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.

From the fact that Hausdorff s-dimensional measure is a regular Carathéodory outer measure follows (see Saks (3), ch. II, §§ 6, 8) the standard result:Theorem A. If {En} is any increasing sequence of sets, then ∧sEn → ∧s(ΣEn) as n → ∞.

Since ∧s X is denned (for every set X) as , the problem arises whether for every positive δ and every increasing sequence of sets one can prove

Now there are four possible ways of defining ; it is the lower bound of taken over coverings ΣUr of X by convex sets, but in most applications it is not important whether the restriction on the convex sets is that they shall be(i) open and of diameter less than δ,(ii) closed and of diameter less than δ,(iii) open and of diameter not exceeding δ, or(iv) closed and of diameter not exceeding δ.

The equations to be considered are of the form

where Xk is to be determined as a function of t, is its ith derivative with respect to t, Fj is a given analytic function of t, a given constant, and i = 0, 1, …, n; j, k = 1,…, m, and they represent a system of simultaneous linear differential equations of order n in m functions Xk (k = 1,…, m).

In numerical solution of a Sturm-Liouville system, it is necessary to determine an eigenvalue by a method of successive approximation. A relation is derived between the estimated accuracy of an approximate eigenvalue and the accuracy at every point of its corresponding eigenfunction. A method is also described whereby the correction to a trial eigenvalue, required for convergence to its true value, can be automatically determined. This method has been successfully used in solving radial wave-function equations, both with and without ‘exchange’, arising from the Hartree-Slater-Fock analysis of Schrödinger's equation.

Let Z, Q, C denote respectively the ring of rational integers, the field of rational numbers and the field of complex numbers. Minkowski (4) solved the problem of minimizing

for x, y ∈ Z(i) or Z(ρ), where a, b, c, d ∈ C have fixed determinant Δ ≠ 0. Here ρ = exp 2/3πi, and Z(i) and Z(p) are the rings of integers in Q(i) and Q(ρ) respectively. In fact he found the best possible results

for Z(i), and

for Z(ρ), where

while Buchner (1) used Minkowski's method to show that

for Z(i√2). Hlawka(3) has also proved (1·2), and Cassels, Ledermann and Mahler (2) have proved both (1·2) and (1·3). In a paper being prepared jointly by H. P. F. Swinnerton-Dyer and the author, general problems of the geometry of numbers in complex space are discussed and a systematic method given for solving the above problem for all complex quadratic fields Q(ϑ). Here, ϑ is a non-real number satisfying. an irreduc7ible quadratic equation with rational coefficients. The above problem is solved in detail for Q(i√5), for which

and the ‘critical forms’ can be reduced to

Ward(1) has discussed a method, introduced by Tricomi (6), of calculating transients by means of series involving Laguerre functions which in some cases makes it unnecessary to determine poles of the relevant characteristic function. This method is here investigated with special reference to conditions for convergence and adjustments for improving convergence; some of the examples discussed by Ward are reconsidered.

Both in Ward's paper and here the location of poles of the characteristic function is assumed to be approximately known. In some cases the determination of poles of outstandingly small or large modulus and their separation from the remainder may be the most satisfactory procedure. Lin's method (2) for determining a quadratic factor of a polynomial is more widely applicable than has previously been supposed, and this is discussed in bare outline, without proof, here, but in detail, with adequate numerical examples, elsewhere (3).

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.

The solution of a system of simultaneous homogeneous linear differential equations with constant coefficients has been obtained (1) in the form

where the elements of the vector Z are the k functions determined by the system and their derivatives of order up to m – 1, where m is the order of the system. If Zn is the value of Z at t = n, then

Let F(z) be a rational function of z which is regular at z = 0 and so possesses a convergent power series

The problem arises of characterizing those rational functions F(z) that have infinitely many vanishing Taylor coefficientsfh. After earlier and more special results by Siegel(2) and Ward(4) I applied in 1934(1) a p-adic method due to Skolem(3) to the problem and obtained the following partial solution.

1. degenerate iterative processes. A well-known class of iterative processes is specified by the equation

The paper is concerned with linear second-order differential equations in one dimension. The arguments are developed for these equations in general and the examples given are drawn from quantum mechanics, where the accuracies required are in general higher than in classical mechanics and in engineering. An examination is made of the convergence of the eigenvalue Λ(h) of the corresponding finite difference equations towards the eigenvalue λ of the differential equation itself and it is shown that

where h is the size of the interval of the grid covering the range of the independent variable; the constant ν is usually a negative number and consequently Λ(h) may well be a lower bound to λ. This convergence property is used in the numerical calculation of λ by a simple extrapolation technique to a high degree of accuracy. Three examples are given of bounded problems in quantum mechanics. The corresponding eigenfunction can be calculated by a refinement of the familiar relaxation technique by using differences higher than the second, and an example is given.

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.The study of the numerical transformation of slowly convergent series and sequences permits of a certain unity of approach, since the partial sums of the former may be regarded as members of a slowly convergent sequence, while the differences of the latter may be treated as terms in a slowly convergent series. In the ensuing discussion, no essential distinction between the two problems is made, and methods devised for one relate with equal facility to the other.

Suppose that x1, …, xn are jointly distributed in a multivariate normal distribution, f(x1 …, xn), with a given correlation matrix (ρij). Then the evaluation of

is a problem which has been much discussed and has many applications (see David (1)). When n = 3

Let

be a function regular for | z | < 1. With the hypotheses f(0) = 0 and

for some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively by

K(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.

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.

This paper deals with the relationships of the maxima, minima and zeros of two random functions of known autocorrelations and cross-correlation, based on the work of Rice(4). Ratcliffe(3) and Briggs and Spencer(2) discuss a similar problem in connexion with a ‘Phillips Record’ of an experiment in ionospheric reflexion. In this experiment there are two highly correlated reflected signals, their maxima coming close together, and the record shows the time lags between a maximum on one signal and a maximum on the other. Briggs and Page (1) have made an experimental study of the distribution of the differences between the positions of the maxima of two highly correlated random functions, using EDSAC to construct the functions. In §§ 3–7 the frequency distributions of intervals between successive zeros and maxima, and of the lengths of intercepts by a horizontal line, are considered. This has applications to the study of the fading of long-wave radio signals, where the tune differences between successive maxima of the amplitude have been investigated.

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.

Allowance is made for the error terms in the cases where χ has a simple zero at x = 0, or two simple zeros at a; = ± 1 (with χ either positive or negative for − 1 < x < 1). The errors of the approximations given by neglecting the error terms are found to be small in all cases, sometimes much smaller than there was any previous reason to expect.

Many of the statistical properties of a random noise function f(t) have been derived by Rice (4); for example, the mean frequency of zero crossings and the statistical distribution of maxima and minima. These distributions have been shown to apply quite accurately to physical phenomena; in particular, when f(t) represents the height of sea waves at a certain point or the pressure at a fixed point on the sea bed (1, 3, 5).