Dierential-dierence Complex and the Poincar´e Lemma

摘要

Dierential geometry play a fundamental role in discussing partial dierential equations(PDEs) in mathematical physics. Recently discrete dierential geometry is an active mathematical terrain, which aims at the development and application of discrete equivalents of the geometric notions and methods of dierential geometry. In this paper, a discrete theory of exterior dierential calculus and the analogue of the Poincar´e lemma for dierential-dierence complex are proposed. They provide an intrinsic idea for developing the theory to discuss the integrability of dierence equations.