Wavelet and Linear Algebra

Abstract In this paper, a new notion of frames is introduced: $\ast$-operator frame as generalization of $\ast$-frames in Hilbert $C^{\ast}$-modules introduced by A. Alijani and M. A. Dehghan \cite{Ali} and we establish some results.

Abstract ‎‎‎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 $A \in \mathbb{C}^{n\times n}$ is invertible‎, ‎$ B \in \mathbb{C}^{n\times m}‎, ‎C \in \mathbb{C}^{m\times n}$ and $D \in \mathbb{C}^{m\times m}$ 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 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.

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

Abstract ‎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.

Abstract ‎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.