By a ‘representation’ we shall mean throughout a representation by n × n matrices with entries from an arbitrary field. Elsewhere [9] the author has introduced the concept of a principal representation of a semigroup S (see § 3 below for the definition) and has shown that if S satisfies the minimal condition on principal ideals then every irreducible representation is of this type. Moreover, if S satisfies the minimal conditions on both principal left and right ideals, which together imply the minimal condition on principal two-sided ideals [6, Theorem 4], the irreducible representations of S can ultimately be expressed explicitly in terms of group representations.

The roots of these equations are of importance in several theories and various authors have studied certain of their properties. Here we solve the equations in the sense that we define two numbers z(1), z(0), and a sequence {zn} which include the roots of both equations. Except for a small, finite number of values of n, we find a rapidly convergent series for zn whose terms are alternately real and purely imaginary. We give a number of expansions and a variety of practical methods which enable us to calculate the small number of remaining roots to any required degree of accuracy.

The main results of this article have been announced without proof or details in Wright 1960.

The preparation of 1-benzhydrylnaphthalene, 1, 2, 3, 4-tetrahydro-1-benzoylnaphthalene, and 1-diphenylmethylenetetralin is described.

Thanks are expressed to the Department of Scientific and Industrial Research for a maintenance grant (to D. K.); to the Royal Society of London for a grant (to A. J. P.); to Miss E. W. Robertson for help with the experimental work; and to the Education Committee of Stirlingshire for permission to use laboratory facilities.

Algebras which are nilpotent and anti-commutative are studied. Canonical forms are found for all such algebras of dimension n whose centres have dimension n−r (r < 3), and characters are given which enable any two non-isomorphic algebras to be distinguished.

A metrisable Lie algebra is a Lie algebra for which there is a non-singular, symmetric, adjoint-invariant bilinear form a(λ, μ), and such an algebra is reduced if its centre is contained in its derived algebra. The importance of the reduced algebras follows from the fact that every metrisable Lie algebra is the direct sum of a reduced metrisable Lie algebra and an abelian Lie algebra. Tsou (Thesis 1955) introduced metrisable Lie algebras, and obtained canonical forms for all real reduced metrisable Lie algebras whose derived algebras have dimension 3. We conclude this paper by providing an alternative derivation, two of the algebras being nilpotent.

Let f(n) = an2+ bn + c be an irreducible quadratic polynomial with integer coefficients, and let D denote the discriminant b2 – 4ac of f(n).We shall assume that (D, k) = 1, and that for all positive integer n, f(n) is positive and coprime with k, where k is a fixed integer greater than 1.

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.

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.

1. Let f = f(x1, …, xn) be an indefinite quadratic form in n variables with discriminant d = d(f) ¹ 0; and let ξ1, …, ξn be real numbers. We consider how closely the inhomogeneous quadratic polynomial

can be made to approximate to a given real number α by choice of suitable integral values of the variables xi. The best that is known seems to be that the inequalities

can always be satisfied if the implied constant is given a suitable value depending only on n. For α ≥ 0 this is a restatement of a result proved by Dr. D. M. E. Foster.

An exact expression in finite terms is found for the small deflexion at any point of an infinitely large plate clamped along an inner curvilinear edge, with outer edge free, and loaded by a concentrated force at an arbitrary point of the plate. The plate can be mapped on the area outside the unit circle by a rational mapping function involving two parameters. By varying these parameters holes having various shapes and several axes of symmetry are obtained. Infinite plates with holes in the forms of regular and approximately rectilinear polygons are included as special cases.

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.

Let G1,… Gn be groups, let *Gi be their free product, and let ´ Gi be their direct product. A homomorphism may be defined by requiring it to be trivial on Gj and the identity on Gi for i ¹ j. Let [G1 … Gn] = Çker pj. P. J. Hilton [2] proves that [G1, …, Gn] is a free group, and, if HiÌGi, i = 1, …, n, that [H1,… Hn] is a free factor of [G1 … Gn]. He asks whether, if Hλi Ì Gi, λ = 1, …, k, i = 1, …, n, and Hλ = [Hλ1, …, Hλn] the group generated by the Hλ is a free factor of [G1, …, Gn].

In recent papers [1, 2] I have given expressions for the stresses and displacement due to elastic distributions in an infinite isotropic elastic solid bounded internally by a spherical hollow, the boundary of which is either stress or displacement free. This paper gives corresponding results for a semi-infinite elastic solid together with expressions for the stresses and displacement due to thermoelastic distributions in such a solid, the boundary of which is maintained at a constant temperature and is either stress or displacement free.

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.

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.

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: .

Let N be a large positive integer and let n1, …, nN be any N distinct integers. Let

Hardy and Littlewood proposed the problem: to find a lower bound for

in terms of N, this lower bound to be a function of N which tends to infinity with N. It is easily seen, on examining the case when n1, …, nN art in arithmetic progression, that a lower bound of higher order than log N is impossible.