مجله موجکها و جبر خطی
سال پنجم شماره 2 (Winter and Spring 2018)

In this paper, a new notion of frames is introduced: $\ast$-operator frame as generalisation of $\ast$-frames in Hilbert $C^{\ast}$-modules introduced by A. Alijani and M. A. Dehghan \cite{Ali} and we establish some results. we investigate tensor product of Hilbert $C^{\ast}$-modules, we show that tensor product of $\ast$-operator frames for Hilbert $C^{\ast}$-modules $\mathcal{H}$ and $\mathcal{K}$, present $\ast$-operator frames for $\mathcal{H}\otimes\mathcal{K}$, and tensor product of their $\ast$-frame operators is the $\ast$-frame operator of the tensor product of $\ast$-operator frames.

‎‎‎This paper presents a remarkable formula for spectral distance of a given block normal matrix $G_{D_0} = begin{pmatrix}‎ ‎A & B \‎ ‎C & D_0‎ ‎end{pmatrix} $ to set of block normal matrix $G_{D}$ (as same as $G_{D_0}$ except block $D$ which is replaced by block $D_0$)‎, ‎in which $A in mathbb{C}^{ntimes n}$ is invertible‎, ‎$ B in mathbb{C}^{ntimes m}‎, ‎C in mathbb{C}^{mtimes n}$ and $D in mathbb{C}^{mtimes m}$ with $rm {Rank{G_D}} < n+m-1$‎ ‎and given eigenvalues of matrix $mathcal{M} = D‎ - ‎C A^{-1} B $ as $z_1‎, ‎z_2‎, ‎cdots‎, ‎z_{m}$ where $|z_1|ge |z_2|ge cdots ge |z_{m-1}|ge |z_m|$‎. Finally, an explicit formula is proven for spectral distance $G_D$ and $G_D_0$ which is expressed by the two last eigenvalues of $mathcal{M}$.

Abstract. In this paper, we define the notion of C*-affine maps in the unital *-rings and we investigate the C*-extreme points of the graph and epigraph of such maps. We show that for a C*-convex map f on a unital *-ring R satisfying the positive square root axiom with an additional condition, the graph of f is a C*-face of the epigraph of f. Moreover, we prove some results about the C*-faces of C*-convex sets in *-rings. Keywords: C*-affine map, C*-convexity, C*-extreme point, C*-face. MSC(2010): Primary: 52A01; Secondary: 16W10, 46L89.

Global Krylov subspace methods are the most ecient and robust methods to solve generalized coupled Sylvester matrix equation. In this paper, we propose nested splitting conjugate gradient process for solving this equation. This method has inner and outer iterations, which employs the generalized conjugate gradient method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergence and symmetric positive definite splitting of the coecient matrices. Convergence properties of this method are investigated. Finally, the eectiveness of the nested splitting conjugate gradient method is explained by some numerical examples.

‎Many optimization problems can be reduced to a problem with an increasing and co-radiant objective function by a suitable transformation of variables. Functions, which are increasing and co-radiant, have found many applications in microeconomic analysis. In this paper, the abstract convexity of positive valued affine increasing and co-radiant (ICR) functions are discussed. Moreover, the basic properties of this class of functions such as support set, subdifferential and maximal elements of support set are characterized. Finally, as an application, necessary and sufficient conditions for the global minimum of the difference of two strictly positive valued affine ICR functions are presented.

‎In this paper we consider the special classes of Sonnenschein matrices‎, ‎namely the Karamata matrices $K[\alpha,\beta]=\left(a_{n,k}\right)$ with the entries‎
‎\[{a_{n,k}} = \sum\limits_{v = 0}^k {\left( \begin{array}{l}‎
‎n\\‎
‎v‎
‎\end{array} \right){{\left( {1‎ - ‎\alpha‎ - ‎\beta } \right)}^v}{\alpha ^{n‎ - ‎v}}\left( \begin{array}{l}‎
‎n‎ + ‎k‎ - ‎v‎ - ‎1\\‎
‎\,\,\,\,\,\,\,\,\,\,k‎ - ‎v‎
‎\end{array} \right)‎
‎{\beta ^{k‎ - ‎v}}},\] and calculate their row and column sums and give some applications of these sums‎.