Department of Mathematics

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http://hdl.handle.net/20.500.11905/1216

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PARTICIPATION PRIVACY ANALYSIS OF BLOCKCHAIN BASED E-VOTING SYSTEMS
(2022-06-14) ÖZDEMİR, Buse Yeşim; SULAK, Fatih
Technology is advancing almost every day. This progress changes the balance in our lives. One of these changes is that voting systems can now be done remotely. This means that no one needs to be physically present in a specific place to vote for the candidate they choose. One of the new technologies added to remote voting sys tems is blockchain technology. Building an e-voting system is a fundamental issue, and with the integration of blockchain technology into e-voting systems, they can become more decentralized, immutable and transparent. Because of these enhance ments, the e-voting system’s members (voters, authorities, etc.) have more trust in one another, and thus the system’s success increases. In addition, blockchain-based e-voting systems must ensure participation privacy and provide key features such as completeness, robustness, anonymity, non-reusability, fairness, relevance, and verifi ability. This thesis explains the blockchain-based voting systems published by Zhang, Gajek, Lewandowsky, Zhou and Krillov in-depth. It is shown that the participation privacy principle is not provided in the proposed systems.
EXISTENCE OF SOLUTIONS FOR HIGHER ORDER MULTI-POINT IMPULSIVE BOUNDARY VALUE PROBLEMS ON TIME SCALES
(2022-06-14) KUŞ, Murat Eymen; DOĞRU AKGÖL, Sibel; GEORGIEV, Svetlin G.
In this thesis, we investigate the sufficient conditions for existence of solutions of multi-point higher order impulsive boundary value problems on time scales. In par ticular, a class of third order impulsive boundary value problem and a class of 2n+1, n ≥ 1, order impulsive boundary value problem is studied. In chapter 1, we give the definitions and basic notions on time scales calculus. Then, we present some exam ples and give the fixed point theorems that are used in the thesis. Chapter 2 is devoted to existence of solutions of third order multi-point dynamic impulsive boundary value problems. In chapter 3, we focus on existence of solutions of multi-point dynamic impulsive boundary value problems of odd order. Finally, we give a short conclusion in Chapter 4. The results in this thesis are published/accepted for publication in the Georgian Mathematical Journal and Miskolc Mathematical Notes, respectively.
DEVELOPMENT OF CUSTOM LOAD AND RESISTANCE FACTORS FOR DESIGN OF REINFORCED CONCRETE STRUCTURES
(2021-07-10) ELOSTA, İbrahim; MERTOL, Halit Cenan
The load and resistance factors used nowadays in the design of reinforced concrete structures were developed before this century. Using these factors from the past significantly penalizes the design of reinforced concrete structures constructed using materials having better quality control and loads having better predictions of occurrences today. The purpose of this study was to develop a tool that determined the load and resistance factors depending on the statistical data (bias and covariance) related to current materials (concrete and steel) and current prediction of loads (dead, live, etc.) and the target reliability index. The First Order Second Moment (FOSM) and Monte Carlo Simulation (MSC) were the methods used as the structural reliability models. The first method was used to determine the resistance parameters for different failure modes. These resistance parameters (biases and covariances) were calculated using 20 million random variables using MCS Method to determine reliability index values. Finally, a program using Microsoft Excel Software was developed to determine custom load and resistance factors to design reinforced concrete members. Based on the input date for biases and covariances of resistance, dead, and live loads; failure modes of the beams and column members; and the target reliability index, the produced program selects custom load factors for your project.
ON THE DYNAMICS OF A SECOND ORDER NONLINEAR DIFFERENCE EQUATION
(2014-07-09) AKSOY, Aycan; TURAN, Mehmet
In this thesis, a certain second order fractional difference equation containing two ar bitrary parameters is handled. The issue equation is investigated with aspects of some dynamics structures: the boundedness character and semi-cycle analysis of positive solutions are examined; existence of periodic solutions is studied; local and global stability analysis of the fixed point are performed. This thesis consists of four chapters. In the first chapter, historical information about difference equations, some modelings with them, and some recent studies are given. In the second chapter, basic concepts and known results concerning the sequences and difference equations are provided. Main results are presented in Chapter 3. A short conclusion is written down in the last chapter.
COMPUTATIONAL METHODS FOR PRICING AMERICAN OPTIONS
(2014) AYDOĞAN, Burcu; AKSOY, Ümit; UĞUR, Ömür
In financial mathematics, option pricing is a popular problem in theory of finance and mathematics. In option pricing theory, the valuation of American options is one of the most important problems. American options are the most traded option styles in all financial markets. In spite of the recent developments, the valuation of American options continues to exist as a challenging problem. There are no closed-form analytical solutions of American options, so that a usual way to deal with this problem is to develop numerical and approximation techniques. In this thesis, we analyze binomial, finite difference and approximation meth ods, for pricing American options. We first consider the binomial approxima tion which is very easy to implement and assumes that the asset prices follow from geometric Brownian motion. Then, we present American options as a free boundary value problem based on Black-Scholes partial differential equa tion, which leads to a very famous model in finance theory, and formalize it as a linear complementarity problem. We refer to the projected successive over relaxation (PSOR) method to solve this problem. Although there are no closed form solutions for American options, we deal with some analytical approximation methods to approach the option values. We demonstrate the applications of the each method and compare their solutions.
LINEARLY IMPLICIT SCHEMES FOR THE ROSENAU-KORTEWED-DE VRIES REGULARIZED LONG WAVE EQUATIONS
(2015-06-25) AL- OMAİRİ, Salim Mahmood Yaseen; AYDIN, Ayhan
In this thesis, we consider the numerical solution of the Rosenau– Korteweg de Vries– Reularized Long Wave (Rosenau KdV–RLW) equation which couples the general Rosenau–KdV and Rosenau–RLW equations. Two conserved quantities of the Rose nau KdV–RLW equation has been proven, namely the mass and the energy. The aim is to develop numerical methods to preserve these conserved quantities exactly or preserve with a small error. Two linearly implicit schemes for the initial–boundary value problem of the Rosenau KdV–RLW equation are proposed. One method is con servative and the other method is nonconservative. It is proved that the conservative method preserves the energy of the equation exactly. It is also shown that the scheme is second order accurate and unconditionally stable. The second scheme is nonconser vative. It is first order accurate and conditionally stable. Numerical results shown that both scheme well simulates the solitary wave of the equation in long time. Moreover, numerical results verify the exact energy conservation of the conservative scheme.
STATISTICAL RANDOMNESS TESTING OF SECOND ROUND CANDIDATE ALGORITHMS OF THE CAESAR COMPETITION
(2016-12-17) Aşkın Özdemir, Betül; Sulak, Fatih
In order to improve symmetric key research several competitions had arranged by organizations like NIST and IACR. In recent years, the importance of authenticated encryption has rapidly increased because of the necessity of simultaneously enabling integrity, confidentiality and authenticity. Since the necessity of this kind of security rises, at January 2013, the CAESAR Competition announced at the Early Symmetric Crypto workshop. This competition is run by the IACR and this research commu nity will select the final portfolio algorithm. During the competition, a number of algorithms is eliminated in each round. Therefore, analysis of algorithms are very important for the CAESAR Competition. For this purpose, while the competition continues, we apply statistical randomness testing for the algorithms such as AEGIS, Ascon, Joltik, MORUS, Pi-Cipher and Tiaoxin. In this thesis, we have focused on the analysis of the algorithms participated in CAESAR Competition using statistical randomness tests and observe that up to how many rounds, those algorithms, behave random according to statistical randomness tests.
ON PERMUTATION POLYNOMIALS OVER FINITE FIELDS
(2017-03-04) Dabboor Asad, Maha M.M.; Gülmez Temür, Burcu
In this thesis, we study on permutation polynomials defined over finite fields. We have made a survey of some recent research results on constructions and classifications of some types of permutation polynomials over finite fields.
ON PROPERTIES OF q−BERNSTEIN POLYNOMIALS
(2017-04-04) Almesbahi, Manal Mastafa; Turan, Mehmet; Ostrovska, Sofiya
The aim of this thesis is to study the theory of the Bernstein polynomials and its recent extension to the q-calculus. The main focus of the present study is on the q-Bernstein polynomials which appeared twenty years ago and have been attracting many researchers afterward. This work exhibits a review of well-known results on the Bernstein polynomials along with the necessary preliminaries, introduction to the the ory of the q-Bernstein polynomials, and some new developments. The latter include the result on strong operator limit for the sequence of the limit q-Bernstein operators and the proposition that the q-Bernstein operators are weakly Picard.
EXPONENTIAL FINITE DIFFERENCE METHOD FOR NONLINEAR BLACK-SCHOLES EQUATION
(2017-04-04) Omar, Fathia; Aksoy, Ümit; Aydın, Ayhan
In this thesis, we investigate exponential finite difference method for nonlinear Black Scholes equation arising in an illiquid market. Chapter 1 is devoted to the literature survey with some basic definitions and terminology of the option pricing problem. In Chapter 2 we review the Black-Scholes model and finite difference methods for Black Scholes equation. In Chapter 3, an explicit finite difference method for a nonlinear Black-Scholes equation is studied with the monotonicity, stability and consistency re sults. In Chapter 4, we apply the exponential finite difference method to linear and nonlinear Black-Scholes equations. Moreover, we investigate consistency and con vergence of the method. Numerical experiments are performed to verify theoretical results. Exponential finite difference method is compared with an explicit finite dif ference method proposed for linear and nonlinear Black-Scholes equation. Numerical results show that exponential finite difference method exhibits better performance then explicit method. Finally, we give a brief conclusion in Chapter 5.
ON PROPERTIES OF HERMITE AND q−HERMITE I POLYNOMIALS AND THEIR LIMIT RELATIONS
(2017-05-02) Alwhishi, Sakina; Sevinik Adıgüzel, Rezan; Turan, Mehmet
In this thesis, some important properties of the Hermite polynomials and discrete q Hermite I polynomials are presented. Their properties will be considered in the same manner. The discrete q-Hermite I polynomials are the q-analogues of the Hermite polynomials. Such polynomials are an important class of the classical orthogonal polynomials and their q-analogues. The central idea in this thesis is to study the dif ferential and q-difference equation of hypergeometric type, three terms recurrence relations, Rodrigues formulas, orthogonalities and generating functions that the Her mite polynomials and its discrete version have. Hermite polynomials are obtained from the discrete q-Hermite I polynomials in the limiting case as q → 1. Such limit relation between the Hermite polynomials and the discrete q-Hermite I polynomials on each properties that is introduced in the thesis are considered in detailed.
CONNECTEDNESS OF THE CUT-SYSTEM COMPLEX ON SURFACES
(2017-05-02) Ali, Fatema; Atalan Ozan, Ferihe
Let M be a compact, connected, orientable or nonorientable surface of genus g ≥ 1 with n boundary components. In this thesis, we study connectedness of cut-system complex of M. More precisely, in Chapter 3, we study the work of Wajnryb on the connectedness of the cut-system complex of an orientable surface. In the last chapter, we prove that the cut-system complex of a nonorientable surface is connected.
AUTOMORPHISMS OF THE GRAPH OF CURVES
(2017-05-02) Elamin, Amel Omar; Atalan Ozan, Ferihe
In this thesis, we study the automorphisms of a certain graph of nonseparating curves and automorphisms of complexes of two-sided curves on surfaces. In Chapter 3, we deal with the work of P. S. Schaller on mapping class groups of hyperbolic surfaces and automorphism groups of graphs for orientable surfaces of genus g ≥ 1 with n punctures. More precisely, it is shown that the automorphism group of the certain graph is isomorphic to the extended mapping class group of the orientable surface proved by Schaller. In the last chapter of this thesis, we study the automorphisms of the complexes of two-sided simple closed curves on nonorientable surfaces of genus g with n holes.
NEW PRECONDITIONERS FOR STATIONARY ITERATIVE METHODS
(2022-03-01) Atya, Naima Ibrahim; Özban, Ahmet Yaşar
The convergence of an iterative method used for the solution of systems of linear equation depends on the properties of the spectrum of the matrix of the linear sys tem. So, to speed up the convergence, the given linear system is transformed into an equivalent one by linear transformations, known as preconditioners. In this thesis, we introduce two new preconditioners for Jacobi and Gauss-Seidel (GS) iterative methods for the solution of linear systems with strictly columnwise diago nally dominant (SCDD) L− matrices and SCDD positive matrices. The new precon ditioners can be applied on a single row, on a limited number of rows, called partial preconditioning, or on all rows, called complete preconditioning, of the system ma trix. First of all, the properties of the preconditioned matrices are determined for systems with SCDD L− matrices and SCDD positive matrices. Then convergence analysis of the Jacobi and GS iterative methods are performed for the preconditioned systems. For systems with SCDD L− matrices, it is shown that the spectral radii of Jacobi and GS iteration matrices for preconditioned systems are smaller than the ones associated with unpreconditioned systems. Although, for systems with SCDD positive matrices, we prove that the spectral radii of Jacobi iteration matrices for preconditioned systems are smaller than the ones associated with unpreconditioned systems, no such result is available for GS iteration matrices. Numerical results show that for systems with SCDD L−matrices, the new precondi tioners are quite competitive with the ones existing in the literature in the sense of spectral radii and the number of iterations. Nevertheless, for systems with SCDD positive matrices, numerical results demonstrate that although new preconditioners are still competitive with some other preconditioners, usually they are not preferable on many of the already existing ones. Finally, the performances of new precondition ers for CDD L−matrices and non-CDD L−matrices and even for non-SCDD positive matrices deserve further research.
PSEUDOSPECTRAL METHODS WITH VARIOUS BASIS FUNCTIONS AND APPLICATIONS TO QUANTUM MECHANICS
(2022-02-28) Wlie, Saeida; Erhan, İnci
In this thesis, we studied the pseudospectral methods and their application to the so lution of eigenvalue problems associated with ordinary differential equations. In par ticular, we considered second order differential equations and a specific example, the Schrodinger equation for quantum dynamical systems with polynomial potentials. ¨ After an introduction to self adjoint eigenvalue problems and the Schrodinger equation ¨ for particles, in the presence of polynomial potentials, we recollected some impor tant properties of Lagrange interpolation and orthogonal polynomials. We presented a method to compute the zeros of an orthogonal polynomial of arbitrary degree by means of a symmetric tridiagonal matrix eigenvalue problem. We constructed the particular symmetric tridiagonal matrices for computation of the zeros of Hermite, Associated Laguerre, Chebyshev and Legendre polynomials. After that, we explained in details the pseudospectral schemes using Hermite and Associated Laguerre polynomials by studying some published articles. We also made substitutions on the independent variable in order to transform infinite interval to a finite one and derived pseudospectral formulations using Chebyshev and Legendre polynomials. As a specific example, we applied the pseudospectral methods using the four types of orthogonal polynomials mentioned above to the Schrodinger equation for quantum ¨ dynamical systems with polynomial potentials. We compared our numerical results with the numerical results obtained previously by other authors and made comments about the efficiency of our method.
STIELTJES CLASSES FOR KUMMER TYPE PROBABILITY DISTRIBUTIONS
(2022-02-21) Khalleefah, Mohammed Ahmed Saad; Ostrovska, Sofiya; Turan, Mehmet
The moment problem is one of the classical directions in Probability Theory, which studies whether or not a probability distribution is uniquely determined by its mo ments. The problem originated in XIX century and is still drawing attention of re searches both in mathematics and applied disciplines. During the last decades, the subject of finding families of different probability distributions with the same mo ment sequences has gained a popularity and a large number of papers in this area has been published. Special classes of such families, called the Stieltjes classes, have become an area of intensive research. In this thesis, after background information on the transform methods, a review of both classical and present-day results on the moment problems is presented. The re view includes a general description of the moment problem, a list of checkable criteria for the moment (in)determinacy, and some methods to construct Stieltjes classes for probability densities. All notions and results are illustrated by examples. In addition, recently introduced power Lindley distribution has been studied and new Stieltjes classes for the power Lindley density has been constructed.
SERIES SOLUTIONS OF DYNAMIC EQUATIONS ON TIME SCALES
(2022-02-21) Alusta, Fatma; Erhan, İnci
In this thesis, we study the series solution method for dynamic equations on time scales. We propose a series expansion for the solution of a given dynamic equation and derive a very general recurrence relation formula for the computation of the co efficients in this series. The importance of time scales and dynamic equations on time scales shows itself in the fact that time scales unify the continuous and discrete analysis and therefore, dynamic equations cover both the differential and difference equations. In Chapter 1 we give the definition of time scales, some basic notions on time scales and present some examples. We introduce basic calculus concepts such as delta derivative and integral of function defined on time scales in Chapter 2. In the same chapter we also define some elementary functions on time scales. Chapter 3 is devoted to basic thery of linear dynamic equation of first and higher order. The Series solu tion method is presented in details in Chapter 4. In Chapter 5 we apply the method to some specific examples of linear dynamic equations including both constant and nonconstant coefficients equations. Finally, we discuss the conclussion in Chapter 6.
INTERPOLATION ON TIME SCALES
(2022-02-16) Jaddoa, Najlaa; Sevinik Adıgüzel, Rezan; Erhan, İnci
In this thesis, we investigate the interpolation on time scales. We define the Lagrange, σ-Lagrange, Hermite, σ-Hermite, Newton and σ-Newton interpolation polynomials on arbitrary time scales. We define the divived and σ-divided differences and con struct a divided difference table to be used to obtain the Hermite polynomial for a given data set in a very easy way. The interpolation of a data set by means of the so called σ-polynomials is an unusual one where the interpolating functions may not be polynomials depending on the form of the time scales under consideration. In this way, we provide a different aspect to the interpolation on time scales. We consider numerous examples on various types of time scales. The examples are supported by numerical computation and relevant figures obtained with Matlab.
FINITE ELEMENT SOLUTION OF THE POISSON EQUATION
(2022-02-15) Mohammed, Taha Yousif Mohammed; Eid, Rajeh
The finite element method (FEM) has been implemented in this thesis to solve the two dimensional (2D) Poisson equation (see [9]). We used the shape function procedure to construct our algorithm. The shape function considered in this study employs two types of interpolation functions, which are linear and quadratic shape functions (see [2] and [10]). We made use of linear shape function on 3-node triangular elements and 4-node rectangular elements, while the quadratic shape function has been used on 6- node triangular and rectangular elements. Then we applied the finite element method to compute the approximate solution of the problem under consideration. Numerical experiments were performed for both linear and quadratic shape functions to show the accuracy of the solution. The efficiency of the method has been observed by comparing the numerical results obtained by using finite element method with the exact solution.
AVERAGE VECTOR FIELD METHOD FOR HAMILTONIAN SYSTEMS
(2022-01-11) Sabawe, Bahaa Ahmed Khalaf; Aydın, Ayhan
In this thesis, we present and analyze four energy preserving methods for the numer ical solution of initial value problems of Hamiltonian type. In particular, the average vector field (AVF) and partitioned AVF (PAVF) methods are used to drive energy preserving methods. In addition to these two energy preserving methods, two en ergy persevering PAVF composition (PAVF-C) and PAVF plus (AVF-P) methods are presented. The thesis accompanied numerical result for Zakharov system that demon strate remarkable properties of the proposed energy persevering methods. In this the sis, this is the first time that energy persevering AVF, PAVF, PAVF-C and PAVF-P methods are proposed for Zakharov system. It is shown that PAVF and PAVF-C meth ods for Zakharov system are linearly implicit methods that have remarkable lower cost than the original AVF method. In addition, we further show that the PAVF meth ods preserve the mass conservation of the Zakharov system while the AVF method cannot.

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