Applied Mathematics and Computer Science
95 theses under this subject heading
Szasz and Phillips Operators Based on q-Integers
ABSTRACT: In this thesis, q-Szász-Durrmeyer (0 < q < 1) and q-Phillips (q > 0) operators are defined and some properties of these operators are studied. More precisely, local approximation results for continuous functions in terms of modulus of continuity are proved and Voronovskaja type asymptotic results are investigated. Keywords: q-Szász-Durrmeyer operators, k-functional, modulus of continuity, qcalculus, q-Phillips operators, q-integers, q-gamma functions, rate of convergence, qintegral. ……………………………………………………………………………………………………………………………………………………………………………………………………………………
Association Rule Mining Using k-Map Model in Data Mining
In data mining, many algorithms were suggested to define the frequent rules within the data set. One of the problems is to choose a correct algorithm for the problem and the determination of the efficiency of the algorithm has important role during the investigation of hidden knowledge. The thesis describes how to handle data set with Association Rules Analysis/ Market Basket Analysis with the popular Apriori algorithm and k -Map algorithm of data mining. The goal of this thesis is to find the most frequent patterns within the data set and then using different measurements to do further investigation on the obtained frequent patterns. Keywords: Data Mining, Association Rules Analysis, Market-Basket Analysis
A Statistical Analysis on the Visits to EMU Health Center by the Students
There are different statistical techniques in estimating and predicting future events or outcome given a set of independent factors influencing such an event. Regression analysis is one of the modern statistical tools used for such purpose. Knowing the outcome of an event given a set of independent variable will help make proper decisions regarding the scenario. Here, regression analysis was used to predict the number of visitors visiting some key department of the Eastern Mediterranean University Health Center. This will help the school management to know the area where the health center is shorting man power and to also carry out a research or study on the reason why visitors are faced with such illness relating to the department they visit often. A solution has been detected and discussed to help in the prediction of the number of visitors visiting some key department of the school health center. A regression analysis has been carried out on the data set of the visitors who visited the health center in the past 22 months (January, 2014 to October, 2015) this involves the number of visitors in each month and the department they visited. This is done by the use of statistical software called SPSS. It is use for regression, and prediction measure especially when one is dealing with large numbers. Keywords: Estimating, Department, Health Center, Predicting, Regression Analysis, SPSS, Statistical Techniques
Smoothing Techniques for Time Series Forecasting
There are many forecasting techniques available, and selecting the appropriate technique is very important issue to achieve a good forecasting performance. This thesis intends to present the smoothing techniques for time series forecasting. The forecasting process using simple moving average and weighted moving average methods is investigated. The exponential smoothing forecasting method is analyzed. The simple exponential smoothing method is described. Some error measures - Mean Absolute Deviation, Mean Absolute Percentage Error, and Mean Square Error are calculated for above forecasting techniques to define the forecast accuracy of these methods. The double exponential smoothing method is discussed.
Multivariate Regression Compared with Moving Average Smoothing
Regression analysis is a statistical method having application in all fields of scientific and technological studies. Theoretical concepts lead to the development of regression theory are examined in some detail to lay down the foundation for the application of the theory. In statistics regression is mainly used to establish the kind of relationship between dependent and independent variables, i.e. linear or any other type. Moving average is a statistical method widely used for smoothing out raw data trajectories to obtain trends by filtering out the noise from the random fluctuations. The trend is an estimation of the functional behavior of the variable under study. This thesis is first centered on the theoretical characteristics of linear regression in chapter 3, examining the abstract concepts behind the regression theory, and the least squares method for establishing the model to be fitted from available data. Chapter 4 is allocated for the moving average technique used as a smoother of the trajectory for a variable. That smooth trend can be generated for every variable. The fitted regression model itself can be considered a smooth functional representation of the response variable in relation to the predictor/s. In Chapter 5 a case study of a data set is undertaken, where moving average technique was implemented for smoothing with 2 different orders, using m = 3 and m = 6 values for averaging of a real life data. It became evident that the smoother the data, the lower the error measures will be in a regression analysis. However, too much smoothing of a variable will runs the risk of obtaining close to a perfect regression fit, which will not be realistic. Based on the results obtained in the case study, it was then recommended that where large data sets are used for regression study, some smoothing can be beneficial as it will result in reduced estimation errors. Some software programs like Excel, Minitab, and S.P.S.S were all used to help in data processing to find the needed outputs.
2-Head Pushdown Automata
Finite state automata recognize regular languages which can be used in text processing, compilers, and hardware design. 2-head finite automata accept linear context-free languages. In addition, pushdown automata are able to recognize context-free languages which can be used in programming languages and artificial intelligence. We distinguish between deterministic and nondeterministic finite automata, 2-head automata and also pushdown automata. The deterministic version of these machines is such that there is no choice of move in any situation while the non-deterministic version may have a choice of move. The present thesis describes 2-head pushdown automata which is more powerful than the pushdown automata and it is able to recognize some non-context-free languages as well. Throughout the thesis we try to focus on characterization of aforementioned machines. Keywords: 2-head pushdown automata, non-context-free languages, deterministic automata, non-deterministic automata.
The Block-Hexagonal Grid Method for Laplace’s Equation with Singularities
A fourth order accurate matching operator is constructed on a hexagonal grid, for the interpolation of the mixed boundary value problem of Laplace’s equation, by using the harmonic properties of the solution. With the application of this matching operator for the connection of the subsystems, the Block-Grid method (BGM), which is a difference-analytical method, has been analysed on a hexagonal grid, for the solution of both the Dirichlet and mixed boundary value problems of Laplace’s equation with singularities. First of all, BGM is considered on staircase polygons and it is justified that when the boundary functions outside the finite neighbourhood of the singular points are from the Hölder classes C6;l ; 0 < l < 1; the error of approximation has an accuracy of O ��� h4 ; where h is the mesh size. The analysis of this method is extended to special polygons whose interior angles are a jp; a j 2 1 3 ; 2 3 ;1;2 ; and for the Dirichlet problem of Laplace’s equation it is proved that, with the application of BGM, it is possible to lower the smoothness requirement on the boundary functions to C4;l ; 0 < l < 1; outside the finite neighbourhood of the singular points, in order to obtain an accuracy of O ��� h4 . For the demonstration of the theoretical results on staircase polygons, BGM has been applied on an L-shaped domain for two examples, which has a singularity at the vertex with an interior angle of 3p 2 ; where Dirichlet and mixed boundary conditions are assumed respectively. The slit problem, which has the strongest singularity due to the interior angle of 2p at the vertex of the slit, has been considered on a parallelogram with a slit, in order to illustrate the results obtained on polygons with interior angles of a jp; a j 2 1 3 ; 2 3 ;1;2 : The second example on a parallelogram demonstrates the application of BGM on a domain with two singularities as it is assumed that the vertices with interior angles of 2p 3 are singular points. Solutions of the numerical examples are consistent with the theoretical results obtained. Keywords: Hexagonal grids, Laplace’s equation, singularity problem, block-grid method.
Quantum Error Correction Methods
This study surveys the mathematical structure of a quantum error correcting codes and the way they are developed through certain stages of error correction. In particular, the families of Calderbank-Shor-Steane codes (CSS) and the stabilizer codes are discussed and through elaborative examples it will be shown that the CSS codes are in the family of the stabilizer codes. Since the study of the CSS codes depends on a firm knowledge of classical coding theory, a rigorous mathematical review of the linear codes is done separately. Analysing the structure of the stabilizer formalism is highly depended on the effective use of some group theoretic notions. This structure is discussed in more detail and examples will be given. As the ultimate application of the quantum error correction the rules of the fault-tolerant quantum computing is explored and finding the threshold condition of an example will be done. Keywords: QEC, Coding theory, Stabilizer formalism, CSS codes, Fault-tolerant quantum computing, Threshold condition
Some Properties of Hypergeometric Functions
ABSTRACT: This thesis consists of five chapters. The first chapter gives brief information about the thesis. In the second chapter, we give some preliminaries and auxilary results which we will use in thesis. In chapter three, the extension of beta function containing an extra parameter, which proved to be useful earlier, was used to extend Appell’s hypergeometric functions of two variables and extend Lauricella’s hypergeometric function of three variables. Furthermore, linear and bilinear generating relations for these extended hypergeometric functions are obtained by defining the extension of fractional derivative operator. Some properties of the extended fractional derivative operator are also presented. In chapter four, we consider generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new generalizations. In chapter five, we present various families of generating functions for a class of polynomials in two variables. Furthermore, several general classes of bilinear, bilateral or mixed multilateral generating functions are obtained for these polynomials. Keywords: Generating functions, Hypergeometric function, Fractional derivative operator, Gamma function, Beta function. ……………………………………………………………………………………………………………………………………………………………………………………………………………………
Adomian’s Decomposition of Multi-Order Fractional Differential Equations
Adomian's Decomposition Method (ADM) was introduced about three decades ago, it has proven to be efficient, reliable and easy to compute the solutions of non-linear and linear differential equations. It can also be used to compute various types of equations such as Boundary value problems, Integral equations, Equations arising in fluid flow e.t.c. This thesis work presents the derivation of Adomian's decomposition algorithms and the possible solution of fractional differential equations of the multi-order type in the Caputo sense. It consist of four chapters, Chapter 1 contains a brief introduction of Adomian's Decomposition Method(ADM) and definitions, while the second chapter deals with basis proofs and methodology with respect to Adomian's Decomposition Method(ADM). In Chapter 3, we applied the method of solution to multi-order fractional differential equations. We then discuss the results and make conclusion in Chapter 4.
q-Polynomials and Location of Their Zeros
ABSTRACT: In this thesis, we define the q-Bernoulli numbers and polynomials, q-Euler numbers and polynomials, q-Frobenius-Euler numbers and polynomials and q-Genocchi numbers and polynomials of higher order in two variables x and y, by using two q-exponential functions. We also prove some properties and relationships of these polynomials and q-analogue of the Srivastava and Pinter addition theorem. Furthermore, we represent the figures of the q-Bernoulli, q-Euler and q-Genocchi numbers and polynomials. We find the solutions of these q-polynomials, for n 2 N, x and q 2 C by using a computer package MathematicaR ⃝ software. Finally, we discuss the reflection symmetries of these q-polynomials. Keywords: q-analogues of Bernoulli - Euler - Genocchi - Frobenius-Euler numbers and Polynomials, Srivastava Pinter addition Theorems, shapes and roots of q-polynomials …………………………………………………………………………………………………………………………
Using Rough Set Theory to Find Minimal Log with Rule Generation
The current research aims to enhance real-time analysis and speed of security algorithms by focusing on creating a minimal security log with the same efficiency as the original one through keeping core knowledge. Hence, manipulating this created minimal log with any security algorithm will surely enhance execution time and accordingly push towards real-time analysis. Rough set concepts are being used to reduce log size by using algorithms of the RoughSets package in R language, these algorithms guarantee to find a minimum log at least, and to find a minimal log at most depending on hardware resources available. Embedded knowledge is being extracted from minimum and minimal logs in the form of IF...THEN statements, and finally conclusions were discussed. Keywords: hate speech, text classification, classifier, classifier ensembles, stacking ensemble, text mining, genetic programming, pattern classification.
Quantum Encryption Algorithm Based on Modified BB84 and Authentication DH Algorithm
This thesis will describe two main contributions to quantum cryptography. First of all the well-known BB84 protocol for quantum key exchange is modified to eliminate the originally high bit error rate completely without compromising on the security. As the classical channel is the only place where eavesdropping could be possible, any attempt to eavesdropping is prevented. Secondly, the Quantum Linear Feedback Shift Register is proposed as a tool to generate the key matrix for the classical Hill-Cipher algorithm. Finally, all proposed methods are combined in an encryption process, proving to be secure, i.e. the resulting quantum encrypted message is less susceptible to attacks as shown in the detailed security. Keywords : Network security, Quantum cryptography, Quantum computing, BB84 Protocol, Quantum encryption algorithm , polarization filter, Multi-qubit Quantum shift register matrices,.
Fuzzy Utility Based Decision Analysis in the Credit Scoring Problem
A method that uses fuzzy c-means (FCM) is proposed for credit scoring based on unsupervised learning of a set training data. Data vectors are composed of significant applicant attributes and corresponding expert decisions. Two new statistical cost functions Jm and Jσ are introduced to evaluate the candidate models by k -fold cross validation based on the mean and the standard deviation of the decision attributes. A linguistic approach based on the fuzzy-valued Choquet integral is suggested to rank the consumer loan applicants. The lower and upper imprecise probabilities are used as a capacity measure in Choquet integral to determine the utility ranking of the consumer loan applicants. This thesis proposes an algorithm to calculate the applicant’s non-expected utility by using imprecise probabilities of accepted cases over the Fuzzy C-Means clusters for fuzzy Choquet integral. The method is applied on consumer loan evaluations for a financial institution to verify expert decisions in parallel to extracting linguistic rules of decision making. In the suggested approach linguistic fuzzy valued Choquet integral is used as measure of fuzzy utility. The results indicate that the proposed method is successful in ranking the consumer loan applications with only six fails in total of 135 applications. Keywords: Fuzzy c-means, Fuzzy clustering, Sugeno integral, Fuzzy valued Choquet integral, imprecise probability …………………………………………………………………………………………………………………………………………………………………………………………………………………… ÖZ: Bulanık- c -ortalaması (FCM) kullanan ve anlamlı basvuru nitelikleri ve karsılıgı uzman kararından olusan bir modelleme veri kümesiyle yönlendirilmemis ögrenmeye dayanan kredi puanlama metodu önerilmektedir. Degerlendirmeye aday modelleri k -kat çapraz saglamayla karar niteliginin ortalama ve standard sapmasına dayanan Jm ve Jσ adında iki yeni istatistiksel maliyet fonksiyonu tanımlanmaktadır. Tüketici kredisi için basvuran müsterileri sıralamak üzere bulanık degerli Choquet integrale dayanan sözel bir yaklasım önerilmektedir. Choquet integralde kapasite ölçüsü olarak tüketici kredisi basvurularının fayda sıralamasını belirlemek üzere alt ve üst belirsizlik olasılıkları kullanılmaktadır. Bu tez basvuranların umulmadık faydasını hesaplamak üzere Choquet integralin kapasite ölçüsü olarak bulanık- c -ortalaması kümelerindeki uzmanlarca onaylananların belirsiz olabilirliligi kullanan bir algoritma önermektedir. Önerilen metod bir finans kurulusunun tüketim kredisi degerlendirmelerinde bir yandan karar vermenin sözel kurallarını bulurken, diger taraftan uzman kararlarını sınamak üzere uygulanmıstır. Önerilen yaklasımda bulanık-sayı degerli ölçüt kullanan Choquet integrali kullanılmıstır. Sonuçlar ileri sürülen metodun tüketici kredisi basvurularını sıralamada toplam 135 basvurudan yalnızca altı yanılmayla basarılı oldugunu göstermektedir. Anahtar Kelimeler: Bulanık- c -ortalaması, bulanık sınıflandırma, Sugeno integral, bulanık degerli Choquet integral, kesin olmayan ihtimal.
Martingale Theory
From ages to ages there had been expectation of individuals on a specific predictions and future occurrences. So also in a game, different participant that involves in those specified game have their various expectations of the results or the output of the game they are involved in. That is why we need a mathematical theory that helps in prediction of the future expectations in our day to day activities. Therefore the Martingale Theory is a very good theory that explains and dissects the expectation of a gamer in a given game of chance. So in this thesis, we shall talk about the Martingale Theory expressing the expectations of a gamer in a game of chance, and also discuss the gaming strategies so as to enlighten everyone involved in a specific game their required expectation after proper understanding of the Martingale Theory. Keywords: Martingale, Game of chance, Random walk, Stopping time.
Stochastic Processes and Markov Chain
Andrey Andreyevich Markov is the founder of the Markov Chain. The Markov Chain is a stochastic process involving modeling over time and space. In sciences or randomize sciences in particular, it is usually important to predict an outcome based on the acquired or previous knowledge of a process. There exits various random processes. The Markov Chain appears as a key technique to deal and model such processes. Keywords: Stochastic Matrix, Probability Vector, Markov Chain.
Existence of Solutions of Fractional Differential Equations
This thesis is aimed to study the existence of mild solutions of a class of fractional differential equations with different boundary conditions. By using fixed point theorems, the existence results about mild solutions are expected to obtain. A strong motivation for studing fractional differential equations comes from the fact, that is essential in various fields of science, engineering and economics. Keywords: Differential equations, integral boundary conditions, irregular boundary conditions, p-laplacian operator.
Fuzzy Intelligent Traffic Control System
ABSTRACT: The aim of this thesis is to develop the fuzzy intelligent traffic control system for the optimal controlling of the traffic flow at the traffic intersections. The proposed fuzzy control system is used to effectively manage the urban traffic junction of the intersections of the city Famagusta (Gazimagusa), North Cyprus. The importance of the proposed fuzzy intelligent traffic control system consists in consideration of uncertainty and vagueness of information about the values of the input and output parameters of the system. Using the input parameters and based on the inferences from the fuzzy rules, the fuzzy traffic controller decides how to adjust the extension time of the green phase of traffic lights. The computer simulation is carried out using Matlab software. The optimal extension time of the green phase is determined using the Mamdani inference engine. The effectiveness of the fuzzy traffic controller with four input parameters is explained. Keywords: Fuzzy System, Traffic Control, Fuzzy Logic Controller. ……………………………………………………………………………………………………………………………………………………………………………………………………………………
A Survey on Mathematical Modeling of Cancer Incidence Rates
ABSTRACT: Bioinformatics is a novel interdisciplinary field which attempts to response the biological questions by the assist of other basic sciences as well as computer sciences. Cancer modeling is a real example of such these endeavors in order to help oncologists to find new ways to cure and prevent cancer diseases or predict, estimate, and analyze this hazard in order to step forward to a better future. In this arena mathematics and statistics have played great roles and enabled biology, oncology, and epidemiology to achieve new results by applying some mathematical and statistical methods such as various graphs and tools to compare the different criteria, curve fitting as well as analyzing and predicting the future data, time series and Markov processes to model the natural phenomena and study their behaviors. However this field is still like a young sapling which requires enough patience and care of scientists to bring forth. This thesis is mainly to make a survey generally on bioinformatics and specifically investigations on cancer cases as an application. First chapter provides a general overview about bioinformatics and its application. Then some preliminary concepts are explained including required biological information about cancer and its causes to comprehend the next concepts and implications better. Many scientists have tried to offer an applicable model for cancer incidence rate which can be acceptable and interpretable biologically. Second chapter provides some past findings about cancer incidence models as a background for readers with little biological information. In fact, the focus of this research is mainly on mathematical modeling of cancer incidence rates. In addition, chapter three is discussing about the cancer incidence. For instance some factors affecting on the process of cancer incidence such as the place and time period of living, sex, race, and the amount of development are checked. In chapter four some curve fittings are performed by MATLAB software, and also special mathematical model which is called Furrier model has been fitted to the real cancer incidence rates data with the best goodness of fit. Keywords: bioinformaics, biomathematics, cancer, incidence rate, mathematical modeling, curve fitting, Furrier model. ……………………………………………………………………………………………………………………………………………………………………………………………………………………
A Survey on The Methods of Spatial Statistics
ABSTRACT: In this thesis the kriging approach is presented for interpolating spatial data points. The mathematical model of the kriging method is mentioned first. Then a small example is provided about the steps of this approach. ArcGIS software package is an important tool for creating maps and also for performing geostatistics about large data points. A kriging example is also tried by this package and the predicted data points are listed. Keywords: Geostatistics, spatial data, kriging, Arcgis. …………………………………………………………………………………………………………………………
Strategies to Fast Evaluation of Expression Trees
Expression trees are well-known tools to visualize the syntactical structure of the expressions. They are helpful also in evaluations, e.g., decision trees are widely used. Games and game theory form an important field in Artificial Intelligence and it has several connections to Business and Economy. Short circuit, short cut, or by other name, lazy evaluations play important roles in various fields of computer science including logic, hardware design, programming, decision making. In this thesis, different types of trees are considered including extensions of game trees using operations, e.g., multiplication, (constrained) addition and the usual minimum and maximum, and three of the best known and used fuzzy logic systems, (Gödel, Lukasiewicz, and product logics). The evaluation of lots of formulae can be speeded up by various pruning techniques by discovering which remaining part of the formulae has no influence on the final result for various reasons. The presented techniques can be seen as generalizations of short circuit evaluations in Boolean logic and also of alpha-beta pruning of game trees. Simulation results show the efficiency of the presented techniques. Keywords: expression trees, game trees, formula trees, fast evaluation, fuzzy logic, many valued logic, pruning techniques, short circuit evaluation, lazy evaluations.
Numerical Methods for Solving Systems of Ordinary Differential Equations
ABSTRACT: This thesis concantrates on numerical methods for solving ordinary differential equations. Firstly, we discuss the concept of convergence, local-truncation error, globaltruncation error, consistency, types of stability, zero-stability, and weak-stability. Afterwards, we inform some materials for Euler and Runge-Kutta method. The given ordinary differential equation is analyzed on Euler and Runge-Kutta method to find the approximated solution with the given initial conditions. Then, the stability of each method is examined briefly. We also focus on numerical methods for systems. Then, the reason of the stiff system is discussed. After investigating the numerical methods, we gave advantages and disadvantages of Euler method and Fourth Order Runge-Kutta method. Finally, numerical experiments is applied on Explicit Euler method and Explicit Fourth Order Runge-Kutta method. The approximated solutions with different step-size and analytical solutions of methods are computed in Matlab software. The computation of approximated solutions of methods are compared with analytical solutions. Then we discussed the accuracy of these methods when they are applied to the specified system in Chapter 7. Finally, we conclude that Explicit Fourth Order Runge-Kutta method is more accurate than the Explicit Euler method. Keywords: Ordinary Differential Equations, Numerical solutions, Euler’s method, Runge-Kutta method, Stiff System ……………………………………………………………………………………………………………………………………………………………………………………………………………………
Decision Making Under Fuzzy Environment
High competition between universities has been increasing over the years, and stimulates higher education institutions to attain higher positions in the ranking list. Ranking is an important performance indicator of university status evaluation, and therefore plays an essential role in students’ university selection. In this thesis, analytic hierarchy process (AHP) and fuzzy analytic hierarchy process (FAHP) are applied for the comparison and ranking of performances of five UK universities, according to four criteria. The criteria used for the evaluation of universities’ performances are teaching, research, citations, and international outlook. The comparison matrix is used to compare criteria as well as alternatives with respect to each criterion. Consistency index and consistency ratio are calculated by using eigenvalues to check the consistency of comparison matrix and by obtaining the final priorities, ranking is possible with AHP. Eigenvalues are also calculated to check the consistency for FAHP as well as eigenvectors are calculated for ranking process. Besides these, by the calculation of coefficient of variation for all alternatives, it becomes possible to rank the universities in prioritized order with FAHP. Keywords: Ranking, Analytic Hierarchy Process (AHP); Fuzzy Analytic Hierarchy Process (FAHP); ranking; consistency; eigenvalue; eigenvector; coefficient of variation.
Numerical Solutions of Fractional Differential Equations
In this thesis, we focus on numerical solutions of general linear multi-term fractional differential equations (FDEs) with fractional derivatives defined in the Caputo sense. Multi-term fractional order differential equations are involving both ordinary and fractional derivative operators. Numerical methods plays very crucial role for solving fractional differential equations, since analytical solutions are not always possible for solving them. Memory trait of fractional calculus is one of the main reason for difficulty of developing analytical techniques for such a equations. Therefore, there has been considerable interest in solving FDEs numerically in recent years and many powerful schemes have been developed. Essentially, most of the developed methods are modified from original versions for classical differential equations and applied to FDEs. In this study, we introduce a numerical technique based on the fractional Taylor vector and we construct fractional Taylor operational matrix of fractional integration to solve multi-term FDEs. The main characteristic of this technique is to reduce the given IVP of fractional order to a system of algebraic equations by employing the fractional Taylor operational matrix of fractional integration. Finally, this set of algebraic equations can be solved easily and efficiently for unknown coefficients by using computer programming. Consequently, by using these coefficients, the approximate solution of the given problem can be obtained. Some numerical examples are presented to demonstrate the accuracy and applicability of given method. The approximate solutions obtained by use of given technique are compared with numerical results of some other methods in literature and exact solutions of given problems. From these results, we can conclude that the presented technique is efficient and applicable for solving high order multi-term fractional order differential equations numerically. Keywords: numerical solutions, fractional Taylor vector,fractional differential equations, spectral method, Caputo fractional derivative, Riemann-Liouville fractional integral, operational matrices.