Here we will provide an overview of the method of nding symmetries of partial di erential equations. First we will discuss some basic facts regarding local groups of transformations, symmetries of algebraic equations and group actions on functions. We will then proceed to a discussion of how to interpret the task of solving a PDE in terms of jet spaces, and the meaning of group actions in this ...

Solvability of forward–backward stochastic di erential equations with nonsmooth coe cients is considered using the Four-Step Scheme and some approximation arguments. For the onedimensional case, the existence of an adapted solution is established for the equation which allows the di usion in the forward equation to be degenerate. As a byproduct, we obtain the existence of a viscosity solution t...

In this paper, a numerical method is presented for proving the existence and inclusion of connecting orbits of continuous dynamical systems described by parameterized nonlinear ordinary di erential equations. Taking a certain second order nonlinear ordinary di erential equaiton as an example, the existence of homoclinic bifurcation points is proved by the method.

We study a new class of backward stochastic di€erential equations, which involves the integral with respect to a continuous increasing process. This allows us to give a probabilistic formula for solutions of semilinear partial di€erential equations with Neumann boundary condition, where the boundary condition itself is nonlinear. We consider both parabolic and elliptic equations.

In this paper, we investigate the almost surely asymptotic stability for the nonlinear stochastic di erential delay equations with Markovian switching. Some su0cient criteria on the controllability and robust stability are also established for linear stochastic di erential delay equations with Markovian switching. ? 2003 Elsevier Ltd. All rights reserved.

In this paper we give necessary design principles to be used, when constructing secure Feistel ciphers. We introduce a new concept, practical security against linear and di erential attacks on Feistel ciphers. We give examples of such Feistel ciphers (practically) resistant to di erential attacks, linear attacks and other attacks.

A stability theorem is proved for nonmonotone waves in a reaction-di usion system modeling three competing species, where few stability results currently exist for systems with more than two species. Asymptotic stability with respect to the nonlinear equations is established by showing that the spectrum of the linearized operator has no unstable eigenvalues and that the zero eigenvalue associat...

The e¢ cient and accurate calculation of sensitivities of the price of …nancial derivatives with respect to perturbations of the parameters in the underlying model, the so called ‘Greeks’, remains a great practical challenge in the derivative industry. This is true regardless of whether methods for partial di¤erential equations or stochastic di¤erential equations (Monte Carlo techniques) are be...

Voronoi cells and the notion of natural neighbours are used to develop a nite di erence method for the di usion operator on arbitrary unstructured grids. Natural neighbours are based on the Voronoi diagram, which partitions space into closest-point regions. The Sibson and the Laplace (non-Sibsonian) interpolants which are based on natural neighbours have shown promise within a Galerkin framewor...

Collocation schemes are presented for solving linear fourth order di erential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalen...