Publication: Ultrametrics, Banach's fixed point theorem and the Riordan group
Loading...
Files
Full text at PDC
Publication Date
2008-07-28
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Citations
Exportar
Abstract
We interpret the reciprocation process in K[[x]] as a fixed point problem related to contractive functions for certain adequate ultrametric spaces. This allows us to give a dynamical interpretation of certain arithmetical triangles introduced herein. Later we recognize, as it special case of our construction, the so-called Riordan group which is a device used in combinatorics. In this manner we give a new and alternative way to construct the proper Riordan arrays. Our point of view allows us to give a natural metric on the Riordan group turning this group into a topological group. This construction allows us to recognize a countable descending chain of normal subgroups.
Description
UCM subjects
Unesco subjects
Keywords
Citation
M.A. Amstrong, Groups and Symmetry, Springer-Verlag, 1988.
J. Dugundji, A. Granas, Fixed Point Theory, in: Monografie Matematyczne. Tom 61, vol. I, PWN- Polish Scientific Publishers, 1982.
R.W. Easton, Geometric Methods for Discrete Dynamical Systems, in: Oxford Engineering Science Series, vol. 5, Oxford University Press,1998.
G.P. Egorychev, E.V. Zima, Decomposition and group theoretic characterization of pairs of inverse relations of the Riordan type, Acta Appl.Math. 85 (2005) 93–109.
Hirsch, Pugh, Stable Manifolds and Hyperbolic Sets, in: Global Analysis, vol. XIV, Amer. Math. Soc., 1970, pp. 133–164.
I.C. Huang, Inverse relations and Schauder bases, J. Combin. Theory S.A 97 (2002) 203–224.
D. Kalman, R. Mena, The Fibonacci numbers-exposed, Math. Mag. 76 (2003) 167–181.
D. Merlini, D.G. Rogers, R. Sprugnoli, M.C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math. 49 (2) (1997) 301–320.
D. Merlini, D.G. Rogers, R. Sprugnoli, M.C. Verri, Underdiagonal lattice paths with unrestricted steps, Discrete Appl. Math. 91 (1999) 197–213.
D. Merlini, M.C. Verri, Generating trees and proper Riordan arrays, Discrete Math. 218 (2000) 167–183.
D. Merlini, R. Sprugnoli, A Riordan Arrays proof of a curious identity, Electron. J. Combin. Number Theory 2 (2002).
D. Merlini, R. Sprugnoli, M.C. Verri, Waiting patterns for a printer, Discrete Appl. Math. 144 (2004) 359–373.
D. Merlini, R. Sprugnoli, M.C. Verri, The Akiyama–Tanigawa transformation, Electron. J. Combin. Number Theory 5 (2005).
G. Rangan, Non-archimedian metrizability of topological groups, Fund. Math. LXVIII (1970) 179–182.
J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, 1978.
Alain M. Robert, A course in p-adic analysis, in: Graduate Texts in Mathematics, vol. 198, Springer, NY, 2000.
D.G. Rogers, Pascal triangles, Catalan numbers and renewal arrays, Discrete Math. 22 (1978) 301–310.
L.W. Shapiro, S. Getu, W.J. Woan, L. Woodson, The Riordan group, Discrete Appl. Math. 34 (1991) 229–239.
L.W. Shapiro, Bijections and the Riordan group, Theoret. Comput. Sci. 307 (2003) 403–413.
J. Sotomayor, Lic¸oes de equac¸oes diferenciais ordin´arias, in: Projeto Euclides, IMPA, Brasil, 1979.
R. Sprugnoli, Riordan arrays and combinatorial sums, Discrete Math. 132 (1994) 267–290.
R. Sprugnoli, Riordan arrays and the Abel–Gould identity, Discrete Math. 142 (1995) 213–233.
X. Zhao, S. Ding, T. Wang, Some summation rules related to Riordan arrays, Discrete Math. 281 (2004) 295–307.