Special Stochastic Systems Seminar
Abstract: The typical behavior of a queueing system leading to a rare event can be described by minimizing paths. Such paths are defined as the solution of variational problems. Large deviations theory justifies that they represent the most probable conditional behavior in an asymptotic sense.
New analytical and algorithmical developments allow the automatic calculation of minimizing paths of queueing systems in a number of new and interesting situations. As examples we present several minimizing paths of multi-class feedforward queueing networks, generalized processor sharing disciplines and networks with priority service disciplines. The calculated minimizing paths allow new and deep insights to the complicated behavior of such queueing systems.
The common root of these examples is the possibility to model them by a Skorohod map. Based on sample path large deviation principles, contraction principles and exponential tightness arguments we obtain the variational problems which define the minimizing paths. A branch and bound algorithm is used to approximate a minimizing path by restricting the search space to piecewise linear paths. Numerically we observe that very few pieces are sufficient to obtain a minimizing path.
Brown Mathematics Department Distinguished Lecture
Applied Mathematics Colloquium
Abstract: Though oil and water do not mix, the addition of a small amount of surfactant or amphiphile can play havoc with their interface, giving rise to a wealth of self-assembled complex structures with length scales of tens of nanometers. This simple chemical means of nanofabrication has a wide range of applications, from the familiar behavior of soaps and detergents, to industrial efforts to remove oil from polluted aquifers or petroleum reservoirs, to the transport of nutrients in and among biological cells.
A variety of simple models have been used to study the equilibrium properties of such amphiphilic fluids for over a decade now. Many interesting questions about them, however, center on their behavior far from equilibrium. Droplet growth rates, fluctuations of interfaces loaded with surfactant, vesicle formation, and rheology in the presence of shearing flow, for example, all involve hydrodynamics, and therefore require a model with a conserved momentum.
In this talk, I describe three approaches to modeling and simulating the hydrodynamics of amphiphilic fluids: (i) Microscopic, particulate "vector models" endow molecules of like/unlike species with attractive/repulsive potentials and treat amphiphile molecules as "dipoles." (ii) Mesoscopic field-theoretical approaches use mean-field theory to derive partial differential equations for the oil/water/amphiphile order parameters which are treated as continuous fields. (iii) At the largest scales, macroscopic models treat interfaces as sharp and possess an energy functional that depends on local curvature.
Finally, I shall present results from an on-lattice microscopic model in both two and three dimensions. We have used this model to study domain growth, interfacial fluctuations, and emulsion rheology.
PDE Seminar
Brown Mathematics Department Distinguished Lecture
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