On the surjectivity of quadratic stochastic operators acting on the simplex
It is well-known that a linear stochastic operator (a Markov operator) associated with a square stochastic matrix is a surjection of the simplex if and only if it is bijective. The similar problem was open for nonlinear stochastic operators (nonlinear Markov operators) associated with stochastic hy...
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| Format: | Article |
| Language: | English English English |
| Published: |
Pleiades Publishing
2016
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| Online Access: | http://irep.iium.edu.my/51164/ http://irep.iium.edu.my/51164/ http://irep.iium.edu.my/51164/ http://irep.iium.edu.my/51164/1/Surjective_QSO_---_MN_%28Eng%29.pdf http://irep.iium.edu.my/51164/4/51164-On_the_surjectivity_of_quadratic_stochastic_operators_SCOPUS.pdf http://irep.iium.edu.my/51164/5/51164-On_the_surjectivity_of_quadratic_stochastic_operators_WOS.pdf |
| Summary: | It is well-known that a linear stochastic operator (a
Markov operator) associated with a square stochastic matrix is a surjection of the simplex if and only if it is bijective. The similar problem was open for nonlinear stochastic operators (nonlinear Markov operators) associated with stochastic hyper-matrices (higher dimensional matrices). In this paper, we solved this problem for quadratic stochastic operators acting on the simplex. Namely, we showed that a quadratic stochastic operator associated with a cubic stochastic matrix is a surjection of the simplex if and only if
it is bijective. Moreover, we also described all surjective quadratic stochastic operators of the simplex. |
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