1. The object of this note is to prove the restricted Burnside conjecture for exponent 5, that is, to prove, for n = 5, the proposition:

Rn: For each positive integer k there is an integer rn, k such that every finite group of exponent n that can be generated by k elements has order at most rn, k.

Let an element of a group be called an FC element if it has only a finite number of conjugates in the group. Baer(1) and Neumann (8) have discussed groups in which every element is FC, and called them FC-groups. Both Abelian and finite groups are trivially FC-groups; Neumann has studied the properties common to FC-groups and Abelian groups, and Baer the properties common to FC-groups and finite groups. Baer has also shown that, for an arbitrary group G, the set H1 of all FC elements is a characteristic subgroup. Haimo (3) has defined the FC-chain of a group G by

Hi/Hi−1 is the subgroup of all FC elements in G/Hi−1.

Isomorphisms are, in many ways, the generalizations of isometrics to uniform spaces. Yet some theorems on isometries of metric spaces only generalize to uniform spaces in terms of more restricted transformations of the uniform space. In § 1, in the course of a discussion of a theorem on transitive groups of automorphisms, we define such a transformation and call it an isobasism. It appears that in many respects isobasisms, rather than isomorphisms, are the generalizations of isometries to uniform spaces. The results of Freudenthal and Hurewicz (7) on contractions, expansions and isometries of totally bounded metric spaces are generalized, in § 2, to contractions, expansions and isobasisms of totally bounded uniform spaces. These results, together with generalizations of some theorems of Eilenberg (6) on compact groups of homeomorphisms of metric spaces which are obtained in §3, give a characterization of isobasisms. The language of Bourbaki (2,3,4) is used throughout this note.

1. Let (x1, x2, …, xn) be Cartesian coordinates in Rn. We identify the point X = (x1, x2, …, xn) with these coordinates with the vector X of components x1, x2, …, xn and use the usual vector notation.

General shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

Some properties are given of the incomplete Bessel and Struve functions defined by a Poisson-type integral. These functions are tabulated for the orders 0 and 1.

Consider the integral

where x1, x2, …, xN are jointly distributed in a multivariate normal distribution f(x1, x2, …, xN) with (pij) as the correlation matrix. The integral has been expressed in an infinite series of tetrachoric functions for N≥2. The infinite series is not only complicated, but also is very slowly convergent and is consequently not of much practical use. Plackett (8) obtains a reduction formula for expressing normal integrals in four variates as a finite sum of single integrals of tabulated functions. These integrals have then to be evaluated by a rather awkward numerical quadrature.

As we have stressed in a previous paper (9), the main concern in Monte Carlo work is to achieve without inordinate labour a respectably small standard error in the final result. Mere replication of the Monte Carlo results is unrewarding; for, to reduce a standard error by a factor k, the labour must be increased k2-fold, and this will be beyond the resources of even electronic computers when k = 1000, say. The remedy lies in a skilful choice of sampling technique and the substitution of analytical methods for random processes wherever possible. The efficiency of a Monte Carlo process may be taken as inversely proportional to the product of the sampling variance of the final estimate and the amount of labour expended in obtaining this estimate; and it is profitable to allow some increase in the labour if that produces an overwhelming decrease in the variance. For instance, in the last example quoted below (Table 2), we reduce the variance by a factor of four million at the expense of only multiplying the labour sixteenfold, thereby attaining a 250,000-fold gain of efficiency.

1. In (2) the idea of antithetic variates in Monte Carlo work is introduced and applied to the estimation of integrals. The present paper studies the general theoretical structure of antithetic variates. We have made only limited progress and present various unsolved problems as a challenge to the reader.

Exact tests for serial independence in vector Markoff processes have been obtained by considering the system of regressions of the vectors observed at time 2t on the vectors observed at times (2t − 1) and (2t + 1). The tests then reduce to those obtained from canonical correlation procedures in multivariate analysis. Two particular cases are

(1) The test for serial independence of a vector of residuals from a system of regressions in which the regressors are all independent of the residuals. At the same time a test of the hypothesis that the regressor and regrediend vectors are independent is obtained.

(2) The test for serial correlation or partial serial correlation in a multiple Markoff process (autoregressive process).

An investigation of the efficiency of the test so obtained, of the hypothesis that a process is a simple Markoff process (against the alternative that the process is a second-order auto-regression) suggests that the efficiency of all of these tests will be low.

The paper contains some remarks on the problems treated by Hammersley(9). With the aid of a new and simple method the exact solutions of Hammersley's problems and the asymptotic behaviour of the solutions are found. The problems in question are: A Poisson process is transformed into a new process by a Type II counter, and this new process is transformed again by a Type I counter. The distribution of the number of events occurring in each transformed sequence is to be determined.

The statistic is applied in several different situations to detect departure from uniform dispersion. The given sample points may lie on a square lattice (without repetition), a rectangular lattice (repetition being allowed), or within rectangular or circular plane boundaries. The problems considered include an analysis of cosmic ray shower structure, and the counting of dust particles and bacterial colonies.

A disjunctive combination of a finite set of two-person games Γ1, Γ2, …, Γk may be defined thus: The players play alternately, each in turn making a move in one and only one of the individual games. If, in addition, the conditions are imposed that

(i) a player loses if unable to move (in any game),

(ii) the games are impartial, i.e. the allowable moves from any position do not depend on which player is about to play (or on the previous moves, though these can be ‘built in’ to the position if necessary),

(iii) the games are of bounded play, i.e. for each game Γi corresponding to any initial position Pj there is an integer bij such that the game must terminate after at most bij moves, then Grundy (6) has shown that there is a function G(P) (called by him Ω(P)) of the positions P with the following properties:

(1a) G(P) = 0 for a terminal position, from which no move is possible; for other positions G(P) is the smallest non-negative integer different from all values of G(Qi), where there is a permissible move from P to Qi,

(1b) for a disjunctive combination of games, G for the combined position is the nim-sum of the G's the individual positions. By the nim-sum, we mean that each separate G is to be written in the scale of 2, as Σar 2r, and then in forming the sum, the ar's are to be added mod 2 for each value of r, as in the theory of Nim((1), (7), (8), pp. 36–8).

Various authors have discussed the ‘disjunctive compound’ of games with the last player winning (Grundy(2); Guy and Smith (3); Smith (4)). We attempt here to analyse the disjunctive compound with the last player losing, though unfortunately with less complete success. The problem was explored tentatively by the first named author, but the rigorous development is due to the second.

By a generalization of methods recently applied in electrodynamics, it is proved that the ‘bare-particle’ coupling constant in renormalizable meson theories becomes imaginary at infinite cut-off. Analytical assumptions made are discussed.

The Pilot ACE has been used to solve integro-differential equations appearing in the theory of nuclear scattering. The application of finite-difference approximations results in a set of simultaneous algebraic equations which are solved to obtain the wave function and from this the phase is evaluated.

The thermodynamic behaviour of a body in a static electromagnetic field depends only on the field over the region of the body itself, whence its own thermodynamic behaviour is determined even in the presence of other polarizable media. Its behaviour depends only on the entropy density, for in fact no real energy density need exist. Also new formulae are developed for the energy and for the entropy of a body in electric and magnetic fields, including those due to permanent magnets.

The scattering of the dominant transverse electric mode in an infinite perfectly conducting rectangular wave-guide by a semi-infinite resistive strip, centrally placed and parallel to the electric field, is calculated by the use of Laplace transforms. Formulae are derived for the amplitude of the scattered waves, and the numerical results, obtained using a digital computer, are given for various values of the surface resistivity of the strip.

This paper discusses a problem in traffic flow by the method of kinematic waves developed by Lighthill and Whitham(1,2). The theory of kinematic waves introduced by Lighthill and Whitham has been extended here to include the case when the flow q varies with the time, and it is seen that the expression for the wave velocity is of the same form as before, namely, ∂q/∂k, where q is the flow (quantity passing a given point in unit time) and k is the concentration (quantity per unit distance). The theory is applied to the problem of estimating how a uniform oncoming flow behaves on entering a bottleneck, the capacity of which varies with time. This capacity has initially a higher value than the oncoming flow but falls at a uniform rate to a lower value, where it remains constant for a time, and again rises at a uniform rate to the original value. A shock wave is found to move back from the bottleneck, and later forward again and through it, much as in the case of a bottleneck of constant capacity with varying oncoming flow studied by Lighthill and Whitham.

The problem of estimating the entry length over which the laminar motion of an electrically conducting, viscous fluid in a pipe under a transverse field adjusts to a steady state from some initial state is linearized by a Rayleigh approximation adapted for cases where there is a pressure gradient. Solutions are found which express the difference between the initial and ultimate velocity profiles as a series of orthogonal, exponentially decaying terms. Approximate entry lengths for the most persistent terms are calculated for flow between two infinite parallel planes and in rectangular tubes when magnetic effects are present. Entry lengths in circular and rectangular pipes and between infinite planes are calculated in the case where magnetic effects are negligible to permit a comparison of the results with those derived by other methods. Pressure losses at entry are considered in the simpler cases.