Taiwanese Journal of Mathematics

MULTIPLE COMBINATORIAL STOKES’ THEOREM WITH BALANCED STRUCTURE

Shyh-Nan Lee, Chien-Hung Chen, and Mau-Hsiang Shih

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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, $\pi$-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.

Article information

Source
Taiwanese J. Math., Volume 14, Number 3B (2010), 1169-1200.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405912

Digital Object Identifier
doi:10.11650/twjm/1500405912

Mathematical Reviews number (MathSciNet)
MR2674603

Zentralblatt MATH identifier
1209.05019

Subjects
Primary: 05A19: Combinatorial identities, bijective combinatorics 52B05: Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx] 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Keywords
pseudomanifold triangulation orientable $\pi$-balanced $\pi$-subbalanced general position map multiple combinatorial Stokes' theorem multiple combinatorial Sperner's lemma

Citation

Lee, Shyh-Nan; Chen, Chien-Hung; Shih, Mau-Hsiang. MULTIPLE COMBINATORIAL STOKES’ THEOREM WITH BALANCED STRUCTURE. Taiwanese J. Math. 14 (2010), no. 3B, 1169--1200. doi:10.11650/twjm/1500405912. https://projecteuclid.org/euclid.twjm/1500405912


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