Wavelet and Linear Algebra

Abstract A Banach algebra $\A$ is said to be zero Lie product determined Banach algebra if for every continuous bilinear functional $\phi:\A \times \A\rightarrow \mathbb{C}$ the following holds: if $\phi(a,b)=0$ whenever $ab=ba$, then there exists some $\tau \in \A^*$ such that $\phi(a,b)=\tau(ab-ba)$ for all $a,b\in \A$. We show that any finite nest algebra over a complex Hilbert space is a zero Lie product determined Banach algebra.

Abstract Let $X$ be a measure space and let $E$ be a measurable subset of $X$ with finite positive measure. In this paper, we investigate frame and Riesz basis properties of a family of functions multiplied by another measurable function in $L^2(E)$. Also, we study the equivalent conditions for a system of translates to be a Bessel family in $L^2(G)$ and to be a frame for $P_E$ (the space of the band limited functions). Finally, we study the properties of frames of translates that preserved by convolution.

Abstract In this paper, we consider the class of 4-convex functions and we obtain  some inequalities  related to 4-convex functions. Moreover, for $k\leq n$,  we present a  majorization $\prec_k $ on $\mathbb{R}_n$ and we give some equivalent conditions for $\prec_4 $ on $\mathbb{R}_4$.

Abstract Chi-Kwang Li, Bit-Shun Tam and Nam-Kiu Tsing have obtained necessary and sufficient condition  for a linear operator on linear space of generalized doubly stochastic matrices to be strong preserver of doubly stochastic matrices and permutation matrices.
    We show if a linear operator $T:M_m\rightarrow M_n$ is a (strong) preserver of doubly stochastic matrices, then $T$ is a (strong) preserver of the linear manifold of r-generalized doubly stochastic matrices and the linear space of generalized doubly stochastic matrices. Also we give necessary and sufficient condition for a linear operator $T:M_m\rightarrow M_n$ to be (strong) preserver of doubly stochastic matrices and permutation matrices.

Abstract In this paper,  the notion of $n$-weak biamenability of Banach algebras is introduced and for every $n\geq 3$, it is shown that $n$-weak biamenability of the second dual $A^{**}$ of a Banach algebra $A$ implies $n$-weak biamenability of $A$ and this  is true for $n=1, 2$ under some mild conditions.   As a concrete example,  it is  shown that for every abelian locally compact group $G$, $L^1(G)$ is $1$-weakly biamenable and  $\ell^1(G)$ is $n$-weakly biamenable for every odd integer $n$.

Abstract In this paper, we introduce new notions of $I$-biflatness and $I$-biprojectivity, for a Banach algebra $A$, where $I$  is a closed ideal of $A$. We show that $M(G)$ is $L^{1}(G)$-biprojective ($I$-biflat) if and only if $G$ is a compact group (an amenable group), respectively. Also, we show that, for a non-zero ideal $I$, if the Fourier algebra  $A(G)$ is $I$-biprojective, then $G$ is a discrete group. Some examples are given to show the differences between these new notions and the classical ones.

Abstract DNA microarray datasets suffer scaling and uncertainty problems. This paper develops a model that manages DNA microarray datasets challenges more precisely by using the advantages of Wavelet decomposition and fuzzy numbers. For this aim, the proposed method is utilized to classify linguistic DNA microarray datasets set, where datasets can be given as linguistic genes. Linguistic genes are represented by using triangular fuzzy numbers provided as LR (left-right) fuzzy numbers. Then the WABL method is applied as the defuzzification method. Also, a set of orthogonal wavelet detail coefficients based on wavelet decomposition at different levels is extracted to specify the localized genes of DNA microarray datasets. Three DNA microarray datasets are used to evaluate this method. The experiments are shown that the proposed model has better diagnostic accuracy than other methods.