The first result of this paper was proved in January 1975 in order to engage the interest of Professor J. E. Littlewood, who was in hospital at the time. This theorem is of some independent interest and we present it here with some related results.

It has long been known, after Wiener (e.g. see (11), vol. 1, p. 108, (5), (8), §5·6)) that a measure μ whose Fourier transform vanishes at infinity is continuous, and generally, that μ is continuous if and only if is small ‘on the average’. Baker (1) has pursued this theme and obtained concise necessary and sufficient conditions for the continuity of μ, again expressed in terms of the rate of decrease of . On the other hand, for continuous μ, Rudin (9) points out the difficulty in obtaining criteria based solely on the asymptotic behaviour of by which one may determine whether μ has a singular component. The object of this paper is to show further that any such criteria must be complicated indeed. We shall show that the absolutely continuous measures on T = [0, 2π) whose Fourier transforms are the most well-behaved (namely, those of the form (1/2π)f(x)dx, where f has an absolutely convergent Fourier series) are such that one may modify their transforms on ‘large’ subsets of Z so that they become the transforms of singular continuous measures. Moreover, the singular continuous measures in question may be chosen so that their Fourier transforms do not vanish at infinity.

Let G be a finite group and n a natural number. Set Hn(G) = ≺g ∈ G|gn ≠ 1≻. The aim of this paper is to prove the following result.

In (1), Burgoyne and Fong have shown that the Schur multiplier of the Mathieu group M12 is of order 2. It is shown in Theorem 2·4 that the Schur multiplier of Aut M12, the automorphism group of M12, is also of order 2. It is therefore possible to choose a complex 2-cocycle α of Aut M12, taking only the values 1 and − 1, such that the cohomology class of α is of order 2 and the cohomology class of the restriction of α to M12 is of order 2. The characters of the irreducible α-projective representations of Aut M12 are calculated in § 2.

A unital C*-algebra A is said to have unitary 1-stable range (8) if for all pairs a, b of elements of A satisfying aA + bA = A there exists a unitary u in A such that a + bu is invertible. This concept is somewhat stronger than the usual stable range condition of algebraic K-theory ((3), chapter V). Handelman(8) shows among other things that finite AW*-algebras have unitary 1-stable range and uses this fact to study the algebraic K1 of a finite AW*-algebra. We prove below that a unital C*-algebra has unitary 1-stable range if and only if its group of invertible elements is dense. In addition we give some consequences of this fact and consider the related question of (unitary) polar decomposition in C*-algebras.

The aim of this paper is to generalize a result of Carter and Lusztig(1) proving the existence of certain non-zero homomorphisms between Weyl modules for the special linear group SLn(K) over an infinite field of characteristic p, and between Specht modules for the symmetric group Sr over K.

In (4) the classification of (complex, projective) cubic surfaces by the number and nature of their singularities is carried out. This gives a natural partition of the vector space of cubic surfaces (which we denote by H3(4, 1)). In this paper we investigate the differential geometric properties of this partition; we show that it provides a finite constructible stratification of H3(4,1) which, in the notation of (10), is Whitney (A) regular. In fact Whitney (B) regularity holds over each stratum other than E6, but this stratum of cubic cones has an exceptional (equianharmonic) orbit at which (B) regularity fails. It remains to be seen whether or not this is the only exceptional orbit.

Every closed connected orientable three-manifold can be constructed by surgery along an appropriately chosen link in the three-sphere, S3 ((10), (15)). Such a link-surgery description is never unique, but the equivalence between different descriptions has been explicitly identified ((8),(3)). However, the situation with regard to manifolds that can be obtained by surgery along a knot in S3 is less clear ((6), p. 47): some manifolds are known to have at least two knot-surgery descriptions ((12), (14), p. 270, (1), (11)), whilst certain manifolds (e.g. S3 and S2 × S1) are widely suspected to have just one (see problems 1·15, 1·16 of (9)).

In this paper we prove some results about the stable homotopy and cohomotopy of spaces related to the infinite real protective space RP∞. These include M. E. Mahowald's conjecture on the limit of stable homotopy of stunted real projective spaces RP2N+m/RP2N−m as N, m → ∞, G. Segal's Burnside ring conjecture for

and the stable analogue of a conjecture of D. Sullivan on RP∞.

The first-named author has proved interesting results about the stable homotopy and cohomotopy of spaces related to real projective space RP∞; these are presented in an accompanying paper (6). His proof is by the Adams spectral sequence, and so depends on the calculation of certain Ext groups. The object of this paper is to prove the required result about Ext groups. The proof to be given is not Lin's original proof, which involved substantial calculation; it follows an idea of the second and third authors. The version to be given incorporates modifications suggested later by the fourth author.

Let X and Y be normed spaces and let L(X, Y) denote the set of linear transformations from X into Y, with domain D(T) and range R(T). For a given T ∈ L(X, Y) we investigate the existence and properties of a closed densely defined operator S ∈ L(Y, X) such that ST − I/D(T) (or TS − I/D(S)) is a bounded operator of finite dimensional range. These results were previously announced without proof in (2).

Classical differential geometry defines two symmetrical fundamental forms, the first dealing with length measurement, the second essentially with curvature. In applications, particularly in the mechanics of plates and shells, certain unsymmetrical features of the fundamental forms are of interest and are here investigated.

The purpose of this paper is to prove a theorem giving conditions yielding global bifurcation of the solutions of a family of parameterized nonlinear equations, the domain and the range of which lie in Banach spaces, where the parameter is allowed to be a vector in , k a positive integer. The basic contribution is that the parameter is vector valued and that the nonlinearities allowed are very general; however, even for scalar parameters, our results extend those of previous authors.

In this paper we investigate the connection between strong ellipticity and the regularity of weak solutions to the equations of nonlinear elastostatics and other nonlinear systems arising from the calculus of variations. The main mathematical tool is a new characterization of continuously differentiable strictly convex functions. We first describe this characterization, and then explain how it can be applied to the calculus of variations and to elastostatics.

In a recent paper, hereafter referred to as I (1) we derived two alternate forms for the fundamental solution of the axially symmetric wave equation. We demonstrated that for α > 0, the fundamental solution (the so-called free space Green's function) of the partial differential equation

could be written as

if b > r

or

if r > b.

In (5), Schmidt devised a procedure for completing a space-time M by adjoining a boundary ∂M called the b-boundary. Bosshard(1) and Johnson(4) have shown that certain Friedmann and Schwarzschild space-times have non-Hausdorff b-completions.

In (2), (3), the first author considered a modification of the b-completion which gives a Hausdorff completion in the Friedmann case. The modification used a parallelization of the space-time and can be given in terms of the linear connexion determined by the parallelization. We show that the completion obtained is the Cauchy completion in a particular Riemannian metric on the manifold and so is always Hausdorff.

For a manifold with a generic symmetric affine connection, explicit necessary and sufficient conditions for the local existence of metric compatible with the connection are obtained in terms of the Riemann tensor and its first-order covariant derivatives. If these conditions are satisfied, the solutions for metric are unique up to a constant scale factor and the absolute value of the signature is uniquely determined. Explicit formulae for the solutions are given in terms of integrals.

On behalf of the authors and the Press, the editor has to notify the following amendments.

Volume 48 (1952), 72–86 and 519–20

J. W. S. Cassels. ‘The inhomogeneous minimum of binary quadratic, ternary cubic and quaternary quartic forms.’

Professor H. W. Lenstra Jr has pointed out that there is a mistake in the proof of Theorem 6, and that the constant 5300 must be replaced by the worse constant 16730. In (iii) of Lemma 16, k should be replaced by k2 and consequently k¼ by k½ in the display at the bottom of page 85. The new estimate results then from putting k = (2·35)2.

Volume 87, part 3, pages 459–69

W. H. Lin, D. M. Davis, M. E. Mahowald and J. F. Adams. ‘Calculation of Lin's Ext groups.’

The diagram (1·5) at the foot of page 460 should be interchanged with that on page 462.