QUADRATIC STOCHASTIC OPERATORS AND PROCESSES: RESULTS AND OPEN PROBLEMS

被引：175
作者：
Ganikhodzhaev, Rasul [1 ]
Mukhamedov, Farrukh [2 ]
Rozikov, Utkir [3 ]
机构：
[1] Natl Univ Uzbekistan, Dept Mech & Math, Tashkent 100174, Uzbekistan
[2] Int Islamic Univ Malaysia, Dept Computat & Theoret Sci, Fac Sci, Kuantan 25710, Pahang, Malaysia
[3] Inst Math & Informat Technol, Tashkent 100125, Uzbekistan
来源：
INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS | 2011年 / 14卷 / 02期
关键词：
Quadratic stochastic operator; quadratic stochastic process; quantum quadratic stochastic operator; quantum quadratic stochastic process; fixed point; trajectory; Volterra and non-Volterra operators; ergodic; simplex; LYAPUNOV FUNCTIONS; ERGODIC PROPERTIES; MARKOV; SYSTEMS; POINTS; MAP;
D O I：
10.1142/S0219025711004365
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
The history of the quadratic stochastic operators can be traced back to the work of Bernshtein (1924). For more than 80 years, this theory has been developed and many papers were published. In recent years it has again become of interest in connection with its numerous applications in many branches of mathematics, biology and physics. But most results of the theory were published in non-English journals, full text of which are not accessible. In this paper we give all necessary definitions and a brief description of the results for three cases: (i) discrete-time dynamical systems generated by quadratic stochastic operators; (ii) continuous-time stochastic processes generated by quadratic operators; (iii) quantum quadratic stochastic operators and processes. Moreover, we discuss several open problems.
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收藏
页码：279 / 335
页数：57
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