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Multiple Combinatorial Stokes' Theorem with Balanced Structure

By 陳建宏 and Chien-Hung Chen


[[abstract]]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, -balanced and -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 dissertation, 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.

Topics: 擬流形;一般位置映射;組合型Stokes定理;組合型Sperner引理;平衡集;次平衡集;多重集值標號;三角分割, pseudomanifold;general position map;combinatorial Stokes' theorem;combinatorial Sperner's lemma;balanced set;subbalanced set;multiple set-valued labelling;triangulation, [[classification]]37
Year: 2011
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