Towards Geometric Integration of Rough Differential Forms

Differential Forms and Integration

a f(x) dx (which one would use for instance to compute the work required to move a particle from a to b). For simplicity we shall restrict attention here to functions f : R → R which are continuous on the entire real line (and similarly, when we come to differential forms, we shall only discuss forms which are continuous on the entire domain). We shall also informally use terminology such as “i...

Differential Forms in Geometric Calculus

Geometric calculus and the calculus of differential forms have common origins in Grassmann algebra but different lines of historical development, so mathematicians have been slow to recognize that they belong together in a single mathematical system. This paper reviews the rationale for embedding differential forms in the more comprehensive system of Geometric Calculus. The most significant app...

Geometric Numerical Integration of Differential Equations

Geometric Numerical Integration of Differential Equations

What is geometric integration? ‘Geometric integration’ is the term used to describe numerical methods for computing the solution of differential equations, while preserving one or more physical/mathematical properties of the system exactly (i.e. up to round–off error)1. What properties can be preserved in this way? A first aspect of a dynamical system that is important to preserve is its phase ...

A Differential-geometric View of Normal Forms of Contractions

A version of non-stationary normal forms theory was recently obtained by M. Guysinsky and A. Katok, which has become an important ingredient in several recent investigations of rigidity of group actions. Our main aim is to explain their results, the sub-resonance normal forms and centralizer theorems, from a differential-geometric perspective.