Exchangeable Random Orders and Almost Uniform Distributions

Random orders on invariant under permutations are called exchangeable. The compact and convex set of all random total orders is shown to be a Bauer simplex whose set of extreme points, the socalled totally ordered paintbox processes, is homeomorphically parametrized by almost uniform distributions on the unit interval, i.e. by probability measures w on [0, 1] whose distribution functions are w-almost surely the identity.     

High brightness 735 nm tapered lasers – optimisation of the laser geometry

High brightness tapered laser diodes with different resonator geometries were fabricated and analysed. The devices consist of an index-guided straight section and a gain-guided tapered section. Lasers with a total length L = 2 and 4 mm and different length of the ridge waveguide L _RW (500 μm ≤ L _RW ≤ 1250 μm for L = 2 mm and 500 μm ≤ L _RW ≤ 2000 μm for L = 4 mm) were processed to study the influence of the straight section on the spatial mode filtering. The power–voltage–current-characteristics, the beam waist, the far field, and the beam propagation factor M ² were measured. From the experiments, it can be stated that the lasers with a small L _RW reach higher output powers compared to those with larger L _RW. Concerning the beam quality the length L _RW should exceed a minimal value to guarantee efficient spatial mode filtering. Devices optimised concerning maximum output power and excellent beam quality reach a beam propagation factor smaller than 2.1 at an output power P = 2 W.     

Random Partitions by Semigroup Methods

P of all partitions of {1,2,3,...}, or rather its distribution. There is a natural compact metrizable topology on P taking care of measurability questions. With respect to the maximum operation P becomes an abelian semigroup, and our first theorem characterizes random partitions as normalized positive definite functions on the subsemigroup of partitions "with finite support". We then present a new proof of Kingman's theorem stating that the exchangeable random partitions form a simplex whose extreme points are the so-called "paint-box distributions". An interesting moment problem which is still open arises in this connection.