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Uspekhi Mat. Nauk, 2003, Volume 58, Issue 4(352), Pages 3–28 (Mi umn641)  

This article is cited in 14 scientific papers (total in 15 papers)

Topology and statistics of formulae of arithmetics

V. I. Arnol'd

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This paper surveys some recent and classical investigations of geometric progressions of residues that generalize the little Fermat theorem, connect this topic with the theory of dynamical systems, and estimate the degree of chaotic behaviour of systems of residues forming a geometric progression and displaying a distinctive mutual repulsion. As an auxiliary tool, the graphs of squaring operations for the elements of finite groups and rings are studied. For commutative groups the connected components of these graphs turn out to be attracting cycles homogeneously equipped with products of binary rooted trees, the algebra of which is also described in the paper. The equipping with trees turns out to be homogeneous also for the graphs of symmetric groups of permutations, as well as for the groups of even permutations.

DOI: https://doi.org/10.4213/rm641

Full text: PDF file (1073 kB)
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English version:
Russian Mathematical Surveys, 2003, 58:4, 637–664

Bibliographic databases:

Document Type: Article
UDC: 51
MSC: Primary 11B50, 11K99; Secondary 37A45, 05C05
Received: 05.01.2003

Citation: V. I. Arnol'd, “Topology and statistics of formulae of arithmetics”, Uspekhi Mat. Nauk, 58:4(352) (2003), 3–28; Russian Math. Surveys, 58:4 (2003), 637–664

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. I. Arnol'd, “The Topology of Algebra: Combinatorics of Squaring”, Funct. Anal. Appl., 37:3 (2003), 177–190  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Aicardi F., “Empirical estimates of the average orders of orbits period lengths in Euler groups”, C. R. Math. Acad. Sci. Paris, 339:1 (2004), 15–20  crossref  mathscinet  zmath  isi  scopus  scopus
    3. V. I. Arnol'd, “Fermat Dynamics, Matrix Arithmetics, Finite Circles, and Finite Lobachevsky Planes”, Funct. Anal. Appl., 38:1 (2004), 1–13  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Arnold V., “Number-theoretical turbulence in Fermat-Euler arithmetics and large young diagrams geometry statistics”, J. Math. Fluid Mech., 7, Suppl. 1 (2005), S4–S50  crossref  mathscinet  zmath  isi  scopus
    5. A. V. Zarelua, “On matrix analogs of Fermat's little theorem”, Math. Notes, 79:5 (2006), 783–796  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. Uribe-Vargas R., “Topology of dynamical systems in finite groups and number theory”, Bull. Sci. Math., 130:5 (2006), 377–402  crossref  mathscinet  zmath  isi  scopus
    7. Arnold V.I., “Complexity of finite sequences of zeros and ones and geometry of finite spaces of functions”, Funct. Anal. Other Math., 1:1 (2007), 1–15  crossref  mathscinet
    8. “Vladimir Igorevich Arnol'd (on his 70th birthday)”, Russian Math. Surveys, 62:5 (2007), 1021–1030  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    9. Shparlinski I.E., “On some dynamical systems in finite fields and residue rings”, Discrete Contin. Dyn. Syst., 17:4 (2007), 901–917  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    10. A. V. Zarelua, “On Congruences for the Traces of Powers of Some Matrices”, Proc. Steklov Inst. Math., 263 (2008), 78–98  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    11. Fel L.G., “Weak asymptotics in the 3-dim Frobenius problem”, Funct. Anal. Other Math., 2:2-4 (2009), 179–202  crossref  mathscinet  zmath
    12. Marcin Mazur, Bogdan V. Petrenko, “Generalizations of Arnold’s version of Euler’s theorem for matrices”, Jpn J Math, 5:2 (2010), 183  crossref  mathscinet  zmath  isi  scopus
    13. A. A. Evdokimov, A. L. Perezhogin, “Discrete dynamical systems of a circulant type with linear functions at vertices of network”, J. Appl. Industr. Math., 6:2 (2012), 160–166  mathnet  crossref  mathscinet  zmath  elib
    14. Yu. V. Merekin, “The Shannon function of computation of the Arnold complexity of length $2^n$ binary words”, J. Appl. Industr. Math., 7:2 (2013), 229–233  mathnet  crossref  mathscinet
    15. Yu. V. Merekin, “The Shannon function for calculating the Arnold complexity of length $2^n$ binary words for arbitrary $n$”, J. Appl. Industr. Math., 9:1 (2015), 98–109  mathnet  crossref  mathscinet
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