On the mathematics of emergence

Japanese Journal of Mathematics

Volumn 2, Issue 1, 2007, Pages 197-227

  Cucker, Felipe ^a   Smale, Steve ^b  

^a CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^b TOYOTA TECHNOLOGICAL INSTITUTE AT CHICAGO   (United States)

Author keywords

Consensus reaching; Emergence; Flocking

Indexed keywords

EID: 34047112216     PISSN: 02892316     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11537-007-0647-x     Document Type: Article

References (19)

1. R. Agaev and P. Chebotarev, On the spectra of nonsymmetric Laplacian matrices,Linear Algebra Appl., 399 (2005), 157-168.
2. Y. L. Chuang, Y. R. Huang, M. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, preprint, 2006.
3. F. Chung, Spectral Graph Theory, Amer. Math. Soc., Providence, RI., 1997.
4. F. Chung, Laplacians and the Cheeger inequality for directed graphs, Ann. Comb., 9 (2005), 1-19.
5. F. Cucker and S. Smale, Best choices for regularization parameters in learning theory, Found. Comput. Math., 2 (2002), 413-428.
6. F. Cucker and S. Smale, Emergent behavior in flocks, to appear at IEEE Trans. Automat. Control (2007).
7. F. Cucker, S. Smale and D. X. Zhou, Modeling language evolution, Found. Comput. Math., 4 (2004), 315-343.
8. G. Flierl, D. Grünbaum, S. Levin and D. Olson, From individuals to aggregations: The interplay between behavior and physics, J. Theoret. Biol., 196 (1999), 397-454.
9. M. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra, Pure Appl. Math., 60, Academic Press, 1974.
10. A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988-1001.
11. J. Ke, J. Minett, C.-P. Au and W. S.-Y. Wang, Self-organization and selection in the emergence of vocabulary, Complexity, 7 (2002), 41-54.
12. B. Mohar, The Laplacian spectrum of graphs, In: Graph Theory, Combinatorics, and Applications, (eds. Y. Alavi, G. Chartrand, O. R. Oellermann and A. J. Schwenk), 2, John Wiley & Sons, 1991, pp. 871-898.
13. P. Niyogi, The Computational Nature of Language Learning and Evolution, The MIT Press, 2006.
14. H. G. Tanner, A. Jadbabaie and G. J. Pappas, Stable flocking of mobile agents, Part I: Fixed topology, In: Proceedings of the 42nd IEEE Conference on Decision and Control, 2003, pp. 2010-2015.
15. H. G. Tanner, A. Jadbabaie and G. J. Pappas, Stable flocking of mobile agents, Part II: Dynamic topology, In: Proceedings of the 42nd IEEE Conference on Decision and Control, 2003, pp. 2016-2021.
16. C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.
17. J. N. Tsitsiklis, Problems in decentralized decision making and computation, PhD thesis, Department of EECS, MIT, 1984.
18. J. N. Tsitsiklis, D. P. Bertsekas and M. Athans, Distributed asynchronous deterministic and stochastic gradient optimization algorithms, IEEE Trans. Automat. Control, 31 (1986), 803-812.
19. T. Vicsek, A. Czirók, E. Ben-Jacob and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.