Wavelet and Linear Algebra

Abstract In this paper we develop the notions of Connes amenability for certain product of Banach algebras. We give necessary and sufficient conditions for the existence of an invariant mean on the predual of $\Theta$-Lau product  $\mathcal{A}\times_{\Theta}\mathcal{B}$, module extension Banach algebra $\mathcal{A}\oplus\mathcal{X}$ and projective tensor product $\mathcal{A} \widehat{{\otimes}} \mathcal{B}$, where $\mathcal{A}$ and $\mathcal{B}$ are dual Banach algebras  with preduals $\mathcal{A}_*$ and $\mathcal{B}_*$ respectively and $\mathcal{X}$ is a normal Banach $\mathcal{A}$-bimodule with predual $\mathcal{X}_*$.

Abstract In this paper, a problem whose cost function and constraints are increasing convex along rays is considered. For solving such problems, an algorithm is presented that is inspired by the generalized Cutting Angle Method. A set that contains the optimal solution of the mentioned problem is defined. Some numerical examples are presented to confirm the validity and accuracy of the algorithm.

Abstract Let $\mathcal{B(X)}$ be the algebra of all bounded linear operators on a complex Banach space $\mathcal{X}$. In this paper, we determine the form of a surjective additive map $\phi: \mathcal{B(X)} \rightarrow \mathcal{B(X)}$ preserving the fixed points of Jordan products of operators, i.e., $F(AoB) \subseteq F(\phi(A) o\phi(B))$, for every $A,B \in \mathcal{B(X)}$, where $AoB=AB+BA$, and $F(A)$ denotes the set of all fixed points of operator $A$.

Abstract  In this paper, the space  $ c (I) $  is introduced and some of its properties examined. Then with the help of a diameter norm  on the space  $ c_{0}(I)$, a norm is defined on the space $ c (I) $ called as D- norm, which  is an extension of  the $ d-$norm. It is also shown that the D- norm is equivalent to the supremum norm.  The extreme points of the unit ball of the spaces $ c_{0}(I) $  and  $ c (I) $ are also specified. In addition we find  some orthogonal vectors in the space $ c (I) $.

Abstract First, we define and investigate some new classes of interval tensors, called interval exceptionally regular tensors  ($ER-$tensor) and interval $wP-$tensors which is relevant to interval strictly semi-positive tensors. Also, we show that $ER-$tensor is a wide class of interval tensors, which includes many important structured tensors. Second, some classes of interval matrices are extended to interval tensors, such as interval $R(R_0)$-tensor and column sufficient interval tensor. We discuss their relationships with interval positive semi-definite tensors and some other structured interval tensors. In addition,  necessary and sufficient conditions for interval (strictly) copositive and interval $E_0$-tensors are presented and investigated. Finally, we extend the concept of the column sufficient interval matrix to the column sufficient interval tensor.

Abstract Let $G$ be a locally compact group, $H$ and $K$ be two closed subgroups of $G$, and $N$ be the normalizer group of $K$ in $G$. In this paper, the existence and properties of a rho-function for the triple $(K, G, H)$ and an $N$-strongly quasi-invariant measure of double coset space $K \backslash G /H$ is investigated. In particular, it is shown that any such measure arises from a rho-function. Furthermore, the conditions under which an $N$-strongly quasi-invariant measure arises from a rho-function are studied.