Pub Date : 2023-09-28DOI: 10.4153/s0008439523000759
Rui Gao, Yingqing Xiao, Zhanqi Zhang
Abstract In this note, we bound the metric dimension of the circulant graphs $C_n(1,2,ldots ,t)$ . We shall prove that if $n=2tk+t$ and if t is odd, then $dim (C_n(1,2,ldots ,t))=t+1$ , which confirms Conjecture 4.1.1 in Chau and Gosselin (2017, Opuscula Mathematica 37, 509–534). In Vetrík (2017, Canadian Mathematical Bulletin 60, 206–216; 2020, Discussiones Mathematicae. Graph Theory 40, 67–76), the author has shown that $dim (C_n(1,2,ldots ,t))leq t+left lceil frac {p}{2}right rceil $ for $n=2tk+t+p$ , where $tgeq 4$ is even, $1leq pleq t+1$ , and $kgeq 1$ . Inspired by his work, we show that $dim (C_n(1,2,ldots ,t))leq t+left lfloor frac {p}{2}right rfloor $ for $n=2tk+t+p$ , where $tgeq 5$ is odd, $2leq pleq t+1$ , and $kgeq 2$ .
{"title":"On the Metric Dimension of Circulant Graphs","authors":"Rui Gao, Yingqing Xiao, Zhanqi Zhang","doi":"10.4153/s0008439523000759","DOIUrl":"https://doi.org/10.4153/s0008439523000759","url":null,"abstract":"Abstract In this note, we bound the metric dimension of the circulant graphs $C_n(1,2,ldots ,t)$ . We shall prove that if $n=2tk+t$ and if t is odd, then $dim (C_n(1,2,ldots ,t))=t+1$ , which confirms Conjecture 4.1.1 in Chau and Gosselin (2017, Opuscula Mathematica 37, 509–534). In Vetrík (2017, Canadian Mathematical Bulletin 60, 206–216; 2020, Discussiones Mathematicae. Graph Theory 40, 67–76), the author has shown that $dim (C_n(1,2,ldots ,t))leq t+left lceil frac {p}{2}right rceil $ for $n=2tk+t+p$ , where $tgeq 4$ is even, $1leq pleq t+1$ , and $kgeq 1$ . Inspired by his work, we show that $dim (C_n(1,2,ldots ,t))leq t+left lfloor frac {p}{2}right rfloor $ for $n=2tk+t+p$ , where $tgeq 5$ is odd, $2leq pleq t+1$ , and $kgeq 2$ .","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135344649","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 : 2023-09-21DOI: 10.4153/s0008439523000747
Alfredo E. Calderón, Edgardo Villar-Sepúlveda
Abstract We consider the attractor $Lambda $ of a piecewise contracting map f defined on a compact interval. If f is injective, we show that it is possible to estimate the topological entropy of f (according to Bowen’s formula) and the Hausdorff dimension of $Lambda $ via the complexity associated with the orbits of the system. Specifically, we prove that both numbers are zero.
{"title":"Piecewise contracting maps on the interval: Hausdorff dimension, entropy and attractors","authors":"Alfredo E. Calderón, Edgardo Villar-Sepúlveda","doi":"10.4153/s0008439523000747","DOIUrl":"https://doi.org/10.4153/s0008439523000747","url":null,"abstract":"Abstract We consider the attractor $Lambda $ of a piecewise contracting map f defined on a compact interval. If f is injective, we show that it is possible to estimate the topological entropy of f (according to Bowen’s formula) and the Hausdorff dimension of $Lambda $ via the complexity associated with the orbits of the system. Specifically, we prove that both numbers are zero.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136236047","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 : 2023-09-19DOI: 10.4153/s0008439523000735
KEI FUNANO
Abstract We obtain a new upper bound for Neumann eigenvalues of the Laplacian on a bounded convex domain in Euclidean space. As an application of the upper bound, we derive universal inequalities for Neumann eigenvalues of the Laplacian.
{"title":"A UNIVERSAL INEQUALITY FOR NEUMANN EIGENVALUES OF THE LAPLACIAN ON A CONVEX DOMAIN IN EUCLIDEAN SPACE","authors":"KEI FUNANO","doi":"10.4153/s0008439523000735","DOIUrl":"https://doi.org/10.4153/s0008439523000735","url":null,"abstract":"Abstract We obtain a new upper bound for Neumann eigenvalues of the Laplacian on a bounded convex domain in Euclidean space. As an application of the upper bound, we derive universal inequalities for Neumann eigenvalues of the Laplacian.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135015937","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 : 2023-09-12DOI: 10.4153/s0008439523000723
Heng Huat Chan, Song Heng Chan, Teoh Guan Chua, Cheng Yeaw Ku
Abstract In this article, we give generalizations of the well-known Fermat’s Little Theorem, Wilson’s theorem, and the little-known Gegenbauer’s theorem.
摘要本文给出了著名的费马小定理、威尔逊定理和鲜为人知的Gegenbauer定理的推广。
{"title":"On Theorems of Fermat, Wilson and Gegenbauer","authors":"Heng Huat Chan, Song Heng Chan, Teoh Guan Chua, Cheng Yeaw Ku","doi":"10.4153/s0008439523000723","DOIUrl":"https://doi.org/10.4153/s0008439523000723","url":null,"abstract":"Abstract In this article, we give generalizations of the well-known Fermat’s Little Theorem, Wilson’s theorem, and the little-known Gegenbauer’s theorem.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135831385","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 : 2023-09-12DOI: 10.4153/s000843952300070x
L. Bayón, P. Fortuny Ayuso, J.M. Grau, M.M. Ruiz
Abstract We settle the question of how to compute the entry and leaving arcs for turnpikes in autonomous variational problems, in the one-dimensional case using the phase space of the vector field associated with the Euler equation, and the initial/final and/or the transversality condition. The results hinge on the realization that extremals are the contours of a well-known function and that the transversality condition is (generically) a curve. An approximation algorithm is presented, and an example is included for completeness.
{"title":"Entry and leaving arcs of turnpikes: their exact computation in the calculus of variations","authors":"L. Bayón, P. Fortuny Ayuso, J.M. Grau, M.M. Ruiz","doi":"10.4153/s000843952300070x","DOIUrl":"https://doi.org/10.4153/s000843952300070x","url":null,"abstract":"Abstract We settle the question of how to compute the entry and leaving arcs for turnpikes in autonomous variational problems, in the one-dimensional case using the phase space of the vector field associated with the Euler equation, and the initial/final and/or the transversality condition. The results hinge on the realization that extremals are the contours of a well-known function and that the transversality condition is (generically) a curve. An approximation algorithm is presented, and an example is included for completeness.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135879285","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 : 2023-09-12DOI: 10.4153/s0008439523000711
Charalampos Magiatis
Abstract We characterize the ideals of the semicrossed product $C_0(X)times _phi {mathbb Z}_+$ , associated with suitable sequences of closed subsets of X , with left (resp. right) approximate unit. As a consequence, we obtain a complete characterization of ideals with left (resp. right) approximate unit under the assumptions that X is metrizable and the dynamical system $(X,phi )$ contains no periodic points.
{"title":"Ideals with approximate unit in semicrossed products","authors":"Charalampos Magiatis","doi":"10.4153/s0008439523000711","DOIUrl":"https://doi.org/10.4153/s0008439523000711","url":null,"abstract":"Abstract We characterize the ideals of the semicrossed product $C_0(X)times _phi {mathbb Z}_+$ , associated with suitable sequences of closed subsets of X , with left (resp. right) approximate unit. As a consequence, we obtain a complete characterization of ideals with left (resp. right) approximate unit under the assumptions that X is metrizable and the dynamical system $(X,phi )$ contains no periodic points.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135831400","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 : 2023-09-11DOI: 10.4153/s0008439523000656
Cristian Enache, Giovanni Porru
Abstract This paper deals with some Monge–Ampère type equations involving the gradient that are elliptic in the framework of convex functions. First, we show that such equations may be obtained by minimizing a suitable functional. Moreover, we investigate a P-function associated with the solution to a boundary value problem of our generalized Monge–Ampère equation in a bounded convex domain. It will be shown that this P-function attains its maximum value on the boundary of the underlying domain. Furthermore, we show that such a P-function is actually identically constant when the underlying domain is a ball. Therefore, our result provides a best possible maximum principles in the sense of L. E. Payne. Finally, in case of dimension 2, we prove that this P-function also attains its minimum value on the boundary of the underlying domain. As an application, we will show that the solvability of a Serrin’s type overdetermined problem for our generalized Monge–Ampère type equation forces the underlying domain to be a ball.
在凸函数的框架下，研究了一类椭圆型的含有梯度的monge - ampantere型方程。首先，我们证明这样的方程可以通过最小化一个合适的泛函得到。此外，我们研究了与有界凸域上广义monge - amp方程边值问题解相关的p函数。结果表明，该p函数在基础域的边界处达到最大值。进一步，我们证明了当基础域是球时，这样的p函数实际上是相同常数。因此，我们的结果提供了L. E. Payne意义上的最佳可能最大值原则。最后，在维数为2的情况下，我们证明了该p函数在基础域的边界上也达到了最小值。作为一个应用，我们将证明Serrin型超定问题对于我们的广义monge - ampantere型方程的可解性迫使底层区域是一个球。
{"title":"Problems for generalized Monge-Ampère equations","authors":"Cristian Enache, Giovanni Porru","doi":"10.4153/s0008439523000656","DOIUrl":"https://doi.org/10.4153/s0008439523000656","url":null,"abstract":"Abstract This paper deals with some Monge–Ampère type equations involving the gradient that are elliptic in the framework of convex functions. First, we show that such equations may be obtained by minimizing a suitable functional. Moreover, we investigate a P-function associated with the solution to a boundary value problem of our generalized Monge–Ampère equation in a bounded convex domain. It will be shown that this P-function attains its maximum value on the boundary of the underlying domain. Furthermore, we show that such a P-function is actually identically constant when the underlying domain is a ball. Therefore, our result provides a best possible maximum principles in the sense of L. E. Payne. Finally, in case of dimension 2, we prove that this P-function also attains its minimum value on the boundary of the underlying domain. As an application, we will show that the solvability of a Serrin’s type overdetermined problem for our generalized Monge–Ampère type equation forces the underlying domain to be a ball.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135981508","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 : 2023-09-11DOI: 10.4153/s0008439523000668
EMMANUEL CHETCUTI, CURT HEALEY
Abstract In this article, the question of whether the Löwner partial order on the positive cone of an operator algebra is determined by the norm of any arbitrary Kubo–Ando mean is studied. The question was affirmatively answered for certain classes of Kubo–Ando means, yet the general case was left as an open problem. We here give a complete answer to this question, by showing that the norm of every symmetric Kubo–Ando mean is order-determining, i.e., if $A,Bin mathcal B(H)^{++}$ satisfy $Vert Asigma XVert le Vert Bsigma XVert $ for every $Xin mathcal {A}^{{++}}$ , where $mathcal A$ is the C*-subalgebra generated by $B-A$ and I , then $Ale B$ .
摘要本文研究了算子代数正锥上的Löwner偏阶是否由任意Kubo-Ando均值的范数决定的问题。对于某些类别的久保安藤手段，这个问题得到了肯定的回答，但对于一般情况，这个问题仍然是一个悬而未决的问题。我们在这里给出了这个问题的完整答案，通过证明每一个对称Kubo-Ando均值的范数是序决定的，即，如果$A,Bin mathcal B(H)^{++}$满足$Vert Asigma XVert le Vert Bsigma XVert $对于每一个$Xin mathcal {A}^{{++}}$，其中$mathcal A$是由$B-A$和I生成的C*-子代数，则$Ale B$。
{"title":"EVERY SYMMETRIC KUBO-ANDO CONNECTION HAS THE ORDER-DETERMINING PROPERTY","authors":"EMMANUEL CHETCUTI, CURT HEALEY","doi":"10.4153/s0008439523000668","DOIUrl":"https://doi.org/10.4153/s0008439523000668","url":null,"abstract":"Abstract In this article, the question of whether the Löwner partial order on the positive cone of an operator algebra is determined by the norm of any arbitrary Kubo–Ando mean is studied. The question was affirmatively answered for certain classes of Kubo–Ando means, yet the general case was left as an open problem. We here give a complete answer to this question, by showing that the norm of every symmetric Kubo–Ando mean is order-determining, i.e., if $A,Bin mathcal B(H)^{++}$ satisfy $Vert Asigma XVert le Vert Bsigma XVert $ for every $Xin mathcal {A}^{{++}}$ , where $mathcal A$ is the C*-subalgebra generated by $B-A$ and I , then $Ale B$ .","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135981353","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 : 2023-09-04DOI: 10.4153/s0008439523000681
S. Sakata
{"title":"Convexity of the radial sum of a star body and a ball","authors":"S. Sakata","doi":"10.4153/s0008439523000681","DOIUrl":"https://doi.org/10.4153/s0008439523000681","url":null,"abstract":"","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46947687","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 : 2023-09-01DOI: 10.4153/s0008439523000632
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"BCM volume 66 issue 3 Cover and Front matter","authors":"","doi":"10.4153/s0008439523000632","DOIUrl":"https://doi.org/10.4153/s0008439523000632","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135304916","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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