A general group-theoretic procedure is indicated for representing rational integers as products of other integers. A detailed example is given.

Given a finite graph H and G, a subgraph of it, we define σ (G, H) to be the largest integer such that every pair of subgraphs of H, both isomorphic to G, has at least σ(G, H) edges in common; furthermore, R(G, H) is defined to be the maximum number of subgraphs of H, all isomorphic to G, such that any two of them have σ(G, H) edges common between them. We are interested in the values of σ(G, H) and R(G, H) for general H and G. A number of combinatorial problems can be considered as special cases of this question; for example, the classical set-packing problem is equivalent to evaluating R (G, H) where G is a complete subgraph of the complete graph H and σ(G, H) = 0, and the decomposition of H into subgraphs isomorphic to G is equivalent to showing that σ(G, H) = 0 and R(G, H) = ε(H)/ε(G) where ε(H), ε(G) are the number of edges in H, G respectively.

A result of S. M. Johnson (1962) gives an upper bound for R(G, H) in terms of σ(G, H). As a corollary of Johnson's result, we obtain the upper bound of McCarthy and van Rees (1977) for the Cordes problem. The remainder of the paper is a study of σ (G, H) and R(G, H) for special classes of graphs; in particular, H is a complete graph and G is, in most instances, a union of disjoint complete subgraphs.

Sharp results for the cofficient estimates, distortion theorems, radius of convexity, arc-length and area of the image curve are obtained for the class R(A, B) of regular functions whose derivative is subordinate to (1+AZ)/(1+Bz), -1 ≤ B ≤ A ≤ 1, in the unit disc E = {z:|z| < 1}. We also establish a convolution theorem for this class.

Let R be an associative ring which is not necessarily commutative. For any torsion theory τ on the category of left R-modules and for any nonnegative integer n we define and study the notion of the nth local cohomology functor with respect to τ. For suitably nice rings a bound for the nonvanishing of these functors is given in terms of the τ-dimension of the modules.

Let X1, X2, …, Xn be identically distributed independent random variables belonging to the domain of attraction of the normal law, have zero means and Pr{Xr ≠ 0} > 0. Suppose a0, a1, …, an are non-zero real numbers and max and εn is such that as n → ∞, εn. If Nn be the number of real roots of the equation then for n > n0, Nn > εn log n outside an exceptional set of measure at most provided limn→∞ (kn/tn) is finite.

On page 354 of James (1975) (in the proof of Theorem 5.3(b)) there appear the following two sentences:

‘When n≥7, replacewe may suppose δ=0.’

Unfortunately, the second sentence does not follow from the first. In fact, the first sentence also forces us to replace remains unaltered. Thus the case δ=1 is omitted in the paper.

We examine the concepts of nowhere simplicity in a wide class of abstract dependence systems. Initially we examine how many of the existing results valid for L(ω), the lattice of r.e. sets, have analogues valid for more general lattices. For example, we show that any r. e. subspace of V∞ can be decomposed into a pair of nowhere simple subspaces.

We identify a large class of rings over which locally free modules are determined by their endomorphism rings. We characterize these endomorphism rings and consider under what circumstances the conditions on the locally free modules can be relaxed, for example by requiring that only one of the rings need be in the special class, or by replacing ‘free' by “projective”.

Let S be a regular semigroup and A a D(S)-module. We proved in a previous paper that the set Ext(S, A) of equivalence classes of extensions of A by S admits an abelian group structure and studied its functorial properties. The main aim of this paper is to describe Ext(S, A) as a second cohomology group of certain chain complex.

A regular double p-algebra L satisfying (i) ∩(xn(+*); n < ω) for every 1 ≠ x ∈ L and (ii) L is not subdirectly irreducible, is constructed. The construction is purely topological and the desired result is obtained via the known Priestly duality. The notion of an auxiliary regular double p-algebra is introduced and the algebras having this property are characterized.

It is well-known that in any near-ring, any intersection of prime ideals is a semi-prime ideal. The aim of this note is to prove that any ideal is a prime ideal if and only if it is equal to its prime radical. As a consequence of this we have any semi-prime ideal I in a near-ring N is the intersection of minimal prime ideals of I in N and that I is the intersection of all prime ideals containing I.

By using pairwise relatively complete families of functions, which we define here, we obtain two characterizations of quasi-pseudometrizable bitopological spaces from which some known theorems can be derived as easy corollaries.

If the second dual of a Banach space E is smooth at each point of a certain norm dense subset, then its first dual admits a long sequence of norm one projections, and these projections have ranges which are suitable for a transfinite induction argument. This leads to the construction of an equivalent locally uniformly rotund norm and a Markuschevich basis for E*.

This is the first of three papers (the others by the first author alone) which determine all varieties of nilpotent groups of class (at most) four. The initial step is to reduce the problem to two cases: varieties whose free groups have no elements of order 2, and varieties whose free groups have no nontrivial elements of odd order. The varieties of the first kind form a distributive lattice with respect to order by inclusion (which is not a sublattice in the lattice of all group varieties). We give an embedding of this lattice in the direct product of six copies of the lattice which consist of 0 (as largest element) and the odd positive integers ordered by divisibility. The six integer parameters so associated with a variety directly match a (finite) defining set of laws for the variety. We also show that the varieties of the second kind do form a sublattice in the lattice of all varieties. That (nondistributive) sublattice will be treated, in a similarly conclusive manner, in the subsequent papers of this series.

In this paper, the questions about bitopological spaces proposed by S. Lal are solved and one of his counterexamples rectified.

Given an ordered Banach space ℬ equipped with an order-norm we construct a larger space ℬ¯ with an order-norm and order-identity such that B is isometrically order-isomorphic to a Banach subspace of B. We also discuss the extension of positive operators from ℬto ℬ¯.

This note establishes estimates for lower bounds of linear forms at algebraic points of E-functions and G-functions defined over a general algebraic number field.

The first paper (written jointly with L. G. Kovács) of this three-part series reduced the problem of determining all varieties of the title to the study of the varieties of nilpotent groups of class (at most) four whose free groups have no nontrivial elements of odd order. The present paper deals with these under the additional assumption that the variety contains all nilpotent groups of class three. We label each such variety by a vector of eleven parameters, each parameter a nonnegative integer or ∞, subject to numerous but simple conditions. Each vector satisfying these conditions is in fact used, and matches directly a (finite) defining set of laws for the variety it labels. Moreover, one can readily recognize from the parameters whether one variety is contained in another. The third paper will complete the determination of all varieties of nilpotent groups of class four.

In this paper we complete the investigation of those varieties of nilpotent groups of class (at most) four whose free groups have no nontrivial elements of odd order. Each such variety is labelled by a vector of sixteen parameters, each parameter a nonnegative integer or ∞, subject to numerous but simple conditions. Each vector satisfying these conditions is in fact used and directly yields a defining set of laws for the variety it labels. Moreover, one can easily recognise from the parameters whether one variety is contained in another. In view of the reduction carried out in the first paper of this series (written jointly with L. G. Kovács) this completes the determination of all varieties of nilpotent groups of class four.

An undirected simple graph G is called chordal if every circle of G of length greater than 3 has a chord. For a chordal graph G, we prove the following: (i) If m is an odd positive integer, Gm is chordal. (ii) If m is an even positive integer and if Gm is not chordal, then none of the edges of any chordless cycle of Gm is an edge of Gr, r < m.