{"title":"Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals","authors":"Hiroyoshi Mitake, Hung V. Tran","doi":"10.1112/blms.13227","DOIUrl":"https://doi.org/10.1112/blms.13227","url":null,"abstract":"<p>Here, we study a level-set forced mean curvature flow with evolving spirals and the homogeneous Neumann boundary condition, which appears in a crystal growth model. Under some appropriate conditions on the forcing term, we prove that the solution is globally Lipschitz. We then study the large time average of the solution and deduce the asymptotic growth rate of the crystal. Some large time behavior results of the solution are obtained.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"748-770"},"PeriodicalIF":0.8,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13227","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143582073","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":"Fast Goodstein walks","authors":"David Fernández-Duque, Andreas Weiermann","doi":"10.1112/blms.13210","DOIUrl":"https://doi.org/10.1112/blms.13210","url":null,"abstract":"<p>We introduce a family <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>A</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo><</mo>\u0000 <mi>ω</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$(mathbb {A}_k)_{k<omega }$</annotation>\u0000 </semantics></math> of fast-growing functions based on <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ε</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <annotation>$varepsilon _0$</annotation>\u0000 </semantics></math> and use these to define a variant of the Goodstein process. We show that this variant terminates and that this fact is not provable in Kripke–Platek set theory (or other theories of Bachmann–Howard strength). We, moreover, show that this Goodstein process is of maximal length, so that any alternative Goodstein process based on the same fast-growing functions will also terminate.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 2","pages":"510-533"},"PeriodicalIF":0.8,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13210","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143363023","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":"Mabuchi Kähler solitons versus extremal Kähler metrics and beyond","authors":"Vestislav Apostolov, Abdellah Lahdili, Yasufumi Nitta","doi":"10.1112/blms.13222","DOIUrl":"https://doi.org/10.1112/blms.13222","url":null,"abstract":"<p>Using the Yau–Tian–Donaldson type correspondence for <span></span><math>\u0000 <semantics>\u0000 <mi>v</mi>\u0000 <annotation>$v$</annotation>\u0000 </semantics></math>-solitons established by Han–Li, we show that a smooth complex <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>-dimensional Fano variety admits a Mabuchi soliton provided it admits an extremal Kähler metric whose scalar curvature is strictly less than <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$2(n+1)$</annotation>\u0000 </semantics></math>. Combined with previous observations by Mabuchi and Nakamura in the other direction, this gives a characterization of the existence of Mabuchi solitons in terms of the existence of extremal Kähler metrics on Fano manifolds. An extension of this correspondence to <span></span><math>\u0000 <semantics>\u0000 <mi>v</mi>\u0000 <annotation>$v$</annotation>\u0000 </semantics></math>-solitons is also obtained.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"692-710"},"PeriodicalIF":0.8,"publicationDate":"2024-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13222","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143582010","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":"Spaces of maps between real algebraic varieties","authors":"Wojciech Kucharz","doi":"10.1112/blms.13220","DOIUrl":"https://doi.org/10.1112/blms.13220","url":null,"abstract":"<p>Given two real algebraic varieties <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math>, we denote by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>,</mo>\u0000 <mi>Y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {R}(X,Y)$</annotation>\u0000 </semantics></math> the set of all regular maps from <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> to <span></span><math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math>. The set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>,</mo>\u0000 <mi>Y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {R}(X,Y)$</annotation>\u0000 </semantics></math> is regarded as a topological subspace of the space <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>,</mo>\u0000 <mi>Y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {C}(X,Y)$</annotation>\u0000 </semantics></math> of all continuous maps from <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> to <span></span><math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math> endowed with the compact-open topology. We prove, in a much more general setting than previously considered, that each path component of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>,</mo>\u0000 <mi>Y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {C}(X,Y)$</annotation>\u0000 </semantics></math> contains at most one path component of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>,</mo>\u0000 <mi>Y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {R}(X,Y)$</annotation>\u0000 </semantics></m","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"669-680"},"PeriodicalIF":0.8,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143582033","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":"Torsion-free connections of second-order maximally superintegrable systems","authors":"Andreas Vollmer","doi":"10.1112/blms.13213","DOIUrl":"https://doi.org/10.1112/blms.13213","url":null,"abstract":"<p>Second-order (maximally) conformally superintegrable systems play an important role as models of mechanical systems, including systems such as the Kepler–Coulomb system and the isotropic harmonic oscillator. This paper is dedicated to understanding non- and semi-degenerate systems. We obtain “projective flatness” results for two torsion-free connections naturally associated to such systems. This viewpoint sheds some light onto the interrelationship of properly and conformally (second-order maximally) superintegrable systems from a geometrical perspective. It is shown that the semi-degenerate secondary structure tensor can be viewed as the Ricci curvature of a natural torsion-free connection defined by the primary structure tensor (and similarly in the non-degenerate case). It is also shown that properly semi-degenerate systems are characterised, similar to the non-degenerate case, by the vanishing of the secondary structure tensor.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 2","pages":"565-581"},"PeriodicalIF":0.8,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13213","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143363054","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":"Associahedra as moment polytopes","authors":"Michael Gekhtman, Hugh Thomas","doi":"10.1112/blms.13212","DOIUrl":"https://doi.org/10.1112/blms.13212","url":null,"abstract":"<p>Generalized associahedra are a well-studied family of polytopes associated with a finite-type cluster algebra and a choice of starting cluster. We show that the generalized associahedra constructed by Padrol, Palu, Pilaud, and Plamondon, building on ideas from Arkani-Hamed, Bai, He, and Yan, can be naturally viewed as moment polytopes for an open patch of the quotient of the cluster <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math>-variety with universal coefficients by its maximal natural torus action. We prove our result by showing that the construction of Padrol, Palu, Pilaud, and Plamondon can be understood on the basis of the way that moment polytopes behave under symplectic reduction.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 2","pages":"551-564"},"PeriodicalIF":0.8,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143363059","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}
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Wen Zhang
{"title":"Multiple solutions with sign information for double-phase problems with unbalanced growth","authors":"Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Wen Zhang","doi":"10.1112/blms.13218","DOIUrl":"https://doi.org/10.1112/blms.13218","url":null,"abstract":"<p>We consider a double-phase Dirichlet problem with unbalanced growth and a reaction term which is <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>p</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(p-1)$</annotation>\u0000 </semantics></math>-sublinear and has partial interaction with the first eigenvalue of the weighted differential operator <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Δ</mi>\u0000 <mi>p</mi>\u0000 <mi>a</mi>\u0000 </msubsup>\u0000 <annotation>$Delta _{p}^{a}$</annotation>\u0000 </semantics></math> (nonuniform nonresonance). Using the Nehari method, we show that the problem has at least three nontrivial bounded solutions, positive, negative and nodal (sign-changing). This paper extends and complements the main results in the recent paper Papageorgiou–Pudelko–Rădulescu (<i>Math. Annalen</i> <b>385</b> (2023) 1707–1745). </p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 2","pages":"638-656"},"PeriodicalIF":0.8,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143363061","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":"Estimates for smooth Weyl sums on minor arcs","authors":"Jörg Brüdern, Trevor D. Wooley","doi":"10.1112/blms.13219","DOIUrl":"https://doi.org/10.1112/blms.13219","url":null,"abstract":"<p>We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <msup>\u0000 <mi>n</mi>\u0000 <mi>k</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$alpha n^k$</annotation>\u0000 </semantics></math>. In particular, when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>⩾</mo>\u0000 <mn>6</mn>\u0000 </mrow>\u0000 <annotation>$kgeqslant 6$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ρ</mi>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$rho (k)$</annotation>\u0000 </semantics></math> is defined via the relation <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ρ</mi>\u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mo>=</mo>\u0000 <mi>k</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>log</mi>\u0000 <mi>k</mi>\u0000 <mo>+</mo>\u0000 <mn>8.02113</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$rho (k)^{-1}=k(log k+8.02113)$</annotation>\u0000 </semantics></math>, then for all large numbers <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> there is an integer <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>⩽</mo>\u0000 <mi>n</mi>\u0000 <mo>⩽</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$1leqslant nleqslant N$</annotation>\u0000 </semantics></math> for which <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>∥</mo>\u0000 <mi>α</m","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"657-668"},"PeriodicalIF":0.8,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13219","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581949","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":"Fixed points for group actions on 2-dimensional affine buildings","authors":"Jeroen Schillewaert, Koen Struyve, Anne Thomas","doi":"10.1112/blms.13197","DOIUrl":"https://doi.org/10.1112/blms.13197","url":null,"abstract":"<p>We prove a local-to-global result for fixed points of finitely generated groups acting on 2-dimensional affine buildings of types <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mover>\u0000 <mi>A</mi>\u0000 <mo>∼</mo>\u0000 </mover>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$tilde{A}_2$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mover>\u0000 <mi>C</mi>\u0000 <mo>∼</mo>\u0000 </mover>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$tilde{C}_2$</annotation>\u0000 </semantics></math>. Our proofs combine building-theoretic arguments with standard results for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>CAT</mo>\u0000 <mo>(</mo>\u0000 <mn>0</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{CAT}(0)$</annotation>\u0000 </semantics></math> spaces.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 1","pages":"285-301"},"PeriodicalIF":0.8,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143116295","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}
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