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

{"title":"The structure of finite groups whose elements outside a normal subgroup have prime power orders","authors":"Changguo Shao, Qinhui Jiang","doi":"10.1017/prm.2024.71","DOIUrl":"https://doi.org/10.1017/prm.2024.71","url":null,"abstract":"The structure of groups in which every element has prime power order (CP-groups) is extensively studied. We first investigate the properties of group <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline3.png\"/> </jats:alternatives> </jats:inline-formula> such that each element of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Gsetminus N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline30a.png\"/> </jats:alternatives> </jats:inline-formula> has prime power order. It is proved that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline10a.png\"/> </jats:alternatives> </jats:inline-formula> is solvable or every non-solvable chief factor <jats:inline-formula> <jats:alternatives> <jats:tex-math>$H/K$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline10.png\"/> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline11.png\"/> </jats:alternatives> </jats:inline-formula> satisfying <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Hleq N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline12.png\"/> </jats:alternatives> </jats:inline-formula> is isomorphic to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$PSL_2(3^f)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline13.png\"/> </jats:alternatives> </jats:inline-formula> with <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=\"S0308210524000714_inline14.png\"/> </jats:alternatives> </jats:inline-formula> a 2-power. This partially answers the question proposed by Lewis in 2023, asking whether <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Gcong M_{10}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline15.png\"/> </jats:alternatives> </jats:inline-formula>? Furthermore, we prove that if each element <jats:inline-formula> <jats:alternatives> <jats:tex-math>$xin Gbackslash N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline16.png\"/> </jats:alternatives> </jats:inline-formula> has prime powe","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"17 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142257696","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":"A unified characterization of convolution coefficients in nonlocal differential equations","authors":"Christopher S. Goodrich","doi":"10.1017/prm.2024.77","DOIUrl":"https://doi.org/10.1017/prm.2024.77","url":null,"abstract":"In loving memory of my beloved miniature dachshund Maddie (16 March 2002 – 16 March 2020). We consider nonlocal differential equations with convolution coefficients of the form <jats:disp-formula> <jats:alternatives> <jats:tex-math>[{-}MBig(big(a*(gcirc |u|)big)(1)Big)u''(t)=lambda fbig(t,u(t)big),quad tin(0,1), ]</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0308210524000775_eqnU1.png\"/> </jats:alternatives> </jats:disp-formula>in the case in which <jats:inline-formula> <jats:alternatives> <jats:tex-math>$g$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000775_inline3.png\"/> </jats:alternatives> </jats:inline-formula> can satisfy very generalized growth conditions; in addition, <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=\"S0308210524000775_inline4.png\"/> </jats:alternatives> </jats:inline-formula> is allowed to be both sign-changing and vanishing. Existence of at least one positive solution to this equation equipped with boundary data is considered. We demonstrate that the nonlocal coefficient <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=\"S0308210524000775_inline5.png\"/> </jats:alternatives> </jats:inline-formula> allows the forcing term <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=\"S0308210524000775_inline6.png\"/> </jats:alternatives> </jats:inline-formula> to be free of almost all assumptions other than continuity.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"114 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142257744","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":"On a supersonic-sonic patch arising from the two-dimensional Riemann problem of the compressible Euler equations","authors":"Yanbo Hu, Guodong Wang","doi":"10.1017/prm.2024.76","DOIUrl":"https://doi.org/10.1017/prm.2024.76","url":null,"abstract":"We are interested in the two-dimensional four-constant Riemann problem to the isentropic compressible Euler equations. In terms of the self-similar variables, the governing system is of nonlinear mixed-type and the solution configuration typically contains transonic and small-scale structures. We construct a supersonic-sonic patch along a pseudo-streamline from the supersonic part to a sonic point. This kind of patch appears frequently in the two-dimensional Riemann problem and is a building block for constructing a global solution. To overcome the difficulty caused by the sonic degeneracy, we apply the characteristic decomposition technique to handle the problem in a partial hodograph plane. We establish a regular supersonic solution for the original problem by showing the global one-to-one property of the partial hodograph transformation. The uniform regularity of the solution and the regularity of an associated sonic curve are also discussed.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"37 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142257746","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":"Duality pairs, phantom maps, and definability in triangulated categories","authors":"Isaac Bird, Jordan Williamson","doi":"10.1017/prm.2024.73","DOIUrl":"https://doi.org/10.1017/prm.2024.73","url":null,"abstract":"<p>We define duality triples and duality pairs in compactly generated triangulated categories and investigate their properties. This enables us to give an elementary way to determine whether a class is closed under pure subobjects, pure quotients and pure extensions, as well as providing a way to show the existence of approximations. One key ingredient is a new characterization of phantom maps. We then introduce an axiomatic form of Auslander–Gruson–Jensen duality, from which we define dual definable categories, and show that these coincide with symmetric coproduct closed duality pairs. This framework is ubiquitous, encompassing both algebraic triangulated categories and stable homotopy theories. Accordingly, we provide many applications in both settings, with a particular emphasis on silting theory and stratified tensor-triangulated categories.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"5 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217205","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":"Dual formulation of constrained solutions of the multi-state Choquard equation","authors":"Gershon Wolansky","doi":"10.1017/prm.2024.69","DOIUrl":"https://doi.org/10.1017/prm.2024.69","url":null,"abstract":"<p>The Choquard equation is a partial differential equation that has gained significant interest and attention in recent decades. It is a nonlinear equation that combines elements of both the Laplace and Schrödinger operators, and it arises frequently in the study of numerous physical phenomena, from condensed matter physics to nonlinear optics.</p><p>In particular, the steady states of the Choquard equation were thoroughly investigated using a variational functional acting on the wave functions.</p><p>In this article, we introduce a dual formulation for the variational functional in terms of the potential induced by the wave function, and use it to explore the existence of steady states of a multi-state version the Choquard equation in critical and sub-critical cases.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"60 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217204","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 roles of nonlinear diffusion, haptotaxis and ECM remodelling in determining the global solvability of a cancer invasion model","authors":"Chunhua Jin","doi":"10.1017/prm.2024.70","DOIUrl":"https://doi.org/10.1017/prm.2024.70","url":null,"abstract":"<p>In this paper, we consider the following PDE-ODE system modelling cancer invasion with slow diffusion and ECM remodelling,<span><span data-mathjax-type=\"texmath\"><span>[ begin{cases} u_t=Delta u^m-chinablacdot(unabla v)-xinablacdot(unablaomega)+mu u(1-u-omega), v_t=Delta v+u-v, omega_t={-}vomega+eta omega(1-u-omega). end{cases} ]</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114635338-0220:S0308210524000702:S0308210524000702_eqnU1.png\"/></span>For the special case <span><span><span data-mathjax-type=\"texmath\"><span>$eta =0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114635338-0220:S0308210524000702:S0308210524000702_inline3.png\"/></span></span>, fruitful results have been achieved since Tao and Winkler's work in 2011. However, there is no any progress for the general case <span><span><span data-mathjax-type=\"texmath\"><span>$eta &gt;0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114635338-0220:S0308210524000702:S0308210524000702_inline4.png\"/></span></span> in the past ten years. In this paper, we analysed some commonly used research methods when <span><span><span data-mathjax-type=\"texmath\"><span>$eta =0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114635338-0220:S0308210524000702:S0308210524000702_inline5.png\"/></span></span>, and found that these methods are completely unsuitable for situations where <span><span><span data-mathjax-type=\"texmath\"><span>$eta &gt;0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114635338-0220:S0308210524000702:S0308210524000702_inline6.png\"/></span></span>. By introducing some new forms of functionals, we reconstruct the relationship between the haptotactic term and the nonlinear diffusion term, and ultimately prove the global existence of weak solutions. This result improves and perfects a series of works previously presented in the literature.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"138 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217206","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":"Maximizing weighted sums of binomial coefficients using generalized continued fractions","authors":"S.P. Glasby, G.R. Paseman","doi":"10.1017/prm.2024.46","DOIUrl":"https://doi.org/10.1017/prm.2024.46","url":null,"abstract":"Let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$m,,rin {mathbb {Z}}$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000465_inline1.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$omega in {mathbb {R}}$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000465_inline2.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; satisfy &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$0leqslant rleqslant m$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000465_inline3.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$omega geqslant 1$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000465_inline4.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. Our main result is a generalized continued fraction for an expression involving the partial binomial sum &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$s_m(r) = sum _{i=0}^rbinom{m}{i}$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000465_inline5.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. We apply this to create new upper and lower bounds for &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$s_m(r)$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000465_inline6.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and thus for &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$g_{omega,m}(r)=omega ^{-r}s_m(r)$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000465_inline7.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. We also bound an integer &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$r_0 in {0,,1,,ldots,,m}$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000465_inline8.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; such that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$g_{omega,m}(0)&lt;cdots &lt; g_{omega,m}(r_0-1)leqslant g_{omega,m}(r_0)$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000465_inline9.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$g_{omega,m}(r_0)&gt;cdots &gt;g_{omega,m}(m)$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000465_inline10.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"3 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217230","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":"Note on almost isometric ideals and local retracts in Banach and metric spaces","authors":"Leandro Candido, Marek Cúth, Ondřej Smetana","doi":"10.1017/prm.2024.68","DOIUrl":"https://doi.org/10.1017/prm.2024.68","url":null,"abstract":"We exhibit a new approach to the proofs of the existence of a large family of almost isometric ideals in nonseparable Banach spaces and existence of a large family of almost isometric local retracts in metric spaces. Our approach also implies the existence of a large family of nontrivial projections on every dual of a nonseparable Banach space. We prove three possible formulations of our results are equivalent. Some applications are mentioned which witness the usefulness of our novel approach.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"54 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217207","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":"Rigidity for the perimeter inequality under Schwarz symmetrization","authors":"Georgios Domazakis","doi":"10.1017/prm.2024.59","DOIUrl":"https://doi.org/10.1017/prm.2024.59","url":null,"abstract":"<p>In this paper, we give necessary and sufficient conditions for the rigidity of the perimeter inequality under Schwarz symmetrization. The term <span>rigidity</span> refers to the situation in which the equality cases are only obtained by translations of the symmetric set. In particular, we prove that the sufficient conditions for rigidity provided in M. Barchiesi, F. Cagnetti and N. Fusco [Stability of the Steiner symmetrization of convex sets. J. Eur. Math. Soc. 15 (2013), 1245-1278.] are also necessary.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"22 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258253","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":"On a class of self-similar sets which contain finitely many common points","authors":"Kan Jiang, Derong Kong, Wenxia Li, Zhiqiang Wang","doi":"10.1017/prm.2024.66","DOIUrl":"https://doi.org/10.1017/prm.2024.66","url":null,"abstract":"For <jats:inline-formula> <jats:alternatives> <jats:tex-math>$lambda in (0,,1/2]$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000660_inline1.png\"/> </jats:alternatives> </jats:inline-formula> let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$K_lambda subset mathbb {R}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000660_inline2.png\"/> </jats:alternatives> </jats:inline-formula> be a self-similar set generated by the iterated function system <jats:inline-formula> <jats:alternatives> <jats:tex-math>${lambda x,, lambda x+1-lambda }$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000660_inline3.png\"/> </jats:alternatives> </jats:inline-formula>. Given <jats:inline-formula> <jats:alternatives> <jats:tex-math>$xin (0,,1/2)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000660_inline4.png\"/> </jats:alternatives> </jats:inline-formula>, let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Lambda (x)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000660_inline5.png\"/> </jats:alternatives> </jats:inline-formula> be the set of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$lambda in (0,,1/2]$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000660_inline6.png\"/> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$xin K_lambda$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000660_inline7.png\"/> </jats:alternatives> </jats:inline-formula>. In this paper we show that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Lambda (x)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000660_inline8.png\"/> </jats:alternatives> </jats:inline-formula> is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any <jats:inline-formula> <jats:alternatives> <jats:tex-math>$y_1,,ldots,, y_pin (0,,1/2)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000660_inline9.png\"/> </jats:alternatives> </jats:inline-formula> there exists a full Hausdorff dimensional set of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$lambda in (0,,1/2]$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000660_inline10.png\"/> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"31 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141196797","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}