Pub Date : 2023-12-13DOI: 10.4153/s0008439523000978
P. Ho
{"title":"A RIGIDITY RESULT FOR THE PRODUCT OF SPHERES","authors":"P. Ho","doi":"10.4153/s0008439523000978","DOIUrl":"https://doi.org/10.4153/s0008439523000978","url":null,"abstract":"","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"79 3‐4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138976892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-11DOI: 10.4153/s0008439523000954
Zhongmin Shen, Runzhong Zhao
The characterization of projectively flat Finsler metrics on an open subset in $R^n$ is the Hilbert’s fourth problem in the regular case. Locally projectively flat Finsler manifolds form an important class of Finsler manifolds. Every Finsler metric induces a spray on the manifold via geodesics. Therefore, it is a natural problem to investigate the geometric and topological properties of manifolds equipped with a spray. In this paper, we study the Pontrjagin classes of a manifold equipped with a locally projectively flat spray and show that such manifold must have zero Pontrjagin classes.
{"title":"On the Pontrjagin classes of spray manifolds","authors":"Zhongmin Shen, Runzhong Zhao","doi":"10.4153/s0008439523000954","DOIUrl":"https://doi.org/10.4153/s0008439523000954","url":null,"abstract":"<p>The characterization of projectively flat Finsler metrics on an open subset in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104114708564-0645:S0008439523000954:S0008439523000954_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$R^n$</span></span></img></span></span> is the Hilbert’s fourth problem in the regular case. Locally projectively flat Finsler manifolds form an important class of Finsler manifolds. Every Finsler metric induces a spray on the manifold via geodesics. Therefore, it is a natural problem to investigate the geometric and topological properties of manifolds equipped with a spray. In this paper, we study the Pontrjagin classes of a manifold equipped with a locally projectively flat spray and show that such manifold must have zero Pontrjagin classes.</p>","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139104806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-11DOI: 10.4153/s0008439523000966
Bin Shu
Suppose $g=g_0+g_1$ is a finite-dimensional restricted Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$. In this article, we propose a conjecture for maximal dimensions of irreducible modules over the universal enveloping algebra $U(g)$ of $g$, as a super generalization of the celebrated first Kac-Weisfeiler conjecture. It is demonstrated that the conjecture holds for all basic classical Lie superalgebras and all completely solvable restricted Lie superalgebras. In this process, we investigate irreducible representations of solvable Lie superalgebras.
{"title":"IRREDUCIBLE MODULES OF MODULAR LIE SUPERALGEBRAS AND SUPER VERSION OF THE FIRST KAC-WEISFEILER CONJECTURE","authors":"Bin Shu","doi":"10.4153/s0008439523000966","DOIUrl":"https://doi.org/10.4153/s0008439523000966","url":null,"abstract":"Suppose $g=g_0+g_1$ is a finite-dimensional restricted Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$. In this article, we propose a conjecture for maximal dimensions of irreducible modules over the universal enveloping algebra $U(g)$ of $g$, as a super generalization of the celebrated first Kac-Weisfeiler conjecture. It is demonstrated that the conjecture holds for all basic classical Lie superalgebras and all completely solvable restricted Lie superalgebras. In this process, we investigate irreducible representations of solvable Lie superalgebras.","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"10 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138979804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-29DOI: 10.4153/s0008439523000917
Carlos Arnoldo Morales, Nguyen Thanh Nguyen
We prove that every topologically stable homeomorphism with global attractor of $mathbb {R}^n$ is topologically stable on its global attractor. The converse is not true. On the other hand, if a homeomorphism with global attractor of a locally compact metric space is expansive and has the shadowing property, then it is topologically stable. This extends the Walters stability theorem (Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems, 1978, pp. 231–244).
我们证明，$mathbb {R}^n$ 的每一个具有全局吸引子的拓扑稳定同构在其全局吸引子上都是拓扑稳定的。反之则不成立。另一方面，如果局部紧凑度量空间中具有全局吸引子的同态是膨胀的，并且具有阴影性质，那么它就是拓扑稳定的。这就扩展了沃尔特斯稳定性定理（沃尔特斯，《论伪轨道追踪特性及其与稳定性的关系》，伦敦：伦敦大学出版社，2006 年）。The structure of attractors in dynamical systems, 1978, pp.）
{"title":"Topological stability for homeomorphisms with global attractor","authors":"Carlos Arnoldo Morales, Nguyen Thanh Nguyen","doi":"10.4153/s0008439523000917","DOIUrl":"https://doi.org/10.4153/s0008439523000917","url":null,"abstract":"<p>We prove that every topologically stable homeomorphism with global attractor of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231221035053529-0023:S0008439523000917:S0008439523000917_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {R}^n$</span></span></img></span></span> is topologically stable on its global attractor. The converse is not true. On the other hand, if a homeomorphism with global attractor of a locally compact metric space is expansive and has the shadowing property, then it is topologically stable. This extends the Walters stability theorem (Walters, <span>On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems</span>, 1978, pp. 231–244).</p>","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"89 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139029953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-29DOI: 10.4153/s0008439523000942
Igor Petkov, Ruslan Salimov, Mariia Stefanchuk
We study a nonlinear Beltrami equation $f_theta =sigma ,|f_r|^m f_r$ in polar coordinates $(r,theta ),$ which becomes the classical Cauchy–Riemann system under $m=0$ and $sigma =ir.$ Using the isoperimetric technique, various lower estimates for $|f(z)|/|z|, f(0)=0,$ as $zto 0,$ are derived under appropriate integral conditions on complex/directional dilatations. The sharpness of the above bounds is illustrated by several examples.
{"title":"Nonlinear Beltrami equation: lower estimates of Schwarz lemma’s type","authors":"Igor Petkov, Ruslan Salimov, Mariia Stefanchuk","doi":"10.4153/s0008439523000942","DOIUrl":"https://doi.org/10.4153/s0008439523000942","url":null,"abstract":"<p>We study a nonlinear Beltrami equation <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231223041908670-0003:S0008439523000942:S0008439523000942_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$f_theta =sigma ,|f_r|^m f_r$</span></span></img></span></span> in polar coordinates <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231223041908670-0003:S0008439523000942:S0008439523000942_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$(r,theta ),$</span></span></img></span></span> which becomes the classical Cauchy–Riemann system under <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231223041908670-0003:S0008439523000942:S0008439523000942_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$m=0$</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:20231223041908670-0003:S0008439523000942:S0008439523000942_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$sigma =ir.$</span></span></img></span></span> Using the isoperimetric technique, various lower estimates for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231223041908670-0003:S0008439523000942:S0008439523000942_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$|f(z)|/|z|, f(0)=0,$</span></span></img></span></span> as <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231223041908670-0003:S0008439523000942:S0008439523000942_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$zto 0,$</span></span></img></span></span> are derived under appropriate integral conditions on complex/directional dilatations. The sharpness of the above bounds is illustrated by several examples.</p>","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139055623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-29DOI: 10.4153/s0008439523000929
Ole Christensen, Pablo Garcia Alvarez, Rae Young Kim
We provide conditions under which a generalized shift-invariant (GSI) system can be approximated by a GSI system for which the generators have compact support in the Fourier domain. The approximation quality will be measured in terms of the Bessel bound (upper frame bound) for the difference between the two GSI systems. In particular, this leads to easily verifiable conditions for a perturbation of a GSI system to preserve the frame property.
{"title":"Truncations of generalized shift-invariant systems","authors":"Ole Christensen, Pablo Garcia Alvarez, Rae Young Kim","doi":"10.4153/s0008439523000929","DOIUrl":"https://doi.org/10.4153/s0008439523000929","url":null,"abstract":"<p>We provide conditions under which a generalized shift-invariant (GSI) system can be approximated by a GSI system for which the generators have compact support in the Fourier domain. The approximation quality will be measured in terms of the Bessel bound (upper frame bound) for the difference between the two GSI systems. In particular, this leads to easily verifiable conditions for a perturbation of a GSI system to preserve the frame property.</p>","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139104762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-28DOI: 10.4153/s0008439523000899
Sergiusz Kużel
Although Naimark dilation theorem was originally stated in 1940, it still finds many important applications in various areas. The objective of this paper is to introduce a method for explicitly constructing the vectors of complementary frames in the Naimark dilation theorem, specifically in cases where the initial Parseval frame contains a Riesz basis as a subset. These findings serve as a foundation for the construction of dual frames.
{"title":"Remarks on Naimark dilation theorem","authors":"Sergiusz Kużel","doi":"10.4153/s0008439523000899","DOIUrl":"https://doi.org/10.4153/s0008439523000899","url":null,"abstract":"<p>Although Naimark dilation theorem was originally stated in 1940, it still finds many important applications in various areas. The objective of this paper is to introduce a method for explicitly constructing the vectors of complementary frames in the Naimark dilation theorem, specifically in cases where the initial Parseval frame contains a Riesz basis as a subset. These findings serve as a foundation for the construction of dual frames.</p>","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"266 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138566913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$R^n$ is the Hilbert’s fourth problem in the regular case. Locally projectively flat Finsler manifolds form an important class of Finsler manifolds. Every Finsler metric induces a spray on the manifold via geodesics. Therefore, it is a natural problem to investigate the geometric and topological properties of manifolds equipped with a spray. In this paper, we study the Pontrjagin classes of a manifold equipped with a locally projectively flat spray and show that such manifold must have zero Pontrjagin classes.
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