The following embedding theorem was proved in the paper (7). Let Π be a set of primes and P, H and K periodic Π-groups satisfying

Then there is a period Π-group J generated by isomorphic copies of H and K such that P ≤ J.

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.

The classification of affine cubic functions in the real case is a fairly easy corollary of that in the complex case (9). However as the results can be easily interpreted by diagrams, one can obtain a much richer understanding. For example, the question of which types of cubic curve occur as level curves of which types of function is now much less trivial. This will lead us first to re-examine the classification of cubic curves going back to Newton (4). Next the ‘dynamic’ approach of considering these curves as members of families leads to the diagrams associated with the umbilic catastrophes of Thorn (8). However the consideration of functions rather than of curves gives a 1-dimensional foliation of these diagrams which we describe next. We conclude by placing the results back in a protective setting.

Let π denote a finite group. It is well known that every element of the projective class group K0 ℤπ may be realized as Wall obstruction of a finitely dominated complex with fundamental group π (cf. (13)). We will study two subgroups N0ℤπ and Nℤπ of K0ℤπ, which are closely related to the Wall obstruction of nilpotent spaces. If the group π is nilpotent and if S denotes the set of elements x ∈ K0ℤπ which occur as Wall obstructions of nilpotent spaces, then

It turns out that in many instances one has N0,ℤπ = Nℤπ (cf. Section 3) and one obtains hence new information on S. The main theorem (2·4) provides a systematic way of constructing finitely dominated nilpotent (or even simple) spaces with non-vanishing Wall obstructions.

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 N be an integer ≥ 1. The affine modular curve Y0(N) parameterizes isomorphism classes of pairs (E; F), where E is an elliptic curve defined over ℂ, the field of complex numbers, and F is a cyclic subgroup of order N. The compacti-fication X0(N) is an algebraic curve defined over ℚ.

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.

We show that if {Gi}J ε I is a generating set for an (elementary) topos ℰ then {P(Gi)}iεI is a cogenerating set for x2130;. From this we show that if topos ℰ contains an object G whose subobjects generate ℰ, then ΩG is a cogenerator for ℰ. Let denote the topos of finite sets and functions. We also show that if ℰ1 is a topos and ℰ2 is a bounded -topos then every geometric morphism ℰ1 → ℰ2 is essential.

Any ribbon knot K in S3 can be obtained from the unlink of r + 1 components {U0, U1,…, Ur}, for some positive integer r, by the simultaneous performance of r band operations (fusions), using disjoint bands. These bands may be assumed to run from U0 to U1, U2, …, Ur respectively, and labelled b1, b2…, br accordingly. This paper investigates the question of what conditions on these bands guarantee that the knot K so constructed is non-trivial. Besides being of independent interest (see 1·2 (A) of (2), and the example below), this problem is related to the ‘slice implies ribbon’ conjecture (1·33 of (2)), to the triviality question for 2-spheres in the 4-sphere (section 10 of (4)), and other unknotting questions for surfaces in 4-space (see, for example, 4·30 of (2)). The main result proved here is that K is non-trivial if the union of the bands ‘geometrically links’ (in a sense made precise below) every Ui at most once, and at least one U0 exactly once. The linking of the bands with U0 does not affect the argument, so may be completely arbitrary.

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.

The purpose of this note is to prove the following result.

Theorem 1. let M be a connected matroid. Suppose that C is a circuit of M and p and q are elements of M. Then M has circuits cp and Cq such that .

Work by L.S.Pontrjagin(18) and M.W.Hirsch(7) allows us to identify the stable n-stem with the bordism group of oriented compact closed smooth n-manifolds immersed in ℝn+1. In a recent paper (11), U. Koschorke discusses invariants thereby defined on by analysing the self-intersections of immersed manifolds. In particular he discusses the homomorphism

defined by assigning to a generic immersion Mn → ℝn+1 number (modulo 2) of its (n+ 1)-fold points.

Let Π be a k-dimensional subspace of Rn(n ≥ 2) and let Π┴ denote its orthogonal complement. If x ∈ Rn we shall write x = x0 + x′ with x0 ∈ Πand x′ ∈ Π ┴. If f(x) is a real measurable function on Rn, the k-plane integral F(Π,x′ )is defined as the integral of f over the affine subspace Π + x′ with respect to k-dimensional Lebesgue measure (assuming that the integral exists). If k = 1 we get the x-ray transform that arises in the problem of radiographic reconstruction, and if k = n − 1, the k-plane integral is the usual projection or Radon transform. The paper by Smith, Solmon and Wagner (4) contains a survey of results on k-plane integrals. Here we shall be interested in the behaviour of the F (Π, x′ ) regarded as a function of x′ for fixed Π for various classes of function f. We shall obtain some surprisingly strong results on the continuity and differentiability of F(Π,x′) with respect to x′ for almost all Π (in the sense of the appropriate Haar measure). As will be seen the dimensions n and k have a crucial effect on what may be said, and most of our results will be confined to the cases where k > ½n.

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.

Knuth (12) seems to have been the first to carry out non-trivial matroid operations on a computer. However, as he remarks, there are considerable computational difficulties. In this paper we examine in detail the computational complexity of some fundamental matroid properties. The model of computation is the Hausmann–Korte oracle machine introduced in (10), (11) to deal with properties of independence spaces and fixed point problems. It was these papers (10), (11) and (12) which motivated the present work. Basically, the problems we will be discussing are matroid analogues of low-level complexity problems for graphs. For an excellent survey of this we refer to the recent monograph of Bollobés ((3), chapter 8). We shall use the approach of (3), and Bollobaé;s and Eldridge(4) rather than the more general treatment of Hausmann and Korte (11).

It is shown that the functions which are analytic in tubular radial domains and satisfy certain growth conditions have distributional boundary values in the weak topology of (WΩ)′-space. Representation of analytic functions in terms of distributional boundary values are given. Converse results are also obtained. An analytic decomposition theorem is proved. The main theorems are established by means of a number of lemmas concerning WM, WΩ spaces and their dual spaces. Several new lemmas are proved for K{Mp} spaces from which results for WM-spaces can be easily deduced.

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.

Let J be a complex Banach space and a complex Jordan algebra equipped with an algebra involution *. Then J is a Jordan C*-algebra if the following conditions are satisfied:

(Ua is defined on page 3).

It was recently shown in (1), corollary 1, that if (I) = (N, q) then I and (N, q) were consistent. By I we denote the unit infinite matrix, and (N, q) denotes a general complex Nörlund mean. The summability field (A) of any infinite matrix A = (ank) is defined as

where c is the space of convergent sequences. The summability field of (N, q) is, for simplicity, written as (N, q) instead of ((N, q)).

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.