The object of this note is to draw attention to a family of eight uniform compounds of regular polyhedra which arise out of an action of the group on the regular icosahedron.

Let A1 denote the algebra generated by one pair of canonical operators. It is shown that the realizations of sl(2) in A1 divide into two non-empty classes determined by the spectrum of the semisimple element of sl(2) in A1.

Let S be a finite p-group and let σ be a p'-automorphism of S. A well-known result of Huppert ((2), IV, (5·12), Satz) states that if σ acts trivially onΩ1(S) (ω2(S) if p = 2) then σ is the identity. In this paper, we give a short proof of this result and also give some applications of it. Our proof depends on a somewhat surprising Lemma (Lemma 1) which is of some interest in its own right.

The modular group, Γ, is the group of linear fractional transformations with integral coefficients and determinant 1. The group is generated by the transformations

which have orders 2 and 3 respectively. The transformation

is of infinite order. Abstractly, the group can be viewed as the free product of cyclic groups of order 2 and 3.

A Petrie matrix (Kendall (1), (2)) is a matrix of O's and l's such that the l's in each column occur consecutively. An example follows.

The following Conjecture was made in (6).

CONJECTURE A. Let Xn be a closed manifold of type K(π, 1). Then X˜n is homeornorphic to.

In section 4, we prove the conjecture when n ≥ 5 and π is a non-trivial direct product. More general than Conjecture A would be

CONJECTURE B. Let Xn be a manifold of type K(π, 1) with ∂X = Ø. Then X˜n is homeomorphic to.

In section 4 we give an example to show that Conjecture B is false in each dimension n ≥ 4.

Kirby and Siebenmann (in (7)) proved the annulus conjecture in dimensions five and above, by proving the stable homeomorphism conjecture in those dimensions, that is, by proving that every orientation-preserving homeomorphism of Rn is stable, if n ≥ 5. Although this result apparently lessens the importance of stable homeomorphisms, the concept of stability should still be of use in settling the fate of the annulus conjecture in dimension 4, or for uncovering a simple (i.e. non-surgical) proof in other dimensions.

All topological spaces considered will be completely regular Hausdorff spaces. The word space will be used to refer to such a topological space.

A theorem of Besicovitch, namely that, assuming the continuum hypothesis, there exists in any uncountable complete separable metric space a set of cardinality the continuum all of whose Hausdorif h-measures are zero, is here deduced by appeal to Martin's Axiom. It is also shown that for measures λ of Hausdorff type the union of fewer than 2ℵ0 sets of λ-measure zero is also of λ-measure zero; furthermore, the union of fewer than 2ℵ0 λ-measurable sets is λ-measurable.

Classes of operators called uniformly bounded projections defined on Banach spaces have been formally introduced (see for example (2), 8·9·28; 8·9·92), and their connexions with semi-groups of operators considered. Notable cases of such operators include the Dirichlet integrals which are connected with the semi-group of Poisson operators in Lp In the main conclusions of this paper, there is an exposition of the connexion between classes of uniformly bounded projections and semi-groups of operators. In particular, it is shown that certain uniformly bounded analytic semi-groups of operators give rise to classes of uniformly bounded projections. Various of the known special cases in Lp are also considered as Fourier trans form multiplier operators with characteristic functions of intervals as multipliers.

Let X1, X2, …, Xn, be n (n ≥ 2) independent observations on a one-dimensional random variable X with distribution function F. Let

be the sample mean and

be the sample variance. In 1925, Fisher (2) showed that if the distribution function F is normal then and S2 are stochastically independent. This property was used to derive the student's t-distribution which has played a very important role in statistics. In 1936, Geary(3) proved that the independence of and S2 is a sufficient condition for F to be a normal distribution under the assumption that F has moments of all order. Later, Lukacs (14) proved this result assuming only the existence of the second moment of F. The assumption of the existence of moments of F was subsequently dropped in the proofs given by Kawata and Sakamoto (7) and by Zinger (27). Thus the independence of and S2 is a characterizing property of the normal distribution.

1. The oscillator group. The one-dimensional oscillator group was studied by Streater (7) when he was comparing different methods of calculating representations. It is interesting to generalize this work and consider the group corresponding to the n-dimensional anisotropic oscillator with frequencies ωj In fact, the ideas can be generalized to even larger groups and, in the process of considering their representa tions, several points emerge connected with the harmonic oscillator degeneracies.

The problem of whether there exist static solutions of Einstein's equations for two extended bodies surrounded by vacuum is investigated. Equilibrium conditions for the general static situation are derived and their physical meaning discussed. Special cases of the problem are solved by using these equilibrium conditions. In one of these cases, the essential assumptions are that the space time is axisymmetric and that the matter of which the two bodies are composed consists of perfect fluid. Solu tions are also obtained in which these restrictions are weakened.

A general scheme for the derivation of wave solutions in general relativity is developed. Some solutions describing the flow of gravitational waves are discussed. Singular electromagnetic fields corresponding to one particular solution are also discussed.

The flow from a line source to a line sink in a multiply connected region is discussed. It is shown that the transport from source to sink passes through the interior region and the Stewartson layers as well as through the Ekman layers.

The mass transport velocity field is determined for surface waves which propagate from a region with a clean free surface into a region beneath an inextensible surface film. The waves are assumed to be incident normally on the edge of the film. Determination of this velocity field requires the investigation of a mixed boundary value problem for the bi-harmonic equation, the solution of which is obtained using the Wiener–Hopf technique. Streamlines for the mean motion of the fluid particles are thus obtained. It is found that considerable vertical displacement of fluid is possible due to the presence of the surface film.

A uniformly valid asymptotic solution for waves on a shearing current is presented. This solution is more general than the previous WKB solutions because it is valid at and near a caustic. Far from the caustic line, the solution reduces to the WKB solution. It predicts that an incident wave will be reflected from a straight caustic with the same amplitude and a ½π phase shift.

The efficacy of making a careful choice of strain measure in problems of solid mechanics was recognized and fully exploited by Hill(1–4). He brought coherence to fundamental stress and strain analysis by the introduction of the idea of conjugate pairs of stress and strain measures, and provided a platform for the further development of constitutive relations for solid materials.