The number of distinct types of Abelian group of prime-power order pn is equal to the number of partitions of n. Let (ρ) = (ρ1, ρ2, …, ρr) be a partition of n and let (μ) = (μ1, μ2, …, μs) be a partition of m, with ρ1≧ρ2≧…≧ρr and μ1≧μ2≧…≧μs, ρi≧μi, r≧s, n>m. The number of subgroups of type μ in an Abelian p-group of type (ρ) is a function of the two partitions (μ) and p, and has been determined as a polynomial in p with integer coefficients by Yeh (1), Delsarte (2) and Kinosita (3). Their results differ in form but are equivalent.

The Kontorovich-Lebedev Transform of a function f(r), 0<r<∞, may be written in a general form as fg(μ) where,

This paper is a sequel to a previous paper (1) on axisymmetric potential problems for one or more circular disks situated inside a coaxial cylinder and applies the method used for these problems to the electrostatic potential problem for a perfectly conducting thin spherical cap situated inside an earthed coaxial infinitely long circular cylinder.

The solution to the boundary problem

where r is the distance of point (x, y) from the origin, and h is a given function the arc length s along the unit circle r = 1, is not necessarily unique, Boggio (1), Weinstein (2), Stoker (3), Martin (4). Indeed if h is a positive integer m is known that the only solutions regular analytic for r≦1 are

where r, θ denote polar coordinates and A, B are arbitrary constants.

It has been well known for many years (2) that if Fμ(t) is the Fourier-Stieltjes transform of a bounded measure μ on the real line R, which is bounded away from zero, it does not follow that [Fμ(t)]−1 is also the Fourier-Stieltjes transform of a measure. It seems of interest (as was remarked, in conversation, by J. D. Weston) to consider measures on the half-line R+ = [0, ∞[, instead of on R.

In problems in the mathematical theory of elasticity related to the symmetric deformation of an infinite elastic solid with an external crack we encounter the problem of determining an axisymmetric function φ(ρ, z) which is harmonic in the half-space z>0 and satisfies the mixed boundary conditions

on the plane boundary z = 0, where it is assumed that f(ρ) is continuously differentiable in [1, ∞). Further φ→0 as √(ρ2+z2)→∞.

A representation of degree 3 is used in § 4 to establish the necessary and sufficient condition

for the finiteness of the group (3, p | q, r) defined by

We present in this paper a straight-forward method of deducing integral transform pairs directly from second order differential equations. In (1) this subject has been very thoroughly treated and in view of this we believe that the method described here has its greatest justification in its simplicity.

This paper concludes the investigation of axially symmetric stress distributions in elastic solids containing penny-shaped cracks, commenced in previous papers (1), (2), by considering the stress distribution in a circular beam containing a crack opened by internal pressure or by uniform tension. The method of analysis, developed in the previous papers, is to first seek a representation of the displacement at a point of the beam as a sum of two terms, one of which is a representation of the displacement due to the crack in an otherwise unbounded infinite solid whilst the second is a general representation of the displacement in an undamaged beam, and then to show that this representation satisfies the conditions on the crack and the curved surface of the beam provided an unknown function occurring in it is the solution of a certain Fredholm integral equation. This equation holds whatever the ratio of the radius of the crack to that of the beam, but is most readily solved by iteration when this ratio is small, this solution being a perturbation on that for a crack in an infinite solid.

For positive integral n and positive real x1, … xn let

Where

and let

The reciprocity, , has been shown to hold when g(x) is in B* [Proc. Edin. Math. Soc., 12 (1960), 85]. The purpose of the present note is to show that it holds more generally and in particular when g(x) is in a subclass B″, of functions of bounded variation B, such that B* ⊂ B″ ⊂ B′ ⊂ B; it is assumed that f(x) is any function in D1 and that f(x) and g(x) are both continuous on the left and possibly with simultaneous points of discontinuity.

In the solution of boundary value problems in mathematical physics by means of integral transforms, we often find that the solution of a particular problem can be expressed in terms of integrals of the type

where r and z are positive, and v and n are integers satisfying the convergencecondition v+n>−1.