Wavelet and Linear Algebra

Abstract Let $G$ and $H$ be simple graphs and $\left| {V(G)} \right| = n$. The corona of two graphs, denoted by $G \circ H$, is the graph obtained by taking one copy of graph $G$ and $n$ copies of $H$ and joining the ${i^{th}}$ vertex of $G$ to every vertex of the ${i^{th}}$ copy of $H$. Let $S(G)$ be the subdivision of graph $G$. In this paper we define four new subdivision coronas of two graphs and find the characteristic and Laplacian polynomials of them in case of regularity.

Abstract Let $A$ be a non-zero normed vector space and let $\varphi$ be a non-zero element of $A^*$ such that $\Vert \varphi \Vert \leq 1$. Assume that $K=\overline{B_1^{(0)}}$ is the closed unit ball of $A$. According to the our recent studies on the spaces of $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)$, generated by $C^b(K)$ and equipped with a new product `` $ \cdot $ '' and different norms $\Vert \cdot \Vert_\infty $ and $\Vert \cdot \Vert_\varphi$, the $n-$weak amenability of $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)$ are investigated. 

Abstract In this paper, for a given set of real numbers such as $\sigma$ with only one positive number and zero summation, we find a distance matrix in which the given set $\sigma$ is its spectrum.
Finally,  we solve special cases of the inverse eigenvalue problem in which the matrix solution is a regular spherical distance matrix.

Abstract Let $A$ and $B$ be Banach algebras. In this paper, we investigate the structure of $2$-cocycles and $2$-coboundaries on $A\oplus B$, when  $A$ and $B$ are unital.  Actually, we provide a specific criterion for each $2$-cocycle map and establish a connection between $2$-cocycles and $2$-coboundaries on $A\oplus B$ and $2$-cocycles and $2$-coboundaries on $A$ and $B$. Finally, our results lead to a connection between  $\mathcal{H}^2(A, A^{*})$,$ \mathcal{H}^2(B, B^{*})$ and $\mathcal{H}^2(A\oplus B, A^{*}\oplus B^{*})$. 

Abstract A toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant. A matrix $R$ is called integral row stochastic, if each row has exactly a nonzero entry, $+1$, and other entries are zero. In this  paper, we present $L$-rays of integral row stochastic toeplitz matrices, and  we provide an algorithm for constructing these matrices.

Abstract This study concerns a detailed analysis of composition operators $C_\varphi$ on the classical Bergman spaces, as well as on the Hardy and Bergman spaces with variable exponents. Here, $\varphi$ is an analytic self-map of the open unit disk in the complex plane.
Accordingly, conditions for the boundedness of these operators are obtained. It is worth mentioning that the Littlewood subordination theorem plays a fundamental role in proving the stated theorems in which we use the Rubio de Francia extrapolation theorem.

Abstract In this note, we give a simple method for computing the column sums of the Sonnenschein summability matrices.