The relationship between certain non-associative algebras and the deterministic theory of population genetics was first investigated by Etherington (3)-(8), who defined the concepts of baric, train and special train algebras. Gonshor (10) dealt with, among other topics, algebras corresponding to autopolyploidy, on the assumption that chromosome segregation operated. In this paper [ discuss algebras corresponding to more general systems of inheritance among polyploids, which have been discussed without using algebras by Haldane (11), Geiringer (9), Moran (13) and Seyffert (16). These algebras are special cases of what I have defined as segregation algebras, and mixtures of them. All the algebras corresponding to a fixed ploidy have a relationship which I have called special isotopy. An example shows that algebras arise in other genetic systems which are not isotopic to segregation algebras.

Let X be an infinite dimensional normed linear space over the complex field Z. X will not be complete, in general, and its completion will be denoted by . If ℬ(X) is the algebra of all bounded linear operators in X then T ∈ ℬ(X) has a unique extension and . The resolvent set of T ∈ ℬ(X) is defined to be

and the spectrum of T is the complement of ρ(T) in Z.

Throughout this note it will be assumed that

(i) unless otherwise stated, k; is a positive integer,

(ii) a ≧ 0; m > − 1,

(iii) q ≧ 1; q′ is the conjugate number to q, and is defined by

(iv) the functions φ(w), ψ(w) are defined in [0, ∞) with absolutely continuous kth derivatives in every interval [a, W],

(v) φ(w) is non-negative and unboundedly increasing,

(vi) λ = {λn} is an unboundedly increasing sequence with λ1 > 0.

Consider the n-dimensional wave equation

Recently Widder (3) has obtained inversion integrals for a convolution transform whose kernel is a Laguerre polynomial. Buschman (1) has considered convolution equations withgeneralised Laguerre polynomial kernel. His result includes that of Widder. However no attempt has so far been made towards the study of singular integral equations involving Laguerre polynomials as kernel. Here the author obtains an inversion integral for such an equation.

Let A be a finite dimensional algebra with identity element over a field. A is generalised uniserial if every primitive left ideal and every primitive right ideal of A has only one compositions series. In the previous papers in this series (6, 7) generalised uniserial algebras have been characterised as algebras all of whose residue class algebras are of certain types. The purpose of this paper is to extend the earlier results by showing that in order that A be generalised uniserial it is sufficient to require weaker conditions on merely a finite sequence of residue class algebras of A.

Many problems in mathematical analysis require a knowledge of the asymptotic behaviour of Γ(z + α)/Γ(z + β) for large values of |z|, where α and β are bounded quantities. Tricomi and Erdélyi in (1), gave the asymptotic expansion

where the are the generalised Bernoulli polynomials, see (2), defined by

In this note, we show that if, instead of considering z to be the large variable, we consider a related large variable, (1) can be improved from a computational viewpoint.

It is well known that two mutually related summability methods for a sequence sn(n = 0, 1, 2, …) are any Cesaro method (C, γ) of order γ > 0 and the Abel method (A). The notation used in this statement is that of Hardy ((1), pp. 96-7, 71) and the statement itself can be amplified as follows. Summability (C, γ), γ > 0, of sn implies (i) sn = o(nγ), (ii) summability (A) of sn. Also, as a conditional converse of this result, we have Offord's result ((3), first part of Theorem 2), that hypothesis (i), and hypothesis (ii) suitably strengthened, together imply summability (C, γ), γ > 0, of sn. It is the object of this note to bring to light a second pair of summability methods mutually related like the methods (C, γ) and (A), by following an argument which is essentially similar to Offord's but differs sufficiently from Offord's in details to justify a separate and self-contained treatment of the second pair of methods.

Let G be any finite group, and p any prime number. (All groups to be considered here are finite, and we assume this without further comment.) We denote by Kp(G) the unique smallest normal subgroup of G for which the quotient G/Kp(G) is a p-group. G/Kp(G) is called the p-residual of G. W. Gaschütz (2, Satz 7) has proved the following

Theorem. Set K = Kp(G). If the Sylow p-subgroups of K are abelian, then G splits over K.

An analysis is made of the motion of a simple undamped pendulum which performs small oscillations in one plane while its string is raised or lowered through the point of support at a constant rate. The behaviour of the system and the variation of its angular and linear amplitudes are expressible in terms of Bessel functions of orders 0, 1 and 2.

In (7), Wright gives an enumerative proof of an identity algebraically equivalent to that of Jacobi, namely

Here, and in the sequel, products run from 1 to oo and sums from - oo to oo unless otherwise indicated. We give here a simplified version of his argument by working directly with (1), the substitution leading to equation (3) of his paper being omitted. We then supply an alternative proof of (1) by means of a generalisation of the Durfee square concept utilising the rectangle of dimensions v by v + r for fixed r and maximal v contained in the Ferrers graph of a partition.

Consider the dual equations

where