**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**1642

# World Academy of Science, Engineering and Technology

## [Mathematical and Computational Sciences]

### Online ISSN : 1307-6892

##### 1642 Pure and Mixed Nash Equilibria Domain of a Discrete Game Model with Dichotomous Strategy Space

**Authors:**
A. S. Mousa,
F. Shoman

**Abstract:**

**Keywords:**
Game theory,
Nash Equilibrium,
coherent strategy,
split strategy,
pure strategy,
mixed
strategy

##### 1641 Matching on Bipartite Graphs with Applications to School Course Registration Systems

**Authors:**
Zhihan Li

**Abstract:**

Nowadays, most universities use the course enrollment system considering students’ registration orders. However, the students’ preference level to certain courses is also one important factor to consider. In this research, the possibility of applying a preference-first system has been discussed and analyzed compared to the order-first system. A bipartite graph is applied to resemble the relationship between students and courses they tend to register. With the graph set up, we apply Ford-Fulkerson (F.F.) Algorithm to maximize parings between two sets of nodes, in our case, students and courses. Two models are proposed in this paper: the one considered students’ order first, and the one considered students’ preference first. By comparing and contrasting the two models, we highlight the usability of models which potentially leads to better designs for school course registration systems.

**Keywords:**
Graph Theory,
bipartite graph,
maximum matching,
Ford-Fulkerson Algorithm

##### 1640 Extended Arithmetic Precision in Meshfree Calculations

**Authors:**
Edward J. Kansa,
Pavel Holoborodko

**Abstract:**

Continuously differentiable radial basis functions (RBFs) are meshfree, converge faster as the dimensionality increases, and is theoretically spectrally convergent. When implemented on current single and double precision computers, such RBFs can suffer from ill-conditioning because the systems of equations needed to be solved to find the expansion coefficients are full. However, the Advanpix extended precision software package allows computer mathematics to resemble asymptotically ideal Platonic mathematics. Additionally, full systems with extended precision execute faster graphical processors units and field-programmable gate arrays because no branching is needed. Sparse equation systems are fast for iterative solvers in a very limited number of cases.

**Keywords:**
Partial Differential Equations,
Extended arithmetic precision,
Meshless spectrally convergent,
no branching

##### 1639 Radiation Effects on the Unsteady MHD Free Convection Flow Past in an Infinite Vertical Plate with Heat Source

**Authors:**
Tusharkanta Das,
Tumbanath Samantara,
Sukanta Kumar Sahoo

**Abstract:**

Unsteady effects of MHD free convection flow past in an infinite vertical plate with heat source in presence of radiation with reference to all critical parameters that appear in field equations are studied in this paper. The governing equations are developed by usual Boussinesq’s approximation. The problem is solved by using perturbation technique. The results are obtained for velocity, temperature, Nusselt number and skin-friction. The effects of magnetic parameter, prandtl number, Grashof number, permeability parameter, heat source/sink parameter and radiation parameter are discussed on flow characteristics and shown by means of graphs and tables.

**Keywords:**
Heat Transfer,
Radiation,
MHD,
porous medium,
free convection,
suction

##### 1638 Characterizing the Geometry of Envy Human Behaviour Using Game Theory Model with Two Types of Homogeneous Players

**Authors:**
A. S. Mousa,
R. I. Rajab,
A. A. Pinto

**Abstract:**

**Keywords:**
Game theory,
Nash Equilibrium,
geometric tilings,
bifurcation thresholds,
envy Nash
Equilibrium

##### 1637 A Methodology for Characterising the Tail Behaviour of a Distribution

**Authors:**
Serge Provost,
Yishan Zang

**Abstract:**

Following a review of various approaches that are utilized for classifying the tail behavior of a distribution, an easily implementable methodology that relies on an arctangent transformation is presented. The classification criterion is actually based on the difference between two specific quantiles of the transformed distribution. The resulting categories enable one to classify distributional tails as distinctly short, short, nearly medium, medium, extended medium and somewhat long, providing that at least two moments exist. Distributions possessing a single moment are said to be long tailed while those failing to have any finite moments are classified as having an extremely long tail. Several illustrative examples will be presented.

**Keywords:**
heavy-tailed distributions,
arctangent transformation,
tail classification,
change of variables

##### 1636 A Non-Standard Finite Difference Scheme for the Solution of Laplace Equation with Dirichlet Boundary Conditions

**Authors:**
Khaled Moaddy

**Abstract:**

In this paper, we present a fast and accurate numerical scheme for the solution of a Laplace equation with Dirichlet boundary conditions. The non-standard finite difference scheme (NSFD) is applied to construct the numerical solutions of a Laplace equation with two different Dirichlet boundary conditions. The solutions obtained using NSFD are compared with the solutions obtained using the standard finite difference scheme (SFD). The NSFD scheme is demonstrated to be reliable and efficient.

**Keywords:**
laplace equation,
Dirichlet boundary conditions,
standard finite difference schemes,
non–standard schemes

##### 1635 The Martingale Options Price Valuation for European Puts Using Stochastic Differential Equation Models

**Authors:**
H. C. Chinwenyi,
H. D. Ibrahim,
F. A. Ahmed

**Abstract:**

In modern financial mathematics, valuing derivatives such as options is often a tedious task. This is simply because their fair and correct prices in the future are often probabilistic. This paper examines three different Stochastic Differential Equation (SDE) models in finance; the Constant Elasticity of Variance (CEV) model, the Balck-Karasinski model, and the Heston model. The various Martingales option price valuation formulas for these three models were obtained using the replicating portfolio method. Also, the numerical solution of the derived Martingales options price valuation equations for the SDEs models was carried out using the Monte Carlo method which was implemented using MATLAB. Furthermore, results from the numerical examples using published data from the Nigeria Stock Exchange (NSE), all share index data show the effect of increase in the underlying asset value (stock price) on the value of the European Put Option for these models. From the results obtained, we see that an increase in the stock price yields a decrease in the value of the European put option price. Hence, this guides the option holder in making a quality decision by not exercising his right on the option.

**Keywords:**
monte carlo method,
martingales,
equivalent martingale measure,
European put option,
girsanov theorem,
option price valuation formula,
option price valuation

##### 1634 Several Spectrally Non-Arbitrary Ray Patterns of Order 4

**Authors:**
Ling Zhang,
Feng Liu

**Abstract:**

A matrix is called a ray pattern matrix if its entries are either 0 or a ray in complex plane which originates from 0. A ray pattern *A *of order *n *is called spectrally arbitrary if the complex matrices in the ray pattern class of *A* give rise to all possible *n*th degree complex polynomial. Otherwise, it is said to be spectrally non-arbitrary ray pattern*.* We call that a spectrally arbitrary ray pattern *A *of order *n *is minimally spectrally arbitrary if any nonzero entry of *A* is replaced, then *A *is not spectrally arbitrary. In this paper, we find that is not spectrally arbitrary when n equals to 4 for any θ which is greater than or equal to 0 and less than or equal to n. In this article, we give several ray patterns A(θ) of order n that are not spectrally arbitrary for some θ which is greater than or equal to 0 and less than or equal to n. by using the nilpotent-Jacobi method. One example is given in our paper.

**Keywords:**
spectrally arbitrary,
nilpotent matrix,
ray patterns,
sign patterns

##### 1633 Bidirectional Discriminant Supervised Locality Preserving Projection for Face Recognition

**Abstract:**

**Keywords:**
Face Recognition,
Dimension Reduction,
locality
preserving projection,
discriminant information,
bidirectional
projection

##### 1632 Metric Dimension on Line Graph of Honeycomb Networks

**Authors:**
M. Hussain,
Aqsa Farooq

**Abstract:**

**Keywords:**
Resolving set,
Metric dimension,
Honeycomb network,
Line graph

##### 1631 The Non-Stationary BINARMA(1,1) Process with Poisson Innovations: An Application on Accident Data

**Authors:**
N. Mamode Khan,
V. Jowaheer,
Y. Sunecher

**Abstract:**

**Keywords:**
non-stationary,
CML,
BINARMA(1,
Poisson
Innovations

##### 1630 Analysis of One Dimensional Advection Diffusion Model Using Finite Difference Method

**Authors:**
Ravneet Kaur,
Vijay Kumar Kukreja

**Abstract:**

**Keywords:**
Stability,
Consistency,
Crank-Nicolson scheme,
Lax-Richtmyer theorem,
Peclet number,
Gerschgorin
circle

##### 1629 An Automated Stock Investment System Using Machine Learning Techniques: An Application in Australia

**Authors:**
Carol Anne Hargreaves

**Abstract:**

A key issue in stock investment is how to select representative features for stock selection. The objective of this paper is to firstly determine whether an automated stock investment system, using machine learning techniques, may be used to identify a portfolio of growth stocks that are highly likely to provide returns better than the stock market index. The second objective is to identify the technical features that best characterize whether a stock’s price is likely to go up and to identify the most important factors and their contribution to predicting the likelihood of the stock price going up. Unsupervised machine learning techniques, such as cluster analysis, were applied to the stock data to identify a cluster of stocks that was likely to go up in price – portfolio 1. Next, the principal component analysis technique was used to select stocks that were rated high on component one and component two – portfolio 2. Thirdly, a supervised machine learning technique, the logistic regression method, was used to select stocks with a high probability of their price going up – portfolio 3. The predictive models were validated with metrics such as, sensitivity (recall), specificity and overall accuracy for all models. All accuracy measures were above 70%. All portfolios outperformed the market by more than eight times. The top three stocks were selected for each of the three stock portfolios and traded in the market for one month. After one month the return for each stock portfolio was computed and compared with the stock market index returns. The returns for all three stock portfolios was 23.87% for the principal component analysis stock portfolio, 11.65% for the logistic regression portfolio and 8.88% for the K-means cluster portfolio while the stock market performance was 0.38%. This study confirms that an automated stock investment system using machine learning techniques can identify top performing stock portfolios that outperform the stock market.

**Keywords:**
Machine Learning,
stock market trading,
automated stock investment system,
logistic principal component analysis

##### 1628 Relation between Roots and Tangent Lines of Function in Fractional Dimensions: A Method for Optimization Problems

**Authors:**
Ali Dorostkar

**Abstract:**

In this paper, a basic schematic of fractional dimensional optimization problem is presented. As will be shown, a method is performed based on a relation between roots and tangent lines of function in fractional dimensions for an arbitrary initial point. It is shown that for each polynomial function with order N at least N tangent lines must be existed in fractional dimensions of 0 < α < N+1 which pass exactly through the all roots of the proposed function. Geometrical analysis of tangent lines in fractional dimensions is also presented to clarify more intuitively the proposed method. Results show that with an appropriate selection of fractional dimensions, we can directly find the roots. Method is presented for giving a different direction of optimization problems by the use of fractional dimensions.

**Keywords:**
optimization problem,
root,
tangent line,
fractional dimension

##### 1627 Multi-Objective Optimization of Combined System Reliability and Redundancy Allocation Problem

**Authors:**
Vijaya K. Srivastava,
Davide Spinello

**Abstract:**

This paper presents established 3** ^{n}** enumeration procedure for mixed integer optimization problems for solving multi-objective reliability and redundancy allocation problem subject to design constraints. The formulated problem is to find the optimum level of unit reliability and the number of units for each subsystem. A number of illustrative examples are provided and compared to indicate the application of the superiority of the proposed method.

**Keywords:**
Multi-objective optimization,
Integer Programming,
mixed integer programming,
Reliability Redundancy Allocation

##### 1626 Characterizations of Γ-Semirings by Their Cubic Ideals

**Authors:**
Debabrata Mandal

**Abstract:**

**Keywords:**
intra-regular,
Cartesian product,
Γ-semiring,
cubic ideal,
normal cubic ideal,
cubic
bi-ideal,
cubic quasi-ideal,
regular

##### 1625 Monte Carlo Estimation of Heteroscedasticity and Periodicity Effects in a Panel Data Regression Model

**Authors:**
Nureni O. Adeboye,
Dawud A. Agunbiade

**Abstract:**

This research attempts to investigate the effects of heteroscedasticity and periodicity in a Panel Data Regression Model (PDRM) by extending previous works on balanced panel data estimation within the context of fitting PDRM for Banks audit fee. The estimation of such model was achieved through the derivation of Joint Lagrange Multiplier (LM) test for homoscedasticity and zero-serial correlation, a conditional LM test for zero serial correlation given heteroscedasticity of varying degrees as well as conditional LM test for homoscedasticity given first order positive serial correlation via a two-way error component model. Monte Carlo simulations were carried out for 81 different variations, of which its design assumed a uniform distribution under a linear heteroscedasticity function. Each of the variation was iterated 1000 times and the assessment of the three estimators considered are based on Variance, Absolute bias (ABIAS), Mean square error (MSE) and the Root Mean Square (RMSE) of parameters estimates. Eighteen different models at different specified conditions were fitted, and the best-fitted model is that of within estimator when heteroscedasticity is severe at either zero or positive serial correlation value. LM test results showed that the tests have good size and power as all the three tests are significant at 5% for the specified linear form of heteroscedasticity function which established the facts that Banks operations are severely heteroscedastic in nature with little or no periodicity effects.

**Keywords:**
Periodicity,
heteroscedasticity,
audit fee,
lagrange multiplier test

##### 1624 Three-Dimensional Generalized Thermoelasticity with Variable Thermal Conductivity

**Authors:**
Hamdy M. Youssef,
Mowffaq Oreijah,
Hunaydi S. Alsharif

**Abstract:**

In this paper, a three-dimensional model of the generalized thermoelasticity with one relaxation time and variable thermal conductivity has been constructed. The resulting non-dimensional governing equations together with the Laplace and double Fourier transforms techniques have been applied to a three-dimensional half-space subjected to thermal loading with rectangular pulse and traction free in the directions of the principle co-ordinates. The inverses of double Fourier transforms, and Laplace transforms have been obtained numerically. Numerical results for the temperature increment, the invariant stress, the invariant strain, and the displacement are represented graphically. The variability of the thermal conductivity has significant effects on the thermal and the mechanical waves.

**Keywords:**
Fourier Transforms,
Thermal Conductivity,
thermoelasticity,
three-dimensional,
Laplace transforms

##### 1623 On the Efficiency of Five Step Approximation Method for the Solution of General Third Order Ordinary Differential Equations

**Authors:**
N. M. Kamoh,
M. C. Soomiyol

**Abstract:**

In this work, a five step continuous method for the solution of third order ordinary differential equations was developed in block form using collocation and interpolation techniques of the shifted Legendre polynomial basis function. The method was found to be zero-stable, consistent and convergent. The application of the method in solving third order initial value problem of ordinary differential equations revealed that the method compared favorably with existing methods.

**Keywords:**
convergent,
shifted legendre polynomials,
third order block method,
discrete method

##### 1622 Jeffrey's Prior for Unknown Sinusoidal Noise Model via Cramer-Rao Lower Bound

**Authors:**
Rasaki O. Olanrewaju,
Samuel A. Phillips,
Emmanuel A. Ayanlowo,
Olayode Fatoki

**Abstract:**

This paper employs the Jeffrey's prior technique in the process of estimating the periodograms and frequency of sinusoidal model for unknown noisy time variants or oscillating events (data) in a Bayesian setting. The non-informative Jeffrey's prior was adopted for the posterior trigonometric function of the sinusoidal model such that Cramer-Rao Lower Bound (CRLB) inference was used in carving-out the minimum variance needed to curb the invariance structure effect for unknown noisy time observational and repeated circular patterns. An average monthly oscillating temperature series measured in degree Celsius (0C) from 1901 to 2014 was subjected to the posterior solution of the unknown noisy events of the sinusoidal model via Markov Chain Monte Carlo (MCMC). It was not only deduced that two minutes period is required before completing a cycle of changing temperature from one particular degree Celsius to another but also that the sinusoidal model via the CRLB-Jeffrey's prior for unknown noisy events produced a miniature posterior Maximum A Posteriori (MAP) compare to a known noisy events.

**Keywords:**
sinusoidal,
Maximum A Posteriori (MAP),
Cramer-Rao lower bound (CRLB),
Jeffrey's prior,
Markov Chain Monte
Carlo (MCMC),
Periodograms

##### 1621 Comparing the Efficiency of Simpson’s 1/3 and 3/8 Rules for the Numerical Solution of First Order Volterra Integro-Differential Equations

**Authors:**
N. M. Kamoh,
D. G. Gyemang,
M. C. Soomiyol

**Abstract:**

This paper compared the efficiency of Simpson’s 1/3 and 3/8 rules for the numerical solution of first order Volterra integro-differential equations. In developing the solution, collocation approximation method was adopted using the shifted Legendre polynomial as basis function. A block method approach is preferred to the predictor corrector method for being self-starting. Experimental results confirmed that the Simpson’s 3/8 rule is more efficient than the Simpson’s 1/3 rule.

**Keywords:**
Collocation shifted Legendre polynomials,
Simpson’s rule and Volterra integro-differential equations

##### 1620 A Study of Two Disease Models: With and Without Incubation Period

**Authors:**
H. C. Chinwenyi,
H. D. Ibrahim,
J. O. Adekunle

**Abstract:**

The incubation period is defined as the time from infection with a microorganism to development of symptoms. In this research, two disease models: one with incubation period and another without incubation period were studied. The study involves the use of a mathematical model with a single incubation period. The test for the existence and stability of the disease free and the endemic equilibrium states for both models were carried out. The fourth order Runge-Kutta method was used to solve both models numerically. Finally, a computer program in MATLAB was developed to run the numerical experiments. From the results, we are able to show that the endemic equilibrium state of the model with incubation period is locally asymptotically stable whereas the endemic equilibrium state of the model without incubation period is unstable under certain conditions on the given model parameters. It was also established that the disease free equilibrium states of the model with and without incubation period are locally asymptotically stable. Furthermore, results from numerical experiments using empirical data obtained from Nigeria Centre for Disease Control (NCDC) showed that the overall population of the infected people for the model with incubation period is higher than that without incubation period. We also established from the results obtained that as the transmission rate from susceptible to infected population increases, the peak values of the infected population for the model with incubation period decrease and are always less than those for the model without incubation period.

**Keywords:**
asymptotic stability,
incubation period,
Routh-Hurwitz criterion,
Runge Kutta method

##### 1619 Comparison of Methods of Estimation for Use in Goodness of Fit Tests for Binary Multilevel Models

**Authors:**
I. V. Pinto,
M. R. Sooriyarachchi

**Abstract:**

It can be frequently observed that the data arising in our environment have a hierarchical or a nested structure attached with the data. Multilevel modelling is a modern approach to handle this kind of data. When multilevel modelling is combined with a binary response, the estimation methods get complex in nature and the usual techniques are derived from quasi-likelihood method. The estimation methods which are compared in this study are, marginal quasi-likelihood (order 1 & order 2) (MQL1, MQL2) and penalized quasi-likelihood (order 1 & order 2) (PQL1, PQL2). A statistical model is of no use if it does not reflect the given dataset. Therefore, checking the adequacy of the fitted model through a goodness-of-fit (GOF) test is an essential stage in any modelling procedure. However, prior to usage, it is also equally important to confirm that the GOF test performs well and is suitable for the given model. This study assesses the suitability of the GOF test developed for binary response multilevel models with respect to the method used in model estimation. An extensive set of simulations was conducted using MLwiN (v 2.19) with varying number of clusters, cluster sizes and intra cluster correlations. The test maintained the desirable Type-I error for models estimated using PQL2 and it failed for almost all the combinations of MQL. Power of the test was adequate for most of the combinations in all estimation methods except MQL1. Moreover, models were fitted using the four methods to a real-life dataset and performance of the test was compared for each model.

**Keywords:**
Power,
multilevel modelling,
goodness-of-fit test,
marginal quasi-likelihood,
penalized quasi-likelihood,
quasi-likelihood,
type-I error

##### 1618 An Attack on the Lucas Based El-Gamal Cryptosystem in the Elliptic Curve Group Over Finite Field Using Greater Common Divisor

**Authors:**
Tze Jin Wong,
Nik Mohd Asri Nik Long,
Lee Feng Koo,
Pang Hung Yiu

**Abstract:**

Greater common divisor (GCD) attack is an attack that relies on the polynomial structure of the cryptosystem. This attack required two plaintexts differ from a fixed number and encrypted under same modulus. This paper reports a security reaction of Lucas Based El-Gamal Cryptosystem in the Elliptic Curve group over finite field under GCD attack. Lucas Based El-Gamal Cryptosystem in the Elliptic Curve group over finite field was exposed mathematically to the GCD attack using GCD and Dickson polynomial. The result shows that the cryptanalyst is able to get the plaintext without decryption by using GCD attack. Thus, the study concluded that it is highly perilous when two plaintexts have a slight difference from a fixed number in the same Elliptic curve group over finite field.

**Keywords:**
Encryption,
decryption,
elliptic curve,
greater common divisor

##### 1617 Cryptographic Attack on Lucas Based Cryptosystems Using Chinese Remainder Theorem

**Authors:**
Tze Jin Wong,
Lee Feng Koo,
Pang Hung Yiu

**Abstract:**

_{4,6}) cryptosystem under the Lenstra’s attack as compared to the other two Lucas based cryptosystems such as LUC and LUC

_{3}cryptosystems. All the Lucas based cryptosystems were exposed mathematically to the Lenstra’s attack using Chinese Remainder Theorem and Dickson polynomial. Result shows that the possibility for successful Lenstra’s attack is less against LUC

_{4,6}cryptosystem than LUC

_{3}and LUC cryptosystems. Current study concludes that LUC

_{4,6}cryptosystem is more secure than LUC and LUC

_{3}cryptosystems in sustaining against Lenstra’s attack.

**Keywords:**
congruence,
Lucas sequence,
Dickson polynomial,
faulty signature,
corresponding signature

##### 1616 Autonomous Vehicle Navigation Using Harmonic Functions via Modified Arithmetic Mean Iterative Method

**Authors:**
Jumat Sulaiman,
Azali Saudi

**Abstract:**

**Keywords:**
Path Planning,
Laplace’s equation,
Modified Arithmetic Mean method,
Harmonic
functions

##### 1615 The Influence of Beta Shape Parameters in Project Planning

**Authors:**
Dimitra Alexiou,
Stefanos Katsavounis,
Αlexios Kotsakis

**Abstract:**

Networks can be utilized to represent project planning problems, using nodes for activities and arcs to indicate precedence relationship between them. For fixed activity duration, a simple algorithm calculates the amount of time required to complete a project, followed by the activities that comprise the critical path. Program Evaluation and Review Technique (PERT) generalizes the above model by incorporating uncertainty, allowing activity durations to be random variables, producing nevertheless a relatively crude solution in planning problems. In this paper, based on the findings of the relevant literature, which strongly suggests that a Beta distribution can be employed to model earthmoving activities, we utilize Monte Carlo simulation, to estimate the project completion time distribution and measure the influence of skewness, an element inherent in activities of modern technical projects. We also extract the activity criticality index, with an ultimate goal to produce more accurate planning estimations.

**Keywords:**
Monte Carlo Simulation,
Skewness,
PERT,
beta distribution,
project completion time distribution

##### 1614 An IM-COH Algorithm Neural Network Optimization with Cuckoo Search Algorithm for Time Series Samples

**Authors:**
Wullapa Wongsinlatam

**Abstract:**

**Keywords:**
Artificial Neural Networks,
Time series,
local minima problem,
back propagation
algorithm,
metaheuristic
optimization

##### 1613 Natural Emergence of a Core Structure in Networks via Clique Percolation

**Authors:**
A. Melka,
N. Slater,
A. Mualem,
Y. Louzoun

**Abstract:**

**Keywords:**
Networks,
Cliques,
percolation,
core structure,
phase
transition