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.

If X is a matrix with non-negative entries then X′X is positive semi-definite with non-negative entries. Conversely, if A is positive semi-definite then there exist matrices Y, not necessarily with non-negative entries, such that Y′Y = A. In the present paper we investigate whether, given a positive semidefinite matrix A with non-negative entries, the equation X′X = A has a solution X with non-negative entries. An equivalent statement of the problem is: Can a positive semi-definite matrix with non-negative entries be expressed as a sum of rank 1 positive semi-definite matrices with non-negative entries? We answer the question in the affirmative for n≦4 and quote the following example due to M.

When a shock wave moving through an isentropic ideal gas catches up with, and passes into a simple expansion wave, the shock decays. Because of this the gas will not be isentropic in the region behind the shock. The problem of determining the motion of the gas in this region is as yet unsolved. In this paper we introduce a simple compression wave behind the shock which catches up with it at the instant of its entry into the leading expansion wave. This second wave is chosen so as to counteract the decaying effect of the first, and keep the shock strength constant throughout the motion. We assume the first wave to be point-centred, and caused by the withdrawal of a piston at a finite velocity from a gas at rest in a shock tube. After a finite time the piston is halted causing the shock. The problem is then to determine the subsequent motion of the piston to produce a compression wave with the desired property.

In the series of papers in the early forties summarised in (1), (2), A. I. Lur'e showed how to utilise Liapunov's second or direct method in the investigation of the stability of linear automatic control systems with a single nonlinear actuator. His approach consists of

1. the transformation of the original system of differential equations via the so-called Lur'e transformation into canonical coordinates in which the construction of the Liapunov function is direct, and

2. the conversion of the differential problem into a purely algebraic problem.

We will be concerned here with the questions of the existence and construction of the Lur's transformation.

The theorem which I propose to establish first attracted my attention while I was turning over the pages of a volume of Cayley's Collected Mathematical Papers (Cayley, 1). The enunciation of the theorem (with no attempt towards a proof) had been published earlier by Kirkman (3) in a lengthy paper on combinatorial analysis (one of the three-score papers of which Kirkman was the author); among the topics discussed in this paper was the enumeration of the total number of different ways D(r, k) in which a (convex) polygon of r sides can be dissected into k+l parts by drawing k non-intersecting diagonals (i.e., no two diagonals may cross each other except at a vertex or outside the polygon).

If letters a, b, c, … are used to denote points of a nondegenerate plane cubic curve, other than the singular point if any, and if the product ab is defined as the third point of the curve collinear with a and b, we obtain an algebraic system having nonassociative multiplication (ab . c ≠ a . bc in general). It is in fact a totally symmetric entropic quasigroup (these terms are defined below). This idea, which was put forward at a meeting of the Edinburgh Mathematical Society a few years ago, will be exploited in a forthcoming paper. Such quasigroups have many properties which can be interpreted geometrically. Or, conversely, known properties of cubic curves suggest theorems about totally symmetric entropic quasigroups, which if established will involve a gain in generality, since not every such quasigroup can be “placed” on a cubic.

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 m and n are integers satisfying the convergence condition m+n>−2.

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

Jacobsthal (4)has proved that the n×n matrix

is invertible with the inverse,

Here μ(x) denotes the Möbius function for positive integral x and is assumed to be 0 for other values; [x] has its usual meaning as the number of positive integers ≦x.

Given a partial automorphism of a group G, i.e. an isomorphic mapping μ of a subgroup A of G onto a second subgroup B of G, it is known (2, Theorem I) that there always exists a group H containing G and an inner automorphism of H which extends µ; i.e. there exists an element t of H, such that the transform by t of any element of A is its image under µ.

Oldroyd (1) considered non-Newtonian liquids for which the stress tensor sik and the rate of strain tensor eik=½(υk,i+υi,k)are related as follows

Here υi denotes the velocity vector, p an isotropic pressure, gik the metric tensor, and t time; η0 is a constant having the dimensions of viscosity and λ1, λ2, μ0, ν1, ν2, are constants having the dimension of time. Covariant suffixes are written below, contravariant above and the usual summation convention is employed. A suffix i following a comma indicates covariant differentiation with respect to the space variable xi.

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.

If µ is a bounded regular Borel measure on a locally compact group G, and L1(G) denotes the class of complex-valued functions which are integrable with respect to the left Haar measure m of G, then, for each f∈L1(G),

defines almost everywhere (a.e.) with respect to m a function μ*f which is again in L1(G). The measure μ will be called isotone on G mapping f→μ*f is isotone, i.e. f≧0 a.e. (m) if and only if μ*f≧0 a.e. (m).

This paper concludes a series of papers (1) on a group of axisymmetric boundary value problems in potential and diffraction theory by considering some potential problems for a circular annulus. The Dirichlet problem for an annulus has recently been considered by Gubenko and Mossakovskiǐ (2), who, by a somewhat complicated method, show it to be governed by either one of two Fredholm integral equations of the second kind. The purpose of the present paper is to show how the method developed in previous papers, by which certain integral representations of the potentials in problems for circular disks arid spherical caps are used to reduce such problems to the solutions of either single Abel integral equations or Abel and Fredholm equations, can be applied to both the Dirichlet and Neumann problems for the annulus to give reasonably straightforward derivations of the governing Fredholm equations.

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.

Throughout this paper, unless otherwise stated, n and L stand for positive integers and α, t, x, x1, x2, … for positive real numbers. Let

where

and

By a partial endomorphism of a group G we mean a homomorphic mapping μ of a subgroup A of G onto a subgroup B of G. If μ is denned on the whole of G then it is called a total endomorphism. We call a partial endomorphism totally extendable (or extendable) if there exists a supergroup G*⊇G with a total endomorphism μ* which extends μ in the sense that gμ* = gμ, whenever the right-hand side is defined (3).

A particular solution of the equations of one-dimensional anisentropic flow of a polytropic gas is linked by a shock to gas at rest in which the density is non-uniform. The approach is inverse in that the density distribution is derived from the position of the shock and the prescribed flow behind it. The velocity and strength of the shock each vary with time. The result is an example of the propagation of a shock through an inhomogeneous gas.

In (1) I obtained † an asymptotic formula for the number of zeros of an arbitrary canonical product II(z) of integral order but not of mean type, all of whose zeros lie on a single radius, from a knowledge of the asymptotic behaviour of (i)log | П(z)| as | z | = r→ ∞ along another radius l, with certain side conditions. After proving the analogous theorem in which log | П(z)| in (i) is replaced by , I show in this note that, at a cost of replacing l by two radii l1 and l2, both of these theorems may be generalised to include a class of canonical products of integral order whose zeros lie along a whole line. In one of the resulting theorems ‡ (Theorem II) I find the asymptotic number of zeros on each half of the line of zeros; another theorem (Theorem III) includes a previous result of mine.§

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.