The results on the local existence and continuation criteria obtained by G. Rein are extended to the case with a non-zero cosmological constant. It is also shown that for the spherically symmetric case and a positive cosmological constant there is a large class of initial data with global existence and inflationary asymptotics in the future as in the case of plane or hyperbolic symmetry. Furthermore we analyze the behaviour of the energy-momentum tensor at late times.

Let G(θ) denote the limit points of the fractional parts of θn (n = 1, 2, 3, …), and T the set of numbers θ for which G(θ) contains an infinity of points. In an earlier note it was proved that if θ is a rational fraction greater than unity then θ belongs to T. The results of this note concern the limit points of the powers of algebraic irrationals.

The velocity distribution in steady rectilinear plastic flow of a Bingham solid between coaxial circular cylinders in relative motion can be obtained exactly by simple arguments. On the basis of the general equations of plastic flow, an approximate velocity distribution is derived below for the case of eccentric circular cylinders as rigid boundaries and zero pressure gradient by making use of a conformal transformation. This approximation is valid only if a dimensionless parameter S (which is proportional to the yield value) is sufficiently small, and there is flow in the whole region between the cylinders. The differences in velocity distribution in the cases of Bingham solid (S finite) and Newtonian liquid (S zero) are illustrated by a particular example. It is shown that the effect of a finite yield value is to cause the contours of equal velocity in a normal section to become more nearly concentric with the inner boundary as the yield value increases from zero.

In this paper we will study the homological properties of the enveloping algebra U = U (Sl2(ℂ)), with particular reference to the homological dimension of simple U-modules and the global dimension of the primitive factor rings of U.

A typical Riesz product on the circle is the weak* limit

where – 1 ≤ rk ≤ 1, øk ∈ R, λT is Haar measure, and the positive integers nk satisfy nk+1/nk ≥ 3. A classical result of Zygmund (11) implies that either µ is absolutely continuous with respect to λT (when ) or µ is purely singular (when ).

Suleῐmanova (3) describes a geometrical technique* for discussing the conditions under which a given set of n + 1 numbers can be a set of characteristic roots of a positive stochastic matrix of order n + 1. Suleῐmanova states that, using this technique, it is possible to prove that a set of n + 1 real numbers 1, λ1, λ2, …, λn, where | λi | < 1 for i = 1, 2, …, n, is a set of characteristic roots of a positive stochastic matrix provided that the sum of the moduli of the negative numbers of the set is less than unity. No indication of the method to be employed is given in (3), and it would seem that the problem is a difficult one. In the present paper I give a full discussion of how Suleῐmanova's technique may be applied to solve the problem for third-order matrices, and simply an indication of the solution for fourth-order matrices; and in, the latter case I point out that Suleῐmanova's result may be replaced by a stronger one. For second-order matrices the problem is almost trivial. As a preliminary I prove two general theorems.

We give a short proof of a slightly weaker version of the multilinear Kakeya inequality proven by Bennett, Carbery and Tao.

We consider the propagation of thermo-elastic Rayleigh waves in a half space, which has been subjected to large uniform extensions, at constant temperature, in three mutually perpendicular directions, one being perpendicular to the surface, and the extension ratios in the other two directions being equal. Rayleigh waves, assuming (a) isentropic conditions and (b) isothermal conditions, are studied, and the relation of the approximation (a) to the exact thermo-elastic theory is considered.

The following work was undertaken in connexion with a device for counting blood cells electronically, which has been developed in the Clinical Pathology Department of the Radcliffe Infirmary with financial assistance from the Medical Research Council and the Nuffield Foundation. Although the results will apply to other types of counter, it will help to fix the ideas if we consider the problem for this specific device. A large number of blood cells, contained in a shallow chamber, are scanned by a photoelectric cell. The depth of the chamber and the concentration of blood cells in solution therein allow blood cells (supposed distributed at random throughout the chamber) to overlap when viewed from above by the scanner. The field of view of the scanner at any instant is somewhat larger than the size of a blood cell, but is, nevertheless, of much the same order of magnitude. With passage of time the chamber moves underneath the photocell so that the field of view traces out a long narrow path not crossing or overlapping itself and only embracing a portion of. the whole chamber. The blood cells have no motion relative to the chamber. As each blood cell comes under the photocell it produces an electrical impulse, whose duration depends upon the size and shape and orientation of the blood cell. These impulses go to a counter, which counts them except that it will not count any impulse which is overlapped by a previous impulse. The problem is to determine the number of blood cells in the chamber from a knowledge of the recorded count and the distribution of the lengths of individual impulses. A further complication arises because it is inadvisable to count two impulses separated only by a very short interval of time, and therefore the counter itself generates some self-paralysing impulses additional to the blood-cell impulses and in a manner which is correlated with them. Until § 3, however, we shall neglect paralysis, because a proper understanding of its effects cannot be reached unless we first know how the system behaves in the absence of paralysis.

If (X, g) is a paranormed space, with paranorm g (see (2)), then we denote by X* the continuous dual of X, i.e. the set of all continuous linear functionals on X. If E is a set of complex sequences x = (xk) then E† will denote the generalized Köthe–Toeplitz dual of E

A long thin plate of finite width (a finite dock) extends over part of the otherwise free surface of a semi-infinite body of water under gravity and is given a forced heaving motion of small amplitude about this position. The solution of the boundary-value problem for the velocity potential of the resulting forced (two-dimensional) surface-wave motion may be reduced to that of an integral equation of the second kind for the value of the potential on the dock by means of an appropriate Green's function. There is an infinite number of choices for such a Green's function, and we show below how to construct one for which the kernel of the corresponding integral equation tends to zero with the ratio wavelength/dock-width. The integral equation may then be solved by iteration to calculate the short-wave asymptotics of the original wave-making problem.

1. We denote by the class of all functions regular in |z| < 1, and never taking the values 0 or 1 in |z| < 1. Then Schottky's(1) theorem states:

Theorem A. Let f(z) = a0 + a1z + … belong to . Then

and hence

where

and the constantsK(a0), K(a0,ρ) depend only on the quantities indicated.

A curve has increasing chords if AD ≥ BC for any four points A, B, C, D lying on the curve in that order. The length of such a curve that connects two points at distance 1 is at most 2π/Z in two dimensions, which is the optimal bound, and less than 30 in three dimensions.

We observe that if is a compatible totally bounded quasi-uniformity on a T0-space (X,), then the bicompletion of (X, ) is a strongly sober, locally quasicompact space. It follows that the b-closure S of (X, ) in is homeomorphic to the sobrification of the space (X, ). We prove that S is equal to if and only if (X, ) is a core-compact space in which every ultrafilter has an irreducible convergence set and is the coarsest quasi-uniformity compatible with . If is the Pervin quasi-uniformity on X, then S is equal to if and only if X is hereditarily quasicompact, or equivalently, is the Pervin quasi-uniformity on .

In 1980 Leedham-Green and Newman made a series of conjectures on the structure of pro-p groups and finite p-groups of a given coclass [10]. These insightful conjectures have gradually been proved, in a series of papers (some of which are still unpublished) by Leedham-Green, Donkin, McKay and Plesken (see, e.g. [11, 2, 8, 9, 13, 14]). Some simplifications (as well as additional information) have recently been given in [12, 15].

Let F(x) be the remainder term in the mean square formula of the error term Δ(t) in the Dirichlet divisor problem. We improve on the upper estimate of F(x) obtained by Preissmann around twenty years ago. The method is robust, which applies to the same problem for the error terms in the circle problem and the mean square formula of the Riemann zeta-function.

Formulae for the mean radius and mean square radius of a relativistic Coulomb wave function are obtained. Screening constants for the energy, mean radius and mean square radius are defined relative to non-relativistic wave functions and the results of numerical calculations given. It is shown that if the screening constants so determined are added to the screening constants due to the presence of other electrons as found by the s.c.f. method, good agreement is obtained in a case where both effects have been considered together. The value of solving the relativistic wave equation in a Thomas-Fermi field is also shown.

This paper is a sequel to an earlier paper in these Proceedings by one of us ((5); this will be referred to as [I]). The question considered there was that of determining the wave front set of the solution of the boundary value problem

where x∈+, y∈n, and n > 1; the precise meaning of the boundary condition at x = 0 is explained in section 1 below. The principal result of [I] can be expressed concisely by saying that singularities do not propagate along the boundary; a detailed statement is given in Theorem 1·9 of the present paper.

We describe some properties of a stationary, isolated, axially symmetric, rotating body of perfect fluid, according to general relativity. We first specialize to the case of constant specific entropy and constant angular velocity. The latter condition is equivalent to rigidity in the Born sense; both conditions are consequences of a simple variational principle. The hydrodynamic equations can then be integrated completely. Analogous first integrals are given also for the case of differential rotation. No use is made of the full field equations.