Let k be a positive integer and G = (V (G), E(G)) a graph. A subset S of V (G) is a k-independent set of G if the subgraph induced by the vertices of S has maximum degree at most k−1. The maximum cardinality of a k-independent set of G is the k-independence number βk(G). A graph G is called β − k -stable if βk(G − e) = βk(G) for every edge e of E(G). First we give a necessary and sufficient con...

Given a 3-colourable graph G and two proper vertex 3-colourings α and β of G, consider the following question : is it possible to transform α into β by recolouring vertices of G one at a time, making sure that all intermediate colourings are proper 3-colourings? We prove that this question is answerable in polynomial time. We do so by characterising the instances G,α, β where the transformation...

A. (Forgive my choice of) Notation. Let G be a complex, connected, semisimple (or reductive) Lie group, T a maximal torus in G, and K the identity component of the fixed points of an involution in G. Equivalently, K is the complexification of a maximal compact subgroup of the identity component of a real form of G. Of course we assume T ⊂ K. We let ΛG denote the lattice of T -characters (the we...

Let G be a reductive connected algebraic group over an algebraic closure of a finite field of characteristic p. Let F be a Frobenius endomorphism on G and write G := G F for the corresponding finite group of Lie type. We consider projective characters of G in characteristic p of the form St ·β, where β is an irreducible Brauer character and St the Steinberg character of G. Let M be a rational G...

This work is a part of a program to describe all 2–generator Kleinian groups with real parameters. We study RP groups, that is, marked 2–generator subgroups Γ = 〈f , g〉 of PSL(2,C) for which the generating pair (f , g) has real parameters β = β(f ) = tr 2f − 4, β′ = β(g) = tr 2g−4 and γ = γ(f , g) = tr [f , g]−2 (see Section 1 for exact definitions). Since discreteness questions were answered f...

The asymptotics of the Gell-Mann – Low function in QED can be determined exactly, β(g) = g at g → ∞, where g = e is the running fine structure constant. It solves the problem of pure QED at small distances L and gives the behavior g ∼ L. According to Landau, Abrikosov, Khalatnikov [1], relation of the bare charge e0 with the observable charge e in quantum electrodynamics (QED) is given by expre...

We consider the frame property of the Gabor system G(g, α, β) = {eg(t − αm) : m,n ∈ Z} for the case of rational oversampling, i.e. α, β ∈ Q. A ’rational’ analogue of the Ron-Shen Gramian is constructed, and prove that for any odd window function g the system G(g, α, β) does not generate a frame if αβ = n−1 n . Special attention is paid to the first Hermite function h1(t) = te −πt .

The synthesis of galacto-oligosaccharides (GOS) catalyzed by a novel commercial preparation of β-galactosidase from Bacillus circulans (Biolactase) was studied, and the products were characterized by MS and NMR. Using 400 g/L lactose and 1.5 enzyme units per milliliter, the maximum GOS yield, measured by HPAEC-PAD analysis, was 165 g/L (41% w/w of total carbohydrates in the mixture). The majo...