Proceedings of the London Mathematical Society最新文献

{"title":"The nonlinear Schrödinger equation on the half-line with homogeneous Robin boundary conditions.","authors":"Jae Min Lee,&nbsp;Jonatan Lenells","doi":"10.1112/plms.12493","DOIUrl":"https://doi.org/10.1112/plms.12493","url":null,"abstract":"<p><p>We consider the nonlinear Schrödinger equation on the half-line <math> <mrow><mrow><mi>x</mi> <mo>⩾</mo> <mn>0</mn></mrow> </mrow> </math> with a Robin boundary condition at <math> <mrow><mrow><mi>x</mi> <mo>=</mo> <mn>0</mn></mrow> </mrow> </math> and with initial data in the weighted Sobolev space <math> <mrow> <mrow><msup><mi>H</mi> <mrow><mn>1</mn> <mo>,</mo> <mn>1</mn></mrow> </msup> <mrow><mo>(</mo> <msub><mi>R</mi> <mo>+</mo></msub> <mo>)</mo></mrow> </mrow> </mrow> </math> . We prove that there exists a global weak solution of this initial-boundary value problem and provide a representation for the solution in terms of the solution of a Riemann-Hilbert problem. Using this representation, we obtain asymptotic formulas for the long-time behavior of the solution. In particular, by restricting our asymptotic result to solutions whose initial data are close to the initial profile of the stationary one-soliton, we obtain results on the asymptotic stability of the stationary one-soliton under any small perturbation in <math> <mrow> <mrow><msup><mi>H</mi> <mrow><mn>1</mn> <mo>,</mo> <mn>1</mn></mrow> </msup> <mrow><mo>(</mo> <msub><mi>R</mi> <mo>+</mo></msub> <mo>)</mo></mrow> </mrow> </mrow> </math> . In the focusing case, such a result was already established by Deift and Park using different methods, and our work provides an alternative approach to obtain such results. We treat both the focusing and the defocusing versions of the equation.</p>","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"126 1","pages":"334-389"},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10091827/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9323383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Issue Information","authors":"","doi":"10.1112/plms.12449","DOIUrl":"https://doi.org/10.1112/plms.12449","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41564299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Various subgroups of Aut(Cn)${rm{Aut}}({mathbb{C}}^{n})$","authors":"Zhang Lin, Xiangyu Zhou","doi":"10.1112/plms.12501","DOIUrl":"https://doi.org/10.1112/plms.12501","url":null,"abstract":"In this paper, using ideas of Andersén and Lempert on the group of holomorphic automorphisms Aut(Cn)${rm{Aut}}({mathbb{C}}^{n})$ ( n⩾2$ngeqslant 2$ ), we prove that the subgroups in Aut(Cn)${rm{Aut}}({mathbb{C}}^{n})$ and Autsp(C2n)${rm{Aut}}_{rm{sp}}({mathbb{C}}^{2n})$ ( n⩾2$ngeqslant 2$ ) generated by different types of ‘shears’ are all distinct, which answer affirmatively to two conjectures posed by Forstnerič.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"126 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41460777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Issue Information","authors":"","doi":"10.1112/plms.12418","DOIUrl":"https://doi.org/10.1112/plms.12418","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47126922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Least energy solutions to quasilinear subelliptic equations with constant and degenerate potentials on the Heisenberg group","authors":"Lu Chen, Guozhen Lu, Maochun Zhu","doi":"10.1112/plms.12495","DOIUrl":"https://doi.org/10.1112/plms.12495","url":null,"abstract":"Let Hn=Cn×R$mathbb {H}^{n}=mathbb {C}^{n}times mathbb {R}$ be the n$n$ ‐dimensional Heisenberg group, Q=2n+2$Q=2n+2$ be the homogeneous dimension of Hn$mathbb {H}^{n}$ . In this paper, we investigate the existence of a least energy solution to the Q$Q$ ‐subLaplacian Schrödinger equation with either a constant V=γ$V=gamma$ or a degenerate potential V$V$ vanishing on a bounded open subset of Hn$mathbb {H}^n$ : 0.1 −divH∇HuQ−2∇Hu+V(ξ)uQ−2u=fu$$begin{equation} -mathrm{div}_{mathbb {H}}{left({left|nabla _{mathbb {H}}uright|}^{Q-2} nabla _{mathbb {H}}uright)} +V(xi ) {left|uright|}^{Q-2}u=f{left(uright)} end{equation}$$with the non‐linear term f$f$ of maximal exponential growth exp(αtQQ−1)$exp (alpha t^{frac{Q}{Q-1}})$ as t→+∞$trightarrow +infty$ . Since the Pólya–Szegö‐type inequality fails on Hn$mathbb {H}^n$ , the coercivity of the potential has been a standard assumption in the literature for subelliptic equations to exclude the vanishing phenomena of Palais–Smale sequence on the entire space Hn$mathbb {H}^n$ . Our aim in this paper is to remove this strong assumption. To this end, we first establish a sharp critical Trudinger–Moser inequality involving a degenerate potential on Hn$mathbb {H}^n$ . Second, we prove the existence of a least energy solution to the above equation with the constant potential V(ξ)=γ>0$V(xi )=gamma >0$ . Third, we establish the existence of a least energy solution to the Q$Q$ ‐subelliptic equation (0.1) involving the degenerate potential which vanishes on some open bounded set of Hn$mathbb {H}^{n}$ . We develop arguments that avoid using any symmetrization on Hn$mathbb {H}^n$ where the Pólya–Szegö inequality fails. Fourth, we also establish the existence of a least energy solution to (0.1) when the potential is a non‐degenerate Rabinowitz type potential but still fails to be coercive. Our results in this paper improve significantly on the earlier ones on quasilinear Schrödinger equations on the Heisenberg group in the literature. We note that all the main results and their proofs in this paper hold on stratified groups with the same proofs.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48573329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Overview of the retina and imaging in patients with severe acute respiratory syndrome coronavirus 2.","authors":"Solmaz Abdolrahimzadeh, Manuel Lodesani, Daria Rullo, Alberto Mariani, Gianluca Scuderi","doi":"10.1007/s10792-022-02338-x","DOIUrl":"10.1007/s10792-022-02338-x","url":null,"abstract":"<p><strong>Introduction: </strong>The role of the human eye in severe acute respiratory syndrome coronavirus 2 (SARS-COV-2) is still under investigation. The pathophysiology of the ocular findings is arduous when dealing with critically ill Covid-19 patients with comorbidities. Multiorgan involvement and the effects of inflammation, infection and systemic treatment on the retina are complex, and comparison of studies is difficult. Most studies in human patients have investigated the anterior segment, whereas few reports deal with the posterior segment of the eye. The present review aims to evaluate the retinal manifestations and imaging features in COVID-19 patients.</p><p><strong>Methods: </strong>Studies on the retinal manifestations and retinal imaging in COVID-19 patients published through June 2021 were reviewed. We included cross-sectional and case-control studies, case series, case reports and correspondence in the analysis.</p><p><strong>Results: </strong>Flame-shaped hemorrhages, cotton wool spots, augmented diameter and tortuosity of retinal vessels were found on funduscopic examination. Peripapillary, macular retinal nerve fiber layer and ganglion cell layer thickness alterations were reported on spectral domain optical coherence tomography. Reduced vessel density of the superficial and deep retinal capillary plexus on optical coherence tomography angiography was reported.</p><p><strong>Conclusions: </strong>Retinal complications may arise in COVID-19 patients. Although no consensus on presentation is currently available, retinal funduscopy and imaging has shown neuronal and vascular alterations. Systemic neurological complications and microangiopathy are associated with SARS-COV-2; thus, as the retina has a neuronal and vascular component, funduscopy and retinal imaging on COVID-19 patients can provide further insight to SARS-COV-2 disease and the follow-up of patients.</p>","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"1 1","pages":"3601-3610"},"PeriodicalIF":1.4,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9094133/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85566942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rational points on complete intersections over Fq(t)${mathbb {F}}_q(t)$","authors":"P. Vishe","doi":"10.1112/plms.12496","DOIUrl":"https://doi.org/10.1112/plms.12496","url":null,"abstract":"A 2‐dimensional version of Farey dissection for function fields K=Fq(t)$K=mathbb {F}_q(t)$ is developed and used to establish the quantitative arithmetic of the set of rational points on a smooth complete intersection of two quadrics X⊂PKn−1$Xsubset mathbb {P}^{n-1}_{K}$ , under the assumption that q$q$ is odd and n⩾9$ngeqslant 9$ .","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49024648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local types of (Γ,G)$(Gamma ,G)$ ‐bundles and parahoric group schemes","authors":"Chiara Damiolini, Jiuzu Hong","doi":"10.1112/plms.12544","DOIUrl":"https://doi.org/10.1112/plms.12544","url":null,"abstract":"Let G$G$ be a simple algebraic group over an algebraically closed field k$k$ . Let Γ$Gamma$ be a finite group acting on G$G$ . We classify and compute the local types of (Γ,G)$(Gamma , G)$ ‐bundles on a smooth projective Γ$Gamma$ ‐curve in terms of the first nonabelian group cohomology of the stabilizer groups at the tamely ramified points with coefficients in G$G$ . When char(k)=0$text{char}(k)=0$ , we prove that any generically simply connected parahoric Bruhat–Tits group scheme can arise from a (Γ,Gad)$(Gamma ,G_{mathrm{ad}})$ ‐bundle. We also prove a local version of this theorem, that is, parahoric group schemes over the formal disc arise from constant group schemes via tamely ramified coverings.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48932594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Heegaard Floer homology for manifolds with torus boundary: properties and examples.","authors":"Jonathan Hanselman,&nbsp;Jacob Rasmussen,&nbsp;Liam Watson","doi":"10.1112/plms.12473","DOIUrl":"https://doi.org/10.1112/plms.12473","url":null,"abstract":"<p><p>This is a companion paper to earlier work of the authors (Preprint, arXiv:1604.03466, 2016), which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We establish a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston norm, and the Turaev torsion. We also give a geometric description of the gradings package from bordered Heegaard Floer homology and establish a symmetry under <math> <mrow><msup><mo>Spin</mo> <mi>c</mi></msup> </mrow> </math> conjugation; this symmetry gives rise to genus one mutation invariance in Heegaard Floer homology for closed three-manifolds. Finally, we include more speculative discussions on relationships with Seiberg-Witten theory, Khovanov homology, and <math> <mrow><msup><mi>HF</mi> <mo>±</mo></msup> </mrow> </math> . Many examples are included.</p>","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"125 4","pages":"879-967"},"PeriodicalIF":1.8,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9826536/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10578675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Issue Information","authors":"","doi":"10.1112/plms.12416","DOIUrl":"https://doi.org/10.1112/plms.12416","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44430426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}