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We consider the problem of computing ratings using the results of games (such as chess) played between a set of n players, and show how this problem can be reduced to computing the positive eigenvectors corresponding to the dominant eigenvalues of certain n by n matrices. There is a close connection with the stationary probability distributions of certain Markov chains. In practice, if n is large, then the matrices involved will be sparse, and the power method may be used to solve the eigenvalue problems efficiently.

It is well-known that the only space-time scaling limits of Galton-Watson processes are continuous-state branching processes. Their genealogical structure is most explicitly expressed by discrete trees and R-trees, respectively. Weak limit theorems have been recently established for some of these random trees. We study here a Markovian forest growth procedure that allows to construct the genealogical forest of any continuous-state branching process with immigration as an a.s. limit of Galton-Watson forests with edge lengths. Furthermore, we are naturally led to continuous forests with edge lengths. Another strength of our method is that it yields results in the general supercritical case that was excluded in most of the previous literature.

/notices/events/abstracts/stochastic-analysis/ht05/draief.shtml

We discuss the application of gradient methods to calibrate mean reverting

stochastic volatility models. For this we use formulas based on Girsanov

transformations as well as a modification of the Bismut-Elworthy formula to

compute the derivatives of certain option prices with respect to the

parameters of the model by applying Monte Carlo methods. The article

presents an extension of the ideas to apply Malliavin calculus methods in

the computation of Greek's.

Many industrial flow problems, expecially in the minerals and process

industries, are very complex, with strong interactions between phases

and components, and with very different length and time scales. This

presentation outlines the algorithms used in the CFX-5 software, and

describes the extension of its coupled solver approach to some

multi-scale industrial problems. including Population Balance modelling

to predict size distributions of a disperse phase. These results will be

illustrated on some practical industrial problems.

It became apparent during the last decade that in several questions in classical complex analysis extremal configurations are fractal, making them very difficult to attack: it is not even clear how to construct or describe extremal objects. We will argue that the most promising approach is to consider conformally self-similar random configurations, which should be extremal almost surely.

Fragmentation processes model the evolution of a particle that split as time goes on. When small particles split fast enough, the fragmentation is intensive and the initial mass is reduced to dust in finite time. We encode such fragmentation into a continuum random tree (CRT) in the sense of Aldous. When the splitting times are dense near 0, the fragmentation CRT is in turn encoded into a continuous (height) function. Under some mild hypotheses, we calculate the Hausdorff dimension of the CRT, as well as the maximal H

/notices/events/abstracts/applied-analysis/ht05/ogden.shtml

/notices/events/abstracts/topology/lackenby.shtml

Given a countable set of sites S an a transition matrix p(x,y) on that set, we consider a process of particles evolving on S according to the following rule: each particle waits an exponential time and then jumps following a Markov chain governed by p(x,y); the particle keeps jumping until it reaches an empty site where it remains for another exponential time. Unlike most interacting particle systems, this process fails to

have the Feller property. This causes several technical difficulties to study it. We present a method to prove that certain measures are invariant for the process and exploit the Kolmogorov zero or one law to study some of its unusual path properties.

According to the Stokes-Einstein law, microscopic particles subject to intense bombardment by (much smaller) gas molecules perform Brownian motion with a diffusivity inversely proportion to their radius. Smoluchowski, shortly after Einstein's account of Brownian motion, used this model to explain the behaviour of a cloud of such particles when, in addition their diffusive motion, they coagulate on collision. He wrote down a system of evolution equations for the densities of particles of each size, in particular identifying the collision rate as a function of particle size.

We give a rigorous derivation of (a spatially inhomogeneous generalization of) Smoluchowski's equations, as the limit of a sequence of Brownian particle systems with coagulation on collision. The equations are shown to have a unique, mass-preserving solution. A detailed limiting picture emerges describing the ancestral spatial tree of particles making up each particle in the current population. The limit is established at the level of these trees.

Spectral projections enjoy high order convergence for globally smooth functions. However, a single discontinuity introduces O(1) spurious oscillations near the discontinuity and reduces the high order convergence rate to first order, Gibbs' Phenomena. Although a direct expansion of the function in terms of its global moments yields this low order approximation, high resolution information is retained in the global moments. Two techniques for the resolution of the Gibbs' phenomenon are discussed, filtering and reprojection methods. An adaptive filter with optimal joint time-frequency localization is presented, which recovers a function from its N term Fourier projection within the error bound \exp(-Nd(x)), where d(x) is the distance from the point being recovered to the nearest discontinuity. Symmetric filtering, however, must sacrifice accuracy when approaching a discontinuity. To overcome this limitation, Gegenbauer postprocessing was introduced by Gottlieb, Shu, et al, which recovers a function from its N term Fourier projection within the error bound \exp(-N). An extension of Gegenbauer postprocessing with improved convergence and robustness properties is presented, the robust Gibbs complements. Filtering and reprojection methods will be put in a unifying framework, and their properties such as robustness and computational cost contrasted. This research was conducted jointly with Eitan Tadmor and Anne Gelb.