We offer a new definition of $varepsilon$-orthogonality in normed spaces, and we try to explain some properties of which. Also we introduce some types of $varepsilon$-orthogonality in an arbitrary  $C^ast$-algebra $mathcal{A}$, as a Hilbert $C^ast$-module over itself, and investigate some of its properties in such spaces. We state some results relating range-kernel orthogonality in $C^*$-algebras.

In this paper a general theory of semi-classical matrix orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distri-butional equation D(uA) = uB, where A and B are matrix polynomials. Several characterizations for these semi-classical functionals are given in terms of the corresponding (left) matrix orthogonal polynomials sequence. They involve a qu...

The standard a priori error analysis of discontinuous Galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the Galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm. In this paper, a new error analysis of discontinuous Galerkin methods is developed using only ...

We present a theory of nonequilibrium long range charge transfer between donor and acceptor centers in a model polymer mediated by magnetic exciton (Kondo) bound states. Our model produces electron tunneling lengths easily exceeding 10 A, as observed recently in DNA and organic charge transfer systems. This long ranged tunneling is effective for weak to intermediate donor-bridge coupling, and i...

The main purpose of this paper is to study weight-semi-greedy Markushevich bases, and in particular, find conditions under which such bases are weight-almost greedy. In context, we prove that, for a large class weights, the two notions equivalent. We also show that all weight semi-greedy truncation quasi-greedy weight-superdemocratic. above cases, bring context weights weak greedy Chebyshev alg...

In this paper first we introduce the notion of weak $Gamma$-(semi)hypergroups, next some classes of equivalence relations which are called good regular and strongly good regular relations are defined.  Then we investigate some properties of this kind of relations on weak $Gamma$-(semi)hypergroups.

1 Linear algebra 2 1.1 Inner product, norm, distance, and orthogonality . . . . . . . . . 2 1.2 Angle and inequality . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Vector projection . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Basics of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . ....

Let $\mathbb{B}(\mathcal{H})$ denote the $C^{\ast}$-algebra of all bounded linear operators on a Hilbert space $\big(\mathcal{H}, \langle\cdot, \cdot\rangle\big)$. Given positive operator $A\in\B(\h)$, and number $\lambda\in [0,1]$, seminorm ${\|\cdot\|}_{(A,\lambda)}$ is defined set $\B_{A^{1/2}}(\h)$ in $\B(\h)$ having an $A^{1/2}$-adjoint. The combination sesquilinear form ${\langle \cdot, \...