1. This paper is a sequel to (2). The notation is the same and we assume the reader to be familiar with that paper. We there proved various theorems which established conditions under which ‘almost all’ labelled (n, q) graphs contained a k-cycle, i.e. under which the proportion of all labelled (n, q) graphs which contained a k-cycle tended to 1 as n → ∞. We here consider a different but related problem. If a graph contains a cycle of length k for every k such that 3 ≤ k ≤ k0, we say that the graph is pancyclic up to k0. We find conditions on k0, n and q such that almost all (n, q) graphs are pancyclic up to k0. If k0 = 0(1), the result follows trivially from Theorem 3a of (1) (Theorem 1 of (2)), but if k0 → ∞with n the matter is not so simple. We shall prove the following theorem.

The Vidav–Palmer theorem [(11), (5), (2) (p. 65)] characterizes C*-algebras among Banach algebras in terms of the algebra and norm structure alone, without reference to an involution, in the following way. Let B denote a complex unital Banach algebra, and let Her (B) denote the set of Hermitian elements of B, that is the elements of B with real numerical ranges. In this notation, the Vidav–Palmer theorem tells us that if

then B is isometrically isomorphic to a C*-algebra of operators on a Hilbert space, with the Hermitian elements corresponding to the self-adjoint operators in the C*-algebra.

Given any embedding f:Sr × Bn−r → Sn, f ∣ Sr × 0 can be extended to a homeomorphism of Sn, except when r = n − 2 and f is actually knotted (10). Is there any sense in which this extension can be chosen to vary continuously with respect to f? If r = n − 1, there certainly is: this is simply the canonical Schoenflies theorem, proved in varying (but equivalent) forms in (6), (5) and (1). One of the principal results of this paper provides an affirmative answer to this question when r < n − 1. In the usual fashion of such canonical extension theorems, we look for a local version; that is attempt to find a neighbourhood of the inclusion in E(Sr × Bn−r, Sn) on which a map F can be defined with each F(f) a homeomorphism of Sn which extends f and F (inclusion) = identity. Now if the neighbourhood is sufficiently small, any element of it can be viewed as an embedding Sr × Bn−r → Sr × Rn−r, still close to the inclusion, by removing a standard Sn−r−1 from the image space. There are known methods for extending such embeddings. It would not be hard to adapt Kister's methods (8) for extending embeddings of Bk in Rk canonically to homeomorphisms (approximately the method that is sketched for the r = n − 1 case in (5)). Alternatively the existence of a canonical extension in this situation follows explicitly from the more general Theorem 5 of (1). Unfortunately both methods extend the embedding in a ‘radial’ fashion, with the result that there is little control over the extension towards infinity, and the final homeomorphism cannot necessarily be further extended over the missing Sn−r−1 to yield a homeomorphism of Sn. In fact this failure is due to the extension map F, derived in either of these ways, being continuous with respect to the compact-open topology on the image space, H(Sr × Rn−r). The extension map to be constructed here is continuous with respect to the uniform topology on the image space, and so the homeomorphisms F(f) of Sr × Rn−r are bounded and can thus be extended over all of Sn by setting F(f) ∣ Sn−r−1 equal to the identity.

Two collections of locally convex space topologies are shown to have the intertwined completeness property. This is done by relating their completions with sets of sequentially continuous functionals on the dual.

A Riesz space E has the (sequential) order-continuity property if every positive linear map from E to an Archimedean Riesz space is (sequentially) order-continuous. This is the case if and only if the canonical maps from E to its Archimedean quotient spaces are all (sequentially) order-continuous. I relate these properties to others that have been described elsewhere.

Using Pitt's Tauberian classes and a theorem of Pitt (3), Krishnan(2) has recently obtained a gap Tauberian theorem for (Aα) summability. We recall briefly the notation introduced in (2). Let

and assume that the series a(t) and A(t) converge for all t > 0.

Suppose that f is analytic in the unit disk D. If its range f(D) is contained in a simply connected proper subdomain of the plane, then the principle of subordination and the distortion theorem for univalent functions show that

where M(r, f) denotes the maximum modulus of f. Cartwright (2) studied functions which, instead of omitting all values on a continuum stretching to infinity, omit only a sequence of values. She assumed that the sequence {wn} satisfies

and

and proved that if f(D) contains none of the points {wn} thenm

for every ε > 0. Cartwright's proof was based on the Ahlfors Distortion Theorem, and is quite complicated. A much simpler proof was given by Pommerenke in (10). The key idea in his proof will also be used in the present paper.

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.

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.

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.

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.

A generalization of the notion of Riemannian submersion is given for submersions of a Lorentzian manifold onto a Riemannian manifold of one lower dimension. These maps are called comersions. A comersion induces a surjection of the space of causal curves in the domain manifold onto curves of bounded length in the image manifold. Thus if the source space is causally 1-connected, the target space is 1-connected.

The general multipole solutions obeying the appropriate boundary conditions are presented for stationary non-axisymmetric electromagnetic fields in the Kerr background geometry.

In this paper, a free-boundary problem of the steady plane flow of an ideal fluid through a slot is solved. The fluid flows under the effect of gravity between rigid boundaries, and then out of a slot as a jet which becomes horizontal at infinity downstream. A constructive proof of the existence of solutions to a non-linear integral equation is given for a parameter range 0 < ε < ε* ≃ 1·02 (where ε, the inverse Froude number, measures the effect of gravity). Approximations to the solution are then found for ε → 0.