Capstone Projectshttp://nur.nu.edu.kz:80/handle/123456789/17012021-07-28T03:18:52Z2021-07-28T03:18:52ZWell-Posedness of the Nonlinear Schrödinger EquationOrumbayeva, Sarahttp://nur.nu.edu.kz:80/handle/123456789/40942019-12-26T09:20:43Z2019-08-07T00:00:00ZWell-Posedness of the Nonlinear Schrödinger Equation
Orumbayeva, Sara
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.
2019-08-07T00:00:00ZNonlinear Schrodinger EquationKazbek, Moldirhttp://nur.nu.edu.kz:80/handle/123456789/40932019-12-26T09:17:51Z2019-08-08T00:00:00ZNonlinear Schrodinger Equation
Kazbek, Moldir
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.
2019-08-08T00:00:00ZFrobenius Singularities of Algebraic Sets of MatricesYerlanov, Madihttp://nur.nu.edu.kz:80/handle/123456789/40922019-12-26T09:15:19Z2019-08-07T00:00:00ZFrobenius Singularities of Algebraic Sets of Matrices
Yerlanov, Madi
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.
2019-08-07T00:00:00ZAnalytic Solutions for a Nonlinear Transport EquationBiyar, Magzhanhttp://nur.nu.edu.kz:80/handle/123456789/40912019-12-26T09:09:06Z2019-08-07T00:00:00ZAnalytic Solutions for a Nonlinear Transport Equation
Biyar, Magzhan
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.
2019-08-07T00:00:00ZNonlinear Regression Analysis of the generalized Logistic Model as an Actuarial life contingency modelKadenova, Aidahttp://nur.nu.edu.kz:80/handle/123456789/40902019-12-26T09:17:45Z2019-08-08T00:00:00ZNonlinear Regression Analysis of the generalized Logistic Model as an Actuarial life contingency model
Kadenova, Aida
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.
2019-08-08T00:00:00ZCopula functions in Credit Metrics’ VaR estimationMagzanov, Shynggyshttp://nur.nu.edu.kz:80/handle/123456789/40892019-12-26T09:09:42Z2019-08-08T00:00:00ZCopula functions in Credit Metrics’ VaR estimation
Magzanov, Shynggys
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.
2019-08-08T00:00:00Z