We consider two related classes of groups. For any group G of the first of these — the smallest class containing all finitely generated free groups and closed under cyclic amalgamations — we show that for any U, W ∈ G we can “effectively describe” the set of all x and y with U* and W* conjugate in G. For the second class, which consists of all groups obtained from the first class through cyclic HNN constructions, we solve the conjugacy and power-conjugacy problems.

The Boolean ring B{M} universal over a meet semilattice M is examined. It is the vector space over the two element field Z2 with base M\{0}. The Z2 linear independence of a meet subsemilattice of a Boolean ring is characterized in order theoretic terms and some ramifications of this on B[M] are considered. The space (ℱpM) of proper filters of M is shown homeomorphic to the Stone space S(B[M]) of B[M] if M has no least element, with ℱP(M) ∪ {M} and S(B[M]) homeomorphic otherwise. The congruence lattice θ(M) of M is compared to the ideal lattice ℱ(B [M]) of B [M] with best results coming if M is a tree with zero when θ (M) = ℐ (B [M]).

We establish a version of Bochner Theorem due to S. Boylan for Banch spaces with a basis.

If is a model with dimension and finite closure, then T() is No-categorical. If is atomic, has dimension and finitely many algebraic elements, then has finite closure or a finite basis. If has finite closure, satisfies the Exchange Lemma, and one-one maps between independent subsets are elementary, then has dimension.

A generalization of Karamata's theorem on integrals of regularly varying functions is proved. Using Laplace-Stieltjes transforms it is shown that any regularly varying function with exponent α (α + 1 ∉ N) is asymptotic to another regularly varying function all of whose derivations are regularly varying.

Cordes (1976) introduced the problem of determining the maximum number of resolution classes of a finite set partitioned into equicardinal subsets such that the number of pairs common to any 2 classes is minimized. A later paper of Mullin and Stanton (1976) investigated those conditions under which the configurations were actually BIBD's. They obtained a bound for these special configurations and conjectured it applied in general. We prove this in the present paper. A recursive and a direct construction are also given for a special class of configurations.

In the present paper we prove that every finite dimensional non-atomic measure ν is open and monotone (viz. ν–1 preserves connected sets) relative to the usual Fréchet-Nikodým topology on its domain and the relative topology on its range. An arbitrary finite dimensional measure is found on the other hand to be biquotient.

Given a vector measure ν, we further investigate the properties of its integral map Tν: φ → ∫ φdν defined on the set of functions φ in L1(|ν|) for which φ(s) ∈ [0,1] |ν|-almost everywhere. When ν is finite dimensional, Tν is found to be always open. In general, when Tν is open, the set of extreme points of the closed convex hull of the range of ν is proved to be closed, and when ν and Tν are both open, the range of ν in itself is closed.

Riesz polynomials are employed to give a sufficient condition for a non-abelian compact group G to have an infinite uniformly approximable Sidon set. This condition is satisfied if G admits infinitely many pairwise inequivalent continuous irreducible unitary representations of the same degree. Consequently a compact Lie group admits an infinite Sidon set if and only if it is not semi-simple.

In this paper, several characterizations of minimal Hausdorff spaces are given.

Let X be a finite set and S a semigroup of transformations of X. We investigate the trace on X of a random walk on S. We relate the structure of the trace process, which turns out to be a Markov chain, to that of the random walk. We show, for example, that all periods of the trace process divide the period of the random walk.

Let G be a group which has exactly one normal subgroup N with G/N infinite cyclic; clearly such a group is an HNN group with base N and stable letter t, where t induces an automorphism θ, of N under conjugation. We call such a group G a unique base group with automorphism θ, and base N.

For a normed linear space X we investigate conditions for the existence of support maps under which the image of X is a dense subset of the dual. In the case of finite-dimensional spaces a complete answer is given. For more general spaces some sufficient conditions are obtained.