An ordered group (o-group for short) is a group endowed with a linear (i.e. total) order such that for all x, y, z, xz ≤ yz and zx ≤ zy whenever x ≤ y. A group for which such an order exists is called an orderable group. A group G is said to be divisible if for each positive integer m and each g ε G, there is x ε G such that xm = g.

In this paper we classify the maximal subgroups of the smallest Fischer group Fi22, which is a simple group of order 64561 751 654 400 = 217. 39. 52. 7. 11. 13. Although this is not a complete classification into conjugacy classes, it is for most practical purposes almost as good, since the exceptional cases are very small groups.

Groups with length functions were introduced by Lyndon [5], and those with integer valued length were shown by Chiswell [2] to be equivalent to groups acting on trees, investigated by Serre[6]. Watatani[7] showed that for a group with integer valued length function the edges of the tree form an orthonormal set in a Hilbert space on which the group acts as a group of isometries.

The model K() presented in this paper is a new inner model of ZFC which can contain measurable cardinals of high order. Like the model L() of [14], from which it is derived, K() is constructed from a sequence of filters such that K() satisfies for each (α, β) ε domain () that (α,β) is a measure of order β on α and the only measures in K() are the measures (α,β). Furthermore K(), like L(), has many of the basic properties of L: the GCH and ⃟ hold and there is a definable well ordering which is on the reals. The model K() is derived from L() by using techniques of Dodd and Jensen [2–5] to build in absoluteness for measurability and related properties.

The function c(p) is defined for positive integers p ≥ 4 by

where > denotes contraction. Random graph examples show

In 1968 Mader showed that c(p) ≤ 8(p − 2) [log2 (p − 2)] and more recently Kostochka showed that p√(log p) is the correct order for c(p). We present a simple argument showing c(p) ≤ 2.68p √(log2p)(l + ο(l)).

In computational matrix algebra, Gershgorin's Theorem [8] and Ostrowski's Theorem[14] play key roles in establishing regularity conditions for complex n × n matrices. As a result of this importance as well as possible applications in other areas, several extensions of these results have been made (Kotelyanski[9], Ky Fan [5], Brauer[2], Nwokah[10]), for point set matrices. More recently, further extensions have been made to include regularity conditions for composite matrices (matrices partitioned into blocks) (Brenner [3], Van der Sluis[16], Feingold and Varga[6], Cook [4]). Cook's result was motivated by engineering applications but is rather restrictive since, like Feingold and Varga, it requires that the matrices under consideration be diagonally dominant. In this paper, the diagonal dominance condition is replaced by a more general condition (the H-matrix condition). Estimates for the diagonal blocks of block-partitioned matrix are then obtained. The results on inclusion regions for eigenvalues unifiex the previous results of Feingolg and Varga [6], kotelyanski [9], Brauer [2], and generalizes the earlier results of Nwokah [10], from point matrices to block-partitioned matrices. In a sense, these results complement those of brenner [3] Van der Sluis [16], but like those of Cook [4], have a more immediate direct application to control engineering in the area of computer-aided design of composite feedback control systems (Nwokah [12]), which will be briefly reviewed towards the end of the paper.

The study of solutions to special examples of elliptic differential equations, using Clifford algebras, has been developed by a number of authors [4–12, 14–16, 18, 19, 21–30, 32]. This study has also been applied [3, 13, 17, 28] within a number of areas of theoretical physics, including Yang–Mills field theory, and the Kähler equation. Most of the function theories associated to the solutions of these elliptic differential equations, referred to as generalized Cauchy–Riemann equations [21,32], generalize in a natural manner many aspects of classical one-variable complex analysis [1]. For instance, each of these function theories contain analogues of the Cauchy theorem, Cauchy integral formula and Laurent expansion theorem. However, no information has yet been obtained on the local topological behaviour of a general solution to any of these equations.

An aspherical manifold is a connected manifold whose universal cover is contractible. It has been conjectured that the Whitehead groups Whj (π1 M) (including the projective class group, the original Whitehead group of π1M, and the higher Whitehead groups of [9]) vanish for any compact aspherical manifold M. The present paper considers this conjecture for twelve hyperbolic 3-manifolds constructed from regular hyperbolic polyhedra. Hyperbolic manifolds are of special interest in this regard since so much is known about their topology and geometry and very little is known about the algebraic K-theory of hyperbolic manifolds whose fundamental groups are not generalized free products.

A classical knot in S3 is said to be slice if it is the boundary of a smooth (or PL locally unknotted) disc in B4. The first obstructions to sliceness were introduced in [3] (the Alexander polynomial is of the form f(t). f(t−1)) and [7] (the signature is zero). Levine defined the notion of algebraically slice knot in [5]. This property implies the first two obstructions vanish.

Throughout this paper all scalars and vector spaces will be assumed to be complex unless there are specific indications to the contrary.

In this paper we examine the existence of extreme points in unbounded, closed and convex subsets K of a Banach space X.

In this note we prove that every nearly uniformly convex space has normal structure and that K-uniformly convex spaces are super-reflexive.

We recall [1] that a Banach space is said to be Kadec–Klee if whenever xn → x weakly and ∥n∥ = ∥x∥ = 1 for all n then ∥xn −x∥ → 0. The stronger notions of nearly uniformly convex spaces and uniformly Kadec–Klee spaces were introduced by R. Huff in [1]. For the reader's convenience we recall them here.

In an article on the philosophy of chance, Poincaré[7] showed that the distribution of the stopping position of a needle pivoted about its centre tends to the uniform distribution on the circle as the distribution of the initial push becomes spread out along the real line. This result was formalized by Feller [2] and strengthened by Mardia [6] as follows.

A random submatroid ωr of the projective geometry PG(r − 1, q) is obtained from PG(r − 1, q) by deleting elements so that each element has, independently of all other elements, probability 1 − p of being deleted and probability p of being retained. The properties of such random structures were studied in [5] and [6]. In the first of these papers, p was kept fixed, while in the second, motivated by Erdös and Rényi's work ([3), [4]) on random graphs, p was taken to be a function of r. A recent paper of Bollobás[2] strengthens and extends a number of the results of Erdös and Rényi. In this paper we prove matroid analogues of several of Bollobàs's results thereby extending some of the results of [6].

This paper presents an alternative analysis of the behaviour of the Moran model. Using some of the techniques developed in the study of interactive particle systems, a process which is dual to the Moran model is constructed. Exact analysis of this dual process is considerably easier than for the original model. Because of the relationship between the two processes it is then possible to find the distribution of the Moran model.

We consider second order differential equations of Liénard type:

Such equations have been very widely studied and arise frequently in applications. There is an extensive literature relating to the existence and uniqueness of periodic solutions: the paper of Staude[6] is a comprehensive survey. Our interest is in the number of periodic solutions of such equations.

Suppose that a function f(z, n) depends on a large parameter n. A proposed expression g(z, n) is an asymptotic approximation for f(z, n) if it can be shown that the error (i.e. the difference between f(z, n) and g(z, n)) is small of an appropriate order when n → ∞. Effective error bounds are particularly useful in numerical work with asymptotic expansions. Most of the existing derivations of error bounds involve complicated calculations.