We continue to study the arrays of nonlinear coupled oscillators. The networks of associative memory based on limit-cycle oscillators connected via complex-valued Hermitian matrices were previously designed. Another class of networks consisting of locally connected nonlinear oscillators, closely related to so-called cellular neural networks, is the subject of study in the present paper. In spat...

We consider di usion processes on a class of R{trees. The processes are de ned in a manner similar to that of Le Gall's Brownian snake. Each point in the tree has a real{valued \height" or \generation", and the height of the di usion process evolves as a Brownian motion. When the height process decreases the di usion retreats back along a lineage, whereas when the height process increases the d...

We study the large time behavior of solutions of the Cauchy problem for a fast di usion equation with a singular powerlike source. It is shown that the large time behavior is described by the similarity solution of related problem.

We study variational formulas for maximizers for domain functionals F (x0; u (x0)), x0 2 , and R F (x; u (x)) dx over all Lipschitz domains satisfying the constraint R g (x) dx = 1. Here u is the solution of a di usion equation in . Functional variations are computed using domain variations which preserve the constraint exactly. We show that any maximizer solves a moving boundary problem for th...

Water pollution by industrial and agricultural waste is an increasingly major public health issue. It is therefore important for water engineers and managers to be able to predict accurately the local behaviour of water-borne pollutants. This paper describes the novel and ecient coupling of dynamically adaptive hierarchical grids with standard solvers of the advection±di€usion equation. Adapti...

We motivate and analyze a reaction-advection-di usion model for the dynamics of a phytoplankton species. The reproductive rate of the phytoplankton is determined by the local light intensity. The light intensity decreases with depth due to absorption by water and phytoplankton. Phytoplankton is transported by turbulent di usion in a water column of given depth. Furthermore, it might be sinking ...

We present the mathematical analysis for the convergence of an h version Finite Element Method (FEM) with piecewise polynomials of degree p ≥ 1, de ned on an exponentially graded mesh. The analysis is presented for a singularly perturbed reaction-di usion and a convection-di usion equation in one dimension. We prove convergence of optimal order and independent of the singular perturbation param...

The completion of interrupted lines or the enhancement of ow-like structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diiusion lter is steered by the so-called interest operator (second-moment matrix, structure tensor). An m-dimensional formulation of this method is analysed with ...

In this paper, a nonlinear inverse problem of parabolic type, is considered. By reducing this inverse problem to a system of Volterra integral equations the existence, uniqueness, and stability of the solution will be shown.

We consider an optimal stopping problem for a Hilbert-space valued di usion. We prove that the value function of the problem is the unique viscosity solution of an obstacle problem for the associated parabolic partial di erential equation in the Hilbert space. The results are applied to investigate the pricing of American interest rate options in the lognormal Heath-Jarrow-Morton model of yield...