It is well known that if, in a continued fraction

there is a subsequence of the an which increases very rapidly, then ξ is a transcendental number. A result of this kind follows from Liouville's Theorem on rational approximations to algebraic numbers, but the most precise result so far established is that which was deduced from Roth's Theorem by Davenport and Roth [1]. They proved (Theorem 3) that if ξ is algebraic, then

where qn is the denominator of the nth convergent to (1). Thus if

ξ is transcendental.

Let ℋ be the system of all continuous increasing functions h(t), denned for t ≥ 0, with h(0) = 0 and h(t)>0 for t > 0. Let Ω be a separable metric space. Then, for each h of ℋ, we may introduce a Hausdorff measure into Ω, by taking

where d(Fi) denotes the diameter of Fi, and where the infimum is taken over all sequences {Fi} of closed sets, covering E and having diameters less than δ. We introduce a natural partial order in the system of these Hausdorff measures by writing j < h, if j, h are functions of ℋ and

Let G be a graph with b+v vertices, each of the b vertices P1, …, Pb having valency k and each of the remaining v vertices Q1 …, Qv having valency r, and each edge joining a vertex Pm to a vertex Qn. Suppose also that b≥v; then r≥k since bk = vr.

The Local Uniformisation Theorem was proved by O. Zariski [5] in 1940, and, for the general case, it is so far the only existing proof of the theorem.

Let V be an irreducible manifold defined over a ground field of characteristic zero, and let Σ be its function field. Suppose V lies in an affine space An, and D is any subvariety of V not at infinity. Let J be the integral domain of V and ρ be the prime ideal in J defining D. Then we denote the quotient ring of D by Q(D|V), and by this we shall mean the quotient ring Jρ [1; p. 99]. Thus when we deal with subvarieties of two birationally equivalent manifolds V and V', then the quotient rings will always be subrings of the same representation of the function field of V and V'. Let B be any valuation of Σ whose centre on V is C. The Local Uniformisation Theorem states that there exists a birational transform V' of V such that the centre C' of B on V' is simple and Q(C/V) ⊆Q(C'/V').

This paper is concerned with infinitesimal transverse displacements of homogeneous isotropic elastic plates. The method uses moments of the fundamental equations of orders 0, 1, 2, 3. Assuming a form for the shear stresses tα3, these equations enable one to determine the mean values of the transverse displacements instead of the weighted mean values associated with plate theories of all but the classical type. The relevant moments of the stresses and displacements are expressed in terms of three functions satisfying three differential equations of the fourth order, the solutions of which may be expressed in terms of six independent functions. Thus six boundary conditions may be satisfied. Equating two, three and four of the above functions to zero in turn gives plate theories involving four, three and two boundary conditions respectively. The method is illustrated by assuming that the shear stresses are quadratic functions of the distance from the mid-plane of the material.

Suppose we have a set of n points P1, …, Pn in a unit square. Let

and let Sn be the upper bound of d1+…dn taken over all positions of P1, …, Pn in the square. We prove that

Let k be a finite field of q elements. The equation f(x, y) = 0, where f(x, y) is a polynomial with coefficients in k, may be construed to represent a curve, C, in a plane in which x, y are affine coordinates. On the other hand, this equation can be thought of as denning y as an algebraic function of x, where x is transcendental over k. The purpose of this paper is to show that, for a certain class of curves, corresponding in the classical case to curves having n distinct branches at x = ∞, if the degree, n (in y), of the polynomial f is large compared with q, then the genus† of C cannot be too small. We infer this result from a theorem about the genus of a function field; for we can think of C as being a model of such a field.

This paper is a sequel to a previous paper [1] on axially symmetric torsion-free stress distributions in isotropic elastic solids and applies the methods used to investigate these distributions to distributions in solids under torsion. The basis of these methods is that in a solid of revolution containing a symmetrically located crack the stresses set up in the neighbourhood of the crack by forces applied over the crack can be found by perturbing on their values in an infinite solid containing a crack of the same radius and under the same applied forces, provided the radius of the crack is small compared with a typical length of the solid of revolution. The problem of determining the stresses in the solid of revolution is shown to be governed by a Fredholm integral equation of the second kind, which holds whatever the ratio of the crack radius to the typical length, but which, when this ratio is small, is readily solved by iteration to give stresses perturbing on those in an infinite solid. A similar method can be applied to an infinite solid containing two or more cracks when the crack radii are small compared with a typical length of the crack array.

Let I be a homogeneous ideal in the polynomial ring R = Λ[X1, X2, …, XN], where Λ is a field or, more generally, an Artin ring†. Then R|I has an induced structure as a graded R-module and its homogeneous elements of degree n form a Λ-module of finite length. If this length is denoted by H(n, R|I), then H(n, R|I), considered as a function of n, is often known as the Hilbert function of the ideal I although, in other contexts, it is called the Hilbert function of the graded module R|I. We shall adopt the latter terminology.

A formal solution of the coupled linear thermoelastic equations representing oscillatory motion of an elastic body is obtained, and the decay of the normal modes of vibration examined in detail. The main results of Zener's theory of the thermal damping of a vibrating body are then derived, and somewhat amplified, and consistency with the foregoing normal mode analysis established.

This note is concerned with an inconsistency in the assumptions made in the Reissner theory of elastic plates. An expression is derived for the stress component t33 and a form is suggested for the shear stress components tα3 which form a suitable basis for an approximate theory of elastic plates.

If f(s) is the analytic function defined by the Dirichlet series and if where 0 ≤ b < 1, then the series converges for Re s > 1 and f(s) is regular in the half plane Re s > b except for a simple pole with residue C ≠ 0 at a s = 1. Thus f(s) has a Laurent expansion at s = 1 and it has been shown [1] that under these conditions

where

In this note we prove:

Theorem. Let Λ be a normal number field of absolute degree n. Then there exist infinitely many normal number fields K with the following properties.

(i) The Galois group of K over the rational field Q is the symmetric group Sn on n symbols.

The purpose of the study of Diophantine equations is to find out as much as one can about the rational solutions of a given indeterminate set of equations. In geometrical language, this is the investigation of the rational points on a given variety. The simplest type of problem for which no satisfactory theory is known is that of the cubic surface—the homogeneous cubic equation in four variables. Throughout this note we shall exclude the case of a cubic cone or cylinder, whose theory follows trivially from that of a cubic curve; however, we do not assume that the cubic surface is non-singular.

During the past seven years Northcott has published several papers (see, for example, [6, 7, 8, 9]) in which he has investigated the local aspect of the theory of dilatations. In a similar manner we shall develop in a later paper a local theory of monoidal transformations of which the global analogue appears in [2]. The present note is concerned with such a theory in the one-dimensional case and closely follows the development given in [8] for local dilatations. Indeed the theorems of the present note are all natural generalizations of theorems which have previously been given by Northcott, and for the most part the proofs are essentially Northcott's proofs.

In this paper we shall consider some axisymmetrical problems involving the determination of a harmonic function which satisfies prescribed conditions on two given spherical surfaces. The latter may be exterior to one another or one inside the other. The classical problems which come under this heading are the electrostatic two sphere condenser and the motion of two spheres along the line of centres in an unbounded inviscid fluid. The early investigators of these problems used the well known method of images (Kelvin [1], Hicks [2]). In the electrostatic case (Dirichlet boundary conditions) the images are sets of point monopoles, and in the hydrodynamic case (Neumann boundary conditions) sets of dipole sources. Later Neumann [3] and Jeffery [4] gave solutions using a bipolar coordinate system.

Let X be a metric space and τ a non-negative function on the subsets of X. By the well-known Carathéodory process, we generate outer measures μδ(τ), for δ > 0, and (see §3). When, for every A ⊂ X, τA = (diamA)s for s ≥ 0, μ(τ) is the Hausdorff s-dimensional measure, and, if τA = h(diam A) for a monotone continuous function h with h(0) = 0, μ(τ) is the Hausdorff h-measure. In both of these cases, μ(τ) has been extensively studied.

Archbold [1] has shown how a “distance” can be denned in an affine plane over the field GF(2n) of 2n elements. In terms of this distance, he has shown how to define a group, R(2, 2n), of 2×2 “rotational” matrices which have certain properties of ordinary orthogonal matrices. In the present note we find a standard form for such matrices. Using this standard form, we show that the order of R(2, 2n) is 2n+1+2 and that it has a “proper rotational” subgroup, R+(2, 2n), of index 2. The multiples of R+(2, 2n) by elements of GF(2n) are shown to form a field, which is necessarily isomorphic to GF(22n). The groups R+(2, 2n) and R(2, 2n) are then shown to be cyclic and dihedral groups respectively.

Several recent papers † have been devoted to the problem of the solubility in integers of a homogeneous equation (or of simultaneous equations), or the representation of an integer by a form (homogeneous polynomial). Such problems can be regarded as extensions of Waring's Problem: that of representing an integer as x1k+…+x8k with positive integral x1 …, x8. The methods used are developments of the Hardy-Littlewood method, or if not they employ auxiliary results proved by that method. The number of variables needed to ensure success is usually very large when the degree k of the homogeneous form is large. In this note we draw attention to a somewhat special problem for which a comparatively small number of variables, namely 2k+1, suffices.

Siegel defined an E-function to be a function which admits a power series expansion

with complex coefficients βn, belonging to some algebraic number field K (of finite degree over the rationals Q), satisfying the following conditions:

(i) Given ε > 0, the maximum of the absolute values of the conjugates of βn, satisfies