Bifurcation from a finitely deformed circular cylindrical configuration of a rotating circular cylindrical elastic membrane is examined. It is found (for a physically realistic choice of elastic strain-energy function) that the angular speed attains a maximum followed by a minimum relative to the increasing radius of the cylinder for either a fixed axial extension or fixed axial force.

At fixed axial extension (a) a prismatic mode of bifurcation (in which the cross-section of the cylinder becomes uniformly non-circular) may occur at a maximum of the angular speed provided the end conditions on the cylinder allow this; (b) axisyim-metric modes may occur before, at or after the angular speed maximum depending on the length of the cylinder and the magnitude of the axial extension; (c) an asymmetric or ‘wobble’ mode is always possible before either (a) or (b) as the angular speed increases from zero for any length of cylinder or axial extension. Moreover, ‘wobble’ occurs at lower angular speeds for longer cylinders.

At fixed axial force the results are similar to (a), (b) and (c) except that an axisym-metric mode necessarily occurs between the turning points of the angular speed.

A vector identity associated with the Dirichlet tessellation is proved as a corollary of a more general result. The identity has applications in interpolation and smoothing problems in data analysis, and may be of interest in other areas.

In a recent paper, hereafter referred to as I (1) we derived two alternate forms for the fundamental solution of the axially symmetric wave equation. We demonstrated that for α > 0, the fundamental solution (the so-called free space Green's function) of the partial differential equation

could be written as

if b > r

or

if r > b.

Let be a standard d-dimensional Brownian motion and L a smooth functional on the space C = Cd[0, 1] of continuous functions from [0,1] to Rd, such that Denote Then

is a square integrable martingale which can be represented as a stochastic integral; consequently the random variable has the representation

For Fréchet differentiable functions L, Clark(1) gave the following formula for the integrand e:

In (5), Schmidt devised a procedure for completing a space-time M by adjoining a boundary ∂M called the b-boundary. Bosshard(1) and Johnson(4) have shown that certain Friedmann and Schwarzschild space-times have non-Hausdorff b-completions.

In (2), (3), the first author considered a modification of the b-completion which gives a Hausdorff completion in the Friedmann case. The modification used a parallelization of the space-time and can be given in terms of the linear connexion determined by the parallelization. We show that the completion obtained is the Cauchy completion in a particular Riemannian metric on the manifold and so is always Hausdorff.

This paper is concerned with a particular method used in the calculation of asymptotic expansions of solutions to problems which contain a small parameter ∈. The physical idea underlying the method is that the parameter introduces new scales (of length and/or time) into the problem, and that the solution should incorporate these (see, for example, Cole (1)). This may be done in various ways: two-timing (two-scaling, multiple-scaling) does it by demanding that the solution is a function of all relevant scales. Consider, for example, a linear oscillator whose frequency is slowly adjusted, governed by

and subject to some initial condition. Account of the slowly varying frequency is taken by the dependence of Ω on ∈t, where 0 < ∈ ≪ 1, and we assume that Ω > 1 for all arguments. The rapid oscillatory behaviour anticipated in the solution is customarily described by a phase function Θ(isin;t)/∈, a generalization of Ωt the case that Ω is constant, while slow changes due to frequency variation occur on a scale ∈t.

For a manifold with a generic symmetric affine connection, explicit necessary and sufficient conditions for the local existence of metric compatible with the connection are obtained in terms of the Riemann tensor and its first-order covariant derivatives. If these conditions are satisfied, the solutions for metric are unique up to a constant scale factor and the absolute value of the signature is uniquely determined. Explicit formulae for the solutions are given in terms of integrals.

Let {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r > 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such that

as n → ∞, where v0 is the p.d.f. of Vo.

On behalf of the authors and the Press, the editor has to notify the following amendments.

Volume 48 (1952), 72–86 and 519–20

J. W. S. Cassels. ‘The inhomogeneous minimum of binary quadratic, ternary cubic and quaternary quartic forms.’

Professor H. W. Lenstra Jr has pointed out that there is a mistake in the proof of Theorem 6, and that the constant 5300 must be replaced by the worse constant 16730. In (iii) of Lemma 16, k should be replaced by k2 and consequently k¼ by k½ in the display at the bottom of page 85. The new estimate results then from putting k = (2·35)2.

Volume 87, part 3, pages 459–69

W. H. Lin, D. M. Davis, M. E. Mahowald and J. F. Adams. ‘Calculation of Lin's Ext groups.’

The diagram (1·5) at the foot of page 460 should be interchanged with that on page 462.