In a series of recent papers ((10), (9) and (11)) Rosenthal and Odell have given a number of characterizations of Banach spaces that contain subspaces isomorphic (that is, linearly homeomorphic) to the space l1 of absolutely summable series. The methods of (9) and (11) are applicable only in the case of separable Banach spaces and some of the results there were established only in this case. We demonstrate here, without the separability assumption, one of these characterizations:

a Banach space B contains no subspace isomorphic to l1 if and only if every weak* compact convex subset of B* is the norm closed convex hull of its extreme points.

Consider the classes OHX of noncompact Riemannian manifolds M of dimension N ≥ 2 carrying no non-constant harmonic functions with properties X = P (positive), B (bounded), D (Dirichlet finite), or BD (B and D). Denote by OG the class of parabolic manifolds of dimension N ≥ 2. The following complete string of strict inclusion relations is known:

The investigation of a topological, uniform or proximity space endowed with a preordering is by now well established. A topological space along with a preordering related to the topology in varying degrees has been studied, for example, in (3), (4), (5) and (6). ‘Uniform preordered spaces’ and ‘Proximity preordered spaces’ have been considered in (6) and (8) respectively. In each of these cases generalizations of varying proportions of the classical theories have been obtained.

If R is a Banach algebra and ø ∈ R′, the dual space, then we may define a bounded linear map by

We shall show that for suitable p the requirement that each be p-absolutely summing constrains R to be an operator algebra, or even, in certain cases, a uniform algebra. In this way we are able to give generalizations of results of Varopoulos (12) and Kaijser (4).

Consider a single-server queueing system with a Poisson arrival process at rate λ and positive service requirements independently distributed with common distribution function B(z) and finite expectation

where βλ < 1, i.e. an M/G/1 system. When the queue discipline is first come first served, or last come first served without pre-emption, the stationary departure process is Poisson if and only if G = M (i.e. B(z) = 1 − exp (−z/β)); see (8), (4) and (2). In this paper it is shown that when the queue discipline is last come first served with pre-emption the stationary departure process is Poisson whatever the form of B(z). The method used is adapted from the approach of Takács (10) and Shanbhag and Tambouratzis (9).

The notion of basic topological binary relations is introduced, and corresponding to each such relation we define -regularity, -connectedness, and a separation axiom . The family of separation axioms so obtained is shown to include all the standard axioms T0, T1, …, T5, while familiar properties such as hyperconnectedness and ultraconnectedness are among the class of connectedness conditions. The regularity conditions include R0, R1, and regularity itself.

This paper will show that after localization at any given prime p, the infinite loop space structure on the space BSO is essentially unique. If the word ‘localization’ is replaced by ‘completion’, the result continues to hold; and both results continue to hold if the space BSO is replaced by the space BSU.

The work in this paper arises from observing a simple example of a potential in the plane, as the potential varies with changes in a ‘control’ parameter. To fix ideas, we first discuss the example in intuitive terms, and the rest of the paper is a detailed technical discussion. The example aids understanding of bifurcation theory and suggests interesting generalizations. We begin by describing the example, as follows.

Consider a long, finite sequence xl, x2, …, xN of repeated measurements on the same physical quantity. The usual model is that the data consists of observed values on a sequence X1, X2, …, XN of independent and identically distributed random variables. It is customary to require that the Xi have a finite mean and variance: µ = E(Xi) is the ‘true value’ of the quantity being measured; and

measures the variability in the measuring process.

Arnol'd (1) remarks that the partition by Segre symbol of the endomorphisms of the complex vector space is a finite constructible stratification: in this paper we verify this statement, and show moreover that it is a Whitney stratification. Some of the arguments are by no means special to the case in hand; however, there seems little point in attempting a general formulation. It is a pleasure to acknowledge helpful conversations with Professor C. T. C. Wall and Dr E. J. N. Looijenga.

Let Xm and Yn be C1 manifolds embedded in , m < n < p, and let . In (4) C. T. C. Wall considered the following conditions:

(as) For any local C1 retraction at x, π: , x has a neighbourhood U such that is a submersion.

The present paper is primarily concerned with the structure of certain random sets of flats in n-dimensional Euclidean space Rn. To be more specific, let be the space of k-dimensional affine subspaces of Rn (to be called k-flats below), equipped with its natural topology ((14), p. 132). is a topological homogeneous space under the action of the group of Euclidean displacements in Rn and our purpose is to give conditions under which a stochastic point process on , stationary under , is a doubly stochastic Poisson process. The results will be seen to apply equally well to abstract spaces sharing some of the structural properties of .

Various elegant properties have been found for the waiting time distribution G for the queue GI/G/1 in statistical equilibrium, such as infinite divisibility ((1), p. 282) and that of having an exponential tail ((11), (2), p. 411, (1), p. 324). Here we derive another property which holds quite generally, provided the traffic intensity ρ < 1, and which is extremely simple, fitting in with the above results as well as yielding some useful properties in the form of upper and lower stochastic bounds for G which augment the bounds obtained by Kingman (5), (6), (8) and by Ross (10).

1. Introduction. Throughout this paper we shall take S to be a separately continuous, completely regular and Hausdorff, topological semigroup. We denote by Cb(S) the space of continuous and bounded complex-valued functions on S, and by βS the Stone-Čech compactification of S.

The W.K.B. solution for radio waves in a plane stratified ionosphere was obtained by Budden ((4), §§ 18.10 and 18.11) by a process using approximate diagonalization of the governing equations. Alternatively the solution may be obtained by specializing to the one-dimensional case the result of Bennett (1) which employs frequency scaling. Formally similar results were obtained by Ludwig (8) without explicitly introducing frequency scaling.

In the well-known deterministic model for the spread of an epidemic, one considers a population of uniform density along a line and divides the population into three classes: susceptible but uninfected, infected and infectious, infected but removed. If we denote space and time variables by s, t and let x(s, t), y(s, t), z(s, t) be the proportions of the population at (s, t) in these three classes, then x + y + z = 1 and we suppose that

Here Ῡ(s, t) denotes a space average ∫ y(s + σ) p(σ) dσ, where p is a probability density function; b is the removal rate; the scale of t has been adjusted to remove a constant that would otherwise occur in (1).

In (1) Akemann and Rosenfeld introduced a property for Banach algebras which they called (*). A Banach algebra satisfies (*) if every maximal one-sided ideal in is closed. They proved that certain classes of Banach algebra with satisfy (*), and they mentioned a conjecture that if is a Banach algebra with , then satisfies (*). In this paper we show that, if is a Banach algebra with a bounded right (left) approximate identity, then maximal left (right) ideals in are closed, and we give a counter-example to the above conjecture. We also give an independent proof that C*-algebras satisfy (*).

This paper considers the behaviour of infinite haploid genetic populations under the influence of mutation and selection depending on a single locus. Under wide conditions the Perron–Frobenius theory of non-negative matrices and its generalization by Vere-Jones are used to prove that there is a single globally stable state of the population when there is a finite or, under more restrictive conditions, an infinite set of possible alleles.

The analysis of the interactions between the electrons of different atoms is complicated by many effects. Unless the atoms are free and well separated, account has to be taken of covalency, overlap, exchange, and superexchange. But, although such effects are often important, it is sometimes useful to regard the electrons as being located on separate atoms. Two electrons, labelled 1 and 2, on respective atoms A and B, then interact principally through the Coulomb energy e2/r12. Neighbouring atoms L, M, N, …, which we suppose are spherically symmetric, make their presence felt by acting as polarizable spheres. The electric fields thereby produced modify the simple Coulombic expression for the energy of interaction of the two electrons. It is the purpose of this article to examine such modifications in detail.

The response of a supersonic laminar boundary layer is considered when fluid is blown obliquely into it from a slot in the boundary. The velocity and length scales associated with the slot blowing are chosen in such a way that the triple-deck theory of Stewartson (9) may be exploited. It is shown that, depending upon the blowing angle, there can be either a compressive or expansive interaction between the boundary layer and free stream upstream from the slot.