Journal of the Australian Mathematical Society最新文献

{"title":"ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS TO FRACTIONAL NEUMANN PROBLEMS","authors":"SOMNATH GANDAL, JAGMOHAN TYAGI","doi":"10.1017/s1446788724000107","DOIUrl":"https://doi.org/10.1017/s1446788724000107","url":null,"abstract":"<p>We study the asymptotic behaviour of the least energy solutions to the following class of nonlocal Neumann problems: <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*} begin{cases} { d(-Delta)^{s}u+ u= vert uvert^{p-1}u } & text{in } Omega, {u>0} & text{in } Omega, { mathcal{N}_{s}u=0 } & text{in } mathbb{R}^{n}setminus overline{Omega}, end{cases} end{align*} $$</span></span></img></span></p><p>where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$Omega subset mathbb {R}^{n}$</span></span></img></span></span> is a bounded domain of class <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$C^{1,1}$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$1<p<({n+s})/({n-s}),,n>max {1, 2s }, 0<s<1, d>0$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {N}_{s}u$</span></span></img></span></span> is the nonlocal Neumann derivative. We show that for small <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$d,$</span></span></img></span></span> the least energy solutions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$u_d$</span></span></img></span></span> of the above problem achieve an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$L^{infty }$</span></span></img></span></span>-bound independent of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/ve","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"2 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"KRONECKER COEFFICIENTS FOR (DUAL) SYMMETRIC INVERSE SEMIGROUPS","authors":"VOLODYMYR MAZORCHUK, SHRADDHA SRIVASTAVA","doi":"10.1017/s1446788724000119","DOIUrl":"https://doi.org/10.1017/s1446788724000119","url":null,"abstract":"<p>We study analogues of Kronecker coefficients for symmetric inverse semigroups, for dual symmetric inverse semigroups and for the inverse semigroups of bijections between subquotients of finite sets. In all cases, we reduce the problem of determination of such coefficients to some group-theoretic and combinatorial problems. For symmetric inverse semigroups, we provide an explicit formula in terms of the classical Kronecker and Littlewood–Richardson coefficients for symmetric groups.</p>","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"GEOMETRY OF CLAIRAUT CONFORMAL RIEMANNIAN MAPS","authors":"KIRAN MEENA, HEMANGI MADHUSUDAN SHAH, BAYRAM ŞAHIN","doi":"10.1017/s1446788724000090","DOIUrl":"https://doi.org/10.1017/s1446788724000090","url":null,"abstract":"&lt;p&gt;This article &lt;span&gt;introduces&lt;/span&gt; the Clairaut conformal Riemannian map. This notion includes the previously studied notions of Clairaut conformal submersion, Clairaut Riemannian submersion, and the Clairaut Riemannian map as particular cases, and is well known in the classical theory of surfaces. Toward this, we find the necessary and sufficient condition for a conformal Riemannian map &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913065017866-0657:S1446788724000090:S1446788724000090_inline1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$varphi : M to N$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; between Riemannian manifolds to be a Clairaut conformal Riemannian map with girth &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913065017866-0657:S1446788724000090:S1446788724000090_inline2.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$s = e^f$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. We show that the fibers of &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913065017866-0657:S1446788724000090:S1446788724000090_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$varphi $&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; are totally umbilical with mean curvature vector field the negative gradient of the logarithm of the girth function, that is, &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913065017866-0657:S1446788724000090:S1446788724000090_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$-nabla f$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. Using this, we obtain a local splitting of &lt;span&gt;M&lt;/span&gt; as a warped product and a usual product, if the horizontal space is integrable (under some appropriate hypothesis). We also provide some examples of the Clairaut conformal Riemannian maps to confirm our main theorem. We observe that the Laplacian of the logarithmic girth, that is, of &lt;span&gt;f&lt;/span&gt;, on the total manifold takes the special form. It reduces to the Laplacian on the horizontal distribution, and if it is nonnegative, the universal covering space of &lt;span&gt;M&lt;/span&gt; becomes a product manifold, under some hypothesis on &lt;span&gt;f&lt;/span&gt;. Analysis of the Laplacian of &lt;span&gt;f&lt;/span&gt; also yields the splitting of the universal covering space of &lt;span&gt;M&lt;/span&gt; as a warped product under some appropriate conditions. We calculate the sectional curvature and mixed sectional curvature of &lt;span&gt;M&lt;/span&gt; when &lt;span&gt;f&lt;/span&gt; is a distance function. We also find the relationships between the total manifold and the fibers being symmetrical and, in particular, having constant sectional curvature, and from there, we compare their universal covering spaces, if fibers are also complete, provided &lt;span&gt;f&lt;/span&gt; is a distance function. We als","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"71 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"CONGRUENCE SUBGROUPS OF BRAID GROUPS AND CRYSTALLOGRAPHIC QUOTIENTS. PART I","authors":"PAOLO BELLINGERI, CELESTE DAMIANI, OSCAR OCAMPO, CHARALAMPOS STYLIANAKIS","doi":"10.1017/s1446788724000089","DOIUrl":"https://doi.org/10.1017/s1446788724000089","url":null,"abstract":"<p>This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.</p>","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"124 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"EVALUATION FUNCTIONS AND REFLEXIVITY OF BANACH SPACES OF HOLOMORPHIC FUNCTIONS","authors":"GUANGFU CAO, LI HE, JI LI, SHUQING ZHANG","doi":"10.1017/s1446788724000077","DOIUrl":"https://doi.org/10.1017/s1446788724000077","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000077_inline1.png\"/> <jats:tex-math> $B(Omega )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a Banach space of holomorphic functions on a bounded connected domain <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000077_inline2.png\"/> <jats:tex-math> $Omega $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000077_inline3.png\"/> <jats:tex-math> ${{mathbb C}^n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we establish a criterion for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000077_inline4.png\"/> <jats:tex-math> $B(Omega )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> to be reflexive via evaluation functions on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000077_inline5.png\"/> <jats:tex-math> $B(Omega )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, that is, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000077_inline6.png\"/> <jats:tex-math> $B(Omega )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is reflexive if and only if the evaluation functions span the dual space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000077_inline7.png\"/> <jats:tex-math> $(B(Omega ))^{*} $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"45 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"TWISTED ACTIONS ON COHOMOLOGIES AND BIMODULES","authors":"VLADIMIR SHCHIGOLEV","doi":"10.1017/s1446788724000065","DOIUrl":"https://doi.org/10.1017/s1446788724000065","url":null,"abstract":"For closed subgroups <jats:italic>L</jats:italic> and <jats:italic>R</jats:italic> of a compact Lie group <jats:italic>G</jats:italic>, a left <jats:italic>L</jats:italic>-space <jats:italic>X</jats:italic>, and an <jats:italic>L</jats:italic>-equivariant continuous map <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000065_inline1.png\"/> <jats:tex-math> $A:Xto G/R$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we introduce the twisted action of the equivariant cohomology <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000065_inline2.png\"/> <jats:tex-math> $H_R^{bullet }(mathrm {pt},Bbbk )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> on the equivariant cohomology <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000065_inline3.png\"/> <jats:tex-math> $H_L^{bullet }(X,Bbbk )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Considering this action as a right action, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000065_inline4.png\"/> <jats:tex-math> $H_L^{bullet }(X,Bbbk )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> becomes a bimodule together with the canonical left action of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000065_inline5.png\"/> <jats:tex-math> $H_L^{bullet }(mathrm {pt},Bbbk )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Using this bimodule structure, we prove an equivariant version of the Künneth isomorphism. We apply this result to the computation of the equivariant cohomologies of Bott–Samelson varieties and to a geometric construction of the bimodule morphisms between them.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"62 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"SPHERICAL REPRESENTATIONS FOR -FLOWS III: WEIGHT-EXTENDED BRANCHING GRAPHS","authors":"YOSHIMICHI UEDA","doi":"10.1017/s1446788724000053","DOIUrl":"https://doi.org/10.1017/s1446788724000053","url":null,"abstract":"We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"3 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140571365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"VISCOSITY SOLUTIONS TO THE INFINITY LAPLACIAN EQUATION WITH SINGULAR NONLINEAR TERMS","authors":"FANG LIU, HONG SUN","doi":"10.1017/s1446788724000041","DOIUrl":"https://doi.org/10.1017/s1446788724000041","url":null,"abstract":"&lt;p&gt;In this paper, we study the singular boundary value problem &lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_eqnu1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$$ begin{align*} begin{cases} Delta_infty^h u=lambda f(x,u,Du) quad &amp;mathrm{in}; Omega, u&gt;0quad &amp;mathrm{in}; Omega, u=0 quad &amp;mathrm{on} ;partialOmega, end{cases} end{align*} $$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/p&gt;&lt;p&gt;where &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$lambda&gt;0$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is a parameter, &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline2.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$h&gt;1$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$Delta _infty ^h u=|Du|^{h-3} langle D^2uDu,Du rangle $&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is the highly degenerate and &lt;span&gt;h&lt;/span&gt;-homogeneous operator related to the infinity Laplacian. The nonlinear term &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$f(x,t,p):Omega times (0,infty )times mathbb {R}^{n}rightarrow mathbb {R}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is a continuous function and may exhibit singularity at &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$trightarrow 0^{+}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. We establish the comparison principle by the double variables method for the general equation &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$Delta _infty ^h u=F(x,u,Du)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; under some conditions on the term &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$F(x,t,p)$&lt;/span&gt;&lt;/s","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"122 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140172514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"BASIC NONARCHIMEDEAN JØRGENSEN THEORY","authors":"MATTHEW CONDER, HARRIS LEUNG, JEROEN SCHILLEWAERT","doi":"10.1017/s1446788724000028","DOIUrl":"https://doi.org/10.1017/s1446788724000028","url":null,"abstract":"We prove a nonarchimedean analogue of Jørgensen’s inequality, and use it to deduce several algebraic convergence results. As an application, we show that every dense subgroup of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000028_inline1.png\" /> <jats:tex-math> ${mathrm {SL}_2}(K)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:italic>K</jats:italic> is a <jats:italic>p</jats:italic>-adic field, contains two elements that generate a dense subgroup of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000028_inline2.png\" /> <jats:tex-math> ${mathrm {SL}_2}(K)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is a special case of a result by Breuillard and Gelander [‘On dense free subgroups of Lie groups’, <jats:italic>J. Algebra</jats:italic>261(2) (2003), 448–467]. We also list several other related results, which are well known to experts, but not easy to locate in the literature; for example, we show that a nonelementary subgroup of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000028_inline3.png\" /> <jats:tex-math> ${mathrm {SL}_2}(K)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> over a nonarchimedean local field <jats:italic>K</jats:italic> is discrete if and only if each of its two-generator subgroups is discrete.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"85 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140172749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"FINITENESS OF CANONICAL QUOTIENTS OF DEHN QUANDLES OF SURFACES","authors":"NEERAJ K. DHANWANI, MAHENDER SINGH","doi":"10.1017/s144678872400003x","DOIUrl":"https://doi.org/10.1017/s144678872400003x","url":null,"abstract":"<p>The Dehn quandle of a closed orientable surface is the set of isotopy classes of nonseparating simple closed curves with a natural quandle structure arising from Dehn twists. In this paper, we consider the finiteness of some canonical quotients of these quandles. For a surface of positive genus, we give a precise description of the 2-quandle of its Dehn quandle. Further, with some exceptions for genus more than 2, we determine all values of <span>n</span> for which the <span>n</span>-quandle of its Dehn quandle is finite. The result can be thought of as the Dehn quandle analogue of a similar result of Hoste and Shanahan for link quandles [‘Links with finite <span>n</span>-quandles’, <span>Algebr. Geom. Topol.</span> <span>17</span>(5) (2017), 2807–2823]. We also compute the size of the smallest nontrivial quandle quotient of the Dehn quandle of a surface. Along the way, we prove that the involutory quotient of an Artin quandle is precisely the corresponding Coxeter quandle, and also determine the smallest nontrivial quotient of a braid quandle.</p>","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"42 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140097987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}