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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Description of the Set of Strictly Regular Quadratic Bistochastic Operators and Examples

Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 425--433 | DOI:10.5890/DNC.2021.09.006

Mirmukhsin Makhmudov

Department of Mathematics, National University of Uzbekistan, Tashkent, 100174, Uzbekistan

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The present paper focuses on the dynamical systems of the quadratic bistochastic operators (QBO) on the standard simplex. In the paper, we show the character of connection of the dynamical systems of a bistochastic operator with the dynamical systems of the extreme bistochastic operators. In addition, we prove that almost all quadratic bistochastic operators are strictly regular and give a description of the strictly regular quadratic bistochastic operators in the convex polytope QBO. Furthermore, the density of the set of strictly regular QBO in the set of QBO is proved and nontrivial examples of strictly regular bistochastic operators are given.


I would like to express deep gratitude to professors U.Rozikov, R.Ghanikhodjaev and U.Jamilov for many useful discussions, M.Saburov for attentive reading of the text and for making many useful comments.


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