Following Robinson(4), we shall use the term Abelian1-group to describe an Abelian group A of finite (Mal'cev special or Prüfer) rank, whose torsion subgroup T is Černikov. These groups were called Abelian groups of type A3 by Mal'cev (3). Let G = Aut A. It is not hard to see that G/CG(A/T) and G/CG(T) are linear groups, and CG(A/T) ∩ CG(T) is Abelian. We improve on this observation by proving

Theorem 1. The group G contains a normal Abelian Černikov subgroup G0, such that G/G0is linear over a field of characteristic zero.

The Frattini subgroup φ(G) of a group G is the intersection of G and all its maximal subgroups. The following results for finite groups are well known:

THEOREM A0. If G is a finite group, then the following three conditions are equivalent:

(i) G is nilpotent,

(ii) G/φ(G) is nilpotent,

(iii) φ(G) ≥ G′.

This paper is a continuation of (2), (3) in the development of a unified theory of the characters of the Weyl groups of the simple Lie algebras using their common structure as reflection groups; compare Carter (1) for a similar development for the conjugacy classes. We look at the Weyl group of type D, which is a subgroup of index two in the Weyl group of type C. It was first studied by Young (4), but rather less is known about the characters of this group than those of types A and C. Indeed, the situation is rather more complicated, but we are able to give, as before, an algorithm to determine irreducible constituents of the principal character of a Weyl subgroup induced up to the whole group. We shall also study the case where the rank of the Weyl group is even, when extra irreducible characters may arise, and after constructing these, we shall state some results on their occurrence in the induced principal character.

Let A, B be real symmetric n × n matrices having orthogonal reductions

where ΛA = diag (λ1(A), …, λn(A)) and ΛB = diag (λ1(B), …, λn(B)) and the sets of latent roots are each in descending order. (Thus we may say that the latent vectors of A, B are ordered with respect to respectively.) Let the multiplicities of the roots and in order of occurrence be respectively and , so that

Mahler (5) proved in 1957 that for any rational a/q, where a, q are relatively prime integers with a > q ≥ 2, and any ε > 0, there exist only finitely many positive integers n such that ∥(a/q)n∥ < e−εn; here ∥x∥ denotes the distance of x from the nearest integer taken positively. In particular there exist only finitely many n such that

and, as Mahler observed, this implies that the number g(k) occurring in Waring's problem is given by

for all but a finite number of values of k. It would plainly be of interest to establish a bound for the exceptional k and this would follow from an upper estimate for the integers n for which (1) holds. But Mahler's work was based on Ridout's generalization of Roth's theorem and, as is well known, the latter result is ineffective.

When k is a 2-bridge knot with group πK, there are parabolic representations (p-reps) θ: πK → PSL(): = PSL(2, ). The most obvious problem that this suggests is the determination of a presentation for an image group πKθ. We shall settle the easiest outstanding case in section 2 below, viz. k the figure-eight knot 41, which has the 2-bridge normal form (5, 3). We shall prove that the (two equivalent) p-reps θ for this knot are isomorphisms of πK on πKθ. Furthermore, the universal covering space of S3\k can be realized as Poincaré's upper half space 3, and πKθ is a group of hyperbolic isometries of 3 which is also the deck transformation group of the covering 3 → S3\k. The group πKθ is a subgroup of two closely related groups that we study in section 3. We shall give fundamental domains, presentations, and other information for all these groups.

1. Wallace's theorem, that the circumcircles of the four triangles formed by the sets of three from four coplanar lines meet in a point, has a generalization to Euclidean affine space of any even dimension n: namely, if we have n + 2 general primes (or hyperplanes) in such a space, the n + 2 circumhyperspheres of the simplexes formed by the sets of n + 1 of the primes meet in a point.

In his work on elementary catastrophes Zeeman(1) has considered what he has named as the double cusp catastrophe. This catastrophe is defined by the unfolding of the two variable polynomial x4 + y4. Using Mather's results (2) on stability of singular germs of C∞ maps we can find an expression for the unfolding. The eight dimensional unfolding can then be considered as a polynomial in two variables with eight parameters.

Let ωn denote a random graph with vertex set {1, 2, …, n}, such that each edge is present with a prescribed probability p, independently of the presence or absence of any other edges. We show that the number of vertices in the largest complete subgraph of ωn is, with probability one,

A generalized metric space is a triple (X, d, G), where X is a set, G an ordered Abelian group and d: X x X → G is a function which satisfies the metric axioms. The generalized metric d defines a topology on X in the same way as an ordinary metric does for a metric space. These spaces are studied by Stevenson & Ton(5) and Sikorski (4) amongst others. A survey of these spaces is the subject of an M.Phil. thesis by the author (6). Generalized metric spaces satisfy most of the separation properties of ordinary metric spaces (T2, regular, etc.). We prove here that these spaces are also paracompact. The proof is a generalization of the proof given by Ruffin (2) for metric spaces. The result given here follows, in fact, from a theorem proved by Hayes (1). He proved that uniformities with totally ordered bases have paracompact topologies. It is known (Stevenson & Ton (5); Shock (3); Souppouris (6), chapter 5) that these spaces are identical to generalized metric spaces. The proof given here uses metric space language.

There are several known results concerning Radonifying and summing properties of linear mappings in terms of their transposed operators. Thus Schwartz has shown that a sufficient condition for a mapping to be p-Radonifying is that its transposed mapping should be p-decomposing, and that this condition is also necessary if the image space has good topological properties. But as Schwartz observes, these properties are seldom observed by biduals of Banach spaces with their weak-topologies, which arise when 0 ≤ p ≤ 1. This suggests that it would be worth looking for a weaker condition than ‘p-decomposing’.

Let f be a real-valued function of two real variables. It is well known(5) that the Lebesgue measurability or even upper semicontinuity of all the sections fx and fy does not imply the measurability of f. Lipiński proved (3) that there exists a non-measurable function f such that each section fx and fy is Baire one, has the Darboux property, and fails to be approximately continuous at no more than one point. Davies (1) noted that the continuum hypothesis implies the existence of a non-measurable function f, such that each fy is measurable and each fx approximately continuous. Ursell (6) proved that if each fy is measurable and each fx continuous (or monotonic), then f is measurable.

Let l be the differential operator of order 2n defined by

where the coefficients are real continuous functions and pn > 0. The formally self-adjoint operator l determines a minimal closed symmetric linear operator L0 in the Hilbert space L2 (0, ∞) with domain dense in L2 (0, ∞) ((4), § 17). The operator L0 has a self-adjoint extension L which is not unique, but all such L have the same continuous spectrum ((4), § 19·4). We are concerned here with conditions on the pi which will imply that the spectrum of such an L is bounded below and discrete.

We shall be concerned with the probability distributions which arise from naive coin tossing experiments and their repetitions. In each case our main question is whether the resulting measure is singular with respect to Lebesgue measure or absolutely continuous. It is a remarkable fact that althoughthese distributions were defined and studied some forty years ago (see (17)for an interesting account), they already provide examples necessary to recent studies of the convolution measure algebra of a locally compact abeliangroup. In order to verify this one is obliged to make substantial improvements on the classical results but, at least at the technical level, this requires no modern apparatus. Accordingly, we shall restrict attention to the circle. Before doing this, however, we summarize the motivational background from abstract harmonic analysis and indicate the applications of our results to that area.

1. In this paper we shall use the methods of (6) to give a short proof of Wunderlich's result (7, 8) on the distribution of twin ‘primes’ for the Hawkins random sieve. We shall also obtain a sharp estimate on the gaps between successive ‘primes’.

In a recent paper(14), Talenti gave a complete characterization of the spectrum of the maximal realization Hop of − Δ in Lp(Rn), for p in the range

Suppose that an elastic half-space, which contains certain surface defects, inclusions and cavities, is in free, two-dimensional, time-harmonic vibration, with the wave field at infinity ‘outgoing’ in character. It is shown that the elastic potentials representing such a ‘standing mode’ can be expressed in the form of contour integrals, for instance

U(t) being an analytic function of t. By considering the far field of these potentials, it is shown that U(t) is zero on a certain arc in the t-plane and is therefore identically zero. It follows that ø(r) is zero everywhere and this proves the non-existence of such standing modes in these configurations.

This uniqueness theorem justifies the solution given by the author (Gregory (2)) for the problem in which time harmonic stresses act on the walls of a cylindrical cavity lying beneath the surface of an elastic half-space. It is also shown that if a Rayleigh surface wave is incident on any system of surface defects, inclusions and cavities, then energy must be transferred from the surface wave to scattered outgoing body waves of both P and S types.

It is shown that, at an arbitrary point in a general elastic material and regardless of the prevailing state of deformation, there exists in every plane at least one direction in which a transverse acceleration wave may propagate. The possibility of a transverse wave actually being able to progress in such a specific direction depends upon the associated local wave speed being real and conditions are obtained under which this requirement is satisfied.

The basic state considered in this paper is a parallel flow of a jet-like character with the centre of the jet being at or near a free surface which is horizontal. Stationary surface gravity waves may exist on such a flow, and a number of examples are looked at for small amplitude waves. Explicit solutions are given for ‘top-hat’ profile jets and for two-dimensional flows. Asymptotic solutions are developed for stationary waves of large wave-number.

A β-plane ocean is considered, and dissipative processes are modelled by using a coefficient of lateral friction. Attention is focused on the singularity which occurs if the angle that the western boundary makes with the north approaches ½π. The dimensions of the singular region are found using boundary layer techniques, and the solution for the transport stream function valid within it is obtained in closed form.