Discrete Asymptotic Reachability via Expansions in Non-integer Bases

Anna Chiara Lai, Paola Loreti

2012

Abstract

Aim of this paper is to show the connection between the theory of expansions in non-integer bases and discrete control systems. This idea is supported by an example of application, in the framework of robotics. We show how a model of multi-phalanx self-similar robot hand may be studied by means of results and techniques coming from non-standard numeration systems and related tools, like Iterated Function Systems (IFS) and, more generally, fractal geometry.

References

  1. Chitour, Y. and Piccoli, B. (2001). Controllability for discrete control systems with a finite control set. Mathematics of Control Signal and Systems, 14:173-193.
  2. Falconer, K. J. (1990). Fractal geometry. John Wiley & Sons Ltd., Chichester. Mathematical foundations and applications.
  3. Hutchinson, J. (1981). Fractals and self-similarity. Indiana Univ. J. Math, 30:713-747.
  4. Komornik, V. and Loreti, P. (2007). Expansions in complex bases. Canad. Math. Bull., 50(3):399-408.
  5. Komornik, V. and Loreti, P. (2010). Universal expansions in negative and complex bases. Integers, 10:A50, 669- 679.
  6. Lai, A. C. (2011). Minimal unique expansions with digits in ternary alphabets. Indagationes Mathematicae, 21(12):1 - 15.
  7. Lai, A. C. and Loreti, P. (2012). Robot's finger and expansions in non-integer bases. Networks and Heterogeneus Media, 7(1):71-111.
  8. Mihail, A. and Miculescu, R. (2009). The shift space for an infinite iterated function system. Math. Rep. (Bucur.), 11:(1), 21-32.
  9. Rényi, A. (1957). Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar, 8:477-493.
  10. Sidorov, N. (2003). Almost every number has a continuum of b-expansions. Amer. Math. Monthly, 110(9):838- 842.
Download


Paper Citation


in Harvard Style

Lai A. and Loreti P. (2012). Discrete Asymptotic Reachability via Expansions in Non-integer Bases . In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO, ISBN 978-989-8565-22-8, pages 360-365. DOI: 10.5220/0004092403600365


in Bibtex Style

@conference{icinco12,
author={Anna Chiara Lai and Paola Loreti},
title={Discrete Asymptotic Reachability via Expansions in Non-integer Bases},
booktitle={Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO,},
year={2012},
pages={360-365},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0004092403600365},
isbn={978-989-8565-22-8},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO,
TI - Discrete Asymptotic Reachability via Expansions in Non-integer Bases
SN - 978-989-8565-22-8
AU - Lai A.
AU - Loreti P.
PY - 2012
SP - 360
EP - 365
DO - 10.5220/0004092403600365