Let A = {a1, …, an} and B={b1, …, bn} be two sets of n elements, and let R be a set of ordered pairs (ai, bi), or in other words a relation defined on A × B. By a map between A and B under R we mean a one-to-one correspondence between A and B such that if bi, corresponds to ai then (ai, bj) is one of the pairs in R.

1. Given a metric space (X, ρ) a family of subsets of X which includes the empty set Ø, and a non-negative function τ on with τ(Ø)=0, an outer measure μ* may be defined by

where empty infimums have value +∞. It is easily seen that μ* is a metric outer measure [i.e., if ρ(A, B)>0 then μ*(A∪B)=μ*(A)+μ*(B)] and from this it follows that all Borel sets in X are μ*-measurable.

Suppose, in n-dimensional Euclidean space, a sequence of disjoint closed spheres is packed into the open unit n-cube In in such a way as to ensure that the residual set has zero volume. Then, of course, is convergent and it can be shown, for n = 2, that is divergent. Let the exponent of convergence of the packing be the supremum of those real numbers t such that is divergent and let tn denote inf, where the infimum is taken over all packings which satisfy the above conditions. In recent work Z. A. Melzak [1] has been interested in finding estimates for in two-dimensional space. He has shown that t may take the value 2 and has produced some estimates for the Apollonius packing of disks. In this note we produce what is, perhaps, the most interesting estimate for tn by showing that tn is greater than n-1. Let sn denote the exact lower bound of the Besicovitch dimensions of every residual set which is formed by a packing of spheres in In.

A number of oscillating systems have been described in the literature which have proved useful in the study of elastico-viscous liquids. The most common type of system in use is the forced oscillating system of the coaxial-cylinder type; in this, the outer cylinder wall is made to oscillate about its axis with a prescribed frequency, and the resulting motion of the inner solid cylinder (constrained by a torsion wire) is recorded (see, for example, [1]). Another type of oscillating system— a, free oscillating system, also of the coaxial-cylinder type—was considered by the present authors in a previous paper [2]; this latter system is one which may prove to be simpler to design and to control in practice.

Professor Kneser has pointed out to me that the results proved in my paper [3] are not new. To be precise, my Theorem 1 is a special case of the result proved in [2], while the routine argument by which I deduced my Theorem 2 is given in substance in [1; 241]. Then my Theorem 3, though perhaps new, follows almost trivially.

By means of all functions regular in a domain †G ⋐ Cn Carathéodory [1,2] defined a metric in G. As usual, using this metric, we can define spherical shells in G. That is, if DG(t, q) denotes the Carathéodory metric between t, q∈G measured in G, then the Caratheéodory spherical shell S(t, q) passing through q and with centre t is defined by:

The object of this paper is to give a simple proof of Menger's famous theorem [1] for undirected and for directed graphs. Proofs of this theorem have been given by D. König [2], G. Nöbeling [3], G. Hajós [4], T. Gallai [5], P. Erdös [6] and 0. Ore [7]. The present proof is shorter, and formulated to apply to directed and undirected graphs equally. The term, graph is to be understood to mean either a finite undirected graph or a finite directed graph throughout. (A: B) denotes either an undirected edge between the two vertices A and B or a directed edge from A to B according to whether undirected or directed graphs are considered. G1 and G2 being two non-empty disjoint graphs, G1: G2-edge denotes an undirected edge between a vertex of G1 and a vertex of G2 or a directed edge from a vertex of G1 to a vertex of G2 as the case may be, and G1: G2-path denotes a path with one end-vertex in G1 and one in G2 having no intermediate vertex in G1 + G2, undirected or directed from the end in G1 to the end in G2, as the case may be. (A set of isolated vertices is a graph.)

1. A study of the recent papers of Roth and Bombieri on the large sieve has led us to the following simple result on the sum of the squares of the absolute values of a trigonometric polynomial at a finite set of points.

With each non-empty compact convex subset K of Ed is associated a Steiner point, s(K), defined by

where u is a variable unit vector, a is a fixed unit vector, H(u, K) is the supporting function of K and dw is an element of surface area of the unit sphere Sd-1 centred at the origin (see [2]). For notational convenience, we put s(Ø) = 0.

1. Certain geometric properties of the valuation theory were considered by O. Zariski in [7]. We have proved some related results in [1] and we consider further similar problems in this paper.

Let V be an irreducible algebraic primal situated in Sd, where d≥3. Throughout the ground field is the field K of complex numbers. For simplicity we assume that V lies in an affine space Ad with coordinates x1,…,xd. Let O be a point on V not at infinity and we take it to be the origin of Ad. Apply a monoidal transformation to V with O as the basis; We obtain thereby a (d−l)-fold V1 lying on a non-singular d-fold U1 situated in an affine space of dimension N1 Since V and V1 are birationally equivalent, we may identify their function fields and thus we denote their common function field by Σ.

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.