Abstract
In this paper we consider two labelling rules used in simplicial fixed point algorithms. The first one is the standard labelling rule from an n-dimensional set to the set of integers {1, …, n+1}. The second one is a labelling to the set {±i|i = 1, …, n}. The main purpose of the paper is to compare the two rules. We define the orientation of a completely labelled simplex and give some generalizations of the lemma of Sperner and the related lemma of Knaster, Kuratowski and Mazurkiewicz. Also, for both labelling rules it is shown that the Brouwer degree can be obtained from the completely labelled simplices.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allgower, E.L. and Georg, K. (1980), Simplicial and continuation methods for approximating fixed points and solutions to systems of equations, SIAM Rev. 22, 28–85.
Eaves, B.C. and Scarf, H.E. (1976). The solution of systems of piecewise linear equations, Math. Oper. Res. 1, 1–27.
Fan, K. (1952), A generalization of Tucker's combinatorial lemma with topological applications, Ann. of Math. 56, 431–437.
Fan, K. (1960), Combinatorial properties of certain simplicial and cubical vertex maps, Arch. Math. 11, 368–377.
Fan, K. (1967), Simplicial maps from an orientable n-pseudomanifold into Sm with the octohedral triangulation, J. Combinatorial Theory 2, 588–602.
Fan, K. (1970), A combinatorial property of pseudomanifolds and covering properties of simplexes, J. Math. Anal. Appl. 31, 68–80.
Forster, W. (1980), An application of the generalized Sperner lemma to the computation of fixed points in arbitrary complexes, in W. Forster (ed.), Numerical Solution of Highly Nonlinear Problems. North-Holland, 219–232.
Freund, R.M. and Todd, M.J. (1979), A constructive proof of Tucker's combinatorial lemma, Cornell University, Ithaca.
Knaster, B., Kuratowski, C. and Mazurkiewicz, S. (1929), Ein Beweis der Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14, 132–137.
Krasnosel'skii, M.A. (1964), Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press.
Kuhn, H.W. (1960), Some combinatorial lemmas in topology. I.B.M. J. Res. Develop. 4, 518–524.
Laan, G. van der (1980), Simplicial fixed point algorithms, Dissertation. Free University, Amsterdam.
Laan, G. van der and Talman, A.J.J. (1978), On the computation of fixed points in the product space of unit simplices and an application to non cooperative N-person games, Free University, Amsterdam.
Laan, G. van der and Talman, A.J.J. (1979), Interpretation of the variable dimension fixed point algorithm with an artificial level, Free University, Amsterdam.
Laan, G. van der and Talman, A.J.J. (1980), Convergence and properties of recent variable dimension algorithms, in: W. Forster (ed.), Numerical Solution of Highly Nonlinear Problems. North-Holland, 3–36.
Laan, G. van der and Talman, A.J.J. (1981), A class of simplicial subdivisions for restart fixed point algorithms without an extra dimension, Mathematical Programming 20, 33–48.
Lefschetz, S. (1949), Introduction to Topology, Princeton University Press.
Lemke, C.E. and Grotzinger, S.J. (1976), On generalizing Shapley's index theory to labelled pseudo manifolds, Math. Prog. 10, 245–262.
Prüfer, M. and Siegberg, H.W. (1979), On computational aspects of topological degree in Rn, in: H.O. Peitgen and H.O. Walther (eds.), Functional Differential Equations and Approximation of Fixed Points. Springer-Verlag, 410–433.
Reiser, P.M. (1978), Ein hybrides Verfahren zur Lösung von nichtlinearen Komplimentaritätsproblemen und seine Konvergenzeigenschaften, Dissertation, Eidgenössischen Technischen Hochschule, Zürich.
Shapley, L.S. (1974), A note on the Lemke-Howson algorithm, Math. Prog. Stud. 1, 175–189.
Siegberg, H.W. (1980), Brouwer degree: history and numerical computation, in: W. Forster (ed.), Numerical Solution of Highly Nonlinear Problems. North-Holland, 389–416.
Sperner, E. (1928), Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Sem. Univ. Hamburg 6, 265–272.
Sperner, E. (1980), Fifty years of further development of a combinatorial lemma, in: W. Forster (ed.), Numerical Solution of Highly Nonlinear Problems, North-Holland, 183–218.
Talman, A.J.J. (1980), Variable dimension fixed point algorithms and triangulations, Dissertation, Free University, Amsterdam.
Todd, M.J. (1976a), The Computation of Fixed Points and Applications, Springer-Verlag.
Todd, M.J. (1976b), Orientation in complementary pivot algorithms, Math. Oper. Res. 1, 54–66.
Todd, M.J. (1980), Global and local convergence and monotonicity results for a recent variable-dimension simplicial algorithm, in: W. Forster (ed.), Numerical Solution of Highly Nonlinear Problems. North-Holland, 43–70.
Todd, M.J. and Wright, A.H. (1979), A variable-dimension simplicial algorithm for antipodal fixed point theorems, Cornell University, Ithaca.
Tucker, A.W. (1945), Some topological properties of disk and sphere, Proc. First Canadian Math. Congress (Montreal), 285–309.
Wolsey, L.A. (1977), Cubical Sperner lemmas as application of generalized complementary pivoting, J. Comb. Theory A 23, 78–87.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1981 Springer-Verlag
About this paper
Cite this paper
Laan, G.V.D., Talman, A.J.J. (1981). Labelling rules and orientation: On Sperner's lemma and brouwer degree. In: Allgower, E.L., Glashoff, K., Peitgen, HO. (eds) Numerical Solution of Nonlinear Equations. Lecture Notes in Mathematics, vol 878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090684
Download citation
DOI: https://doi.org/10.1007/BFb0090684
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-10871-9
Online ISBN: 978-3-540-38781-7
eBook Packages: Springer Book Archive