An atomic decomposition is considered in Banach space.  A method for constructing an atomic decomposition of Banach  space, starting with atomic decomposition of  subspaces  is presented. Some relations between them are established. The proposed method is used in the  study  of the  frame  properties of systems of eigenfunctions and associated functions of discontinuous differential operators.

The aim of this paper is to define cubic Dirac operators for colour Lie algebras. We give a necessary and sufficient condition construct algebra from an $\epsilon$-orthogonal representation $\epsilon$-quadratic algebra. This used prove strange Freudenthal-de Vries formula basic algebras as well Parthasarathy calculate the cohomology induced by operator, analogously algebraic Vogan conjecture pr...

We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...

The O$(N)$ vector model in the presence of a boundary has non-trivial fixed point $(4-\epsilon)$ dimensions and exhibits critical behaviors described by conformal field theory. spectrum operators is investigated at leading order $\epsilon$-expansion diagrammatic axiomatic approaches. In latter, we extend framework Rychkov Tan for bulk theory to case with calculate composite attention analyticit...

CASE SHOWING THE CONTAGIOUSNESS OF PHTHISIS. A man of thirty-four had been sick with phthisis about four years; gradually grew from bad to worse, until the usual end was reached. It was not until the last three or four months of his life that he was confined to his bed. He was fond of pets, and during these three or four mouths, it was hia fancy to have his dog in the room nearly all the time. ...

This work presents a method that permits the characterization, quantification, and 3D visualization of white matter structural information contained within diffusion tensor MR imaging (DT-MRI) data. In this method, regions within the brain are defined as possessing linear, planar, or spherical diffusion. Visualization of this diffusion metric data is realized by generating streamtube and stream...

The g-frame operator for g-frame in Hilbert space is introduced. The results of g-frame operator are presented. A sequence of operators Λ̄j∈J(where Λ̄ = ΛSg −1 ) is a g-frame for the Hilbert space H with frame bounds 1 B and 1 A . Frame identities for g-normalized tight frames are established. Results on direct sum of gframe operators on direct sum of Hilbert spaces are presented. Mathematics Sub...