Let a, b be relatively prime integers with |a| > |b| > 0. For any integer n > 0, let π n denote the nth cyclotomic polynomial, denned by

where ζn is a primitive nth root of unity.

Let X be a compact polyhedron, Xn the topological product of n factors equal to X. The symmetric group Sn operates on Xn by permuting the factors, and hence if G is any subgroup of Sn we have an orbit space Xn/G obtained by identifying each point of Xn with its images under G. In particular Xn/Sn is the nth symmetric product of X, and if G is a cyclic subgroup of order n then Xn/G is the nth cyclic product of X.

We construct various pathological curves and sequences of curves to solve three problems of Steinhaus.

In 1933 Orlicz proved various results concerning unconditional convergence in Banach spaces (4), which were noted by Banach ((l), p. 240) who remarked that absolute and unconditional convergence are equivalent in finite dimensional Banach spaces, but that whether or not the two are non-equivalent in all infinite dimensional spaces was still an open question. MacPhail (3) gave a criterion for the equivalence of the two notions of convergence in a general Banach space and used it to prove non-equivalence in the spaces l1 and L1. In 1950 Dvoretzky and Rogers demonstrated the non-equivalence of the two types of convergence in any infinite dimensional normed linear space, using an elegant and instructive geometrical approach (2). The result has also been proved by a different method by Grothendieck (5).

The object of this paper is to obtain the original function of which the Laplace transform (l) is the product

where, as usual, p is complex, n is any positive integer, and Wk, m(z) is the Whittaker function defined by the equation

In § 2 it will be shown that this original function is

where the symbol Δ(n; α) represents the set of parameters

Consider a characteristic initial value problem of partial differential equations

where the functions E (x) and F (y) are real valued, uniformly Lipschitz continuous on 0 ≤ x ≤ a, 0 ≤ y ≤ b, respectively. Suppose f (x, y, u, p, q) is a real-valued and continuous function defined on 0 ≤ ≤ b. By a solution of (1), we mean a real-valued continuous function u (x, y), having partial derivatives ux (x, y), uy (x, y) and ux, y (x, y) in the domain 0 ≤ x ≤ a, 0 ≤ y ≤ b almost everywhere.

This paper concerns the determination of the smallest eigenvalue of a second order differential equation containing two parameters which arises in problems concerning genic selection under random drift in a population of finite size. A table of values is given, the method of computation is described, and the asymptotic behaviour for large values of one of the parameters is investigated.

This paper describes, in general terms, a Monte Carlo method for estimating statistical parameters of quantum-mechanical systems. In this method, we construct a Markov chain of transitions between finite sequences of indices, and obtain these parameters in terms of parameters of the limit distribution. This is an extension of the method of Metropolis for classical systems, and may have equally wide application.

The decision problem discussed concerns the drift of a Wiener process with known variance per unit time. It is required to find the optimal sequential test for the sign of the drift, when sampling costs and the cost of a wrong terminal decision are specified. Initially, the drift is supposed to have a continuous prior distribution. The optimal procedure is determined by a continuation region in which a certain partial differential equation holds. The boundaries of this region can be found explicitly in a few special cases. In general, methods are developed for obtaining both inner and outer approximations to the optimal continuation region and these techniques are illustrated by application to several examples.

In a recent paper Collins (l) has shown that Copson's method of solution of the problem of determining the electrostatic field due to an electrified disk (2) can be extended to determine the surface charge density induced on an electrified disk when it is situated inside an earthed coaxial infinitely long hollow cylinder. In Collins's solution the problem is reduced to that of solving a Fredholm integral equation of the second kind. The object of the present note is to show that Collins's solution can be obtained more simply by making use of a technique in the theory of dual integral equations developed recently by the author (3) and Copson(4).

It is shown that the problem of the diffraction of an arbitrary electromagnetic wave by a circular disk can be reduced to the solution of independent scalar problems. These latter problems are of a type which can be reduced to the solution of a Fredholm integral equation of the second kind and are particularly suitable for solution by approximate means at low frequencies.

The solutions of the equation

obtained by the author in (3) are applied to determine the Kirchhoff-Helmholtz flow past a wedge in which the free stream Mach number has any value from 0 to 1 inclusive. For small wedge angles the results are developed further and expressions are found for the drag coefficient and for the asymptotic shape of the free streamlines far downstream. While general formulae are obtained for an arbitrary functionk(σ) more detailed results are given for the particular function k(σ) which corresponds to the case of a polytropic gas.

For a gas with a general equation of state the stream function for the flow past a flat plate is determined when the plate is held at an angle ½π–α to the stream. At infinity the Mach number M1 = 1 and free streamlines, on which M1 = 1, extend from the edges of the plate to infinity downstream. The method of solution requires the determination of the correct singularity corresponding to the free stream and the subsequent satisfying of the boundary conditions. The drag coefficient is determined in the form of a contour integral and the computational problems in evaluating the drag are discussed. The integral is evaluated for small values of α for a gas with an equation of state as given by Tomotika and Tamada(6).

Some aspects of generation of water waves by wind and of turbulence in a heterogeneous fluid may be described by the theory of hydrodynamic stability. The technical difficulties of these problems of instability have led to obscurities in the literature, some of which are elucidated in this paper. The stability equation for a basic steady parallel horizontal flow under the influence of gravity is derived carefully, the undisturbed fluid having vertical variations of density and viscosity. Methods of solution of the equation for large Reynolds numbers and for long-wave disturbances are described. These methods are applied to simple models of wind blowing over water and of fresh water flowing over salt water.

The diffraction of a high-frequency plane sound wave by a circular cylinder is investigated when the boundary condition on the cylinder is expressed by means of an equation of the form The special feature of this investigation is that an extended form of the Kirchhoff-Fresnel theory of diffraction is used to find an integral representation for the scattering coefficient. In order to avoid the complicated analysis which would be necessary to evaluate the integrals concerned, the more natural geometrical acoustics approach is used to find the first correction term in the scattering coefficient. Numerical results are given for large and small values of the impedance Z.

The behaviour of a system of two coupled disk dynamos is studied in considerable detail. It is shown that in several cases the stability criteria and general behaviour are closely related to those for the single dynamo studied by Bullard, which are resumed briefly. Extended numerical investigations of the non-linear behaviour of the system are presented: these show that a true ‘reversal of field’ may occur.

The second method of Liapunov is a useful technique for investigating the stability of linear and non-linear ordinary differential equations. It is well known that the second method of Liapunov, when applied to linear differential equations with real constant coefficients, gives rise to sets of necessary and sufficient stability conditions which are alternatives to the well-known Routh-Hurwitz conditions. In this paper a direct proof of the Routh-Hurwitz conditions themselves is given using Liapunov's second method. The new proof is ‘elementary’ in that it depends on the fundamental concept of stability associated with Liapunov's second method, and not on theorems in the complex integral calculus which are required in the usual proofs. A useful by-product of this new proof is a method of determining the coefficients of a linear differential equation with real constant coefficients in terms of its Hurwitz determinants.

We continue below the note ‘Some cyclic and other inequalities’ (3).

For a certain class of differential equations it is shown that the Stokes phenomenon can be related to a basic change in the character of the differential equation.

We consider a random walk defined in the following way. We have a set of states indexed by n where n takes on all negative and positive integral values and zero. When we are at state n, there is a probability per unit time λ of going to n + 1, and a probability per unit time λ of going to n − l. Let us start out at n = 0, and study Wn(t), the probability of being at n at time t. Continuity of probability requires that whence since G(s, 0) = 1, we have It follows from the well-known result .