The first formula to be proved is

where R(z)>0.

The main result in this paper, contained in Theorem 1, is a generalisation of the inequality of the arithmetic-geometric means. A result of a similar character has been proved by Siegel (2). The present result gives an improvement in the inequality in the case when the variables involved are not all distinct, whereas Siegel's result does not. The theorem is used in § 3 to obtain a result in connection with totally real and positive algebraic integers.

The analysis of the intervals which arise between events occurring randomly in time is a problem which is both interesting and important statistically. Two distinct types of data may arise: either the period of time during which the events are observed may be fixed or the number of intervals may be fixed. It may happen that the intervals between pairs of events, close in time, cannot be accurately measured. It is thus necessary to consider the lengths of intervals ordered according to their magnitudes. We derive here functions of these ordered interval lengths which may be used as a basis for tests for randomness of events occurring in a fixed period of time. The mathematical formulation of this problem is in terms of the classical problem of intervals between points on a line.

The idea of a geometry in which the coordinates are elements of a linear algebra, instead of the conventional field, goes back to C. Segre. Most of the subsequent work seems to have been done by N. Spampinato who developed some general results and applied them particularly to the case of an algebra of dual numbers defined over the complex field; in general, his aim appears to have been the study of algebraic varieties in the new kind of space.

On page 2 of my paper “Rational approximations to algebraic numbers” [Mathematika 2 (1955), 1–20], I referred to possible generalizations of my theorem, and mentioned that I had proved the following:

Let α be any algebraic number, not rational. If the inequality

is satisfied by infinitely many algebraic numbers β of degree g, then κ ≤ 2g.

Two finite real functions ƒ(x) and g(x), defined for — ∞ < x < ∞, are said to be Riemann equivalent if |ƒ(x)—g(x)| has a zero Riemann integral over every finite interval; we then write ƒ~g or

N. G. de Bruijn conjectured that if ƒ(x+h)~ƒ(x) for every real number h, then ƒ~c where c is a constant; this was proved by P. Erdös [1]. In this note we associate with an arbitrary function ƒ the additive group (ƒ) of all numbers h which make ƒ(x)~ƒ(x+h), i.e. which make

In a recent book, L. H. Loomis has obtained the “Bohr compactification” of a topological group, in terms of almost periodic functions, by applying the representation theory of commutative B-algebras. It is simpler, and perhaps more natural, to consider this matter from the point of view of comparative topology; we can then obtain a more general result, in that the discussion is no longer restricted to the case of numerically valued (or even vector-valued) functions.

Consider a set of n points lying in a square of side 1. Verblunsky has shown that, if n is sufficiently large, there is some path through all n points whose length does not exceed (2·8n)1/2+2. L. Fejes Tóth has drawn attention to the case when the n points consist of all points of a regular hexagonal lattice lying in the unit square, in which case the length of the shortest path is easily seen to be asymptotically equal to

The main case of Siegel's theorem on algebraic curves may be stated as follows:

THEOREM 1. Let

be an irreducible algebraic curve of genus g≥1, ƒ(x, y) being a polynomial with algebraic coefficients. Let K be an algebraic field of finite degree over the rational field; let o be the ring of integers in K; and let j be a positive rational integer. Then there are at most finitely many points (x, y) on ℭ for which jxεο and yεK.

Let →(X) be a function of the n variables (X) = (X1, …, Xn) defined for all real (X). A fundamental problem in the theory of Diophantine approximation is to prove the existence of real numbers (X) ≡ (x) (mod 1), where (x) = (x1, …, xn) is any given set of real numbers, for which

In several of his papers, Mordell has developed a method which, in certain cases, leads to an inequality connecting the critical determinant of an n dimensional star body with that of a related n—1 dimensional star body. The purpose of this paper is to exhibit the underlying principle in a general form and to show that the same principle can sometimes be carried further.

The method devised by Hardy and Littlewood for the solution of Waring's Problem, and further developed by Vinogradov, applies quite generally to Diophantine equations of an additive type. One particular result that can be proved by this method is that if a1, …, as are integers not all of the same sign, and if s is greater than a certain number depending only on k, the Diophantine equation

has infinitely many solutions in positive integers x1, …, xs, provided that the corresponding congruence has a non-zero solution to every prime power modulus. A result of a similar kind holds if on the right of (1) there stands an arbitrary integer in place of 0.

It was proved in a recent paper that if α is any algebraic number, not rational, then for any ζ > 0 the inequality

has only a finite number of solutions in relatively prime integers h, q. Our main purpose in the present note is to deduce, from the results of that paper, an explicit estimate for the number of solutions.

The object of this note is to construct a set of real three-dimensional Lie groups such that every real three-dimensional Lie group is locally isomorphic with some group in the set. The construction is effected by first finding canonical forms for the constants of structure of real three-dimensional Lie algebras; these canonical forms give rise to certain bilinear forms, and the Lie groups are obtained as linear groups isomorphic with groups of automorphisms which leave these bilinear forms invariant.

Let Sn denote the “surface” of an n-dimensional unit sphere in Euclidean space of n dimensions. We may suppose that the sphere is centred at the origin of coordinates O, so that the points P(x1, x2, …, xn) of Sn satisfy

We suppose that n≥2.

The formulae to be proved are as follows.

If p ≧ q + 1, R(l + m) > 0, R(αr − l − m + n)> −1, R(αr − l − m − n)> 0, r = 1,2,…p,

If p ≧ q + 1, R(l >0, R(l+m >0, R(αr − l − m + n)> −1,