## Taiwanese Journal of Mathematics

### MULTIPLE COMBINATORIAL STOKES’ THEOREM WITH BALANCED STRUCTURE

#### 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

https://projecteuclid.org/euclid.twjm/1500405912

Digital Object Identifier
doi:10.11650/twjm/1500405912

Mathematical Reviews number (MathSciNet)
MR2674603

Zentralblatt MATH identifier
1209.05019

#### 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

#### References

• R. B. Bapat, A constructive proof of a permutation-based generalization of Sperner's lemma, Math. Program., 44 (1989), 113-120.
• K. Fan, Simplicial maps from an orientable $n$-pseudomanifold into $S^m$ with the octahedral triangulation, J. Comb. Theory, 2 (1967), 588-602.
• D. Gale, Equilibrium in a discrete exchange economy with money, Internat. J. Game Theory, 13 (1984), 61-64.
• Y. A. Hwang and M. H. Shih, Equilibrium in a market game, Economic Theory, 31 (2007), 387-392.
• B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für $n$-dimensionale Simplexe, Fund. Math., 14 (1929), 132-137.
• H. W. Kuhn, A new proof of the fundamental theorem of algebra, Math. Program. Study, 1 (1974), 148-158.
• S. N. Lee and M. H. Shih, A counting lemma and multiple combinatorial Stokes' theorem, European J. Combin., 19 (1998), 969-979.
• S. N. Lee and M. H. Shih, A structure theorem for coupled balanced games without side payments (Nonlinear Analysis and Convex Analysis), RIMS Kokyuroku, 1484 (2006), 69-72.
• F. Meunier, Combinatorial Stokes' formulae, European J. Combin., 29 (2008), 286-297.
• H. Scarf, The approximation of fixed points of continuous mapping, SIAM J. Appl. Math., 15 (1967), 1328-1343.
• L. S. Shapley, On balanced games without side payments, in: Mathematical Program. Math. Res. Cent. Publ., (T. C. Hu, M. Robinson, eds.), New York-Academic Press, 30 (1973), 261-290.
• M. H. Shih and S. N. Lee, A combinatorial Lefschetz fixed-point formula, J. Combin. Theory Ser. A, 61 (1992), 123-129.
• M. H. Shih and S. N. Lee, Combinatorial formulae for multiple set-valued labellings, Math. Ann., 296 (1993), 35-61.
• E. Sperner, Neuer Beweis für die Invarianz der Dimensionzahl und des Gebietes, Abh. Math. Sem. Univ. Hamburg, 6 (1928), 265-272.
• A. W. Tucker, Some topological properties of disk and sphere, in: Proc. of the First Canadian Mathematical Congress, Montreal, 1945, pp. 285-309.