Quantum markov chains and related nonlinear dynamical systems /
In this thesis, we study forward quantum Markov chains associated with some quantum models on a Cayley tree. As it usually is, the quantum picture is different from classical one. We obtain some unexpected phenomena in quantum settings. More precisely, we give a construction of forward quantum Marko...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
Kuantan:
Kulliyyah of Science, IIUM
2011
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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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| 008 | 120125t2011 my a g m 000 0 eng d | ||
| 040 | |a UIAM |b eng | ||
| 041 | |a eng | ||
| 043 | |a a-my--- | ||
| 050 | |a QA402.35 | ||
| 100 | 1 | |a Saburov, Mansoor | |
| 245 | 1 | |a Quantum markov chains and related nonlinear dynamical systems / |c by Mansoor Saburov | |
| 260 | |a Kuantan: |b Kulliyyah of Science, IIUM |c 2011 | ||
| 300 | |a xi, 234 leaves : |b ill. ; |c 30cm. | ||
| 500 | |a Abstract in English and Arabic. | ||
| 500 | |a "A thesis submitted in fulfilment of the requirement for the degree of Doctor of Philosophy."--On t.p. | ||
| 502 | |a Thesis (Ph.D)--International Islamic University Malaysia, 2011. | ||
| 504 | |a Includes bibliographical references (leaves 222-228). | ||
| 520 | |a In this thesis, we study forward quantum Markov chains associated with some quantum models on a Cayley tree. As it usually is, the quantum picture is different from classical one. We obtain some unexpected phenomena in quantum settings. More precisely, we give a construction of forward quantum Markov chains and calculate the compatibility condition of boundary conditions for certain interaction operators. We prove that boundary conditions of quantum XY and Ising models can be considered as a trajectory of certain nonlinear dynamical systems. We deeply study asymptotically behaviors of derived dynamical systems. Based on these investigations, we proved an existence of forward quantum Markov chains associated with quantum XY and Ising models. We prove the uniqueness of the quantum Markov chain associated with the quantum model on the Cayley tree of order two. However, we detect an existence of a quantum phase transition on the Cayley tree of order three for the quantum model. Unlike usual quantum phase transition, a phase transition which was observed on the quantum XY model does not exhibit at zero temperature. The phase transition surprisingly occurs between two nonzero temperatures, i.e., there exists a lower bound and an upper bound on the temperature for the existence of the phase transition. In order to show similarities between forward Markov chains and Gibbs measures we study the forward quantum Markov Chain associated with the Ising model on the Cayley tree of any order. Contrary to the quantum XY model, in the Ising model there always exists a phase transition for any order of the Cayley tree. Moreover, we study supplementary properties of this forward quantum Markov chain. Dynamical systems derived from quantum Markov chains have inspired a renewed interest in the theory of nonlinear operators. We study “advanced fixed point theorems” for an implicit and explicit form of nonlinear operators. All presented results generalize and unify many previous results for different class of nonlinear operators. | ||
| 596 | |a 1 6 | ||
| 650 | |a System analysis | ||
| 650 | |a Markov processes | ||
| 650 | |a Nonlinear control theory | ||
| 655 | 7 | |a Theses, IIUM local | |
| 690 | |a Dissertations, Academic |x Kulliyyah of Science |z IIUM | ||
| 710 | 2 | |a International Islamic University Malaysia. |b Kulliyyah of Science | |
| 856 | 4 | |u https://lib.iium.edu.my/mom/services/mom/document/getFile/nfJmU48pbF4clooPQ7T70YwzqdxkEQVZ20130206145948599 |z 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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