This paper is devoted to dualization of paracompactness to the coarse category via the concept of $R$-disjointness. Property A of Yu can be seen as a coarse variant of amenability via partitions of unity and leads to a dualization of paracompactness via partitions of unity. On the other hand, finite decomposition complexity of Guentner, Tessera, and Yu and straight finite decomposition complexity of Dranishnikov and Zarichnyi employ $R$-disjointness as the main concept. We generalize both concepts to that of countable asymptotic dimension and our main result shows that it is a subclass of spaces with Property A. In addition, it gives a necessary and sufficient condition for spaces of countable asymptotic dimension to be of finite asymptotic dimension.

To each idempotent v of a semigroup T, there is associated, in a natural way, a subsemigroup Tv of T. The subsemigroup Tv is simply the collection of all elements of T for which v acts as a two-sided identity. We refer to such a subsemigroup as an I-subsemigroup of T. We first establish some elementary properties of these subsemigroups with no restrictions on the semigroup in which they are contained. Then we turn our attention to the semigroup of all continuous selfmaps of a topological space. The I-subsemigroups of these semigroups are investigated in some detail and so are the a-monomorphisms [3, p. 518] from one such semigroup into another. Among other things, a relationship is established between I-subsemigroups and α-monomorphisms. An analogous theory exists for semigroups of closed selfmaps on topological spaces. A number of results are listed for these semigroups with the proofs often deleted since, in many cases, the situation is much the same as for semigroups of continuous functions.

The setting is a compact Hausfroff space ω. The notion of a Walls class of subsets of Ω is defined via strange axioms—axioms whose justification rests with examples such as the collection of regular open sets or the range of a strong lifting. Avarient of Rosenthal' famous lwmma which applies directly to Banach space-valued measures is esablished, and it is used to obtain, in elementary fashion, the following two uniform boundedness principles: (1)The Nikodym Boundedness Theorem. If K is a family of regular Borel vector measures on Ω which is point-wise bounded on every set of a fixed Wells class, then K is uniformly bounded. (2)The Nikodym Covergence Theorem. If {μn} is a sequence of regular Borel vector measures on Ω which is converguent on every set of a fixed Wells class, then the μn are uniformly countably additive, the sequence {μn} is convergent on every Borel subset of Ω and the pointwise limit constitutes a regular Borel measure.

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.

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.

We give a qualitative description of the set 𝒪G(H) of overgroups in G of primitive subgroups H of finite alternating and symmetric groups G, and particularly of the maximal overgroups. We then show that certain weak restrictions on the lattice 𝒪G(H) impose strong restrictions on H and its overgroup lattice.

For any real number ${\it\beta}$ with ${\it\beta}>1$, let ${\mathcal{M}}(\,{\it\beta})$ (${\mathcal{N}}(\,{\it\beta})$ respectively) denote the class of analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ and satisfying $\text{Re}\,P_{f}<{\it\beta}$ ($\text{Re}\,Q_{f}<{\it\beta}$ respectively) in $\mathbb{D}$, where $P_{f}=zf^{\prime }(z)/f(z)$ and $Q_{f}=1+zf^{\prime \prime }(z)/f^{\prime }(z)$. Also, for ${\it\beta}>1$, let ${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$ (${\mathcal{N}}{\rm\Sigma}(\,{\it\beta})$ respectively) denote the class of analytic functions $g$ of the form $g(z)=z(1+\sum _{n=1}^{\infty }b_{n}z^{-n})$ and satisfying $\text{Re}\,P_{g}<{\it\beta}$ ($\text{Re}\,Q_{g}<{\it\beta}$ respectively) for $z\in {\rm\Delta}=\{z\in \mathbb{C}:1<|z|<\infty \}$. In this paper, we shall determine the coefficient bounds, inverse coefficient bounds, the growth and distortion theorem and the upper bounds for the Fekete–Szegő functional ${\rm\Lambda}_{{\it\lambda}}(f)=a_{3}-{\it\lambda}a_{2}^{2}$ for functions $f$ in the classes ${\mathcal{M}}(\,{\it\beta})$ and ${\mathcal{N}}(\,{\it\beta})$. Further, we shall solve the maximal area problem for functions of the type $z/f(z)$ when $f\in {\mathcal{M}}(\,{\it\beta})$, which is Yamashita’s conjecture for the class ${\mathcal{M}}(\,{\it\beta})$. We shall obtain the radius of convexity for the class ${\mathcal{N}}(\,{\it\beta})$. We shall also determine the coefficient bounds for functions $g$ in the classes ${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$ and ${\mathcal{N}}{\rm\Sigma}(\,{\it\beta})$ and the inverse coefficient bounds for functions $g$ in the class ${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$. All the results are sharp.

We exhibit an interesting Cayley graph X of the elementary abelian group Z26 with the property that Aut(X) contains two regular subgroups, exactly one of which is normal. This demonstrates the existence of two subsets of Z26 that yield isomorphic Cayley graphs, even though the two subsets are not equivalent under the automorphism group of Z26.

In a series of papers, the author has previously investigated the spectra and fine spectra for weighted mean matrices, considered as bounded operators over various sequence spaces. This paper examines the spectra of weighted mean matrices as operators over bνv0, the space of null sequences of bounded variation.

It is shown that a variety ν has distributive congruence lattices if and only if the intersection of two principal congruence relations is definable by equations involving terms with parameters. The nature of the terms involved then provides a useful classification of congruence distributive varieties. In particular, the classification puts into proper perspective two stronger properties. A variety is said to have the Principal Intersection Property if the intersection of any two principal congruence relations is principal, or the Compact Intersection Property if the intersection of two compact congruence relations is compact. For non-congruence-distributive varieties, it is shown that some useful constuctions are nevertheless possible.

We say that a subgroup H is an n-th maximal subgroup of G if there exists a chain of subgroups G = G0 > G1 > … > Gn = H such that each Gi is a maximal subgroup of Gi-1, i = 1, 2, …, n. The purpose of this note is to classify all finite simple groups with the property that every third maximal subgroup is nilpotent.

The authors begin by presenting a brief survey of the various useful methods of solving certain integral equations of Fredholm type. In particular, they apply the reduction techniques with a view to inverting a class of generalized hypergeometric integral transforms. This is observed to lead to an interesting generalization of the work of E. R. Love [9]. The Mellin transform technique for solving a general Fredholm type integral equation with the familiar H-function in the kernel is also considered.

Let K be an open convex domain in n-dimensional Euclidean space, symmetric about the origin O, and of finite Jordan content (volume) V. With K are associated n positive constants λ1, λ2,…,λn, the ‘successive minima of K’ and n linearly independent lattice points (points with integer coordinates) P1, P2, …, Pn (not necessarily unique) such that all lattice points in the body λ,K are linearly dependent on P1, P2, …, Pj-1. The points P1,…, Pj lie in λK provided that λ > λj. For j = 1 this means that λ1K contains no lattice point other than the origin. Obviously

A balanced tournament design, BTD(n), defined on a 2n—set V is an arrangement of the () distinct unordered pairs of the elements of V into an n × 2n − 1 array such that (1) every element of V is contained in precisely one cell of each column, and (2) every element of V is contained in at most two cells of each row. In this paper, we investigate the existence of balanced tournament designs with a pair of almost orthogonal resolutions. These designs can be used to construct doubly resolvable (ν, 3, 2)- BIBD s and, in our smallest applications, have been used to construct previously unknown doubly resolvable (ν, 3, 2)- B I B D s.

The Moore-Penrose inverse of a general bordered matrix is found under various conditions. The Moore-Penrose inverses obtained by Hall and Hartwig (1976) are shown to be special cases of these more general results.

By a fundamental theorem of Brauer every irreducible character of a finite group G can be written in the field Q(εm) of mth roots of unity where m is the exponent of G. Is it always possible to find a matrix representation over its ring Z[εm] of integers? In the present paper it is shown that this holds true provided it is valid for the quasisimple groups. The reduction to such groups relies on a useful extension theorem for integral representations. Iwasawa theory on class groups of cyclotomic fields gives evidence that the answer is at least affirmative when the exponent is replaced by the order, and provides for a general qualitative result.

We establish an extension of the Banach–Stone theorem to a class of isomorphisms more general than isometries in a noncompact framework. Some applications are given. In particular, we give a canonical representation of some (not necessarily linear) operators between products of function spaces. Our results are established for an abstract class of function spaces included in the space of all continuous and bounded functions defined on a complete metric space.

Some results on fixed points of asymptotically regular mappings are obtained in complete metric spaces and normed linear spaces.

The structure of the set of common fixed points is also discussed in Banach spaces. Our work generalizes essentially known results of Das and Naik, Fisher, Jaggi, Jungck, Rhoades, Singh and Tiwari and several others.

Several authors have studied various types of rings of continuous functions on Tychonoff spaces and have used them to study various types of compactifications (See for example Hager (1969), Isbell (1958), Mrowka (1973), Steiner and Steiner (1970)). However many important results and properties pertaining the Stone-Čech compactification and the Hewitt realcompactification can be extended to a more general setting by considering appropriate lattices of sets, generalizing that of the lattice of zero sets in a Tychonoff space. This program was first considered by Wallman (1938) and Alexandroff (1940) and has more recently appeared in Alo and Shapiro (1970), Banachewski (1962), Brooks (1967), Frolik (1972), Sultan (to appear) and others.

In this paper, we establish an extension of the matrix form of the Brunn-Minkowski inequality. As applications, we give generalizations on the metric addition inequality of Alexander.

2000 Mathematics subject classification: primary 52A40.