Throughout this note X will denote a completely regular Hausdorff space and ℬ the σ-algebra of Borel sets in X (see §2 for terminology). For x in X we may define the atomic Borel measure δx to be the unit mass placed at the point x. Observe, however, that the example of Dieudonné [3; §52, example 10] shows that not all atomic measures need have this form. The problem investigated in this note is the existence of finite Borel measures other than the atomic measures. In §3 we show that such a measure necessarily exists on a compact space without isolated points, though we are not able to add to the very meagre supply of examples which are known at the moment. A converse to our result has been given by Rudin [7] and, for completeness, we include a new proof of his result.

In 1953 Erdős† proved in a characteristically ingenious manner that an irreducible‡; integral polynomial f(n) of degree r ≥ 3 represents (r – 1)-free integers (that is to say integers not divisible by an (r – 1)th power other than 1) infinitely often, provided that the obvious necessary condition be given that f(n) have no fixed (r – 1)th power divisors other than 1. His method did not, however, give a means for determining an asymptotic formula for N(x), the number of positive integers n not exceeding x for which f(n) is (r – 1)-free, nor did it even show that the positive integers n for which f(n) is (r – 1)-free had a positive lower density.

1. The purpose of this paper is to give simple proofs for some recent versions of Linnik's large sieve, and some applications.

The first theme of the large sieve is that an arbitrary set of Z integers in an interval of length N must be well distributed among most of the residue classes modulo p, for most small primes p, unless Z is small compared with N. Following improvements on Linnik's original result [1] by Rényi [2] and by Roth [3], Bombieri [4] recently proved the following inequality: Denote by Z(a, p) the number of integers in the set which are congruent to a modulo p.

A well-known theorem of Schauder asserts that a continuous self-map T of a closed convex subset K of a normed linear space X, such that the closure of T(K) is compact, has at least one fixed point. No general method is known for a computation of the fixed point(s) of T.

It was proved in a recent paper† that if au α1, …, αn denote non-zero algebraic numbers and if‡ logαn, …, log αn and and 2πi are linearly independent over the rationals then log α1, …, log αn are linearly independent over the field of all algebraic numbers. Further it was shown that if α1 …, αn are positive real algebraic numbers other than 1 and if β1, …, βn denote real algebraic numbers with 1, β1 …, βn linearly independent over the rationals then is transcendental.

The velocity potential near the boundary of the disturbance is determined for steady supersonic flow past a simple body-and-wing model. It is found that the disturbance decays exponentially as it spreads round the body; the alteration caused by changing the radius of curvature is discussed. A universal formula for the potential away from the fuselage is also derived.

A formal method is developed for deriving a series expansion of the general term in Green-type expansions. The technique is exemplified by detailed calculations for modified Bessel functions of large order.

Until now the chief obstacle to the application of the maximum likelihood method of estimation to factor analysis has been the lack of any really good numerical method of solution. In this paper we give a brief review of recent work which remedies this defect. Two factor analysis models are considered. In each case we derive results which are of use in connection with new methods of solution. Formulae are given for the large-sample variances and covariances of the estimates of parameters in the first model.

The generating function for the number of linear partitions was found by Euler, the method being almost trivial. That for plane partitions is due to Macmahon, but, even in a simplified form found by Chaundy, the proof is far from trivial. The number of solid partitions of n, i.e. the number of solutions of

is denoted by r(n). It has often been conjectured that the generating function of r(n) is , but this is now known to be false. We write η(a, b, c) for the generating function of the number of solutions of (1) subject to the additional condition that

Macmahon 1916 found n(a, 1, 1) for general a. Here we find η(a, b, c) for general a, b. c.

§ 1. The principal object of this note is to establish formula (16) of the preceding paper by H. Jack (1966). This formula, which was conjectured by Jack, evaluates a certain coefficient which is attached to a symbol {p, q, …, r}. In this symbols, p, q, …, r form a partition of m such that o≤p≤q≤…≤r,p+q+…+r=m. The symbol however vanishes if any two of the integers p, q, … r are equal but non-zero. In the remaining cases we have to show that the coefficient in question has the value

Necessary and sufficient conditions are given for the existence of solutions of equations, in any number of variables, on distributive lattices with least and greatest elements, together with an algorithm for determining a solution when these conditions are satisfied.

A multiple integral, whose integrand is an n × n determinant, is evaluated over certain regions of n-dimensional space. Similar integrals are encountered in the theory of Zonal polynomials. In the course of the work a partition problem arises. In the next paper of these Proceedings, Professor Rutherford enumerates these partitions and relates the subject to the theory of the representation of the symmetric group.

The problem considered is that of obtaining solutions of the Helmholz equation ∇2V + k2V = 0, suitable for use in connection with paraboloidal co-ordinates. In these co-ordinates the Helmholz equation is separable, and each of the separated equations is reducible to Hill's equation with three terms (the Whittaker-Hill equation). The properties of solutions of this equation are developed sufficiently to make possible the formal solution of simple boundary-value problems for paraboloidal surfaces, principally for the case k2 < 0.

Fuchsian groups that are unit groups of ternary quadratic forms with rational integer coefficients are studied. By means of the well-known Nielsen classification of finitely generated Fuchsian groups, a complete survey of the unit groups is given. For this, we have to use the arithmetical methods of B. W. Jones. In the second part, the relations between Fuchsian groups arising from different quadratic forms are studied. It turns out that, with a finite number of exceptions, all these Fuchsian groups are subgroups of a particular one.

A necessary and sufficient condition is determined for the modularity of the lattice of congruences on a bisimple inverse semigroup whose semilattice of idempotents is order-anti-isomorphic to the set of natural numbers.

General formulae are derived from first principles for the temporal and spatial autocorrelation functions of stochastic parameters which are defined in terms of superposed, uncorrelated waves with known spectral density distribution. These formulae are first used for obtaining expressions for the autocorrelation functions of the components of the electromagnetic field strength and the electromagnetic energy density in black body radiation fields. The general theory is further applied to compression waves in liquids, and expressions are derived for the temporal and spatial autocorrelation of thermal density fluctuations in liquids, in particular near their critical point. Finally the spectrum of the fluctuations in the total radiation emitted by a thermal source, owing to the fluctuations in the energy supply to the source, is obtained from the appropriate Langevin equation, and the temporal autocorrelation function of the radiation intensity due to this cause is derived from the spectrum.