Mathematics http://nur.nu.edu.kz:80/handle/123456789/4120 2020-05-31T19:45:01Z Finite Element Method for Retaining Walls Subjected to Swelling Pressure http://nur.nu.edu.kz:80/handle/123456789/4752 Finite Element Method for Retaining Walls Subjected to Swelling Pressure Shakipov, Mansur Finite Element Method (FEM) is a widely used method of solving initial boundaryvalue problems from mechanical engineering. It allows addressing irregular domains and force terms, while enabling careful analysis of the approximated solutions. In this capstone project, a standard derivation of FEM from the mechanical engineering standpoint is presented, then all the necessary mathematical machinery is introduced to facilitate the discussion. Once the proper introduction to FEM is given, the paper dives into the two main subjects of the matter. First, the derivation of the EulerBernoulli beam equation is presented. It is a standard model for the analysis of retaining walls. Then a brief derivation of the swelling force, which models the effect of soil swelling onto the retaining walls, is given. Second, the finite element solution to the problem is derived, and the results are presented. 2020-05-25T00:00:00Z Modeling dependencies between exchange rates using timeinvariant and time-varying copulas http://nur.nu.edu.kz:80/handle/123456789/4700 Modeling dependencies between exchange rates using timeinvariant and time-varying copulas Adil, Asem In this project, the bivariate dependence structures between the Japanese Yen, Chinese Yuan, and Hong Kong Dollar exchange rates against the US Dollar are studied by using time-invariant and time-varying copulas. The period from 20.03.2010- 20.03.2020 is used for numerical simulations and marginal distributions are determined by the ARMA-tGARCH approach. The optimal models are chosen based on AIC values. Then the copulas are determined by the optimal choice of marginal distributions and finally, copulas are numerically constructed and used to describe dependencies between these three exchange-rates. Changes in the linear correlation coefficient over time are studied using time-varying copulas. The R script is provided to implement this procedure. The results suggest a positive dependence and greater lower-tail dependence between pairs of Japanese Yen-Chinese Yuan and Chinese Yuan-Hong Kong Dollar exchange rates. 2020-04-30T00:00:00Z (P,Q)-SUB-LAPLACIANS ON THE HEISENBERG GROUP http://nur.nu.edu.kz:80/handle/123456789/4699 (P,Q)-SUB-LAPLACIANS ON THE HEISENBERG GROUP Abdikarim, Aidana; Suragan, Durvudkhan The main purpose of this capstone project is to do some analysis related to (p,q)-sub-Laplacians on the Heisenberg group. In the first part of the project, Green’s identities for (p,q)-sub-Laplacians are given on the Heisenberg group and used further in proof of the uniqueness of a weak solution of a nonlinear Dirichlet boundary value problem for the (p,q)-sub-Laplacian. Moreover, concepts of CC and Kaplan balls are discussed to illustrate the smoothness of the considered domain for the BVP. 2020-08-01T00:00:00Z Simulations of Implied Volatility and Option Pricing using Neural Networks and Finite Difference Methods for Heston Model http://nur.nu.edu.kz:80/handle/123456789/4697 Simulations of Implied Volatility and Option Pricing using Neural Networks and Finite Difference Methods for Heston Model Arziyev, Sukhrat The theory of option pricing made a dramatic step forward when Black and Scholes published a centennial paper with a solution for the prices of European call and put options. However, their solution only deals with the perfect markets. In the realworld, the markets are not perfect and the predictions from their formula deviate significantly from the market prices. This is because of the assumptions on which the model is based. Heston model is one of the newer models which extends the classical Black-Scholes model and can produce better estimates of option prices with non-constant variable volatility. However, this variable volatility needs to be accurately predicted itself, but the price which is often taken as input for finding implied volatility is not always given. In this thesis, two neural network models for learning historical volatility were constructed and compared in order to predict the implied volatility for the option without knowing its price. The results of prediction were tested and evaluated. The better performing model was used to approximate implied volatility and the result was incorporated into the solution grid and finite difference scheme for the Heston model was applied to produce the option price. The generated prices were compared to the prices generated by the classical Black-Scholes model in several different scenarios. 2020-04-30T00:00:00Z Calculation of manifold’s tangent space at a given point from noisy data http://nur.nu.edu.kz:80/handle/123456789/4694 Calculation of manifold’s tangent space at a given point from noisy data Toleubek, Moldir Recently, several studies have been conducted in a field of machine learning to construct manifolds from data in a complex multidimensional space. Therefore manifold learning becomes remarkably attractable among researchers. One of the main tools to identify manifold’s structure is tangent space. In this work, first, we simulate a method for finding tangent space of manifold at some point from noisy data by Principal Component Analysis. In fact, Principal Component Analysis(PCA) provides dimension reduction by its ‘principal components’. Then we introduce concurrent method to PCA that is called Maximum Mean Discrepancy distance. It is based on measuring the distance between smooth distributions. 2020-05-04T00:00:00Z Robust Prediction with Risk Measures http://nur.nu.edu.kz:80/handle/123456789/4684 Robust Prediction with Risk Measures Duisenbay, Yerlan This thesis deals with coherent risk measures and its simulation with respect to different probability distributions. This study gives a numerical scheme to approximate any coherent risk measure via a sum of specific quantiles. We give the theoretical background on coherent risk measures in the first part and in the second part of this thesis we illustrate our findings via several simulations. 2020-04-29T00:00:00Z