Skip to main content
SpringerLink
Log in

Algebraic and Analytic Randomness

  • Conference paper
  • First Online:
Noise, Oscillators and Algebraic Randomness

Part of the book series: Lecture Notes in Physics ((LNP,volume 550))

Abstract

Is it possible to mathematically define words like: randomness, chaos, disorder, irregularity, complexity, or like: determinism, order, periodicity, regularity, simplicity? Are there concepts in between (quasi-periodicity)? How do these concepts fit objects from physics, e.g., glasses, crystals, quasi-crystals? We try to describe and compare various notions used in mathematics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allouche, J.-P. (1987) Automates finis en théorie des nombres. Exposition. Math. 5, 239–266

    MATH  MathSciNet  Google Scholar 

  2. Allouche, J.-P., Davison, L., Queffélec M. (1999) Transcendence of Sturmian or morphic continued fractions. In preparation

    Google Scholar 

  3. Allouche, J.-P. (1994) Sur la complexité des suites infinies. Bull. Belg. Math. Soc. 1, 133–143

    MATH  MathSciNet  Google Scholar 

  4. Allouche, J.-P., Haeseler, F. von, Peitgen, H.-O., Petersen, A., Skordev, G. (1997) Automaticity of double sequences generated by one-dimensional linear cellular automata. Theoret. Comput. Sci. 188, 195–209

    Article  MATH  MathSciNet  Google Scholar 

  5. Allouche, J.-P., Haeseler, F. von, Peitgen, H.-O., Skordev, G. (1996) Linear cellular automata, finite automata and Pascal’s triangle. Disc. Appl. Math. 66, 1–22

    Article  MATH  Google Scholar 

  6. Allouche, J.-P., Mendès France, M. (1995) Automata and automatic sequences in:Beyond quasicrystals, Éd. F. Axel, D. Gratias. Springer/Les Éditions de Physique, pp. 293–367

    Google Scholar 

  7. Allouche, J.-P., Shallit J. (1999) The ubiquitous Prouhet-Thue-Morse sequence. Proceedings of SETA’98, to appear

    Google Scholar 

  8. Allouche, J.-P., Zamboni, L. Q. (1998) Algebraic irrational binary numbers cannot be fixed points of non-trivial constant length or primitive morphisms. J. Number Theory 69, 119–124

    Article  MATH  MathSciNet  Google Scholar 

  9. Berthé, V. (1995) Entropy in deterministic and random systems in: Beyond quasicrystals, Éd. F. Axel, D. Gratias. Springer/Les Éditions de Physique, pp. 441–463

    Google Scholar 

  10. Bertrandias, J.-P. (1966) Espaces de fonctions bornées et continues en moyenne asymptotique d’ordre p. Bull. Soc. Math. France, Suppl., Mém. 5, 1–106

    MathSciNet  Google Scholar 

  11. Champernowne, D. G. (1933) The construction of decimals normal in the scale of ten. J. London Math. Soc. 8, 254–260

    Article  MATH  Google Scholar 

  12. Christol G. (1979) Ensembles presque périodiques k-reconnaissables. Theoret. Comput. Sci. 9, 141–145

    Article  MATH  MathSciNet  Google Scholar 

  13. Christol, G., Kamae, T., Mendès France, M., Rauzy, G. (1980) Suites algébriques, automates et substitutions. Bull. Soc. Math. France 108, 401–419

    MATH  MathSciNet  Google Scholar 

  14. Cobham, A. (1969) On the base-dependence of sets of numbers recognizable by finite automata. Math. Systems Theory 3, 186–192

    Article  MATH  MathSciNet  Google Scholar 

  15. Cobham, A. (1972) Uniform tag sequences. Math. Systems Theory 6, 164–192

    Article  MATH  MathSciNet  Google Scholar 

  16. Dekking, F. M. (1977) Transcendance du nombre de Thue-Morse. C. R. Acad. Sci. Paris Sér. I 285, 157–160

    MATH  MathSciNet  Google Scholar 

  17. Dekking, F. M., Mendès France, M., Poorten, A. J. van der (1982) Folds!. Math. Intelligencer 4, 130–138, 173-181, 190-195

    MATH  Google Scholar 

  18. Ehrenfeucht, A., Lee, K. P., Rozenberg, G. (1975) Subword complexities of various classes of deterministic developmental languages without interaction. Theoret. Comput. Sci. 1, 59–75

    Article  MATH  MathSciNet  Google Scholar 

  19. Korec, I. (1990) Pascal triangles modulo n and modular trellises. Comput. Artif. Intell. 9, 105–113

    MATH  MathSciNet  Google Scholar 

  20. Ferenczi, S., Kása, Z. (1999) Complexity for finite factors of infinite sequences. Theoret. Comput. Sci. 218, 177–195

    Article  MATH  MathSciNet  Google Scholar 

  21. Ferenczi, S., Mauduit, C. (1997) Transcendence of numbers with a low complexity expansion. J. Number Theory 67, 146–161

    Article  MATH  MathSciNet  Google Scholar 

  22. Loxton, J. H., Poorten, A. J. van der (1988) Arithmetic properties of automata: regular sequences. J. Reine Angew. Math. 392, 57–69

    MATH  MathSciNet  Google Scholar 

  23. Mahler, K. (1929) Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Math. Annalen 101, 342–266. (1930) Corrigendum 103, 532

    Article  MathSciNet  Google Scholar 

  24. Mahler, K. (1937) Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen. Proc. Konin. Neder. Akad. Wet. 40, 421–428

    MATH  Google Scholar 

  25. Morse, M. (1921) Recurrent geodesics on a surface of negative curvature. Trans. Amer. Math. Soc. 22, 84–100

    Article  MATH  MathSciNet  Google Scholar 

  26. Morse, M., Hedlund, G. A. (1938) Symbolic Dynamics. Amer. J. Math. 60, 815–866

    Article  MATH  MathSciNet  Google Scholar 

  27. Morse, M., Hedlund, G. A. (1940) Symbolic Dynamics II. Sturmian trajectories. Amer. J. Math. 62, 1–42

    Article  MATH  MathSciNet  Google Scholar 

  28. Parry, W. (1960) On the â-expansions of real numbers. Acta Math. Acad. Sci. Hung. 11, 401–416

    Article  MATH  MathSciNet  Google Scholar 

  29. Prouhet, E. (1851) Mémoire sur quelques relations entre les puissances des nombres. C. R. Acad. Sci. Paris Sér. I 33, 225

    Google Scholar 

  30. Queffélec, M. (1998) Transcendance des fractions continue de Thue-Morse. J. Number Theory 73, 201–211

    Article  MATH  MathSciNet  Google Scholar 

  31. Rényi, A. (1957) Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8, 477–493

    Article  MATH  Google Scholar 

  32. Rudin, W. (1959) Some theorems on Fourier coefficients. Proc. Amer. Math. Soc. 10, 855–859

    MATH  MathSciNet  Google Scholar 

  33. Salon, O. (1987) Suites automatiques à multi-indices et algébricité. C. R. Acad. Sci. Paris Sér. I 305, 501–504

    MATH  MathSciNet  Google Scholar 

  34. Salon, O. (1989) Propriétés arithmétiques des automates multidimensionnels. Thèse, Université Bordeaux I.

    Google Scholar 

  35. Shallit, J. (1988) Ageneralization of automatic sequences. Theoret. Comput. Sci. 61, 1–16

    Article  MATH  MathSciNet  Google Scholar 

  36. Shapiro, H. S. (1951) Extremal problems for polynomials and power series. Thesis, MIT

    Google Scholar 

  37. Thue, A. (1906) Über unendliche Zeichenreihen, Norske vid. Selsk. Skr. Mat. Nat. Kl. 7, 1–22. Reprinted in: (1997) Selected mathematical papers of Axel Thue, T. Nagell, ed., Universitetsforlaget, Oslo, pp. 139-158

    Google Scholar 

  38. Thue, A. (1912) Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske vid. Selsk. Skr. Mat. Nat. Kl. 1, 1–67. Reprinted in: (1997) Selected mathematical papers of Axel Thue, T. Nagell, ed., Universitetsforlaget, Oslo, pp. 413-478

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS, LRI, Bâtiment 490, F-91405, Orsay Cedex, France

    Jean-Paul Allouche

Authors
  1. Jean-Paul Allouche
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratoire de Physique et Merologie, des Oscillateurs du CNRS, 32 Avenue de l’Observatoire, 25044, Besançon Cedex, France

    Michel Planat

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Allouche, JP. (2000). Algebraic and Analytic Randomness. In: Planat, M. (eds) Noise, Oscillators and Algebraic Randomness. Lecture Notes in Physics, vol 550. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45463-2_17

Download citation

  • DOI: https://doi.org/10.1007/3-540-45463-2_17

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67572-3

  • Online ISBN: 978-3-540-45463-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation