In this paper, we develop a novel blind source separation (BSS) method for nonnegative and correlated data, particularly for the nearly degenerate data. The motivation lies in nuclear magnetic resonance (NMR) spectroscopy, where a multiple mixture NMR spectra are recorded to identify chemical compounds with similar structures (degeneracy).

We perform a computationl study of front speeds of G-equation models in time dependent cellular flows. The G-equations arise in premixed turbulent combustion, and are Hamilton-Jacobi type level set partial differential equations (PDEs). The curvature-strain G-equations are also non-convex with degenerate diffusion. The computation is based on monotone finite difference discretization and weighted essentially nonoscillatory (WENO) methods. We found that the large time front speeds lock into the frequency of time periodic cellular flows in curvature-strain G-equations similar to what occurs in the basic inviscid G-equation. However, such frequency locking phenomenon disappears in viscous G-equation, and in the inviscid G-equation if time periodic oscillation of the cellular flow is replaced by time stochastic oscillation.

We consider diffusivity of random walks with transition probabilities depending on the number of consecutive traversals of the last traversed edge, the so called senile reinforced random walk (SeRW). In one dimension, the walk is known to be sub-diffusive with identity reinforcement function. We perturb the model by introducing a small probability \delta of escaping the last traversed edge at each step. The perturbed SeRW model is diffusive for any \delta , with enhanced diffusivity (\delta ) in the small \delta regime. We further study stochastically perturbed SeRW models by having the last edge escape probability of the form \delta with \delta 's being independent random variables. Enhanced diffusivity in such models are logarithmically close to the so called residual diffusivity (positive in the zero \delta limit), with diffusivity between \delta and \delta . Finally, we generalize our results to higher dimensions where the unperturbed model is already diffusive. The enhanced diffusivity can be as much as \delta .

We study a Maxwell-Bloch system describing the dynamics of single-longitudinal Raman lasers in the two transverse space dimensions. Raman lasing is generated by a coherent external pump laser, as a three-wave interaction involving two optical and one material wave. On the other hand, a two-level laser involves incoherent external pumping through, for example, an electrical discharge or flashlamp. Raman lasers have the advantage of being tunable and display a novel explicit nonlinear detuning between the pump and laser emission frequencies. Consequently these lasers exhibit much richer nonlinear dynamics. We establish the global existence of classical solutions, and show that for periodic domains the dynamics is governed by a global smooth attractor of finite dimensions. We explain the structures of nonlinear interactions and couplings that lead to the time asymptotic smoothing. We also construct

We study waves in convex scalar conservation laws under noisy initial perturbations. It is known that the building blocks of these waves are shock and rarefaction waves, both are invariant under hyperbolic scaling. Noisy perturbations can generate complicated wave patterns, such as diffusion process of shock locations. However we show that under the hyperbolic scaling, the solutions converge in the sense of distribution to the unperturbed waves. In particular, randomly perturbed shock waves move at the unperturbed velocity in the scaling limit. Analysis makes use of the Hopf formula of the related Hamilton-Jacobi equation and regularity estimates of noisy processes.