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

{"title":"L\u0000 ∞\u0000 \u0000 $L^infty$\u0000 compactness of solutions of quasilinear problems and applications","authors":"D. Arcoya, M. C. M. Rezende, E. A. B. Silva","doi":"10.1112/jlms.70695","DOIUrl":"https://doi.org/10.1112/jlms.70695","url":null,"abstract":"<p>For a (not necessarily smooth) bounded domain <span></span><math>\u0000 <semantics>\u0000 <mi>Ω</mi>\u0000 <annotation>$Omega$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>N</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^N$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$N geqslant 2$</annotation>\u0000 </semantics></math> and a Carathéodory vector-valued function <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>:</mo>\u0000 <mi>Ω</mi>\u0000 <mo>×</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>N</mi>\u0000 </msup>\u0000 <mo>→</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>N</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$a:Omega times mathbb {R}^N rightarrow mathbb {R}^N$</annotation>\u0000 </semantics></math>, we study the compactness of the inverse of the Leray–Lions operator <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>(</mo>\u0000 <mi>u</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mo>−</mo>\u0000 <mtext>div</mtext>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mo>∇</mo>\u0000 <mi>u</mi>\u0000 <mo>)</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$A(u)=-text{div}(a(x, nabla u))$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>u</mi>\u0000 <mo>∈</mo>\u0000 <msubsup>\u0000 <mi>W</mi>\u0000 <mn>0</mn>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Ω</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$uin W_0^{1, p}(Omega)$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 3","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70695","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148942416","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":"Equidistribution in 2-nilpotent Polish groups and triple restricted sumsets","authors":"Ethan Ackelsberg,&nbsp;Asgar Jamneshan","doi":"10.1112/jlms.70693","DOIUrl":"https://doi.org/10.1112/jlms.70693","url":null,"abstract":"<p>The aim of this paper is to establish a Ratner-type equidistribution theorem for orbits on homogeneous spaces associated with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$hskip.001pt 2$</annotation>\u0000 </semantics></math>-nilpotent locally compact Polish groups under the action of a countable discrete abelian group. We apply this result to establish the existence of triple restricted sumsets in subsets of positive density in arbitrary countable discrete abelian groups, subject to a necessary finiteness condition.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 3","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70693","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148942417","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":"A Buchsbaum theory for Frobenius closure","authors":"Kriti Goel,&nbsp;Kyle Maddox,&nbsp;Lance Edward Miller,&nbsp;Pham Hung Quy,&nbsp;Austyn Simpson","doi":"10.1112/jlms.70661","DOIUrl":"https://doi.org/10.1112/jlms.70661","url":null,"abstract":"&lt;p&gt;We give a partial characterization for when the difference &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;e&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/msub&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;msup&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$e(mathfrak {q})-ell _R(R/mathfrak {q}^F)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is independent of the choice of parameter ideal &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;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathfrak {q}subseteq R$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in an excellent equidimensional local ring &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;mi&gt;R&lt;/mi&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;annotation&gt;$(R,mathfrak {m})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of prime characteristic &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;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$p&gt;0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Here, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathfrak {q}^F$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is the Frobenius closure of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathfrak {q}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;e&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$e(mathfrak {q})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; denotes the Hilbert–Samuel multiplicity of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathfrak {q}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In addition to ideal-theoretic equivalences, our characterization involves the derived category and is motivated by Schenzel's criterion of the Buchsbaum property as well as similar results of Ma–Quy in the ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 3","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148942351","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 probability of generating finite and profinite groups","authors":"Scott Harper,&nbsp;Martyn Quick","doi":"10.1112/jlms.70670","DOIUrl":"https://doi.org/10.1112/jlms.70670","url":null,"abstract":"&lt;p&gt;Famously, every finite simple 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; can be generated by a pair of elements. Moreover, Liebeck and Shalev (1995) proved that the probability that a pair of elements generate &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; tends to 1 as &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;mi&gt;G&lt;/mi&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$|G| rightarrow infty$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In this paper, we generalize this theorem of Liebeck and Shalev. Work of Lucchini and Menegazzo (1997) implies that 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; can be generated by a pair of elements if it has a unique chief series. As a consequence of our main theorem, the probability that a pair of elements generate such a 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; tends to 1 as &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;mi&gt;S&lt;/mi&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$|S| rightarrow infty$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &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 the unique simple quotient 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;. We also prove that a profinite 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; with finitely many chief series has a topological generating set of size &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;&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$d &lt; infty$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, and for any such &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;annotation&gt;$d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, the probability that a &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 3","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70670","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148942221","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":"Uniformization of cofat domains on metric two-spheres","authors":"Chengxi Li,&nbsp;Kai Rajala","doi":"10.1112/jlms.70685","DOIUrl":"https://doi.org/10.1112/jlms.70685","url":null,"abstract":"<p>We extend <i>Schramm's cofat uniformization theorem</i> to cofat domains on upper Ahlfors 2-regular metric two-spheres <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math>. Specifically, we show that if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Ω</mi>\u0000 <mo>⊂</mo>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation>$Omega subset X$</annotation>\u0000 </semantics></math> is a cofat domain, then there exists a <span></span><math>\u0000 <semantics>\u0000 <mfrac>\u0000 <mi>π</mi>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 <annotation>$frac{pi }{2}$</annotation>\u0000 </semantics></math>-quasiconformal homeomorphism <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>:</mo>\u0000 <mi>Ω</mi>\u0000 <mo>→</mo>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation>$f: Omega rightarrow D$</annotation>\u0000 </semantics></math> onto a circle domain <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 <mo>⊂</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$D subset mathbb {S}^2$</annotation>\u0000 </semantics></math>. Moreover, <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> preserves the point-components and non-trivial complementary components. We also construct examples which show that the above conclusions are not true for countably connected <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ℓ</mi>\u0000 <mi>α</mi>\u0000 </msup>\u0000 <annotation>$ell ^{alpha }$</annotation>\u0000 </semantics></math>-subdomains of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$mathbb {S}^2$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 3","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70685","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148942150","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":"Hecke equivariance of the divisor map","authors":"Daeyeol Jeon,&nbsp;Soon-Yi Kang,&nbsp;Chang Heon Kim,&nbsp;Toshiki Matsusaka","doi":"10.1112/jlms.70692","DOIUrl":"https://doi.org/10.1112/jlms.70692","url":null,"abstract":"<p>We study the multiplicative Hecke operators acting on the space of meromorphic modular forms, and show that the divisor map to divisors on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>X</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>N</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$X_0(N)$</annotation>\u0000 </semantics></math> is a Hecke equivariant map. As applications, we investigate the divisor sum formula of Bruinier–Kohnen–Ono and more general Rohrlich-type divisor sums for polyharmonic Maass forms, discussing several implications for the Hecke action and its relation to the self-adjointness of the Hecke operators.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 3","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148942139","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":"Cauchy identities for staircase matrices","authors":"Evgeny Feigin,&nbsp;Anton Khoroshkin,&nbsp;Ievgen Makedonskyi","doi":"10.1112/jlms.70680","DOIUrl":"https://doi.org/10.1112/jlms.70680","url":null,"abstract":"&lt;p&gt;The well-known Cauchy identity expresses the product of terms &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&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;msub&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;y&lt;/mi&gt;\u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;${(1-{x}_{i}{y}_{j})}^{-1}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for &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;mi&gt;i&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(i,j)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; indexing entries of a rectangular &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mrow&gt;&lt;/mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mensuremath{times{}}n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-matrix as a sum over partitions &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;annotation&gt;$lambda $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of products of Schur polynomials: &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;y&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;${s}_{lambda}(x){s}_{lambda}(y)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Algebraically, this identity comes from the decomposition of the symmetric algebra of the space of rectangular matrices, considered as a &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;gl&lt;/mi&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;${mathfrak{gl}}_{m}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 3","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70680","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148943663","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":"Grushin-type degenerate parabolic equations with applications to mean field games","authors":"Yiming Jiang,&nbsp;Jingchuang Ren,&nbsp;Yawei Wei,&nbsp;Jie Xue","doi":"10.1112/jlms.70690","DOIUrl":"https://doi.org/10.1112/jlms.70690","url":null,"abstract":"<p>In this paper, we establish a weak maximum principle (WMP) and global a priori Schauder estimates for a class of linear degenerate Cauchy problems associated with Grushin-type operators. These regularity results yield the existence and uniqueness of solutions to the Cauchy problems on the unbounded domain. To derive the WMP, we construct a subsolution tailored to the specific structure of Grushin-type operators, rather than relying on fundamental solutions. To address the non-commutativity and inhomogeneity of the Hörmander vector fields, we separate the coefficients via a Taylor expansion at the degenerate points and analyze each component individually. This technique provides an alternative method for the regularity analysis of degenerate operators, which is distinct from the classical framework based on fundamental solutions. As an application, we obtain the existence of classical solutions to a class of degenerate mean field games, in which a typical player may encounter a “forbidden” direction at certain points.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 3","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148943639","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":"Multiplicative irreducibility of shifted multiplicative subgroups","authors":"Seoyoung Kim,&nbsp;Chi Hoi Yip,&nbsp;Semin Yoo","doi":"10.1112/jlms.70688","DOIUrl":"https://doi.org/10.1112/jlms.70688","url":null,"abstract":"<p>In a recent breakthrough, Kalmynin resolved conjectures of Lev–Sonn and Sárközy on additive decompositions of multiplicative subgroups of prime fields. In this paper, inspired by a related conjecture of Sárközy, we prove multiplicative analogues of Kalmynin's results. We show that for every proper multiplicative subgroup <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>, the shifted set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 <mo>∖</mo>\u0000 <mo>{</mo>\u0000 <mn>0</mn>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$(G-1)setminus lbrace 0rbrace$</annotation>\u0000 </semantics></math> cannot be written as a product set nontrivially, addressing a conjecture of Sárközy. In addition, we prove that no nonzero shift of any coset of a proper multiplicative subgroup is a ratio set of the form <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>/</mo>\u0000 <mi>A</mi>\u0000 </mrow>\u0000 <annotation>$A/A$</annotation>\u0000 </semantics></math>. Our results substantially sharpen previous theorems of Shkredov and the authors.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 3","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70688","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148816636","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":"Isometric immersions into three-dimensional unimodular metric Lie groups","authors":"Ildefonso Castro,&nbsp;José M. Manzano,&nbsp;José S. Santiago","doi":"10.1112/jlms.70682","DOIUrl":"https://doi.org/10.1112/jlms.70682","url":null,"abstract":"<p>We study isometric immersions of surfaces into simply connected three-dimensional unimodular Lie groups endowed with either Riemannian or Lorentzian left-invariant metrics, assuming that Milnor's operator is diagonalizable in the Lorentzian case. We provide global models in coordinates for all these <i>metric Lie groups</i> that depend analytically on the structure constants and establish some fundamental theorems characterizing such immersions. In this sense, we study up to what extent we can recover the immersion from (a) the tangent projections of the natural left-invariant ambient frame, (b) the left-invariant Gauss map, and (c) the shape operator. In particular, we show that an isometric immersion is determined by its left-invariant Gauss map up to certain well-controlled <i>angular companions</i>. We also classify totally geodesic surfaces and introduce four Lorentzian analogs of the Daniel correspondence within two families of Lorentzian homogeneous 3-manifolds with four-dimensional isometry group. We also classify isometric immersions in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^3$</annotation>\u0000 </semantics></math> or <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$mathbb {S}^3$</annotation>\u0000 </semantics></math> whose left-invariant Gauss maps differ by a direct isometry of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$mathbb {S}^2$</annotation>\u0000 </semantics></math>. Finally, we show that Daniel's is the furthest extension of the classical Lawson correspondence for constant mean curvature surfaces in Riemannian unimodular metric Lie groups.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 3","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70682","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148816668","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}