For $0leqalphaleq 1,$ let $H_{alpha}(x,y)$ be the convex weighted harmonic mean of $x$ and $y.$ We establish differential subordination implications of the form begin{equation*} H_{alpha}(p(z),p(z)Theta(z)+zp'(z)Phi(z))prec h(z)Rightarrow p(z)prec h(z), end{equation*} where $Phi,;Theta$ are analytic functions and $h$ is a univalent function satisfying some special properties. Further, we prove differential subordination implications involving a combination of three classical means. As an application, we generalize many existing results and obtain sufficient conditions for starlikeness and univalence.
{"title":"Application of Pythagorean means and Differential Subordination","authors":"S. S. Kumar, Priyanka Goel","doi":"10.36045/j.bbms.210605","DOIUrl":"https://doi.org/10.36045/j.bbms.210605","url":null,"abstract":"For $0leqalphaleq 1,$ let $H_{alpha}(x,y)$ be the convex weighted harmonic mean of $x$ and $y.$ We establish differential subordination implications of the form begin{equation*} H_{alpha}(p(z),p(z)Theta(z)+zp'(z)Phi(z))prec h(z)Rightarrow p(z)prec h(z), end{equation*} where $Phi,;Theta$ are analytic functions and $h$ is a univalent function satisfying some special properties. Further, we prove differential subordination implications involving a combination of three classical means. As an application, we generalize many existing results and obtain sufficient conditions for starlikeness and univalence.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89507962","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":"New bounds for the solution and derivatives of the Stein equation for the generalized inverse Gaussian and Kummer distributions","authors":"E. Konzou, A. Koudou, K. Gneyou","doi":"10.36045/j.bbms.200504","DOIUrl":"https://doi.org/10.36045/j.bbms.200504","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87395771","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}
We prove that every connected locally finite regular graph has a double cover which is isomorphic to a Schreier graph. The aim of this short note is to provide a proof of the following result. Proposition 1. Let G be a d-regular connected graph. Then either G is isomorphic to a Schreier graph or G has a double-cover H which is isomorphic to a Schreier graph. While we were not able to find a reference to the above result in the literature, we do not claim any priority on it. In fact, this note was inspired by the following remark, which can be found at the end of Section 7 of [4]: “In fact up to the cover of degree 2 any regular graph can be realized as a Schreier graph [8]”. However, [8] seems to treat only the case of graphs of even degree. Section 1 contains all the definitions as well as a discussion on the unusual concept of degenerated loop. The short Section 2 contains more materials on coverings and perfect matchings, as well as the proof of Proposition 1. Acknowledgements The author is thankful to A. Georgakopoulos, R. Grigorchuk and M. de la Salle for comments on a previous version of this note. The author was supported by NSF Grant No. 200021_188578.
证明了每一个连通的局部有限正则图都有一个与Schreier图同构的双盖。这篇短文的目的是为以下结果提供证明。命题1。设G是一个d正则连通图。要么G与Schreier图同构要么G有一个双盖H与Schreier图同构。虽然我们无法在文献中找到对上述结果的参考，但我们不认为它具有任何优先权。实际上，这篇笔记的灵感来自于文献[4]第7节末尾的一句话:“事实上，直到2度的覆盖，任何正则图都可以被实现为Schreier图[8]”。然而，[8]似乎只处理偶数次图的情况。第1节包含了所有的定义，并讨论了退化环路的不同寻常的概念。较短的第2节包含更多关于覆盖物和完美匹配的材料，以及命题1的证明。作者感谢a . Georgakopoulos、R. Grigorchuk和M. de la Salle对本文前一版本的评论。作者获得国家自然科学基金资助(200021_188578)。
{"title":"Up to a double cover, every regular connected graph is isomorphic to a Schreier graph","authors":"P. Leemann","doi":"10.36045/j.bbms.210416","DOIUrl":"https://doi.org/10.36045/j.bbms.210416","url":null,"abstract":"We prove that every connected locally finite regular graph has a double cover which is isomorphic to a Schreier graph. The aim of this short note is to provide a proof of the following result. Proposition 1. Let G be a d-regular connected graph. Then either G is isomorphic to a Schreier graph or G has a double-cover H which is isomorphic to a Schreier graph. While we were not able to find a reference to the above result in the literature, we do not claim any priority on it. In fact, this note was inspired by the following remark, which can be found at the end of Section 7 of [4]: “In fact up to the cover of degree 2 any regular graph can be realized as a Schreier graph [8]”. However, [8] seems to treat only the case of graphs of even degree. Section 1 contains all the definitions as well as a discussion on the unusual concept of degenerated loop. The short Section 2 contains more materials on coverings and perfect matchings, as well as the proof of Proposition 1. Acknowledgements The author is thankful to A. Georgakopoulos, R. Grigorchuk and M. de la Salle for comments on a previous version of this note. The author was supported by NSF Grant No. 200021_188578.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84003919","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":"Isomorphisms between certain sublattices of continuous functions","authors":"Vahid Ehsani, F. Sady","doi":"10.36045/j.bbms.200410","DOIUrl":"https://doi.org/10.36045/j.bbms.200410","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82203692","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}
Let L be a zero-dimensional frame and Z L be the ring of integer-valued continuous functions on L . We associate with each sublocale of ζL , the Banaschewski compactification of L , an ideal of Z L , and show the behaviour of these types of ideals. The socle of Z L is shown to be always the zero ideal, in contrast with the fact that the socle of the ring R L of continuous real-valued functions on L is not necessarily the zero ideal. The ring Z L has been shown by B. Banaschewski to be (isomorphic to) a subring of R L , so that the ideals of the larger ring can be contracted to the smaller one. We show that the contraction of the socle of R L to Z L is the ideal of Z L associated with the join (in the coframe of sublocales of ζL ) of all nowhere dense sublocales of ζL . It also appears in other guises.
{"title":"On ideals of rings of continuous integer-valued functions on a frame","authors":"T. Dube, O. Ighedo, Batsile Tlharesakgosi","doi":"10.36045/j.bbms.210412","DOIUrl":"https://doi.org/10.36045/j.bbms.210412","url":null,"abstract":"Let L be a zero-dimensional frame and Z L be the ring of integer-valued continuous functions on L . We associate with each sublocale of ζL , the Banaschewski compactification of L , an ideal of Z L , and show the behaviour of these types of ideals. The socle of Z L is shown to be always the zero ideal, in contrast with the fact that the socle of the ring R L of continuous real-valued functions on L is not necessarily the zero ideal. The ring Z L has been shown by B. Banaschewski to be (isomorphic to) a subring of R L , so that the ideals of the larger ring can be contracted to the smaller one. We show that the contraction of the socle of R L to Z L is the ideal of Z L associated with the join (in the coframe of sublocales of ζL ) of all nowhere dense sublocales of ζL . It also appears in other guises.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76614468","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":"Cohomology and deformations of crossed homomorphisms","authors":"A. Das","doi":"10.36045/j.bbms.200513","DOIUrl":"https://doi.org/10.36045/j.bbms.200513","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87104891","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":"On set-star-$K$-Hurewicz spaces","authors":"Sumit Singh","doi":"10.36045/j.bbms.200926","DOIUrl":"https://doi.org/10.36045/j.bbms.200926","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81350281","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}
In this paper, we introduced the mildly version of the Hurewicz basis covering property, studied by Babinkostova, Kov{c}inac, and Scheepers. A space $X$ is said to have mildly-Hurewicz property if for each sequence $langle mathcal{U}_n : nin omega rangle$ of clopen covers of $X$ there is a sequence $langle mathcal{V}_n : nin omega rangle$ such that for each $n$, $mathcal{V}_n$ is a finite subset of $mathcal{U}_n$ and for each $xin X$, $x$ belongs to $bigcupmathcal{V}_n$ for all but finitely many $n$. Then we characterized mildly-Hurewicz property by mildly-Hurewicz Basis property and mildly-Hurewicz measure zero property for metrizable spaces.
{"title":"Mildly version of Hurewicz basis covering property and Hurewicz measure zero spaces","authors":"M. Bhardwaj, A. Osipov","doi":"10.36045/j.bbms.210114a","DOIUrl":"https://doi.org/10.36045/j.bbms.210114a","url":null,"abstract":"In this paper, we introduced the mildly version of the Hurewicz basis covering property, studied by Babinkostova, Kov{c}inac, and Scheepers. A space $X$ is said to have mildly-Hurewicz property if for each sequence $langle mathcal{U}_n : nin omega rangle$ of clopen covers of $X$ there is a sequence $langle mathcal{V}_n : nin omega rangle$ such that for each $n$, $mathcal{V}_n$ is a finite subset of $mathcal{U}_n$ and for each $xin X$, $x$ belongs to $bigcupmathcal{V}_n$ for all but finitely many $n$. Then we characterized mildly-Hurewicz property by mildly-Hurewicz Basis property and mildly-Hurewicz measure zero property for metrizable spaces.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87799193","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}
Let $C(X)$ be the set of all real valued continuous functions on a metric space $(X,d)$. Caserta introduced the topology of strong Whitney convergence on bornology for $C(X)$ in [A. Caserta, Strong Whitney convergence, Filomat, 2012], which is a generalization of the topology of strong uniform convergence on bornology introduced by Beer-Levi in [Beer-Levi, Strong uniform continuity, J. Math. Anal. Appl., 2009]. The purpose of this paper is to study various cardinal invariants of the function space $C(X)$ endowed with the topologies of strong Whitney and Whitney convergence on bornology. In the process, we present simpler proofs of a number of results from the literature. In the end, relationships between cardinal invariants of strong Whitney convergence and strong uniform convergence on $C(X)$ have also been studied.
{"title":"Cardinal Functions, Bornologies and Strong Whitney convergence","authors":"T. Chauhan, V. Jindal","doi":"10.36045/j.bbms.220204","DOIUrl":"https://doi.org/10.36045/j.bbms.220204","url":null,"abstract":"Let $C(X)$ be the set of all real valued continuous functions on a metric space $(X,d)$. Caserta introduced the topology of strong Whitney convergence on bornology for $C(X)$ in [A. Caserta, Strong Whitney convergence, Filomat, 2012], which is a generalization of the topology of strong uniform convergence on bornology introduced by Beer-Levi in [Beer-Levi, Strong uniform continuity, J. Math. Anal. Appl., 2009]. The purpose of this paper is to study various cardinal invariants of the function space $C(X)$ endowed with the topologies of strong Whitney and Whitney convergence on bornology. In the process, we present simpler proofs of a number of results from the literature. In the end, relationships between cardinal invariants of strong Whitney convergence and strong uniform convergence on $C(X)$ have also been studied.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85397931","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}
The Borel problem for Denjoy--Carleman and Braun--Meise--Taylor classes has well-known optimal solutions. The unified treatment of these ultradifferentiable classes by means of one-parameter families of weight sequences allows to compare these optimal solutions. We determine the relations among them and give conditions for their equivalence in the Roumieu case.
{"title":"On optimal solutions of the Borel problem in the Roumieu case","authors":"David Nicolas Nenning, A. Rainer, G. Schindl","doi":"10.36045/j.bbms.220322","DOIUrl":"https://doi.org/10.36045/j.bbms.220322","url":null,"abstract":"The Borel problem for Denjoy--Carleman and Braun--Meise--Taylor classes has well-known optimal solutions. The unified treatment of these ultradifferentiable classes by means of one-parameter families of weight sequences allows to compare these optimal solutions. We determine the relations among them and give conditions for their equivalence in the Roumieu case.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77434699","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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