# WSEAS Transactions on Mathematics

**Contents:**

2016 | 2015 | 2014 | 2013 | 2012 | 2011 | 2010 | 2009 | 2008 | Pre-2008

**Print ISSN: 1109-2769
E-ISSN: 2224-2880**

**Volume 13, 2014**

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of WSEAS Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

**Volume 13, 2014**

Title of the Paper: **Application of Homo-Separation of Variables Method on Nonlinear System of PDEs**

Authors: **Abdolamir Karbalaie, Mohammad Mehdi Montazeri, Maryam Shabani, Bjorn-Erik Erlandsson**

Abstract: In this study, the Homo-Separation of Variables method is introduced to find the exact solution of some coupled nonlinear partial differential equations. This analytical method is a new combination of two powerful and popular methods; namely the homotopy perturbation method (HPM) and the separation of variables method. The exact solution is constructed by choosing an appropriate initial approximation in addition to retaining only the first term of the series obtained by HPM. The proposed method is capable of reducing the computational load considerably when compared to other classical method.

Keywords: Homo-separation of variables , Homotopy perturbation, Separation of variables, coupled Burgers’ equations

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #97, pp. 992-1000

Title of the Paper: **Research on the Triopoly Dynamic Game Model Based on Different Rationalities and Its Chaos Control**

Authors: **Junhai Ma, Ying Wang, Wei Xu**

Abstract: Based on the earlier research results, and combined with the actual competition in Chinese insurance market, the triopoly dynamic game model is built under the hypothesis that the oligarchs make adaptive decision, delayed bounded rational decision and bounded rational decision respectively. The existence of only Nash equilib- rium in the system is analyzed theoretically, and its stability and the existence of Hopf bifurcation are also studied. The further numerical simulation result verified the accuracy of theoretical study and displayed more intuitively the dynamic behaviour of the system. From this we can conclude that in the process of considering delayed price game, the insurance company must control the value of delay parameter well and decrease his price adjustment speed properly. Only in this way they can make the system stable to its equilibrium state more rapidly. At last, the variable feedback control method is used to make control to the system, it makes the unstable system to the Nash equilibrium state again.

Keywords: Insurance market, Oligopoly, Dynamic game, Self-adaptive, Delay, Bounded rationality

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #96, pp. 983-991

Title of the Paper: **Global Asymptotic Stability for Second-Order Neutral Type Cohen-Grossberg Neural Networks with Time Delays**

Authors: **Chunfang Miao, Yunquan Ke**

Abstract: In this paper, the global asymptotic stability of second-order neutral type Cohen-Grossberg neural networks with time-varying delays are investigated. First, by choosing properly variable substitution the system is transformed to first-order differential equation. Second, some sufficient conditions which can ensure the existence and global asymptotic stability of equilibrium point for the system are obtained through using homeomorphism and the differential mean value theorem, constructing suitable Lyapunov functional and applying the positive matrix. Finally, two examples are given to illustrate the effectiveness of the results.

Keywords: second-order neutral type, Cohen-Grossberg neural networks, Lyapunov functional, Homeomorphism theory, differential mean value theorem, asymptotic stability

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #95, pp. 973-982

Title of the Paper: **Generalized Inverses of Convolution Type Operators on Bounded Intervals**

Authors: **A. M. Simoes**

Abstract: In this paper, we introduce a study of generalized inverses for convolution type operators defined in bounded intervals Ω. The main results of this paper are organized as follows: First, the construction of equivalence relations between the convolution type operators and newWiener-Hopf operators (operators of convolution type defined on the half-line) based on a decomposition method of higher order Wiener-Hopf operators. Second, we introduce the generalized inverses of these operators, when they exist.

Keywords: Convolution equations, Wiener-Hopf operators, Convolution type operators, Equivalence after extension, Projection methods, Generalized inverse

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #94, pp. 962-972

Title of the Paper: **Exact Solutions for Fractional Partial Differential Equations by an Extended Fractional Riccati Sub-Equation Method**

Authors: **Hongxia Wang, Bin Zheng**

Abstract: In this paper, based on the fractional Riccati equation, we propose an extended fractional Riccati sub-equation method for solving fractional partial differential equations. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. By a proposed variable transformation, certain fractional partial differential equations are turned into fractional ordinary differential equations, whose solutions can be expressed in certain forms composed of the solutions of the fractional Riccati equation. As for applications of this method, we apply it to the space-time fractional Whitham-Broer- Kaup (WBK) equations and the space-time fractional Fokas equation. With the aid of the mathematical software Maple, some new exact solutions for the two equations are successfully obtained.

Keywords: Fractional Riccati sub-equation method, Fractional partial differential equations, Exact solutions, Modified Riemann-Liouville derivative, Fractional Whitham-Broer-Kaup (WBK) equations, Fractional Fokas equation

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #93, pp. 952-961

Title of the Paper: **Improving the Stability Properties of Sampling Zeros of Multivariable Discrete Model via Taylor Method and Multirate Fast Sampling**

Authors: **Cheng Zeng, Yingying Su**

Abstract: It is well known that multirate input and hold, such as a generalized sample hold function (GSHF), can be used to shift the sampling zeros of a discrete-time model for a continuous-time system. This paper deals with the stability of sampling zeros, as the sampling period tends to zero, of discrete-time models that are composed of a GSHF, a continuous-time multivariable plant and a sampler in cascade. We propose a hold design that places the sampling zeros asymptotically to the stable region by deriving the approximate expressions of the sampling zeros as power series expansions with respect to a sampling period. The focus is on the evolution of the complex plant for the sampling zeros with the design parameters αj of the GSHF. These parameters are only determined to propose the GSHF that obtains sampling zeros as stable as possible, or with improved stability properties even when unstable, for a given continuous-time multivariable system. The research is extended by obtaining the optimum value of αj for sufficiently small sampling periods and a continuous-time plant, and meanwhile a new stability condition of the sampling zeros is obvious weaker than that of ZOH or FROH case.

Keywords: Stability properties, sampled-data multivariable models, sampling zeros, generalized sample hold function, fast sampling, Taylor series expansions

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #92, pp. 941-951

Title of the Paper: **Four Positive Almost Periodic Solutions to Two Species Parasitical Model with Impulsive Effects and Harvesting Terms**

Authors: **Zhouhong Li**

Abstract: By applying Mawhin’s continuation theorem of coincidence degree theory and some skills of inequal- ities, sufficient conditions are obtained for the existence of at least four positive almost periodic solutions to two species parasitical system with impulsive effects and harvesting terms.

Keywords: Almost periodic solutions, Parasitical system, Coincidence degree, Harvesting term, Time delay, Impulsive effects

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #91, pp. 932-940

Title of the Paper: **Operational Methods for Hermite Polynomials with Applications**

Authors: **Clemente Cesarano, Gerardo Maria Cennamo, Luca Placidi**

Abstract: We exploit methods of operational nature to derive a set of new identities involving families of polynomials associated with operators providing different realizations of the Weyl group. The identities, we will deal with, extend the Nielsen formulae, valid for ordinary Hermite to families of Hermite-like polynomials. It will also be shown that the underlying formalism yields the possibility of obtaining further identities relevant to multi-variable and multi-index polynomials. Applications of higher order Hermite polynomials have been underlined for purpose of numerical simulation in continuous damage mechanics.

Keywords: Orthogonal Polynomials, Hermite, Weyl group, monomiality principle, generating functions

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #90, pp. 925-931

Title of the Paper: **The Performance of Robust Latent Root Regression Based on MM and Modified GM Estimators**

Authors: **Habshah Midi, Mohammed A. Mohammed**

Abstract: It is now evident that the Ordinary Least Squares (OLS) estimator suffers a huge set back in the presence of multicollinearity. As an alternative, the Latent Root Regression (LRR) is put forward to remedy this problem. Nevertheless, it is now evident that the LRR performs poorly when outliers exist in a data. In this paper, we propose an improved version of the LRR to rectify the problem of multicollinearity which comes together with the existence of outliers. The proposed method is formulated by incorporating robust MM - estimator and modified generalized M- estimator (MGM) in the LRR algorithm. We call these methods the Latent Root MM-based (LRMMB) and the Latent Root MGM-based (LRMGMB) methods. The performance of our developed methods are compared with some existing methods such as the OLS, LRR, and the Latent Root M-based (LRMB). The numerical results indicate that the LRR performs very well in the presence of multicollinearity, but performs poorly in the presence of outliers. The proposed methods (LRMMB and LRMGMB) are more efficient than the OLS and the LRR estimators for data having both problems of multicollinearity and outliers.

Keywords: Multicollinearity, OLS, Latent Root Regression, MM-estimator, GM-estimator

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #89, pp. 916-924

Title of the Paper: **Bifurcation in a Discrete Two Patch Logistic Metapopulation Model**

Authors: **Li Xu, Zhongxiang Chang, Fengxia Mai**

Abstract: In this paper, local bifurcation of a discrete two patch logistic metapopulation is discussed. By the central manifold method, flip bifurcation can be analyzed at the positive fixed point from the viewpoint of the dynamical system, and the system can not undergo a fold bifurcation. Simulations on this model show the discrete model can have rich dynamical behaviors. The state feedback control is done to stabilize chaotic orbits at an unstable fixed point. Then the eigenvalues of the corresponding diffusion system with Dirichlet boundary conditions are found for future bifurcation analysis.

Keywords: Logistic metapopulation, Flip bifurcation, Fold bifurcation, Central manifold, State feedback control, Eigenvalue

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #88, pp. 907-915

Title of the Paper: **Two Positive Almost Periodic Solutions of Harvesting Predator-Prey Model with Holling III Type Functional Response and Time Delays**

Authors: **Liyan Pang, Lijun Xu, Tianwei Zhang**

Abstract: By means of Mawhin’s continuation theorem of coincidence degree theory, some sufficient conditions are obtained for the existence of at least two positive almost periodic solutions of harvesting predator-prey model with Holling III type functional response and time delays. An example and numerical simulations are employed to illustrate the main result of this paper.

Keywords: Almost periodicity, Coincidence degree, Predator-prey model, Holling III, Harvesting

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #87, pp. 895-906

Title of the Paper: **A Statistical Study to Develop a Reliable Scale to Evaluate Instructors within Higher Institutions**

Authors: **Said Taan El Hajjar**

Abstract: It is not clear yet to many scientists in different fields that using any scale to test a certain data must have this scale at least reliable and valid. The purpose of this study is to develop a reliable and valid scale that will pro- mote the measurement of the level of instructor evaluation within higher institutions. To satisfy this measurement, Cronbach’s alpha is used to obtain a thorough understanding of remarkable questions/constraints that is initiated in the instructor evaluation form to evaluate the instructor properly. Discussions during focus groups of students and interviews recorded. Then, a model is developed, and collected data classified based on the questions arising dur- ing the discussions with the participants. Accordingly, the collected qualitative data are analyzed using a thematic approach in order to identify the basic constructs/variables of the model. There are five main constructs or an- tecedents representing the independent variables that have great influence on the instructor’s assessment. They are Teacher’s Behaviour (TB), Type of Courses (TC), Level of Students (LS), Mood of Students (MS), and Accuracy of Faculty’s Evaluation (AFE). Each variable is explained by six to eight constraints. Such a scale is being tested through a pilot study in terms of reliability and content validity is being purified further through the collection of quantitative data. To further investigate and verify our theoretical framework, the study employs ordinary least square (OLS) test. OLS technique as a linear regression model is implemented in this study to examine the extent to which the independent variables influence the dependant variable (Faculty Evaluation ( FE)), and the results of the regression analysis reveals that the model is significant.

Keywords: Instructor evaluation, correlation, performance, reliability, validity, construct, scale

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #86, pp. 885-894

Title of the Paper: **Robust Guaranteed Cost Control of Discrete-Time System with Mixed Time-Varying Delays and Nonlinear Perturbations**

Authors: **Yan Zhang, Yali Dong, Tianrui Li**

Abstract: This paper is concerned with the problem of robust static output feedback guaranteed cost control for a class of nonlinear discrete systems with mixed time-varying delays. The proposed controller uses the output measurement to robustly exponentially stabilize the closed-loop system and guarantee an adequate level of system performance. By constructing a set of improved Lyapunov-Krasovskii functionals, novel criteria for the existence of robust output feedback guaranteed cost controller are established in terms of linear matrix inequality (LMI). Two numerical examples are given to illustrate the effectiveness of the proposed methods.

Keywords: Guaranteed cost control, Mixed time-varying delays, Discrete-time system, Nonlinear perturbations, Lyapunov-Krasovskii functions, Linear matrix inequality

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #85, pp. 873-884

Title of the Paper: **The Existence of Positive Solutions for a Class of Singular Third-Order Three-Point Boundary Value Problem**

Authors: **Li Li, Jianqiang Gao**

Abstract: The existence of positive solutions for a class of singular third-order three-point boundary value problem is considered by using Guo-Krasnosel’skii fixed point theorem of cone expansion-compression type. In this class of problem, the nonlinear term is allowed to be singular. Main results show that this class of problem can have positive solutions provided that the conditions on the nonlinear term on some bounded sets are appropriate.

Keywords: Third-order ordinary differential equation, Three-point boundary value problem, Singular nonlinearity, Fixed point theorem on cone, Existence of positive solutions

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #84, pp. 863-872

Title of the Paper: **Ridge-Type Kalman Filter and Its Algorithm**

Authors: **Yongming Li, Qingming Gui, Yongwei Gu, Songhui Han, Kai Du**

Abstract: Kalman filter is one of common ways of dealing with dynamic data under linear minimum variance estimation. However, in practice, the observation matrix maybe has multicollinearity. Therefore, the impact of multicollinearity on Kalman filter is studied in this paper. Firstly, by analyzing the normal equation of Kalman filter, we proposed sufficient conditions about how the multicollinearity affects the estimator. Secondly, a ridge- type Kalman filter algorithm is designed under the mean square error (MSE), and then the characters of the new algorithm are analyzed. Thirdly, six specific methods for determining the ridge parameters of ridge-type Kalman filter are proposed based on the canonical form of normal equation. Finally, examples illustrate the new algorithms can overcome the influence of the ill-condition on Kalman filter effectively, which improves the accuracy of the estimates of parameters.

Keywords: Kalman filter, Ill-condition, Ridge-Type Kalman filter, Ridge parameter

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #83, pp. 852-862

Title of the Paper: **Modelling of Drying Tropical Fruits Using Multiphase Model**

Authors: **N. Shahari, S. Hibberd**

Abstract: Single-phase heat and mass models have been widely used to describe the movement of moisture and heat during drying of food products. This model often taken food as a homogeneous medium and does not allow to incorporate more detailed ongoing processes inside the fruits tissues during drying. Hence, this paper focuses on the development of a model to describe the movement of moisture and heat during drying of fruits that is based on their cellular structure and model are derived using multi-phase approach. Specifically, mathematical model is developed to utilizing COMSOL Multiphysics to solve the equations governing the transport of moisture and heat in order to predict the moisture lost during drying of fruit. Simulation of multi-phase model yield result similar with single-phase model and also generates result not evident in homogeneous model such as water vapour in intercellular space. The results show that the moisture decreased due to the effect of the temperature, water vapour and bound water. However, bound water has no influence at the beginning of drying, while plays a significant role at the later stage of drying. Moreover, pressure plays a significant role on transport of liquid water from one cell to another cell which associates by plasmodesmata transport. The results obtained herein can provide some new theoretical guidelines for the development of drying model.

Keywords: Drying, food, multiphase, heat, mass

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #82, pp. 840-851

Title of the Paper: **Convergence of Explicit Iterative Algorithms for Solving a Class of Variational Inequalities**

Authors: **Yang Caiping, He Songnian**

Abstract: Tian proposed a general iterative algorithm for finding a solution for variational inequalities over the set of fixed points of a nonexpansive mapping on Hilbert spaces and obtained the strong convergence theorem [M. Tian, Nonlinear Analysis, 73 (2010) 689-694]. Zhou et al. proposed a simpler explicit iterative algorithm for finding a solution for variational inequalities over the set of common fixed points of a finite family of nonexpansive mappings on Hilbert spaces and proved the strong convergence [H. Y. Zhou, P. Y.Wang, J. Optim. Theory Appl.09 November 2013]. In this paper, we firstly give a new proof of Tian’s convergence theorem, which is much more simpler than Tian’s original proof. Then we improve the main convergence result of Zhou et al., more precisely, using a recent new lemma, we prove the strong convergence of this algorithm under more weaker conditions (indeed, one of the original conditions is removed). Based on the two results, a more general algorithm is then proposed for solving a more general class of variational inequalities over the set of common fixed points of a finite family of nonexpansive mappings on Hilbert spaces and its strong convergence is proved. Finally, some extensions to our main results have been obtained. Our results in this paper extend and improve ones of Tian and Zhou et al.

Keywords: Variational inequalities, Hybrid steepest-descent method, Nonexpansive mappings, Common fixed points

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #81, pp. 830-839

Title of the Paper: **Some New Generalized Gronwall-Bellman Type Inequalities Arising In The Theory Of Fractional Differential-integro Equations**

Authors: **Hongling Fan, Bin Zheng**

Abstract: In this paper, motivated by the research on qualitative properties of solutions for fractional differential-integro equations, some new Gronwall-Bellman type inequalities in two independent variables are established. Based on these inequalities, explicit bounds for unknown functions concerned are derived. As for applications, we apply the inequalities established to research boundedness, uniqueness, and continuous dependence on the initial value and parameter for the solution to a certain fractional differential-integro equation.

Keywords: Gronwall-Bellman type inequality, Fractional differential-integro equation, Boundedness, Qualitative analysis

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #80, pp. 820-829

Title of the Paper: **Notes on a Particular Class of Perfect Cuboids**

Authors: **Valeriu Dragan**

Abstract: The current paper deals with the problem of the perfect cuboid, in particular with cuboids having only two of its edges even numbers. It is shown that in this particular class, a discussion can be made and some conclusions can be drawn to limit the existence of the perfect cuboid. In this paper Euclids formalism is used to generate Pythagorean triangles that define the faces of the cuboid. Because the pairs, (a; b); (c; d); (u; v), used to generate the triangles were not assumed to be relative primes, the triangles are not necessarily primitives. A computerized algebraic solver is used to solve the equations which were then compared with the numerical solutions obtained independently. Further generalization is required in order to extend the conclusions of this paper to a broader class of possible cuboids.

Keywords: Perfect cuboid, Euler brick, Euclid equation, Pythagorean triangles

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #79, pp. 811-819

Title of the Paper: **Oscillatory Criteria For Two Fractional Differential Equations**

Authors: **Qinghua Feng**

Abstract: In this paper, by use of the properties of fractional derivative and certain inequality technique, some new oscillatory criteria are established for a certain functional fractional differential equation with damping. The fractional differential equation is defined in the sense of the modified Riemann-Liouville fractional derivative. Using a similar analytical method, we also research oscillation of a higher order fractional differential equation, and obtain some oscillatory criteria for it. These oscillatory criteria are new results so far in the literature. As for applications of the oscillatory criteria established, some examples are presented.

Keywords: Oscillation, Modified Riemann-Liouville derivative, Fractional differential equation, Qualitative analysis, Damping, Interval oscillatory criteria

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #78, pp. 800-810

Title of the Paper: **A Toxoplasmosis Spread Model between Cat and Oocyst Populations with Independent Stochastic Perturbations**

Authors: **Qianyong Wu, Yongzhen Pei**

Abstract: A toxoplasmosis spread model between cat and oocyst populations with independent stochastic pertur- bations is proposed, the existence of global positive solution is derived. By the method of stochastic Lyapunov functions, we study their asymptotic behavior in terms of the intensity of the stochastic perturbations and the re- productive number. When the perturbations about the susceptible and infective cats are sufficiently small, as well as magnitude of the reproductive number is less than one, the infective cats, recovered cats and population oocysts decay to zero whilst the susceptible components converge to a class of explicit stationary distributions regardless of the perturbations on the recovered cats and population oocysts. When all the perturbations are small and the reproductive number is larger than one, we construct a new class of stochastic Lyapunov functions to show the positive recurrence, and our results reveal some cycling phenomena of recurrent diseases. These results mean that stochastic system has the similar property with the corresponding deterministic system when the white noise is small. Finally, numerical simulations are carried out to support our findings.

Keywords: Stochastic model, Brownian motion, Asymptotically stable in the large, Lyapunov function

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #77, pp. 790-799

Title of the Paper: **An Analysis on Two Different Data Sets by using Ensemble of k-Nearest Neighbor Classifiers**

Authors: **Nor Azuana Ramli, Mohd Tahir Ismail, Hooy Chee Wooi**

Abstract: An ensemble of classifiers is proposed in this study where we combined the k-nearest neighbor classifier with LAD Tree through stacking. Two different data sets which are macroeconomic and the International Risk Country Risk are used for application. Both of the data are taken for 27 countries from the first quarter of 1984 until the fourth quarter of 2011. Before we applied those two data sets on the ensemble of classifiers, we have done the multicollinearity test to see if the predictor variables are highly correlated. By using the variance inflation factor, the results showed that some predictor variables are highly correlated in the macroeconomic data set. To solve the multicollinearity problem, we used principal component analysis that is available in SPSS as it can reduce a set of factors. The remaining variables are used in the analysis of the proposed method to observe the prediction accuracy. It is proven that different type of data does affect the prediction accuracy, but then it is different for every country.

Keywords: Currency crisis, principal component analysis, multicollinearity test, stacking, macroeconomic

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #76, pp. 780-789

Title of the Paper: **Oleinik Line Method and Its Application**

Authors: **Huashui Zhan**

Abstract: The paper studies the existence of the solution to the equation θηηw+wθξw−θtw=f(η,ξ,t,w). The equation comes from mathematics finance. With the help of Fichera-Oleinik theory, we find the suitable boundary value conditions to assure the posedness of the equation. By modifying Oleinik’s line method, we translate the mathematics finance equation to a system of ordinary differential equations, then by the uniformly estimates of the solution of the system, using Arzela Theorem, we can extract a convergent subsequence, which is convergent to the solution of the mathematics finance equation itself. Some constraint conditions of known functions f, w0, w1 and w2, that appearing in the initial boundary value conditions, are imposed.

Keywords: Mathematics finance, Oleinik’s line method, Fichera-Oleinik theory, boundary condition, t-global solution

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #75, pp. 768-779

Title of the Paper: **Solvability of a Coupled System of a Fractional Boundary Value Problem with Fractional Integral Condition**

Authors: **Dehong Ji, Weigao Ge**

Abstract: In this paper, a system of a fractional boundary value problem with fractional integral condition is considered. By using Schauder fixed point theorem, we establish sufficient conditions for the existence of solutions for the system.

Keywords: Integral condition, Fractional Caputo derivative, Coupled system, Schauder fixed point theorem

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #74, pp. 757-767

Title of the Paper: **Identifying Change Point in Production Time-Series Volatility Using Control Charts and Stochastic Differential Equations**

Authors: **Martin Kovarik**

Abstract: The article focuses on volatility change point detection using SPC (Statistical Process Control) meth- ods, specifically time-series control charts and stochastic differential equations (SDEs). Contribution will review recent advances in change point detection for the volatility component of a process satisfying stochastic differen- tial equation (SDE) based on discrete observations, and also by using time-series control charts. Theoretical part will discuss methodology of time-series control charts and SDEs driven by a Brownian motion. Research part will demonstrate the methodologies in a simulation study focusing on analysis of the AR(1) process by means of time-series control charts and SDEs. The aim is to make use of change point detection in time series of production processes and highlight versatility of control charts not only in manufacturing but also in managing financial cash flow stability.

Keywords: Autocorrelation, Change point, Control charts, Outliers, Stochastic Differential Equations (SDE), Average Run Length (ARL), Statistical Process Control (SPC)

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #73, pp. 747-756

Title of the Paper: **Anti-periodic Solutions for A Shunting Inhibitory Cellular Neural Networks with Distributed Delays and Time-varying Delays in the Leakage Terms**

Authors: **Changjin Xu, Qiming Zhang**

Abstract: This paper is concerned with the existence and exponential stability of anti-periodic solutions of a class of shunting inhibitory cellular neural networks with continuously distributed delays and time-varying delays in the leakage terms. By using differential inequality techniques, some verifiable and practical delay-dependent criteria on the existence and exponential stability of anti-periodic solutions for the model are obtained. Numerical simulations are carried out to illustrate the theoretical findings.

Keywords: Shunting inhibitory cellular neural networks, Anti-periodic solution, Exponential stability, Distributed delay, Time-varying delay, Leakage term

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #72, pp. 736-746

Title of the Paper: **Almost Periodic Solution of Schoener’s Competition Model with Pure-Delays**

Authors: **Liyan Pang, Lijun Xu, Yaqin Li, Tianwei Zhang**

Abstract: In this paper, we consider an almost periodic Schoener’s competition model with time-varying delays. By means of Mawhin’s continuation theorem of coincidence degree theory, some new sufficient conditions are obtained for the existence of at least one positive almost periodic solution for a kind of Schoeners competition model with time-varying delays. The result of this paper complements previous results. Finally, two examples and numerical simulations are given to illustrate the feasibility and effectiveness of our main results.

Keywords: Positive almost periodic solution, Coincidence degree, Schoener, Competition model, Time-varying delay

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #71, pp. 726-735

Title of the Paper: **Three Dimensional Visco-Elastic Flow with Heat and Mass Transfer Past a Vertical Porous Plate in Presence of Variable Suction**

Authors: **Rita Choudhury, Bandita Das**

Abstract: An analysis is carried out to study the unsteady three -dimensional flow of a visco-elastic fluid past a vertical porous plate in presence of variable suction .The effects of heat and mass transfer are taken into account .The fluid is characterized by Rivlin-Ericksen second - order fluid model .The analytical expressions for the velocity ,temperature and species concentration have been obtained .The non -dimensional shearing stress, rate of heat transfer in terms of Nusselt number and the rate of mass transfer in terms of Sherwood number at the plate have been demonstrated graphically in possible cases. It is found that the flow field is significantly affected by the visco-elastic parameter in combination with other flow parameters.

Keywords: Visco-elastic, Porous, Rivlin-Ericksen tensors, Grashof number, Schmidt number, Nusselt number, Sherwood number

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #70, pp. 714-725

Title of the Paper: **A Continuum Model for Condensation of DNA in Solution**

Authors: **Yongzhao Wang, Qichang Zhang, Wei Wang, Dongwei Huang**

Abstract: As a coarse-gained model, an anisotropic super-thin elastic rod subjected to interfacial interactions is used to investigate the condensation of DNA segment. The interfacial pressure distributes on the surface of rod, and is determined in terms of the Young-Laplace equation. The influences of the interfacial factor, initial curvature, anisotropic bend ability and salt solution concentration on the formation of DNA segment are discussed, and DNA loop under different solution environment is studied. The simulated results show that DNA segment will undergo a series of alteration with the change of the solution environment, initial restriction, the shape of DNA rod cross- section and ionic strength. The total size of DNA segments under the interfacial energy and elastic strain reduce many times from the original shape, which is used to explain why several millimeter lengths of DNA can fit inside a cell of diameter 1 μm. The DNA loop exerts diverse forces upon the protein clamp at the end under different solution environment.

Keywords: DNA configuration, Interfacial traction, Solution environment, Anisotropic modulus of DNA bending, DNA loop

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #69, pp. 704-713

Title of the Paper: **Weighted Generalized LDA for Undersampled Problems**

Authors: **Jing Yang, Liya Fan, Quansen Sun**

Abstract: Linear discriminant analysis (LDA) is a classical approach for dimensionality reduction. It aims to max imize between-class scatter and minimize within-class scatter, thus maximize the class discriminant. However, for undersampled problems where the data dimensionality is larger than the sample size, all scatter matrices are singular and the classical LDA encounters computational difficulty. Recently, many LDA extensions have been developed to overcome the singularity, such as, Pseudo-inverse LDA (PILDA), LDA based on generalized singular value decomposition (LDA/GSVD), null space LDA (NLDA) and range space LDA (RSLDA). Moreover, they endure the Fisher criterion that is nonoptimal with respect to classification rate. To remedy this problem, weighted schemes are presented for several LDA extensions in this paper and called them weighted generalized LDA algo rithms. Experiments on Yale face database and AT&T face database are performed to test and evaluate effective of the proposed algorithms and affect of weighting functions.

Keywords: Feature extraction, dimensionality reduction, undersampled problem, weighting function, misclassification rate

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #68, pp. 694-703

Title of the Paper: **A New Yang-type Estimator of Distribution Function at Quantal Response over Indirect Data**

Authors: **Mikhail Tikhov, Maxim Ivkin**

Abstract: In the present work we consider the problem of the kernel estimation of the distribution function and the quantiles of this distribution in dose-effect relationship when the data are observed with error. We make both theoretical and computer research of estimations of the multivariate estimators of the distribution function in dose-effect rela-tionship and in a case when measurements are made with an error having normal distribution. We offer also procedure of reduction of a measurement error.

Keywords: Response model, multivariate data, nonparametric kernel estimates, measurement error

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #67, pp. 684-693

Title of the Paper: **On Construction Third Order Approximation Using Values of Integrals**

Authors: **I. G. Burova, T. O. Evdokimova**

Abstract: Sometimes results of experiments contain values of integrals of a function over sub-intervals however one needs to construct a continuous approximation of the function on the interval. This problem may be solved by the polynomial or by the trigonometrical integro-differential splines that are constructed in this paper. The integro- differential splines are useful for construction the approximation of the function on each sub-interval separately. We use the values of the integrals of the function over sub-intervals and construct a continuous approximation of the function on the interval by the polynomial integro-differential spline, and by the trigonometric integro- differential spline in two steps. First we obtain discontinuous third order approximation of the function in form of the polynomial integro-differential spline, or the trigonometrical integro-differential spline. Approximation on sub-interval uses only the values of the integrals of the function to be approximated over three intervals and basic functions which we obtain here. Then we construct continuous approximation of the function on the interval by solving the system of equations. After that we compare the properties of solutions in form of the polynomial and in form of the trigonometric integro-differential splines. One can see that for trigonometric function sometimes the trigonometric integro-differential spline gives better approximation then the polynomial integro-differential spline. Finally we construct approximation of the function in case we use quadrature formula of the third order instead of the value of the integral over sub-interval.

Keywords: interpolation, splines, polynomial splines, trigonometric splines, polynomial integro-differential splines, trigonometric integro-differential splines, approximation, error of approximation

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #66, pp. 676-683

Title of the Paper: **Statistical Functionals Consistent with a Weak Relative Majorization Ordering: Applications to the Mimimum Divergence Estimation**

Authors: **Tommaso Lando, Lucio Bertoli-Barsotti**

Abstract: Most of the statistical estimation procedures are based on a quite simple principle: find the distribution that, within a certain class, is as similar as possible to the empirical distribution, obtained from the sample observations. This leads to the minimization of some statistical functionals, usually interpreted ad measures of distance or divergence between distributions. In this paper we study the majorization pre-order of the distance between distributions. This concept, known in literature as relative majorization, is extended to the weak definition of majorization, which is more relevant in many practical contexts such as estimation problem. Providing mathematical proofs, we study under which conditions statistical functionals are consistent with respect to the relative weak majorization (from above) pre-order.

Keywords: Majorization; statistical functional; divergence measure; estimation; convex function; inequality; dissimilarity

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #65, pp. 666-675

Title of the Paper: **â„“p-Norm Minimization Method for Solving Nonlinear Systems of Equations**

Authors: **Andre A. Keller**

Abstract: Solving nonlinear systems of equations also refers to an optimization problem. Moreover, this equiva- lence can be interpreted as an optimal design problem. We must determine the design variables needed to reduce the deviation between an actual vector of valued functions and a vector of desired constants. This study focuses on these two characteristic features for solving nonlinear systems of equations. One variety of numerical two- dimensional systems with multiple solutions helps demonstrate the effectiveness of the transformation process.

Keywords: Nonlinear system, least-squares-problem, regularized least-squares, â„“p-norm, Newton’s method

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #64, pp. 654-665

Title of the Paper: **Efficient Evaluation of Sparse Jacobians by Matrix Compression Part II: Implementation and Experiments**

Authors: **Nikolai Strogies, Andreas Griewank**

Abstract: The accurate and efficient calculations of Jacobians matrices at a sequence of arguments is a key ingredient of numerical methods for nonlinear problems in scientific computing. It has been known since the seminal work of Curtis Powell and Reid [1] that once their sparsity pattern is known Jacobians can be estimated on the basis of divided differences for a set carefully chosen directions. The number p of such seed directions and thus extra function evaluations can often be limited a priori to a smallish number, which is typically much smaller than the number of independent variables and unaffected by grid sizes and other discretization parameters. The cost factor p is bounded below by the maximal number of nonzeros per row, which is actually sufficient for Jacobian estimation using Newsam-Ramsdell compression. This NR approach is numerically less stable than the CPR method, which was therefore preferred in practice as divided differences are strongly affected by truncation and round off errors. However now, using automatic or algorithmic differentiation, one obtains directional derivatives with working accuracy and can thus utilize the more economical NR approach.

Keywords: Matrix Compression, Algorithmic Differentiation, Graph Coloring, Traveling Salesman

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #63, pp. 646-653

Title of the Paper: **nr-Excellent Graphs**

Authors: **S. Suganthi, V. Swaminathan, G. Jothi Lakshmi, A. P. Pushpalatha**

Keywords: Neighbourhood resolving set, neighbourhood resolving number, nr-excellent, nr-bad vertex

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #62, pp. 636-645

Title of the Paper: **NLS-TSTM: A Novel and Fast Nonlinear Image Classification Method**

Authors: **Xi Zhan Gao, Liya Fan, Haitao Xu**

Abstract: This paper is devoted to study nonlinear image classification methods based on support tensor machine (STM). Firstly, a new linear method named linear least squares twin support tensor machine (LLS-TSTM) is proposed, which is an improvement of linear STM. The utility of twin skill and least squares technology aims to speed up the computation time (sum of training time and testing time). Secondly, in order to study nonlinear version of LS-TSTM, a new matrix kernel function is introduced and then based on which, a nonlinear LS-TSTM (NLS-TSTM) classification method is suggested with detailed theoretical derivation. Finally, in order to examine the effectiveness of LLS- and NLS-TSTM, we perform a series of comparative experiments with linear STM and linear TSTM on ORL and Yale face databases. Experiment results show that the proposed methods are effective and efficient.

Keywords: Least squares twin support tensor machine; image classification; matrix kernel function; iterative algorithm; classification accuracy

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #61, pp. 626-635

Title of the Paper: **The Fourier Transform with Piecewise Trigonometric Kernels and its Applications**

Authors: **Oleg Yaremko, Vladimir Selutin, Natalia Yaremko**

Abstract: In this paper, we develop a theory of the Fourier integral transforms with piecewise-homogeneous trigonometric kernels. Formulas for Integral transforms are obtained in a Hermite-type orthogonal polynomials series form. The resulting formulas are new for both the classical and piecewise- homogeneous cases. In the sec- ond part of the article method of Hermite-type orthogonal polynomials series expansion is used to solve the direct and inverse Cauchy problems for the heat equation in a piecewise- homogeneous medium. The inverse Cauchy problem is ill-posed and requires regularization. Formulas for solving direct and inverse Cauchy problems are obtained by the developed method and have advantages: first, they do not contain derivatives, can serve as a basis for regularizing algorithms , and secondly, these formulas are mutually symmetrical.

Keywords: Fourier integral transform, heat equation, Cauchy problem, Hermite polynomials

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #60, pp. 615-625

Title of the Paper: **Qualitative Analysis for A Lotka-Volterra Model with Time Delays**

Authors: **Changjin Xu, Qiming Zhang**

Abstract: This paper considers a Lotka-Volterra model with time delays and delay dependent parameters. The linear stability conditions are obtained with characteristic root method. The Hopf bifurcation is demonstrated. Using normal form theory and center manifold theory, Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived. Finally, numerical simulations are carried out to support the theoretical predictions.

Keywords: Time delay; Lotka-Volterra model, Stability, Hopf bifurcation, Delay dependent parameter

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #59, pp. 603-614

Title of the Paper: **Humbert Polynomials and Functions in Terms of Hermite Polynomials Towards Applications to Wave Propagation**

Authors: **Clemente Cesarano, Gerardo Maria Cennamo, Luca Placidi**

Abstract: By starting from the standard definitions of the incomplete two-variable Hermite polynomials, we propose non-trivial generalizations and we show some applications to the Bessel-type functions as the Humbert functions. We also present a generalization of the Laguerre polynomials in the same context of the incomplete-type and we use these to obtain relevant operational techniques for the Humbert-type functions. Final considerations are inserted to include the problem of wave propagation in the present theoretical framework.

Keywords: Hermite Polynomials, Laguerre polynomials, Generating Functions, Orthogonal Polynomials, Humbert polynomials, Humbert Functions, Bessel Functions

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #58, pp. 595-602

Title of the Paper: **A Characterization of Some Groups by their Orders and Degree Patterns**

Authors: **Shitian Liu**

Keywords: Order component, Element order, Symmetric group, Degree pattern, Prime graph, Simple group

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #57, pp. 586-594

Title of the Paper: **Some Iterative Algorithms for k-Strictly Pseudo-Contractive Mappings in a CAT(0) Space**

Authors: **Aynur Sahin, Metin Basarir**

Abstract: In this paper, we prove the Δ-convergence theorems of the cyclic algorithm and the new multi-step iteration for k-strictly pseudo-contractive mappings and give also the strong convergence theorem of the modified Halpern’s iteration for these mappings in a CAT(0) space. Our results extend and improve the corresponding recent results announced by many authors in the literature.

Keywords: CAT(0) space, fixed point, strong convergence, Δ-convergence, k-strictly pseudo-contractive mapping, iterative algorithm

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #56, pp. 577-585

Title of the Paper: **Finite Groups with Three or Four Conjugacy Class Sizes of Primary and Biprimary Elements**

Authors: **Qingjun Kong, Ping Kang**

Abstract: Let G be a finite group. In this paper, we determine completely the structure of G with three or four conjugacy class sizes of elements of orders divisible by at most two primes.

Keywords: Conjugacy class sizes; Finite groups; Nilpotent groups; Solvable groups; Sylow p-subgroup.

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #55, pp. 567-576

Title of the Paper: **Models and Methods of Multiobjective Optimization in Problems of Quarry Design and Planning**

Authors: **Andrey M. Valuev**

Abstract: In the paper existing approaches are practice of quarry design and long-term planning are analyzed. Mathematical model of a quarry contour represented with a set of curves, namely bench edges (a SC model) is formulated for a quarry on a flat territory and an on-slope quarry; its likelihood to dynamic systems is shown. Most attention is given to optimization problems setup and solution on the basis of continuous-valued models resulting from piecewise-linear approximation of SC models. Sectoral and contour models aimed at solution of problems in question are presented; they are compared from viewpoints of adequacy and accuracy. Characteristic features of optimization problems based on presented models are shown resulting in specific features of direct methods application. Practical significance of multiobjective optimization for problems in question is substantiated. The solution of that type of problems is exemplified by multiobjective optimization of mining works for “Neriungrinskii” coal quarry as well as by a design stage contour optimization for Ekibastuz coal quarry. Main features of intelligent optimization software developed by the author and used in these computations are presented as well.

Keywords: Multiobjective optimization, mining work development, mathematical modelling, feasible direction method, gradient-restoration method, decomposition, intelligent software

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #54, pp. 557-566

Title of the Paper: **Existence and Longtime Behavior of Global Solutions for a Nonlinear Damping Petrovsky Equation**

Authors: **Yaojun Ye**

Abstract: The initial boundary value problem for a class of nonlinearly damped Petrovsky equation utt + Δ2u + a(1 + |ut|r)ut = b|u|pu in a bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set in H2 0 (Ω), and obtain the energy decay result through the use of an important lemma of V.Komornik. Meanwhile, under the conditions of the positive initial energy, it is proved that the solution blows up in the finite time and the lifespan estimates of solutions are also given.

Keywords: Damped Petrovsky equation; Stable sets; Global solutions; Energy decay estimate

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #53, pp. 547-556

Title of the Paper: **Fredholm Characterization for a Wave Diffraction Problem with Higher Order Boundary Conditions: Impedance Case**

Authors: **A. M. Simoes**

Abstract: In this paper, a Fredholm characterization for operators related to a wave diffraction problem with higher order impedance boundary conditions is developed. We consider an impedance boundary transmission problem for the Helmholtz equation. The problem will be analysed in an operator theory viewpoint and is considered within a framework of Bessel potential spaces. Relations between operators and extension methods are built to deal with the problem and, as consequence, a transparent interpretation of the problem in an operator theory framework are associated to the problem. Different types of operator relations are exhibited for different types of operators acting between Lebesgue and Bessel potential spaces on a finite interval and on the positive half-line. At the end, we describe when the operators associated with the problem enjoy the Fredholm property with Fredholm index equal to zero in terms of the initial space order parameters. In addition, an operator normalization method is applied.

Keywords: Higher order impedance boundary condition, wave diffraction, Helmholtz equation, Bessel potential space, convolution type operator, Wiener-Hopf operator, Fredholm property, normalization

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #52, pp. 535-546

Title of the Paper: **Weak Greedy Algorithms for Nonlinear Approximation with Quasi-Greedy Bases**

Authors: **Long Jingfan, Ye Peixin**

Abstract: We study weak greedy approximation with respect to quasi-greedy bases. For a Hilbert space, we prove that the error of the m-th weak greedy approximation is bounded by the error of best m-term approximation mul- tiplied by an extra factor of order lnm. However we show weak greedy algorithm realizes the best expansional m-term approximation for individual element and best m-term approximation for some sparse classes. Further- more we establish sharp Lebesgue-type inequality for expansional m-term approximation with quasi-greedy bases in Banach spaces.

Keywords: Quasi-greedy basis, Weak greedy algorithm, m-term approximation, Sparse classes, Optimality, Hilbert spaces

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #51, pp. 525-534

Title of the Paper: **Two Iterative Algorithms for k -Strictly Pseudo-Contractive Mappings in a CAT(0) Space**

Authors: **Aynur Åžahin, Metin BaÅŸarir**

Abstract: In this paper, we prove the Δ -convergence of the cyclic algorithm for k -strictly pseudo-contractive mappings and give also the strong convergence theorem of the modified Halpern iteration for these mappings in a CAT(0) space. Our results extend and improve the corresponding recent results announced by many authors in the literature.

Keywords: CAT(0) space, fixed point, strong convergence, Δ-convergence, k -strictly pseudo-contractive mapping, iterative algorithm

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #50, pp. 516-524

Title of the Paper: **Power-Law Index and its Application based on the Degree of Distribution and Scale-Free Network Evolution**

Authors: **Yaning Zhang, Junhai Ma**

Abstract: In realistic networks, many scholars have found that the connection between nodes does not only depend on the size of the node degree probability. This paper sets up a kind of changing scaling exponent scale-free network evolution model(NABA) based on the BA model. In NABA model, the preferential probability depends on sites’ degree and the growth rate of degree. Using mean field theory, we calculate the degree distribution function and give the range of scaling exponent. Furthermore we perform some simulations to testify the point of view. Simulation results verify that the distribution function is correct, and scaling exponent will vary in the range (2;+∞] with the variation of parameters. The NABA model can degenerate into BA model when = 1 and r = 0, and hence the BA model can be regarded as its special form.

Keywords: Scale-free, degree distribution, evolution model, computer simulation, mean field theory

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #49, pp. 505-515

Title of the Paper: **Planar Graph Characterization - Using γ - Stable Graphs**

Authors: **M. Yamuna, K. Karthika**

Keywords: Stable graph, planar, nonplanar, dominating set

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #48, pp. 493-504

Title of the Paper: **Research on the Revenue-Sharing Mechanism based on the Price Game of Retailers**

Authors: **Junhai Ma, Aiwen Ma**

Abstract: We set up a two level supply chain model which consists of one manufacturer and two retailers in this paper. Necessities as the main research object, we develop the nonlinear demand function of products which is more related to the reality. In order to increase the performance of whole supply chain in the decentralized decision, we introduce revenue sharing mechanism in this paper. Through numerical simulations, we analyze the optimal retail prices of products in the basic model and revenue-sharing model respectively, study the impacts of price sensitive coefficient and price adjustment speed parameter on the performance of supply chain. Investigation results from the analysis reveal that, the revenue-sharing mechanism can effectively improve the profits of retailers, and make the competitive market environment more stable.

Keywords: Revenue sharing mechanism, game, equilibrium state, price sensitive coefficient, price adjustment, speed parameter

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #47, pp. 484-492

Title of the Paper: **New Study of Classes of Hurwitz-Zeta Function Related to Integral Operator**

Authors: **F. Ghanim**

Keywords: Analytic function; Convex function; Meromorphic function; Hurwitz Zeta function; Linear operator; Hadamard product; Functions with positive real part; Integral operator

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #46, pp. 477-483

Title of the Paper: **Completeness and Separability of the Space of a Class of Integrable Fuzzy Valued Functions Based on the tK-Integral Norm Metric**

Authors: **Weiwei Song, Guijun Wang, Quanjie Han**

Abstract: When a class of fuzzy value functions constitute a metric space, the completeness and separability is an important problem that must be considered to discuss the approximation of fuzzy systems. In this paper, Firstly, a new tK-integral norm is defined by introducing two induced operators, and prove that the class of tK-integrable fuzzy value functions is a metric space. And then, the integral transformation theorems and tK-integrable Borel- Cantelli Lemma are applied to study the completeness of the space, furthermore, its separability is discussed by means of the approximation of fuzzy valued simple functions and fuzzy valued Bernstein polynomials. The results show that the space of the tK-integrable fuzzy valued functions constitutes a complete separable metric space in the sense of the tK-integral norm.

Keywords: K-Additive Measure, tK-Integrable Function, tK-Integral Norm, Completeness, Separability

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #45, pp. 466-476

Title of the Paper: **An Extended Newton’s Method with Free Second-order Derivatives**

Authors: **Young Hee Geum, Young Ik Kim**

Abstract: We propose the cubic-order numerical method free of second derivatives and derive the asymptotic error constant in terms of control parameters. Applying this proposed scheme to various test functions, numerical results show a good agreement with the theory analyzed in this paper and are proven using Mathematica with its highprecision computability.

Keywords: Newton’s method, Multiple root, Nonlinear equation, Order of convergence, Root finding

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #44, pp. 462-465

Title of the Paper: **Some New Hermite-Hadamard-Type Inequalities for Geometric-Arithmetically S-Convex Functions**

Authors: **Meihui Qu, Wenjun Liu, Jaekeun Park**

Abstract: In this paper, motivated by the concept of “geometric-arithmetically s-convex function”, we establish some new Hermite-Hadamard-type inequalities for geometric-arithmetically s-convex functions, which not only recapture the recent results about Hermite-Hadamard-type inequalities for convex functions, but also give some new results as special cases.

Keywords: Hermite-Hadamard-type inequalities, H¨older’s inequality, s-GA-convexity

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #43, pp. 452-461

Title of the Paper: **Solution of Nonlinear Riccati Differential Equation using Chebyshev Wavelets**

Authors: **S. Balaji**

Abstract: A generalized Chebyshev wavelet operational matrix (CWOM) is presented for the solution of nonlin- ear Riccati differential equations. The operational matrix together with suitable collocation points converts the fractional order Riccati differential equations into a system of algebraic equations. Accuracy and efficiency of the proposed method is verified through numerical examples and comparison with the recently developed approaches. The obtained results reveal that the performance of the proposed method is more accurate and reliable.

Keywords: Riccati equation, Nonlinear ODE, Chebyshev wavelet, convergence, operational matrix

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #42, pp. 441-451

Title of the Paper: **Hybrid Cuckoo Search Algorithm based on Powell Search for Constrained Engineering Design Optimization**

Authors: **Wen Long, Jianjun Jiao**

Abstract: eCuckoo search (CS) has been recently proposed as a population-based optimization algorithm and it is has so far been successfully applied in a variety of fields. An efficient hybrid cuckoo search algorithm (HCSA) based on Powell direct search method is proposed for solving constrained engineering design optimization prob- lems. The inertia weight of Levy flights is introduced to balance the ability of global and local search. The Powell local search technique is used to improve the best approximation found by the cuckoo search algorithm. The hy- bridization of cuckoo search algorithm and Powell search method may provide a more effective trade-off between exploitation and exploration of the search space. The proposed hybrid method is tested based on three widely used benchmark constrained functions and three well-known constrained engineering design optimization problems, which shows that HCSA is of better or competitive performances when compared with several existing algorithms.

Keywords: Cuckoo search algorithm, Powell search, constrained optimization, engineering design optimization

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #41, pp. 431-440

Title of the Paper: **Point of Infinite Order on an Elliptic Curve over a Quadratic Field**

Authors: **S. Abdelalim, A. Chillali, S. Elhajji**

Abstract: Let an elliptic curve over the quadratic field given by Weierstrass equation: , where in . We introduce some fundamental results of the elliptic curve. After we create an elliptic curve with an element of infinite order [2,3,4].

Keywords: Elliptic Curves, Quadratic Fields, Infinite Order

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #40, pp. 428-430

Title of the Paper: **The Weak Solutions to Doubly Nonlinear Diffusion Equation with Convection Term**

Authors: **Xin Si, Huashui Zhan**

Keywords: Doubly nonlinear diffusion equation, weak solution, convection term

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #39, pp. 416-427

Title of the Paper: **Curve Fitting for Coarse Data Using Artificial Neural Network**

Authors: **C. Balasubramanyam, M. S. Ajay, K. R. Spandana, Amogh B. Shetty, K. N. Seetharamu**

Abstract: This paper demonstrates the capability of curve fitting using Artificial Neural Network (ANN), not only for a moderate set of input data but also for a coarse set of input. When appropriate number of neurons is chosen for the training purpose, accurate graphs can be obtained, despite having a coarse data. The effect of number of neurons used for curve fitting and the accuracy obtained is also studied. This aspect of ANN has been illustrated through 2 examples, Weibull distribution and another complex sinusoidal system. This curve fitting technique has been applied to a real world problem i.e. mechanism of a deep drawing press, for both slider displacement and slider velocity.

Keywords: Curve fitting, ANN, accurate, coarse data, best fit, neurons

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #38, pp. 406-415

Title of the Paper: **An Algorithm for Storage Tanks Deflection Identification Based on Pattern Search**

Authors: **Yucheng Chen, Huashui Zhan**

Abstract: By analyzing the function relationship between the retained volume of oil and numerical readings of oil probe in storage tanks with two spherical crowns, a mathematical model is given for identifying the horizontal and vertical deflection angles. According this model, a Matlab program, which is based on Hooke-Jeeves Pattern Search Algorithm (HJPSA), is implemented for precisely establishing the deflection angles. A set of data about the volume of oil pumped out, the numerical readings of the oil probe and the retained oil, which is obtained by experiment, indicates that the storage tanks may occurs the horizontal and vertical deflections 6.69 degrees and 13.33 degrees respectively. Moreover, for the retained oil in the storage tanks, a recalibrated table is shown at the end of the paper.

Keywords: Deflection identification, Least Square Method, Hooke-Jeeves Pattern Search Algorithm

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #37, pp. 395-405

Title of the Paper: **The Reduction Method for Approximative Solution of Systems of Singular Integro-Differential Equations in Lebesgue Spaces (Case γ ≠ 0)**

Authors: **Feras M. Al Faqih, Iurie Caraus, Nikos E. Mastorakis**

Abstract: In this article we have elaborated the numerical schemes of reduction methods for approximate solution of system of singular integro-differential equations when the kernel has a weak singularity. The equations are defined on the arbitrary smooth closed contour of complex plane. We suggest the numerical schemes of the reduction method over the system of Faber-Laurent polynomials for the approximate solution of weakly singular integro- differential equations defined on smooth closed contours in the complex plane. We use the cut-off technique kernel to reduce the weakly singular integro-differential equation to the continuous one. Our approach is based on the Krykunov theory and Zolotarevski results. We have obtained the theoretical background for these methods in classical Lebesgue spaces.

Keywords: Faber- Laurent Polynomials, systems of singular integro-differential equations, reduction methods

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #36, pp.385-394

Title of the Paper: **The Stability, Bifurcation and Chaos of a Duopoly Game in the Market of Complementary Products with Mixed Bundling Pricing**

Authors: **Fang Wu, Junhai Ma**

Abstract: This paper constructs a non-competitive duopoly price model with mixed bundling pricing in com- plementary product market. The system is modeled by nonlinear difference equations. The existence of Nash equilibrium point and its stability are studied. The effects of complementarity degree and discount of bundling on the stability are investigated. The effects of the price adjustment parameters on the producer surplus are also investigated, especially when the parameters are used with the discount of bundling. The results show: increasing the degree of complementarity or the discount of bundling can enlarge the stable region of the system. Firms can get higher producer surplus with suitable discount of bundling in stable state than in unstable state of the market.

Keywords: Complex Dynamics, Duopoly game, Complementary products, Mixed Bundling pricing

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #35, pp.374-384

Title of the Paper: **The Periodic Solution of Hypersonic Functionally Graded Plate Subjected to Aero-Thermal Load**

Authors: **Xiaona Yin, Jing Li, Bin He**

Abstract: In this paper, the periodic solution of hypersonic functionally graded plate subjected to aero-thermal load with two degrees of freedom is investigated. Firstly, the average equations in 1:2 internal resonances are obtained by using multi-scale method. Based on the Poincare mapping and the Melnikov function, we get the sufficient condition for the existence of periodic solution about the system. Then we obtain the phase diagrams of the solution which were under different parameters by using Matlab. The simulation results demonstrate the sufficient condition which we have got is correct. Finally, by using the integral manifold theory and average method, we analyze the stability of the periodic orbit.

Keywords: Poincare mapping, Periodic Solution, Stability, Numerical simulations

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #34, pp.363-373

Title of the Paper: **Square-Mean Almost Periodic Solutions of Neutral Stochastic Functional Differential Equations on Time Scales**

Authors: **Meng Hu, Lili Wang**

Abstract: In this paper, based on the properties of almost periodic function and exponential dichotomy of linear system on time scales as well as Krasnoselskii’s fixed point theorem, some sufficient conditions are established for the existence of square-mean almost periodic solutions of neutral stochastic functional differential equations on time scales. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.

Keywords: Neutral stochastic differential equation, Square-mean almost periodic solution, Exponential dichotomy, Krasnoselskii’s fixed point theorem, Time scale

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #33, pp.353-362

Title of the Paper: **New Periodic Exact Solutions of the Kuramoto-Sivashinsky Evolution Equation**

Authors: **Ognyan Yordanov Kamenov**

Abstract: In the present paper, three families of exact periodic localized solutions of the popular Kuramoto- Sivashinsky model evolution partial differential equation have been obtained. Similar exact solutions have not been published so far. The exact solutions found are cnoidal, sinusoidal and a solitary-wave one, which were established to be dynamically equivalent. To obtain them a spatial modification of the Hirota-Matsuno bilinear transformation method has been applied. The non-integrability of the evolution equation under consideration generates specific dynamic phenomena – the individual spatial displacements, defined exactly for each separate harmonics in the localized periodic solutions.

Keywords: Hirota bilinear operators, Hirota-Matsuno bilinear transformation method, Weierstrass elliptic functions, Jacobi elliptic functions, Jacobi Theta functions, phase modulations of elliptic functions

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #32, pp.345-352

Title of the Paper: **A Cellular Automaton Model with Varied Walk Velocity**

Authors: **Yanfang Zhang, Lijuan Zhang, Yibin Zhao, Jianqiong Zhou**

Abstract: This paper presents a new cellular automaton model in which the neighborhood is the extended Moore’s neighborhood(r = 2) and each cell’s walk velocity is the function of inter-person distance. This model can simulate pedestrian movement with different velocity through letting pedestrian move more than one cell. In the evacuation process, the moving rules are determined by three functions: exit function, direction function and velocity func- tion. An emergency evacuation example is used to illustrate the validity of the new presented varied walk velocity cellular automaton. Comparing the results obtained by the non-varied walk velocity cellular automaton model and the new model, it is demonstrated that the new model is more valid to simulate evacuation and the evacuation time is shorter.

Keywords: Cellular automaton, evacuation, varied walking velocity, updating function, extended Moore’s neighborhood, simulation

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #31, pp.335-344

Title of the Paper: **Spectral Analysis of Transport Operator in Lebowitz-Rubinow Model**

Authors: **Senghua Wang, Guofei Cheng, Dengbin Yuan**

Abstract: In this paper, we deal with the Lebowitz-Rubinow model of an age structured proliferating cell population in Lp-space (1 ≤ p < +∞). It is to prove that the C0 semigroups generated by the transport operator is compact for the boundary operator is compact, and it is to obtain that the spectrum of the transport operators is countable and consists of, at most, isolate eigenvalues with finite algebraic multiplicity with −∞ as the only possible limit point, we are to obtain that the spectrum of the transport operators only consists of finitely isolate eigenvalues with finite algebraic multiplicities in the right half plane trip for the boundary operator is not compact, also we show that the asymptotic behavior of the transport equation’s solution.

Keywords: Transport operator, Cell population, Compactness, Spectrum, Asymptotic behavior

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #30, pp.324-334

Title of the Paper: **Some Comparison Results with New Effective Preconditioners for L-Matrices**

Authors: **Cui-Xia Li, Shi-Liang Wu**

Abstract: Based on the work of Wang and Li [A new preconditioned AOR iterative method for L-matrices, J. Comput. Appl. Math. 229 (2009) 47-53], in this paper, a new preconditioner for the AOR method is proposed for solving linear systems whose coefficient matrix is an L-matrix. Several comparison theorems are shown for the proposed method with two preconditioners. It follows from the comparison results that the proposed new method can achieve faster convergence than the preconditioner introduced by Wang and Li.

Keywords: Preconditioner, L-matrix, AOR mehtod, SOR mehtod

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #29, pp.314-323

Title of the Paper: **Laminar and Turbulent Simulations of Several TVD Schemes in Two-Dimensions – Part I**

Authors: **Edisson Sávio De Góes Maciel**

Abstract: This work, first of this study, describes five numerical tools to perform perfect gas simulations of the laminar and turbulent viscous flow in two-dimensions. The Van Leer, Harten, Frink, Parikh and Pirzadeh, Liou and Steffen Jr. and Radespiel and Kroll schemes, in their first- and second-order versions, are implemented to accomplish the numerical simulations. The Navier-Stokes equations, on a finite volume context and employing structured spatial discretization, are applied to solve the supersonic flow along a ramp in twodimensions. Three turbulence models are applied to close the system, namely: Cebeci and Smith, Baldwin and Lomax and Sparlat and Allmaras. On the one hand, the second-order version of the Van Leer scheme is obtained from a “MUSCL” extrapolation procedure, whereas on the other hand, the second order version of the Harten scheme is obtained from the modified flux function approach. The convergence process is accelerated to the steady state condition through a spatially variable time step procedure, which has proved effective gains in terms of computational acceleration (see Maciel). The results have shown that, with the exception of the Harten scheme, all other schemes have yielded the best result in terms of the prediction of the shock angle at the ramp. Moreover, the wall pressure distribution is also better predicted by the Van Leer scheme.

Keywords: Laminar and turbulent flows, TVD algorithms, Cebeci and Smith turbulence model, Baldwin and Lomax turbulence model, Sparlat and Allmaras turbulence model

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #28, pp.291-313

Title of the Paper: **On a Fuzzy Logistic Difference Equation**

Authors: **Qianhong Zhang, Wenzhuan Zhang, Jingzhong Liu, Yuanfu Shao**

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #27, pp.282-290

Title of the Paper: **A General Iterative Method for Constrained Convex Minimization Problems in Hilbert Spaces**

Authors: **Ming Tian, Minmin Li**

Abstract: It is well known that the gradient-projection algorithm plays an important role in solving constrained convex minimization problems. In this paper, based on Xu’s method [Xu, H. K.: Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl. 150, 360-378(2011)], we use the idea of regularization to establish implicit and explicit iterative methods for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our methods converge strongly to a solution of the constrained convex minimization problem. Such a solution is also a solution of a variational inequality defined over the set of the solutions of the constrained convex minimization problem. As application, we apply our main result to solve the split feasibility problem in Hilbert spaces.

Keywords: Variational inequality, Regularization algorithm, Constrained convex minimization, Fixed point, Averaged mapping, Nonexpansive mappings

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #26, pp.271-281

Title of the Paper: **Hybrid Proximal-Point Methods for Systems of Generalized Equilibrium Problems and Maximal Monotone Operators in Banach Spaces**

Authors: **Jiawei Chen, Zhongping Wan, Yeol Je Cho, Shih-Sen Chang**

Abstract: In this paper, by using Bregman’s technique, we introduce and study the hybrid proximal-point methods for finding a common element of the set of solutions to a system of generalized equilibrium Problems and zeros of a finite family of maximal monotone operators in reflexive Banach spaces. Strong convergence results of the proposed hybrid proximal-point algorithms are also established under some suitable conditions. As applications, the existence of solutions for a class of bilevel variational inequalities are established and some numerical examples are reported.

Keywords: Equilibrium problem, maximal monotone operator, bilevel variational inequalities, Bregman distance, Bregman projection, totally convex function, Legendre function.

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #25, pp.256-270

Title of the Paper: **Implementing Control Charts to Corporate Financial Management**

Authors: **Martin Kovarik, Libor Sarga**

Abstract: In the paper, corporate financial management using statistical process control (SPC), especially She- whart’s control charts operating with the constant mean, control charts with non-constant mean, and process ca- pability indices will be introduced. The center line, UCL and LCL for the control charts will be defined with the regulated process not allowed to cross the UCL and LCL boundaries. Altman’s model (the so called Z-score), the most popular corporate financial stability index, will be used. We will demonstrate benefits of SPC on two case studies: the first will focus on corporate financial flow control, the second will include six companies. Special types of control charts, i.e., CUSUM and EWMA, will be discussed due to their mean shift sensitivity and practi- cal applications demonstrated on additional two case studies. The results prove control charts can be successfully implemented not only in manufacturing processes but in corporate financial management as well.

Keywords: Altman’s Z-score, Statistical Process Control, Shewhart’s control charts, process capability indices, EWMA control chart, CUSUM control chart

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #24, pp.246-255

Title of the Paper: **Equitable Colorings of Cartesian Product Graphs of Wheels with Complete Bipartite Graphs**

Authors: **Liancui Zuo, Fanglan Wu, Shaoqiang Zhang**

Abstract: By the sorting method of vertices, the equitable chromatic number and the equitable chromatic threshold of the Cartesian products of wheels with bipartite graphs are obtained.

Keywords: Cartesian product, Equitable coloring, Equitable chromatic number, Equitable chromatic threshold

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #23, pp.236-245

Title of the Paper: **Globally Exponential Synchronization of Diffusion Recurrent FNNs with Time-Delays and Impulses on Time Scales**

Authors: **Kaihong Zhao**

Abstract: The globally exponential synchronization of diffusion recurrent fuzzy neural networks (FNNs) with time-delays and impulses on time scales is investigated. By applying Lyapunov function and inequality skills, we establish some sufficient conditions to guarantee the globally exponential synchronization of diffusion recurrent FNNs with time-delays and impulses on time scales. One example is given to illustrate the effectiveness of our results.

Keywords: Globally Exponential synchronization; Diffusion recurrent FNNs; Lyapunov functional; Time scales

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #22, pp.224-235

Title of the Paper: **Thermochemical Non-Equilibrium Entry Flows in Mars in Two- Dimensions – Part II**

Authors: **Edisson Sávio De Góes Maciel, Amilcar Porto Pimenta**

Abstract: This work, second part of this study, describes the numerical results obtained by thermochemical non-equilibrium simulations of reactive flow in two-dimensions. The Van Leer and Liou and Steffen Jr. schemes, in their first- and second-order versions, are implemented to accomplish the numerical simulations. The Euler and Navier-Stokes equations, on a finite volume context and employing structured spatial discretization, are applied to solve the “hot gas” hypersonic flows around a double ellipse, around an entry capsule, and around a pathfinder in two-dimensions. The second-order version of the Van Leer and Liou and Steffen Jr. schemes are obtained from a “MUSCL” extrapolation procedure in a context of structured spatial discretization. The convergence process is accelerated to the steady state condition through a spatially variable time step procedure, which has proved effective gains in terms of computational acceleration (Maciel). The reactive simulations involve a Mars atmosphere chemical model of nine species: N, O, N2, O2, NO, CO2, C, CO, and CN, based on the work of Kay and Netterfield. Fifty-three chemical reactions, involving dissociation and recombination, are simulated by the proposed model. The Arrhenius formula is employed to determine the reaction rates and the law of mass action is used to determine the source terms of each gas species equation. The results have indicated the Van Leer scheme as the most accurate one, both inviscid and viscous cases.

Keywords: Thermochemical non-equilibrium, Mars entry, Nine species model, Hypersonic “hot gas” flow, Finite volume, Euler and Navier-Stokes equations, Two-dimensions

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #21, pp.201-223

Title of the Paper: **The Riccati Sub-ODE Method for Fractional Differential-difference Equations**

Authors: **Bin Zheng**

Abstract: In this paper, we are concerned with seeking exact solutions for fractional differential-difference equa- tions by an extended Riccati sub-ODE method. The fractional derivative is defined in the sense of the modified Riemann-liouville derivative. By a combination of this method and a fractional complex transformation, the it- erative relations from indices n to n 1 are established. As for applications, we apply this method to solve the two-component fractional Volterra lattice equations and the fractional m-KdV lattice equation. Some new exact solutions for the two fractional differential-difference equations are obtained.

Keywords: Fractional differential-difference equations, Exact solutions, Riccati sub-ODE method, Fractional complex transformations, Traveling wave solutions, Nonlinear evolution equations

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #20, pp.192-200

Title of the Paper: **Dynamic Behaviors of an Almost Periodic Volterra Integro Dynamic Equation on Time Scales**

Authors: **Lili Wang, Meng Hu**

Abstract: This paper is concerned with an almost periodic Volterra integro dynamic equation on time scales. Based on the theory of calculus on time scales, by using differential inequality theory and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the permanence and the global attractivity of the system are obtained. Then, by using the properties of almost periodic functions and Razumikhin type theorem, sufficient conditions which guarantee the existence of a positive almost periodic solution of the system are obtained. Finally, an example and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

Keywords: Permanence, Global attractivity, Almost periodic solution, Time scale

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #19, pp.182-191

Title of the Paper: **Forecasting Cargo Throughput with Modified Seasonal ARIMA Models**

Authors: **Ming-Hung Shu, Thanh-Lam Nguyen, Bi-Min Hsu, Chunwei Lu, Jui-Chan Huang**

Abstract: The importing-and-exporting goods normally requires organizational cooperation among different inter- national sectors ranging from purchasing, manufacturing, transporting, inventory, to distribution centers. For hav- ing their smooth coordination in the international trade, accurately forecasting the volume of imported-exported goods has become the core issue. This paper aims to obtain efficient forecasting models for estimating the cargo throughput at sea ports. We comprehensively compare the predicting performance of traditional autore- gressive integrated moving average (ARIMA) models, Grey models, and their residual Fourier-modified models. The forecasting accuracy of the conventional models combined with the Fourier modification in residual is dis- covered to be significantly boosted. Two empirical studies of cargo and container throughputs collected from the Hong Kong and Kaohsiung ports were investigated, where two Fourier-modified seasonal ARIMA models, FSARIMA(4; 1; 4)(1; 1; 0)4 and FSARIMA(4; 1; 4)(0; 1; 0)12, are strongly suggested due to their superior forecasting powers. Finally, both models are further employed to provide the valid forecasts of two-port cargo and container throughputs in 2013.

Keywords: ARIMA model, Grey Model, Cargo throughput, Container throughput, Fourier modification, Hong Kong port, Kaohsiung port

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #18, pp.171-181

Title of the Paper: **The 3-Rainbow Index of Graph Operations**

Authors: **Tingting Liu, Yumei Hu**

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #17, pp.161-170

Title of the Paper: **Hyperchaotic Dynamic of Cournot-Bertrand Duopoly Game with Multi-Product and Chaos Control**

Authors: **Fang Wu, Junhai Ma**

Abstract: In real market, firms usually produce multi-product for some reasons, such as cost-saving advantages, catering for the diversity of consumer tastes and providing a barrier to entry. In this paper, Cournot-Bertrand duopoly game with two-product is presented. According to the different statuses of the products, the bounded rationality is expanded as the ordinary bounded rationality and the cautious bounded rationality. The bifurcation diagrams, the Lyapunov exponents, the attractors and the initial condition sensitiveness are investigated by numer- ical simulations. The simulation results show that the economic model is a novel four-dimensional hyperchaotic system which may appear flip-hopf bifurcation and wave shape chaos. The system is more complex and uncer- tain than ordinary chaos system, and it is disadvantage to firm’s decision. At last, the nonlinear feedback control method is applied to the system, and achieves good effect. The results of this paper provide reference to practical industrial structure of this type, and are of theoretical and practical significance.

Keywords: Cournot, Bertrand, multi-product, hyperchaos, bifurcation, complexity

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #16, pp.152-160

Title of the Paper: **Exact Solution of Time-Fractional Partial Differential Equations Using Sumudu Transform**

Authors: **Abdolamir Karbalaie, Mohammad Mehdi Montazeri, Hamed Hamid Muhammed**

Abstract: In this study, we propose a new algorithm to find exact solution of nonlinear time- fractional partial differential equations. The new algorithm basically illustrates how two powerful algorithms, the homotopy pertur- bation method and the Sumudu transform method can be combined and used to get exact solutions of fractional partial differential equations. We also present four different examples to illustrate the preciseness and effectiveness of this algorithm.

Keywords: Mitting-Leffler functions, biological population model, Caputo derivative, Sumudu transform, Fokker-Plank equation

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #15, pp.142-151

Title of the Paper: **A Final Bifurcation in a Simple Duopoly Model**

Authors: **Natascia Angelini, Massimiliano Ferrara**

Abstract: We study global dynamic behavior of a simple duopoly model already existing in literature, when a supercritical Neimark-Sacker bifurcation occurs. In particular, through the study of the critical lines, we provides numerical examples in order to explain the evolution of a closed invariant curve coexisting with periodic attractors into a ”weakly chaotic ring”.

Keywords: Cournot duopoly, critical lines, weakly ring

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #14, pp.132-141

Title of the Paper: **Blow-up of Solution to Cauchy Problem for the Generalized Damped Boussinesq Equation**

Authors: **Cai Donghong, Ye Jianjun**

Abstract: In this paper we study the local solution, global solution and the Blow-up of the solution for a generalized damped Boussinesq equation. By employing various methods such as the energy method, the Fourier transform method, contraction mapping principle and the concavity method, we proved the existence of local and global solutions under certain conditions. Finally, we gave three the conditions about the Blow-up of the solution.

Keywords: Cauchy problem, local solution, global solution, Blow-up of solution

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #13, pp.122-131

Title of the Paper: **A Projective Separating Plane Method with Additional Clipping for Non-Smooth Optimization**

Authors: **Evgeniya A. Vorontsova**

Abstract: This paper is devoted to constructing a definite efficient scheme for non-smooth optimization. A sepa- rating plane algorithm with additional clipping is proposed. The algorithm is used for solving the unconstrained non-smooth convex optimization problem. The latter problem can be reformulated as the computation of the value of a conjugate function at the origin. The algorithm convergence is proved. Encouraging results of numerical experiments are presented.

Keywords: Non-smooth optimization, convex optimization, separating plane algorithms, black-box minimization

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #12, pp.115-121

Title of the Paper: **The Well-posedness of A SARS Epidemic Model**

Authors: **Ruixiao Hao, Lingling Zhang, Lina Guo**

Abstract: In this paper, a class of age-structured SARS epidemic model described by a set of partial differential equations is discussed. The existence and uniqueness of global solution to the model is obtained by using the fixed point theory. Also, the continuous dependence of the solution on the initial value and the regularity of the solution are discussed.

Keywords: Partial differential equations, Global solution, Fixed point theory, Regularity of solution

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #11, pp.105-114

Title of the Paper: **Forecasting Inbound Tourism Demand in Thailand with Grey Model**

Authors: **Kuo-Min Ko, Jung-Fang Chen, Thanh-Lam Nguyen, Bi-Min Hsu, Ming-Hung Shu**

Abstract: Tourism has been a “smoke-free” and vital industry in many countries because it not only generates a plenty of quality employments but also offers a significant economic contribution to the country’s GDP. In order to make the tourism industry stably grown, understanding of its key determinants becomes critical. Various studies have argued that certain determinants were either biasedly measured or incompletely collected due to their intrinsically subjectivity and/or availability; up to date, no standard models have been generally applied for accurate forecasting the tourism demand in the industry. In this study, we propose Grey forecasting approaches to tackle forecasting tourist-demand accurate problems under the sparsely available tourism-demand data. In the empirical study of annual inbound-tourism demand in Thailand, the forecasting performance of the traditional Grey model joined with a modified Fourier series has been verified to be satisfactory with a relatively low mean absolute percentage error.

Keywords: GM(1, 1), Fourier modification, FGM(1, 1), Tourism demand, Grey forecasting model

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #10, pp.95-104

Title of the Paper: **Complexity Analysis in Evolutionary Game System in the Real Estate Market**

Authors: **Yujing Yang, Junhai Ma**

Abstract: In this paper, an evolutionary game model is constructed by introducing reciprocity theory which based on behavioral economics. The complexity and dynamic characteristics are analyzed by mapping out the bifurcation diagrams, Lyapunov index, the strange attractors, power spectrum and sensitivity depending on the initial conditions which mark the chaotic phenomena. The conclusion can be drawn as disorderly chaotic phenomena is as a result of specific parameter conditions in evolutionary game model. The conditions push the evolutionary game system into a specific area in the form of exponential convergence, and inhibit the evolutionary game from evolving towards the desired direction. Finally a linear feedback controlled method is adopted to achieve the goal of driving the system to a steady state, and the goal of evolving to the desired direction is realized.

Keywords: Behavioral economics, Evolutionary game,Chaos, Linear feedback control

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #9, pp.79-94

Title of the Paper: **Support Vector Regression with Missing Data Treatment Based Variables Selection for Water Level Prediction of Galas River in Kelantan Malaysia**

Authors: **Noraini Ibrahim, Antoni Wibowo**

Abstract: Rising in water level becomes an important issue in the state of Galas River in Kuala Krai- Kelantan Malaysia since it is one of important indicator toward to flooding when it achieves a certain level. The increasing of water level is influenced by some factors which called the predictor variables such as month, rainfall, temperature, relative humidity and surface wind. The data for this analysis including the predictors and water level as response were collected from Water Resources Management and Hydrology Division Department of Irrigation and Drainage Malaysia and Malaysian Meteorological Department. However, we noticed there are missing values in the collected data. The selection of suitable predictor variables useful for developing prediction model since the analysis data uses many variables. The suitable predictor variables are selected using Support Vector Regression (SVR) and Cross Validation to obtain an appropriate predictive water level of Galas River Kuala Krai. We take into account the K-fold cross-validation for determining of the dominant variables and best model. However, we need to perform pre-processing data of the datasets since the original data contain missing values. We perform two types of pre-processing data using mean (type I pre-processing data) and Ordinary Linear Regression (type II pre-processing data) to overcome the existing of the missing values. Our experimental result shows that the Gaussian kernel is the suitable kernel function with type I pre-processing data for the predicting water level in Galas River.

Keywords: Galas River, Kelantan, support vector regression, missing value, water level, nonlinear regression, cross validation, variables selection

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #8, pp.69-78

Title of the Paper: **Constructions for Key Distribution Pattern Using Resolvable Designs**

Authors: **Chen Shangdi, Wei Huihui**

Abstract: Key distribution patterns (KDPs) are finite incidence structures satisfying a certain property which makes them widely used in minimizing the key storage and ensuring the security of communication between users in a large network. In this paper we discuss the close connection between resolvable designs and KDPs, and convert the constructions of KDPs into the constructions of resolvable designs. Finally, we give a construction of (q^2, q, 1)- ARBIBD and generalize it to constructions of resolvable design with qn (n is a integer and n ≥ 2) points by mathematical induction.

Keywords: Key distribution pattern, finite incidence structure, resolvable design

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #7, pp.59-68

Title of the Paper: **Diophantine Equations and Fuzzy Adaptive Simulated Annealing**

Authors: **Hime Aguiar e O. Jr.**

Abstract: This work uses the Fuzzy Adaptive Simulated Annealing algorithm to solve many examples of Diophantine equations, offering a global optimization alternative for those ones that did not have a general method or even a closed form for their solution. The proposed method uses the built-in integer programming capabilities of Fuzzy ASA and the candidate solutions evolve in Z^n, in contrast to the best known applications of ASA, exposing an almost unexplored feature of that paradigm. Some significant numerical results are introduced, and empirical relationships between number of independent variables and necessary number of cost function evaluations to get to solutions are presented.

Keywords: Global optimization, Equation solving, Diophantine equations, Adaptive Simulated Annealing

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #6, pp.48-58

Title of the Paper: **Robust Stability of Uncertain Discrete-time State-Delayed Systems with State Saturation Nonlinearities**

Authors: **Chen Dongyan, Wu Hongfei, Shao Chen**

Abstract: This paper is concerned with the robust stability of a class of uncertain discrete-time systems with time- varying delays and state saturation nonlinearities. The saturation nonlinearities are assumed to be the standard saturation function, and the delay is allowed to be time-varying with known bounds. By applying the Lyapunov sta- bility theorem and the delay-fractioning approach, a suitable Lyapunov-Krasovskii functional and a non-negative scalar are constructed respectively. A delay-dependent criterion of the robust stability is given for the addressed systems. Finally, a numerical example is given to illustrate the effectiveness of the presented criterion.

Keywords: Robust stability, Time-varying delay, State saturation nonlinearities, Lyapunov-Krasovskii functional, Delay-fractioning approach

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #5, pp.39-47

Title of the Paper: **The Chebyshev Collection Method for Solving Fractional Order Klein-Gordon Equation**

Authors: **M. M. Khader, N. H. Swetlam, A. M. S. Mahdy**

Abstract: In this paper, we are implemented the Chebyshev spectral method for solving the non-linear fractional Klein-Gordon equation (FKGE). The fractional derivative is considered in the Caputo sense. We presented an approximate formula of the fractional derivative. The properties of the Chebyshev polynomials are used to re- duce FKGE to the solution of system of ordinary differential equations which solved by using the finite difference method. Special attention is given to study the convergence analysis and estimate an upper bound of the error of the derived formula. The numerical results of applying this method to FKGE show the simplicity and the efficiency of the proposed method.

Keywords: Fractional Klein-Gordon equation, Caputo derivative, Chebyshev spectral method, Convergence analysis

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #4, pp.31-38

Title of the Paper: **Contract Design for 4PL Service Supply Chain and Its Pareto Improvement**

Authors: **Qiuxiang Li, Junhai Ma**

Abstract: Considering the probability of the logistics business successfully completed is influenced by the effort level of 4PL and 3PL, it will cause bilateral moral risk in the case that both sides of the factors of production cannot be checked. Using the principal-agent theory, the paper discusses both sides contracts in the centralized decision and distributed decision, founding that short-term contract in distributed decision can’t make both sides realize system revenue maximization; In this paper, we develop a long-term development contract programming model to improve the system benefits by introducing earnings discount factor and designing contract on reservation revenue of 3PL which is called secondary revenue-sharing contract, where improving benefits is important. Numerical simulations are presented to show that long-term development contract programming model have good effects and verify the related conclusions.

Keywords: 4PL, principal-agent, 3PL, Contract design, Pareto improvement

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #3, pp.22-30

Title of the Paper: **Forecasting with Fourier Residual Modified ARIMA Model - An Empirical Case of Inbound Tourism Demand in New Zealand**

Authors: **Ming-Hung Shu, Wei-Ju Hung, Bi-Min Hsu, Thanh-Lam Nguyen, Chunwei Lu**

Abstract: Tourism, one of the gigantic industries in a country, has been considered as a complexly integrated and self-contained economic activity. Key determinants of the tourism demand have not been being fully identified in literature, so varied levels of the forecast accuracy existing in distinct formations of forecasting models, such as the autoregressive integrated moving average (ARIMA) and its joint Fourier modified model, are investigated in this paper. With a certain degree of Fourier-modification factors joined, the model performance is found to be significantly boosted. In an empirical study for the inbound-tourism demand forecasting in New Zealand, the Fourier-modified seasonal ARIMA model, named FSARIMA(1 ; 0 ; 1 )(1 ; 1 ; 1 )12 , is highly recommended due to its satisfactory forecasting power for the historical data. We further employ this model to provide the New Zealand’s tourism projecting demand in 2013 so as to assist policy makers as well as related organizations in early establishing their appropriate strategies for sustaining growth in this extremely-intensified competitive industry.

Keywords: ARIMA model, Tourism demand, Fourier modification, New Zealand tourism, Tourism forecasting

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #2, pp.12-21

Title of the Paper: **The Dangers of Precipitous Adjustment Speed of Recovery Price to the Closed-loop Supply Chain and Improvement Measure**

Authors: **Lei Xie, Junhai Ma**

Abstract: This paper analyzes the recycling price game in waste household appliance market in China. Based on the analysis of recycling behavior game of two retailers and a manufacturer in closed-loop supply chain, we propose that precipitous adjustment speed of recovery price will lead to the system go into a chaotic state. In this state, the recycling price of each recovery party will enter the chaotic state, and it will be significantly affected by the initial price, the profit of each corporate will be apparent fluctuations, and thus the chaotic state caused by the precipitous speed of price adjustment will make competition be invalid and vicious. In this regard, this paper introduces delay decision-making to control the chaotic state and gives the Economics of chaos control.

Keywords: supply chain, recycling household appliance, Nash equilibrium, chaos, delay decision

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #1, pp.1-11