Dynamics of positive quadratic stochastic operators on 2D simplex /
The Perron–Frobenius theorem states that a linear operator associated with a positive square stochastic matrix has a unique fixed point in the simplex and it is regular to that fixed point. Inspired by this classical result, in this thesis, we study a set of fixed points and the regularity of so-cal...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
Kuantan, Pahang :
Kulliyyah of Science, International Islamic University Malaysia,
2016
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| Subjects: | |
| Online Access: | Click here to view 1st 24 pages of the thesis. Members can view fulltext at the specified PCs in the library. |
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| Summary: | The Perron–Frobenius theorem states that a linear operator associated with a positive square stochastic matrix has a unique fixed point in the simplex and it is regular to that fixed point. Inspired by this classical result, in this thesis, we study a set of fixed points and the regularity of so-called positive quadratic stochastic operators (PQSO) associated with positive cubic stochastic matrices on 2D simplex. We show that, in general, the analogue of Perron–Frobenius theorem does not hold true for PQSO. Namely, it may have more than one fixed point in the simplex. Moreover, the uniqueness of fixed points does not imply its regularity. We study the structure of the fixed point set of PQSO on 2D simplex and provide a uniqueness criterion for fixed points of PQSO. Moreover, by introducing a new class of PQSO so-called r-majorizing PQSO, we also provide some sufficient conditions in which the positivity implies the uniqueness of its fixed points as well as its regularity. Some supporting examples are also presented. |
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| Physical Description: | x, 103 leaves : ill. ; 30cm. |
| Bibliography: | Includes bibliographical references (leaves 65-67). |