>We study a critical problem for an operator of mixed order obtained by the superposition of a Laplacian with a fractional Laplacian.</p><p>The main novelty is that we consider a mixed operator of the form <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mi>Δ</mi>\u0000 <mo>−</mo>\u0000 <mi>γ</mi>\u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mo>−</mo>\u0000 <mi>Δ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mi>s</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$-Delta - gamma (-Delta)^s$</annotation>\u0000 </semantics></math>, namely we suppose that the fractional Laplacian has the “wrong sign” and can be seen as a nonlocal perturbation of the purely local case, which is needed to produce a nontrivial solution of the critical problem.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 9","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148784831","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}

{"title":"A spectral volume comparison for manifolds with weakly convex boundary","authors":"Jia Li","doi":"10.1112/blms.70473","DOIUrl":"https://doi.org/10.1112/blms.70473","url":null,"abstract":"&lt;p&gt;We establish the Bonnet–Myers theorem and the Bishop–Gromov volume comparison theorem in the spectral sense for manifolds with weakly convex boundary. For &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$ngeqslant 3$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, let &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;msup&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&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;$(M^n,g)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a connected compact smooth &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-manifold with weakly convex boundary &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$partial M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. If there exists a positive function &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;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$win C^{infty }(M)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; that satisfies:\u0000 &lt;/p&gt;&lt;p&gt;If, in addition, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;w&lt;/mi&gt;\u0000 &lt;annotation&gt;$w$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; attains its minimum on the boundary &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$partial M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, we obtain a sharp upper bound for the volume of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&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;mo&gt;Vol&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;⩽&lt;/mo&gt;\u0000 &lt;mo&gt;Vol&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"58 9","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148784953","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}