08. School of Science & Technology (2015-2019)No Descriptionhttps://nur.nu.edu.kz/handle/123456789/81922024-10-16T04:05:31Z2024-10-16T04:05:31Z701QUANTUM EVOLUTIONARY ALGORITHM FOR QUANTUM CIRCUIT SYNTHESISKrylov, Georgiyhttps://nur.nu.edu.kz/handle/123456789/69542024-08-13T12:10:49Z2018-06-01T00:00:00Zdc.title: QUANTUM EVOLUTIONARY ALGORITHM FOR QUANTUM CIRCUIT SYNTHESIS
dc.contributor.author: Krylov, Georgiy
dc.description.abstract: Quantum computing area has a lot research attention due to opportunities that possessing
such device could provide. For example, quantum computers could deliver
new insights to previously unsolvable problems. The reason for that is higher parallel
capabilities of such devices. In addition, since quantum computers are naturally
reversible, no heat dissipation occurs during computation [21]. This property could
serve as a viable solution to the problem that computer chip production industry
faces. Moreover, since the chip manufacturing industry reaches nanometer scale of
size of elements, the effects that could cause unexpected information behavior in
classical paradigm are part of the technology of quantum devices [31, 14].
Considering possible benefits that could be achieved by quantum computing devices,
the new areas of Quantum Information Theory, Quantum Cryptography, Quantum
Algorithms and Logic Design and many others emerged at the end of the twentieth
century [31]. These areas are concentrating their efforts on solving problems of
designing communication protocols, ensuring the security of the new systems, constructing
appropriate algorithms. Computers that could be advancing in finding
solutions in problems listed above require quantum circuits that have optimal structure
and could implement error correction. This is the main motivation for this thesis
work to explore the problem of circuit design. The approach that we investigate is
circuit construction by the means of Quantum Evolutionary Algorithms. We propose
a version of an algorithm that accounts with specificity and constraints of quantum
paradigm. We use its Graphic Processing Unit (GPU) accelerated classical implementation
to evaluate the behavior and performance of the proposed algorithm. Later
we discuss additional complexity introduced by accounting with these constraints.
We support our ideas with results of synthesis of small circuits and compare the
performance with classical genetic algorithm on similar task.
2018-06-01T00:00:00ZKNOTTED OPTICAL VORTEX LINES IN NONLINEAR SATURABLE MEDIUMIssakhanov, Alfarabihttps://nur.nu.edu.kz/handle/123456789/59022024-08-13T11:17:34Z2018-06-01T00:00:00Zdc.title: KNOTTED OPTICAL VORTEX LINES IN NONLINEAR SATURABLE MEDIUM
dc.contributor.author: Issakhanov, Alfarabi
dc.description.abstract: In last 50 years, a significant progress was noticed in medicine, communications
and entertainment. Such advanced development of these fields was directly related to
ability of controlling light. Photonics is exactly about this ability. At the present time,
photonics is walking together with a fundamental physical concept, optical soliton.
Optical solitons are shape-preserving laser beams. They are found potentially useful
in data transmission, which is very significant nowadays. Hence, research in the field
of optical solitons is still a vital issue.
When optical soliton is perturbed in a specific manner, there appear zeros of
optical field around the soliton, which are called optical vortices. In general optical
vortices are lines in space. Hence, we might expect them to become knotted.
Knotting optical vortices around perturbed seems spontaneous and cannot be directly
predicted. To explain this phenomenon, a similar system is constructed based on perturbation
theory. In this system, however, we have a mathematical problem which
yet lacks a full understanding. We address this problem by introducing concepts from
three disciplines: laser physics, knot theory and singular optics.
We believe that understanding the mechanism underlying spontaneous knotting
of optical vortices will be a step forward in other systems too, such as quantum
turbulence in superfluids and formation of optical vortices around other types of
optical solitons.
2018-06-01T00:00:00ZSynchronization of Coupled Nonlinear Oscillators with Applications to Photonic ArraysZharas, Banuhttps://nur.nu.edu.kz/handle/123456789/42002024-08-13T12:12:02Z2019-05-01T00:00:00Zdc.title: Synchronization of Coupled Nonlinear Oscillators with Applications to Photonic Arrays
dc.contributor.author: Zharas, Banu
dc.contributor.editor: Bountis, Anastasios
dc.description.abstract: In recent years, the study of synchronization of coupled oscillators have been the subject of intense research interest, leading to many new and unexpected phenomena. Our research is first focused on the analysis of a network of coupled nonlinear oscillators exhibiting the breakdown of synchronization into fascinating “chimera states” exhibiting the coexistence of synchronized and unsynchronized parts. We then apply these ideas to laser arrays of photonic “oscillators”, which have numerous applications in optical communications, sensing and imaging. First of all, we demonstrate the occurrence of synchronization and chimera states in a simpler problem, consisting of a ring of coupled 4D simplified Lorenz systems, in which each oscillator is described by a Li-Sprott oscillator [1]. An interesting feature of each oscillator is the coexistence of a limit cycle and two symmetric strange attractors for some specific range of parameters, which influences the global synchronization dynamics and leads to the formation of chimera states. Inspired
by this model, we study some fascinating oscillatory phenomena of coupled photonic oscillators consisting of dimers of semiconductor lasers, each of which is capable of performing limit cycle oscillations. Coupling in an appropriate way a large number of dimers in long arrays we find that they can exhibit combinations of oscillatory patterns involving long amplitude oscillations (LAO) and also localized oscillations of very small amplitude close to the fixed points (LOCFP). As preliminary results of this investigation, we show the coexistence of LOA and LOCFP patterns reminiscent of “chimera–like” states and LOCFP “breather– like” phenomena. Both of these behaviors are shown to be spatially robust, when we calculate the Discrete Laplacian of their amplitudes for long times.
dc.description: Master of Science Thesis in Applied mathematics
Department of Mathematics, School of Science and Technology Nazarbayev University
2019
2019-05-01T00:00:00ZThe Dynamics of Hamiltonian Lattices With Application to Hollomon OscillatorsZholmaganbetova, Aigerimhttps://nur.nu.edu.kz/handle/123456789/41992024-08-13T12:12:32Z2019-05-29T00:00:00Zdc.title: The Dynamics of Hamiltonian Lattices With Application to Hollomon Oscillators
dc.contributor.author: Zholmaganbetova, Aigerim
dc.contributor.editor: Bountis, Anastasios
dc.description.abstract: Many problems in theoretical physics are expressed in the form of Hamiltonian systems. Among these the first to be extensively studied were low-dimensional, possessing as few as two (or three) degrees of freedom. In the last decades, however, great attention has been devoted to Hamiltonian systems of high dimensionality. The most famous among them are the ones that deal with the dynamics and statistics of a large number N of mass particles connected with nearest neighbor interactions. At low energies E,
these typically execute quasiperiodic motions near some fundamental stable periodic orbits (SPOs) which
represent nonlinear continuations of the N normal mode solutions of the corresponding linear system.
However, as the energy is increased, these solutions destabilize causing the motion in their vicinity to drift into chaotic domains, thus giving rise to important questions concerning the systems behavior in the thermodynamic limit, where E and N diverge with E=N = constant. One of the open problems in Hamiltonian dynamics, therefore, examines the relation between local (linear) stability properties of simple periodic solutions of Hamiltonian systems, and the more “global” dynamics. In this thesis, after reviewing the main results on these topics for the case of N-particle Fermi-Pasta-Ulam Hamiltonians, I proceed to apply the corresponding methods to a lattice of Hollomon oscillators, which are of interest to applications in problems of nonlinear elasticity.
dc.description: Master of Science Thesis in Applied Mathematics
Department of Mathematics, School of Science and Technology Nazarbayev University
Astana 010000, Kazakhstan
2019-05-29T00:00:00ZConvergence Rate of Fourier Neural NetworksZhumekenov, Abylayhttps://nur.nu.edu.kz/handle/123456789/41982024-08-13T12:13:29Z2019-04-26T00:00:00Zdc.title: Convergence Rate of Fourier Neural Networks
dc.contributor.author: Zhumekenov, Abylay
dc.contributor.editor: Assylbekov, Zhenisbek
dc.description.abstract: The paper investigates a convergence rate for 2-layer feedforward Fourier Neural Network
(FNN). Such networks are motivated by the approximation properties of wellknown
Fourier series. Several implementations of FNNs were proposed since 1990’s:
by Gallant and White; A. Silvescu; Tan, Zuo and Cai; Liu. The main focus of this
research is Silvescu’s FNN, because such activation function does not fit into the category
of networks, where the linearly transformed input is exposed to activation. The
latter ones were extensively described by Hornik in 1989. In regard to non-trivial
Silvescu’s FNN, its convergence rate is proven to be of order 𝑂(1/𝑛). The paper
continues investigating classes of functions approximated by Silvescu FNN, which
appeared to be from Schwartz space and space of positive definite functions.
2019-04-26T00:00:00ZExplorations on chaotic behaviors of Recurrent Neural NetworksMyrzakhmetov, Bagdathttps://nur.nu.edu.kz/handle/123456789/41972024-08-13T12:14:01Z2019-04-29T00:00:00Zdc.title: Explorations on chaotic behaviors of Recurrent Neural Networks
dc.contributor.author: Myrzakhmetov, Bagdat
dc.contributor.editor: Assylbekov, Zhenisbek; Takhanov, Rustem
dc.description.abstract: In this thesis work we analyzed the dynamics of the Recurrent Neural Network architectures.
We explored the chaotic nature of state-of-the-art Recurrent Neural Networks:
Vanilla Recurrent Network, Recurrent Highway Networks and Structurally
Constrained Recurrent Network. Our experiments showed that they exhibit chaotic
behavior in the absence of input data. We also proposed a way of removing chaos
chaos from Recurrent Neural Networks. Our findings show that initialization of the
weight matrices during the training plays an important role, as initialization with
the matrices whose norm is smaller than one will lead to the non-chaotic behavior
of the Recurrent Neural Networks. The advantage of the non-chaotic cells is stable
dynamics. At the end, we tested our chaos-free version of the Recurrent Highway
Networks (RHN) in a real-world application.
In a sequence-to-sequence modeling experiments, particularly in the language
modeling task, chaos-free version of RHN perform on par with the original version by
using the same hyperparameters.
dc.description: Submitted to the Department of Mathematics on Apr 29, 2019, in partial fulfillment of the
requirements for the degree of Master of Science in Applied Mathematics
2019-04-29T00:00:00ZNumerical computations of complexification of Legendrian knotsYerzhigit, Bauyrzhanhttps://nur.nu.edu.kz/handle/123456789/41962024-08-13T12:14:29Z2019-04-29T00:00:00Zdc.title: Numerical computations of complexification of Legendrian knots
dc.contributor.author: Yerzhigit, Bauyrzhan
dc.contributor.editor: Lawrence, Mark
dc.description.abstract: With the recent interest in knots, it is interesting to study their complexification. We
have chosen to study Legendrian knots as they have the property that we can reconstruct
the original knot from its projection. This property is especially useful in the case of the
complexification of a knot as in this case the diagram of the projection of the knot is no
longer real. In this paper we show a way to compute complex rational functions that have a Legendrian knot as an image under unit circle.
2019-04-29T00:00:00ZMultigrid method for Mild-Slope equation in Coastal Wave ModellingTabarek, Rysbergenhttps://nur.nu.edu.kz/handle/123456789/41952024-08-13T12:14:56Z2019-04-19T00:00:00Zdc.title: Multigrid method for Mild-Slope equation in Coastal Wave Modelling
dc.contributor.author: Tabarek, Rysbergen
dc.contributor.editor: Erlangga, Yogi
dc.description.abstract: In this thesis we propose and study an efficient iterative multigrid method for the time independent modified mild slope equation with and without energy dissipation term.
The algorithm relies on a multigrid method preconditioned with shifted-Laplacian
preconditioner and solved by Bi-CGSTAB algorithm. Multigrid analysis results are
shown by numerical experiments. Numerical experiments are conducted in depth
sloped elliptic shoal introduced by Berkhoff et. al
dc.description: Submitted to the Department of Mathematics on Apr 19, 2019, in partial fulfillment of the
requirements for the degree of Master of Applied Mathematics
2019-04-19T00:00:00ZPulse vaccination of a time-delayed SIRS epidemic model with nonlinear incidence rateYeleussinova, Meruyerthttps://nur.nu.edu.kz/handle/123456789/41942024-08-13T12:16:28Z2019-05-03T00:00:00Zdc.title: Pulse vaccination of a time-delayed SIRS epidemic model with nonlinear incidence rate
dc.contributor.author: Yeleussinova, Meruyert
dc.contributor.editor: Kashkynbayev, Ardak
dc.description.abstract: This work deals with an application of pulse vaccination for a varying size of the population of time-delayed 𝑆𝐼𝑅𝑆 epidemic model. The dynamics of the infectious disease
depends on the threshold value, 𝑅0, known as the basic reproduction number. In the classical epidemic models, this value is evaluated by means of the next generation matrix. However, this method does not work for non-autonomous systems. Since we consider the pulse vaccination strategy for epidemic models our system is naturally non-autonomous. We follow the general approach to derive 𝑅0 in terms of spectral radii of Poincare maps. Further, we show the existence of an infectious-free periodic solution and its global attractiveness for 𝑅0 < 1 and the persistence of infectious disease for 𝑅0 > 1.
dc.description: Submitted to the Department of Mathematics on May 3, 2019, in partial fulfillment of the
requirements for the degree of Master of Science in Applied Mathematics
2019-05-03T00:00:00ZGlobal stability analysis for a tick-borne modelKoptleuova, Daianahttps://nur.nu.edu.kz/handle/123456789/41932024-08-13T12:17:03Z2019-05-06T00:00:00Zdc.title: Global stability analysis for a tick-borne model
dc.contributor.author: Koptleuova, Daiana
dc.contributor.editor: Kashkynbayev, Ardak
dc.description.abstract: This thesis consider three type of epidemiological models: SIR, SIS and SIRS with nonlinear incidence rate and piecewise constant delay of generalized type. In this paper the total population size is varied with time elapse. We study the global asymptotic stability of the disease-free and endemic equilibrium states of models by constructing suitable Lyapunov functions and Lyapunov–LaSalle technique. The main contribution of this master thesis is to develop more realistic compartmental models by extending the literature of models with piecewise constant delay. The theoretical findings are illustrated through numerical simulations.
dc.description: Submitted to the Department of Mathematics on May 6, 2019, in partial fulfillment of the
requirements for the degree of Master of Science in Applied Mathematics
2019-05-06T00:00:00Z