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In this paper, the positive invariance of sets defined by nonlinear inequalities with respect to nonlinear maps from a partially ordered set into itself is investigated. The results of this paper extend former results relative to the positive invariance of polyhedral sets with respect to linear systems defined in ordered semi-rings as well as former results relative to nonlinear dynamical systems defined in the real n-space. The v-monotonicity of nonlinear maps is also been investigated. It is shown that, opposite to the case of linear maps, the v-monotonicity of a nonlinear map is not a necessary condition for a set defined by nonlinear inequalities to be positively invariant. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the comparison of expected rewards for discrete-time reward Markov chains with different state spaces. Necessary and sufficient conditions for such a comparison are derived. Due to the special nature of the introduced binary relation, a criterion may be formulated in terms of an inclusion of polyhedral sets. Then, algebraic and geometric forms are easily obtained from Haar's Lemma. Our results allow us to discuss some earlier results on the stochastic comparison of functions of Markov chains. Keywords: Linear discrete-time systems; Invariant sets; Haar's lemma. [ABSTRACT FROM AUTHOR]

In this paper, the positive invariance of sets defined by nonlinear inequalities with respect to nonlinear maps from a partially ordered set into itself is investigated. The results of this paper extend former results relative to the positive invariance of polyhedral sets with respect to linear systems defined in ordered semirings as well as former results relative to nonlinear dynamical systems defined in the real n-space. The v-monotonicity of nonlinear maps is also been investigated. It is shown that, opposite to the case of linear maps, the v-monotonicity of a nonlinear map is not a necessary condition for a set defined by nonlinear inequalities to be positively invariant. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the comparison of expected rewards for discrete-time reward Markov chains with different state spaces. Necessary and sufficient conditions for such a comparison are derived. Due to the special nature of the introduced binary relation, a criterion may be formulated in terms of an inclusion of polyhedral sets. Then, algebraic and geometric forms are easily obtained from Haar's Lemma. Our results allow us to discuss some earlier results on the stochastic comparison of functions of Markov chains. [ABSTRACT FROM AUTHOR]