Correcting some “proofs” given in an earlier paper of the same title, we prove here, among other things, that, if S is a subgroup of a topological group that is complete in a left invariant metric or locally compact, then every weakly almost periodic function on S is (left and right) uniformly continuous. We also prove a theorem related to results of R. B. Burckel and of W. W. Comfort and K. A. Ross: a topological group is pseudocompact if and only if WAP(G) = C(G).

We are concerned with the problem of the existence and uniqueness of regularly varying (in Karamata's sense) solutions ϕ of the linear functional equation in a right neighbourhood of x = 0. Under suitable conditions on the given functions f and h, the uniqueness of solutions depends essentially on whether the series Σh ∘ f1 converges or diverges; here fi denotes the i-th functional iterate of f. The existence of solutions may be proved under further assumptions.

The case of the more general linear functional equation may be reduced to that of equation (*).

The aim of this paper is to enumerate all equivalence classes of perfect septenary forms with Δ4 = 2. This is an important section of the complete enumeration of perfect septenary forms by the method which was outlined in Stacey (1975). There are nine equivalence classes of these forms, four of which were announced for the first time in Stacey (1975). This work forms part of the author's D. Phil, thesis and was done at Oxford under the supervision of Dr. B. J. Birch.

The objective of the paper Aczél and Ostrowski (1973) is to show in an elementary way that any real-valued function f, defined on (0,1) and satisfying he inquality , where n ≧ 3 is fixed, is necessary of the form .

A general metric is considered. Some solutions correspoding to the field equations Rik λgik = −8πEik are obtained as particular cases. Details of these solutions are also discussed. An Einstein space corresponding to the field equations Rik = λ gik is also constructed as a particular case of the general metric.

Let f(z) denote a rational or entire function of the complex variable z and fn(z), n = 1,2, …, the n−th iterate of f. Provided f is not rational of order 0 or 1, the set of those points where {fn(z)} forms a normal family is a proper subset of the plane and is invariant under the map z → f(z). A component G of is a wandering domain of f if fn(G)∩fn(G) = Ø for all k ≧ 1, n ≧ 1, k ≠ n. The paper contains the construction of a transcendental entire function which has wandering domains.

A technique for the estimation of domains of local stability for difference equations is discussed. A Liapunov function is used in the estimation. Sharper results are possible if there is only one type of nonlinearity. when open Liapunov surfaces are possible. An example of the technique is given.

Let ν = ef + 1 be a prime power, and consider G the cyclic group of order ν − 1 with e cosets Ci, of order f defined as Ci = {xej+i: 0 ≦ j ≦ f − 1} and 0 ≦ i ≦ e − 1, where x is a primitive element of GF(pα) and a generator of G. By using these cosets we give a simple construction for certain classes of Supplementary Difference Sets, Difference Sets, and Szekeres Difference Sets. These classes are not new, but the simple method of construction is original.

An inverse semigroup S is called E-unitary if the equations ea = e = e2 together imply a2 = a. In a previous paper the author showed that every inverse semigroup is an idempotent separating homomorphic image of an E-unitary inverse semigroup. The main question considered in this paper is the following. Given an inverse semigroup S give a method for constructing E-unitary inverse semigroups P and idempotent separating homomorphisms φ P → S in such a way that the structure of P as a P-semigroup is evident.

We consider the effect on the stability properties of a graph G, of the presence in the automorphism group of G of automorphisms (uv)h, where u and v are vertices of G, and h is a permutation of vertices of G excluding u and v. We find sufficient conditions for an arbitrary graph and a cartesian product to have stability index one, and conjecture in the latter case that they are necessary. Finally we exhibit explicitly a large class of graphs which have stability index one.

All such nonabelian finite p-groups are classified. They coincide with the class of nonabelian finite p-subgroups of GL(p, F), where F is a field, not of characteristic p, which contains all p power roots of 1, or agian with the class of all nonabelian finite subgroups of Zp= wr Zp. Various automorphism groups associated to them and their representations are calculated. Two such subgroups of GL(p, F) are conjugate as subgroups of GL(p, F) iff they are isomorphic.

Any congruence on an orthodox semigroup S induces a partition of the set E of idempotents of S satisfying certain normality conditions. Meakin (1970) has characterized those partitions of E which are induced by congruences on S as well as the largest congruence ρ and the smallest congruence σ on S corresponding to such a partition of E. In this paper a more precise description of ρ and σ is given.

A class of algebras A is said to have the strong amalgamation property if for any indexed set of algebras {Ai: i ∈ J} from A, each having an algebra U ∈ A as a subalgebra, there exist an algebra B ∈ A and monomorphisms φi: Ai → B (for each i ∈ J) such that (i) φi|U = φi| U for all i,j∈J, (ii) φi(Ai) ∩ φi(Aj) =φi(U) for atl i, j ∈ J with i ≠ j, where φi|U denotes the restriction of φi to U. Omitting condition (ii) gives us the definition of the weak amalgamation property.

Conditions are investigated under which a product G = HK of soluble subgroups H, K, each permutable in G, is soluble. It is shown in particular that this is true if one of the subgroups is nilpotent or if H ∩ K is subnormal in both H and K.