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

{"title":"Regular matrices of unbounded linear operators","authors":"Paolo Leonetti","doi":"10.1017/prm.2024.1","DOIUrl":"https://doi.org/10.1017/prm.2024.1","url":null,"abstract":"<p>Let <span><span><span data-mathjax-type=\"texmath\"><span>$X,,Y$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130151803585-0502:S0308210524000015:S0308210524000015_inline1.png\"/></span></span> be Banach spaces and fix a linear operator <span><span><span data-mathjax-type=\"texmath\"><span>$T in mathcal {L}(X,,Y)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130151803585-0502:S0308210524000015:S0308210524000015_inline2.png\"/></span></span> and ideals <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {I},, mathcal {J}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130151803585-0502:S0308210524000015:S0308210524000015_inline3.png\"/></span></span> on the nonnegative integers. We obtain Silverman–Toeplitz type theorems on matrices <span><span><span data-mathjax-type=\"texmath\"><span>$A=(A_{n,k}: n,,k in omega )$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130151803585-0502:S0308210524000015:S0308210524000015_inline4.png\"/></span></span> of linear operators in <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {L}(X,,Y)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130151803585-0502:S0308210524000015:S0308210524000015_inline5.png\"/></span></span>, so that<span><span data-mathjax-type=\"texmath\"><span>[ mathcal{J}text{-}lim Aboldsymbol{x}=T(mathcal{I}text{-}lim boldsymbol{x}) ]</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130151803585-0502:S0308210524000015:S0308210524000015_eqnU1.png\"/></span>for every <span><span><span data-mathjax-type=\"texmath\"><span>$X$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130151803585-0502:S0308210524000015:S0308210524000015_inline6.png\"/></span></span>-valued sequence <span><span><span data-mathjax-type=\"texmath\"><span>$boldsymbol {x}=(x_0,,x_1,,ldots )$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130151803585-0502:S0308210524000015:S0308210524000015_inline7.png\"/></span></span> which is <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {I}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130151803585-0502:S0308210524000015:S0308210524000015_inline8.png\"/></span></span>-convergent (and bounded). This allows us to establish the relationship between the classical Silv","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"32 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647939","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":"Karamata's theorem for regularized Cauchy transforms","authors":"Matthias Langer, Harald Woracek","doi":"10.1017/prm.2023.128","DOIUrl":"https://doi.org/10.1017/prm.2023.128","url":null,"abstract":"We prove Abelian and Tauberian theorems for regularized Cauchy transforms of positive Borel measures on the real line whose distribution functions grow at most polynomially at infinity. In particular, we relate the asymptotics of the distribution functions to the asymptotics of the regularized Cauchy transform.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"101 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139585133","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 generalized eigenvalue problems of fractional (p, q)-Laplace operator with two parameters","authors":"Nirjan Biswas, Firoj Sk","doi":"10.1017/prm.2023.134","DOIUrl":"https://doi.org/10.1017/prm.2023.134","url":null,"abstract":"<p>For <span><span><span data-mathjax-type=\"texmath\"><span>$s_1,,s_2in (0,,1)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240120165804277-0267:S0308210523001348:S0308210523001348_inline2.png\"/></span></span> and <span><span><span data-mathjax-type=\"texmath\"><span>$p,,q in (1,, infty )$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240120165804277-0267:S0308210523001348:S0308210523001348_inline3.png\"/></span></span>, we study the following nonlinear Dirichlet eigenvalue problem with parameters <span><span><span data-mathjax-type=\"texmath\"><span>$alpha,, beta in mathbb {R}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240120165804277-0267:S0308210523001348:S0308210523001348_inline4.png\"/></span></span> driven by the sum of two nonlocal operators:<span><span data-mathjax-type=\"texmath\"><span>[ (-Delta)^{s_1}_p u+(-Delta)^{s_2}_q u=alpha|u|^{p-2}u+beta|u|^{q-2}u text{in }Omega, quad u=0 text{in } mathbb{R}^d setminus Omega, quad mathrm{(P)} ]</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240120165804277-0267:S0308210523001348:S0308210523001348_eqnU1.png\"/></span>where <span><span><span data-mathjax-type=\"texmath\"><span>$Omega subset mathbb {R}^d$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240120165804277-0267:S0308210523001348:S0308210523001348_inline5.png\"/></span></span> is a bounded open set. Depending on the values of <span><span><span data-mathjax-type=\"texmath\"><span>$alpha,,beta$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240120165804277-0267:S0308210523001348:S0308210523001348_inline6.png\"/></span></span>, we completely describe the existence and non-existence of positive solutions to (P). We construct a continuous threshold curve in the two-dimensional <span><span><span data-mathjax-type=\"texmath\"><span>$(alpha,, beta )$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240120165804277-0267:S0308210523001348:S0308210523001348_inline7.png\"/></span></span>-plane, which separates the regions of the existence and non-existence of positive solutions. In addition, we prove that the first Dirichlet eigenfunctions of the fractional <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:20240120165804277-0267:S0308210523001348:S0308210523001348_inline8.png\"/></span></span>-Laplace and fractiona","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"57 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139517566","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 gradient estimate, rigidity and almost rigidity of Green functions on non-parabolic RCD(0, N) spaces","authors":"Shouhei Honda, Yuanlin Peng","doi":"10.1017/prm.2024.131","DOIUrl":"https://doi.org/10.1017/prm.2024.131","url":null,"abstract":"<p>Inspired by a result in T. H. Colding. (16). <span>Acta. Math.</span> <span>209</span>(2) (2012), 229-263 [16] of Colding, the present paper studies the Green function <span><span><span data-mathjax-type=\"texmath\"><span>$G$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116102450072-0592:S0308210523001312:S0308210523001312_inline2.png\"/></span></span> on a non-parabolic <span><span><span data-mathjax-type=\"texmath\"><span>$operatorname {RCD}(0,,N)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116102450072-0592:S0308210523001312:S0308210523001312_inline3.png\"/></span></span> space <span><span><span data-mathjax-type=\"texmath\"><span>$(X,, mathsf {d},, mathfrak {m})$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116102450072-0592:S0308210523001312:S0308210523001312_inline4.png\"/></span></span> for some finite <span><span><span data-mathjax-type=\"texmath\"><span>$N&gt;2$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116102450072-0592:S0308210523001312:S0308210523001312_inline5.png\"/></span></span>. Defining <span><span><span data-mathjax-type=\"texmath\"><span>$mathsf {b}_x=G(x,, cdot )^{frac {1}{2-N}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116102450072-0592:S0308210523001312:S0308210523001312_inline6.png\"/></span></span> for a point <span><span><span data-mathjax-type=\"texmath\"><span>$x in X$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116102450072-0592:S0308210523001312:S0308210523001312_inline7.png\"/></span></span>, which plays a role of a smoothed distance function from <span><span><span data-mathjax-type=\"texmath\"><span>$x$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116102450072-0592:S0308210523001312:S0308210523001312_inline8.png\"/></span></span>, we prove that the gradient <span><span><span data-mathjax-type=\"texmath\"><span>$|nabla mathsf {b}_x|$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116102450072-0592:S0308210523001312:S0308210523001312_inline9.png\"/></span></span> has the canonical pointwise representative with the sharp upper bound in terms of the <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:20240116102450072-0592:S0308210523001312:S0308210523001312_","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"23 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139481144","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":"Nonexpansive and noncontractive mappings on the set of quantum pure states","authors":"Michiya Mori, Peter Šemrl","doi":"10.1017/prm.2024.133","DOIUrl":"https://doi.org/10.1017/prm.2024.133","url":null,"abstract":"<p>Wigner's theorem characterizes isometries of the set of all rank one projections on a Hilbert space. In metric geometry, nonexpansive maps and noncontractive maps are well-studied generalizations of isometries. We show that under certain conditions Wigner symmetries can be characterized as nonexpansive or noncontractive maps on the set of all projections of rank one. The assumptions required for such characterizations are injectivity or surjectivity and they differ in the finite and the infinite-dimensional case. Motivated by a recently obtained optimal version of Uhlhorn's generalization of Wigner's theorem, we also give a description of nonexpansive maps which satisfy a condition that is much weaker than surjectivity. Such maps do not need to be Wigner symmetries. The optimality of all presented results is shown by counterexamples.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"52 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139481093","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":"Classification of simple smooth modules over the Heisenberg–Virasoro algebra","authors":"Haijun Tan, Yufeng Yao, Kaiming Zhao","doi":"10.1017/prm.2024.132","DOIUrl":"https://doi.org/10.1017/prm.2024.132","url":null,"abstract":"<p>In this paper, we classify simple smooth modules over the mirror Heisenberg–Virasoro algebra <span><span><span data-mathjax-type=\"texmath\"><span>${mathfrak {D}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116103651863-0511:S0308210523001324:S0308210523001324_inline1.png\"/></span></span>, and simple smooth modules over the twisted Heisenberg–Virasoro algebra <span><span><span data-mathjax-type=\"texmath\"><span>$bar {mathfrak {D}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116103651863-0511:S0308210523001324:S0308210523001324_inline2.png\"/></span></span> with non-zero level. To this end we generalize Sugawara operators to smooth modules over the Heisenberg algebra, and develop new techniques. As applications, we characterize simple Whittaker modules and simple highest weight modules over <span><span><span data-mathjax-type=\"texmath\"><span>${mathfrak {D}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116103651863-0511:S0308210523001324:S0308210523001324_inline3.png\"/></span></span>. A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg–Virasoro vertex algebras. We also present a few examples of simple smooth <span><span><span data-mathjax-type=\"texmath\"><span>${mathfrak {D}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116103651863-0511:S0308210523001324:S0308210523001324_inline4.png\"/></span></span>-modules and <span><span><span data-mathjax-type=\"texmath\"><span>$bar {mathfrak {D}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116103651863-0511:S0308210523001324:S0308210523001324_inline5.png\"/></span></span>-modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over <span><span><span data-mathjax-type=\"texmath\"><span>$mathfrak {D}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116103651863-0511:S0308210523001324:S0308210523001324_inline6.png\"/></span></span> and <span><span><span data-mathjax-type=\"texmath\"><span>$bar {mathfrak {D}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116103651863-0511:S0308210523001324:S0308210523001324_inline7.png\"/></span></span> which are always tensor products of simple Virasoro modules and simple Heisenberg m","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"19 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139480944","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":"Branching patterns of wave trains in mass-in-mass lattices","authors":"Ling Zhang, Shangjiang Guo","doi":"10.1017/prm.2023.130","DOIUrl":"https://doi.org/10.1017/prm.2023.130","url":null,"abstract":"<p>We investigate the existence and branching patterns of wave trains in the mass-in-mass (MiM) lattice, which is a variant of the Fermi–Pasta–Ulam (FPU) lattice. In contrast to FPU lattice, we have to solve coupled advance-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction and invariant theory. We establish a link between the MiM lattice and the monatomic FPU lattice. That is, the monochromatic and bichromatic wave trains persist near <span><span><span data-mathjax-type=\"texmath\"><span>$mu =0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110161445970-0774:S0308210523001300:S0308210523001300_inline1.png\"/></span></span> in the nonresonance case and in the resonance case <span><span><span data-mathjax-type=\"texmath\"><span>$p:q$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110161445970-0774:S0308210523001300:S0308210523001300_inline2.png\"/></span></span> where <span><span><span data-mathjax-type=\"texmath\"><span>$q$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110161445970-0774:S0308210523001300:S0308210523001300_inline3.png\"/></span></span> is not an integer multiple of <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:20240110161445970-0774:S0308210523001300:S0308210523001300_inline4.png\"/></span></span>. Furthermore, we obtain the multiplicity of bichromatic wave trains in <span><span><span data-mathjax-type=\"texmath\"><span>$p:q$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110161445970-0774:S0308210523001300:S0308210523001300_inline5.png\"/></span></span> resonance where <span><span><span data-mathjax-type=\"texmath\"><span>$q$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110161445970-0774:S0308210523001300:S0308210523001300_inline6.png\"/></span></span> is an integer multiple of <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:20240110161445970-0774:S0308210523001300:S0308210523001300_inline7.png\"/></span></span>, based on the singular theorem.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139423410","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":"Quasilinear duality and inversion in Banach spaces","authors":"Jesús M. F. Castillo, Manuel González","doi":"10.1017/prm.2023.120","DOIUrl":"https://doi.org/10.1017/prm.2023.120","url":null,"abstract":"<p>We present a unified approach to the processes of inversion and duality for quasilinear and <span><span><span data-mathjax-type=\"texmath\"><span>$1$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110161145945-0202:S0308210523001208:S0308210523001208_inline1.png\"/></span></span>-quasilinear maps; in particular, for centralizers and differentials generated by interpolation methods.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"119 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139423464","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":"Cavitation of a spherical body under mechanical and self-gravitational forces","authors":"Pablo V. Negrón–Marrero, Jeyabal Sivaloganathan","doi":"10.1017/prm.2023.125","DOIUrl":"https://doi.org/10.1017/prm.2023.125","url":null,"abstract":"<p>In this paper, we look for minimizers of the energy functional for isotropic compressible elasticity taking into consideration the effect of a gravitational field induced by the body itself. We consider two types of problems: the displacement problem in which the outer boundary of the body is subjected to a Dirichlet-type boundary condition, and the one with zero traction on the boundary but with an internal pressure function. For a spherically symmetric body occupying the unit ball <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {B}in mathbb {R}^3$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240106111107785-0833:S0308210523001257:S0308210523001257_inline1.png\"/></span></span>, the minimization is done within the class of radially symmetric deformations. We give conditions for the existence of such minimizers, for satisfaction of the Euler–Lagrange equations, and show that for large displacements or large internal pressures, the minimizer must develop a cavity at the centre. We discuss a numerical scheme for approximating the minimizers for the displacement problem, together with some simulations that show the dependence of the cavity radius and minimum energy on the displacement and mass density of the body.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"45 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139397124","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":"Adhesion and volume filling in one-dimensional population dynamics under Dirichlet boundary condition","authors":"Hyung Jun Choi, Seonghak Kim, Youngwoo Koh","doi":"10.1017/prm.2023.129","DOIUrl":"https://doi.org/10.1017/prm.2023.129","url":null,"abstract":"<p>We generalize the one-dimensional population model of Anguige &amp; Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by Müller &amp; Šverák [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity. TE check: Please check the reference citation in abstract.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"33 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139398466","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}