A constructive fixed point theorem for min-max functions

Dynamics and Stability of Systems

Volumn 14, Issue 4, 1999, Pages 407-433

  Cochet Terrasson, Jean ^a,b   Gaubert, Stéphane ^a   Gunawardena, Jererny ^a  

^a INRIA   (France)

^b HEWLETT PACKARD LABORATORIES   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0347667444     PISSN: 02681110     EISSN: None     Source Type: Journal    
DOI: 10.1080/026811199281967     Document Type: Article

References (41)

1. Baccelli, F. and Mairesse, J. (1998) Ergodic theorems for stochastic operators and discrete event systems. Idempotency (ed. J. Gunawardena; Publications of the Isaac Newton Institute, Cambridge University Press, Cambridge).
2. Baccelli, F., Cohen, G., Olsder, G. J. and Quadrat, J.-P. (1992) Synchronization and Linearity (Wiley Series in Probability and Mathematical Statistics, John Wiley, New York).
3. Burns, S. M. (1990) Performance analysis and optimization of asynchronous circuits. PhD thesis, California Institute of Technology.
4. Cao, Z.-Q., Kirn, K. H. and Roush, R. W. (1984) Incline algebra and applications. Mathematics and its Applications (Ellis-Horwood).
5. Cochet-Terrasson, J. (1996) Modélisation et Méthodes Mathématiques en Économie. Rapport de Stage de DEA, Université de Paris I.
6. Cochet-Terrasson, J. and Gaubert, S. (1996) A min-max analogue of the Howard algorithm. Privately circulated draft. Preliminary version appears as an Appendix to Cochet-Terrasson (1996).
7. Cohen, G. and Quadrat, J.-P. ( 1994) 11th International Conference on Analysis and Optimization of Systems (Springer LNCIS 199).
8. Crandall, Ai. G. and Tartar, L. (1980) Some relations between nonexpansive and order preserving maps. Proceedings oftheAMS 78(3), 385-390.
9. Cuninghame-Green, R. A. (1962) Describing industrial processes with interference and approximating their stead-state behaviour. Operational Research Quarterly 13(1), 95-100.
10. Cuninghame-Green, R. A. (1979) Minimax Algebra (Vol. 166 of Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin).
11. Gaubert, S. (1996) On the Burnside problem for semigroups of matrices in the (max,+ ) algebra. Semigroup Forum 52, 271-292.
12. Gaubert, S. and Gunawardena, J. (1998a) A non-linear hierarchy for discrete event dynamical systems. Proceedings of WODES'98, IEE, Cagliari, Italy, August.
13. Gaubert, S. and Gunawardena, J. (1998b) The duality theorem for min-max functions. Comptes rendu l'Académie des Sciences 326, 43-48.
14. Gaubert, S. and Plus, M. (1997) Methods and applications of (max,+ ) linear algebra. Proceedings STACS'97 (Springer LNCS 1200).
15. Glasserman, P. and Yao, D. D. (1994) Monotone Structure in Discrete Event Systems (Wiley Series in Probability and Mathematical Statistics, John Wiley, New York).
16. Gondran, M. and Minoux, M. (1984) Graphs and Algorithms (Wiley-Interscience Series in Discrete Mathematics, John Wiley, New York).
17. Gunawardena, J. (1993) Timing analysis of digital circuits and the theory of min-max functions. Digest of Technical Papers of the ACAl International Workshop on Timing Issues in the Specification and Synthesis of Digital Systems (ACM).
18. Gunawardena, J. (1994a) Cycle times and fixed points of min-max functions, llth International Conference on Analysis and Optimization of Systems (eds G. Cohen and J.-P. Quadrat, Springer LNCIS 199), pp. 266-272.
19. Gunawardena, J. (1994b) A dynamic approach to timed behaviour. CONCUR'94: Concurrency Tlieory (eds B. Jonsson and J. Farrow, Springer LNCS 836), pp. 178-193.
20. Gunawardena, J. (1994c) Min-max functions. Discrete Event Dynamic Systems 4, 377-406.
21. Gunawardena, J. (1998a) Idempoteney (Publications of the Isaac Newton Institute, Cambridge University Press, Cambridge).
22. Gunawardena, J. (1998b) An introduction to idempotency. Idempoteney (ed. J. Gunawardena; Publications of the Isaac Newton Institute, Cambridge University Press, Cambridge).
23. Gunawardena, J. and Keane, M. (1995) On the existence of cycle times for some nonexpansive maps. Technical Report HPL-BRIMS-95-003, Hewlett-Packard Labs.
24. Ho, Y. C. (ed.) (1989) Special issue on Dynamics of Discrete Event Systems. Proceedings of the IEEE 77(1).
25. Johnstone, P. T. (1982) Stone Spaces (Vol. 3 of Studies in Advanced Mathematics, Cambridge University Press, Cambridge).
26. Karp, R. M. (1978) A characterization of the minimum cycle mean in a digraph. Discrete Mathematics 23,309-311.
27. Kolokoltsov, V. N. (1992) On linear, additive and homogenous operators in idempotent analysis. Idempotent Analysis (eds V. P. Maslov and S. N. Samborskii; Vol. 13 of Advances in Soviet Mathematics, American Mathematical Society).
28. Maslov, V. P. and Samborskiï, S. N. (eds) (1992) Idempotent Analysis (Vol. 13 of Advances in Soviet Mathematics, American Mathematical Society).
29. Mine, H. (1988) Nonnegative Matrices (Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley, New York).
30. Nussbaum, R. D. (1990) Hubert's projective metric and iterated nonlinear maps. Memoirs of the AMS 75(391).
31. Nussbaum, R. D. (1991) Convergence of iterates of a nonlinear operator arising in statistical mechanics. Nonlinearity 4, 1223-1240.
32. Olsder, G. J. (1991) Eigenvalues of dynamic max-min systems. Discrete Event Dynamic Systems 1, 177-207.
33. Pin, J.-E. (1998) Tropical semirings. Idempoteney (ed. J. Gunawardena; Publications of the Isaac Newton Institute, Cambridge University Press, Cambridge).
34. Puterman, Ai. L. (1990) Markov decision processes. Handbook on Operations Research and Management Science (North-Holland, Amsterdam), pp. 331-434.
35. Rajsbaum, S. and Sidi, M. (1994) On the performance of synchronized programs in distributed networks with random processing times and transmission delays. IEEE Transactions on Parallel and Distributed Systems 5, 939-950.
36. Sakallah, K. A., Mudge, T. N. and Olukotun, O. A. (1990) checkTc and minTc: timing verification and optimal clocking of synchronous digital circuits. Digest of Technical Papers of the IEEE International Conference on Computer-Aided Design of Integrated Circuits (IEEE Computer Society), pp. 552-555.
37. Simon, I. (1994) On semigroups of matrices over the tropical semiring. Theoretical Informatics and Applications 28, 277-294.
38. Szymanski, T. and Shenoy, N. (1992) Verifying clock schedules. Digest of Technical Papers of the IEEE International Conference on Computer-Aided Design of Integrated Circuits (IEEE Computer Society), pp. 124-131.
39. Vincent, J. M. (1997) Some ergodic results on stochastic iterative discrete even systems. Discrete Event Dynamic Systems 7, 209-233.
40. Whittle, P. (1986) Optimization over Time (John Wiley, New York).
41. Zimmermann, U. (1981) Linear and Combinatorial Optimization in Ordered Algebraic Structures (Vol. 10 of Annals of Discrete Mathematics, North-Holland, Amsterdam).