Let be a positive definite quadratic form of determinant D, and let M be the minimum of f(x) for integral x ≠ 0. Then we set and the maximum being over all positive forms f in n variables. f is said to be extreme if y γn(f) is a local maximum for varying f, absolutely extreme if y γ(f) is an absolute maximum, i.e. if y γ(f) = γn.

Expectation values of one-particle and two-particle operators are evaluated in the quasi-chemical equilibrium (pair correlation) approximation to statistical mechanics. Certain reductions, corresponding to the “quenching” of interactions by the Pauli exclusion principle, are carried out quite generally. More specific reductions, which lead to immediately useful expressions, are possible on the assumption of extreme Bose-Einstein condensation of the correlated pairs.

In this paper f(z) will always stand for an entire transcendental function of the complex variable z. For p= 1, 2, … the natural iterate fD(z) of f(z) is defined by These natural iterates are themselves entire transcendental functions; they have been studied by various writers, notably Fatou [3]. References to many papers on iterated will be found in [1].

This note is concerned with a question arising from some work of D. R. Smart (2). In the introduction we indicate the nature of the problem. Notations will be explained only in as far as they differ from those of (2).

R. A. Fisher (1930) has obtained approximate expressions for the probability of survival of a new mutant in a finite population of haploid individuals in which the generations are non-overlapping. Suppose that we have M haploid individuals which are either of genotype a or A, and suppose that a has a small selective advantage over A so that the relative numbers of offspring have expectations proportional to 1 + s and 1 respectively, where s is small and positive. If each generation is produced by binomial sampling with probabilities proportional to the numbers of a and A individuals in the previous generation multiplied by their respective selective values, and if initially there is only one individual of type a, the probability of the population ultimately becoming entirely of this type is approximately so long as s2M is small. This also holds when s is small and negative.

A mathematical theory of the separating flow past a cascade of aerofoils is developed. The flow is assumed to be inviscid, incompressible and twodimensional. The wakes are represented by regions of stationary fluid, which could, in the general case, maintain a pressure gradient, although much of the theory is developed for the case of constant wake pressure.

Euler's criterion states that if p is a prime then if and only if D is a k-th power residue of p.

However if (1) does not hold then (2)

where ακ is some κ-th root of unity modulo ρ.

An investigation has been made into the numerical solution of non-singular linear integral equations by the direct expansion of the unknown function f(x) into a series of Chebyshev polynomials of the first kind. The use of polynomial expansions is not new, and was first described by Crout [1]. He writes f(x) as a Lagrangian-type polynomial over the range in x, and determines the unknown coefficients in this expansion by evaluating the functions and integral arising in the equation at chosen points xi. A similar method (known as collocation) is used here for cases where the kernel is not separable. From the properties of expansion of functions in Chebyshev series (see, for example, [2]), one expects greater accuracy in this case when compared with other polynomial expansions of the same order. This is well borne out in comparison with one of Crout's examples.

This is a sequel to a recent paper [1] on the construction by the hodograph method of trans-sonic nozzle-flows of a perfect gas. At the end of that paper it was shown how we can obtain regular flows that are ultimately uniform (as required in the test section of a supersonic wind tunnel), and the object now is to give some quantitative examples of such flows. The gas is supposed to have the polytropic equation of state Pρ−γ = constant, and the calculations have been made for the case γ = 1.4, with the Mach number M = 2.25 at the test section. The results, which are exhibited graphically, are indicative of what may be expected for other supersonic values of M, and it is hoped that they may be significant for the design of wind tunnels.

An aggregate consisting of a large number of small, equal, hard spheres might be expected to behave like ordinary sand; in this article therefore an attempt is made to find the porperties that will be exhibited by this model, and in particular to find what stresses the sand will withstand without collapsing.

Bivariate distributions, subject to a condition of φ2 boundedness to be defined later, can be written in a canonical form. Sarmanov [4] used such a form to deduce that two random variables are independent if and only if the maximal correlation of any square summable function, ξ (x1), of the first variable with any square summable function, η(x2), of the second variable is zero. This is equivalent to the condition that the canonical correlations are all zero. The theorem of Sarmanov [4] was proved without any restriction in Lancaster [2] and the proof is now extended to an arbitrary number of dimensions.

Let represent the deviations from expectation of a set of multinomial or independent Poisson variables, and Η be a positive definite matrix. A lower bound is obtained for Pr in terms of Pr, Where is a vector of normal variables with the same mean and covariance matrix as Δ.

There are a number of well known theorems on the mutual independence of forms, either linear or quadratic, in normal variables. Some of these theorems can only hold when the system of variables is normal or degenerate and so the possibility of certain forms being independent characterises the normal distribution. The theorems on characterisation have usually been proved by consideration of the necessary properties of the characteristic function. Here we shall be considering the characteristic function of the variables but we shall make more use of cumulant theory than previous authors. To do so we have first to prove that the existence of cumulants of all orders is implied by the independence conditions. The basic theorems we use are those from general cumulant theory and the special theorems of Cramer and of Marcinkiewicz. An advantage of the methods of this paper is that it is possible to show that some of the characterisation theorems require neither of these special theorems. For example, spherical symmetry is a very strong condition and so neither theorem is required whereas both theorems are needed for the most general theorems on the independence of two linear forms. Throughout we take the class of normal distributions to include the degenerate normal, that is, the distribution of a sure variable.

For investigating the steady irrotational isentropic flow of a perfect gas in two dimensions, the hodograph method is to determine in the first instance the position coordinates x, y and the stream function ψ as functions of velocity compoments, conveniently taken as q (the speed) and θ (direction angle). Inversion then gives ψ, q, θ as functions of x, y. The method has the great advantage that its field equations are linear, so that it is practicable to obtain exact solutions, and from any two solutions an infinity of others are obtainable by superposition. For problems of flow past fixed boundaries the linearity of the field equations is usually offset by non-linearity in the boundary conditions, but this objection does not arise in problems of transsonic nozzle design, where the rigid boundary is the end-point of the investigation.

1. One of the best known theorems on the finite Fourier transform is:—The integral function F(z) is of the exponential type C and belongs to L2 on the real axis, if and only if, there exists an f(x) belonging to L2 (—C, C) such that ( Additionally, if f(x) vanishes almost everywhere in a neighbourhood of C and also in a neighbourhood of —C, then F(z) is of an exponential type lower than C.

Introductory. This paper considers a canonical form, or rather a class of canonical forms, for three dimensional probability distributions subject to a rather mild restriction. These canonical forms are used to develop suitable tests of independence and lead to a consideration of the partition of χ2 in the analysis of complex contingency tables. Where these methods and Bartlett's are both applicable it is shown that they give comparable results; but the partitioning methods are more general.