1. Introduction. All spaces in this paper are Hausdorff. We recall that a space X is sequentially compact, if every countable subset of X contains a convergent sequence. Let us consider the three statements:

(1) Every compact space of cardinality ≤ ʗ contains a point of countable character.

(2) Every compact space of cardinality ≤ ʗ is sequentially compact.

(3) Every infinite compact space of cardinality ≤ ʗ contains a convergent sequence.

Recently, Good discovered the following curious identity (1). Let θ be a real-valued random variable having cumulants Km ≡ Km(θ) (m = 1, 2, …). Take a set of m statistically independent variables θ(1), θ(2), …, θ(m), each with the same distribution as θ, and form the random variable

where ω is any primitive mth root of unity (e.g. ω = e2πi/m). Then

where 〈.〉 denotes expectation.

In this paper, which assumes a familiarity with (1), we produce a short proof of the following theorem. The proof uses MOG techniques to extend a powerful observation of Conway (2) on intransitive subgroups of M24.

In this paper, the duality theory for vector lattices developed by the author in (2) is applied to the study of vector lattices freely generated by distributive lattices.

If L is a finite distributive lattice defined by a set of generators X, subject to a finite set of relations R, then the vector lattice FVL(L) freely generated by the elements of L subject to the relations governing intersection and union in L is a finitely generated projective vector lattice, being a quotient of the vector lattice freely generated by elements of X by the principal ideal generated by the relations R. Thus, there is a polyhedron Q, such that FVL(L) is isomorphic with the vector lattice of real valued polyhedral functions on Q, under pointwise operations, and the isomorphism class of FVL(L) determines and is determined by the polyhedral type of Q. The principal result of this paper shows that the polyhedral class to which Q belongs can be simply characterised in terms of the nerve of the poset P of join-irreducible elements of L. Explicitly, up to polyhedral equivalence, Q is a suspension of the nerve of P.

1. Introduction. It is well known that every finite-dimensional irreducible representation of a nilpotent group over an algebraically closed field is monomial, that is induced from a 1-dimensional representation of some subgroup. However, even a finitely generated nilpotent group in general has infinite-dimensional irreducible representations, and as a first step towards an understanding of these one wants to discover whether they too are necessarily monomial. The main point of this note is to show how far they can fail to be so.

We prove the following theorem:

Theorem A. Let G be a finite simple group of order divisible by 3. Suppose

(i) G contains no element of order 6;

(ii) G has cyclic Sylow 3-subgroups;

(iii) G does not involve A4;

(iv) Every 3′-simple section of G is a Suzuki group.

Then G ≅ SL(2, 2n) for some odd n ≥ 3.

1. Introduction. We introduce the notion of a weakly prime compact convex set, and we develop a reduction theory for spaces A(K). The notion is less restrictive in general than the prime compact convex sets of Chu, but gives a finer reduction than the Bishop and Silov decompositions forA(K) (12). The natural analogue for uniform algebras is related to the concept of weakly analytic sets due to Arenson, but unlike maximal weakly analytic sets the maximal weakly prime sets are always generalized peak sets; the uniform algebra can always be retrieved from the restrictions of the algebra to the maximal weakly prime sets.

We investigate here the question of uniqueness of best approximation to operators in von Neumann algebras by elements of certain linear subspaces. Recall that a linear subspace V of a Banach space X is called a Chebyshev subspace if each vector in X has a unique best approximation by vectors in V. Our first main result characterizes the one-dimensional Chebyshev subspaces of a von Neumann algebra. This may be regarded as a generalization of a result of Stampfli [(4), theorem 2, corollary] which states that the scalar multiples of the identity operator form a Chebyshev subspace. Alternatively it may be regarded as a generalization of the commutative situation in which a continuous complex-valued function f on a compact Hausdorff space X spans a Chebyshev subspace of C(X) if and only if f does not vanish on X [(3), p. 215]. Our second main result is that a finite dimensional * subalgebra, of dimension > 1, of an infinite dimensional von Neumann algebra cannot be a Chebyshev subspace. This imposes limits to further generalization of Stampfli's result.

In section 3 we shall prove the following results: Let G be a separable locally compact abelian group, R a von Neumann algebra acting on a separable Hilbert space, and α a weakly continuous representation of G by inner *-automorphisms of R, say α(g) = ad Wg with Wg ∈ U(R). Then there is a weakly continuous unitary representation of G, by unitaries in R, implementing α if and only if the Wg's commute with each other. The result was motivated by the proof of (7), theorem 1. Suppose now Gis a discrete amenable group of *-automorphisms of a countably decomposable von Neumann algebra R. In section 3 we give a necessary and sufficient condition for the existence of a faithful normal G-invariant state on R. This generalizes a result of Hajian and Kakutani on invariant measures (2).

Let A be a *-algebra of operators on a Hilbert space H, and let Ah, A+ denote respectively the sets of self-adjoint and positive operators in A. A+ is a positive cone in Ah and it induces a partial ordering in Ah. The lattice properties of Ah were studied by R. Archbold in (1) and (2), and Chu Cho-Ho gave a different proof of some of his results in (3).

Let µ be a probability measure on the Borel subsets of R∞. If D is a countable subgroup of R∞ we say that µ is D-ergodic if (1) for any D invariant Borel subset A of R we have µ(A) = 0 or 1 and (2) if µ*δx ≈ µ for all x ∈ D (where δx stands for unit mass at x while the equivalence relation ≈ signifies that the two measures have the same null sets.) We say that x is an admissible translate for µ if µ*δx ≈ µ. We say that µ is D-smooth if sx is an admissible translate for µ for all x ∈ D and all s ∈ R. We say that µ is a smooth ergodic measure if µ is D-ergodic and D-smooth for some countable subgroup D as above. In this paper we show that any two smooth ergodic probability measures µl, µ2 are either equivalent or singular (where the latter means that there exist disjoint Borel sets Al, A2 ⊂ R∞ such that µi(Ai) = 1 and is signified by µ1 ┴ µ2). It is important to note that the countable subgroup D1 associated with µl need not be the same as the subgroup D2 associated with µ2.

In this paper we continue the study of Orlicz sequence spaces initiated by Lindberg (5) and Lindenstrauss and Tzafriri (7), (8) and (9). Our main concern is to investigate features of the theory which occur when the restriction of local convexity is lifted. It is clear that some results will hold with identical proofs, at least when the space is locally bounded. However, we are chiefly interested in the differences which arise. We always assume that the Orlicz function F satisfies the Δ2-condition.

It is shown that the differentiability of a space-time is implied by that of its Riemann tensor, assuming a priori only boundedness of the first derivations of the metric. Consequently all the results on space-time singularities proved in earlier papers by the author hold true in C2− space-times.

1. Introduction. In studying the elastic behaviour of inhomogeneous systems certain inclusion and inhomogeneity problems are fundamental. In the ‘transformation problem’, a region (the ‘inclusion’) of an unbounded homogeneous anisotropic elastic medium would undergo some prescribed infinitesimal uniform strain (because of some spontaneous change in its shape) if it were not for the constraint imposed by the surrounding matrix. When the inclusion has an ellipsoidal shape, Eshelby (3, 4) was able to show that the stress and strain fields within the constrained inclusion are uniform and that calculations could be completed when the medium was isotropic. A generally anisotropic medium seemed to raise forbidding analyses, but Eshelby (3) did point the way to an evaluation of the uniform strain which several authors (referred to later) developed into an expression amenable to numerical computation. Here we offer an elementary and immediate route to this expression of the uniform strain, which has been accessible hitherto only by the circuitous procedures of Fourier transforms. It is available as soon as the uniform state of strain in the inclusion is perceived and before an alternative evaluation is commenced. First, we appeal to a theorem (not it seems previously known) which reveals (in particular) the vanishing of the mean strain in the infinitesimally thin ellipsoidal homoeoid lying just outside the inclusion. Secondly, we need only reflect that at each point of the interface there is an immediate algebraic expression of the strain just outside the inclusion in terms of the uniform strain just inside.

In this paper we investigate a time harmonic plane strain problem in a homogeneous isotropic semi-infinite elastic solid with mixed boundary conditions on its surface. The boundary is free from applied stress apart from on an infinite strip where the tangential components of stress are zero, and the normal component of displacement is a prescribed function. By reducing the mixed boundary value problem to the solution of a Fredholm integral equation of the second kind, an existence theorem is proved and a simple low frequency asymptotic formula relating the normal stress to the prescribed displacement is derived rigorously. The results are applied to the forced heaving and rocking motion of a rigid cylindrical body in smooth contact with an elastic half space. Some further applications to scattering problems are suggested.

For an elastic solid the constitutive law can be written in terms of the deformation gradient α and its conjugate nominal stress s ≡ s(α), and also in terms of the right stretch u and its conjugate stress τ ≡ τ(u). It is shown that for isotropic elastic solids s(α) is invertible, in the local sense, for all u in the domain of u-space where τ(u) is locally invertible, with the exception of certain configurations which correspond to planes in τ-space.In the global sense a given s corresponds to four distinct τ's, and s is invertible to give four distinct α's when the corresponding τ's are uniquely invertible. That there are four branches of the inversion α(s) is of fundamental importance in that it clarifiesthe extent to which non-uniqueness of solution of boundary-value problems can be expected.

The implications of these results in respect of the complementary variational principle are discussed, and the controversy surrounding the use of nominal stress in this principle resolved.

Consequences of the required restrictions on τ(u) are examined and discussed in relation to inequalities which may be regarded as entailing physically reasonable response. It is intimated that τ(u) is invertible in the domain of elastic response.

A laboratory model simulating aspects of oceanic or atmospheric circulationconsists of a rotating fluid annulus which is subject to a temperature gradient on the lower surface and is insulated on the inner and outer curved walls and on the top surface. A theoretical study of the resulting steady axi-symmetric fluid motion in the limit of small Ekman number, E, by Daniels (2) indicated that if an appropriate convective parameter is sufficiently large, oscillatory solutions are generated in the layer which is located on the hotter of the two side-walls, which do not decay into the geostrophic interior. The present paper resolves this difficulty by using the method of multiple scales to trace the development of the oscillatory solutions in the interior of the fluid. This leads to a consistent overall flow pattern throughout the annulus, except at an infinite set of discrete values of the convective parameter at which resonance is predicted.