JIMS Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 Iranian Mathematical Society 144305 10.30504/jims.2022.305123.1039 47-XX Operator theory The \$n\$-th residual relative operator entropy \$mathfrak{R}^{[n]}_{x,y}(A|B)\$ and the \$n\$-th operator valued divergence The \$n\$-th residual relative operator entropy \$mathfrak{R}^{[n]}_{x,y}(A|B)\$ and the \$n\$-th operator valued divergence Tohyama H. Department of Life Science and Informatics, Faculty of Engineering, Maebashi Institute of Technology, Maebashi, Japan Kamei E. 1-1-3, Sakuragaoka, Kanmakicho, Kitakaturagi-gun, Nara, 639-0202, Japan. Watanabe M. Maebashi Institute of Technology, 460-1, Kamisadori, Maebashi, Gunma, 371-0816, Japan. 01 06 2021 2 2 71 79 16 09 2021 04 02 2022 Copyright © 2021, Iranian Mathematical Society. 2021 http://jims.ims.ir/article_144305.html

We 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 \$xle y\$ if \$Ale B\$ or \$n\$ is odd.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 \$ngeq 2\$ while \$mathfrak{R}^{}_{x,y}(A|B)= mathfrak{R}^{}_{y,x}(A|B)\$.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)\$.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.

the n-th relative operator entropy the n-th residual relative operator entropy the n-th operator valued divergence
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JIMS Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 Iranian Mathematical Society 147063 10.30504/jims.2022.321037.1050 54-XX General topology Connectifying a topological space by adding one point Connectifying a topological space by adding one point Koushesh M. R. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran 01 06 2021 2 2 81 110 22 12 2021 20 03 2022 Copyright © 2021, Iranian Mathematical Society. 2021 http://jims.ims.ir/article_147063.html

P. 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=3frac{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}+negmathbf{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=3frac{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--\$rm‎check{C}\$‎ech remainders of components of the space.

One-point connectification one-point compactification Stone-Cech compactification Local connectedness Component
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JIMS Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 Iranian Mathematical Society 148449 10.30504/jims.2022.319293.1049 05-XX Combinatorics Some Cayley graphs with propagation time 1 Some Cayley graphs with propagation time 1 Rameh Z. Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University Vatandoost E. Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University 01 06 2021 2 2 111 122 11 12 2021 19 04 2022 Copyright © 2021, Iranian Mathematical Society. 2021 http://jims.ims.ir/article_148449.html

In 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\$.‎

Cayley graph Zero forcing number Propagation time
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