N. N. Ganikhodzhaev, U. A. Rozikov, “On quadratic stochastic operators generated by Gibbs distributions”, Regul. Chaotic Dyn., 11:4 (2006), 467

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Regular and Chaotic Dynamics, 2006, Volume 11, Issue 4, Pages 467–473
DOI: https://doi.org/10.1070/RD2006v011n04ABEH000364 (Mi rcd687)

This article is cited in 19 scientific papers (total in 19 papers)

On quadratic stochastic operators generated by Gibbs distributions

N. N. Ganikhodzhaev^a, U. A. Rozikov^b

^a International Islamic University Malaysia, 53100 Kuala Lumpur, Malaysia
^b Institute of Mathematics, 29, F. Hodjaev str., 700125 Tashkent, Uzbekistan

Citations (19)

DOI: https://doi.org/10.1070/RD2006v011n04ABEH000364

Abstract: We give a constructive description of quadratic stochastic operators which act to the set of all probability measures on some measurable space. Our construction depends on a probability measure

$\mu$ and cardinality of a set of cells (configurations) which here can be finite or continual. We study behavior of trajectories of such operators for a given probability measure

$\mu$ which coincides with a Gibbs measure. For the continual case we compare the quadratic operators which correspond to well-known Gibbs measures of the Potts model on

$Z^d$ . These investigations allows a natural introduction of thermodynamics in studying some models of heredity. In particular, we show that any trajectory of the quadratic stochastic operator generated by a Gibbs measure

$\mu$ of the Potts model converges to this measure

Keywords: quadratic stochastic operator, Gibbs distribution, Potts model.

Received: 12.10.2005
Accepted: 24.04.2006

Bibliographic databases:

Document Type: Article

MSC: 37C20, 37C25, 82B26

Language: English

Citation: N. N. Ganikhodzhaev, U. A. Rozikov, “On quadratic stochastic operators generated by Gibbs distributions”, Regul. Chaotic Dyn., 11:4 (2006), 467–473

Citation in format AMSBIB

\Bibitem{GanRoz06}

\by N. N. Ganikhodzhaev, U. A. Rozikov

\paper On quadratic stochastic operators generated by Gibbs distributions

\jour Regul. Chaotic Dyn.

\yr 2006

\vol 11

\issue 4

\pages 467--473

\mathnet{http://mi.mathnet.ru/rcd687}

\crossref{https://doi.org/10.1070/RD2006v011n04ABEH000364}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2292205}

\zmath{https://zbmath.org/?q=an:1164.37309}

Linking options:

https://www.mathnet.ru/eng/rcd687

https://www.mathnet.ru/eng/rcd/v11/i4/p467

This publication is cited in the following 19 articles:

Z. S. Boxonov, “On Dynamical Systems of Quadratic Stochastic Operators Constructed for Bisexual Populations”, Math. Notes, 114:6 (2023), 1134–1150
Abd El-Qa H., Ab Ghani A.T., Qaralle I., “Classification and Dynamics of Class of Xi((as))-Qsos”, J. Math. Fundam. Sci., 52:2 (2020), 143–173
M. Ladra, U.A. Rozikov, “Flow of finite-dimensional algebras”, Journal of Algebra, 470 (2017), 263
Nasir Ganikhodjaev, “Quadratic Stochastic Operators with Countable State Space”, J. Phys.: Conf. Ser., 697 (2016), 012016
B. J. Mamurov, U. A. Rozikov, “On cubic stochastic operators and processes”, J. Phys.: Conf. Ser., 697 (2016), 012017
Farrukh Mukhamedov, Nasir Ganikhodjaev, Lecture Notes in Mathematics, 2133, Quantum Quadratic Operators and Processes, 2015, 7
M. Ladra, U.A. Rozikov, “Evolution algebra of a bisexual population”, Journal of Algebra, 378 (2013), 153
Farrukh Mukhamedov, Mansoor Saburov, Izzat Qaralleh, “Onξ(s)-Quadratic Stochastic Operators on Two-Dimensional Simplex and Their Behavior”, Abstract and Applied Analysis, 2013 (2013), 1
NASIR GANIKHODJAEV, “LATTICE GAS AND THERMODYNAMICS IN MODELS OF HEREDITY”, Int. J. Mod. Phys. Conf. Ser., 09 (2012), 157
U. A. Rozikov, A. Zada, “On a class of separable quadratic stochastic operators”, Lobachevskii J Math, 32:4 (2011), 385
U. A. Rozikov, J. P. Tian, “Evolution algebras generated by Gibbs measures”, Lobachevskii J Math, 32:4 (2011), 270
U. A. Rozikov, “System of recursive equations for the partition functions of 1D models”, Lobachevskii J Math, 32:2 (2011), 109
RASUL GANIKHODZHAEV, FARRUKH MUKHAMEDOV, UTKIR ROZIKOV, “QUADRATIC STOCHASTIC OPERATORS AND PROCESSES: RESULTS AND OPEN PROBLEMS”, Infin. Dimens. Anal. Quantum. Probab. Relat. Top., 14:02 (2011), 279
U. A. Rozikov, S. Nazir, “Separable quadratic stochastic operators”, Lobachevskii J Math, 31:3 (2010), 215
U. A. ROZIKOV, A. ZADA, “ON DYNAMICS OF ℓ-VOLTERRA QUADRATIC STOCHASTIC OPERATORS”, Int. J. Biomath., 03:02 (2010), 143
U. U. Zhamilov, U. A. Rozikov, “The dynamics of strictly non-Volterra quadratic stochastic operators on the 2-simplex”, Sb. Math., 200:9 (2009), 1339–1351
U. A. Rozikov, N. B. Shamsiddinov, “On Non-Volterra Quadratic Stochastic Operators Generated by a Product Measure”, Stochastic Analysis and Applications, 27:2 (2009), 353
U. A. Rozikov, A. Zada, “On $l$-Volterra quadratic stochastic operators”, Dokl. Math., 79:1 (2009), 32–34
U. A. Rozikov, U. U. Zhamilov, “$F$-Quadratic Stochastic Operators”, Math. Notes, 83:4 (2008), 554–559

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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