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 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.

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.

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).

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.

A solution of a triad of integral equations involving Bessel functions is given. This, like earlier ones, is in the form of a pair of Fredholm integral equations, which may be solved by iteration in certain cases. In spite of a slightly more general formulation of the problem, the kernels of these equations are simpler than those given in earlier solutions. Certain extensions are considered and a formal solution given. Application is made to the problem of incompressible inviscid flow normal to an annular disc, and to the flow due to the slow rotation of such a disc in a viscous fluid.

One of the simplest three part boundary value problems is the electrostatic problem for the circular annulus and, at present, there seems to be no method available for obtaining the solution in a closed form. It has recently been shown by the author (1) and Cooke (2) that this problem can be reduced to the solution of a Fredholm integral equation of the second kind. The equation obtained in (1, 2) is fairly simple and is suitable for obtaining a numerical solution but, unfortunately, it cannot be solved iteratively to give a simple form of solution valid for small values of the ratio (inner radius/outer radius).

In the present paper we determine the Laplace transforms of the modified Bessel function of the second kind Kn(t±mx), where m is any positive integer. The Laplace transforms are given in (2) and (4) below, p being the transform parameter and having positive real part.

Let pn denote the nth prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n,

where, for k≧1, logk+1x = log (logk x) and log1x = log x. In a recent paper (4) Schönhage has shown that the constant ⅓ may be replaced by the larger number ½eγ, where γ is Euler's constant; this is achieved by means of a more efficient selection of the prime moduli used. Schönhage uses an estimate of mine for the number B1 of positive integers n≦u that consist entirely of prime factors p≦y, where

Herer x is large and α and δ are positive constants to be chosen suitably.

The purpose of this article is to describe two problems which involve drawing graphs in the plane. We will discuss both complete graphs and complete bicoloured graphs. The complete graphKn with n points or vertices has a line or edge joining every pair of distinct points, as shown in fig. 1 for n = 2, 3, 4, 5, 6.