In (1), Allan introduced the concept of the spectrum of an element of a locally convex algebra, and developed a spectral theory for pseudo-complete algebras. In a commutative Banach algebra the spectral radius is a continuous seminorm, and so it is natural to investigate continuity properties of the spectral radius in various classes of locally convex algebra. Continuity is too strong a condition to be expected in any general case, and an interesting property to investigate appears to be lower semi-continuity. We shall show easily that in a commutative pseudo-complete locally m-convex algebra (in the sense of (4)) the spectral radius is lower semi-continuous. We shall then exhibit a commutative complete metrizable algebra in which lower semi-continuity fails to hold.

0. Introduction. It has long been known that the space of all smooth homeomorphisms of a smooth manifold with the Cl topology can be given the structure of a smooth manifold. A path in this space corresponds to a diffeotopy. It has been shown more recently (1) that the space of all homeomorphisms of a topological manifold is locally contractible when given the C–0 topology. A path in this space corresponds to an isotopy.

Fine topologies are discussed for Minkowski space. The conformal group and projective group have representations as C1 groups of homeomorphisms of various topologies.

In this paper a Fuchsian group G shall be a discrete group of Möbius transformations each of which maps the unit disc △ in the complex plane onto itself. We shall also assume throughout this paper that G is both finitely generated and of the first kind.

0. Introduction. Let X be a normed space and let T be an operator on X. Let S(X) denote its unit sphere, {x ∈ X: ∥x∥ = 1}, B(X) = {x ∈ X: ∥x∥ ≤ 1} its unit ball, X′ its dual and ℬ(X) the normed algebra of bounded linear operators on X. Let II be the subset of the Cartesian product X × X′ defined by

1. Introduction. This paper is concerned with the problem of obtaining theoretical estimates for the number of arithmetical operations required to factorize a large integer n or test it for primality. One way of making these problems precise uses a multi-tape Turing machine (e.g. (1), although we require a version with an input tape). At the start of the calculation n is written in radix notation on one of the tapes, and the machine is to stop after writing out the factors in radix notation or after writing one of two symbols denoting ‘prime’ or ‘composite’. There are, of course, other definitions which could be used; but the differences between these are unimportant for our purpose.

A proof of the Theorem of Carathéodory is exhibited in a covariant formalism.

The problem of completing the mathematical structures abstracted from the time ordering of the events in the space-times of general relativity is motivated and solved. The new construction is compared with that of Geroch, Kronheimer and Penrose (5).

The problem of wave generation by a line source of sinusoidally varying strength situated in water above a beach of arbitrary angle α(0 < α ≤ π) is solved by the use of a Laplace-integral representation of the solution. It is shown that a solution can be constructed which is regular at the shoreline and gives an outgoing wave-train at infinity.

An expression is obtained for the amplitude of the waves radiated to infinity by a line source of sinusoidally-varying strength situated in water over a beach of angle α (0 < α ≤ π). It is shown that, for certain positions of the source, this amplitude is zero. Equations for the loci of these positions, and approximations to their solutions in particular cases, are derived.

A double series solution of the biharmonic equation is derived in the rectangular domain. The solution utilizes the finite double Fourier sine transform and the analysis shows how the transform may be evaluated for quite arbitrary boundary conditions, the solution then being found from the corresponding inversion formula.

The method is applied to some classical boundary value problems which have been investigated by other means by earlier workers. It is felt that some of the series solutions given here are more amenable to computation than some of the earlier methods.

In some of the problems considered, notably the elastostatics of the rectangular plate and the bending of clamped plates, the method reduces to solving an infinite set of linear equations. In such cases, a finite leading matrix of the coefficients has to be considered. In the examples considered, this technique produced results which converged to known solutions obtained by other methods. However, the mathematical criteria which permit the infinite system to be curtailed, have not been established.

An isotropic elastic material containing a random distribution of identical spherical particles of another elastic material is considered. The bonding between the spheres and the matrix is imperfect, so that slip may occur at interfaces when stress is applied to the medium. The shear stresses at the interface is assumed to be proportional to the amount of slip. The velocity and attenuation of the average harmonic elastic waves propagating through such a medium are calculated. The results are valid to the lowest order in frequency for wave lengths long compared with the radius of the sphere. The dynamic elastic moduli are obtained from these results and are compared with available results for welded contact. The variations in the P and S wave velocities for propagation across earthquake faults is discussed.

1. Introduction. The optimum design of plane reinforced concrete slabs frequently requires consideration of deflected shapes for the slab such that through every point there is at least one principal curvature (κ1) direction on the deflected surface which is a constant curvature direction. The most interesting cases are those where only one principal direction has this property.

It is found that the correlation functions of quantum stochastic processes are very similar to the corresponding functions for classical stochastic processes. It is shown also that if the state for the process satisfies the KMS boundary condition then this is reflected in the analyticity of the continuation of the correlation.