A series ∑an is said to be summable (C, — 1) to s if it converges to s and nan = o(1) [8]. It is well known that this definition is equivalent to tn→s (n→∞), where tn = sn + nan, sn = ao + … + an. The series is summable | C, — 1 | to s if the sequence t = {tn} is of bounded variation (t ∈ B.V.), i.e. ∑ |; ▲tn |; = ∑ | tn - tn-1 | < ∞, and ∑ ▲tn = lim tn = s. An equivalent condition is ∑ | an |; < ∞, ∑an = s and ∑ | ▲(nan) | < ∞. For, suppose that ∑an = s | C, - 1 |. Since {sn} is the sequence of (C, 1)-means of {tn} and since | C, 0 | ⊂ | C, 1 |, we have ∑ | an | < ∞ and ∑an = s whence ∑ | ▲(nan) | < ∞. Conversely, ∑ | an | < ∞, ∑an = s and ∑ | ▲(nan) | < ∞ imply t ∈ B.V. and ∑▲n = s + lim nan. But lim nan = 0, since ∑ | an | < ∞.

In a recent paper [2] one of the authors has introduced the concept of module type of a ring, for rings with unit. The object of this paper is to generalize this concept to arbitrary rings, without assuming the existence of a unit. This is easily accomplished for rings with one-sided unit, and we shall define the type of such a ring. Theorem 2.5 gives a relation between this type and the module type of [2], and permits the immediate extension of all results in [2] to rings with one-sided unit.

In an unpublished, dissertation Cleaver [1] proved the following

Theorem 1. If L is a lattice in euclidean four-space R4 of determinant d(L) = 1 and with no pair of its points within unit distance apart then any four-sphere of radius 1 contains a point of L.

This paper is concerned with (a) a new simple method of solution of a wide variety of problems of elastic strips by means of Fourier transforms in the complex plane and (b) a direct solution of the elastic annulus. Continuation of functions into adjacent regions of the plane and the solution of differential-difference equations are seen to be unnecessary complications.

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.

This paper is concerned with marginal convection in a self-gravitating sphere of uniform incompressible fluid containing a uniform distribution of heat sources. Its purpose is twofold. The first aim is to present the mathematical argument in a form which, the author believes, is more succinct than that which has been given heretofore. The second aim is to determine the effect of the convective motions upon the moments of inertia of the body and, in the light of the results obtained, examine briefly the hypothesis that the moon is in a state of convection.

Let p be a prime, t a positive integer, and P = P(x1, …, xn) a polynomial over the rational field K, in any number n of variables, of degree k = 2, 3, or 4. We shall consider the congruence

Let p be an odd prime and denote by [p], the finite field of residue classes, mod p. In Euclidean n-space, let n denote the lattice of points x = (x1, …, xn) with integral coordinates and C = C(n, p), the set of points of n satisfying

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.

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.

In a recent paper I. J. Schoenberg [1] considered relations

where the av are rational integers and the ζv are roots of unity. We may in (1) replace all negative coefficients av by −av replacing at the same time ζv by −ζv so that we may, if it is convenient, assume that all av are positive. If we do this and arrange the ζr so that their arguments do not decrease with v then (1) can, as suggested by Schoenberg (oral communication) be interpreted as a convex polygon with integral sides whose angles are rational when measured in degrees. Accordingly we shall call a relation (1) a polygon if all av are non-negative. We shall call a polygon (1) k-sided if all av are positive. The polygon is called degenerate if two of the ζv are equal. Schoenberg calls these polygons polar rational polygons (abbreviated prp) because the vectors composing them have rational coordinates in their polar representations. Schoenberg showed that every prp can be obtained as a linear combination with integral positive or negative coefficients of regular p-gons where p is a prime.

Our purpose in this note is to present a natural geometrical definition of the dimension of a graph and to explore some of its ramifications. In §1 we determine the dimension of some special graphs. We observe in §2 that several results in the literature are unified by the concept of the dimension of a graph, and state some related unsolved problems.

Let L be a lattice in euclidean n-space Rn of determinant d(L). A well-known problem in the geometry of numbers is whether

(1) if d(L) = l and there exists a sphere centred at the origin containing n linearly independent points of L on its boundary and none other than the origin inside, then any sphere in Rn of radius ½√ contains a point of L.

If an is a sequence of numbers between 0 and 1, then

has infinitely many integral solutions n, l either for almost all real x or for almost no real x[1,4]. Duffin and Schaeffer [2], improving on an earlier theorem of Khintchine [7], proved that for decreasing sequences an, (1) has infinitely many solutions a.e. if and only if Σan diverges. They also gave an example of a sequence an for which Σan diverges, but for which (1) has only finitely many solutions a.e. No general necessary and sufficient condition for (1) to have infinitely many solutions a.e. is known.

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.

The flows induced by an oscillating cylinder and by a torsionally oscillating disk are considered. For the case of the cylinder attention is restricted to the high frequency case whilst for the disk both the high and low frequency cases are discussed.

Let K be a bounded, open convex set in euclidean n-space Rn, symmetric in the origin 0. Further let L be a lattice in Rn containing 0 and put extended over all positive real numbers ui for which uiK contains i linearly independent points of L. Denote the Jordan content of K by V(K) and the determinant of L by d(L). Minkowski's second inequality in the geometry of numbers states that Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].

The purpose of this paper is to prove that the altitudes of an n-simplex (a simplexin an n-space) S form an associated set of n+1 lines (see Baker, [4] for n = 4) such that any (n–2)-space meeting n of them meets the (n+1)th too. As an immediate consequence 2 quadrics are associated with S, one touching its primes at the respective feet of its altitudes and the other touching n(n+1) primes, n parallel to each of its altitudes and 2 through each of its (n−2)-spaces. Certain special cases are also mentioned.