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.

We propose to study a topological property which is not new, but seems not to have been systematically investigated.

In this paper we derive solutions of the field equations of general relativity for a compressible fluid sphere which obeys density-temperature and pressure-temperature relations which allow for a variation of the polytropic index throughout the sphere.

If we think of the input to a queueing system as arising from some process and depending on the history of that process, we might well expect the duration of inter-arrival intervals to depend mostly on the recent history and to a much smaller extent on that which is more remote.

Journal of the Australian Mathematical Society 5 (1965), 169–195

Theorem 9 (p. 183) should read: The indices of any closed curve inD have the form

n+ = hp/s, n− = hq/s

for some integer h, where s (1 ≦ s ≦ n) is the largest number of identical blocks into which the signature σ can be partitioned. Moreover, for any integer h there exists a closed curve in d with these indices.

The concept of critical group was introduced by D. C. Cross (as reported by G. Higman in [5]): a finite group is called critical if it is not contained in the variety generated by its proper factors. (The factors of a group G are the groups H/K where K H ≦ G, and H/K is a proper factor of G unless H = G and K =1). Some investigations concerning finite groups and varieties depend on the investigator's ability to decide whether a given group is critical or not. (For instance, one of the crucial points in the important paper [9] of Sheila Oates and M. B. Powell is a necessary condition of criticality: their Lemma 2.4.2.) An obvious necessary condition is that the group should have only one minimal normal subgroup: the group is then called monolithic, and the minimal normal subgroup its monolith. This is, however, far from being a sufficient condition, and it is the purpose of the present paper to give some sufficient conditions for the criticality of monolithic groups. (We consider the trivial group neither monolithic nor critical.) The basis of our results is an analysis of the following situation.

Let K be a bounded open convex set in euclidean n-space Rn symmetric in the origin 0. Further let L be a discrete point set in Rn containing 0 and at least n linearly independent points of Rn. Put mi = inf ui extended over all positive real numbers ui for which uiK contains i linearly independent points of L, i = 1, 2, …, n.

The theory of Cosserats' couple-stresses is briefly described in a cartesian system of coordinates, and is applied to the problem of stress distribution in a semi-infinite medium which possesses a non-homogeneous elastic property of an exponential type. Effects of couple-stresses on the stress concentration factors are determined both in homogeneous and non-homogeneous materials.

In this note, we draw attention to a natural connection between a group closely related to the homogeneous Lorentz group, and the most general set of measurements possible on particles with only two discrete states. We may think of these two states as “spin up” and “spin down’, represented by the vectors α = (1, 0) and β = (0, 1), respectively.

Let Λ be a lattice in n-dimensional Euclidean space En. For any lattice there is a unique minimal positive number μ such that if spheres of radius μ are placed at the points of the lattice then the entire space is covered, i.e. every point in En lies in at least one of the spheres. The density of this covering is defined to be θn(Λ) = Jnμn/d(Λ), where Jn is the volume of an n-dimensional unit sphere and d(Λ) is the determinant of the lattice.

In this paper a solution of the Einstein field equations for a spherically symmetric distribution of a perfect fluid of variable density has been obtained.

Let S be a compact topological semigroup, and let be the collection of all normalized non-negative Borel measures on S. It is well-known that , under convolution and the topology induced by the weak-star topology on the dual of the Benach space C(S) of all complex valued continuous functions on S, forms a compact topological semigroup which is known as the convolution semigroup of measures (see for instance, Glicksberg [3], Collins [1], Schwarz [5] and the author [4]). [1], Schwarz [5] and the author [4]). Professor A. D. Wallace asked if the process of forming the convolution semigroup of measures might be generalized to a more general class of set functions, the so-called “modular functions.” The purpose of the present note is to settle this question in the affirmative under a slight restriction. Before we are able to state the Wallace problem precisely, some preliminaries are necessary.

In a previous paper [1] we considered those conformally-flat Riemannian spaces which satisfy the tensorial characterisation where, as usual, gij, Rhijk, Rij are the fundamental tensor, the curvature tensor, the Ricci tensor and E ≠ 0, F are certain scalars. The tensor g is always supposed to be real and analytic. A special form of the metrics of these spaces was seen to be where f is any real analytic function, subject to a restriction, of the argument θ. Writing f, f′, f″,… for f(θ), df|dθ, d2f|dθ2, … the quantities E, F and the scalar curvature R of the type of spaces (1.2) were seen to be

This paper is a sequel to T. G. Room's “Self-polar double configurations in projective geometry, I and II” ([2]). I would like to thank Professor Room for supervising and inspiring my work, and to acknowledge the financial assistance of the C.S.I.R.O.

Definition 1. A real algebra A is a real vector space in which an operation of multiplication is defined satisfying the following conditions: for arbitrary x, y, z ∈ A and any real number α.