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We consider analytic maps f j : D → D of a domain D into itself and ask when does the sequence f 1 •· · ·•f n converge locally uniformly on D to a constant. In the case of one complex variable, we are able to show that this is so if there is a sequence {w 1 , w 2 , . . . } in D whose values are not taken by any f j in D, and which is homogeneous in the sense that it comes within a fixed hyperbolic distance of any point of D. The situation for several complex variables is also discussed.

A study of the structure of glutathione transferase B (ligandin) has been made with a view to understanding the relationship between the structures of the subunits of which it is composed. It consists of a mixture of a homodimer (Ya Ya) and a heterodimer (Ya Yc) in which the monomers are denned by their apparent molecular weights, that of Ya being 22000 and Yc 25000. Soluble tryptic peptides from the native homodimer Ya Ya have been compared with those from an artificial homodimer Yc Yc produced by rehybridization of native Ya Yc. Approximately 10 peptides specific to Ya Ya, 12 specific to Yc Yc and 21 common to both have been detected. Some of the above peptides are derived from variants of the monomers themselves. Ya Ya and Yc Yc have two C termini which are the same in both dimers, namely phenylalanine and lysine. Also there are four cysteinyl peptides, of which three are common to Ya Ya and Yc Yc and one specific to each. These results suggest that Ya and Yc are derived from at least two different but related genes.

In [2] D. G. Kendall introduced shape spaces which classsified TV-tuples of points in U K and explained their role for statistics. The aim of this paper is to study such shape spaces and, in particular, to explore the corresponding spaces for points on spheres. The ultimate purpose of the work is to be able to apply statistical techniques to observations where only the shape of the observations is significant. It is apparent that the geometry of the shape space is critical in determining the type of statistical technique which one should use. Therefore, the major part of this paper is devoted to an examination of the geometry, and the statistics is only introduced in a simple case when considering random processes on the shape space. We refer the reader to the surveys of the statistical background to this work in [2] and [5]. Further details of the background to this work, and related ideas are contained in the references [3] to [13] and [16] to [18].The general situation which we wish to study is as follows. Let G be a Lie group which acts smoothly on the differentiable manifold M. Then G acts smoothly on the space M N of TV-tuples from M. The shape space ~Z N (M, G) is defined to be the set of orbits for this action. So two TV-tuples (x n ) and (y n ) determine the same shape if there is a T e G with Tx n = y n for each n. There is a natural quotient map q: M N^1 . N (M, G) which sends an TV-tuple to its orbit. We wish to give the shape space more structure. It can certainly be given a quotient topology. We would like to give it a differentiable structure so that q were a submersion. This is not possible everywhere, for it can fail at points where the action of G is not free. However, for the cases which we will consider, there is a dense open set of regular points in H N (M, G) which can be made into a differentiable manifold so that q is a submersion above it. The remaining points are singularities of the shape space; they are often of importance in the related statistical problems. If M is a Riemannian manifold and G a Lie subgroup of the isometries, then we may find a Riemannian metric on the regular points of the shape space for which q is a Riemannian submersion. Furthermore, the Brownian motion on M N maps under q to a Markov process and we will wish to explore how this process compares with Brownian motion on the shape space. (By considering the densities of such a process at fixed times we may obtain information, in a particularly symmetric case, about the image of a probability density under q. See the references [3,7,8,10,11,12] for related results on the images of distributions.)We will only consider very special examples of shape spaces where M is either an Euclidean space IR* or a sphere S K . In the first section we examine in detail the case where M = U K and G is the orthogonal group O(K). This is simple because M N is flat. In the second section we consider the, largely superficial, changes that are necessary when we replace O(K) by the special orthogonal group SO(K). D. G. Kendall has studied...

Abstract.Let A(B) be the uniform algebra on the unit ball of a dual Banach space Z = ff* generated by the weak-star continuous linear functionals. We focus on three related problems: (i) to determine when A(B) is a tight uniform algebra; (ii) to describe which functions in H°°(B) are approximate pointwise on B by bounded nets in A(B) ; and (iii) to describe the weak topology of B regarded as a subset of the dual of A(B). With respect to the second problem, we show that any polynomial in elements of y* can be approximated pointwise on B by functions in A(B) of the same norm. This can be viewed as a generalization of Goldstine's theorem. In connection with the third problem, we introduce a class of Banach spaces, called A-spaces, with the property that if (Xj) is a bounded sequence in ä? such that P(x¡) -> 0 for any mhomogeneous analytic function P on 3?, m > 1, then Xj -► 0 in norm. We show for instance that a Banach space has the Schur property if and only if it is a A-space with the Dunford-Pettis property.