Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite dimensional random ...

Transformation groups figure prominently in Lie’s theory of symmetry groups of differential equations, which we discuss in Chapter 4. They will act on the basic space coordinatized by the independent and dependent variables relevant to the system of differential equations under consideration. Since we are dealing with differential equations we must be able to handle the derivatives of the depen...

We present the method of group foliation for constructing non-invariant solutions of partial differential equations on an important example of the Boyer-Finley equation from the theory of gravitational instantons. We show that the commutativity constraint for a pair of invariant differential operators leads to a set of non-invariant solutions of this equation. In the second part of the paper we...

Annals of Probability 31(2003), 2109-2135. Invariant man-ifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invarian...

Let A be a Hopf algebra and Γ be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra Γ = Γ/J that contains Γ as its first order part. Corresponding to the three choices of the ideal J , we distinguish the ‘universal’ exterior algebra, the ‘second antisymmetrizer’ exterior algebra, and Woronowicz’ externa...