The Čech compactification of the set of integers is known [6] to have the remarkable property that it has no closed subsets of cardinal ℵ0 or c, every infinite closed subset of it having 2c points. The main object of the present paper is to investigate whether similar gaps in the cardinals of closed subsets can occur in metric spaces. We shall see that the situation there is rather different; if the generalized continuum hypothesis is assumed, there are no gaps, and in any case the missing cardinals, if any, must be big rather than small. The main results are obtained in §3; in particular, we completely determine the cardinals of the closed subsets of complete metric spaces, and also how many closed subsets of each cardinal there are. The methods depend on a study of the discrete subsets of metric spaces, which is carried out in §2, and which may be of independent interest. In conclusion, we briefly consider some fragmentary results for non-metric spaces, in §4. Throughout, we assume the axiom of choice but not the continuum hypothesis.

In the present paper we consider a one-dimensional local ring Q with maximal ideal tn and residue field K = Q/m. It will be assumed that not every element of mis a zero-divisor but no other restricting hypothesis will be made. In particular Q and K may have unequal characteristics and K may be finite.

In a recent paper Segedin [1] has derived a solution of the problem in which a perfectly rigid punch in the form of a solid of revolution of prescribed shape with axis along the z-axis bears normally on the boundary z = 0 of the semi-infinite elastic body z ≥ 0, so that the area of contact is a circle whose radius is a. Segedin solves the problem by building up the solution in a direct way which avoids both the use of dual integral equations and the introduction of an awkward system of curvilinear coordinates. By introducing a kernel function K(ξ), Segedin derives new potentials of the form

where U(r, z, a) is the solution of the simplest punch problem (namely that of a flat-ended punch) satisfying the mixed boundary conditions

on the boundary z = 0. It can then be easily shown that, under wide conditions on K, the function Φ (r, z, a) satisfies the boundary conditions

provided that K(ξ) is chosen so that

The receptance function is defined and constructions for it are given both for the general case when ω, the frequency of excitation, is not a natural frequency and for the special case when it is.

Green and Zerna [1] have given a method of determining the electrostatic potential due to a circular disc maintained at a given axisymmetric potential, their method depending on the solution of a Volterra integral equation of the first kind and being a generalization of a method given by Love [2] for the determination of the electrostatic potential due to two equal co-axial circular discs maintained at constant potentials. In a recent paper [3], henceforth referred to as Part I, the author applied this method to the corresponding problem for a hollow spherical cap and also to the determination of the Stokes' stream-functions for perfect fluid flows past a cap and a disc. The method consists of expressing the potential as the real part of a complex integral of a real variable t, the integrand involving an unknown function g(t). The boundary condition on the disc or cap gives a Volterra integral equation of the first kind for g(t), it being possible to solve this equation and hence determine the potential by integration.

In a recent paper Northcott [3] introduced the notion of the reduction number of a one-dimensional local ring, and demonstrated its importance in the theory of abstract dilatations. In the present paper we define the reduction number of an ideal which is primary for the maximal ideal of a one-dimensional local ring, and show that under certain necessary and sufficient conditions the reduction numbers can take only a finite number of values.

The two centred expansion of the Coulomb Green's function arises naturally in discussing the static interaction energy of two charge distributions ρ1, and ρ2. This is given by the well-known expression

In a recent paper Rogers [13] has discussed packings of equal spheres in n-dimensional space and has shown that the density of such a packing cannot exceed a certain ratio σn. In this paper, we discuss coverings of space with equal spheres and, by using a method which is in some respects dual to that used by Rogers, we show that the density of such a covering must always be at least

A triangle, in the sense of Schubert [1], is an entity in the plane S2 consisting of

(a) an ordered triad of points (P1, P2, P3);

(b) an ordered triad of lines (l1, l2, l3) connected with the points Pi by the incidence relations Pi ⊂ lj (i ≠ j); and

(c) a three base-point net Φof conies with the Pi as base-points.

A polygon, with all sides and all diagonals of rational length, will be called a rational polygon.

Prof. I. Schoenberg has set a problem whether rational polygons are everywhere dense in the class of all polygons, that is, given an arbitrary polygon whether there exists a rational polygon whose sides and diagonals are of length arbitrarily near to those of the given one. Once set, the problem becomes very interesting both for its simplicity and for its fundamental nature. For obvious reasons the likely answer to the problem is in the negative. In this note I consider two problems simply related to the above problem.

A positive quadratic form , of determinant and minimum M for integral , is said to be extreme if the ratio is a (local) maximum for small variations in the coefficients .

Minkowski [3] has given a criterion for extreme forms in terms of a fundamental region (polyhedral cone) in the coefficient space. This criterion, however, involves a complete knowledge of the edges of the region and is therefore of only theoretical value.

Let be a positive definite quadratic form of determinant D, and let M be the minimum of f(x) for integral x ≠ 0. Then we set and the maximum being over all positive forms f in n variables. f is said to be extreme if y γn(f) is a local maximum for varying f, absolutely extreme if y γ(f) is an absolute maximum, i.e. if y γ(f) = γn.

The problem of the survival of a single mutant in a haploid genetic population when there exists selection is considered for a type of population model in which the generations ar overlapping. The results are compared with the previous work of Fisherand others for other models. The need is stressed for a solution of the same problem in a diploid population with general phenotypic selection coefficients.

For investigating the steady irrotational isentropic flow of a perfect gas in two dimensions, the hodograph method is to determine in the first instance the position coordinates x, y and the stream function ψ as functions of velocity compoments, conveniently taken as q (the speed) and θ (direction angle). Inversion then gives ψ, q, θ as functions of x, y. The method has the great advantage that its field equations are linear, so that it is practicable to obtain exact solutions, and from any two solutions an infinity of others are obtainable by superposition. For problems of flow past fixed boundaries the linearity of the field equations is usually offset by non-linearity in the boundary conditions, but this objection does not arise in problems of transsonic nozzle design, where the rigid boundary is the end-point of the investigation.

We consider the finite dam model due to Moran, in which the storage {Zt} is known to be a Markov chain. The method of generating functions is used to derive stationary distributions of Zt in the two particular cases where the input is of geometric and negative binomial types.

Suppose a preliminary set of m independent observations are drawn from a population in which a random variable x has a continuous but unknown cumulative distribution function F(x). Let y be the largest observation in this preliminary sample. Now suppose further observations are drawn one at a time from this population until an observation exceeding y is obtained. Let n be the number of further drawings required to achieve this objective. The problem is to determine the distribution function of the random variable n. More generally, suppose y is the r-th from the largest observation in the preliminary sample and let n denote the number of further trials required in order to obtain k observations which exceed y. What is the distribution function of n?

The main result obtained in this paper is Theorem 1. The symplectic group on the skew matrix T of 2m rows and columns over GF (2)** can be generated by the two matrices Q, R, whereTi,j being the substitution matrix which interchanges the elements numbered i and j,.