We define a class of completely positive maps, closed under composition, on a von Neumann algebra. We show that when the algebra has no atomic part, the correspondences associated to this class of completely positive maps are disjoint from the identity correspondence. This enables one simultaneously to generalize the statement and simplify the proof of a theorem of A. Connes and V. F. R. Jones on factors of type II1 with property T.

Theorems on random permutations are translated into distribution-free results about record times and greatest convex minorants, by defining them together on appropriate probability spaces. The Bernoulli random variables that appear in the standard representation of the number of sides of the greatest convex minorant of a random walk are identified.

A construction of Weierstrass-like functions using recurrent sets is described, and the Hausdorff dimensions of the graphs computed. An important part of the proof is the notion of a globally random recurrent set. The Hausdorff dimension of a class of such sets is calculated using techniques of random matrix products.

For random walks on finite graphs, we record some equalities, inequalities and limit theorems (as the size of graph tends to infinity) which hold for vertex-transitive graphs but not for general regular graphs. The main result is a sharp condition for asymptotic exponentiality of the hitting time to a single vertex. Another result is a lower bound for the coefficient of variation of hitting times. Proofs exploit the complete monotonicity properties of the associated continuous-time walk.

We classified the set of Fm-cobordism classes of Fm-links by their Seifert matrices in [5]. On the other hand Cappell and Shaneson identified them with essentially a quotient group of their homology surgery obstruction group [2]. In this paper, we will find a description of their surgery obstruction in terms of a Seifert matrix. In relation to Ledimet's recent results [7], we hope this might provide some clue to whether Fm-cobordism or boundary cobordism is stronger than ordinary link cobordism. It also seems to be an interesting algebraic question to find an algorithm for obtaining a Seifert matrix from their surgery obstruction.

The non-linear Volterra integral equation

has been studied recently in connection with non-linear diffusion and percolation problems [4, 6, 10]. The existence, uniqueness and qualitative behaviour of non-negative, non-trivial solutions are the questions of physical interest.

The family of Brownian bridge processes (ba)a∈R has a number of characterizations, the most fundamental being that ba: [0,1] → ℝ is Brownian motion conditioned to be at the point a at time 1. Equivalently, ba is a continuous process whose law Wa is that of Wiener measure conditioned on the set of paths with x1 = a. These ideas are not so easy to make precise, so that more down to earth and workable characterizations of the Brownian bridge such as the following are often used in practice (see [6] for example)

We first examine the class of far field patterns for the scalar Helmholtz equation in ℝ2 corresponding to incident time harmonic plane waves subject to an impedance boundary condition where the impedance is piecewise constant with respect to the incident direction and continuous with respect to x ε ∂ D where ∂ D is the scattering obstacle. We then examine the class of far field patterns for Maxwell's equations in subject to an impedance boundary condition with constant impedance. The results obtained are used to derive optimization algorithms for solving the inverse scattering problem.

We provide a concise account of the influence of design variables on the convergence rate in an L1 regression problem. In particular, we show that the convergence rate may be characterized precisely in terms of third and fourth moments of the design variables. This result leads to necessary and sufficient conditions on the design for the Berry-Esseen rate to be achieved. We also show that a moment condition on the error distribution is necessary and sufficient for a non-uniform Berry-Esseen theorem, and that an Edgeworth expansion is possible if the design points are not too clumped.

This paper continues the study initiated in [10] of the Markov branching process (MBP) subject to an independent Poisson process of catastrophe times at which the population size is reduced by random decrements whose distribution is independent of the current population size. More generally, let (Xt:t ≥ 0) be the Feller process on the non-negative integers ℕ+ having the generator