This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form φ(x) = g(x) + ∑ α∈Zs a(α)φ(Mx − α), x ∈ R, where φ = (φ1, . . . , φr) is the unknown, g = (g1, . . . , gr) is a given vector of functions on Rs, M is an s× s dilation matrix, and a is a finitely supported refinement mask such that each a(α) is an r × r (complex) matrix. Let φ0 be an initial vector i...

[82] provides a proof of Jensen's remarkable Coding Theorem , which demonstrates that the universe can be included in L[R] for some real R, via class forcing. The purpose of this article is to present a simpler proof of Jensen's theorem, obtained by implementing some changes first developed for the theory of strong coding (Friedman [87]). The basic idea is to first choose A ⊆ ORD so that V = L[...

[82] provides a proof of Jensen's remarkable Coding Theorem , which demonstrates that the universe can be included in L[R] for some real R, via class forcing. The purpose of this article is to present a simpler proof of Jensen's theorem, obtained by implementing some changes first developed for the theory of strong coding (Friedman [87]). The basic idea is to first choose A ⊆ ORD so that V = L[...

We give a criterion for an HNN extension of a finite p-group to be residually p. 1. Statement of the Main Results By an HNN pair we mean a pair (G,φ) where G is a group and φ : A → B is an isomorphism between subgroups A and B of G. Given such an HNN pair (G,φ) we consider the corresponding HNN extension G∗ = 〈G, t | t−1at = φ(a), a ∈ A〉 of G, which we denote, by slight abuse of notation, as G∗...

Let A be a strong independence algebra of finite rank with at most one constant, and let G be the group of automorphisms of A. Let α be a singular endomorphism of A and α = 〈{α} ∪G〉. We describe the elements of α and give additional characterisations when A is a proper independence algebra and G is a periodic group. Mathematics subject classification: 20M20, 20M10, 08A35.

conclusions the results indicated that there was a positive association between susceptibility to chronic hbv infection and tnf-α polymorphism. in addition, hbv patients carrying -308 gg, -857 cc, and -863 aa genotypes with lower serum levels of tnf-α had an increased risk of infection. background the host genetic background regulates the natural history of chronic hepatitis b virus (hbv) infec...

The stability number of a graph G, denoted by α(G), is the cardinality of a stable set of maximum size in G. If its stability number remains the same upon the addition of any edge, then G is called α-stable. G is a König-Egervary graph if its order equals α(G) + μ(G), where μ(G) is the cardinality of a maximum matching in G. In this paper we characterize α-stable König-Egervary graphs, generali...

For a graph G and a real α / = 0, we study the graph invariant sα(G) – the sum of the αth power of the non-zero Laplacian eigenvalues of G. The cases α = 2, 1 2 and −1 have appeared in different problems. Here we establish some properties for sα with α / = 0, 1. We also discuss the cases α = 2, 1 2 . © 2008 Elsevier Inc. All rights reserved. AMS classification: 05C50; 05C90

We consider the existence of positive solutions of the nonlinear two point boundary value problem u′′ + λf(u) = 0, u(−1) = u(1) = 0, where f(u) = u(u − a)(u− b)(u− c)(1−u), 0 < a < b < c < 1, as the parameter λ varies through positive values. Every solution u(x) is an even function, and when it exists, it is uniquely identified by α = u(0). We study how the number of solutions changes when the ...

Let α(G) and β(G) be the independent number and vertex covering number of G, respectively. The Kronecker Product G1 ⊗ G2 of graph of G1 and G2 has vertex set V (G1 ⊗ G2) = V (G1) × V (G2) and edge set E(G1 ⊗ G2) = {(u1v1)(u2v2)|u1u2 ∈ E(G1) and v1v2 ∈ E(G2)}. In this paper, let G is a simple graph with order p, we prove that, α(Km,n⊗G)= max {(m+n)α(G),p max{m,n}} and β(Km,n⊗G) =min {(m + n)β(G)...