MULTIPLE COMBINATORIAL STOKES' THEOREM WITH BALANCED STRUCTURE

作者：Lee Shyh Nan; Chen Chien Hung; Shih Mau Hsiang^*

来源：Taiwanese Journal of Mathematics, 2010, 14(3B): 1169-1200.

• Pseudomanifold
• Triangulation
• Orientable
• pi-balanced
• pi-subbalanced
• General position map
• Multiple combinatorial Stokes' theorem
• Multiple combinatorial Sperner's lemma

Combinatorics of complexes plays an important role in topology, nonlinear analysis, game theory, and mathematical economics. In 1967, Ky Fan used door-to-door principle to prove a combinatorial Stokes' theorem on pseudomanifolds. In 1993, Shih and Lee developed the geometric context of general position maps, pi-balanced and pi-subbalanced sets and used them to prove a combinatorial formula for multiple set-valued labellings on simplexes. On the other hand, in 1998, Lee and Shih proved a multiple combinatorial Stokes' theorem, generalizing the Ky Fan combinatorial formula to multiple labellings. That raises a question : Does there exist a unified theorem underlying Ky Fan's theorem and Shih and Lee's results? In this paper, we prove a multiple combinatorial Stokes' theorem with balanced structure. Our method of proof is based on an incidence function. As a consequence, we obtain a multiple combinatorial Sperner's lemma with balanced structure.

• 出版日期2010-6