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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)

- 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)

- I Fatkullin, V Slastikov
- 2004

We study Onsager's free energy functional for nematic liquid crystals with orientation parameter on a unit circle. For a class of interaction potentials we obtain explicit expressions for all critical points, analyze their stability, and construct the corresponding bifurcation diagram. We also derive asymptotic expansions of the equilibrium density of… (More)

We present a theory of orientational order in nematic liquid crystals which interpolates between several distinct approaches based on the director field (Oseen and Frank), order parameter tensor (Landau and de Gennes), and orientation probability density function (Onsager). As in density-functional theories, the suggested free energy is a functional of… (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 study the Keller–Segel model of chemotaxis and develop a composite particle-grid numerical method with adaptive time stepping which allows us to resolve and propagate singular solutions. We compare the numerical findings (in two dimensions) with analytical predictions regarding formation and interaction of singularities obtained through analysis of the… (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)

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)

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)

- Ibrahim Fatkullin, Gregor Kovačič, Eric-Vanden-Eijnden
- 2009

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)