In this note a simple application of the Kontorovich-Lebedev transform is described. The problem to which the transform is applied is the one considered in a recent paper (1).

The well-known “basis theorem” of elementary algebra states that in a finite-dimensional vector space, any two bases have the same number of elements; or, equivalently, that a vector space is n-dimensional if it has a basis consisting of n vectors (where the dimension of a vector space is defined to be the least upper bound of the numbers k for which there exist k linearty independent vectors, and a basis is defined to be a maximal set of linearly independent vectors). This theorem has an analogue in the theory of groups : if an Abelian group has a finite maximal set of independent non-cyclic elements, the number of elements in one such set being n, then no set of independent non-cyclic elements can have more than n members.

It is well known that every monic polynomial of degree n with coefficients in a field Φ is the characteristic polynomial of some n × n matrix A with elements in in Φ . However, it is clear that this result is an extremely weak one, and that it should be possible to impose considerable restrictions upon the matrix A. In this note we prove two results in this direction. In section 2, we show that it is possible to prescribe all but one of the diagonal elements of A. This result was first proved by Mirsky (2) when the ground field Φ is the field of complex numbers. In section 3, we see that we can require A to have any prescribed non-derogatory n–l × n–1 matrix in the top left-hand corner.

In many branches of applied mathematics there exists a class of problems which depend for their solution upon the integration of a set of simultaneous linear partial differential equations subject to certain boundary conditions. In all but the simplest cases it is not practicable to deal with these equations by standard methods. For problems involving infinite regions, solutions can often be found by the use of integral transforms. However, in many problems we are concerned with media of finite extent, so that if we are to make a direct application of this method, we shall have to use finite transforms, and under certain conditions these are much more difficult to apply than transforms over an infinite range.

Longitudinal elastic waves in solid circular cylinders were first investigated by Pochhammer (1) and Chree (2) but they did not take into account the effects that thermal properties have on the propagation of these waves. In this paper we shall consider waves in solid and hollow cylinders as well as in the infinite medium with a cylindrical cavity, and in each case we shall take account of the thermoelastic effects.

In this note generalisations of certain integrals involving Legendre functions including the Mehler-Dirichlet integral for Legendre functions of the first kind are given, these new results expressing associated Legendre functions of the first or second kinds as integrals involving corresponding functions of the same degree but different order. These integrals appear to be analogous to Sonine's integral in the theory of Bessel functions.

One of the outstanding theorems in the theory of continued fractions is the result described by O. Perron, Die Lehre von den Kettenbrüchen (1912, 1929, 1954–7) as the transformation of Bauer and Muir (for brevity I shall call it the BM theorem); this theorem and limiting cases of it give rise to numerous extremely interesting consequences.

Agnew (1) has defined a binary transformation T(α), with α real, as one which takes the sequence {Si}, i = 0, 1,…, into the sequence {si(1,α)} where