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- I. Fatkullin, V. Slastikov
- 2005

We study Onsager's model of nematic phase transitions with orientation parameter on a sphere. We consider two interaction potentials: the antisymmetric (with respect to orientation inversion) dipolar potential and symmetric Maier-Saupe potential. We obtain a complete classification and explicit expressions of all critical points, analyze their stability,… (More)

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)

We study a two-dimensional model describing spatial variations of orientational ordering in nematic liquid crystals. In particular, we show that the spatially extended Onsager-Maier-Saupe free energy may be decomposed into Landau-de Gennes-type and relative entropy-type contributions. We then prove that in the high concentration limit the states of the… (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)

- Agostino Capponi, Ibrahim Fatkullin, Ling Shi
- IEEE Trans. Automat. Contr.
- 2011

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)

We consider stochastically perturbed gradient flows in the limit when the amplitude of random fluctuations is small relative to the typical energy scale in the system and the minima of the energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics may be described in terms of a diffusion process on these manifolds.… (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)