Combinatorial Integer Labeling Theorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations

Arbeitspapier

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.

Angaben zum Objekt

• Sprache: Englisch

Verweise und Beziehungen

• Erschienen in: Series: Tinbergen Institute Discussion Paper ; No. 07-084/1

Klassifikation und Themen

• Klassifikation: Wirtschaft ; Optimization Techniques; Programming Models; Dynamic Analysis ; Existence and Stability Conditions of Equilibrium ; Computable General Equilibrium Models ; Noncooperative Games ; Financial Econometrics
• Thema: Sperner lemma ; Tucker lemma ; integer labeling ; simplicial algorithm ; discrete nonlinear equations ; Nichtlineare Optimierung ; Nichtkooperatives Spiel ; Theorie

Beteiligte, Orts- und Zeitangaben

• Ereignis: Geistige Schöpfung
• (wer): van der Laan, Gerard ; Talman, Dolf ; Yang, Zaifu
• Ereignis: Veröffentlichung
• (wer): Tinbergen Institute
• (wo): Amsterdam and Rotterdam
• (wann): 2007

Weitere Informationen

• Handle: http://hdl.handle.net/10419/86411
• Letzte Aktualisierung: 10.03.2025, 11:43 MEZ