For the solubility of an inhomogeneous polynomial Diophantine equation, there is one well-known necessary, but not sufficient condition; namely the necessary congruence condition (NCC) explained in §2, below. Till recently, no progress had been made with the general cubic equation, because no one knew what else to assume. Examples given here, see (4.3), (5.4), indicate that some rather subtle hypothesis is needed. The first such hypothesis, see Davenport and Lewis [1], was very far from being necessary for the solubility of the equation. It would seem that any supplementary hypothesis which (loosely) is somewhere near necessary and also (together with the NCC) somewhere near sufficient deserves separate detailed investigation before one proceeds to use it.

Write ‖θ‖ for the distance from the real number θ to the nearest integer. An n-tuple of real numbers (β1, …, βn) will be called badly approximable, if there is constant C > 0 such that

for all positive integers q. As is well known, a single number β is badly approximable if and only if the partial quotients in its continued fraction are bounded.

In many investigations into the properties of convex bodies, authors have made use of distance functions ρ(K1K2) which give a measure of the “nearness” of two convex bodies K1 and K2. Sometimes they have introduced new functions to deal with particular problems. The purpose of this paper is to compare and contrast the properties of four of these functions, namely all those (so far as we are aware) which occur in the literature and have the property that they are metrics on the set of all convex bodies of some given dimension.

Throughout this paper a ring will mean a commutative ring with identity element. If A is an ideal of the ring R and P is a minimal prime ideal of A, then the intersection Q of all P-primary ideals which contain A is called the isolated primary component of A belonging to P. The ideal Q can also be described as the set of all elements x∈R such that xr∈A for some r∈R\P. If {Pα} is the collection of all minimal prime ideals of A and Qα is the isolated primary component of A belonging to Pα, then is called the kernel of A.

A general theory of an elastic-plastic continuum which is valid for non-isothermal deformation and which includes explicit restrictions derived from thermodynamics has been given recently by Green and Naghdi [2]. In the development of this theory, the analysis was carried out for a symmetric plastic strain tensor, although it was noted that it is possible to use instead a plastic strain tensor which is nonsymmetric and this would require only a slight modification of the results.

The paper discusses the advantages of solving boundaryvalue problems by the use of eigen-function expansions of suitable fourth order differential equations instead of those of second order equations. Some such expansions are constructed, their convergence properties studied and their use in different types of boundary-value problems are discussed.

Let PGL(2, F) denote the group of all Moebius transformations over a field F. In a recent paper [2], the author has given a characterisation of the groups PGL(2, F), F finite, char F ≠ 2. It is the purpose of this paper to give a similar characterisation of the group PGL (2, F), char F = 2, F finite or infinite.

Let α(n) be a multiplicative arithmetic function. H. Delange [1] has proved that if |α(n)| ≦ 1 for all n and for a certain constant ρ, , where if ρ = 1 then then . He applied this result to several problems such as uniform distribution (mod 1) of certain types of sequences.

By a linear canonical system we mean a system of linear differential equations of the form where J is an invertible skew-Hermitian matrix and H(t) is a continuous Hermitian matrix valued function. We reserve the name Hami1tonia for real canonical systems with where Ik denotes the k × k unit matrix. In recent years the stability properties of Hamiltonian systems whose coefficient matrix H(t) is periodic have been deeply investigated, mainly by Russian authors ([2], [3], [5], [7]). An excellent survey of the literature is given in [6]. The purpose of the present paper is to extend this theory to canonical systems. The only work which we know of in this direction is a paper by Yakubovič [9].

The boundary value problem of the infinite wedge in plane elastostatics is reduced to the solution of a differential-difference equation. The complementary function of this equation is determined in the form of a Fourier integral, which, on expansion by residue theory, gives the complete eigenfunction expansion for the wedge. The properties of the eigenfunctions are discussed in some detail, and orthogonality property is derived.

The main concern of this paper is with the solution of infinite linear systems in which the kernel k is a continuous function of real positive variables m, n which is homogeneous with degree –1, so that If k is a rational algebraic function it is supposed further that the continuity extends up to the axes m = 0, n > 0 and n = 0, m > 0; the possibly additional restriction when k is not rational is discussed in § 1,2.

Nevile [2] has shown that if Rt is a certain measure of the rate of growth of the national income in Harrod's growth model of an economy, then Rt satisfies the non-linear recurrence relation , where 0 < k < and −1 < c < 1. The definition of Rt ([2] p. 369) is such that Rt > 0 for all t. Nevile has pointed out features of the model that indicate that it may be unstable. In this paper I propose to show that the model is, in general. unstable, but that proper choice of the initial values R0R1 apparently leads to stability. In order to do this, we require the conditions (if any) under which Rt converges.

Throughout this paper, G denotes a Hausdorff locally compact Abelian group, X its character group, and Lp(G) (1 ≦ p ≦ ∞) the usual Lebesgue space formed relative to the Haar measure on G. If f ∈ Lp(G), we denote by Tp[f] the closure (or weak closure, if p = ∞) in Lp(G) of the set linear combinations of translates of f.

Consider a positive regular Markov chain X0, X1, X2,… with s(s finite) number of states E1, E2,… E8, and a transition probability matrix P = (pij) where = , and an initial probability distribution given by the vector p0. Let {Zr} be a sequence of random variables such that and consider the sum SN = Z1+Z2+ … ZN. It can easily be shown that (cf. Bartlett [1] p. 37), where λ1(t), λ2(t)…λ1(t) are the latent roots of P(t) ≡ (pijethij) and si(t) and t′i(t) are the column and row vectors corresponding to λi(t), and so constructed as to give t′i(t)Si(t) = 1 and t′i(t), si(o) = si where t′i(t) and si are the corresponding column and row vectors, considering the matrix .

Our present view of the universe suggest that the set of mutually receding galaxies may provide a natural substratum for the propagation of light. It is shown that this assumption leads to a consistent derivation and interpretation of special relativity, along the lines evvisaged by Lorentz but requiring also the employment of Einstein's measurement definitions. The time-dilatation and Fitzgerald contraction effects emerge as intelligible consequences of this approach, and their interaction with an associated anisotropy effect produces the relativity of simultaneity, the reciprocity phenomenon and the results described by Einstein's principles; the approach provides a definitive resolution of the “clock paradox” within the framework of Special Relativity.

One of the elementary applications of the Rankine-Hugoniot shock relations which relate conditions on the two sides of a plane shock wave is that of determining the flow when a piston is pushed with constant velocity ū into a tube containing gas at rest. A shock wave races into the undisturbed gas at a constant speed Ū whose value can easily be found in terms of ū and the constants which specify the uniform condition of the gas at rest. If, however, the piston is suddenly brought to rest after a finite time the subsequent behaviour of the shock wave is very difficult to determine. A rarefaction wave is generated at the piston, and, as the velocity of the shock is subsonic relative to the gas behind it, this eventually overtakes the shock wave causing it to weaken. Since the energy supplied is finite the ultimate speed of the shock will tend to that of a sound wave. The analytical treatment of the flow behind the shock is made difficult by the entropy gradients which arise because of the variation in shock strength. It is further complicated by the disturbances which are reflected off the piston and give rise to a secondary interaction with the shock. Indeed, it seems safe to say that a complete description of the motion would certainly depend on some form of numerical integration.

The purpose of this note is to establish the following characterisation of the radical: Theorem. Let R be a ring with the minimum condition for left ideals. Then the radical of R is the intersection of the maximal nilpotent subrings of R.