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,.

An aggregate consisting of a large number of small, equal, hard spheres might be expected to behave like ordinary sand; in this article therefore an attempt is made to find the porperties that will be exhibited by this model, and in particular to find what stresses the sand will withstand without collapsing.

This paper investigates the existence and equality of the double and repeated integrals of a real function on a plane set. The main result (Theorem 2) is that if a function on a plane Lebesgue measurable set is continuous in one variable and measurable in the other then it is measurable in the plane.

1. One of the best known theorems on the finite Fourier transform is:—The integral function F(z) is of the exponential type C and belongs to L2 on the real axis, if and only if, there exists an f(x) belonging to L2 (—C, C) such that ( Additionally, if f(x) vanishes almost everywhere in a neighbourhood of C and also in a neighbourhood of —C, then F(z) is of an exponential type lower than C.

Euler's criterion states that if p is a prime then if and only if D is a k-th power residue of p.

However if (1) does not hold then (2)

where ακ is some κ-th root of unity modulo ρ.

The most elementary problem of the calculus of variations consists in finding a single-valued function y(x), defined over an interval [a, b] and taking given values at the end points, such that the integral is stationary relative to all small weak variations of the function y(x) consistent with the boundary conditions. Since y′ occurs in the integrand, it is clear that I is only defined when y(x) is differentiable and accordingly when y(x) is continuous. Usually y′(x) is also continuous. Occasionally, however, the boundary conditions can only be satisfied and a stationary value of I found, by permitting y′ (x) to be discontinuous at a finite number of points. The arc y = y(x) will then possess ‘corners’ and the well-known Weierstrass-Erdmann corner conditions [1]must be satisfied at all such points by any function y (x) for which I is stationary. Arcs y = y (x) for which y′(x) is continuous except at a finite number of points, are referred to as admissible arcs. In this paper, we shall extend the range of admissible arcs to include those for which y(x) is discontinuous at a finite number of points.

The trace T of the metrical energy-momentum tensor Tkl of fields associated with particles of zero rest mass may be zero, either identically or as a consequence of the field equations. This property of Tkl is correlated here with the behaviour of the Lagrangian of the field under arbitrary conformal transformations. Certain classes of special fields are considered explicitly. It is shown in particular that T vanishes for all non-zero spin fields which correspond respectively to the two-component neutrino field or the photon field.

Let F(x) be a distribution function and denote by its characteristic function and by its moment of order k H. Milicer-Grużewska [2]has derived the following theorem: Suppose that F(x) has moments of all orders and that they satisfy the relations. where while C is a positive constant. Then f(t) is an entire function of order not exceeding two. The proof given by H. Milicer-Gruzewska is rather complicated, moreover the restriction (1.2) seems to be artificial and motivated only by the particular method of proof. We note that condition (1.1) means that the moments of F (x) are majorated by the moments of a normal distribution and we use this remark to generalize the problem.