مجله موجکها و جبر خطی
سال هفتم شماره 2 (Autumn and Winter 2020)

‎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 random process. Also we obtain upper bound and lower bound for Shannon's entropy of information sources.

Convex sets and convex functions play a fundamental role in the development of various fields of pure and applied mathematics. Recently, many new general- izations 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.

‎An 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}‎.

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‎We consider $C^{bvarphi}(K)$ to be the space $C^b(K)$ equipped with the product $fcdot g=fvarphi g$ for all $f‎, ‎gin C^b(K)$ 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 $Vert varphi Vertleq 1$‎. ‎We define $Vert f Vert_varphi=Vert fvarphi Vert_infty$ for all $fin C^{bvarphi}(K)$‎. ‎Some relations between the dual spaces of $(C^{bvarphi}(K)‎, ‎Vert cdot Vert_infty)$ and $(C^{bvarphi}(K)‎, ‎Vert cdot Vert_varphi)$ are investigated‎. ‎Also we characterize the derivations from $(C^{bvarphi}(K)‎, ‎Vert cdot Vert_infty)$ and $(C^{bvarphi}(K)‎, ‎Vert cdot Vert_varphi)$ into $(C^{bvarphi}(K)‎, ‎Vert cdot Vert_infty)^*$ and $(C^{bvarphi}(K)‎, ‎Vert cdot Vert_varphi)^*$ respectively‎. ‎Finally we investigate the weak and cyclic amenability of $(C^{bvarphi}(K)‎, ‎Vert cdot Vert_infty)$ and $(C^{bvarphi}(K)‎, ‎Vert cdot Vert_varphi)$‎.

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In this paper, for every unital ∗-ring S, we investigate the Jensen‘s inequality preserving maps on C*-convex subsets of S, which we call them C*-convex maps on 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 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.