複體組合學在拓撲，非線性分析，賽局論和數理經濟學中扮演非常重要的角色。1967年樊土畿教授使用挨門挨戶原則證明了在擬流形上組合型的 Stokes 定理，1993年施茂祥教授與李是男教授從幾何觀點發展一般位置映射，pi平衡集、pi次平衡集之相關理論，並且利用這些結果來證明在單純形上具有多重集值標號之組合公式，1998年李是男教授與施茂祥教授將樊土畿的組合公式一般化到多重標號，證明具有多重標號組合型的 Stokes 定理，於是我們提出下列問題:是否存在一個統合的定理能包含樊教授與施教授和李教授的結果呢? 在這篇論文中，我們利用關聯函數的方法證明在擬流形上具有平衡結構的多重組合型 Stokes 定理，我們也獲得具有平衡結構的多重組合型 Sperner 引理。

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.

[1] R.B. Bapat, A constructive proof of a permutation-based generalization of Sperner's lemma, Math. Program. 44 (1989), 113-120.
[2] K. Fan, Simplicial maps from an orientable n-pseudomanifold into Sm with the octahedral triangulation, J. Comb. Theory 2 (1967), 588-602.
[3] D. Gale, Equilibrium in a discrete exchange economy with money, Internat. J. Game Theory 13 (1984), 61-64.
[4] Y.A. Hwang and M.H. Shih, Equilibrium in a market game, Economic Theory 31 (2007), 387-392.
[5] B. Knaster, C.Kuratowski, S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe, Fund. Math. 14 (1929), 132-137.
[6] H.W. Kuhn, A new proof of the fundamental theorem of algebra, Math. Program. Study 1 (1974), 148-158.
[7] S.N. Lee and M.H. Shih, A counting lemma and multiple combinatorial Stokes' theorem, European J. Combin. 19 (1998), 969-979.
[8] 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.
[9] F. Meunier, Combinatorial Stokes' formulae, European J. Combin. 29 (2008), 286-297.
[10] H. Scarf, The approximation of fixed points of continuous mapping, SIAM J. Appl. Math. 15 (1967), 1328-1343.
[11] L.S. Shapley, On balanced games without side payments. In Hu, T.C., Robinson, M.(eds.) Mathematical Program. Math. Res. Cent. Publ. (New York: Academic Press) 30 (1973), 261-290.
[12] M.H. Shih and S.N. Lee, A combinatorial Lefschetz fixed-point formula, J. Combin. Theory Ser. A 61 (1992), 123-129.
[13] M.H. Shih and S.N. Lee, Combinatorial formulae for multiple set-valued labellings, Math. Ann. 296 (1993), 35-61.
[14] E. Sperner, Neuer Beweis fur die Invarianz der Dimensionzahl und des Gebietes, Abh. Math. Sem. Univ. Hamburg 6 (1928), 265-272.
[15] A.W. Tucker, Some topological properties of disk and sphere, in: Proc. of the First Canadian Mathematical Congress, Montreal (1945), 285-309.