Recently K. Mahler [1] introduced a set of φ(2n) matrices of n rows and columns which form under multiplication the abelian group of the residue classes prime to 2n modulo 2n. These remarkable matrices whose elements 0, 1 and −1, have latent roots and determinants which can be given explicitly. Thus we have new examples of matrices with given elements whose powers, roots, inverses and determinants can be written down precisely. Such matrices are often useful in testing the efficacy of methods for finding these functions for a general matrix.

We consider real-valued functions ƒ(x) which are defined for all sufficiently large real numbers x. In discussing the behaviour of such functions as x → + ∞, it is useful to compare ƒ with the functions of some “comparison scale”. The early work in this field was due to Du Bois-Reymond (see, for example, (2), (3)). This was elaborated by Hardy, (6), who was mainly concerned with what he calls the “logarithmico-exponential” scale of functions. This “scale of Hardy” may be defined as the smallest class ℋ of functions ƒ with the following properties:

(i) if ƒ ∈ ℋ, ƒ, is defined and continuous for all sufficiently large values of x;

(ii) (a) the function ƒ(x) ≡ α, where α is any real constant, is in ℋ;

(b) the function ƒ(x) ≡ x is in ℋ;

(iii) if ƒ and g are in ℋ and g is non-zero for all sufficiently large values of x, then

are also in ℋ.

In [2] we studied parametric n-surfaces (f, Mn), where Mn was a compact, oriented, topological n-manifold and f a continuous mapping of Mn into the real euclidean k-space Rn (k≧n). A definition of bounded variation was given and, for each surface with bounded variation and each projection P from Rk to Rn, a signed measure: Was constructed. This measure was used to define a linear type of surface integral: over a “measurable” subset A of Mn, as the Lebesgue-Stieltjes integral: .

The ultraspherical polynomial (x) of degree n and order λ is defined by for n = 0, 1, 2, …. Of these polynomials, the most commonly used are the Chebyshev polynomials Tn(x) of the first kind, corresponding to λ = 0; the Legendre polynomials Pn(x) for which λ = ½; and the Chebyshev polynomials Un(x) of the second kind (λ = 1). In the first case the standardisation is different from that given in equation (1), since.

Uniform asymptotic formulae are obtained for solutions of the differential equation for large positive values of the parameter u. Here p is a positive integer, θ an arbitrary parameter and z a complex variable whose domain of variation may be unbounded. The function ƒ (u, θ, z) is a regular function of ζ having an asymptotic expansion of the form for large u.

The results obtained include and extend those of earlier writers which are applicable to this equation.

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.

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.

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.