A graph is said to be k-chromatic if its vertices can be split into k classes so that two vertices of the same class are not connected (by an edge) and such a splitting is not possible for k−1 classes. Tutte was the first to show that for every k there is a k-chromatic graph which contains no triangle [1].

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.

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').

The results given here represent an extension of previous work [1, 2] in which the author considered the oscillations of a plane current-vortex sheet in an ideal perfectly conducting fluid. In this paper we consider the effects of curvature of the sheet in a direction transverse to the velocity and magnetic field direction. This problem may be regarded as that of finding longitudinal small oscillations on a jet of fluid which moves along the lines of force of an impressed magnetic field. For oscillations, whose wavelength is small by comparison with the radius of curvature of the section of the jet, it is to be expected that the criterion for stable or unstable oscillations will be the same as for the plane case examined previously, and this is verified. When one considers the other extreme, in which the wavelength of the oscillations is large, the analysis shows that the magnetic field aligned to the jet has the effect of stabilising the jet, irrespective of the magnetic field strength. The magnetic field thus behaves for large wavelengths in the same way as a surface tension does for small wavelengths. For values of the applied magnetic field which would make the current-vortex sheet without curvature unstable, it is seen that there is a single transition from instability to stability as the wavelength increases. It is shown also that when small wavelengths are stable, in addition to large wavelengths, it does not necessarily mean that the jet will be stable for all wavelengths. Criteria are deduced to distinguish this case from another in which the jet remains unstable for a simple bounded range of intermediate wavelengths.

The continuous-time behaviour of a model which represents certain queues and infinite dams with correlated inputs is considered. It is shown how the transient behaviour may be investigated, and the asymptotic behaviour is obtained. Finally the methods are illustrated for a queue whose input consists of two superimposed renewal processes.

This paper considers a generalisation of the queueing system M/G/I, where customers arriving at empty and non-empty queues have different service time distributions. The characteristic function (c.f.) of the stationary waiting time distribution and the probability generating function (p.g.f.) of the queue size are obtained. The busy period distribution is found; the results are generalised to an Erlangian inter-arrival distribution; the time-dependent problem is considered, and finally a special case of server absenteeism is discussed.

A method is proposed for obtaining a uniformly valid perturbation expansion of the solution of a non-linear partial differential equation, involving either a large or small parameter, when the solution exhibits boundary layer type dependence on the parameter. The method differs from those previously in use in that it is not based on drawing a distinction between points in the boundary layer and points in the remainder of the field. Each point is treated as belonging to both regimes and this enables a stricter control to be maintained on the error terms in the expansions. The method is devised so as to ensure that all forms of error terms are reduced in order at each step in the expansion and not merely those error terms which are mathematically most significant for limiting values of the parameter. The perturbation series can then be used for a wider range of the parameter and provides a solution even when the boundary layer is not particularly thin.

The method is presented through its application to a problem which arises in the theory of the large deflexion of thin elastic plates but the principles underlying the method are more widely applicable.

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.

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.

About 14 years ago A. C. Zaanen [7] published a series of papers on compact symmetrisable linear operators in Hilbert space. Four years later I was encouraged by Dr. F. Smithies to study the spectral properties of general symmetrisable operators in Hilbert space and the resulting research formed the basis of part of a dissertation I submitted to the University of Cambridge in 1952 [4]. For various personal reasons I have not previously been able to, publish these results more widely, although I believe some of them, at least, to be of general interest.

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

In this paper we shall discuss the boundary value problem consisting of the nonlinear ordinary differential equation of the second order, and the boundary conditions.

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.

In continuation of the two previous papers (10; 11), this paper was originally written at the Indian Institute of Technology, Kharagpur and revised at the University of Sydney under the advice of Prof. T. G. Room. Although the altitudes of a general simplex S(A) in n-space (n > 2) do not concur as they do for a triangle (n = 2), yet we observe that its Monge point, M (1; 5), is an appropriate analogue of the orthocentre of a triangle such that M coincides with its orthocentre when it is orthogonal (or orthocentric). In consistency with the previous papers (10; 11; 13; 15) we shall call M as the S-point of S(A) and denote it as S as explained in § 1.2. The altitudes of S(A) are all met by the (n − 2)-spaces normal to its plane faces at their orthocentres, each parallel to of them, thus indicating the associated character of the altitudes as discussed separately in 2 other papers (12; 16). Before we introduce an orthogonal simplex and develop its properties in regard to its γ-altitudes and associated hyperspheres, we come across a number of intermediate ones of special interest. Two special types are treated here and the other two are developed in 2 other papers (13; 15).

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

Over a field of characteristic p the group algebra of a finite group has a non-trivial radical if and only if the order of the group is divisible by the prime p. It would be of interest to determine the powers of the radical in the non-semi-simple case [2, p. 61]. In the particular case of p-groups the solution to the problem is known through the work of Jennings [6]. We here consider the special case of group algebras whose radicals have square zero and we relate this condition to the structure of the group itself.

1. Suppose throughout that a, k are positive numbers and that p is the integer such that k—l≦p<k. Suppose also that φ(w), ψ(w) are functions with absolutely continuous (p+1)th derivatives in every interval [a, W] and that φ(w) is positive and unboundedly increasing. Let λ ={λn} be an unboundedly increasing sequence with λ1 > 0.

Given a series and a number m ≧0, we write

otherwise,

and A(w) = AO(w).