Abbas Edalat

Domain Theory and Differential Calculus

We introduce domain theory in differential calculus. Based on a new structure in domain theory, we define the derivative of a Scott continuous function on the domain of intervals, which is itself a Scott continuous function. This leads to a domain-theoretic generalization of the fundamental theorem of calculus. We then construct a domain for differentiable real valued functions of a real variable. The classical C^1 functions, equipped with their C^1 norm, is embedded into the set of maximal elements of this domain, which is a countably based bounded complete continuous domain. The construction can be generalized to C^k and C^infty functions and to real-valued functions of several variables. It can also be extended to analytic functions. As an immediate application, we present a domain-theoretic generalization of Picard's theorem, which provides a data type for solving differential equations.

Prof Edalat will also be speaking at the Pure and Applied Logic Colloquium on Thursday, 28 September, at 5409 Wean Hall, 4:30 PM (refreshments at 3:30) on the topic "A Data Type for Solid Modeling and Computational Geometry". [Please note: owing to a misunderstanding the POP and PAL talks were originallly scheduled in the other order. The talks are related, but the first on Thursday will be more introductory.] For appointments with Edalat on Thursday and Friday, please contact Phyllis Pomerantz , x8-7897.

September 29, 2000
Wean 8220