Let V be a smooth complex algebraic threefold with k(V) = 0. Consider the space of holomorphic 2-forms on V, and let s denote the rank of the subsheaf of generated by these global forms. It was conjectured by Uenoin[8] that ≤ 3-here as always, hi denotes the dimension of the corresponding cohomology group Hi.

The quadratic functional equation f(f(x)) *–Tf(x) + Dx = 0 is equivalent to the requirement that the graph be invariant under a certain linear map The induced projective map is used to show that the equation admits a rich supply of continuous solutions only when L is hyperbolic (T2 > 4D), and then only when T and D satisfy certain further conditions. The general continuous solution of the equation is given explicitly in terms of either (a) an expression involving an arbitrary periodic function, function additions, inverses and composites, or(b) suitable limits of such solutions.

Let C be a set with at least two, and at most ℵ0, members, and for any set X let [X]2 denote the set of its 2-element subsets. If Γ is a countable set, and Fc is a function from [Γ]2 into C, then the structure Γc = (Γ, Fc) is called the countable universal C-coloured graph if the following condition is satisfied:

Whenever α is a map from a finite subset of Γ into C there is xεΓ–dom α such that (∀yεdom α) Fc {x, y} = α(y).

In this paper I shall define a deformation of hyperbolic structures on n-manifolds, called ‘bending’, by constructing a family of quasiconformal homeomorphisms of hyperbolic (n+ l)-space. The deformation can be applied to manifolds which contain totally geodesic submanifolds of codimension 1.

Exact octagons, i.e. circular eight-term exact sequences, have cropped up recently in a few places in the literature. Papers of the author [11], and implicitly [10], the book of M. Warshauer[14], and the notes [5], all contain exact octagons. The first three references involve octagons of Witt groups of quadratic and other kinds of forms, the last reference extending the octagons to the setting of L-groups, i.e. surgery obstruction groups.

We prove that the Poincaré conjecture implies that the 4-dimensional version of the realization theorem for Whitehead torsion is false. We show that infinitely many examples may be constructed to demonstrate this.

Let be a closed 3-braid in S3 with axis Lγ. The exterior of γ U Lγ in S3 is a fibre bundle over S1 with fibre a disc with 3 holes. The monodromy is given by a matrix in PSL(2,) and the braid γ is called hyperbolic if its matrix is hyperbolic.

In [5] we proved two results: theorem 1, which said that if k was a simple (2q – 1)-knot, q 1, then it was equivalent to the m-fold branched cyclic cover of another knot if and only if there existed an isometry u of its Blanchfield pairing 〈,〉, whose mth power was the map induced by a generator t of the group of covering translations associated with the infinite cyclic cover of k; and theorem 2, which showed that if k were the m-fold b.c.c. of two such knots, then these would be equivalent if and only if the corresponding isometries were conjugate by an isometry of 〈,〉. Using this second result, we present two cases where k may only be the m-fold b.c.c. of finitely many knots.

In the early sixties Effros[9] and Prosser[14] studied, in independent work, the duality of the faces of the positive cones in a von Neumann algebra and its predual space. In an implicit way, this work was generalized to certain ordered Banach spaces in papers of Alfsen and Shultz [3] in the seventies, the duality being given in terms of faces of the base of the cone in a base norm space and the faces of the positive cone of the dual space. The present paper is concerned with the facial structure of the unit balls in an ordered Banach space and its dual as well as the duality that reigns between these structures. Specifically, the main results concern the sets of norm-exposed and norm-semi-exposed faces of the unit ball V1 in a GL-space or complete base norm space V and the sets of weak*-exposed and weak*-semi-exposed faces of the unit ball in its dual space V* which forms a unital GM-space or a complete order unit space.

Let X be a compact Hausdorff space, let E be a (real or complex) Banach space, and let C(X, E) stand for the Banach space of all continuous E-valued functions defined on X under the supremum norm. If A is an arbitrary linear subspace of C(X, E), then it is shown that each bounded linear functional l on A can be represented by a boundary E*-valued vector measure μ on X that has the same norm as l. This result constitutes an extension to vector-valued functions of the so-called analytic version of Choquet's integral representation theorem.

In this paper we shall consider the class Hm of subharmonic functions u, unbounded in, which have a thick tract where u is bounded. We recall here that non-constant subharmonic functions in space (i.e. m ≥ 3) are not necessarily unbounded as in the two-dimensional case. For basic concepts in this area we refer the reader to [3].

Let B(t) be a two-dimensional Brownian motion. For 0 < α < 2π, set

and, for 0 ≥ β< 2π, let F(α,β) be F(α) rotated through an angle β about the origin.

In § 33 of [2], Professor Williams asks whether there are L1 random variables† X, Y, Z (not identically zero) such that

1. Main results and notations. We consider Cauchy's problem for the classical wave equation in :

with initial conditions u(0, x) = uo(x) and ∂tu(0, x) = u1 (x). In (1·1) δ and m2 stand for the Laplacian in and a constant respectively. Note that a second order hyper-bolic differential equation with real constants can be reduced to (1·1) ([1], p. 183). Let uj (j = 0,1) be C∞ funotions on whose support is contained in the closed ball BR = {xεl; |x| ≥R} for some R > 0. In this note we shall show that the solution u(t, x) of (1·1) possesses the following properties.

In this paper a quantum-mechanical variational principle is proposed, in which the functional varied is the precision, with which the time-dependent Schrödinger equation is satisfied by the wave functions of the given class. Another distinctive feature of our approach is that the independently varied functions are the time derivatives of the one-particle wave functions, of which the wave function of the system is constructed.

Volume 97 (1985), 219–223

‘Finite Bol loops: III’

By R. P. BURN

Homerton College, Cambridge