ORIGINAL_ARTICLE The $n$-th residual relative operator entropy $\mathfrak{R}^{[n]}_{x,y}(A|B)$ and the $n$-th operator valued divergence 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 $x\le y$ if $A\le B$ 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 $n\geq 2$ 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. http://jims.ims.ir/article_144305_9e853377ecb86fb4bc9ccfcf7be09985.pdf 2021-06-01 71 79 10.30504/jims.2022.305123.1039 the n-th relative operator entropy the n-th residual relative operator entropy the n-th operator valued divergence H. Tohyama tohyama@maebashi-it.ac.jp 1 Department of Life Science and Informatics, Faculty of Engineering, Maebashi Institute of Technology, Maebashi, Japan LEAD_AUTHOR E. Kamei ekamei1947@yahoo.co.jp 2 1-1-3, Sakuragaoka, Kanmakicho, Kitakaturagi-gun, Nara, 639-0202, Japan. AUTHOR M. Watanabe masayukiwatanabe@maebashi-it.ac.jp 3 Maebashi Institute of Technology, 460-1, Kamisadori, Maebashi, Gunma, 371-0816, Japan. AUTHOR  J. I. Fujii and E. Kamei, Relative operator entropy in noncommutative information theory, Math. Jpn. 34 (1989), 3, 341--348. 1  T. Furuta, Parametric extensions of Shannon inequality and its reverse one in Hilbert space operators, Linear Algebra Appl. 381 (2004) 219--235. DOI:10.1016/j.laa.2003.11.017 2  H. Isa, M. Ito, E. Kamei, H. Tohyama and M. Watanabe, Relative operator entropy, operator divergence and Shannon inequality. Sci. Math. Jpn. 75 (2012), no. 3, 289--298. DOI:10.32219/isms.75.3_289 3  H. Isa, E. Kamei, H. Tohyama and M. Watanabe, The n-th relative operator entropies and operator divergences, Ann. Funct. Anal. 11 (2020), no. 2, 298--313. DOI:10.1007/s43034-019-00004-5 4  F. Kubo and T. Ando, Means of positive linear operators, Math Ann. 246 (1979/80), no. 3, 205--224. DOI:10.1007/BF01371042 5  H. Tohyama, E. Kamei and M. Watanabe, The n-th operator valued divergences Δ[n]i;x(AjB), Sci. Math. Jpn. 84 (2021), no. 1, 51--60. DOI:10.32219/isms.84.1_51 6  H. Tohyama, E. Kamei and M. Watanabe, The n-th residual relative operator entropy R[n]x;y(AjB), Adv. in Oper.Theory (2021), no. 1, paper no. 18, 11 pages. DOI:10.1007/s43036-020-00120-3 7  K. Yanagi, K. Kuriyama and S. Furuichi, Generalized Shannon inequalities based on Tsallis relative operator entropy,Linear Algebra Appl. 394 (2005) 109--118. DOI:10.1016/J.LAA.2004.06.025 8
ORIGINAL_ARTICLE Connectifying a topological space by adding one point 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 ($i=3\frac{1}{2},4,5$) 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.pdf 2021-06-01 81 110 10.30504/jims.2022.321037.1050 One-point connectification one-point compactification Stone-Cech compactification Local connectedness Component M. R. Koushesh koushesh@iut.ac.ir 1 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran LEAD_AUTHOR  M. Abry, J.J. Dijkstra and J. van Mill, On one-point connectications, Topology Appl. 154 (2007), no. 3, 725--733. 1  O.T. Alas, M.G. Tkacenko, V.V. Tkachuk and R.G. Wilson, Connectifying some spaces, Topology Appl. 71 (1996), 2 3, 203--215. 3  P. Alexandroff, Uber die Metrisation der im Kleinen kompakten topologischen Raume, (German) Math. Ann. 92 4 (1924), no. 3-4, 294--301. 5  D.K. Burke, Covering Properties, Handbook of Set-Theoretic Topology, 347-422, North-Holland, Amsterdam, 1984. 6  W.W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Springer--Verlag, New York-Heidelberg, 1974. 7  J.J. Dijkstra and J. van Mill, Erd}os space and homeomorphism groups of manifolds, Mem. Amer. Math. Soc. 208 8 (2010), no. 979, vi+62 pp. 9  J.J. Dijkstra, J. van Mill and K.I.S Valkenburg, On nonseparable Erd}os spaces, J. Math. Soc. Japan 60 (2008), no. 10 3, 793--818. 11  J.J. Dijkstra and D. Visser, On generalized Erd}os spaces, Topology Appl. 155 (2008), no. 4, 233--251. 12  J.M. Domnguez, A generating family for the Freudenthal compactification of a class of rimcompact spaces, Fund. 13 Math. 178 (2003), no. 3, 203--215. 14  A. Dow and V.V. Tkachuk, Connected compactifications of countable spaces, Houston J. Math. 37 (2011), no. 2, 15  R. Engelking, General Topology, Second edition, Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 16  L. Gillman and M. Jerison, Rings of Continuous Functions, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto- 17 London-New York 1960. 18  G. Gruenhage, J. Kulesza and A. Le Donne, Connectifications of metrizable spaces, Topology Appl. 82 (1998), no. 19 1-3, 171--179. 20  M. Henriksen, L. Janos and R.G. Woods, Properties of one-point completions of a noncompact metrizable space, 21 Comment. Math. Univ. Carolin. 46 (2005), no. 1, 105--123. 22  B. Honari and Y. Bahrampour, Cut-point spaces, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2797--2803. 23  B. Knaster, Sur un probleme de P. Alexandroff, (French) Fund. Math. 33 (1945) 308--313. 24  M.R. Koushesh, On one-point metrizable extensions of locally compact metrizable spaces, Topology Appl. 154 25 (2007), no. 3, 698--721. 26  M.R. Koushesh, On order structure of the set of one-point Tychonoff extensions of a locally compact space, Topology 27 Appl. 154 (2007), no. 14, 2607--2634. 28  M.R. Koushesh, Compactification-like extensions, Dissertationes Math. 476 (2011) 88 pages. 29  M.R. Koushesh, The partially ordered set of one-point extensions, Topology Appl. 158 (2011), no. 3, 509--532. 30  M.R. Koushesh, One-point extensions and local topological properties, Bull. Aust. Math. Soc. 88 (2013), no. 1, 31  M.R. Koushesh, Topological extensions with compact remainder, J. Math. Soc. Japan 67 (2015), no. 1, 1--42. 32  M.R. Koushesh, One-point connectifications, J. Aust. Math. Soc. 99 (2015), no. 1, 76--84. 33  A. Leiderman and M. Tkachenko, Some properties of one-point extensions, Topology Proc. 59 (2022) 195--208. 34  J. Mack, M. Rayburn and R.G. Woods, Local topological properties and one point extensions, Canad. J. Math. 24 35 (1972) 338--348. 36  J. Mack, M. Rayburn and R.G. Woods, Lattices of topological extensions, Trans. Amer. Math. Soc. 189 (1974) 37  K.D. Magill, Jr. The lattice of compactifications of a locally compact space, Proc. London Math. Soc. (3) 18 (1968) 38  F. Mendivil, Function algebras and the lattice of compactifications, Proc. Amer. Math. Soc. 127 (1999), no. 6, 39 1863--1871. 40  S. Mrowka, On local topological properties, Bull. Acad. Polon. Sci. Cl. III 5 (1957) 951--956. 41  S. Mrowka, On the unions of Q-spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 6 (1958) 365--368. 42  S. Mrowka, Some comments on the author's example of a non-R-compact space, Bull. Acad. Polon. Sci. Ser. Sci. 43 Math. Astronom. Phys. 18 (1970), no. 8, 443--448. 44  A. Mysior, A union of realcompact spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. 29 (1981), no. 3-4, 169--172. 45  J.R. Porter and C. Votaw, H-closed extensions, II Trans. Amer. Math. Soc. 202 (1975) 193--209. 46  J.R. Porter and R.G. Woods, Extensions and Absolutes of Hausdorff Spaces. Springer, New York, 1988. 47  J.R. Porter and R.G. Woods, The poset of perfect irreducible images of a space, Canad. J. Math. 41 (1989), no. 2, 48  J.R. Porter and R.G. Woods, Subspaces of connected spaces, Topology Appl. 68 (1996), no. 2, 113--131. 49  M.C. Rayburn, On Hausdorff compactifications, Pacific J. Math. 44 (1973) 707--714. 50  R.M. Stephenson, Jr., Initially κ-compact and related spaces, Handbook of Set-Theoretic Topology, 603--632, North-- 51 Holland, Amsterdam, 1984. 52  J.E. Vaughan, Countably compact and sequentially compact spaces, Handbook of Set-Theoretic Topology, 569--602, 53 North--Holland, Amsterdam, 1984. 54  S. Watson and R.G. Wilson, Embeddings in connected spaces. Houston J. Math. 19 (1993), no. 3, 469--481. 55  W. Weiss, Countably compact spaces and Martin's axiom, Canad. J. Math. 30 (1978), no. 2, 243--249. 56  R.G. Woods, Zero-dimensional compactifications of locally compact spaces. Canad. J. Math. 26 (1974) 920--930. 57
ORIGINAL_ARTICLE Some Cayley graphs with propagation time 1 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 $\Omega$.‎ http://jims.ims.ir/article_148449_87528e4f55ec1f900582e73234ca32bd.pdf 2021-06-01 111 122 10.30504/jims.2022.319293.1049 Cayley graph Zero forcing number Propagation time Z. Rameh z.rameh@edu.ikiu.ac.ir 1 Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University AUTHOR E. Vatandoost vatandoost@sci.ikiu.ac.ir 2 Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University LEAD_AUTHOR  A.Abdollahi, E. Vatandoost, Which Cayley graphs are integral? Electron. J. Combin. 16 (2009), no. 1, 1--17. 1  AIM Minimum Rank-Special GraphsWork Group, Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl. 428 (2008), no. 7, 1628--1648. 2  J.S.Alameda, E.Curl, A. Grez, L. Hogben , A.Schulte, D.Young and M.Young, Families of graphs with maximum nullity equal to zero forcing number, Spec. Matrices 6 (2018) 56--67. 3  A. Berman,S. Friedland,L. Hogben,U.G. Rothblum and B.Shader, An upper bound for the minimum rank of a graph, Linear Algebra Appl. 429 (2008), no. 7, 1629--1638. 4  D. Burgarth, and V. Giovannetti, Full control by locally induced relaxation, Physical Review Letters 99 (2007), no. 10, p100501. 5  C.J. Edholm, L. Hogben, M. Huynh, J. LaGrange and D.D. Row, Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph, Linear Algebra Appl. 436 (2012), no. 12, 4352--4372. 6  L. Eroh, C.X. Kang and E. Yi, A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs, Acta Math. Sin. (Engl. Ser.) 33 (2017), no. 6, 731--747. 7  L. Hogben, M.Huynh, N. Kingsley, S.Meyer S. Walker and M. Young, Propagation time for zero forcing on a graph, Discrete Appl. Math. 160 (2012), no. 13, 1994--2005. 8  F.Ramezani, E. and Vatandoost, Domination and Signed Domination Number of Cayley Graphs, Iran. J. Math. Sci. Inform. 14 (2019), no. 1, 35--42. 9  S.Severini, Nondiscriminatory propagation on trees, J. Phys. A 41 (2008), no. 48, p.482002. 10  E. Vatandoost and Y. Golkhandy Pour, On the zero forcing number of some Cayley graphs, Algebraic Structures and Their Applications 4 (2017), no. 2, 15--25. 11  E.Vatandoost, F. Ramezani and S. Alikhani, On the zero forcing number of generalized Sierpinski graphs, Trans. Comb. 6 (2019), no. 1, 41--50. 12