{"title":"PRM volume 153 issue 4 Cover and Front matter","authors":"","doi":"10.1017/prm.2023.60","DOIUrl":"https://doi.org/10.1017/prm.2023.60","url":null,"abstract":"","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"12 1","pages":"f1 - f2"},"PeriodicalIF":1.3,"publicationDate":"2023-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80092464","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":"PRM volume 153 issue 4 Cover and Back matter","authors":"","doi":"10.1017/prm.2023.61","DOIUrl":"https://doi.org/10.1017/prm.2023.61","url":null,"abstract":"","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"11 1","pages":"b1 - b2"},"PeriodicalIF":1.3,"publicationDate":"2023-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89819665","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}
Let $fcolon (mathbb {R}^{3},0)to (mathbb {R}^{4},0)$ be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of $f$ with a small enough sphere $S^{3}_epsilon$ centred at the origin in $mathbb {R}^{4}$. If $f$ is of fold type, we define a tree, that we call dual tree, that contains all the topological information of the link and we prove that in this case it is a complete topological invariant. As an application we give a procedure to obtain normal forms for any topological class of fold type.
{"title":"The dual tree of a fold map germ from $mathbb {R}^{3}$ to $mathbb {R}^{4}$","authors":"J. A. Moya-Pérez, J. J. Nuño-Ballesteros","doi":"10.1017/prm.2022.27","DOIUrl":"https://doi.org/10.1017/prm.2022.27","url":null,"abstract":"Let $fcolon (mathbb {R}^{3},0)to (mathbb {R}^{4},0)$ be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of $f$ with a small enough sphere $S^{3}_epsilon$ centred at the origin in $mathbb {R}^{4}$. If $f$ is of fold type, we define a tree, that we call dual tree, that contains all the topological information of the link and we prove that in this case it is a complete topological invariant. As an application we give a procedure to obtain normal forms for any topological class of fold type.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"40 1","pages":"958 - 977"},"PeriodicalIF":1.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73085657","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":"PRM volume 153 issue 3 Cover and Front matter","authors":"","doi":"10.1017/prm.2023.47","DOIUrl":"https://doi.org/10.1017/prm.2023.47","url":null,"abstract":"","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"26 1","pages":"f1 - f2"},"PeriodicalIF":1.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73826025","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":"PRM volume 153 issue 3 Cover and Back matter","authors":"","doi":"10.1017/prm.2023.48","DOIUrl":"https://doi.org/10.1017/prm.2023.48","url":null,"abstract":"","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"8 1","pages":"b1 - b2"},"PeriodicalIF":1.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84069993","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}
The Schrödinger–Poisson system describes standing waves for the nonlinear Schrödinger equation interacting with the electrostatic field. In this paper, we are concerned with the existence of positive ground states to the planar Schrödinger–Poisson system with a nonlinearity having either a subcritical or a critical exponential growth in the sense of Trudinger–Moser. A feature of this paper is that neither the finite steep potential nor the reaction satisfies any symmetry or periodicity hypotheses. The analysis developed in this paper seems to be the first attempt in the study of planar Schrödinger–Poisson systems with lack of symmetry.
{"title":"Groundstates of the planar Schrödinger–Poisson system with potential well and lack of symmetry","authors":"Zhisu Liu, Vicentiu D. Rădulescu, Jianjun Zhang","doi":"10.1017/prm.2023.43","DOIUrl":"https://doi.org/10.1017/prm.2023.43","url":null,"abstract":"The Schrödinger–Poisson system describes standing waves for the nonlinear Schrödinger equation interacting with the electrostatic field. In this paper, we are concerned with the existence of positive ground states to the planar Schrödinger–Poisson system with a nonlinearity having either a subcritical or a critical exponential growth in the sense of Trudinger–Moser. A feature of this paper is that neither the finite steep potential nor the reaction satisfies any symmetry or periodicity hypotheses. The analysis developed in this paper seems to be the first attempt in the study of planar Schrödinger–Poisson systems with lack of symmetry.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"9 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88408413","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}
We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem.
{"title":"Second-kind symmetric periodic orbits for planar perturbed Kepler problems and applications","authors":"A. Alberti, C. Vidal","doi":"10.1017/prm.2023.46","DOIUrl":"https://doi.org/10.1017/prm.2023.46","url":null,"abstract":"We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"15 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72781203","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}
We study anti-symmetric solutions about the hyperplane ${x_n=0}$ for the following fractional Hardy–Hénon system: [ left{begin{array}{@{}ll} (-Delta)^{s_1}u(x)=|x|^alpha v^p(x), & xinmathbb{R}_+^n, (-Delta)^{s_2}v(x)=|x|^beta u^q(x), & xinmathbb{R}_+^n, u(x)geq 0, & v(x)geq 0, xinmathbb{R}_+^n, end{array}right. ]where $0< s_1,s_2<1$, $n>2max {s_1,s_2}$. Nonexistence of anti-symmetric solutions are obtained in some appropriate domains of $(p,q)$ under some corresponding assumptions of $alpha,beta$ via the methods of moving spheres and moving planes. Particularly, for the case $s_1=s_2$, one of our results shows that one domain of $(p,q)$, where nonexistence of anti-symmetric solutions with appropriate decay conditions at infinity hold true, locates at above the fractional Sobolev's hyperbola under appropria
{"title":"Nonexistence of anti-symmetric solutions for fractional Hardy–Hénon system","authors":"Jiaqian Hu, Zhuoran Du","doi":"10.1017/prm.2023.40","DOIUrl":"https://doi.org/10.1017/prm.2023.40","url":null,"abstract":"<jats:p>We study anti-symmetric solutions about the hyperplane <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:tex-math>${x_n=0}$</jats:tex-math>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523000409_inline1.png\" />\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> for the following fractional Hardy–Hénon system:\u0000<jats:disp-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:tex-math>[ left{begin{array}{@{}ll} (-Delta)^{s_1}u(x)=|x|^alpha v^p(x), & xinmathbb{R}_+^n, (-Delta)^{s_2}v(x)=|x|^beta u^q(x), & xinmathbb{R}_+^n, u(x)geq 0, & v(x)geq 0, xinmathbb{R}_+^n, end{array}right. ]</jats:tex-math>\u0000\t\t<jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0308210523000409_eqnU1.png\" />\u0000\t </jats:alternatives>\u0000\t </jats:disp-formula>where <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:tex-math>$0< s_1,s_2<1$</jats:tex-math>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523000409_inline2.png\" />\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:tex-math>$n>2max {s_1,s_2}$</jats:tex-math>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523000409_inline3.png\" />\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. Nonexistence of anti-symmetric solutions are obtained in some appropriate domains of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:tex-math>$(p,q)$</jats:tex-math>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523000409_inline4.png\" />\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> under some corresponding assumptions of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:tex-math>$alpha,beta$</jats:tex-math>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523000409_inline5.png\" />\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> via the methods of moving spheres and moving planes. Particularly, for the case <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:tex-math>$s_1=s_2$</jats:tex-math>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523000409_inline6.png\" />\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, one of our results shows that one domain of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:tex-math>$(p,q)$</jats:tex-math>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523000409_inline7.png\" />\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, where nonexistence of anti-symmetric solutions with appropriate decay conditions at infinity hold true, locates at above the fractional Sobolev's hyperbola under appropria","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"12 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87750359","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}
This paper is focused on spreading dynamics for a discrete Nicholson's blowflies model with time convolution kernel. This problem arises in the invasive activity of blowflies scattered in discrete spatial environment and has distributed maturated age. We found that for a general convolution kernel, the model can exhibit travelling wave phenomena in a discrete spatial habitat. In particular, we determine the minimal wave speed of travelling waves by deriving the non-existence of travelling waves, and we demonstrate that the minimal wave speed can determine the long time behaviour of solutions with compact initial function. Moreover, we prove that all travelling waves are strictly increasing, which implies that the waveforms remain monotone in the propagation process. Some numerical simulations are also presented to confirm the analytical results.
{"title":"Spreading dynamics of a discrete Nicholson's blowflies equation with distributed delay","authors":"Ruiwen Wu, Zhaoquan Xu","doi":"10.1017/prm.2023.34","DOIUrl":"https://doi.org/10.1017/prm.2023.34","url":null,"abstract":"This paper is focused on spreading dynamics for a discrete Nicholson's blowflies model with time convolution kernel. This problem arises in the invasive activity of blowflies scattered in discrete spatial environment and has distributed maturated age. We found that for a general convolution kernel, the model can exhibit travelling wave phenomena in a discrete spatial habitat. In particular, we determine the minimal wave speed of travelling waves by deriving the non-existence of travelling waves, and we demonstrate that the minimal wave speed can determine the long time behaviour of solutions with compact initial function. Moreover, we prove that all travelling waves are strictly increasing, which implies that the waveforms remain monotone in the propagation process. Some numerical simulations are also presented to confirm the analytical results.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"20 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89499652","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}
A. Ballester-Bolinches, R. Esteban-Romero, V. Pérez-Calabuig
Skew left braces arise naturally from the study of non-degenerate set-theoretic solutions of the Yang–Baxter equation. To understand the algebraic structure of skew left braces, a study of the decomposition into minimal substructures is relevant. We introduce chief series and prove a strengthened form of the Jordan–Hölder theorem for finite skew left braces. A characterization of right nilpotency and an application to multipermutation solutions are also given.
{"title":"A Jordan–Hölder theorem for skew left braces and their applications to multipermutation solutions of the Yang–Baxter equation","authors":"A. Ballester-Bolinches, R. Esteban-Romero, V. Pérez-Calabuig","doi":"10.1017/prm.2023.37","DOIUrl":"https://doi.org/10.1017/prm.2023.37","url":null,"abstract":"Skew left braces arise naturally from the study of non-degenerate set-theoretic solutions of the Yang–Baxter equation. To understand the algebraic structure of skew left braces, a study of the decomposition into minimal substructures is relevant. We introduce chief series and prove a strengthened form of the Jordan–Hölder theorem for finite skew left braces. A characterization of right nilpotency and an application to multipermutation solutions are also given.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"112 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84771141","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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