This paper is a continuation of (F3). In its first part we shall expand and extend the general theory of the earlier paper, while in the second part we specialize to number fields. The theory of resolvents and of the trace form, presented here, complements the more arithmetic theory of module conductors and module resolvents as described elsewhere (cf. (F4)). Both these papers will be applied in work on the connexion, for tame extensions, between Galois module structure of algebraic integers on the one hand, and Artin conductors and root numbers on the other hand (cf. (F5)). The results of the present paper are however not restricted to the tame case and, it is hoped, will subsequently be applied in a more general context.

P. M. Cohn has given a characterization of pseudovaluation on a field in (1). There he defines the notion of an integral element over a multiplicatively closed set and states that the association of the set A to the set A* of all elements that are integral over it is a closure operator. There is an error in the statement that A* = (A*)*. In this note we show, by an example, that A* and (A*)* are not equal in general. We also give an example to show that even for the special type of sets that Cohn needs, namely the gauge sets, this property fails to hold. However, since the other conditions for a closure operator (A ⊆ A* and A ⊆ B implies A* ⊆ B*) are satisfied, one obtains a Kuratowski closure operator by iterating *. If we denote by Ā the closure of A by iterating * then theorem 10·3, its corollary and theorem 10·4 in (1) that have a bearing on this error, are valid as stated with the same proof replacing only K* by . Further theorem 13·3 is valid without modification since Cohn considers only a special class of pseudovaluations namely regular pseudo valuations for which K* = .

It is a well known theorem of Gaschütz (4) and Schenkman (12) that if G is a finite group whose nilpotent residual A is Abelian, then G splits over A and the complements to A in G are conjugate. Following Robinson (10) we describe this situation by saying that G splits conjugately over A. A number of generalizations of this result have since been obtained, some of them being in the context of the formation theory of finite or locally finite groups (see, for example, (1), (3)) and others, for example, the recent and far-reaching results of Robinson (10, 11) being concerned with groups which are not necessarily periodic. Our results here are of the latter type.

When two triangles, ABC and A′B′C′, are perspective from a point 0, their pairs of corresponding sides meet on a line 0, the axis of perspective. The line OA passes through A′ and some point on 0. These four points have a certain cross ratio which is the same if B or C is used instead of A. The reciprocal cross ratio arises if A′B′C′ is regarded as the first triangle and ABC the second. The complete figure contains ten pairs of perspective triangles, yielding twenty cross ratios. In section 6 a technique, suggested by D. W. Babbage and John Rigby, is used to express these cross ratios in the form

It is shown that the internal and external angles at the faces of a polyhedral cone satisfy various bilinear relations. The first two of these are related to the Gauss–Bonnet and Steiner parallel formulae for spherical polytopes, while the third is completely new. However, the proofs are basically combinatorial in nature, rather than differential geometric, as in the more classical treatments. These relations lead to inversion formulae, analogous to Euler-type relations, for certain functions defined on polytopes and polyhedral cones. As a result, various new relations involving quermassintegrals and Grassmann angles are found; there is also an application to lattice polytopes.

The theorem of R. Rado (12) to which I refer by the name ‘Rado's theorem for matroids’ gives necessary and sufficient conditions for a family of subsets of a finite set Y to have a transversal independent in a given matroid on Y. This theorem is of fundamental importance in both transversal theory and matroid theory (see, for example, (11)). In (3) J. Edmonds introduced and studied ‘polymatroids’ as a sort of continuous analogue of a matroid. I start this paper with a brief introduction to polymatroids, emphasizing the role of the ‘ground-set rank function’. The main result is an analogue for polymatroids of Rado's theorem for matroids, which I call not unnaturally ‘Rado's theorem for polymatroids’.

Let ℳg be the coarse moduli scheme of curves of genus g. For an algebraically closed field k define is a quasiprojective algebraic variety over k, its dimension being 3g – 3 for g ≥ 2, 1 for g = 1, and 0 for g = 0. It can be considered as the moduli variety for the classes of birationally equivalent curves of genus g over k. For 0 < g, g′ and n ≥ 1 let be the subset of those points of whose corresponding curves possess a rational map of degree n into a curve of genus g′ over k.

In this paper, we study the structure of certain infinite-dimensional Jordan algebras admitting an inner product. These algebras, called J*-algebras in the sequel, have already been considered in (4) in connexion with the norm uniqueness problem for non-associative algebras. We deal here with the structure and classification of these algebras. Existence of self-adjoint idempotents plays a central role in the classification problem.

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

1. In (4) Bott and Heitsch discussed certain classifying spaces of importance in the study of foliations of various types. For example they studied the space BГq which helps to classify real foliations of codimension q and proved that a canonical map

induces an injection in integral cohomology for q ≥ 2. Similar results hold for classifying spaces BUГq and BSГq related to complex and oriented foliations respectively. The object of this paper is to demonstrate that the canonical maps referred to above also induce injections in integral K-theory.

Let K be a compact convex set regularly embedded in a locally convex Hausdorff topological vector space E. Let Ab(K) be the GM-space (6) with unit 1K consisting of all real valued bounded of affine functions on K and let A(K) be the GM-subspace of Ab(K) consisting of continuous functions. For properties of K and A(K), see (2). The monotone σ-envelope A(K)μ of A(K) is the smallest subset of Ab(K) containing A(K) and such that the pointwise limit of every uniformly bounded monotone sequence (an) in A(K)μ also lies in A(K)μ. In (1), Alfsen shows that when K is a metrizable simplex A(K)μ is a vector lattice. In this note it is shown conversely that if K is metrizable and A(K)μ is a vector lattice, then K is a simplex.

We show that the only compact subsets of [0, 1] of measure zero are the obvious ones, and extend this idea to show that the only compact subsets of [0, 1] that generate semigroups of ℝ of measure zero are the obvious ones. We apply this last result to prove that any proper Raikov system of subsets of ℝ, with a single generator, is contained in a strictly larger proper Raikov system.

The aim of this note is to construct generators for II*(MU). Generators of a similar type were constructed by various authors; Hazewinkel (2), Kozma (3) and Liulevicius (4). The construction in (4) is in fact for II*(BP) at the prime 2.

The aim of this note is to prove an analogue of P. Hall's theorem on distinct representatives (1) for a family of measurable sets.

We prove a conjecture of (5), namely that the convex set of all infinite doubly stochastic matrices whose entries are all strictly less than θ(0 < θ ≤ 1) possesses extreme points if and only if θ is irrational.

Let θ be an n-dimensional random vector with components θ1, θ2, …, θn and let itscumulants

be defined as usual by the identity

where x1, x2, …, xn are purely imaginary variables, E denotes expectation, and

Let θ(1), θ(2), …, θ(R) be independent and identically distributed (i.i.d.) random vectors each with the same distribution as θ, where R = r1 + r2 + … + rn = |r|, the order of the cumulant Kr. Let θ* have the components

1. Introduction and statement of results. The most substantial contribution in recent years to the literature on the Galton–Watson embedding problem has been that of Karlin and McGregor (4) and (5), to which the readeris referred for a more complete bibliography of the subject. Their results and methods very much involve the analytic properties of the associated probability generating functions (p.g.f.'s), and most of the results they describe as ‘negative’ in the sense that they are sufficient conditions for non-embeddability. The two results here are similarly ‘negative’ but obtained from more direct probabilistic considerations.

1.Introduction. The purpose of this note is to apply the Kostant-Souriau quantization theory (2, 3, 4, 5, 7) to construct representations of a semi-direct product.

The equations satisfied by perturbations of solutions of the Euler-Lagrange equations corresponding to a Lagrangian density L are derived from the first and second variations of L and these new Lagrangian densities are used to construct Noether and Euler identities. It is shown how these identities can be used in general relativity to construct conserved quantities and local uniqueness theorems.