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

{"title":"A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part II: justification as a shallow water approximation","authors":"Vincent Duchêne, Tatsuo Iguchi","doi":"10.1017/prm.2024.30","DOIUrl":"https://doi.org/10.1017/prm.2024.30","url":null,"abstract":"<p>We consider the Kakinuma model for the motion of interfacial gravity waves. The Kakinuma model is a system of Euler–Lagrange equations for an approximate Lagrangian, which is obtained by approximating the velocity potentials in the Lagrangian of the full model. Structures of the Kakinuma model and the well-posedness of its initial value problem were analysed in the companion paper [14]. In this present paper, we show that the Kakinuma model is a higher order shallow water approximation to the full model for interfacial gravity waves with an error of order <span><span><span data-mathjax-type=\"texmath\"><span>$O(delta _1^{4N+2}+delta _2^{4N+2})$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240316153711817-0521:S0308210524000301:S0308210524000301_inline1.png\"/></span></span> in the sense of consistency, where <span><span><span data-mathjax-type=\"texmath\"><span>$delta _1$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240316153711817-0521:S0308210524000301:S0308210524000301_inline2.png\"/></span></span> and <span><span><span data-mathjax-type=\"texmath\"><span>$delta _2$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240316153711817-0521:S0308210524000301:S0308210524000301_inline3.png\"/></span></span> are shallowness parameters, which are the ratios of the mean depths of the upper and the lower layers to the typical horizontal wavelength, respectively, and <span><span><span data-mathjax-type=\"texmath\"><span>$N$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240316153711817-0521:S0308210524000301:S0308210524000301_inline4.png\"/></span></span> is, roughly speaking, the size of the Kakinuma model and can be taken an arbitrarily large number. Moreover, under a hypothesis of the existence of the solution to the full model with a uniform bound, a rigorous justification of the Kakinuma model is proved by giving an error estimate between the solution to the Kakinuma model and that of the full model. An error estimate between the Hamiltonian of the Kakinuma model and that of the full model is also provided.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"16 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140146352","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 index of real polynomials","authors":"Gabriel E. Monsalve, Mihai Tibăr","doi":"10.1017/prm.2024.23","DOIUrl":"https://doi.org/10.1017/prm.2024.23","url":null,"abstract":"<p>We develop two methods for expressing the global index of the gradient of a 2 variable polynomial function <span><span><span data-mathjax-type=\"texmath\"><span>$f$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240316153410714-0444:S0308210524000234:S0308210524000234_inline1.png\"/></span></span>: in terms of the atypical fibres of <span><span><span data-mathjax-type=\"texmath\"><span>$f$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240316153410714-0444:S0308210524000234:S0308210524000234_inline2.png\"/></span></span>, and in terms of the clusters of Milnor arcs at infinity. These allow us to derive upper bounds for the global index, in particular refining the one that was found by Durfee in terms of the degree of <span><span><span data-mathjax-type=\"texmath\"><span>$f$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240316153410714-0444:S0308210524000234:S0308210524000234_inline3.png\"/></span></span>.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"96 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140146505","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":"Eternal solutions to a porous medium equation with strong non-homogeneous absorption. Part I: radially non-increasing profiles","authors":"Razvan Gabriel Iagar, Philippe Laurençot","doi":"10.1017/prm.2024.29","DOIUrl":"https://doi.org/10.1017/prm.2024.29","url":null,"abstract":"<p>Existence of specific <span>eternal solutions</span> in exponential self-similar form to the following quasilinear diffusion equation with strong absorption<span><span data-mathjax-type=\"texmath\"><span>[ partial_t u=Delta u^m-|x|^{sigma}u^q, ]</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240312163119404-0773:S0308210524000295:S0308210524000295_eqnU1.png\"/></span>posed for <span><span><span data-mathjax-type=\"texmath\"><span>$(t,,x)in (0,,infty )times mathbb {R}^N$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240312163119404-0773:S0308210524000295:S0308210524000295_inline1.png\"/></span></span>, with <span><span><span data-mathjax-type=\"texmath\"><span>$m&gt;1$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240312163119404-0773:S0308210524000295:S0308210524000295_inline2.png\"/></span></span>, <span><span><span data-mathjax-type=\"texmath\"><span>$qin (0,,1)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240312163119404-0773:S0308210524000295:S0308210524000295_inline3.png\"/></span></span> and <span><span><span data-mathjax-type=\"texmath\"><span>$sigma =sigma _c:=2(1-q)/ (m-1)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240312163119404-0773:S0308210524000295:S0308210524000295_inline4.png\"/></span></span> is proved. Looking for radially symmetric solutions of the form<span><span data-mathjax-type=\"texmath\"><span>[ u(t,x)={rm e}^{-alpha t}f(|x|,{rm e}^{beta t}), quad alpha=frac{2}{m-1}beta, ]</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240312163119404-0773:S0308210524000295:S0308210524000295_eqnU2.png\"/></span>we show that there exists a unique exponent <span><span><span data-mathjax-type=\"texmath\"><span>$beta ^*in (0,,infty )$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240312163119404-0773:S0308210524000295:S0308210524000295_inline5.png\"/></span></span> for which there exists a one-parameter family <span><span><span data-mathjax-type=\"texmath\"><span>$(u_A)_{A&gt;0}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240312163119404-0773:S0308210524000295:S0308210524000295_inline6.png\"/></span></span> of solutions with compactly supported and non-increasing profiles <span><span><span data-mathjax-type=\"texmath\"><span>$(f_A)_{A&gt;0}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:ca","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"24 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124398","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":"Decay at infinity for solutions to some fractional parabolic equations","authors":"Agnid Banerjee, Abhishek Ghosh","doi":"10.1017/prm.2024.9","DOIUrl":"https://doi.org/10.1017/prm.2024.9","url":null,"abstract":"<p>For <span><span><span data-mathjax-type=\"texmath\"><span>$sin [tfrac {1}{2},, 1)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313135246144-0856:S030821052400009X:S030821052400009X_inline1.png\"/></span></span>, let <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:20240313135246144-0856:S030821052400009X:S030821052400009X_inline2.png\"/></span></span> solve <span><span><span data-mathjax-type=\"texmath\"><span>$(partial _t - Delta )^s u = Vu$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313135246144-0856:S030821052400009X:S030821052400009X_inline3.png\"/></span></span> in <span><span><span data-mathjax-type=\"texmath\"><span>$mathbb {R}^{n} times [-T,, 0]$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313135246144-0856:S030821052400009X:S030821052400009X_inline4.png\"/></span></span> for some <span><span><span data-mathjax-type=\"texmath\"><span>$T&gt;0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313135246144-0856:S030821052400009X:S030821052400009X_inline5.png\"/></span></span> where <span><span><span data-mathjax-type=\"texmath\"><span>$||V||_{ C^2(mathbb {R}^n times [-T, 0])} &lt; infty$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313135246144-0856:S030821052400009X:S030821052400009X_inline6.png\"/></span></span>. We show that if for some <span><span><span data-mathjax-type=\"texmath\"><span>$0&lt;mathfrak {K} &lt; T$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313135246144-0856:S030821052400009X:S030821052400009X_inline7.png\"/></span></span> and <span><span><span data-mathjax-type=\"texmath\"><span>$epsilon &gt;0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313135246144-0856:S030821052400009X:S030821052400009X_inline8.png\"/></span></span><span><span data-mathjax-type=\"texmath\"><span>[ {unicode{x2A0D}}-_{[-mathfrak{K},, 0]} u^2(x, t) {rm d}t leq Ce^{-|x|^{2+epsilon}} forall x in mathbb{R}^n, ]</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313135246144-0856:S030821052400009X:S030821052400009X_eqnU1.png\"/></span>then <span><span><span data-mathjax-type=\"texmath\"><span>$u equiv 0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/i","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"18 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124399","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":"Necessary and sufficient conditions for ground state solutions to planar Kirchhoff-type equations","authors":"Chunyu Lei, Binlin Zhang","doi":"10.1017/prm.2024.26","DOIUrl":"https://doi.org/10.1017/prm.2024.26","url":null,"abstract":"<p>In this paper, we are concerned with the ground states of the following planar Kirchhoff-type problem:<span><span data-mathjax-type=\"texmath\"><span>[ -left(1+bdisplaystyleint_{mathbb{R}^2}|nabla u|^2,{rm d}xright)Delta u+omega u=|u|^{p-2}u, quad xinmathbb{R}^2. ]</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240308141104504-0258:S030821052400026X:S030821052400026X_eqnU1.png\"/></span>where <span><span><span data-mathjax-type=\"texmath\"><span>$b,, omega &gt;0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240308141104504-0258:S030821052400026X:S030821052400026X_inline1.png\"/></span></span> are constants, <span><span><span data-mathjax-type=\"texmath\"><span>$p&gt;2$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240308141104504-0258:S030821052400026X:S030821052400026X_inline2.png\"/></span></span>. Based on variational methods, regularity theory and Schwarz symmetrization, the equivalence of ground state solutions for the above problem with the minimizers for some minimization problems is obtained. In particular, a new scale technique, together with Lagrange multipliers, is delicately employed to overcome some intrinsic difficulties.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"36 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140100117","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":"Optimal inverse problems of potentials for two given eigenvalues of Sturm–Liouville problems","authors":"Min Zhao, Jiangang Qi","doi":"10.1017/prm.2024.28","DOIUrl":"https://doi.org/10.1017/prm.2024.28","url":null,"abstract":"<p>The present paper is concerned with the infimum of the norm of potentials for Sturm–Liouville eigenvalue problems with Dirichlet boundary condition such that the first two eigenvalues are known. The explicit quantity of the infimum is given by the two eigenvalues.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"167 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140057543","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":"Index estimates of compact hypersurfaces in smooth metric measure spaces","authors":"Márcio Batista, Matheus B. Martins","doi":"10.1017/prm.2024.25","DOIUrl":"https://doi.org/10.1017/prm.2024.25","url":null,"abstract":"<p>In this article, we investigate the spectra of the stability and Hodge–Laplacian operators on a compact manifold immersed as a hypersurface in a smooth metric measure space, possibly with singularities. Using ideas developed by A. Ros and A. Savo, along with an ingenious computation, we have obtained a comparison between the spectra of these operators. As a byproduct of this technique, we have deduced an estimate of the Morse index of such hypersurfaces.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"18 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140053735","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":"Invariant measures and large deviation principles for stochastic Schrödinger delay lattice systems","authors":"Zhang Chen, Xiaoxiao Sun, Bixiang Wang","doi":"10.1017/prm.2024.20","DOIUrl":"https://doi.org/10.1017/prm.2024.20","url":null,"abstract":"<p>This paper is concerned with stochastic Schrödinger delay lattice systems with both locally Lipschitz drift and diffusion terms. Based on the uniform estimates and the equicontinuity of the segment of the solution in probability, we show the tightness of a family of probability distributions of the solution and its segment process, and hence the existence of invariant measures on <span><span><span data-mathjax-type=\"texmath\"><span>$l^2times L^2((-rho,,0);l^2)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240302133705906-0511:S0308210524000209:S0308210524000209_inline1.png\"/></span></span> with <span><span><span data-mathjax-type=\"texmath\"><span>$rho &gt;0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240302133705906-0511:S0308210524000209:S0308210524000209_inline2.png\"/></span></span>. We also establish a large deviation principle for the solutions with small noise by the weak convergence method.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025541","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":"Bifurcations and pattern formation in a host–parasitoid model with nonlocal effect","authors":"Chuang Xiang, Jicai Huang, Min Lu, Shigui Ruan, Hao Wang","doi":"10.1017/prm.2024.24","DOIUrl":"https://doi.org/10.1017/prm.2024.24","url":null,"abstract":"<p>In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a <span>(8,0)-mode Turing–Hopf bifurcation</span> and unstable for a <span>(3,0)-mode Turing–Hopf bifurcation</span>. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a <span>(0,1)-mode double-Hopf bifurcation</span>. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"55 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025704","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":"Invariant set generated by a nonreal number is everywhere dense","authors":"Artūras Dubickas","doi":"10.1017/prm.2024.22","DOIUrl":"https://doi.org/10.1017/prm.2024.22","url":null,"abstract":"A set of complex numbers <jats:inline-formula> <jats:alternatives> <jats:tex-math>$S$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000222_inline1.png\" /> </jats:alternatives> </jats:inline-formula> is called invariant if it is closed under addition and multiplication, namely, for any <jats:inline-formula> <jats:alternatives> <jats:tex-math>$x, y in S$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000222_inline2.png\" /> </jats:alternatives> </jats:inline-formula> we have <jats:inline-formula> <jats:alternatives> <jats:tex-math>$x+y in S$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000222_inline3.png\" /> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$xy in S$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000222_inline4.png\" /> </jats:alternatives> </jats:inline-formula>. For each <jats:inline-formula> <jats:alternatives> <jats:tex-math>$s in {mathbb {C}}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000222_inline5.png\" /> </jats:alternatives> </jats:inline-formula> the smallest invariant set <jats:inline-formula> <jats:alternatives> <jats:tex-math>${mathbb {N}}[s]$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000222_inline6.png\" /> </jats:alternatives> </jats:inline-formula> containing <jats:inline-formula> <jats:alternatives> <jats:tex-math>$s$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000222_inline7.png\" /> </jats:alternatives> </jats:inline-formula> consists of all possible sums <jats:inline-formula> <jats:alternatives> <jats:tex-math>$sum _{i in I} a_i s^i$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000222_inline8.png\" /> </jats:alternatives> </jats:inline-formula>, where <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=\"S0308210524000222_inline9.png\" /> </jats:alternatives> </jats:inline-formula> runs over all finite nonempty subsets of the set of positive integers <jats:inline-formula> <jats:alternatives> <jats:tex-math>${mathbb {N}}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000222_inline10.png\" /> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$a_i in {mathbb {N}}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http:","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"3 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007494","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}