We consider the Hardy-type operator (Tf) (x) := v(x) ∫ x a u(t)f(t)dt, x > a, and establish properties of T as a map from L(a, b) into L(a, b) for 1 < p ≤ q ≤ 2, 2 ≤ p ≤ q < ∞ and 1 < p ≤ 2 ≤ q < ∞. The main result is that, with appropriate assumptions on u and v, the approximation numbers an(T ) of T satisfy the inequality c1 ∫ b a |uv|dt ≤ lim inf n→∞ nan(T ) ≤ lim sup n→∞ nan(T ) ≤ c2 ∫ b a ...

An additive and hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be two additive and hereditary graph properties and let r, s be integers such that r ≥ s. Then an r s -fractional (P,Q)-total coloring of a finite graphG = (V,E) is a mapping f , which assigns an s-element subset of the set {1, 2, . . . , r} to each vert...

Consider the equation dx dt +Q(t)G(x(t− σ(t))) = f(t) (∗) where f, σ,Q ∈ C([0,∞), [0,∞)), G ∈ C(R,R),G(−x) = −G(x), xG(x) > o for x 6= 0, G is non-decreasing, t > σ(t), σ(t) is decreasing and t − σ(t) → ∞ as t → ∞. Whenf(t) ≡ 0, a sufficient condition in terms of the constants k = lim inf t→∞ ∫ t t−σ(t) Q(s)ds and L = lim sup t→∞ ∫ t

and Applied Analysis 3 In 2005, C. E. Chidume and C. O. Chidume 3 proved the convergence theorems for fixed points of uniformly continuous generalized Φ-hemicontractive mappings and published in 3 . However, there exists a gap in the proof course of their theorems. The aim of this paper is to show the convergence of the multistep iteration with errors for fixed points of uniformly continuous Φ-...

The tensor product of vector and arbitrary representations of the nonstandard q-deformation U ′ q(son) of the universal enveloping algebra U(son) of Lie algebra son is defined. The Clebsch– Gordan coefficients of tensor product of vector and arbitrary classical or nonclassical type representations of q-algebra U ′ q(son) are found in an explicit form. The Wigner–Eckart theorem for vector operat...

Several ways to embed q-deformed versions of the Heisenberg algebra into the classical algebra itself are presented. By combination of those embeddings it becomes possible to transform between q-phase-space and q-oscillator realizations of the q-Heisenberg algebra. Using these embeddings the corresponding Schrödinger equation can be expressed by various difference equations. The solutions for t...

We introduce a certain cellular algebra Q(n; r) which is a quotient of the q-Schur algebra S q (n; r). This is naturally equipped with a canonical basis which is compatible with Lusztig's canonical bases for certain modules for the quantized enveloping algebra U (sl n). We describe a diagram calculus for Q(n; r) which makes calculations involving the corresponding canonical bases easy to unders...

In this paper, we represent an inexact inverse subspace iteration method for computing a few eigenpairs of the generalized eigenvalue problem Ax = Bx [Q. Ye and P. Zhang, Inexact inverse subspace iteration for generalized eigenvalue problems, Linear Algebra and its Application, 434 (2011) 1697-1715 ]. In particular, the linear convergence property of the inverse subspace iteration is preserved.

In this paper we investigate the dynamics of solitons occurring in the nonlinear Schroedinger equation when a parameter h → 0. We prove that under suitable assumptions, the the soliton approximately follows the dynamics of a point particle, namely, the motion of its barycenter q h (t) satisfies the equation ¨ q h (t) + ∇V (q h (t)) = H h (t) where sup t∈R |H h (t)| → 0 as h → 0.