For integers a and b, 0 ≤ a ≤ b, an [a, b]-graph G satisfies a ≤ deg(x,G) ≤ b for every vertex x of G, and an [a, b]-factor is a spanning subgraph F such that a ≤ deg(x, F ) ≤ b for every vertex x of F . An [a, b]-factor is almost-regular if b = a+1. A graph is [a, b]-factorable if its edges can be decomposed into [a, b]-factors. When both k and t are positive integers and s is a nonnegative in...

In this paper we obtain appropriate necessary and sufficient conditions for |N, pn|k summability to imply that of |N,qn|s for 1< k≤ s<∞. As in [6] we make use of a result of Bennett [1], who has obtained necessary and sufficient conditions for a factorable matrix to map lk → ls. A factorable matrix A is one in which each entry ank = bnck. Weighted mean matrices are factorable. It will not be po...

A finitely supported sequence a that sums to 2 defines a scaling operator Taf = ∑ k∈Z a(k)f(2 ·−k) on functions f, a transition operator Sav = ∑ k∈Z a(k)(2 ·−k) on sequences v, and a unique compactly supported scaling function φ that satisfies φ = Taφ normalized with φ̂(0) = 1. It is shown that the eigenvalues of Ta on the space of compactly supported square-integrable functions are a subset of ...

We consider the problem of separability: decide whether a Hermitian operator on a finite dimensional Hilbert tensor product H = H ⊗ · · ·⊗H is separable or entangled. We show that the tensor convolution ( φ . . . φ ) : G → H defined for mappings φ : G → H μ on an almost arbitrary locally compact abelian group G, give rise to formulation of an equivalent problem to the separability one.

In this paper we deal with the subnormality and the quasinormality of Toeplitz operators with matrix-valued rational symbols. In particular, in view of Halmos’s Problem 5, we focus on the question: Which subnormal Toeplitz operators are normal or analytic ? We first prove: Let Φ ∈ L∞Mn be a matrix-valued rational function having a “matrix pole,” i.e., there exists α ∈ D for which kerHΦ ⊆ (z−α)H...

Let V = B(H) or S(H), where B(H) is the algebra of bounded linear operator acting on the Hilbert space H, and S(H) is the set of self-adjoint operators in B(H). Denote the numerical range of A ∈ B(H) by W (A) = {(Ax, x) : x ∈ H, (x, x) = 1}. It is shown that a surjective map φ : V→ V satisfies W (AB +BA) =W (φ(A)φ(B) + φ(B)φ(A)) for all A,B ∈ V if and only if there is a unitary operator U ∈ B(H...

Letf(n) denote the number of factorizations of the natural number n into factors larger than 1 where the order of the factors does not count. We say n is " highly factorable " if f (m) <f(n) for all m < n. We prove that f (n) = n L(n)-" " " ' for n highly factorable, where L(n) = exp(log n logloglog n/loglog n). This result corrects the 1926 paper of Oppenheim where it is asserted thatf(n) = n ...

Two-dimensional (2-D) compactly supported, orthogonal wavelets and filter banks having linear phase are presented. Two cases are discussed: wavelets with two-fold symmetry (centrosymmetric) and wavelets with four-fold symmetry that are symmetric (or anti-symmetric) about the vertical and horizontal axes. We show that imposing the requirement of linear phase in the case of order-factorable wavel...

We propose extending Alternating-time Temporal Logic (ATL) by an operator 〈i ⊑ Γ〉φ to express that i can distribute its powers to a set of sub-agents Γ in a way which satisfies ATL condition φ on the strategic ability of the coalitions they may form, possibly together with others agents. We prove the decidability of model-checking of formulas whose 〈. ⊑ .〉-subformulas have the form 〈i1 ⊑ Γ1〉 . ...