Capstone Projects
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Item Open Access Multiple Point Compression on Elliptic Curves(Nazarbayev University School of Science and Technology, 2015) Otemissov, AdiletThe paper aims at developing new point compression algorithms which are useful in mobile communication systems where Elliptic Curve Cryptography is employed to achieve secure data storage and transmission. Compression algorithms allow elliptic curve points to be represented in the form that balances the usage of memory and computational power. The two- and three-point compression algorithms developed by Khabbazian, Gulliver and Bhargava [4] are reviewed and extended to generic cases of four and five points. The proposed methods use only basic operations (multiplication, division, etc.) and avoids square root finding. In addition, a new two-point compression method which is heavy in compression phase and light in decompression is developed.Item Open Access Pairwise Overlap and Misclassification in Cluster Analysis(Nazarbayev University School of Science and Technology, 2015) Akynkozhayev, BirzhanSeparation 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 algorithmItem Open Access Boundary Element Method for Stokes Flow in Incompressible Newtonian Fluids(Nazarbayev University School of Science and Technology, 2015-04) Tazhimbetov, Nurbek AkhmetulyIn 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.Item Open Access Stable holomorphic polynomials on the half-plane and generalizations(Nazarbayev University School of Science and Technology, 2015-04) Kairzhan, AdilbekThe study of locations of zeroes of functions became popular among mathematicians many years ago. This investigation contributes a lot to wide range of theories and topics in Mathematics and Physics.Item Open Access Optimization of convex geometries: component quadratic and general(Nazarbayev University School of Science and Technology, 2016) Myrzakul, ZhanbotaIn this Capstone Project, we worked with a class of closure systems called convex geometries, which are closure systems with a closure operator that satisfies the anti-exchange property. We first looked at the result of optimization algorithm of component quadratic systems, which are discussed in [4], and reproved it for the case of convex geometries. We then investigated the following question: if a convex geometry is given by a set of implications, is it possible to find its optimum basis in polynomial time when the convex geometry does not have particular properties (for instance, not component quadratic)?Item Open Access On a stochastic interacting particle system with pushing dynamics(Nazarbayev University School of Science and Technology, 2016) Abdukadyrov, NurlanIn this paper we study a stochastic two-particle system on Z where particles interact each other by pushing dynamics. We derive the explicit formulas of the transition probability and of the probability distributions of each particle's position at time t. Finally, we discuss about the generalization of our works to N-particle system.Item Open Access Statistical Morphological Disambiguation for Kazakh Language(Nazarbayev University School of Science and Technology, 2016) Azamat, DaianaThis paper presents the results of developing a statistical model for morphological disambiguation of Kazakh text. Starting with basic assumptions we tried to cope with the complex morphology of Kazakh language by breaking up lexical forms across their derivational boundaries into inflectional groups and modeling their behavior with statistical methods. We also provide maximum likelihood estimates for the parameters and an effective way to perform disambiguation with the Viterbi algorithm.Item Open Access On logistic-normal distribution(Nazarbayev University School of Science and Technology, 2016-04) Almagambetova, Ayanna; Zakiyeva, NazgulExisting distributions do not always provide an adequate fit to the complex real world data. Hence, the interest in developing more flexible statistical distributions remains strong in statistics profession. In this project, we present a family of generalized normal distributions, the T-normal family. We study in some details a member of the proposed family namely, the logistic-normal (LN) distribution. Some properties of the LN distribution including moments, tail behavior, and modes are examined. The distribution is symmetric and can be unimodal or bimodal. The tail of the LN distribution can be heavier or lighter than the tail of the normal distribution. The performance of the maximum likelihood estimators is evaluated through small simulation study. Two bimodal data sets are used to show the applicability of the LN distributionItem Open Access On the well-posedness of the Boltzmann's moment system of equations in fourth approximation(Nazarbayev University School of Science and Technology, 2016-05) Issagali, AizhanWe study the one-dimensional non-linear non-stationary Boltzmann's moment system of equations in fourth approxi- mation with the tools developed by Sakabekov in [4],[5] and [6]. For the third approximation system Sakabekov proves the mass conservation law (cf. Theorem 2.1 in [4]) and discusses the existence and uniqueness of the solution (cf. Theorem in [6]). We extend the analysis of the existence and uniqueness of the solution to the fourth approximation system. In particular, for the fourth approximation system we discuss the well-posed initial and boundary value problem and obtain the a-priori estimate of the solution belonging to the space of functions, continuous in time and square summable by spatial variable.Item Open Access Finite Element Solution of Thermal Gee-Lyon Flows in a Circular Tube(Nazarbayev University School of Science and Technology, 2016-05) Aimambet, NarkenThe goal of this project is to calculate the temperature distribu- tion of certain pressure-driven non-Newtonian ows inside a circular tube. The rheology under consideration is the type in which the shear stress is an implicit function of the shear rate de ned as the inverse function of an odd function. The uid velocity in the tube is approximated by the steady state velocity pro le along the tube length and the viscosity is assumed to be independent of the temperature. The velocity pro le is computed by using Mathematica's build-in ODE solver semi-analytically. The corresponding steady state temperature pro le at tube length is then calculated taking into account of heat source generated by shear rate from the uid ow by solving an ODE. The temperature distribution from the entrance to the fully developed region is then approximated numerically by using the axisymmet- ric linear triangular nite elements. Material and geometric constants and data for extrusion of chemical Lucite through the tube in literature are used for the numerical example. Comparison of the numerical result with the industrial experimental result is made at a point along the central axis of the tube.Item Open Access Stability Analysis of SEIS model with spatial variations(Nazarbayev University School of Science and Technology, 2016-05) Baki, ZhuldyzayIn this report we present an SEIS model for infectious diseases with latent period and no immune response for spatially heterogeneous environment. Spatial heterogeneity is designed by several metapopulations. It was shown that global dynamics of an epidemics completely depends on basic reproduction number R0. By fxing the number of patches to two, we use next generation matrix method to obtain basic reproduction number and make further analysis on it. Migration rates of individuals are considered as one of the main factors that influence R0. Moreover, some numerical simulations for the dynamics of the system with different initial conditions is presented.Item Open Access Representation of Convex Geometries by Convex Structures on a Plane(Nazarbayev University School of Science and Technology, 2016-05) Bolat, MadinaConvex geometries are closure systems satisfying anti-exchange axiom with combinatorial properties. Every convex geometry is represented by a convex geometry of points in n-dimensional space with a special closure operator. Some convex geometries are represented by circles on a plane. This paper proves that not all convex geometries are represented by circles on a plane by providing a counterexample. We introduce Weak n-Carousel rule and prove that it holds for confgurations of circles on a plane.Item Open Access Modeling of corneal deformation under air-puff by nonlinear differential equations(Nazarbayev University School of Science and Technology, 2016-05) Jangabylova, Aliya; Zhalgas, AidanaIntraocular Pressure (IOP) is a main factor for the diagnosis of glaucoma. In this report, the kinematic viscoelastic corneal models, specifcally the Maxwell and the Kelvin-Voight models, of human eye ball will be proposed for determining the displacement of the cornea during the air-puff tonometry simulations and its relationship to IOP. The purpose of project is to study the in uence of elasticity and viscosity to the corneal deformations under an air puff.Item Open Access Finite element solutions of the nonlinear RAPM Black-Scholes model(Nazarbayev University School of Science and Technology, 2016-05-09) Zhexembay, Laila; Pak, AndreyThe main purpose of our Capstone project is to study the Risk-Adjusted Pricing Methodology (RAMP) Black-Scholes model and to find the finite element solutions of the nonlinear Black-Scholes equation. The RAPM is one of the many nonlinear models in option pricing considering factors which affect the volatility in the original Black-Scholes equation. This model can be simplifed to a nonlinear parabolic equation in a new vari- able which equals the product of Gamma and the price of the underlying asset. Galerkin nite element method is applied to the parabolic equation. Two types of solutions will be presented: one using the linear elements and the other using quadratic elements. Local finite element equations for the linear and quadratic elements are derived with some specifc inter- polations of the nonlinear terms. Numerical solutions are obtained and compared to the results in literature. The explanation of the discrepancies will be given together with the future goals of this study.Item Open Access A Short Note On Solving 1-D Porous Medium Equation by Finite Element Methods(Nazarbayev University School of Science and Technology, 2016-05-20) Matayev, ChingisPorous Medium Equation (PME) is one of the simplest types of of nonlinear evolution equation of parabolic type. It emerges in the description of di erent natural phenomena, and its theory and properties depart strongly from the heat equation, ut = u, its most famous relative. Hence the interest of its study, both for the pure mathematician and the applied scientist (Vazquez, 2006). The aim of this paper is to study the Porous Medium Equation in one dimension.Item Open Access Analysis of Dynamic Pull-in for a Graphene-based MEMS Model(Nazarbayev University School of Science and Technology, 2018) Omarov, DaniyarA 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.Item Open Access Copulas and Correlation in Statistical Risk Theory(Nazarbayev University School of Science and Technology, 2018) Rat, SaraThe 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.Item Open Access Initial Explorations on Regularizing the SCRN Model(Nazarbayev University School of Science and Technology, 2018-05) Kabdolov, OlzhasRecurrent 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.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, AkylWe 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.Item Open Access Actuarial Applications of a two-parameter generalized Logistic model(Nazarbayev University School of Science and Technology, 2018-05-10) Alekberova, NargizThis 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.