Stochastic flows on a countable set: Convergence in distribution

Consider a sequenceF ₁,F ₂,... of i.i.d. random transformations from a countable setV toV. Such a sequence describes a discrete-time stochastic flow onV, in which the position at timen of a particle that started at sitex isM _n(x), whereM _n =F _n F _n–1 F ₁. We give conditions on the law ofF ₁ for the sequence (M _n) to be tight, and describe the possible limiting law. an example called the block charge model is introduced. The results can be formulated as a statement about the convergence in distribution of products of infinite-dimensional random stochastic matrices. In practical terms, they describe the possible equilibria for random motions of systems of particles on a countable set, without births or deaths, where each site may be occupied by any number of particles, and all particles at a particular site move together.     

Ergodicity of a measure-valued Markov chain induced by random transformations

Let Z ₁, Z ₂,... be a sequence of i.i.d. random transformations (possibly discontinuous) of a compact metric space M, and let E denote the space of normalized mass distributions on M. Given in E, let _n denote the random measure °(Z _n °...°Z ₁)^–1 (when well-defined). We construct the transition probability P of the E-valued Markov chain ( _n ), and give a necessary and sufficient condition for P to have a unique invariant measure concentrated on the degenerate mass distributions. Convergence to statistical equilibrium of the associated discrete-time stochastic flow is investigated.Research supported by NSF grants INT-8420360 and DMS-8502802. This work was done while the author was visiting the Universities of Warwick and Minnesota     

Construction of multiporphyrin arrays using ruthenium and rhodium coordination to phosphines

The synthesis of linear multiporphyrin arrays with mono- and bisphosphine-substituted porphyrins as ligand donors and ruthenium(II) or rhodium(III) porphyrins as ligand acceptors is described. With appropriate amounts of the building blocks mixed, linear dimeric and trimeric arrays have been synthesized and analyzed by (1)H NMR and (31)P NMR spectroscopy. The Ru/Rh acceptor porphyrins can be located either at the periphery or in the center of the array. Likewise, the monophosphine porphyrins can be positioned at the periphery, thus allowing a high degree of freedom in the overall composition of the arrays. This way, both donor and acceptor porphyrins can act as chain extenders or terminators. One of the trimeric complexes with two nickel and one ruthenium porphyrin has also been analyzed by X-ray crystallography. Attempts have also been made to synthesize higher order arrays by mixing appropriate amounts of the porphyrins; however, from the NMR data it cannot be concluded if monodisperse five, seven, or nine porphyrin arrays are present or if the solutions are composed of a statistical mixture of smaller and larger arrays.     

Approximating Ito integrals of differential forms and geodesic deviation

Summary Suppose X is a semimartingale on a differential manifold M with a linear connection . The main purpose of this paper is to show that the Ito integral (with respect to ) of a differential form along the path of X is the limit in probability of certain Riemann sums, constructed in a natural way using the exponential map in differential geometry. For this, we study the deviation between the stochastic development of X in the tangent space at some point, and the image of X under the inverse of the exponential map at the point.     

Synthesis of reduced complex oxides of molybdenum by fused salt electrolysis

Fused salt electrolysis has been used to prepare a number of reduced oxides of molybdenum with lanthanum, neodymium, and yttrium in single crystal or oriented polycrystalline form. The average valence of molybdenum in the various compounds ranged from 5.67 to 3.50. Previously unreported compounds include La₅Mo₄O₁₆ (triclinica = 5.64 Å,b = 20.7 0Å,c = 5.64 Å, α = 86.55°, β = 90.0°, γ = 93.45°); La₂Mo₂O₇ (orthorhombic,a = 12.19 Å,b = 6.05 Å,c = 3.87 Å); LaMo₂O₅ (hexagonal,a = 8.378 Å,c = 19.26 Å). In addition, single crystal specimens have been prepared of Y₂MoO₅,Ln₅Mo₃O₁₆(Ln =La, Nd) and metal atom cluster compounds of theA₂Mo₃O₈ type (A = Mg, Co, Ni, Zn).