A complete characterization of Boolean algebras which admit nonatomic charges (i.e. finitely additive measures) is obtained. This also gives rise to a characterization of superatomic Boolean algebras. We also consider the problem of denseness of the set of all nonatomic charges in the space of all charges on a given Boolean algebra, equipped with a suitable topology.

Nikolskii-type inequalities, thus inequalities between different metrics of a function, are established for trigonometric polynomials and pth power integrable functions, 0<p<∞, of several variables having Fourier transform with compact support. It is shown that certain gaps in the spectra of the functions involved may be taken into account. As an immediate consequence it follows that the general results cover the classical inequalities which are concerned with functions of rectangular type. But at the same time one may give applications to functions of type K where K is a symmetric body in Euclidean n–space.

A construction is given for a non-desarguesian projective plane P and an absolute-point free polarity on P such that the group of collineations of P which commute with the polarity is isomorphic to an arbitrary preassigned finite group.

All three-groups with cyclic centre and such that the quotient by the centre has maximal class are listed and a presentation with generators and relations for each is given.

A certain natural extension B of the Borel σ-algebra is studied in generalized weakly θ-refinable spaces. It is shown that a set belongs to B whenever it belongs to B locally. From this it is derived that if ℵωα is more complicated than aunion of less than ℵα weakly θ-refinable subspaces.

A continuum (that is, a compact connected Hausdorff space) is hereditarily locally connected if each of its subcontinua is locally connected. It is shown that a continuum X is hereditarily locally connected if and only if for each connected open set U in X and each point p in the boundary of U, U ∪ {p} is locally connected. This result is used to prove that if X is an hereditarily locally connected continuum, U is a connected open subset of X, p is an element of the boundary of U and X is first countable at p, then p is arcwise accessible from U.

Given a group G, we may ask whether it is the commutator subgroup of some group G. For example, every abelian group G is the commutator subgroup of a semi-direct product of G x G by a cyclic group of order 2. On the other hand, no symmetric group Sn(n>2) is the commutator subgroup of any group G. In this paper we examine the classical linear groups over finite fields K of characteristic not equal to 2, and determine which can be commutator subgroups of other groups. In particular, we settle the question for all normal subgroups of the general linear groups GLn(K), the unitary groups Un(K) (n≠4), and the orthogonal groups On(K) (n≧7).

A sandwich semigroup of continuous functions consists of continuous functions with domains all in some space X and ranges all in some space Y with multiplication defined by fg = foαog where α is a fixed continuous function from a subspace of Y into X. These semigroups include, as special cases, a number of semigroups previously studied by various people. In this paper, we characterize the regular elements of such semigroups and we completely determine Green's relations for the regular elements. We also determine the maximal subgroups and, finally, we apply some of these results to semigroups of Boolean ring homomorphisms.

If f is areally mean p–valent in the unit disc, if λ>0 and if fλ is defined as a single-valued analytic function on the unit disc with finitely many arcs removed, several results in the recent literature suggest that fλ might be areally mean pλ–valent. The purpose of this note is to determine the valence of fλ when f is areally mean p–valent, and also to characterize those functions for which fλ is pλ–valent for all λ>0. Analogous results are obtained for functions which are either s–dimensionally mean p–valent or logarithmically mean p–valent.

If is a saturated formation of finite soluble groups and G is a finite group whose -residual A is abelian then it is well known that G splits over A and the complements are conjugate. Hartley and Tomkinson (1975) considered the special case of this result in which is the class of nilpotent groups and obtained similar results for abelian-by-hypercentral groups with rank restrictions on the abelian normal subgroup. Here we consider the super-soluble case, obtaining corresponding results for abelian-by-hypercyclic groups.

Let R be the incidence algebra of a finite partially ordered set T over a commutative noetherian ring A. Then the spectrum of R is homeomorphic to the product (Spec A) x T, where Spec A has the usual Zariski topology and T has the order topology. An explicit construction is given for the structure sheaf of R over its spectrum.

It is shown that every possible spectrum of a Mal'cev definable class of varieties which should occur does occur. It follows that there are continuum many Mal'cev definable classes, a result also obtained by Taylor (1975) and Baldwin and Berman (1976).

The structure is determined for the existence of some amicable weighing matrices. This is then used to prove the existence and non-existence of some amicable orthogonal designs in powers of two.