#### 03. Bachelor's Thesis

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Item Open Access Well-Posedness of the Nonlinear Schrödinger Equation(Nazarbayev University School of Science and Technology, 2019-08-07) Orumbayeva, SaraShow more The Nonlinear Schrödinger equation (NLSE) is a prototypical example of nonlinear partial differential equation. It is commonly used to describe propagation of light in nonlinear optical fibers and is of great importance in quantum mechanics. In this Capstone Project, we provide a complete proof of well-posedness, that is existence of a unique solution of the NLSE using one of the major mathematical techniques: the Banach fixed-point theorem. Both local and global results for inital data in L2(R) are obtained. Moreover, we briefly discuss possible extensions of the topic in terms of different function spaces, general nonlinearities and higher dimension.Show more Item Open Access Nonlinear Schrodinger Equation(Nazarbayev University School of Science and Technology, 2019-08-08) Kazbek, MoldirShow more Rogue waves are fascinating destructive phenomena in nature that have not been fully explained so far [1-3]. Oceanographers commonly agree that linear theories cannot provide explanations for their existence[6,7]. Only nonlinear theories can explain the dramatic concentration of energy into a single "wall of water" well above the average height of the surrounding waves[3,8,9]. Among nonlinear theories the most fundamental is based on the nonlinear Schr odinger equation (NLSE)[6]. If the fundamental approach allows us to give a basic explanation, then it can be extended to more general ones which take into account the two-dimensional nature of the problem. Which is our main goal.Show more Item Open Access Frobenius Singularities of Algebraic Sets of Matrices(Nazarbayev University School of Science and Technology, 2019-08-07) Yerlanov, Madi; Kadyrsizova, Zhibek; Sica, FrancescoShow more When one studies certain rings, it is natural to classify them according to certain properties. This project focuses on the study of properties of commutative rings associated with algebraic sets. In particular, we consider the algebraic set of pairs of square matrices whose commutator has a zero diagonal. We prove that it is irreducible and F-regular for matrices of all sizes and when the matrix entries are from a eld of positive prime characteristic. In addition, we provide a proof of its F-purity and nd a system of parameters on it. Moreover, we state several conjectures associated to this topic.Show more Item Open Access Analytic Solutions for a Nonlinear Transport Equation(Nazarbayev University School of Science and Technology, 2019-08-07) Biyar, Magzhan; da Silva, Daniel Oliveira; Castro Castilla, Alejandro JavierShow more We prove that the Cauchy problem for a transport equation with algebraic nonlinearity of degree p with initial data in Gevrey spaces is locally well-posed. In particular, we show that the analyticity of solutions persists for a short time and we derive a sufficient condition for solutions to be analytic for all times.Show more Item Open Access Nonlinear Regression Analysis of the generalized Logistic Model as an Actuarial life contingency model(Nazarbayev University School of Science and Technology, 2019-08-08) Kadenova, Aida; Wei, Dongming; Erlangga, YogiShow more The aim of this project is to analyze three different population models such as Gompertz, Logistic and Generalized Logistic based USA population data. Finding the appropriate model is essential in actuarial application. Firstly, the parameters of the two models are estimated using the special function ~nls in the R language program. But, due to some complexities, parameters of the generalized logistic model are evaluated using the new method from Causton`s paper. Secondly, two different comparison methods such as residual plot and AIC are used to analyze what model is appropriate for USA statistical data. Lastly, suitable models are used to estimate the force of mortality.Show more Item Open Access Copula functions in Credit Metrics’ VaR estimation(Nazarbayev University School of Science and Technology, 2019-08-08) Magzanov, Shynggys; Wei, Dongming; Assylbekov, ZhenisbekShow more Credit risk modelling of a portfolio of exposures is essential part of activity of every financial institution. However this procedure is complicated since the joint behavior of chosen exposures must be known. In this paper Value at Risk percentile of the portfolio consisting of three corporate bonds issued by Lukoil, Gazprom and Norilsk Nickel was estimated at three different significance levels within the frame of Credit Metrics approach proposed by J.P.Morgan. Following the Asset value model, Monte-Carlo simulations were performed to obtain possible portfolio values in one year time horizon. Where the joint distribution of asset returns of three companies was constructed by means of pair-copula construction method discussed in Aas, Czado, Frigessi,Bakken (2009). Results reveal that for particular portfolio of bonds at 90%, 95% and 99% confidence levels the value of our portfolio will not fall below 2057.915 ,1798.117 and 1375.011 dollars respectively.Show more Item Open Access Copulas and Correlation in Statistical Risk Theory(Nazarbayev University School of Science and Technology, 2018) Rat, SaraShow more The Financial Risk Management (FRM) aims to identify, measure and manage risks in different sectors. One of the core things during such operations is measuring different dependencies. Linear correlation is known as one of the most popular measure of dependence, however it is known that it is a reasonable mea- sure of dependence only when variables are Normally distributed, but this is not the case with credit and portfolio risks, therefore other measures of dependency are needed. This paper presents a Copula function for bivariate case as one of the tools to analyze dependencies in portfolio risk management. Copulas were first introduced by Sklar in 1959 [8], and in 1999 they were studied in financial context for the first time by Embrechts et al. in 1999 [4]. Motivated by the copula analysis of European stock portfolios [6], this paper aims to analyze portfolio consisting of Asian S&P Asia 50 and S&P BSE 100 indices, and apply copula to this portfolio.Show more Item Open Access Analysis of Dynamic Pull-in for a Graphene-based MEMS Model(Nazarbayev University School of Science and Technology, 2018) Omarov, DaniyarShow more A novel procedure based on the Sturm’s theorem for real-valued polynomials is developed to predict and identify periodic solutions and non-periodic solutions in the pull-in analysis of a graphene-based MEMS lumped parameter model with general initial conditions. It is demonstrated that under specific conditions on the lumped parameters and the initial conditions, the model has certain periodic solutions and otherwise there is no such solutions. This theoretical procedure is made practical by numerical implementations with Python scripts to verify the predicted behaviour of the periodic solutions. Numerical simulations are performed with sample data to justify by this procedure the analytically predicted existence of periodic solutions. Also, Low Order Fourier Approximation is used to find the solution for the linear spring case. Comparison with the highly accurate Runge-Kutta method is done to verify derived values from the new numerical approximation.Show more Item Open Access Power Series Solutions of a NODE Systemin the Complex Domain(Nazarbayev University School of Science and Technology, 2018-05-10) Madiyeva, AigerimShow more In this Capstone Project, we analyze a second order nonlinear ordinary differential equation (NODE), y^" (x)=f(y^',y) that is impossible to solve analytically. First, using the Taylor Power Series method, we obtain a series expansion of the solution y(x) about x = 0 for x ∈R and find that this series diverges for values of x a little above x = 1. This implies that the equation has a singularity in the complex domain. Therefore, we investigate this NODE by using Laurent expansions about the unknown singularity at x =x_*, which is called movable because its location depends on the initial conditions. By finding the general form of these expansions, we obtain approximate expressions for the singularity closest to x = 0 and thus are able to estimate the radius of convergence for different initial conditions. We also integrate numerically the solutions in the real x, y plane and demonstrate the connection of the global form of the solutions of the problem with the predictions of our laurent series expansions in the complex x- plane.Show more Item Open Access Initial Explorations on Regularizing the SCRN Model(Nazarbayev University School of Science and Technology, 2018-05) Kabdolov, OlzhasShow more Recurrent neural networks are very powerful sequence models which are used for language modeling as well. Under correct regularization such as naive dropout these models are able to achieve substantial improvement in their performance. We regularize the Structurally Constrained Recurrent Network (SCRN) model and show that despite its simplicity it can achieve the performance comparable to the ubiquitous LSTM model in language modeling task while being smaller in size and up to 2x faster to train. Further analysis shows that regularizing both context and hidden states of the SCRN is crucial.Show more Item Open Access Solution a Problem of Nonlinear Elasticity Using Power Series in Complex Time(Nazarbayev University School of Science and Technology, 2018-05) Shakir, AkylShow more We apply singularity analysis in complex time to investigate the solutions of a dynamical system of one degree of freedom related to the oscillations of a Micro-Electro-Mechanical System (MEMS) of nonlinear elasticity. This problem is expressed mathematically by a second order differential equation for the position variable x(t ) of the oscillator. Using the fact that this equation is connected with an energy integral (kinetic plus potential energy), we first study its solutions by plotting the graph of its potential function for different values of an important parameter of the problem K>0. Then, we analyze these solutions by expanding x(t using power series in complex time. Our solutions are expanded about a singularity of the equation of motion at x = 1, which constitutes a very important point in the analysis, and complements solutions found by other researchers using real time. As a first application, our complex time expansions can be used to estimate locations in x space where the solutions are bounded and periodic and regions where the solutions are unbounded (and hence non-physical). We also examine our approach as an extension of the well–known Frobenius theory for linear second order ODEs. Finally, we investigate the locations of the singularities near the periodic solutions using a Newton-Raphson method near the singularity of the equation at x =1.Show more Item Open Access Estimation and application of best ARIMA model for forecasting the uranium price.(Nazarbayev University School of Science and Technology, 2018-05-13) Amangeldi, MedeuShow more This paper presents the application of an iterative approach for prediction of uranium price by model identification, parameter estimation and diagnostic checking which are designed by Box and Jenkins. In particular, the autoregressive integrated moving average model is used to predict the future values of monthly uranium price. As the analysis of structural dependence in observations is one of the key features of time series analysis, the past values, which were taken as monthly values from January 2000 to June 2017, are used for forecasting. As a result, ARIMA (2,1,0) became one that met all the criteria and predicted the increase of uranium price over time within 95% confidence.Show more Item Open Access Actuarial Applications of a two-parameter generalized Logistic model(Nazarbayev University School of Science and Technology, 2018-05-10) Alekberova, NargizShow more This project presents a generalization of the classical logistic and the Gompertz model by using two parameter power law exponent. The suggested generalized with two parameters models was shown by numerical example to be potentially a better choice for fitting a certain data. The solutions of these models are used in terms of force of mortality function in actuarial math as insurance contingency models or as hazard function for quality control in science and engineering. The complexity of solution is presented in the form of hypergeometric function and corresponding to it nonlinear implicit regression analysis. Due to the mentioned complexity we develop the approximation of the actual solution and consider only its special cases.Show more Item Open Access Optimization of Liquid Scintillator Composition(Nazarbayev University School of Science and Technology, 2017) Batyrkhanov, AyanShow more Nowadays, many particle detectors use liquid scintillator (LS) as a detection medium. In particular, Water-based Liquid Scintillator (WbLS) that is a new material currently under development. It is based on the idea of dissolving the organic scintillator in water using special surfactants. This material strives to achieve the novel detection techniques by combining the Cherenkov and scintillation light, as well as the total cost reduction compared to pure liquid scintillator.Show more Item Open Access Particle detector calibration and performance for Horizon-T cosmic rays experiment(Nazarbayev University School of Science and Technology, 2017) Yessenov, MuratShow more Due to large variation of particle density in EAS, a response linearity of each detector needs to be assessed. The number of photon in a photo multiplier tube (PMT) pulse allows inferring the deviation from linearity in each case. In order to obtain an approximate photon count of a pulse, a single photo-electron response pulse calibration for the R7723 PMT was conducted using low-level light pulse. Non-linearity measurements of PMT were conducted using the same setup. PMT signals were chosen such that their widths are comparable with the real data. Results included in this work show that the PMT signal is mostly linear and nonlinearity starts only at the upper end of ADC range.Show more Item Open Access NUMERICAL INVESTIGATION OF AN EXTERNALLY DRIVEN SPIN DYNAMICS IN SEARCH OF A CLASSICAL CONDENSATION OF WAVES(Nazarbayev University School of Science and Technology, 2017) Yegovtsev, NikolayShow more The following work is a numerical study of a recently discovered Bose-Einstein condensation of magnons at room temperature by purely classical means. The classical Hamiltonian of the system includes exchange and dipolar interaction as well as interaction with external magnetic field. The system under consideration is a 2-dimentional lattice of classical spins with fixed positions. Classical equations of motions are derived from this Hamiltonian by applying the Poisson-bracket formalism. Equations of motions were solved numerically by the 4th order Runge-Kutta algorithm and obtained spin dynamics is transformed to the frequency space by the means of Fourier transform. This spectral decomposition of the dynamics allows monitoring the occupation of the energy states and observing condensation into the lowest energy state. In this work, we were able to reproduce the dispersion curve with characteristic minimum at non-zero value of the wave-vector by varying parameters of the system. We discuss models for pumping and dissipation, and hope to implement them in the future work.Show more Item Open Access Interaction of Turbulence with Shock Wave(Nazarbayev University School of Science and Technology, 2017) Zhaksylykov, AzamatShow more At the final stages of the lives of massive stars supernova explosion occurs. Before that process nuclear shell burning happens and strong turbulent convection arises in between iron core and silicon shell. Turbulence by amplifying flow in the post-shock region aid supernova explosion. In this work the physical mechanism behind this amplifi cation using a linear perturbation theory was investigated. Shock wave was modeled as one-dimensional planar discontinuity. Vorticity and entropy perturbations in the upstream flow considered as turbulence. Amplifi cation of turbulent kinetic energy of ow crossing shock wave by a factor of 2 was obtained.Show more Item Open Access Simulation activities for particle detectors R&D at Horizon-T and HorizonT-KZ cosmic rays experiments(Nazarbayev University School of Science and Technology, 2017) Duspayev, AlisherShow more A nature of Ultra-high energy Cosmic Ray is one of the remaining open topics in the field of High Energy Physics. The modern detector systems are being designed and constructed to study this phenomenon. "Horizon-T" that is experiment located at Tien Shan high-altitude Science Station (TSHASS) near Almaty, Kazakhstan, is one of such systems, and there is work underway for a novel detector system "HorizonT-KZ" (HT-KZ). The HT-KZ is being developed by Nazarbayev University (NU) in collaboration with TSHASS and is planned to be installed here, at NU, Astana, Kazakhstan. A simulation of a single detector module that is aimed to determine the optimal detector arrangement and parameters is a significant part of the R&D process. A description and a discussion of the results of the simulation runs are presented in this thesis.Show more Item Open Access Boundary Element Method for Stokes Flow in Incompressible Newtonian Fluids(Nazarbayev University School of Science and Technology, 2015-04) Tazhimbetov, Nurbek AkhmetulyShow more In this thesis, I present different discretization techniques for boundary integral method for Stokes flow in case of an incompressible Newtonian fluid. Boundary integral method (BIM) is one of many techniques that are used to solve Partial Differencial Equations (PDE) numerically. However, the basic advantage of the BIM is that it reduces the problem from n-dimensional domain to n - 1; for example, the two-dimensional square-box that contains viscous liquid can be solved by using the values of an unkown function at the boundary of square. Nevertheless, the BIM exhibits some challenges in finding the Green's function for a particular domain or differential operator, solving the integral equations and, especially, in computing the values of a complex domain. The latter one is quite diffcult because the flow diverges at corners (exhibits singularity). The goal of this work is to derive general analytical solution for Stokes equation (in integral equations form) and to compute the discretized integral equations using different quadrature rules for cavity problem.Show more Item Open Access Pairwise Overlap and Misclassification in Cluster Analysis(Nazarbayev University School of Science and Technology, 2015) Akynkozhayev, BirzhanShow more Separation of data into distinct groups is one of the most important tools of learning and means of obtaining valuable information from data. Cluster analysis studies the ways of distributing objects into groups with similar characteristics. Real-world examples of such applications are age separation of a population, loyalty grouping of customers, classification of living organisms into kingdoms, etc. In particular, cluster analysis is an important objective of data mining, which focuses on studying ways of extracting key information from data and converting it into some more understandable form. There is no single best algorithm for producing data partitions in cluster analysis, but many that perform well in various circumstances (Jain, 2008). Many popular clustering algorithms are based on an iterative partitioning method, where single items are moved step-by-step from one cluster to another based on optimization of some parameter. One of such algorithms, which will be mentioned in this paper is K-means algorithm, where data points are partitioned based on optimization of sum of squared distances within clusters (MacQueen, 1967). Another large class of algorithms are based on finite mixture model clustering methods. For example, stochastic emEMclustering method, which will also be covered in this article, is based on maximum likelihood estimation of statistical model parameters (Melnykov & Maitra). Misclassification of data is not a rare situation in cluster analysis. For instance, we can observe that several points have been misclassified on the previous figure (Figure 1) of true partition (a) versus the solution found by the K-means algorithm (b). Various factors lead to misclassification in clustering algorithms. The main goal of this paper is to analyze the effect of pairwise overlap, number of dimensions of data, and number of clusters on misclassification. The simplest case where misclassification can occur is when there are two clusters. The overlap is exact in this case, thus, we proceeded to use one of the simplest algorithms – K-means. At the higher number of clusters, when overlap is estimated, we considered more complex emEM algorithmShow more