In a previous paper a new method, based on Kemmer's β-formalism, of calculating meson processes was given for the case in which the meson interacts with an electromagnetic field. This method is now extended to the nuclear interaction, so that the whole of the meson theory can be given either in tensor or in matrix form, the former being preferable when the wave aspect of the meson is important and the latter when the particle aspect is dominant.

As examples of the matrix method, derivations are given of the cross-sections for the nuclear scattering of mesons and for the production of mesons from nuclei by photons. It is pointed out that the usual non-relativistic theory of the nuclear interaction is inadequate even for very small velocities.

In this paper we use a method due to Williams(1) to discuss the scattering of surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending indefinitely downwards from a finite depth.

A poset model of a topological space X is a poset P such that the subspace Max(P) of the Scott space ΣP is homeomorphic to X, where Max(P) is the set of all maximal points of P. Every T1 space has a (bounded complete algebraic) poset model. It was, however, not known whether every T1 space has a directed complete poset model and whether every sober T1 space has a directed complete poset model whose Scott topology is sober. In this paper we give a positive answer to each of these two problems. For each T1 space X, we shall construct a directed complete poset E that is a model of X, and prove that X is sober if and only if the Scott space Σ E is sober. One useful by-product is a method for constructing more directed complete posets whose Scott topology is not sober.

Let {Xnj, 1 ≤ j ≤ kn} be independent random variables with zero means and satisfying . Let p ≥ 1. We prove that

if and only if, for all ε > 0,

and we use this result to obtain necessary and sufficient conditions for the Lp convergence of sums of non-negative, independent random variables.

We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.

The wave equation for the deuteron in its ground state is solved on the assumption that the mutual potential energy of a neutron and a proton is of the form r−1e−λr. The binding energy of the hydrogen isotope H3 is calculated approximately by the variation method.

The recent attempt at a physical interpretation of non-Riemannian spaces by Einstein (1, 2) has stimulated a study of these spaces (3–8). The usual definition of a non-Riemannian space is one of n dimensions with which is associated an asymmetric fundamental tensor, an asymmetric linear affine connexion and a generalized curvature tensor. We can also consider an n-dimensional space with which is associated a complex symmetric fundamental tensor, a complex symmetric affine connexion and a generalized curvature tensor based on these. Some aspects of this space can be compared with those of a Riemann space endowed with two metrics (9). In the following the fundamental properties of this non-Riemannian manifold will be developed, so that the relation between the geometry and physical theory may be studied.

In this note we consider the axially symmetric indentation of a transversely isotropic layer resting on a rigid foundation. The corresponding half-space problems both for isotropic and transversely isotropic materials have been solved previously (see (l), (2), (3), p. 455) and, more recently, the present problem has been solved for isotropic materials in (4) and (5) using methods different from that given in this paper. In § 2 the problem is reduced to the solution of a Fredholm integral equation of the second kind. In § 3 and subsequently the case of incomplete penetration by the punch, when the radius of the circle of contact is less than that of the end face of the punch, is considered. Finally, in §4 the results are illustrated by considering a hemispherically ended punch.

1. The consequences of assuming that an integral function or its derivatives take integer values for some or all integer values of the variable have been widely investigated. The best-known result of this kind was obtained by Pólya(9,10) and Hardy(6) in 1920 and states that among all transcendental integral functions which assume integer values for all non-negative integer values of the variable that of least increase is the function 2Z (cf. also (7)). Analogous results relating to integers in the Gaussian field were proved by Fukasawa(3)in 1926 and Gelfond(4)in 1929, and theorems relating to the function and its derivatives are included in the works of Gelfond (5), Selberg(14) and Straus (15). (For further results and references see (1,2, 11).) The earlier of these investigations may be regarded as the genesis of the celebrated Gelfond-Schneidersolution to the seventh problem of Hilbert and the much important work arising therefrom.

The problem of the determination of the overall dynamic elastic moduli of an elastic solid permeated by uniformly distributed penny-shaped cracks is considered. The cracks are assumed to be filled with a viscoelastic material. The orientations of the cracks may be either parallel or perfectly random. The overall velocities as well as the specific attenuation coefficients of plane harmonic compressional and shear waves are calculated for low frequencies and dilute concentration of the cracks.

In this paper, a new family of factors of (2, 3, 7) is obtained which contains groups found in three other papers. It is shown that for all n, there exists an m such that there are at least n isomorphism types of Hurwitz groups of order m. Finally, presentations for all groups considered are obtained.

We consider a variational problem for steady axisymmetric vortex-rings, in which kinetic energy is maximised subject to prescribed impulse, with the ratio of vorticity to axial distance belonging to the class of rearrangements of a prescribed function, as proposed by Benjamin. We prove existence of maximisers in an extended constraint set, allowing some loss of vorticity. We then study a particular family of vortex-rings including Hill's spherical vortex, determining the precise range of impulse values for which the maximiser loses vorticity, and show that the maximisers are spherical when this happens.

A laboratory model simulating aspects of oceanic or atmospheric circulationconsists of a rotating fluid annulus which is subject to a temperature gradient on the lower surface and is insulated on the inner and outer curved walls and on the top surface. A theoretical study of the resulting steady axi-symmetric fluid motion in the limit of small Ekman number, E, by Daniels (2) indicated that if an appropriate convective parameter is sufficiently large, oscillatory solutions are generated in the layer which is located on the hotter of the two side-walls, which do not decay into the geostrophic interior. The present paper resolves this difficulty by using the method of multiple scales to trace the development of the oscillatory solutions in the interior of the fluid. This leads to a consistent overall flow pattern throughout the annulus, except at an infinite set of discrete values of the convective parameter at which resonance is predicted.

Bescovitch considered the Hausdorff dimensions of sets related to the sum of digits of real numbers represented in the dyadic expansions. In this paper, we provide an analogy for continued fraction expansions over the field of formal Laurent series. We also calculate the Hausdorff dimensions of sets of Laurent series which have given polynomial or exponential approximation orders. Such approximations converge faster than those of almost all Laurent series (with respect to the Haar measure).

The notion of fiducial probability was introduced by R. A. Fisher and is now widely used in statistical work involving a single unknown parameter. Fisher has also considered the joint fiducial distribution of several parameters for which there exists a sufficient set of statistics, and has derived the fiducial distribution of the two parameters of the one-variate normal law*. Since Fisher's account is brief, we give a more extended description of the fiducial distribution of several independent parameters possessing a sufficient set of statistics.

If the Bing double of a knot K is slice, then K is algebraically slice. In addition the Heegaard–Floer concordance invariants τ, developed by Ozsváth–Szabó, and δ, developed by Manolescu and Owens, vanish on K.

We examine the multifractal spectra of one-sided local dimensions of Ahlfors regular measures on ${\bf R}$. This brings into a natural context a curious property that has been observed in a number of instances, namely that the Hausdorff dimension of the set of points of non-differentiability of a self-affine ‘devil's staircase’ function is the square of the dimension of the set of points of increase.

A special monoid is a monoid presented by generators and defining relations of the form w = e, where w is a non-empty word on generators and e is the empty word. Groups are special monoids. But there exist special monoids that are not groups. Special monoids have been extensively studied by Adjan[1] and Makanin[7] (see also [2]).

The point of this note is to make an observation concerning the variety M(E) parametrizing line subbundles of maximal degree in a generic stable vector bundle E over an algebraic curve C. M(E) is smooth and projective and its dimension is known in terms of the rank and degree of E and the genus of C (see Section 1). Our observation (Theorem 3·1) is that it has exactly the Chern numbers of an étale cover of the symmetric product SδC where δ = dim M(E).

This suggests looking for a natural map M(E) → SδC; however, it is not clear what such a map should be. Indeed, we exhibit an example in which M(E) is connected and deforms non-trivially with E, while there are only finitely many isomorphism classes of étale cover of the symmetric product. This shows that for a general deformation in the family M(E) cannot be such a cover (see Section 4).

One may conjecture that M(E) is always connected. This would follow from ampleness of a certain Picard-type bundle on the Jacobian and there seems to be some evidence for expecting this, though we do not pursue this question here.

Note that by forgetting the inclusion of a maximal line subbundle in E we get a natural map from M(E) to the Jacobian whose image W(E) is analogous to the classical (Brill–Noether) varieties of special line bundles. (In this sense M(E) is precisely a generalization of the symmetric products of C.) In Section 2 we give some results on W(E) which generalise standard Brill–Noether properties. These are due largely to Laumon, to whom the author is grateful for the reference [9].