Let u(x, t), where x ∈ 3, t ∈ , be a C∞ solution of the wave equation on {|x| > a} × for some a > 0, and suppose that u = 0 for |x|> a, t < 0. One then also has u = 0 for |x| > a + t, t > 0, so that u(.,t) can be thought of as a wave expanding into a previously undisturbed medium. One can now ask for a description of the asymptotic behaviour of u as |x| → ∞. It turns out that there is a v0(θ, τ) ∈ C∞ (S2 × ) such that

in the topology of C∞ (S2 × ). This limit may be called the radiation field of the expanding wave u. (See (2) and the earlier papers quoted there.)

In the present article we prove a result characterizing the Jordan analogues of B*-algebras among the complex Banach Jordan algebras in terms of the algebra and norm structures alone.

Stokes and Wilson(1,2) have shown that the integral breadths of the Debye-Scherrer lines produced by small or imperfect crystals of the cubic system are given by

or, what is the same thing, that the apparent particle sizes are given by

In these equations λ is the X-ray wave-length, θ is the Bragg angle, Vt is the volume common to the crystal and its ‘ghost’ shifted a distance t in the hkl direction(1), and Jt is the mean value of the product FF of the structure amplitudes of two cells separated by a distance t in the hkl direction (2). The purpose of this note is to outline an argument which establishes the same result for crystals of any symmetry.

The preceding paper ((1)) dealt with some general properties of the gravitational field of a rotating fluid mass. An interesting example of a vacuum solution that might be the exterior field of some rotating body was recently found by Kerr ((4)). It was natural to apply the preceding theory to the Kerr solution. This paper deals with other aspects of that solution, particularly the behaviour of its bounded geodesics (planetary orbits). It would seem desirable to know what sort of rotating body could be a source of the Kerr field. It will appear that one of the parameters in Kerr's solution can plausibly be related to the angular momentum per unit mass of a uniformly rotating sphere, the other parameter being a measure of the mass of the sphere.

Let A, B be real symmetric n × n matrices having orthogonal reductions

where ΛA = diag (λ1(A), …, λn(A)) and ΛB = diag (λ1(B), …, λn(B)) and the sets of latent roots are each in descending order. (Thus we may say that the latent vectors of A, B are ordered with respect to respectively.) Let the multiplicities of the roots and in order of occurrence be respectively and , so that

We prove that every closed, connected contact 3-manifold can be obtained from $S^3$ with its standard contact structure by contact (${\pm}1$)-surgery along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in a slightly stronger form).

The equilibrium of a layer of a viscous incompressible fluid of density ρ variable in the vertical (z) direction, the fluid extending indefinitely in the horizontal directions and being confined between rigid horizontal boundaries, is examined by the usual method of studying the initial behaviour of a small disturbance. Diffusion effects are ignored. Chandra-sekhar has shown quite generally that a variational procedure for solving the resulting characteristic value problem is admissible, suggesting a method whereby an approximate solution can be obtained analytically. In this paper, by making use of appropriate approximate methods of solution, three types of density configuration, ρ(z), are studied. The first corresponds to two superposed fluids of great depth, the upper and lower fluids haying densities and coefficients of viscosity ρ2 and μ2, and ρ1 and μ1, respectively. Chandrasekhar has given an exact numerical solution to this problem for the case where μ2/ρ2 = μ1/ρ1 The results of the approximate theory, by which not all the boundary conditions can be satisfied, and the exact theory are in good agreement. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (ρ2 > ρ1) are found. The stable case (ρ2 > ρ1) is also discussed in detail; the formulae obtained for the phase and group velocities of gravity waves in deep viscous fluids are especially interesting. There is an upper limit to the natural frequency of oscillation of the system which would be well within the range of experimental investigation.

The second problem considered is that of two superposed fluids of small depth. Formulae for the phase and group velocities of ‘long gravity waves’ (which are not equal when viscous effects are included) are obtained.

Finally, a continuously stratified fluid of finite depth d in which ρ(z) = ρ0eβz is investigated. The character of the equilibrium depends on a Grashof number G = (gβd4)/(π4ν2) where g is the acceleration of gravity and ν is the coefficient of kinematical viscosity, assumed for simplicity to be constant. Expressions for the wave-length and growth rate of the mode of maximum instability in the unstable case (G > 0) are obtained. In the stable case (G < 0) all modes are aperiodically damped if – G < . Otherwise, the possibility of a disturbance being propagated as a horizontal wave does arise, but only within a limited wave-length range.

1. In a previous paper (1) the authors used the theory of the moment generating function to deduce the known random sampling distributions of the estimated variance and co-variance in a system of variables following the normal law of frequency. In this paper we shall go much further. By means of the same general method the simultaneous distribution of the ½p (p + 1) second order moment statistics in a normal system of p mutually correlated variables will be deduced. It will be shown to be completely independent of that of the p sample means, and incidentally the method of proof to be developed will be found, in the special cases p = 1 and 2, to be an improvement on that given in the earlier paper, being independent of previous results reached by other authors.

It was shown by Baber and Hassé(1) that the Schrödinger equation for the single electron in H+2 with fixed nuclei is separable in spheroidal coordinates. The separated equations were integrated by Bates, Ledsham and Stewart (2) and their solutions may be regarded as exact. However, several attempts have been made to derive approximate solutions for this, which is the simplest of all molecular problems, generally with the hope that it might be possible to extend the analysis to larger diatomic systems. Recently a single-centre expansion has been much favoured, since it avoids the difficult two-centre integrals; but unfortunately this advantage has been offset by the fact that the rate of convergence for the lowest states has been rather slow since the deviations of diatomic H+2 from the united-atom H+e are serious. But successful approximations to some excited states suggest that a single-centre approach may still be useful, provided that the trial functions are suitably chosen.

A formula for the generalized Schwarz–Christoffel conformal mapping from a bounded multiply connected circular domain to an unbounded multiply connected polygonal domain is derived. The formula for the derivative of the mapping function is shown to contain a product of powers of Schottky–Klein prime functions associated with the circular preimage domain. Two analytical checks of the new formula are given. First, it is compared with a known formula in the doubly connected case. Second, a new slit mapping formula from a circular domain to the triply connected region exterior to three slits on the real axis is derived using separate arguments. The derivative of this independently-derived slit mapping formula is shown to correspond to a degenerate case of the new Schwarz–Christoffel mapping. The example of the mapping to the triply connected region exterior to three rectangles centred on the real axis is considered in detail.

Equilibrium states in galactic dynamics can be described as stationary solutions of the Vlasov–Poisson system, which is the non-relativistic case, or of the Vlasov–Einstein system, which is the relativistic case. To obtain spherically symmetric stationary solutions the distribution function of the particles (stars) on phase space is taken to be a function Φ(E, L) of the particle energy and angular momentum. We give a new condition on Φ which guarantees that the resulting steady state has finite mass and compact support both for the non-relativistic and the relativistic case. The condition is local in the sense that only the asymptotic behaviour of Φ for E → E0 needs to be prescribed, where E0 is a cut-off energy above which no particles exist.

The theorem of R. Rado (12) to which I refer by the name ‘Rado's theorem for matroids’ gives necessary and sufficient conditions for a family of subsets of a finite set Y to have a transversal independent in a given matroid on Y. This theorem is of fundamental importance in both transversal theory and matroid theory (see, for example, (11)). In (3) J. Edmonds introduced and studied ‘polymatroids’ as a sort of continuous analogue of a matroid. I start this paper with a brief introduction to polymatroids, emphasizing the role of the ‘ground-set rank function’. The main result is an analogue for polymatroids of Rado's theorem for matroids, which I call not unnaturally ‘Rado's theorem for polymatroids’.

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such that

for every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.

1. Introduction. In the study of diophantine equations in two variables, it is often necessary to consider rational functions on a curve with prescribed zeros and poles. Although it is well known that such functions can, in principle, always be effectively constructed, the detailed proof does not appear to have been given. The purpose of the present paper is to give the complete proof of such a construction. Our method, and the statement of our results, are motivated by the applications to diophantine equations which we have in mind. In particular, our results will play an important role in a subsequent paper (1), in which explicit bounds will be established for the integer points on any curve of genus 1.

The triple velocity correlation, in turbulence produced by inserting a square-mesh grid near the beginning of the working section of a wind tunnel, has been measured for mesh Reynolds numbers of RM = 5300, 21,200 and 42,400 (RM = UM/ν, where U is the mean wind speed in the working section of the tunnel and M is the centre to centre spacing of the rods making up the grid; ν is the kinematic viscosity of air). At the lowest Reynolds number the correlation has been measured at distances downstream of the grid varying from 20 to 120M. This range covers practically all of the initial period of the decay of turbulence, where the turbulent intensity varies as t−1.

For each integer m, Rasala [6] has shown how to list all the ordinary irreducible representations of the symmetric group n which have degree less than nm, provided that n is large enough, and in this note we shall prove similar results for the irreducible representations of n over an arbitrary field K. Our estimates are very crude, so although we recover Rasala's results, we get nowhere near his precise information on how large n has to be.

The aim of this paper is to prove some new results towards answering the question: When does Dehn surgery on a knot give a non-prime manifold? This question has been raised on several occasions (see for instance [5] or [4]; concerning the latter see below). Recall that a 3-manifold is prime if, in any connected sum decomposition

one of M1, M2 is S3. (For standard definitions of low-dimensional topology see [2] or [16].)

By an isolated singularity of an algebraic surface in [r] (i.e. space of r dimensions) I shall mean one which not merely is not upon any branch of a multiple curve of the surface, but has also the property that when the surface is projected into [3] from a general space [r − 4] the singular point remains in isolation, i.e. no branch of the double curve created by the projection will of necessity pass through it.

In (3), some of us proved that Brownian paths in n-space have double points with probability 1 if n = 2 or 3; but, for n ≥ 4, there are no double points with probability 1. The question naturally arises as to whether or not Brownian paths in n-space (n = 2 or 3) have triple points. The case of paths in the plane is settled by (4), where it is shown that, with probability 1, Brownian paths in the plane have points of multiplicity k (k = 2, 3,4, …). The purpose of the present paper is to settle the remaining case, n = 3. We prove that, with probability 1, Brownian paths in 3-dimensional space have no triple points. The general idea behind our proof is to show that there are not too many double points. That is, we show that the set of double points has sigma-fmite linear measure, and therefore zero capacity, with probability 1.

Let K be a knot in S3, and let K(a/b) denote the closed oriented 3-manifold obtained by a/b-Dehn surgery on K. (We parametrize the Dehn surgeries on K by ℚ ∩ {1/0} as in [9]; in particular, K(1/0) = S3.) If K is a (p, q)-cable knot, then K(pq) is always reducible (see Section 3), and it is conjectured by González-Acuña and Short in [5] that these are the only examples where Dehn surgery on a knot in S3 yields a reducible manifold. One of the results in this direction proved in [5] is that if π1 (K(a/b)) is a non-trivial free product then |b| ≤ 5. We show that this may be sharpened to the assertion in the title.