S. T. Hedetniemi and P. J. Slater have shown that if G is a triangle-free connected graph with at least three vertices, then

where K(G) is the clique graph of G and K2(G) = K(K(G)) is the first iterated clique graph. In this paper, we generalize the above result to a wider class of graphs.

This paper deals with some asymptotic properties of nonoscillatory solutions of a class of n-th order (n < 1) differential equations with deviationg arguments involving the so called n-th order r-derivative of the unknown function x defined by

where ri (i = 0,1…n) are positive continous functions on [t0, ∞). The fundamental purpose of this paper is to find for any integer m, 0 < m < n – 1, a necessary and sufficient condition (depending on m) in order that three exists at least one (nonoscillatory) solution x so that the exists in R – {0} The results obtained extend some recent ones due to Philos (1978a) and they prove, in a general setting, the validity of a conjecture made by Kusano and Onose (1975).

Two theorems on limit distributions for sums of values sampled from a finite population without replacement are presented. The emphasis is on non-normal limit distributions.

A new necessary and sufficient condition for a distribution of unbounded support to be in a domain of partial attraction is given. This relates the recent work of [5] and [6].

A submanifold of a Riemannian manifold is called a totally umbilical submanifold if its first and second fundamental forms are proportional. In this paper we prove the following best possible result.

Let F(x, θ) be a family of distribution functions indexed by θ ∈ Ω. If G(θ) is a distribution function on Ω H(x) = ƒohm; F(x, θ) dG(θ) is a mixture with respect to G. If there is a unique G yielding H, the mixtures is said to be identifiable.This paper summarises some known results related to identifiability of special types of mixtures and then discusses the general problem of identifiability in terms of mappings. Some new results follow for mappings with special features.

The definition of cyclic maps is recalled and their existence discussed. Among other things, it is shown that cyclicity of maps is closed under product and that if f is cyclic then Ωf is central.Some results of Gottlieb (1972) on homology are applied to investigate the relationship between cyclicity of maps and maps of finite order.

In this paper an ill-posed problem for the heat equation is investigated. Solutions u to the equation ut – uxx = 0, which are approximately known on the positive half-axis t = 0 and on some vertical lines x = x1,…, x = xn, are considered and stability estimates of these solutions are presented. We assume an a priori bound, governing the heat flow across the boundary x = 0.

Necessary and sufficient optimality conditions of Kuhn-Tucker type for a convex programming problem with subdifferentiable operator constraints have been obtained. A duality theorem of Wolfe's type has been derived. Assuming that the objective function is strictly convex, a converse duality theorem is obtained. The results are then applied to a programming problem in which the objective function is the sum of a positively homogeneous, lower-semi-continuous, convex function and a continuous convex function.

From an integer-valued function f we obtain, in a natural way, a matroid Mf on the domain of f. We show that the class of matroids so obtained is closed under restriction, contraction, duality, truncation and elongation, but not under direct sum. We give an excluded-minor characterization of and show that consists precisely of those transversal matroids with a presentation in which the sets in the presentation are nested. Finally, we show that on an n-set there are exactly 2n members of .

Congruences on regular semigroups have been characterized in terms of normal equivalences on sets of idempotents and kernels of congruences. A revised characterization is presented here with considerably simplified expressions for the least and greatest congruences associated with normal equivalences and with a new description of kernels. The results are then applied to characterize congruences on completely regular semigroups.

The purpose of this note is to determine the automorphism group of the doubly regular tournament of Szekeres type, and to use it to show that the corresponding skew Hadamard matrix H of order 2(q + 1), where q ≡5(mod 8) and q > 5, is not equivalent to the skew Hadamard matrix H(2q + 1) of quadratic residue type when 2q + 1 is a prime power.

Minkowski established a lower bound for the determinant D of a Minkowski-reduced quadratic form in terms of the product of its diagonal coefficients aii (i = 1, …n). Oppenheim and Barnes found, for n = 3 and n = 4 respectively, the precise minimum of D in terms of the aii; in each case the minimum is a polynomial in the aii. Here it is shown that no such result exists when n = 5; however a polynomial in all,…, a55 is determined which gives the minimum of D when a55 is sufficiently large.

Suppose that (pn) is an infinitely divisible distribution on the non-negative integers having Lévy measure (vn). In this paper we derive a necessary and sufficient condition for the existence of the limit limn→∞ pn/vn. We also derive some other results on the asymptotic behaviour of the sequence (Pn) and apply some of our results to the theory of fluctuations of random walks. We obtain a necessary and sufficient condition for the first positive ladder epoch to belong to the domain of attraction of a spectrally positive stable law with index α, α ∈ (1,2).