We solve two problems on convex bodies stated on p. 38 of S. M. Ulam's book, A collection of mathematical problems (New York, 1960).

Problem 1. This problem is due to Mazur. In three-dimensional Euclidean space there is given a convex surface W and a point O in its interior. Consider the set V of all points P defined by the requirement that the length of the interval OP is equal to the area of the plane section of W through O and perpendicular to OP. Is the (centrally symmetric) set V a convex surface?

In this note we consider a particular type of covering problem. Let T be given a plane set and Tθ be the set obtained from T by a rotation about some point in the plane through an angle θ in the clockwise sense. If a set K is such that for every θ there is a translation which transforms Tθ into a subset of K then we say that K is a rotation cover of T. The problem considered here is to determine for fixed n, l the triangle of least area which is a rotation cover for an n-sided regular polygon of side length l. In each case the solution is an equilateral triangle the altitude of which is given by

This solution is unique.

Interest in the theory of linear graphs has grown recently in several fields which at first sight seem rather far apart. Recent publications include an excellent review by Bryant(1) of electrical network theory, statistical mechanical work by Sykes(2) and in the field of linear programming a paper by Glicksman, Johnson and Eselson (3). It is in this latter field that the subject of this paper arose, but we shall confine attention here to the graph theoretical aspect alone.

In a recent paper (3) Mordell proposed for solution

PROBLEM 1. To find the constants a1, …, an so that

where xn+i = xi ≥ 0 for all i.

Let f(z) = u(z) + iv(z) be a function regular for |z| < 1 with real and imaginary parts u(z) and u(z) respectively. The following theorems have been proved by Miss M. L. Cartwright(1) and (2).

Theorem A. if v(0) = 0, 0 < α < ∞ and

for 0 ≤ r < 1, 0 ≤ θ < 2π then

for 0 ≤ r < 1, 0 ≤ θ < 2π

The familiar difference equations approximating to Laplace's equation are investigated for exact solutions. Solutions corresponding to the electrostatic potentials for line dipole, point and ring charges are obtained analytically.

Methods for finding complex roots numerically may broadly be classified into the following three types.

(a) Direct methods. These require no knowledge of an initial approximation. The best known are Graeffe's ‘root-squaring’ method (see, for example, Whittaker and Robinson(17)) and Bernoulli's method extended to complex roots (Aitken(1)). Fry (8) gives an interesting method for finding the characteristic roots of a matrix, originally due to Duncan and Collar (5), (6), which contains the above-mentioned methods as special cases.

This paper defines the concept of sequential Monte Carlo and outlines the principal modes of approach which may be expected to yield useful sequential processes. Three workable sequential processes, derived from a non-sequential method of J. von Neumann and S. M. Ulam for solving systems of linear algebraic equations, are described and analysed in detail.

The queue M |G| l is considered in the case in which customers are served in random order. A formal solution is obtained for the waiting time distribution in the steady state, and is used to consider the exponential decay of the distribution. The moments of the waiting time are examined, and the asymptotic form of the distribution in heavy traffic is found. Finally, the problem is related to those of the busy period and the approach to the steady state.

No adequate theory of self-avoiding walks on a regular lattice exists, and the only effective approach at present is numerical in character. An electronic computer has been used to enumerate prescribed sets of walks on the usual regular lattices; the method is described and some results are given.

Møller's theory on the concept of energy and its localization in general relativity is applied to calculate the energy of the interior and exterior Schwarzschild fields. It is found that the energy is equal to the (Newtonian) gravitational mass of the material system associated with the fields and that all the energy resides wholly in the material system.

The electromagnetic energy & of the field of a charged particle is calculated, using Møller's theory summarized in the previous paper. The contribution of & to the gravitational mass of the charged particle is discussed, by studying the behaviour of a neutral test particle in its field. The conclusion is that & gives rise to an ‘effective’ gravitational mass of the charged particle, which is equal to the (Newtonian) gravitational mass of the charged particle, plus the mass-equivalence of &. This is contrary to the currently accepted theory, that what we have called the ‘effective’ gravitational mass is equal to the ‘Newtonian’ gravitational mass of the charged particle.

It has been shown by Landau(8) that in a uniform plasma with a Maxwelliau velocity distribution longitudinal waves are damped. Penrose(9) has considered more general distributions and developed stability criteria. In particular, if the velocity distribution function has only one maximum, then the plasma is stable. In this paper the values of the Landau damping for some stable single maximum distributions are compared in order to assess the sensitivity of Landau damping to the form of the distribution function, and to investigate the usefulness of approximating to the Landau damping in a Maxwellian plasma by using an algebraically simpler velocity distribution function.

It is shown also that, for at least some velocity distributions having a maximum particle velocity, the behaviour of the plasma can no longer be described in terms of exponential damping, since there will be longer lasting perturbations with phase velocity equal to the maximum particle velocity.

In an earlier paper (Cohen and Coulson(3), referred to hereafter as I), it was shown that satisfactory energy eigenvalues and eigenfunctions for various even σ-states of may be obtained using a single-centre expansion, provided that the radial functions are properly determined. In particular, the ground-state energy at the equilibrium internuclear separation of 2 a.u. was found to be within 0·25% of the exact value (Bates, Ledsham and Stewart (2)), and the eigenfunction reproduced all the characteristics of the exact wave-function. The method has now been extended to the odd σ-states, as well as to the two lowest π-states (2pπu, 3dπg), and the results are in good agreement with the calculations of Bates et al.

An approximate solution of the two-dimensional problem of reflexion of plane harmonic P and S V waves by an irregular boundary is obtained using a modification of Rice's perturbation method of approximation on the assumption that the curvature of the surface is everywhere small. The case of a periodic surface is treated in more detail. It is found that the reflected waves are composed of specularly reflected waves and various diffracted waves, propagating in both horizontal directions if the wavelength of the incident waves is long compared with that of the surface. If the wavelength of incident distortional waves is long compared with that of the surface, the amplitudes of some of the scattered waves decrease exponentially with depth. In general the phases of the waves change on reflexion and the phase angles of the reflected waves are functions of the wavelength of the corrugation and the angle of incidence. It is verified, in the case of zero angle of incidence, that the energy going into the scattered radiation is obtained at the expense of the energy of the specularly reflected waves.

Throughout this note the series is a rearrangement (in the order of its terms) of the series of complex terms a1 + a2 + ….

MacMahon (2) makes much use of the following result, which he describes as the “Master Theorem”.

Let

and let m1, m2, … mn be non-integers. Then the coefficient ofinis equal to the coefficient ofin

MacMahon (5) used his “Master Theorem”, which is stated in the preceding note (4), in order to obtain an elegant proof of Dixon's identity

It is usual in the theory of queues to assume that customers are served in the order of their arrival. In some applications (e.g. telephone engineering), however, other forms of queue discipline are more realistic. The precise effect of any such change on the waiting-time distribution of a customer will depend on the procedure envisaged (random service, “last come, first served”, etc.), but it is possible to make certain general statements. Thus it is well known that, under certain conditions, the mean is independent of the queue discipline. The purpose of the present note is to consider the variance of waiting time, and we shall prove that this is a minimum when the customers are served in order of arrival. Thus this is, in a sense, the “fairest” queue discipline. This does not, of course, mean that other procedures may not be justified when different criteria are taken into account.