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Labelling rules and orientation: On Sperner's lemma and brouwer degree

  • G. V. D. Laan
  • A. J. J. Talman
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 878)

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.

Keywords

Sign Vector Boundary Property Standard Labelling Unit Simplex Fixed Point Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Eaves, B.C. and Scarf, H.E. (1976). The solution of systems of piecewise linear equations, Math. Oper. Res. 1, 1–27.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Fan, K. (1952), A generalization of Tucker's combinatorial lemma with topological applications, Ann. of Math. 56, 431–437.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Fan, K. (1960), Combinatorial properties of certain simplicial and cubical vertex maps, Arch. Math. 11, 368–377.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Fan, K. (1967), Simplicial maps from an orientable n-pseudomanifold into Sm with the octohedral triangulation, J. Combinatorial Theory 2, 588–602.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Fan, K. (1970), A combinatorial property of pseudomanifolds and covering properties of simplexes, J. Math. Anal. Appl. 31, 68–80.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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.Google Scholar
  8. Freund, R.M. and Todd, M.J. (1979), A constructive proof of Tucker's combinatorial lemma, Cornell University, Ithaca.zbMATHGoogle Scholar
  9. Knaster, B., Kuratowski, C. and Mazurkiewicz, S. (1929), Ein Beweis der Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14, 132–137.zbMATHGoogle Scholar
  10. Krasnosel'skii, M.A. (1964), Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press.Google Scholar
  11. Kuhn, H.W. (1960), Some combinatorial lemmas in topology. I.B.M. J. Res. Develop. 4, 518–524.MathSciNetzbMATHGoogle Scholar
  12. Laan, G. van der (1980), Simplicial fixed point algorithms, Dissertation. Free University, Amsterdam.zbMATHGoogle Scholar
  13. 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.Google Scholar
  14. 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.zbMATHGoogle Scholar
  15. 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.Google Scholar
  16. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Lefschetz, S. (1949), Introduction to Topology, Princeton University Press.Google Scholar
  18. Lemke, C.E. and Grotzinger, S.J. (1976), On generalizing Shapley's index theory to labelled pseudo manifolds, Math. Prog. 10, 245–262.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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.Google Scholar
  20. Reiser, P.M. (1978), Ein hybrides Verfahren zur Lösung von nichtlinearen Komplimentaritätsproblemen und seine Konvergenzeigenschaften, Dissertation, Eidgenössischen Technischen Hochschule, Zürich.zbMATHGoogle Scholar
  21. Shapley, L.S. (1974), A note on the Lemke-Howson algorithm, Math. Prog. Stud. 1, 175–189.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Siegberg, H.W. (1980), Brouwer degree: history and numerical computation, in: W. Forster (ed.), Numerical Solution of Highly Nonlinear Problems. North-Holland, 389–416.Google Scholar
  23. Sperner, E. (1928), Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Sem. Univ. Hamburg 6, 265–272.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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.Google Scholar
  25. Talman, A.J.J. (1980), Variable dimension fixed point algorithms and triangulations, Dissertation, Free University, Amsterdam.zbMATHGoogle Scholar
  26. Todd, M.J. (1976a), The Computation of Fixed Points and Applications, Springer-Verlag.Google Scholar
  27. Todd, M.J. (1976b), Orientation in complementary pivot algorithms, Math. Oper. Res. 1, 54–66.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 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.Google Scholar
  29. Todd, M.J. and Wright, A.H. (1979), A variable-dimension simplicial algorithm for antipodal fixed point theorems, Cornell University, Ithaca.zbMATHGoogle Scholar
  30. Tucker, A.W. (1945), Some topological properties of disk and sphere, Proc. First Canadian Math. Congress (Montreal), 285–309.Google Scholar
  31. Wolsey, L.A. (1977), Cubical Sperner lemmas as application of generalized complementary pivoting, J. Comb. Theory A 23, 78–87.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • G. V. D. Laan
    • 1
  • A. J. J. Talman
    • 2
  1. 1.Interfaculteit der Actuariële Wetenschappen en EconometrieVrije UniversiteitAmsterdam
  2. 2.Yale School of Organization and ManagementNew HavenUSA

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