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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2003, Volume 37, Issue 3, Pages 20–35 (Mi faa155)

The Topology of Algebra: Combinatorics of Squaring

V. I. Arnol'dab

a Université Paris-Dauphine
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We study the graph each of whose edges connects an element of a given ring with the square of itself. For a finite commutative group (e.g., for the multiplicative group of coprime residue classes modulo a positive integer), we describe this graph explicitly: each of its connected components is an oriented attracting cycle equipped with identical $2^k$-vertex rooted trees of special form whose roots reside on the cycle. We also compute the graphs of permutation groups on not too many elements and of the subgroups of even permutations; the connected components of these graphs are also uniformly equipped cycles.

Keywords: Euler function, Fermat's little theorem, quadratic residues, geometric series, attractor, tree, permutation, Young diagram

DOI: https://doi.org/10.4213/faa155

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English version:
Functional Analysis and Its Applications, 2003, 37:3, 177–190

Bibliographic databases:

Document Type: Article
UDC: 51+515+512+519.1+511+517.938

Citation: V. I. Arnol'd, “The Topology of Algebra: Combinatorics of Squaring”, Funktsional. Anal. i Prilozhen., 37:3 (2003), 20–35; Funct. Anal. Appl., 37:3 (2003), 177–190

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa155
• https://doi.org/10.4213/faa155
• http://mi.mathnet.ru/eng/faa/v37/i3/p20

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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, “Topology and statistics of formulae of arithmetics”, Russian Math. Surveys, 58:4 (2003), 637–664
2. A. V. Zarelua, “On matrix analogs of Fermat's little theorem”, Math. Notes, 79:5 (2006), 783–796
3. “Vladimir Igorevich Arnol'd (on his 70th birthday)”, Russian Math. Surveys, 62:5 (2007), 1021–1030
4. Shparlinski, IE, “On some dynamical systems in finite fields and residue rings”, Discrete and Continuous Dynamical Systems, 17:4 (2007), 901
5. A. V. Zarelua, “On Congruences for the Traces of Powers of Some Matrices”, Proc. Steklov Inst. Math., 263 (2008), 78–98
6. Mazur M., Petrenko B.V., “Generalizations of Arnold's version of Euler's theorem for matrices”, Jpn J Math, 5:2 (2010), 183–189
7. Ramos A.D., Toom A., “Trajectories in Random Monads”, J Stat Phys, 142:1 (2011), 201–219
8. Ramos A.D., Toom A., “Phase Transitions in the Dynamics of Slow Random Monads”, J Stat Phys, 145:5 (2011), 1324–1342
9. R. S. Ismagilov, “A Formula for the Spectra of Differential Operators on Graphs”, Funct. Anal. Appl., 46:2 (2012), 94–99
10. Ramos A.D., Toom A., “Moments and Distributions of Trajectories in Slow Random Monads”, J. Stat. Phys., 147:3 (2012), 623–633
11. Canovas Pena J.S., Linero Bas A., Soler Lopez G., “a Converse Result Concerning the Periodic Structure of Commuting Affine Circle Maps”, J. Nonlinear Sci. Appl., 9:7 (2016), 5041–5060
12. V. S. Kalnitskii, A. N. Petrov, “Lokalnye gladkie sopryazheniya endomorfizmov Frobeniusa”, Geometriya i topologiya. 13, Zap. nauchn. sem. POMI, 476, POMI, SPb., 2018, 111–124
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