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)

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.

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.

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].

We say that a system ∑ of equal spheres S1S2, … covers a proportion θ of n-dimensional space, if the limit, as the side of the cube C tends to infinity, of the ratio

of the volume of C covered by the spheres to the volume of C, exists and has the value θ. We say that such a system ∑ has density δ, if the corresponding ratio

has the limit δ as the side of the cube C tends to infinity. We confine our attention to systems ∑ for which both limits exist. It is clear that δ = θ, if no two spheres of the system overlap, i.e. if we have a. packing; and that, in general, the difference δ-θ is a measure of the amount of overlapping.

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 linearrelations 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)

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.

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.

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.

Let (G) denote the lattice of all subgroups of a group G. By an -isomorphism (lattice isomorphism) of G onto a group H, we mean an isomorphism of (G) onto (H). By an -isomorphism (normaliser preserving -isomorphism) of G onto H, we mean an -isomorphism ø such that (Aø) = (A)ø for all A ∈ (G). In this paper, we study certain properties of groups which remain invariant under -isomorphisms.

Let G be a finite group all of whose proper subgroups are nilpotent. Then by a theorem of Schmidt-Iwasawa the group G is soluble. But what can we say about a finite group G is only one maximal subgroup is nilpotent? Let G be a finite group with a nilpotent maximal subgroup M.

Let L be a finite dimensional Lie algebra over the Field F. We denote by (L) the lattice of all subalgebras of L. By a lattice isomorphiusm (whicn we abbrevite to -isomorphism) of L onto a Lie algebra M over the same field F, we mean an isomorphism of (L) onto (M). It is possible for non-isomorphic Lie algebras to -isomorphic, for example, the algebra of real vectors with product the vector product is -isomorphic to any 2-dimensional Lie algebra over the field of real numbers.

If f(z) is analytic at the origin, f(0)=0, and f′(0)=λ, where 0< λ <1, then Koenigs' [3] solution of Schroeder's equation w(f(z))=λw(z), with multiplier λ, is given by . Here fn(z) denotes the nth iterate of f(z), defined inductively as f0(z)=z, f(fn-1(z)), n=1, 2, 3, …. More generally the solution w(z) of Schroeder's equation is uniquely determined to within a multiplicative constant by the requirement that it be analytic at the origin. From the uniqueness it follows that if g(z) is analytic at the origin, vanishes there, and commutes with f(z), i.e., f(g(z)) = g(f(z)), then w(g(z)) = w(z), for some multiplier a. Since w′(0) = 1 for Koenig's' solution, it has an inverse locally, and we find that g(z) is uniquely determined by its linear part; in fact g(z) = w-1(μw(z)). In particular the integral iterates of f can be put in the form w-1(λw(z)) for integral n. Thus for any α, real or complex, we may define fα(z), consistent with the above definition when α is a positive integer, as w-1(λw(z)). In this manner any function g(z) of the above type can be considered as an iterate of /(z). Also if cc, j9, 0 are any two distinct points sufficiently close to the origin there exists an analytic function g(z) which commutes with f(z) such that g(0) = 0, g(α) = β. In fact g(z)=w-1(χw(z)), where the multiplier χ=w(β)(w(α))-1. These facts are all well known, e.g. [2] [5], and we shall establish analogous results in a more general situation.

Let f(x) and g(x) be two polynomials with arbitrary complex coefficients that are relatively prime. Hence the maximum is positive for all complex x. Since m(x) is continuous and tends to infinity with |x|, the quantity E(f, g) = min m(x) is therefore also positive.

The functions where ξ > 0, are essentially Fourier convolutions of Gaussian and Cauchy type distributions, and have thus found application in many fields. In reactor theory, and no doubt elsewhere, the need has arisen to analyse complicated integrals and functional equations involving ψ and φ, the so-called Voigt profiles.

In connection with Relativity, Kottler [2] introduced the space V4 whose metric tensor is given by xi being space coordinates and Ф being a function of x1 only, and showed that if Where a and b are arbitrary constants, then the V4 is an Einstein space.

In a paper of nearly thirty years ago (Mahler 1937) I first studied approximation properties of algebraic number fields relative to their full system of inequivalent valuations. I now return to these questions with a slightly improved method and establish a number of existence theorems for such fields.

D. A. Edwards has shown [1] that if X is a locally compact Abelian group and f ∈ L∞, then the translate fa of f varies continuously with α if and only if f is (equal l.a.e. to) a bounded, uniformly continuous function. He remarks that this is a sort of dual to part of a result due to Plessner and Raikov which asserts that an element μ of the space Mb of bounded Radon measures on X belongs to L1 (i.e., is absolutely continuous relative to Haar measure) if and only its translates vary continuously with the group element, the relevant topology on Mb being that defined by the natural norm of Mb as the dual of the space of continuous functions vanishing at infinity. The proof he uses (ascribed to Reiter) applies equally well in both cases, and also to the case in which X is non-Abelian. A brief examination shows that in the latter case it is ultimately immaterial whether left- or right-translates are considered; since the extra complexities of this case are principally terminological, we shall direct no further attention to it.

Kendall [4] has given for the distribution of the time to first emptiness in a store with an input process which is homogeneous and has non-negative independent increments and an output of one unit per unit time the formula . In this formula, z is the initial content of the store, g(t, z) is the density function of the time to first emptiness τ(z), defined by and k(t, x) is the density function of the input process ξ(t), defined by .

The present day theory of finite groups might be regarded as the outgrowth of the algebraic theory of equations. In much the same way one might consider the modern theory of infinite groups as stemming from late nineteenth century topology. The groups that crop up in topology are of a particularly simple type in that they are both finitely generated and finitely related. This means that every element in such a group can be expressed in terms of a finite number of elements and their inverses and every relation is an algebraic consequence of a finite number of relations between these elements. In other words the legacy of topology to group theory is the estate of finitely presented groups. This talk is concerned with the seemingly simplest of the finitely presented groups, the so-called groups with a single defining relator.