Ibrahim Fatkullin

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Numerical schemes for systems with multiple spatio-temporal scales are investigated. The multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales. The multiscale schemes are analyzed in general, then illustrated on a specific(More)
The distribution of interval lengths between Brownian walkers on the line is investigated. The walkers are independent until collision; at collision, the left walker disappears, and the right walker survives with probability p. This problem arises in the context of diffusion-limited reactions and also in the scaling limit of the voter model. A systematic(More)
We provide a framework for computing the density of states of a noisy system given the sequence of hitting times of predefined thresholds. Our method relies on eigenfunction expansion corresponding to the Fokker-Planck operator of the diffusion process. For illustration, we present a particular example in which the state and the noise are one-dimensional(More)
All Rights Reserved iii To my cousin Francesco. iv Acknowledgements I would like to express my gratitude to my advisor, Prof. Mani Chandy, for believing in me and giving me the wonderful opportunity to work with him. I would like to thank him for his constant support and guidance during my first two years at the California Institute of Technology. It has(More)
Nodes in a sensor network can generate messages periodically , or when anomalies are detected, or when queried by other nodes. In this paper we propose a strategy called predicate signaling that generalizes these schemes by generating messages when specified predicates-that can deal with both time and anomalies-hold. We show how power consumption, message(More)
The stochastic Ginzburg-Landau equation in one dimension is the simplest continuum model describing the spatio-temporal evolution of a bistable system in the presence of thermal noise. Relaxation to equilibrium in this model proceeds by coarsening of the field during which regions in the two stables phases separated by localized kinks grow on the average.(More)