This paper investigates the structure of general affine subspaces of ( )  L . For a d × d expansive matrix A, it shows that every affine subspace can be decomposed as an orthogonal sum of spaces each of which is generated by dilating some shift invariant space in this affine subspace, and every non-zero and non-reducing affine subspace is the orthogonal direct sum of a reducing subspace and a ...

We introduce the concepts of T-generalized state machines and coverings of products of them. Also some of algebraic properties of them are investigated. Finely some products such as direct sum and sum of Tgeneralized state machines are introduced. An interesting distributive property of cascade product over the sum of Tgeneralized state machines concern to covering of T-generalized state machin...

Let E be a B-complete (B -complete) locally convex algebra and $ the topological direct sum of countably many copies of the scalar field with a jointly continuous algebra multiplication. It has been shown that E is also B-complete (B -complete) for componentwise multiplication on E . B-and Br-completeness of E , the unitization of E, and also of E x for other multiplications on E ...

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...

Let L2 denote the stable homotopy category of v−1 2 BP -local spectra at the prime three. In [2], it is shown that the Picard group of L2 consisting of isomorphic classes of invertible spectra is isomorphic to either the direct sum of Z and Z/3 or the direct sum of Z and two copies of Z/3. In this paper, we conclude the Picard group is isomorphic to the latter group by showing the existence of ...

We define an asymptotic symmetry algebra for three-dimensional Gödel spacetimes supported by a gauge field which turns out to be the semi-direct sum of the diffeomorphisms on the circle with two loop algebras. A class of fields admitting this asymptotic symmetry algebra and leading to well-defined conserved charges is found. The covariant Poisson bracket of the conserved charges is then shown t...

Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule. Then ${mathcal{S}}=A oplus X$, the $l^1$-direct sum of $A$ and $X$ becomes a module extension Banach algebra when equipped with the algebra product $(a,x).(a',x')=(aa',ax'+xa').$ In this paper, we investigate biflatness and biprojectivity for these Banach algebras. We also discuss on automatic continuity of derivations on ${mathcal{S...