It is well known that an indefinite quadratic form with integral coefficients in 5 or more variables always represents zero properly, and this has raised the problem of proving a similar result for forms of higher degree, namely that such a form, of degree r, represents zero properly if the number of variables exceeds some number depending only on r. For a form of odd degree, no condition corresponding to indefiniteness is needed, but for a form of even degree (4 or more) some even stronger condition must be required.

About twenty years ago, in a note of the same title [2], I obtained the following result.

THEOREM 1. Let u and v be relatively prime integers satisfying u> v ≥ 2 and let ε be an arbitrarily small positive number. Suppose the inequality

is satisfied by an infinite sequence of positive integers n1 n2, … Then

J. R. M. Radok [1] has applied complex variable methods to problems of dynamic plane elasticity. The object of this paper is to show that his results may be obtained in a somewhat simpler way by a more systematic use of complex variable analysis.

In fact it is shown that the problems may be reduced to a form similar to that of the static aelotropic plane strain problems considered by Green and Zerna [2].

Let λ be a random variable with the distribution function F(λ). A transform of F which has, in effect, been used in several recent papers ([1], [2], [3], [4]; see also [6]) is

defined formally by the equation

It is the main purpose of this paper to prove the inversion formulae given in the two theorems below.

If p is a prime other than 2, half of the numbers

1, 2, … p—1

are quadratic residues (mod p) and the other half are quadratic non-residues. Various questions have been proposed concerning the distribution of the quadratic residues and non-residues for large p, but as yet only very incomplete answers to these questions are known. Many of the known results are deductions from the inequality

found independently by Pólya and Vinogradov, the symbol being Legendre's symbol of quadratic character.

Let S be an ordered set, i.e. a set with a transitive irreflexive binary relation “<” such that, for any a, bεS, either a = b or a < b or b < a. By an order automorphism of S we mean a one-one mapping α of S onto itself such that

This paper deals with problems of transverse displacements of thin anisotropie plates with the most general type of digonal symmetry [1]. Proofs of uniqueness of solution under certain conditions are given for problems of plates occupying both finite and infinite regions. This is a generalization to anisotropy of the uniqueness theorems given by Tiff en [2] for isotropic plates.

The problem considered here is the determination of the stresses and displacements in a semi-infinite elastic plate which contains a thin notch perpendicular to its edge, and is in a state of plane strain or generalized plane stress under the action of given loads. The axes of x and y are taken along the infinite edge and along the notch, and the scale is chosen so that the depth of the notch is unity (Fig. 1).

Let K be a bounded n-dimensional convex body, with its centroid at the origin o. Let ϑ denote the density of the most economical lattice covering of the whole of space by K (i.e. the lower bound of the asymptotic densities of the coverings of the whole space by a system of bodies congruent and similarly situated to K, their centroids forming the points of a lattice); and let ϑ* denote the density of the most economical covering of the whole space by K (i.e. the lower bound of the asymptotic lower densities of the coverings of the whole space by a system of bodies congruent and similarly situated to K).

Let Q be a local ring and let q be an m-primary ideal of Q, where m is the maximal ideal of Q. With q we may associate a ring F(Q, q), termed the form ring of Q relative to the ideal q. If u1, …, um is a basis of q, and if B denotes the quotient ring Q/q, there is a homomorphism of the ring B[X1, …, Xm] of polynomials over B in indeterminates X1 …, Xm onto F(Q, q). The kernel of this homomorphism is a homogeneous ideal of B[X1 …, Xm]. Finally, if a is an ideal of Q there is a homomorphism of F(Q, q) onto F(Q/a, q+a/a). The kernel of this latter homomorphism will be termed the form ideal relative to q of a and denoted by ā.

Let ƒ = ƒ(x1, …, xk) be a quadratic form in k variables, which has integral coefficients and is not degenerate. Let n ≠ 0 be any integer representable by ƒ, that is, such that the equation

is soluble in integers x1, …, xk. We shall call a solution of (1) a bounded representation of n by ƒ if it satisfies

It is well known that the thinnest covering of the plane by equal circles (of radius 1, say) occurs when the centres of the circles are at the points of an equilateral lattice, i.e. a lattice whose fundamental cell consists of two equilateral triangles. The density of thinnest covering is

Our main object in this note is to establish (Theorem 1) a necessary and sufficient condition to be satisfied by a sequence {εn} so that a series Σ an εnmay be summable | A |whenever the series Σanis summable (C, — 1). We suppose that an and εn are complex numbers. The condition is unchanged if the an are restricted to be real, but our proof is adapted to the case where they may be complex. Theorem 1 has been quoted by Bosanquet and Chow [12] in order to fill a gap in the theory of summability factors. We also obtain some related results, which are discussed in the Appendix.

An expression is found here for the small transverse displacement of a thin elliptic plate due to a force applied at an arbitrary point of the plate. The plate is in the form of a complete ellipse and is clamped along the boundary. The displacement is expressed in terms of infinite series in §§2–4. The convergence of the series is rapid unless the eccentricity of the ellipse is nearly unity. The simplest case in which the force is applied at the centre of the plate is considered in §5; the displacement of the centre due to this force is compared in §6 with the corresponding displacements of a circular plate and of an infinite strip.

The Faroe-Shetland Channel is the threshold from the north-eastern Atlantic Ocean to north-west European seas. Through it passes the main bulk of the oceanic water-mass which is the predominant influx, among several other water-masses, to these seas.

The following research into the dynamics and general hydrography of the region is based on numerous observations of temperature and salinity, from surface to bottom, taken mainly on two vertical cross-sections of the Channel between the years 1927 and 1952 inclusive, excepting the war years 1940 to 1945.

The research reveals very large scale seasonal and long-term variations in the northeastward volume-transport of oceanic water, suggests the existence on occasions of what appear to be horizontal tortional currents within the oceanic water-mass, and demonstrates (a) the intrusion of Gulf of Gibraltar (extra-Mediterranean) water into this mass over a period of years, (b) the formation of heavy oceanic water and (c) of a sub-oceanic watermass. The last-mentioned may sometimes almost entirely displace the bottom Norwegian Sea water-mass which normally underlies the oceanic mass.

One or other, or both, of two types of Arctic water may also sometimes displace bottom Norwegian Sea water as the bottom water-mass of the region, the process, like that of the above-mentioned Gulf of Gibraltar water influx, waxing and waning over a term of years and thus exemplifying the phenomenon of marine climatic change.

A Lie group is said to be metrisable if it admits a Riemannian metric which is invariant under all translations of the group. It is shown that the study of such groups reduces to the study of what are called metrisable Lie algebras, and some necessary conditions for a Lie algebra to be metrisable are given. Various decomposition and existence theorems are also given, and it is shown that every metrisable algebra is the product of an abelian algebra and a number of non-decomposable reduced algebras. The number of independent metrics admitted by a metrisable algebra is examined, and it is shown that the metric is unique when and only when the complex extension of the algebra is simple.

A study is made of faithful representations of the free cyclic (non-associative) groupoid by means of bifurcating root-trees and by means of index polynomials in two indeterminates emphasizing the intimate connection between these representations. The properties of trees and index polynomials are investigated and the concept of lattice of trees is introduced.

In this note we study the asymptotic behaviour of a product of matrices where Pj is a matrix of transition probabilities in a non-homogeneous finite Markov chain. We give conditions that (i) the rows of P(n) tend to identity and that (ii) P(n) tends to a limit matrix with identical rows.