1 Functions Definition 1.1. Given sets S and T , suppose there is a subset G ⊆ S × T with the following properties: • If (s1, t1) ∈ G and (s2, t2) ∈ G and s1 = s2 then t1 = t2; • For each s ∈ S, there is an element (s, t) ∈ G. Then for each s ∈ S, there is exactly one element α(s) ∈ T so that (s, α(s)) ∈ G. This defines a function α, with domain S and target T , which can be denoted α : S → T ....

A 12-week feeding trial was conducted with 87 g rainbow trout to evaluate the effects on growth performances, feed efficiency and nutrient digestibility of adding β-mannanase and α-galactosidase enzymes, solely or in combination. Seven diets were prepared by adding β-mannanase, α-galactosidase and mixed enzyme at two different levels (1 g/kg and 2 g/kg) to control diet (without enzyme) includin...

The stability number of a graph G, denoted by α(G), is the cardinality of a maximum stable set, and µ(G) is the cardinality of a maximum matching in G. If α(G) + µ(G) equals its order, then G is a König-Egerváry graph. In this paper we deal with square-stable graphs, i.e., the graphs G enjoying the equality α(G) = α(G 2), where G 2 denotes the second power of G. In particular, we show that a Kö...

The Chvátal-Erdös theorems imply that if G is a graph of order n ≥ 3 with κ(G) ≥ α(G), then G is hamiltonian, and if κ(G) > α(G), then G is hamiltonian-connected. We generalize these results by replacing the connectivity and independence number conditions with a weaker minimum degree and independence number condition in the presence of sufficient connectivity. More specifically, it is noted tha...

G-proteins couple membrane-bound receptors to intracellular effectors. Each cell has a characteristic complement of G-protein α, β and γ subunits that partly determines the cell ’s response to external signals. Very little is known about the mechanisms that set and maintain cellular levels of G-proteins or about potential points of regulation. We have assayed the steady-state levels of mRNA and...

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...

In this paper we obtain a local limit theorem for elements of a free group G under the abelianization map [·] : G → G/[G, G]. This is obtained via an analysis involving subshifts of finite type, where we obtain a result of independent interest. The case of fundamental groups of compact surfaces of genus ≥ 2 is also discussed. 0. Introduction Let G denote the free group on k ≥ 2 generators {a1, ...

An independent set in a graph G is a set of pairwise non-adjacent vertices, and the independence number α(G) is the cardinality of a maximum independent set. The independence polynomial of G is I(G; x) = s0 + s1x+ s2x 2 + ...+ sαx , α = α(G), where sk equals the number of independent sets of size k in G (Gutman and Harary, 1983). If si = sα−i, 0 ≤ i ≤ α/2 , then I(G; x) is called palindromic. I...

A classical result of Chvátal and Erdős says that the graph G with connectivity κ(G) not less than its independent number α(G) (i.e. κ(G) ≥ α(G)) is hamiltonian. In this paper, we show that the graph G with κ(G) ≥ α(G) − 1 is either supereulerian, or the Petersen graph, or the graphs obtained from K2,3 by adding at most one vertex in one edge of K2,3 and by replacing exactly one vertex whose ne...