Let N be a natural number, and let be a non-empty subset of the set of integers

Let Z be the number of elements of . We denote by Z(p, h) the number of elements of falling into the congruence class h modulo p, so that

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.

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.

Green [8] has shown that a constitutive relation of the form

arises as a special case of an incompressible anisotropic simple fluid, where S is the stress tensor or matrix,

and V is the velocity gradient matrix at time t, all measured in a fixed rectangular cartesian coordinate system. Also, if F is the displacement gradient measured with respect to some curvilinear reference system θi, then

where R is a proper orthogonal matrix, and M and K are positive definite symmetric matrices. In addition

and

1. Let p be a prime, t a positive integer, and ϕ a cubic polynomial with integral coefficients, and an integral constant term. We study the congruence

because of its obvious connection with the equation

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.

A solution is obtained of the problem of diffusion from an elevated point source into a turbulent atmospheric flow over a horizontal ground z = 0. The mechanism of the turbulence is the one considered by D. R. Davies [1[ when he obtained a solution of the same problem in the case when the source is at ground level, the specifications for the mean wind velocity V, and for the across-wind diffusivity Ky, and for the vertical diffusivity Kz, being V αzm, Ky αzm, Kz αz1−m, where m is a constant. Predictions of the solution are that the maximum concentration at ground level of the diffusing matter varies inversely as h1·8 in adiabatic conditions, where h is source height, and that the distance downwind from the source to the point where this maximum concentration is attained varies, in these conditions, as h1·3.

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.

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.

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.

For a subset S of a real linear space, let ck S denote the set of all points from which S is starshaped; that is p∈ ck S if and only if S contains the segment [p, s] for all s∈ S. The set ck S, which is necessarily convex, was introduced by H. Brunn [2[ in 1913 as the Kerneigebeit or convex kernel of the set S. Of course ck S = S if and only if the set S itself is convex. L. Fejes Tóth asked for a characterization of those plane convex bodies which can be realized as the convex kernels of nonconvex plane domains, and it was proved by N. G. de Bruijn and K. Post that every plane convex body can be so realized. Here we establish a stronger result.

No systematic study seems to have been made of so natural a question as the analogue for matrices of quadratic residues. One generalization of x2 (x an integer) is X2 (X an integral matrix). Another is X′ X, where the prime means “transpose”. We study here the solvability for X of the congruence

where p is a prime, r ≥ 1; I (the identity matrix) and X are n-by-n; and a is an integer not divisible by p2.

The two examples of fluid motion in a container which are described in this paper can be easily demonstrated in any kitchen. The first motion was noticed by Professor C. A. Rogers while attempting to dissolve chlorine tablets in water to improve its drinkability. The water nearly filled a cylindrical jar and he had shaken it, with the axis of the jar horizontal, in such a way that the water had a considerable angular momentum about the axis. When the axis of the jar was suddenly moved into the vertical position, he noticed that the water was now rotating about the vertical, which prompted the question of the source of this vertical component of angular momentum. A simplified version of this motion is determined mathematically in §2, and the observations are found to be in general agreement with the theoretical prediction.