Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16122220210601The $n$-th residual relative operator entropy $\mathfrak{R}^{[n]}_{x,y}(A|B)$ and the $n$-th operator valued divergence717914430510.30504/jims.2022.305123.1039ENH.TohyamaDepartment of Life Science and Informatics, Faculty of Engineering, Maebashi Institute of Technology, Maebashi, JapanE.Kamei1-1-3, Sakuragaoka, Kanmakicho, Kitakaturagi-gun, Nara, 639-0202, Japan.M.WatanabeMaebashi Institute of Technology, 460-1, Kamisadori, Maebashi, Gunma, 371-0816, Japan.Journal Article20210916We introduced the $n$-th residual relative operator entropy $\mathfrak{R}^{[n]}_{x,y}(A|B)$ and showed its monotone property, for example, $\mathfrak{R}^{[n]}_{x,x}(A|B) \le \mathfrak{R}^{[n]}_{x,y}(A|B) \le \mathfrak{R}^{[n]}_{y,y}(A|B)$ and $\mathfrak{R}^{[n]}_{x,x}(A|B) \le \mathfrak{R}^{[n]}_{y,x}(A|B) \le \mathfrak{R}^{[n]}_{y,y}(A|B)$ for $x\le y$ if $A\le B$ or $n$ is odd.<br />The $n$-th residual relative operator entropy $\mathfrak{R}^{[n]}_{x,y}(A|B)$ is not symmetric on $x$ and $y$, that is, $\mathfrak{R}^{[n]}_{x,y}(A|B)\neq \mathfrak{R}^{[n]}_{y,x}(A|B)$ for $n\geq 2$ while $\mathfrak{R}^{[1]}_{x,y}(A|B)= \mathfrak{R}^{[1]}_{y,x}(A|B)$.<br />In this paper we compare $\mathfrak{R}^{[n]}_{x,y}(A|B)$ with $\mathfrak{R}^{[n]}_{y,x}(A|B)$ and give the relations between $\mathfrak{R}^{[n]}_{x,y}(A|B)$ and the $n$-th operator divergence $\Delta_{i,x}^{[n]}(A|B)$.<br />In this process, we find another operator divergence ${\overline \Delta}_{i,x}^{[n]}(A|B)$ which is similar to $\Delta_{i,x}^{[n]}(A|B)$ but not the same.http://jims.ims.ir/article_144305_9e853377ecb86fb4bc9ccfcf7be09985.pdfIranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16122220210601Connectifying a topological space by adding one point8111014706310.30504/jims.2022.321037.1050ENM. R.KousheshDepartment of Mathematical Sciences, Isfahan University of Technology, Isfahan, IranJournal Article20211222P. Alexandroff proved that a locally compact $T_2$-space has a $T_2$ one-point compactification (obtained by adding a ``point at infinity'') if and only if it is non-compact. Also he asked for characterizations of spaces which have one-point connectifications. Here, we study one-point connectifications, and in an attempt to answer Alexandroff's question (and in analogy with Alexandroff's theorem) we prove that in the class of $T_i$-spaces ($i=3\frac{1}{2},4,5$) a locally connected space has a one-point connectification if and only if it has no compact component. We extend this theorem to the case $i=6$ by assuming the set-theoretic assumption $\mathbf{MA}+\neg\mathbf{CH}$, and to the case $i=2$ by slightly modifying its statement. We further extended the theorem by proving that a locally connected metrizable (resp. paracompact) space has a metrizable (resp. paracompact) one-point connectification if and only if it has no compact component. Contrary to the case of the one-point compactification, a one-point connectification, if exists, may not be unique. We instead consider the collection of all one-point connectifications of a locally connected locally compact space in the class of $T_i$-spaces ($i=3\frac{1}{2},4,5$). We prove that this collection, naturally partially ordered, is a compact conditionally complete lattice whose order structure determines the topology of all Stone--$\rm\check{C}$ech remainders of components of the space.http://jims.ims.ir/article_147063_b75fa640bd903974143bcd38d88c197f.pdfIranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16122220210601Some Cayley graphs with propagation time 111112214844910.30504/jims.2022.319293.1049ENZ.RamehDepartment of
Pure Mathematics, Faculty of Science, Imam Khomeini International UniversityE.VatandoostDepartment of
Pure Mathematics, Faculty of Science, Imam Khomeini International UniversityJournal Article20211211In this paper we study the zero forcing number as well as the propagation time of Cayley graph $Cay(G,\Omega),$ where $G$ is a finite group and $\Omega \subset G \setminus \lbrace 1 \rbrace$ is an inverse closed generator set of $G$. It is proved that the propagation time of $Cay(G,\Omega)$ is 1 for some Cayley graphs on dihedral groups and finite cyclic groups with special generator set $\Omega$.http://jims.ims.ir/article_148449_87528e4f55ec1f900582e73234ca32bd.pdfIranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16122220210601Hankel operators on Bergman spaces induced by regular weights12313815546010.30504/jims.2022.342003.1067ENE.WangSchool of Mathematics and Statistics, Lingnan Normal University, P.O. Box 524048, Zhanjiang, China.J.XuSchool of Mathematics and Statistics, Lingnan Normal University, P.O. Box 524048, Zhanjiang, China.Journal Article20220613In this paper, given two regular weights $\omega, \Omega$, we characterize these symbols $f\in L^1_\Omega$ for which the induced Hankel operators $H_f^\Omega$ are bounded (or compact) from weighted Bergman space $A_\omega^p$ to Lebesgue space $L^q_\Omega$ for all $1<p, q<\infty$. Moreover, we answer a question posed by X. Lv and K. Zhu [Integr. Equ. Oper. Theory, 91(2019), 91:5] in the case $n=1$.http://jims.ims.ir/article_155460_3a5cc785226fddf774630b3f896b36c7.pdf