Wavelet and Linear Algebra

Abstract Shannon's entropy plays an important role in information theory, dynamical systems and thermodynamics. In this paper we applying Jensen's inequality in information theory and we obtain some results for the Shannon's entropy of random variables and Shannon's entropy of stochastic process. Also we obtain upper bound and lower bound for Shannon's entropy of information sources.

Abstract Convex sets and convex functions play a fundamental role in the development of various fields
of pure and applied mathematics.  Recently, many new generalizations of inequalities with respect to Hermite-Hadamard  have been proposed in the literature. In this paper,  some  new  inequalities of the Hermite-Hadamard type for differentiable convex functions are given. These new inequalities are based on the second derivative functions.

Abstract A Banach algebra $\mathfrak{A}$ has a generalized Matrix representation if there exist the algebras $A, B$, $(A,B)$-module $M$ and $(B,A)$-module $N$ such that $\mathfrak{A}$ is isomorphic to the generalized matrix Banach algebra $\Big[\begin{array}{cc}
A & \ M \\
N & \ B%
\end{array}%
\Big]$.
In this paper, the algebras with generalized matrix representation will be characterized. Then we show that there is a unital permanently weakly amenable  Banach algebra $A$ without generalized matrix representation such that $H^1(A,A)=\{0\}$.
This implies that there is a unital Banach algebra $A$ without any triangular matrix representation such that $H^1(A,A)=\{0\}$  and gives a negative answer to the open question of \cite{D}.

Abstract We consider $C^{b\varphi}(K)$ to be the space $C^b(K)$ equipped with the product $f\cdot g=f\varphi g$ for all $f, g\in C^b(K)$ where, $K=\overline{B_1^{(0)}}$ is the closed unit ball of a non-zero normed vector space $A$ and $\varphi$ is a non-zero element of $A^*$ such that $\Vert \varphi \Vert\leq 1$. We define $\Vert f \Vert_\varphi=\Vert f\varphi \Vert_\infty$ for all $f\in C^{b\varphi}(K)$. Some relations between the dual spaces of $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)$ are investigated. Also we characterize the derivations from $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)$ into $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)^*$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)^*$ respectively. Finally we investigate the weak and cyclic amenability of $(C^{b\varphi}(K), \Vert \cdot \Vert_\infty)$ and $(C^{b\varphi}(K), \Vert \cdot \Vert_\varphi)$.

Abstract In this paper, for every unital $*$-ring $\mathcal{S}$, we investigate the Jensen's inequality preserving maps on $C^*$-convex subsets of $\mathcal{S}$, which we call them $C^*$-convex maps on $\mathcal{S}$. We consider an involution for maps on $*$-rings, and we show that for every $C^*$-convex map $f$ on the $C^*$-convex set $B$ in $\mathcal{S}$, $f^*$ is also a $C^*$-convex map on $B$. We prove that  in the unital commutative $*$-rings, the set of all $C^*$-convex maps ($C^*$-affine maps) on a $C^*$-convex set $B$, is also a $C^*$-convex set. In addition, we prove some results for increasing $C^*$-convex maps. Moreover, it is proved that the set of all $C^*$-affine maps on $B$, is a $C^*$-face of the set of all $C^*$-convex maps on $B$ in the unital commutative $*$-rings. Finally, some examples of $C^*$-convex maps and $C^*$-affine maps in $*$-rings are given.