When Souslin and Lusin initiated and developed the theories of the Souslin operation, of projective sets and of analytic sets, they attached great importance to the constructive nature of their definitions (see [1], [2] and [3]). When Choquet (see [4], [5] and [6]) made his very successful extension of these theories to an arbitrary Hausdorff space Ω, he defined an analytic set in Ω to be a continuous image in Ω of a Kσδ-set in an unspecified compact Hausdorff space X. Thus, a priori, the construction of the analytic sets in Ω requires the preliminary construction of all compact Hausdorff spaces X.

A graph consists of a set of vertices some pairs of which are joined by a single edge. A tree is a graph with the property that each pair of vertices is connected by precisely one path, i.e., a sequence of distinct vertices joined consecutively by edges. The complexity c of a graph G(n, k) with n vertices and k edges is the number of trees with n vertices which are subgraphs of G(n, k). The distribution of c over the class of all graphs G(n, k) is of physical interest because it throws light on the classical many-body problem. (See, e.g. [9].) Ford and Uhlenbeck [3] gave numerical data which suggested that the distribution of c tends to normality for increasing n if k is near No moments higher than the first were known in general and they remarked in [4] that even “the second would be worth knowing”. The main object in this paper is to derive a formula for the second moment of c.

This note is a continuation of the articles [6] and [2]. In [1], trees with a given partition α = (a1; a2, …), where ai is the number of vertices (points) of valency (degree) i were enumerated. After the determination of the number of plane trees in [2], the number of planted plane trees with a given partition α was found explicitly in [6]. In the present note, the number of plane trees with a given partition is expressed as a function of the number of planted trees with a given partition. The method, which is not new, consists of an application of the enumeration techniques of Otter [3] and Pólya [4]; it was used in [1] and also by Riordan [5].

Let K be any convex body in En, and K any given class of convex sets in En. Then we shall say that K is approximable by the class K if there exists a sequence of sets {Ki}, such that, as i→∞, Ki→K in a suitable metric (for example, the Hausdorff metric), where each set Ki is a vector sum of (a finite number of) sets of the class K An approximation problem is to determine necessary and sufficient conditions for K to be approximable by a given class K.

In engineering practice an important class of problems concerns the evaluation of the thermal stresses set up in a heated elastic solid containing cracks. The calculation of the thermal stresses in an infinite space, in which an axially symmetric heat flux across the faces of a penny-shaped crack is prescribed, was first carried out by Olesiak and Sneddon [1], using integral transform techniques. Their solution of the statical equations of thermoelasticity is appropriate to the case of a crack whose faces are stress free and gives zero shear stress on the plane containing the crack. Williams [2] has subsequently shown that the displacement vector in [1 ] can be written in terms of two harmonic functions, one of which is directly related to the temperature field, and has indicated how the analysis of [1] can be reduced to certain simple potential boundary value problems.

Solutions of the boundary-layer equations governing the radial laminar flow of a mixture of two different gases forming a wall jet are obtained. Attention is concentrated on flow in which the concentration of one gas in the mixture is small. The stream function is expanded in terms of a parameter whose magnitude depends upon the concentration of this gas in the mixture.

This paper is concerned with diffusion into a turbulent atmosphere from an infinite ground level line source at right angles to the direction of the mean wind velocity. A solution is obtained for a mechanism which takes into account diffusion in the direction of the velocity, and the predictions of the solution are found to be in good agreement with experimental data in adiabatic atmospheric conditions.

Wave propagation in an elastic medium rotating with a constant angular velocity Ω about an axis is studied by the method of Lighthill [1]. The waves are created by a concentrated periodic force of fixed frequency ω at the origin. The changes in the wave pattern and the decay of amplitude are discussed when 2Ω. increases from zero to a value greater than ω.

Criteria for 2 to be an e-th power residue of a prime

p ≡ 1 (mod e = ef+1,

have been obtained in various forms for e = 2, 3, 5. Euler proved the well known result that 2 is a quadratic residue of a prime p ≡ ± l (mod 8). Dickson [1] showed that 2 is a cubic residue of p ≡ 1 (mod 3) if and only if p = L2+27M2 is soluble in integers L, M.

In a recent paper on a divisor problem the author showed incidentally that there is a certain regularity in the distribution of the roots of the congruence

for variable k, where D is a fixed integer that is not a perfect square. In fact, to be more precise, it was shown that the ratios v/k, when arranged in the obvious way, are uniformly distributed in the sense of Weyl. In this paper we shall prove that a similar result is true when the special quadratic congruence above is replaced by the general polynomial congruence

where f(u) is any irreducible primitive polynomial of degree greater than one. An entirely different procedure is adopted, since the method used in the former paper is only applicable to quadratic congruences.

The n-th roots of unity 1, ω, …, ωn-1, where ω = exp (2πi\n), are linearly dependent in the field Q of rationals since, for instance, their sum vanishes. We are here concerned with the linear relations between them with integral coefficients. Let U denote the vector space of elements u = (u0, …, un−1) over Q and let N be the subspace of elements u defined by the relation u0+u1ω+…+un−1ωn−1=0. (1)

In 1956 Cassels proved the following result, which generalized a theorem of Marshall Hall on continued fractions. Let λ1 …, λr be any real numbers. Then there exists a real number α such that

for all integers u > 0 and for q = 1,…,r, where C = C(r) > 0. Thus all the numbers α+ λ1, …, α+ λr are badly approximable by rational numbers, which is equivalent to saying that the partial quotients in their continued fractions are bounded. In a previous paper I extended Cassels's result to simultaneous approximation. In the simplest case—that of simultaneous approximation to pairs of numbers—I proved that for any real λ1, …, λr and μ1, …, μr there exist α, β such that

for all integers u > 0 and for q=1,…, r, where again C = C(r) > 0. Both the construction of Cassels and my extension of it to more dimensions allow one to introduce an infinity of arbitrary choices, and consequently the set of α for (1) and the set of α, β for (2) may be made to have the cardinal of the continuum.

Throughout this paper, E will denote a finite-dimensional vector space over an ordered field . The real number field will be denoted by ℜ and its rational subfield by . Many of the basic notions in the theory of convexity (convex set, extreme point, hyperplane, etc.) can be defined in the general case just as they are when , but their behaviour may be different from that in the real case. By way of example, we consider the following theorem (due essentially to Minkowski), which is of fundamental importance both for geometric investigations and for the applications of convexity in analysis:

(1) Suppose . If K is a convex subset of E which is linearly closed and linearly bounded, then. K = con ex K; that is, K is the convex hull of its set of extreme points.

A famous problem of Littlewood is whether or not inf u¬¬ux ¬¬u⬬=0, (1) for all real numbers α, β, where the infimum is taken over all positive integers u, and ¬¬ε¬¬, as usual, denotes the distance from ε to the nearest integer. By a well-known transference principle (see [2, p. 78], with an obvious modification), problem (1) is equivalent to whether or not inf ¬xy¬ ¬¬xx+y⬬=0 (2) for all real numbers α, β, with 1, α, β linearly independent over the rationals, where the infimum is taken over all non-zero integers x, y.

1. Throughout this note p is a prime and θ = θ(x1, …, xn) a polynomial of degree 3, with integral coefficients and an integral constant term. The object is to study, by elementary methods, the cubic congruence θ(x1, … xn)≡0 (mod p). (1)

Let K be a finite algebraic extension of the rational number field Q, and let R denote the ring of algebraic integers in K. The algebraic integers in a finite extension field of K form a ring which may be considered as a module over R. The structure of these modules has been entirely determined in Fröhlich [2], where, in particular, necessary and sufficient conditions have been established deciding when such a module will be a free R-module.

A slow steady motion of incompressible viscous liquid, bounded by an infinite rigid plane, which is generated when a rigid sphere of radius a moves steadily without rotation in a direction parallel to, and at a distance d from, the plane is considered. Use is made of bispherical coordinates, which were employed some years ago by G. B. Jeffery [1] and Stimson and Jeffery [2] in solving the axi-symmetrical problems in which the sphere is fixed and rotates about a diameter perpendicular to the plane, or when two spheres move without rotation along their line of centres in infinite liquid. The coordinate system has been used recently by Dean and O'Neill [3] in solving the problem in which the sphere is fixed and rotates about a diameter parallel to the plane.

Summary. This paper is concerned with an infinite plate of homogeneous isotropic elastic material in a state of generalised plane stress and having a circular hole with boundary γ divided into two parts. Over one part of γ the stresses are zero; over the other the shear stress is zero and the normal displacement is specified. The problem corresponds to a smooth loose rigid pin pressed against the edge of a circular hole in an infinite plate.

The problem considered here is that of a torsional impulsive body force within a semi-infinite elastic solid. The surface of the solid is assumed to be either stress-free or rigidly clamped. The basic solution is found to be essentially the same as that for the surface loading of a half space previously considered by Eason [1]. The displacement is determined in terms of elementary functions for one particular type of body force, although using the results in [1] the solution can be obtained for other types of force. Some numerical results for the surface displacement of the stress-free solid are presented in graphical form.

Summary. A rigid circular inclusion, or peg, is symmetrically fixed in an infinite elastic strip of finite width. A simple tension acts on the ends of the strip while the edges are stress free, and no slip takes place between peg and strip. The system is in a state of generalized plane stress.