0. Introduction. Let and be connected cohomology theories on the category of all CW spaces. In this paper we examine the relationship between natural homomorphisms (stable operations) of degree zero from to and natural homomorphisms from to in a number of cases of particular interest in topology – foremost the K-theories and the bundle theories classifying ‘surgery problems’. There are two parts to this. On the one hand we ask under what conditions a homomorphism from to extends to a stable operation and on the other hand when is a stable operation determined by its restriction to .

We investigate here the question of uniqueness of best approximation to operators in von Neumann algebras by elements of certain linear subspaces. Recall that a linear subspace V of a Banach space X is called a Chebyshev subspace if each vector in X has a unique best approximation by vectors in V. Our first main result characterizes the one-dimensional Chebyshev subspaces of a von Neumann algebra. This may be regarded as a generalization of a result of Stampfli [(4), theorem 2, corollary] which states that the scalar multiples of the identity operator form a Chebyshev subspace. Alternatively it may be regarded as a generalization of the commutative situation in which a continuous complex-valued function f on a compact Hausdorff space X spans a Chebyshev subspace of C(X) if and only if f does not vanish on X [(3), p. 215]. Our second main result is that a finite dimensional * subalgebra, of dimension > 1, of an infinite dimensional von Neumann algebra cannot be a Chebyshev subspace. This imposes limits to further generalization of Stampfli's result.

In (1) Atkinson and Reuter prove the existence of wave solutions with velocities strictly greater than some critical velocity ʗ0 of an equation associated with the deterministic model for the spread of an epidemic. The existence of wave solutions with velocity ʗ0 is left as an open question. In this note we show that Atkinson and Reuter's results and a simple limiting argument imply the existence of such wave solutions.

In this paper we return to consider more fully some questions which were touched on in (6). This was the first of a series of papers dealing with the spectral theory of the Laplace–Beltrami operator on a Riemann surface. In the later parts of that series we dealt fully with the case when the fundamental group is finitely generated. In this paper we return to the study of more general questions. In many ways this work continues that of W. Roelcke (7). We shall develop in section 2 the spectral theory for the Laplace operator Δ operating on functions on G\D where D is the unit disc and G is an arbitrary Fuchsian group. The main object is to find generalized eigenfunction expansions which are valid in the sense of absolute convergence rather than L2 convergence. The most convenient form of spectral theory for our purposes appears to be a version due to Gårding for Carleman operators which is presented in ((4), Ch. xvii, §§ 8,9). As a first application we use the results in section 3 to solve a problem of W. K. Hayman. Then we turn to Fuchsian groups G with exponent of convergence δ(G) < ½. Using the results of (6) we first obtain sufficient information on the action of G on the principal circle of G (section 4) and finally determine the spectral resolution in section 5.

Square integrable and local martingales on a family of σ-fields generated by a basic jump process are shown to have representations as stochastic integrals with respect to a family of martingales associated with the jump process by using the idea of an innovation projection and the associated Lévy system, which is a local characterization of the jumps.

An infinite genetic population of haploid particles is considered in which selection is controlled by a single locus at which there are an infinite number of possible alleles. These alleles are arranged in an infinite sequence and mutation occurs only to nearest neighbours. This is the ‘ladder model’ of Ohta and Kimura which was put forward as a possible explanation of the distributions of electromorphs in electrophoretic observations. Following an earlier paper, conditions are obtained on the selection coefficients which ensure that a stationary stable state exists. One such model is solved explicitly. The problem, important in evolutionary theory, of the rate of approach to such stationary states starting from some other state, is also discussed briefly.

In section 3 we shall prove the following results: Let G be a separable locally compact abelian group, R a von Neumann algebra acting on a separable Hilbert space, and α a weakly continuous representation of G by inner *-automorphisms of R, say α(g) = ad Wg with Wg ∈ U(R). Then there is a weakly continuous unitary representation of G, by unitaries in R, implementing α if and only if the Wg's commute with each other. The result was motivated by the proof of (7), theorem 1. Suppose now Gis a discrete amenable group of *-automorphisms of a countably decomposable von Neumann algebra R. In section 3 we give a necessary and sufficient condition for the existence of a faithful normal G-invariant state on R. This generalizes a result of Hajian and Kakutani on invariant measures (2).

Let A be a *-algebra of operators on a Hilbert space H, and let Ah, A+ denote respectively the sets of self-adjoint and positive operators in A. A+ is a positive cone in Ah and it induces a partial ordering in Ah. The lattice properties of Ah were studied by R. Archbold in (1) and (2), and Chu Cho-Ho gave a different proof of some of his results in (3).

The aim of this article is to prove a result which has been thought true for some time. Roughly speaking, if you take a universal unfolding of a germ in finitely many variables, and add to it a non-degenerate quadratic form on an infinite-dimensional space, you still have a universal unfolding.

Moran (7) has shown that some models in population genetics lead to a bilinear recurrence relation which, for a finite number of alleles, has a globally stable equilibrium. When the possible alleles are infinite in number, non-trivial problems of existence and stability of the equilibrium arise, which Moran has resolved in special cases. In this paper a powerful sufficient condition is established for the existence of a globally stable equilibrium, and its consequences are explored for cases of genetical interest. A more speculative final section describes a variant of Moran's model which may possibly have some relevance to the assessment of experimental evidence for or against selective neutrality.

Alexandrov (1), Zeeman (3) and others after them have investigated at length the problem of characterizing as Lorentz transformations those mappings f of Minkowski space M which ‘preserve cones’, i.e. if C(a) denotes the null-cone in M with vertex a ∈ M, then

for all a ∈ M.

Let µ be a probability measure on the Borel subsets of R∞. If D is a countable subgroup of R∞ we say that µ is D-ergodic if (1) for any D invariant Borel subset A of R we have µ(A) = 0 or 1 and (2) if µ*δx ≈ µ for all x ∈ D (where δx stands for unit mass at x while the equivalence relation ≈ signifies that the two measures have the same null sets.) We say that x is an admissible translate for µ if µ*δx ≈ µ. We say that µ is D-smooth if sx is an admissible translate for µ for all x ∈ D and all s ∈ R. We say that µ is a smooth ergodic measure if µ is D-ergodic and D-smooth for some countable subgroup D as above. In this paper we show that any two smooth ergodic probability measures µl, µ2 are either equivalent or singular (where the latter means that there exist disjoint Borel sets Al, A2 ⊂ R∞ such that µi(Ai) = 1 and is signified by µ1 ┴ µ2). It is important to note that the countable subgroup D1 associated with µl need not be the same as the subgroup D2 associated with µ2.

This paper is a sequel to an earlier paper in these Proceedings by one of us ((5); this will be referred to as [I]). The question considered there was that of determining the wave front set of the solution of the boundary value problem

where x∈+, y∈n, and n > 1; the precise meaning of the boundary condition at x = 0 is explained in section 1 below. The principal result of [I] can be expressed concisely by saying that singularities do not propagate along the boundary; a detailed statement is given in Theorem 1·9 of the present paper.

The familiar W.K.B.J. method expresses in terms of elementary functions approximate solutions of second order differential equations in normal form that otherwise have no analytical solutions expressible in terms of the standard elementary or transcendental functions. Proofs of the connexion formulae required to trace these approximate solutions across transition points are usually derived by comparison with known functions such as the Airy integral or Bessel functions, or by an appeal to the Stokes phenomenon in the complex plane. A new device is developed for the synthesis of refractive index profiles with transition points, and yet yielding solutions in terms of the elementary functions. These are then used to derive the W.K.B.J. connexion formula by means of a novel limiting process based on solutions considered only along the real z-axis. The method is obviously more complicated than the more usual approach, but contains special features throwing new light on the connexion formula across a single transition point.

In this paper we continue the study of Orlicz sequence spaces initiated by Lindberg (5) and Lindenstrauss and Tzafriri (7), (8) and (9). Our main concern is to investigate features of the theory which occur when the restriction of local convexity is lifted. It is clear that some results will hold with identical proofs, at least when the space is locally bounded. However, we are chiefly interested in the differences which arise. We always assume that the Orlicz function F satisfies the Δ2-condition.

The transmission of a quantal particle through a barrier obtained by inverting a Morse potential is studied by means of the ‘symmetric’ phase-integral approximations introduced by N. Fröman and further developed by N. Fröman and P. O. Fröman. The calculations are performed up to the fifth-order approximation. Approximate expressions for important quantities such as transmission and reflexion coefficients, phases of the wave function and also for quantities related to the phenomenon of resonances, which may occur when the wave function is given as a standing wave, are compared to the corresponding exact quantities, and this comparison shows the great accuracy of these phase-integral approximations.

The possibility is explored that the general static solution of Einstein's vacuum field equations is given in terms of a single harmonic function, i.e. solution of the flat space Laplace equation. It is shown that coordinate conditions can be chosen such that one of Einstein's equations reduces to Laplace's equation. To examine the extent to which the metric is determined by the resulting harmonic function ø, a power series expansion is considered in which in the first non-trivial order one gets a system of eight equations for six unknowns. This is reduced to a system of three equations for three unknowns. This is some indication that the coordinate conditions are consistent and shows the sense in which, to this order, the metric is determined by ø. This system of three equations is then solved explicitly for a particular choice of ø, thereby giving an approximate non-axisymmetric solution of Einstein's vacuum equations. This solution is asymptotically flat and could represent the approximate gravitational field of a non-axisymmetric bounded rigid body. The form of a class of time-dependent metric related to the general static metric is discussed.

It is shown that the differentiability of a space-time is implied by that of its Riemann tensor, assuming a priori only boundedness of the first derivations of the metric. Consequently all the results on space-time singularities proved in earlier papers by the author hold true in C2− space-times.

Starting from an arbitrary algebra in place of the algebra of smooth functions on a differentiable manifold M, we construct an algebra which generalizes the algebra of polynomial functions on T*M. We define a Poisson bracket on this algebra and show that it can be obtained from a natural one-form which generalizes the fundamental one-form on T*M.

1. Introduction. In studying the elastic behaviour of inhomogeneous systems certain inclusion and inhomogeneity problems are fundamental. In the ‘transformation problem’, a region (the ‘inclusion’) of an unbounded homogeneous anisotropic elastic medium would undergo some prescribed infinitesimal uniform strain (because of some spontaneous change in its shape) if it were not for the constraint imposed by the surrounding matrix. When the inclusion has an ellipsoidal shape, Eshelby (3, 4) was able to show that the stress and strain fields within the constrained inclusion are uniform and that calculations could be completed when the medium was isotropic. A generally anisotropic medium seemed to raise forbidding analyses, but Eshelby (3) did point the way to an evaluation of the uniform strain which several authors (referred to later) developed into an expression amenable to numerical computation. Here we offer an elementary and immediate route to this expression of the uniform strain, which has been accessible hitherto only by the circuitous procedures of Fourier transforms. It is available as soon as the uniform state of strain in the inclusion is perceived and before an alternative evaluation is commenced. First, we appeal to a theorem (not it seems previously known) which reveals (in particular) the vanishing of the mean strain in the infinitesimally thin ellipsoidal homoeoid lying just outside the inclusion. Secondly, we need only reflect that at each point of the interface there is an immediate algebraic expression of the strain just outside the inclusion in terms of the uniform strain just inside.