Proceedings of the Royal Society of Edinburgh Section A-Mathematics最新文献

{"title":"Desingularization of 2D elliptic free-boundary problem with non-autonomous nonlinearity","authors":"Jie Wan","doi":"10.1017/prm.2024.48","DOIUrl":"https://doi.org/10.1017/prm.2024.48","url":null,"abstract":"In this paper, we consider the existence and limiting behaviour of solutions to a semilinear elliptic equation arising from confined plasma problem in dimension two <jats:disp-formula> <jats:alternatives> <jats:tex-math>[ begin{cases} -Delta u=lambda k(x)f(u) &amp; text{in} D, u= c &amp; displaystyletext{on} partial D, displaystyle - int_{partial D} frac{partial u}{partial nu},{rm d}s=I, end{cases} ]</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0308210524000489_eqnU1.png\"/> </jats:alternatives> </jats:disp-formula>where <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Dsubseteq mathbb {R}^2$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000489_inline1.png\"/> </jats:alternatives> </jats:inline-formula> is a smooth bounded domain, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$nu$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000489_inline2.png\"/> </jats:alternatives> </jats:inline-formula> is the outward unit normal to the boundary <jats:inline-formula> <jats:alternatives> <jats:tex-math>$partial D$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000489_inline3.png\"/> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$lambda$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000489_inline4.png\"/> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$I$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000489_inline5.png\"/> </jats:alternatives> </jats:inline-formula> are given constants and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$c$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000489_inline6.png\"/> </jats:alternatives> </jats:inline-formula> is an unknown constant. Under some assumptions on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$f$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000489_inline7.png\"/> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$k$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000489_inline8.png\"/> </jats:alternatives> </jats:inline-formula>, we prove that there exists a family of solutions concentrating near strict local minimum points of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Gamma (x)=({1}/{2})h(x,,x)- ({1}/{8pi })l","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"34 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828465","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":"Spreading primitive groups of diagonal type do not exist","authors":"John Bamberg, Saul D. Freedman, Michael Giudici","doi":"10.1017/prm.2024.53","DOIUrl":"https://doi.org/10.1017/prm.2024.53","url":null,"abstract":"The synchronization hierarchy of finite permutation groups consists of classes of groups lying between <jats:inline-formula> <jats:alternatives> <jats:tex-math>$2$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000532_inline1.png\"/> </jats:alternatives> </jats:inline-formula>-transitive groups and primitive groups. This includes the class of spreading groups, which are defined in terms of sets and multisets of permuted points, and which are known to be primitive of almost simple, affine or diagonal type. In this paper, we prove that in fact no spreading group of diagonal type exists. As part of our proof, we show that all non-abelian finite simple groups, other than six sporadic groups, have a transitive action in which a proper normal subgroup of a point stabilizer is supplemented by all corresponding two-point stabilizers.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"57 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805683","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":"Topologically free actions and ideals in twisted Banach algebra crossed products","authors":"Krzysztof Bardadyn, Bartosz Kwaśniewski","doi":"10.1017/prm.2024.37","DOIUrl":"https://doi.org/10.1017/prm.2024.37","url":null,"abstract":"We generalize the influential <jats:inline-formula> <jats:alternatives> <jats:tex-math>$C^*$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000374_inline1.png\"/> </jats:alternatives> </jats:inline-formula>-algebraic results of Kawamura–Tomiyama and Archbold–Spielberg for crossed products of discrete groups actions to the realm of Banach algebras and twisted actions. We prove that topological freeness is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalized intersection property for a full <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L^p$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000374_inline2.png\"/> </jats:alternatives> </jats:inline-formula>-operator algebra crossed product for any <jats:inline-formula> <jats:alternatives> <jats:tex-math>$pin [1,,infty ]$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000374_inline3.png\"/> </jats:alternatives> </jats:inline-formula>. This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L^p$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000374_inline4.png\"/> </jats:alternatives> </jats:inline-formula>-operator algebras coincide.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"38 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806755","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":"Singer conjecture for varieties with semismall Albanese map and residually finite fundamental group","authors":"Luca F. Di Cerbo, Luigi Lombardi","doi":"10.1017/prm.2024.52","DOIUrl":"https://doi.org/10.1017/prm.2024.52","url":null,"abstract":"<p>We prove the Singer conjecture for varieties with semismall Albanese map and residually finite fundamental group.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"94 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140631136","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 homotopy decomposition of the suspension of a non-simply-connected five-manifold","authors":"Pengcheng Li, Zhongjian Zhu","doi":"10.1017/prm.2024.49","DOIUrl":"https://doi.org/10.1017/prm.2024.49","url":null,"abstract":"In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth <jats:inline-formula> <jats:alternatives> <jats:tex-math>$five$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000490_inline2.png\" /> </jats:alternatives> </jats:inline-formula>-manifolds. As applications, we compute the reduced <jats:inline-formula> <jats:alternatives> <jats:tex-math>$K$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000490_inline3.png\" /> </jats:alternatives> </jats:inline-formula>-groups of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$M$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000490_inline4.png\" /> </jats:alternatives> </jats:inline-formula> and show that the suspension map between the third cohomotopy set <jats:inline-formula> <jats:alternatives> <jats:tex-math>$pi ^3(M)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000490_inline5.png\" /> </jats:alternatives> </jats:inline-formula> and the fourth cohomotopy set <jats:inline-formula> <jats:alternatives> <jats:tex-math>$pi ^4(Sigma M)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000490_inline6.png\" /> </jats:alternatives> </jats:inline-formula> is a bijection.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"75 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140616365","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":"Sharp conditions for the validity of the Bourgain–Brezis–Mironescu formula","authors":"Elisa Davoli, Giovanni Di Fratta, Valerio Pagliari","doi":"10.1017/prm.2024.47","DOIUrl":"https://doi.org/10.1017/prm.2024.47","url":null,"abstract":"<p>Following the seminal paper by Bourgain, Brezis, and Mironescu, we focus on the asymptotic behaviour of some nonlocal functionals that, for each <span><span><span data-mathjax-type=\"texmath\"><span>$uin L^2(mathbb {R}^N)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053606689-0049:S0308210524000477:S0308210524000477_inline1.png\"/></span></span>, are defined as the double integrals of weighted, squared difference quotients of <span><span><span data-mathjax-type=\"texmath\"><span>$u$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053606689-0049:S0308210524000477:S0308210524000477_inline2.png\"/></span></span>. Given a family of weights <span><span><span data-mathjax-type=\"texmath\"><span>${rho _{varepsilon} }$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053606689-0049:S0308210524000477:S0308210524000477_inline3.png\"/></span></span>, <span><span><span data-mathjax-type=\"texmath\"><span>$varepsilon in (0,,1)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053606689-0049:S0308210524000477:S0308210524000477_inline4.png\"/></span></span>, we devise sufficient and necessary conditions on <span><span><span data-mathjax-type=\"texmath\"><span>${rho _{varepsilon} }$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053606689-0049:S0308210524000477:S0308210524000477_inline5.png\"/></span></span> for the associated nonlocal functionals to converge as <span><span><span data-mathjax-type=\"texmath\"><span>$varepsilon to 0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053606689-0049:S0308210524000477:S0308210524000477_inline6.png\"/></span></span> to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"24 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583448","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":"Linear actions of","authors":"Jim Fowler, Courtney Thatcher","doi":"10.1017/prm.2024.36","DOIUrl":"https://doi.org/10.1017/prm.2024.36","url":null,"abstract":"<p>For an odd prime <span><span><span data-mathjax-type=\"texmath\"><span>$p$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053638415-0645:S0308210524000362:S0308210524000362_inline3.png\"/></span></span>, we consider free actions of <span><span><span data-mathjax-type=\"texmath\"><span>$(mathbb {Z}_{/{p}})^2$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053638415-0645:S0308210524000362:S0308210524000362_inline4.png\"/></span></span> on <span><span><span data-mathjax-type=\"texmath\"><span>$S^{2n-1}times S^{2n-1}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053638415-0645:S0308210524000362:S0308210524000362_inline5.png\"/></span></span> given by linear actions of <span><span><span data-mathjax-type=\"texmath\"><span>$(mathbb {Z}_{/{p}})^2$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053638415-0645:S0308210524000362:S0308210524000362_inline6.png\"/></span></span> on <span><span><span data-mathjax-type=\"texmath\"><span>$mathbb {R}^{4n}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053638415-0645:S0308210524000362:S0308210524000362_inline7.png\"/></span></span>. Simple examples include a lens space cross a lens space, but <span><span><span data-mathjax-type=\"texmath\"><span>$k$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053638415-0645:S0308210524000362:S0308210524000362_inline8.png\"/></span></span>-invariant calculations show that other quotients exist. Using the tools of Postnikov towers and surgery theory, the quotients are classified up to homotopy by the <span><span><span data-mathjax-type=\"texmath\"><span>$k$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053638415-0645:S0308210524000362:S0308210524000362_inline9.png\"/></span></span>-invariants and up to homeomorphism by the Pontrjagin classes. We will present these results and demonstrate how to calculate the <span><span><span data-mathjax-type=\"texmath\"><span>$k$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240416053638415-0645:S0308210524000362:S0308210524000362_inline10.png\"/></span></span>-invariants and the Pontrjagin classes from the rotation numbers.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583461","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":"Quasi-hereditary covers of Temperley–Lieb algebras and relative dominant dimension","authors":"Tiago Cruz, Karin Erdmann","doi":"10.1017/prm.2024.35","DOIUrl":"https://doi.org/10.1017/prm.2024.35","url":null,"abstract":"Many connections and dualities in representation theory and Lie theory can be explained using quasi-hereditary covers in the sense of Rouquier. Recent work by the first-named author shows that relative dominant (and codominant) dimensions are natural tools to classify and distinguish distinct quasi-hereditary covers of a finite-dimensional algebra. In this paper, we prove that the relative dominant dimension of a quasi-hereditary algebra, possessing a simple preserving duality, with respect to a direct summand of the characteristic tilting module is always an even number or infinite and that this homological invariant controls the quality of quasi-hereditary covers that possess a simple preserving duality. To resolve the Temperley–Lieb algebras, we apply this result to the class of Schur algebras <jats:inline-formula> <jats:alternatives> <jats:tex-math>$S(2, d)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000350_inline5.png\" /> </jats:alternatives> </jats:inline-formula> and their <jats:inline-formula> <jats:alternatives> <jats:tex-math>$q$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000350_inline6.png\" /> </jats:alternatives> </jats:inline-formula>-analogues. Our second main result completely determines the relative dominant dimension of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$S(2, d)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000350_inline7.png\" /> </jats:alternatives> </jats:inline-formula> with respect to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Q=V^{otimes d}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000350_inline8.png\" /> </jats:alternatives> </jats:inline-formula>, the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$d$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000350_inline9.png\" /> </jats:alternatives> </jats:inline-formula>-th tensor power of the natural two-dimensional module. As a byproduct, we deduce that Ringel duals of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$q$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000350_inline10.png\" /> </jats:alternatives> </jats:inline-formula>-Schur algebras <jats:inline-formula> <jats:alternatives> <jats:tex-math>$S(2,d)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000350_inline11.png\" /> </jats:alternatives> </jats:inline-formula> give rise to quasi-hereditary covers of Temperley–Lieb algebras. Further, we obtain precisely when the Temperley–Lieb algebra is Morita equivalent to the Ringel dual of th","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"52 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583452","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":"Global centres in a class of quintic polynomial differential systems","authors":"Leonardo P. C. da Cruz, Jaume Llibre","doi":"10.1017/prm.2024.43","DOIUrl":"https://doi.org/10.1017/prm.2024.43","url":null,"abstract":"A centre of a differential system in the plane <jats:inline-formula> <jats:alternatives> <jats:tex-math>$ {mathbb {R}}^2$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400043X_inline1.png\" /> </jats:alternatives> </jats:inline-formula> is an equilibrium point <jats:inline-formula> <jats:alternatives> <jats:tex-math>$p$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400043X_inline2.png\" /> </jats:alternatives> </jats:inline-formula> having a neighbourhood <jats:inline-formula> <jats:alternatives> <jats:tex-math>$U$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400043X_inline3.png\" /> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Usetminus {p}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400043X_inline4.png\" /> </jats:alternatives> </jats:inline-formula> is filled with periodic orbits. A centre <jats:inline-formula> <jats:alternatives> <jats:tex-math>$p$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400043X_inline5.png\" /> </jats:alternatives> </jats:inline-formula> is global when <jats:inline-formula> <jats:alternatives> <jats:tex-math>$ {mathbb {R}}^2setminus {p}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400043X_inline6.png\" /> </jats:alternatives> </jats:inline-formula> is filled with periodic orbits. In general, it is a difficult problem to distinguish the centres from the foci for a given class of differential systems, and also it is difficult to distinguish the global centres inside the centres. The goal of this paper is to classify the centres and the global centres of the following class of quintic polynomial differential systems <jats:disp-formula> <jats:alternatives> <jats:tex-math>begin{align*} dot{x}= y,quad dot{y}={-}x+a_{05},y^5+a_{14},x,y^4+a_{23},x^2,y^3+a_{32},x^3,y^2+a_{41},x^4,y+a_{50},x^5, end{align*}</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S030821052400043X_eqnU1.png\" /> </jats:alternatives> </jats:disp-formula>in the plane <jats:inline-formula> <jats:alternatives> <jats:tex-math>$ {mathbb {R}}^2$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400043X_inline7.png\" /> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"119 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583316","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":"Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential","authors":"Tianxiang Gou","doi":"10.1017/prm.2024.44","DOIUrl":"https://doi.org/10.1017/prm.2024.44","url":null,"abstract":"In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, <jats:disp-formula> <jats:alternatives> <jats:tex-math>[ (-Delta)^s u+ left(omega+|x|^2right) u=|u|^{p-2}u quad mbox{in} mathbb{R}^n, ]</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0308210524000441_eqnU1.png\" /> </jats:alternatives> </jats:disp-formula>where <jats:inline-formula> <jats:alternatives> <jats:tex-math>$n geq 1$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline1.png\" /> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$0&lt; s&lt;1$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline2.png\" /> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$omega &gt;-lambda _{1,s}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline3.png\" /> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$2&lt; p&lt; {2n}/{(n-2s)^+}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline4.png\" /> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$lambda _{1,s}&gt;0$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline5.png\" /> </jats:alternatives> </jats:inline-formula> is the lowest eigenvalue of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$(-Delta )^s + |x|^2$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline6.png\" /> </jats:alternatives> </jats:inline-formula>. The fractional Laplacian <jats:inline-formula> <jats:alternatives> <jats:tex-math>$(-Delta )^s$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline7.png\" /> </jats:alternatives> </jats:inline-formula> is characterized as <jats:inline-formula> <jats:alternatives> <jats:tex-math>$mathcal {F}((-Delta )^{s}u)(xi )=|xi |^{2s} mathcal {F}(u)(xi )$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline8.png\" /> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:tex-math>$xi in mathbb {R}^n$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000441_inline9.png\" /> </jats:alternatives> </jats:inli","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"124 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140602284","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}