The purpose of this paper is to give a new and improved version of Linnik's large sieve, with some applications. The large sieve has its roots in the Hardy-Littlewood method, and in its most general form it may be considered as an inequality which relates a singular series arising from an integral where S(α) is any exponential sum, to the integral itself.

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.

We say that a set of closed circular discs of radii r1r2, …, all lying in a Euclidean plane, is saturated if and only if r = inf ri > 0 and any circle of radius r has at least one point in common with a circle of the set. For any set X we use α(X) to denote the area of X. If X denotes the point set union of the discs and X(k) the part of X inside the disc whose centre is the origin and radius k then by the lower density of the covering we mean . The problem is to find the exact lower bound of the lower density for any saturated set of circles. We show that it is φ/(6√3) provided the circles are disjoint. The general case, when they may overlap, remains unsolved.

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.

In boundary wave problems, when there are two or more boundaries where conditions have to be satisfied, it is often necessary to set up a process of successive approximation in order to gain the solution. Invariably the process cannot be continued beyond a few stages, and the error incurred by halting the process cannot be satisfactorily determined. Such seems to be the case in the surface wave problem when there is a streaming motion of constant velocity past a submerged circular cylinder, which is set in liquid of infinite depth with its axis perpendicular to the stream.

Some of the elastic properties of liquids in shear can be detected and measured by observing suitable types of oscillatory motion. Oscillating systems have proved to be fairly simple to design and to control in practice, and can lead, in the case of purely viscous liquids, to accurate measurements of the viscosity [see, for example, 1, 2, 3]. One such system, used for the purpose of investigating the rheological properties of dilute polymer solutions, is the coaxial-cylinder elastoviscometer of Oldroyd, Strawbridge and Toms [4]; in this experimental arrangement, the liquid is contained between two cylinders with a common, vertical axis, the inner cylinder being suspended by a vertical torsion wire. The theory of the motion of such an instrument in the case when the outer cylinder is forced to oscillate about its axis, which is fixed, with a known constant frequency, and the resulting motion of the inner cylinder is constrained by the torsion wire, has been considered by Oldroyd [5], who shows that the experimental results available for some typical polymer solutions can be interpreted in terms of an idealised elastico-viscous liquid characterised by three constants (a viscosity and two relaxation times). A new representation of the relaxation spectrum of a liquid has subsequently been used by Walters [6] in order to develop the theory of oscillatory flow of the most general (linear) visco-elastic liquid. Walters has shown that the experimental results for dilute polymer solutions (previously interpreted in terms of a discrete relaxation spectrum by Oldroyd [5]) can, equally, be interpreted in terms of a simple continuous relaxation spectrum characterised by three constants.