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.

The Fitting class (of finite, soluble, groups), , is said to be Hall π-closed (where π is a set of primes) if whenever G is a group in and H is a Hall π-subgroup of G, then H belongs to . In this paper, we study the Hall π-closure of products of Fitting classes. Our main result is a characterisation of the Hall π-closedFitting classes of the form (where denotes the so-called smallest normal Fitting class), subject to a restriction connecting π with the characteristic of . We also characterise those Fitting classes (respectively, ) such that (respectively, ) is Hall π-closed for all Fitting classes . In each case, part of the proof uses a concrete group construction. As a bonus, one of these construction also yields a “cancellation result” for certain products of Fitting classes.

The representation theory of Clifford algebras has been used to obtain information on the possible orders of amicable pairs of orthogonal designs on given numbers of variables. If, however, the same approach is tried on more complex systems of orthogonal designs, such as product designs and amicable triples, algebras which properly generalize the Clifford algebras are encountered. In this paper a theory of such generalizations is developed and applied to the theory of systems of orthogonal designs, and in particular to the theory of product designs.

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).

The quaternion algebra of degree 2 over a commutative ring as defined by S. Parimala and R. Sridharan is generalized to a separable cyclic extension B[j] of degree n over a noncommutative ring B. A characterization of such an extension is given, and a relation between Azumaya algebras and Galois extensions for B[j] is also obtained.

What we cail here the “construction principle” is a principle on the ground of which some functionals can be defined; the domain and the range of such a functional consist of some “computable” functionals of various finite types. The principle above is considered here as the basis of the functional interpretation of transfinite induction up to ε0. It is concretely repesented as the “term-forms”, where every term-form is shown to be “computable” in some sense.

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.

The spectral duality theory of H.-E. Porst and M. B. Wischnewsky is examined in more generality, and examples based on topological fields are described.

It is shown that if R is a 2-torsion-free semi-prime ring such that [xy, [xy, yx]] = 0 for all x, y ∈ R, then R is commutative.

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].

Two local nilpotent properties of an associative or alternative ring A containing an idempotent are shown. First, if A = A11 + A10 + A01 + A00 is the Peirce decomposition of A relative to e then if a is associative or semiprime alternative and 3-torsion free then any locally nilpotent ideal B of Aii generates a locally nilpotent ideal 〈B〉 of A. As a consequence L(Aii) = Aii ∩ L(A) for the Levitzki radical L. Also bounds are given for the index of nilpotency of any finitely generated subring of 〈B〉. Second, if A(x) denotes a homotope of A then L(A) ⊆ L(A(x)) and, in particular, if A(x) is an isotope of A then L(A) = L(A(x)).

We show that the classical interpretation of H3(G, A) is equivalent to Taylor's solutions of compound extensions of groups. It is also equivalent to the exactness to an eight term sequence. Only halves of the equivalences are fully shown in the paper but the other halves are clear.

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.

The aim of this note is to investigate the structure of general surjectivity problem for a continuous linear map between locally convex spaces. We shall do so by using the method introduced in Yamamuro (1980). Its basic notion is that of calibrations which has been introduced in Yamamuro (1975), studied in detail in Yamamuro (1979) and appliced to several problems in Yamamuro (1978) and Yamamuro (1979a).

A celebrated result, due to Jacobi, says that the number of representations of the positive integer n as a sum of four squares is equal to eight times the sum of the divisors of n which are not divisible by 4. We give a new and simple proof of this result which depends only on Jacobi's triple product identity.

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.

It is well known that in any near-ring, any intersection of prime ideals is a semiprime ideal. The aim of this paper is to prove that any semiprime ideal I in a near-ring N is the intersection of all minimal prime ideals of I in N. As a consequence of this we have any seimprime ideal I is the intersectionof all prime ideals containing I.

Suppose that f(z) is a function of one complex variable satisfying

where ρ is an integer larger than 1 and a(z) and b(z) are rational functions. We consider f evaluated at the algebraic point a and develop a transcendence measure for f(α) under suitable conditions on f and α.

Schmidt has shown that if r and s are positive integers and there is no positive integer power of r which is also a positive integer power of s, then there exists an uncountable set of reals which are normal to base r but not even simply normal to base s. We give a structurally simple proof of this result

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.