참고문헌
- K. Bartoszek, J. Domsta, and M. Pulka, Centred quadratic stochastic operators, arXiv:1511.07506.
- W. Bartoszek and M. Pulka, On mixing in the class of quadratic stochastic operators, Nonlinear Anal. 86 (2013), 95-113. https://doi.org/10.1016/j.na.2013.03.011
-
W. Bartoszek and M. Pulka, Asymptotic properties of quadratic stochastic operators acting on the
$L_1$ space, Nonlin. Anal. 114 (2015), 26-39. https://doi.org/10.1016/j.na.2014.10.032 -
W. Bartoszek and M. Pulka, Prevalence problem in the set of quadratic stochastic operators acting on
$L_1$ , Bull. Malays. Math. Sci. Soc., Accepted (Published online: 05 November 2015). - S. N. Bernstein, The solution of a mathematical problem related to the theory of heredity, Uchn. Zapiski. NI Kaf. Ukr. Otd. Mat. 1 (1924), 83-115.
- P. Billingsley, Probability and Measure, Anniversary Edition, Wiley 2012.
- N. Ganikhodjaev, On stochastic precesses generated by quadratic operators, J Theoret. Probab. 4 (1991), 639-653. https://doi.org/10.1007/BF01259547
- N. Ganikhodjaev, R. Ganikhodjaev, and U. Jamilov, Quadratic stochastic operators and zero-sum game dynamics, Ergodic Theory Dynam. Systems 35 (2015), no. 5, 1443-1473. https://doi.org/10.1017/etds.2013.109
- N. Ganikhodjaev and N. Z. A. Hamzah, On Poisson Nonlinear Transformations, The Scientific World J. 2014 (2014), Article ID 832861, 7 pp.
- N. Ganikhodjaev, M. Saburov, and U. Jamilov, Mendelian and non-Mendelian quadratic operators, Appl. Math. Info. Sci. 7 (2013), no. 5, 1721-1729. https://doi.org/10.12785/amis/070509
- N. Ganikhodjaev, M. Saburov, and A. M. Nawi, Mutation and chaos in nonlinear models of heredity, The Scientific World J. 2014 (2014), 1-11.
- N. Ganikhodjaev and D. Zanin, On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex, Russian Math. Surveys 59 (2004), no. 3, 571-572. https://doi.org/10.1070/RM2004v059n03ABEH000744
-
R. Ganikhodzhaev, A family of quadratic stochastic operators that act in
$S^2$ , Dokl. Akad. Nauk UzSSR 1 (1989), 3-5. - R. Ganikhodzhaev, Quadratic stochastic operators, Lyapunov function and tourna-ments, Acad. Sci. Sb. Math. 76 (1993), no. 2, 489-506.
- R. Ganikhodzhaev, A chart of fixed points and Lyapunov functions for a class of discrete dynamical systems, Math. Notes 56 (1994), no. 5-6, 1125-1131. https://doi.org/10.1007/BF02274660
- R. Ganikhodzhaev and D. Eshmamatova, Quadratic automorphisms of a simplex and the asymptotic behavior of their trajectories, Vladikavkaz. Mat. Zh. 8 (2006), no. 2, 12-28.
- R. Ganikhodzhaev, F. Mukhamedov, and U. Rozikov, Quadratic stochastic operators: Results and open problems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 (2011), no. 2, 279-335. https://doi.org/10.1142/S0219025711004365
- H. Kesten, Quadratic transformations: A model for population growth I, Adv. in App. Probab. 2 (1970), 1-82.
- Yu. Lyubich, Mathematical structures in population genetics, Springer-Verlag, 1992.
- F. Mukhamedov and M. Saburov, On homotopy of volterrian quadratic stochastic op-erator, Appl. Math. Inf. Sci. 4 (2010), no. 1, 47-62.
- F. Mukhamedov and M. Saburov, On dynamics of Lotka-Volterra type operators, Bull. Malays. Math. Sci. Soc. 37 (2014), no. 1, 59-64.
-
F. Mukhamedov, M. Saburov, and I. Qaralleh, On
$\xi^{(s)}$ -quadratic stochastic operators on two-dimensional simplex and their behavior, Abstr. Appl. Anal. 2013 (2013), 1-12. - M. Saburov, On ergodic theorem for quadratic stochastic operators, Dokl. Acad. N. Rep. Uz. 6 (2007), 8-11.
- M. Saburov, The Li-Yorke chaos in quadratic stochastic Volterra operators, Proceedings of International Conference of Application Science & Technology (2012), 54-55.
- M. Saburov, On regularity, transitivity, and ergodic principle for quadratic stochastic Volterra operators, Dokl. Acad. Nauk Rep. Uzb. 3 (2012), 9-12.
- M. Saburov, Some strange properties of quadratic stochastic Volterra operators, World Appl. Sci. J. 21 (2013), 94-97.
- M. Saburov and Kh. Saburov, Mathematical models of nonlinear uniform consensus, Science Asia 40 (2014), no. 4, 306-312. https://doi.org/10.2306/scienceasia1513-1874.2014.40.306
- M. Saburov and Kh. Saburov, Reaching a nonlinear consensus: Polynomial stochastic operators, Internat. J. Control Automation Systems 12 (2014), no. 6, 1276-1282. https://doi.org/10.1007/s12555-014-0061-0
- M. Saburov and Kh. Saburov, Reaching a consensus: a discrete nonlinear time-varying case, Internat. J. Sys-tems Sci. 47 (2016), no. 10, 2449-2457. https://doi.org/10.1080/00207721.2014.998743
- T. A. Sarymsakov and N. N. Ganikhodjaev, Analytic methods in the theory of quadratic stochastic processes, J. Theoret. Probab. 3 (1990), no. 1, 51-70. https://doi.org/10.1007/BF01063328
- S. Ulam, A Collection of Mathematical Problems, New-York-London, 1960.
- M. Zakharevich, On behavior of trajectories and the ergodic hypothesis for quadratic transformations of the simplex, Russian Math. Surveys 33 (1978), no. 6, 265-266. https://doi.org/10.1070/RM1978v033n06ABEH003890
피인용 문헌
- Quadratic stochastic operators on Banach lattices 2017, https://doi.org/10.1007/s11117-017-0522-9