Pub Date : 2024-09-13DOI: 10.1017/s1446788724000107
SOMNATH GANDAL, JAGMOHAN TYAGI
<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
{"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}
Pub Date : 2024-09-13DOI: 10.1017/s1446788724000119
VOLODYMYR MAZORCHUK, SHRADDHA SRIVASTAVA
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.
{"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}
Pub Date : 2024-09-13DOI: 10.1017/s1446788724000090
KIRAN MEENA, HEMANGI MADHUSUDAN SHAH, BAYRAM ŞAHIN
<p>This article <span>introduces</span> 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 <span><span><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"><span data-mathjax-type="texmath"><span>$varphi : M to N$</span></span></img></span></span> between Riemannian manifolds to be a Clairaut conformal Riemannian map with girth <span><span><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"><span data-mathjax-type="texmath"><span>$s = e^f$</span></span></img></span></span>. We show that the fibers of <span><span><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"><span data-mathjax-type="texmath"><span>$varphi $</span></span></img></span></span> are totally umbilical with mean curvature vector field the negative gradient of the logarithm of the girth function, that is, <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913065017866-0657:S1446788724000090:S1446788724000090_inline4.png"><span data-mathjax-type="texmath"><span>$-nabla f$</span></span></img></span></span>. Using this, we obtain a local splitting of <span>M</span> 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 <span>f</span>, 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 <span>M</span> becomes a product manifold, under some hypothesis on <span>f</span>. Analysis of the Laplacian of <span>f</span> also yields the splitting of the universal covering space of <span>M</span> as a warped product under some appropriate conditions. We calculate the sectional curvature and mixed sectional curvature of <span>M</span> when <span>f</span> 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 <span>f</span> is a distance function. We als
本文介绍了克莱劳特共形黎曼图。这一概念包括之前研究过的克莱劳特共形消隐、克莱劳特黎曼消隐和克莱劳特黎曼图等特例，在经典曲面理论中众所周知。为此，我们找到了黎曼流形之间的共形黎曼图 $varphi : M to N$ 成为周长为 $s = e^f$ 的克莱劳特共形黎曼图的必要条件和充分条件。我们证明了 $varphi $ 的纤维是完全脐形的，其平均曲率向量场为周长函数对数的负梯度，即 $-nabla f$。利用这一点，如果水平空间是可积分的（在一些适当的假设下），我们可以得到 M 的局部分裂，即翘曲积和通常积。我们还提供了一些克莱劳特共形黎曼映射的例子，以证实我们的主要定理。我们观察到，总流形上对数周长（即 f 的周长）的拉普拉卡矩具有特殊形式。它还原为水平分布上的拉普拉卡方，如果它是非负的，那么在对 f 的某种假设下，M 的普遍覆盖空间就会成为一个积流形。对 f 的拉普拉卡方进行分析，还可以得到在某些适当条件下，M 的普遍覆盖空间会分裂为一个翘曲积。当 f 是距离函数时，我们计算 M 的截面曲率和混合截面曲率。我们还发现了总流形与纤维对称，特别是具有恒定截面曲率之间的关系，并由此比较了它们的普遍覆盖空间，如果纤维也是完整的，条件是 f 是距离函数。我们还找到了纤维曲率张量为半对称的条件，前提是总流形为半对称且 f 为距离函数。反过来，这又给出了对称半对称空间的翘曲乘积为两个对称半对称子空间（在某种假设下）。同样，如果黎曼曲率张量场的赫塞斯或拉普拉卡矩为零，或具有谐波曲率张量，那么如果 f 也是距离函数，$varphi $ 的纤维也满足同样的性质。通过得到克莱劳特共形黎曼映射的波赫纳（Bochner）式公式，我们建立了纤维和水平空间上的黎奇曲率张量发散与相应标量曲率之间的关系。我们还研究了恒定长度的水平基林向量场，并证明它们在适当的假设条件下是平行的。这反过来又给出了总流形的分裂，如果它允许水平平行基林向量场并且水平空间是可积分的。最后，假定 $nabla f$ 是 M 上的非难梯度利玛窦孤子，我们证明任何垂直向量场都是不可压缩的，因此纤维的体积形式在向量场的流动下是不变的。
{"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":"<p>This article <span>introduces</span> 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 <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$varphi : M to N$</span></span></img></span></span> between Riemannian manifolds to be a Clairaut conformal Riemannian map with girth <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$s = e^f$</span></span></img></span></span>. We show that the fibers of <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$varphi $</span></span></img></span></span> are totally umbilical with mean curvature vector field the negative gradient of the logarithm of the girth function, that is, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913065017866-0657:S1446788724000090:S1446788724000090_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$-nabla f$</span></span></img></span></span>. Using this, we obtain a local splitting of <span>M</span> 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 <span>f</span>, 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 <span>M</span> becomes a product manifold, under some hypothesis on <span>f</span>. Analysis of the Laplacian of <span>f</span> also yields the splitting of the universal covering space of <span>M</span> as a warped product under some appropriate conditions. We calculate the sectional curvature and mixed sectional curvature of <span>M</span> when <span>f</span> 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 <span>f</span> 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}
Pub Date : 2024-09-13DOI: 10.1017/s1446788724000089
PAOLO BELLINGERI, CELESTE DAMIANI, OSCAR OCAMPO, CHARALAMPOS STYLIANAKIS
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.
{"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}
Pub Date : 2024-05-31DOI: 10.1017/s1446788724000077
GUANGFU CAO, LI HE, JI LI, SHUQING ZHANG
Let $B(Omega )$ be a Banach space of holomorphic functions on a bounded connected domain $Omega $ in ${{mathbb C}^n}$ . In this paper, we establish a criterion for $B(Omega )$ to be reflexive via evaluation functions on $B(Omega )$ , that is, $B(Omega )$ is reflexive if and only if the evaluation functions span the dual space $(B(Omega ))^{*} $ .
{"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}
Pub Date : 2024-05-31DOI: 10.1017/s1446788724000065
VLADIMIR SHCHIGOLEV
For closed subgroups L and R of a compact Lie group G, a left L-space X, and an L-equivariant continuous map $A:Xto G/R$ , we introduce the twisted action of the equivariant cohomology $H_R^{bullet }(mathrm {pt},Bbbk )$ on the equivariant cohomology $H_L^{bullet }(X,Bbbk )$ . Considering this action as a right action, $H_L^{bullet }(X,Bbbk )$ becomes a bimodule together with the canonical left action of $H_L^{bullet }(mathrm {pt},Bbbk )$ . 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.
对于紧凑李群 G 的封闭子群 L 和 R、左 L 空间 X 以及 L-变量连续映射 $A:Xto G/R$，我们引入了等变同调 $H_R^{bullet }(mathrm {pt},Bbbk )$ 对等变同调 $H_L^{bullet }(X,Bbbk )$ 的扭曲作用。把这个作用看作右作用，$H_L^{bullet }(X,Bbbk )$ 就变成了一个双模，同时还有$H_L^{bullet }(mathrm {pt},Bbbk )$ 的典型左作用。利用这个双模块结构，我们证明了库奈特同构的等变版本。我们将这一结果应用于计算博特-萨缪尔森（Bott-Samelson）变体的等变同构，以及它们之间的双模态的几何构造。
{"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}
Pub Date : 2024-04-15DOI: 10.1017/s1446788724000053
YOSHIMICHI UEDA
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.
{"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}
Pub Date : 2024-03-20DOI: 10.1017/s1446788724000041
FANG LIU, HONG SUN
<p>In this paper, we study the singular boundary value problem <span><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"><span data-mathjax-type="texmath"><span>$$ begin{align*} begin{cases} Delta_infty^h u=lambda f(x,u,Du) quad &mathrm{in}; Omega, u>0quad &mathrm{in}; Omega, u=0 quad &mathrm{on} ;partialOmega, end{cases} end{align*} $$</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:20240319145321108-0826:S1446788724000041:S1446788724000041_inline1.png"><span data-mathjax-type="texmath"><span>$lambda>0$</span></span></img></span></span> is a parameter, <span><span><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"><span data-mathjax-type="texmath"><span>$h>1$</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:20240319145321108-0826:S1446788724000041:S1446788724000041_inline3.png"><span data-mathjax-type="texmath"><span>$Delta _infty ^h u=|Du|^{h-3} langle D^2uDu,Du rangle $</span></span></img></span></span> is the highly degenerate and <span>h</span>-homogeneous operator related to the infinity Laplacian. The nonlinear term <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline4.png"><span data-mathjax-type="texmath"><span>$f(x,t,p):Omega times (0,infty )times mathbb {R}^{n}rightarrow mathbb {R}$</span></span></img></span></span> is a continuous function and may exhibit singularity at <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline5.png"><span data-mathjax-type="texmath"><span>$trightarrow 0^{+}$</span></span></img></span></span>. We establish the comparison principle by the double variables method for the general equation <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline6.png"><span data-mathjax-type="texmath"><span>$Delta _infty ^h u=F(x,u,Du)$</span></span></img></span></span> under some conditions on the term <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline7.png"><span data-mathjax-type="texmath"><span>$F(x,t,p)$</span></s
{"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":"<p>In this paper, we study the singular boundary value problem <span><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\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*} begin{cases} Delta_infty^h u=lambda f(x,u,Du) quad &mathrm{in}; Omega, u>0quad &mathrm{in}; Omega, u=0 quad &mathrm{on} ;partialOmega, end{cases} end{align*} $$</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:20240319145321108-0826:S1446788724000041:S1446788724000041_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$lambda>0$</span></span></img></span></span> is a parameter, <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$h>1$</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:20240319145321108-0826:S1446788724000041:S1446788724000041_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$Delta _infty ^h u=|Du|^{h-3} langle D^2uDu,Du rangle $</span></span></img></span></span> is the highly degenerate and <span>h</span>-homogeneous operator related to the infinity Laplacian. The nonlinear term <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$f(x,t,p):Omega times (0,infty )times mathbb {R}^{n}rightarrow mathbb {R}$</span></span></img></span></span> is a continuous function and may exhibit singularity at <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$trightarrow 0^{+}$</span></span></img></span></span>. We establish the comparison principle by the double variables method for the general equation <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$Delta _infty ^h u=F(x,u,Du)$</span></span></img></span></span> under some conditions on the term <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319145321108-0826:S1446788724000041:S1446788724000041_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$F(x,t,p)$</span></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}
Pub Date : 2024-03-19DOI: 10.1017/s1446788724000028
MATTHEW CONDER, HARRIS LEUNG, JEROEN SCHILLEWAERT
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 ${mathrm {SL}_2}(K)$ , where K is a p-adic field, contains two elements that generate a dense subgroup of ${mathrm {SL}_2}(K)$ , which is a special case of a result by Breuillard and Gelander [‘On dense free subgroups of Lie groups’, J. Algebra261(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 ${mathrm {SL}_2}(K)$ over a nonarchimedean local field K is discrete if and only if each of its two-generator subgroups is discrete.
{"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}
Pub Date : 2024-03-11DOI: 10.1017/s144678872400003x
NEERAJ K. DHANWANI, MAHENDER SINGH
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 n for which the n-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 n-quandles’, Algebr. Geom. Topol.17(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.
{"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}
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