Let F(eis) denote a homeomorphism of the positively oriented unit circle onto a convex curve Γ and let f (eit) = F(eiΦ(t)), where Φ(t) is a non-decreasing function such that Φ(2π) – Φ(0) ≤ 2πN (N a positive integer). If f (eit) has Fourier coefficients cn, we show that is either constant or an N -valent analytic function in {|z| < 1}. We prove that where d is the distance from 0 to Γ and δ(N) > 0 depends only on N. This settles affirmatively a conjecture of H. S. Shapiro.

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.

Let X be a real Banach space, X* its dual space and ω a measurable space. Let D be a subset of X, L: Ω × D → X* a random operator and η:Ω →X* a measurable mapping. The random equation corresponding to the double [L, η] asks for a measurable mapping ξ: Ω → D such that

Random equations with operators of monotone type have been studied recentely by Kannan and Salehi [7], Itoh [6] and Kravvarits [8].

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.

Gleason's theorem characterizes the totally additive measures on the closed sub-spaces of a separable real or complex Hilbert space of dimension greater than two. This paper presents an elementary proof of Gleason's theorem which is accessible to undergraduates having completed a first course in real analysis.

Volume 97 (1985), 219–223

‘Finite Bol loops: III’

By R. P. BURN

Homerton College, Cambridge

There are many results which discuss ergodicity in terms of approximate product properties. Here we work throughout with stochastically independent σ-algebras (ℰ, ℱ are independent of μ if μ (E ∩ F) =μ (E)μ(F) for all E ∊ ℰ,F ∊ ℱ), to obtain an exact product property characteristic of (a large class of) ergodic measures. The ideas are based on work of A. V. Skorokhod on admissible translates of probabilities on Hilbert space.

In Eagleson and Weber [2] a central limit theorem for weakly exchangeable arrays is given as a consequence of a reverse martingale central limit theorem. As noted in their remarks, a direct application of this is a central limit theorem for the classical U-statistics. Here we give a corollary to the functional law of the iterated logarithm of Scott and Huggins [4] and use this to obtain laws of the iterated logarithm for weakly exchangeable arrays and hence for U-statistics under a finite (2 + δ)th moment condition.

1. Preliminaries. Let G be a non-elementary finitely generated Fuchsian group of the second kind acting on the unit disc Δ, and let Λ(G) be the limit set of G. It is well known that Λ(G) is a perfect non-dense set on the boundary of A. The Hausdorff dimension of the limit set Λ(G) of G is defined as the non-negative number

where mt(Λ(G)) denotes the t-dimensional measure of Λ(G) ([2], [3]). We say G has the type (g; m) if S = Δ/G is obtained from a compact surface of genus g by removing m(≥ 0) conformal discs.

For a random walk on a finite group, the distribution after n steps will converge, as n→∞, to the uniform distribution (under mild conditions). The asymptotic behaviour of such walks can be studied easily using standard Markov chain theory. But there are many natural problems about the non-asymptotic behaviour which have no simple solution in general. For example, what is

the time until a specified element is first visited?

the time until all element have been visited?

the time until the distribution approaches the uniform distribution?

Given a sequence of non-negative independent and identically distributed random variables, we determine conditions on the common distribution such that the sum of appropriately normalized and centred upper kn extreme values based on the first n random variables converges in distribution to a normal random variable, where kn → ∞ and kn/ n → 0 as n → ∞. The probabilistic problem is motivated by recent statistical work on the estimation of the exponent of a regularly varying distribution function. Our main tool is a new Brownian bridge approximation to the uniform empirical and quantile processes in weighted supremum norms.

1. Consider the Cauchy problem for the classical wave equation in ℝn:

with initial conditions u(0, x) = uo(x) and ∂tu(0, x)= u1(x), where uj (j = 0,1) are compactly supported C∞ functions on ℝn. In this paper we shall establish the following result.

A strong approximation for randomly indexed U-statistics is derived. The special case when the indices are stopping times based upon the jackknife estimator of the variance is discussed.

This paper proves the existence of solutions for the discrete coagulation–fragmentation equation over all times, even when source and efflux terms are present. The hypotheses required cover most physical applications. Roughly speaking, the hypotheses ensure a finite flux of mass through the system. The techniques used, which extend those in I of this series, may apply to other infinite systems of differential equations.

A point H is known to lie on a given bounded interval ℋ. A searcher wishes to locate H by making successive guesses g1, g2, …, each with the knowledge of whether the previous guesses were too high or low, or exactly right. Under these circumstances it is easy to devise a search strategy which ensures the convergence of the gi to H. One such strategy is the ‘halving’ strategy which always guesses the midpoint of the interval on which H is currently known to lie. The problem becomes well defined, and more difficult, if the searcher has to minimize a given cost function which in some way measures the speed of convergence of the gi to H.

We present the first anisotropic cosmological solutions of d = 11 supergravity. The solutions given are of Bianchi types III, V and VIh coupled with the seven-dimensional torus. The cosmological implications are briefly discussed.

The theoretical transition between basic properties of elastoplastic media at two levels of description is examined rigorously. At the micro-level the material response is heterogeneous, whereas at the macro-level it appears homogeneous. A broad class of constitutive relations is envisaged, and no restriction is placed on the magnitude of deformations and rotations at the micro-level. The investigation is concerned with quadratic differential forms that feature prominently in constitutive analyses, and is complementary to a previous study of bilinear differential forms. A principal objective is to access the transmissibility of measure-invariant inequalities from one level to the other.