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Autoren: 
van der Laan, Gerard
Talman, Dolf
Yang, Zaifu
Datum: 
2007
Reihe/Nr.: 
Tinbergen Institute Discussion Paper 07-084/1
Zusammenfassung: 
Tucker's well-known combinatorial lemma states that for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set {1,2,...n,-1,-2,....-n} with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set {1,2,...n,-1,-2,....-n}. Using a constructive approach we prove two combinatorial theorems of Tucker type, stating that under some mild conditions there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the same unit cube. These theorems will be used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions.
Schlagwörter: 
Sperner lemma
Tucker lemma
integer labeling
simplicial algorithm
discrete nonlinear equations
JEL: 
C61
C62
C68
C72
C58
Dokumentart: 
Working Paper
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