Browsing by Author "Ahmad Fadillah bin Embong"

In the present thesis, we considered a majorization in term of b-order. Using such majorization, we defined bistochastic QSOs with respect to b-order, namely b-bistochastic QSOs in which it defines a new class of QSOs. Further, we provided some crucial properties of b-bistochastic QSOs in a general setting, and fully described b-bistochastic QSOs on one and two dimensional simplices. In addition, we showed that for every b-bistochastic QSOs, their trajectories were convergent. Besides, it was found that (0,0,…,1) was the fixed point of any b-bistochastic QSOs. Next, we managed to fully study the dynamics of b-bistochatic QSOs on one dimensional simplex, whereas on two dimensional simplex, we limited the study on boundaries only. Generally, it was established that the set of all b-bistochastic QSOs formed a convex set, hence we were interested to describe its extreme points. The results were presented as the list of extreme points on low dimensional simplices. Moreover, we introduced quasi-extremity of b-bistochastic QSO and investiged it on two dimensional simplex. Finally, we associated Markov measures with b-bistochastic QSOs. On some classes of b-bistochastic QSOs, the defined measures were proven to satisfy the mixing property. Moreover, we studied the absolute continuity of Markov measures associated with a class of b-bistochastic QSOs.

In this thesis we consider a class of quadratic stochastic operators (QSOs) namely, orthogonal preserving QSOs (OP QSOs). First chapter is devoted to the literature review, problem statements, objectives, preliminaries and overviews of the whole thesis. We recall some achievements on permutation Volterra QSOs, OP QSOs and surjective QSOs defined over a finite dimensional simplex (i.e., the set of all probability distributions on a finite subset of natural numbers) in the second chapter. Indeed, permutation Volterra QSOs, OP QSOs and surjective QSOs are equivalent classes. Since finite dimensional Volterra QSOs were well-studied, therefore we continue the investigation over infinite dimensional simplex in Chapter 3. We provide a concrete form for any OP QSOs could take. Several examples are given and some properties of such mapping are described. Note that, every Volterra QSO is an OP QSO. Due to the complexness to study the dynamics for the whole set of OP QSOs on infinite dimensional simplex, so we restrict ourselves to some classes of Volterra QSOs. In particular, Chapter 4 is focusing on the study of ω−limit set (i.e., the set contains all the limiting points) of some classes of Volterra QSOs. The technique of Lypunov functions is employed here to estimate such set. The significant difference between finite and infinite case is that the ω−limit set (wrt ℓ1−norm) could be empty. Moreover, we show that there is a class of Volterra QSOs which satisfies weak ergodic but fail to be ergodic (see Defintion 4.2.4). In Chapter 5, we investigate surjectivity of infinite dimensional simplex. Unlike finite case, OP QSOs and surjective QSOs are different classes. Thus, we provide necessary and sufficient conditions for infinite dimensional surjective QSOs. Moreover, if one takes a QSO which is OP and surjective, then the operator must be a permutation Volterra.