It is shown that In113* can be obtained by nuclear excitation of indium by X-rays. The cross-section per nucleus for the production of In113* with 2 MeV. X-rays is of the same order as the cross-section for the production of In115*. The decay period of In115*, for which some discrepancy was apparent in the literature, was definitely proved to be 4·5 hr.

1. The purpose of this note is to prove the uniqueness of the solutions of two closely related differential equations, under suitable boundary conditions. The first is the equation satisfied by the potential in the electrostatics of the new field theory proposed by Born, namely,

where øx stands for etc.

In this paper I prove the following results.

Theorem 1. Let f(x, y) = ax2 + bxy + cy2be an indefinite form. Then there exist real (x0, y0) such that

for all (x, y) ≡ (x0, y0) (mod 1). If f represents 0 there are indenumerably many such incongruent (x0, y0).

We develop a theory of particle scattering in anisotropic media. That is, a medium in which the microstructure causes the mean free paths of the particles to become dependent on their direction of motion with respect to some fixed axis. The equation which results is similar to the normal, one-speed Boltzmann transport equation but has cross-sections which are functions of direction. This equation is solved for arbitrary cross-sectional dependence on direction in plane geometry.

Allowance is made for the error terms in the cases where χ has a simple zero at x = 0, or two simple zeros at a; = ± 1 (with χ either positive or negative for − 1 < x < 1). The errors of the approximations given by neglecting the error terms are found to be small in all cases, sometimes much smaller than there was any previous reason to expect.

We consider a family of dynamical systems (A, α, L) in which α is an endomorphism of a C*-algebra A and L is a transfer operator for α. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.

It is a classical result in the theory of special functions that Bessel functions are limits in an appropriate sense of Legendre polynomials. For example in (11), § 17.4, the following result is attributed to Heine:

such limiting formulae are also known for certain other special functions (cf. (1), (5)). Apart from their intrinsic interest, these formulae have been used in the theory of special functions to obtain product formulae, etc. for the limit function from those of the approximating sequence.

An orientation-preserving, recurrent homeomorphism of the two-sphere which is not the identity is shown to admit exactly two fixed points. A recurrent homeomorphism of a compact surface with negative Euler characteristic is periodic.

1. Ellis(1) has proposed energy level systems for the bodies of the Thorium series, but the values of the γ-ray intensities forming the basis of this system fail to account satisfactorily for the total energy of the γ-rays emitted by Th C and its products. This is a serious discrepancy which needs examination. Since the publication of these level systems, we have obtained new values for the intensities of the γ-rays from Th C and Th Pb. The details of these measurements are given in the second part of this paper. These γ-ray intensities are, of course, deduced from the intensities of the discrete β-groups by dividing by the appropriate internal conversion coefficient. The changes which we have made in the γ-ray intensities are due first to revised values of the β-ray intensities from Th Pb and secondly to a revised set of values of the internal conversion coefficient (2). We have not however found it necessary to make any change in Ellis and Mott's (3) allocation of lines to dipole or quadripole type. The contributions of the two factors are made clear in Table I.

Using a particular parametrization of paths, the free particle and the harmonic oscillator are treated by the Feynman method of summation over histories of the system. The propagators obtained are, apart from an arbitrary phase factor, those of conventional quantum mechanics.

The problem of the conduction of heat in a solid sphere with a concentric core of a different material, the surface kept at a constant temperature, and the initial temperature of the whole zero, has already been solved in these Proceedings.

Let f be a rational function of degree at least two. We shall prove that the Julia set J(f) of f is uniformly perfect. This means that there is a constant c∈(0, 1) depending on f only such that whenever z∈J(f) and 0 < r < diam J(f) then J(f) intersects the annulus .

Spacetimes with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry are shown to be timelike and null geodesically complete in the expanding direction, provided the data satisfy a certain size restriction.

Varying ideas of the mechanism of the propagation of light and of the constitution of matter have led at different times to different formulae for the dependence of the refractive index n of a medium on its density ρ, notably those of Laplace, , and of Lorenz and Lorentz, . Recently Macdonald has given a theory which seems more in accord with the modern concept of atomic structure and finds a formula for a non-magnetic medium. It has therefore seemed worth while to examine a number of experimental results to see whether this new formula gives as satisfactory agreement as that of Lorenz and Lorentz; it is found that it does in most cases, and is occasionally the better.

If L is a symmetric m-linear form on a Banach space and L^ is the associated polynomial then

For special choices of Banach space this inequality can be improved. This has been done by Harris [5] in the case of the Lp(μ) spaces. In this paper we improve his estimates and disprove one of his conjectures.

Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shape

where (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.

We refer the reader to the IHES notes of Zeeman (14) for basic facts about PL (or piecewise-linear) manifolds. If Mm is a locally flat PL-submanifold of Qm+2, our object will be to study the normal structure of M in Q: one of our main results is:

There exists a PL-bundle over M, with fibre a 2-simplex, which is PL-homeomorphic to a neighbourhood of M in Q; moreover, the bundle and homeomorphism are unique up to equivalence. We also make an application to smoothing theory.

A systematic analysis is conducted for refractive index profiles capable of yielding exact solutions in terms of hypergeometric or confluent hypergeometric functions for the electromagnetic field of vertically polarized waves propagating in horizontally stratified isotropic media. Previously known profiles emerge as special cases of the analysis. Profiles suitable for use with isotropic lossless ionospheres and of prescribed form n2 = 1 − T(z)/k2, with T(z) independent of k, do not arise in this paper.

This note is a sequel to two in earlier volumes of the Proceedings, the first by myself and the second by Wilton†.

Suppose that

for |z| <1; that x > 0; that

forr ≥ 0; and that

where τ(x) is to mean 0 if x is not an integer. Thus

where the dash shows that the last term is to be halved when x is an integer;

forr > 1; and