ISRN Applied Mathematics http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. A New Three-Oscillator Model for the Heart System in the Case of Time Delay and Designing Appropriate Controller for Its Synchronization Thu, 05 Jun 2014 06:34:17 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/168247/ If the and oscillators are not synchronized, it may arise some kinds of blocking arrhythmias in the system of heart. In this paper, in order to examine the heart system more precisely, we apply the three-oscillator model of the heart system, and to prevent arrhythmias, perform the following steps. Firstly, we add a voltage with rang and frequency to node. Then, we use delay time factor in oscillators and finally the appropriate control is designed. In this paper, we have explained how simulating and curing these arrhythmias are possible by designing a three-oscillator system for heart in the state of delay and without delay and by applying an appropriate control. In the end, we present the simulation results. Siroos Nazari, Aghileh Heydari, Mahboobeh Tavakoli, and Javad Khaligh Copyright © 2014 Siroos Nazari et al. All rights reserved. Exact Solutions for the KdV Equation with Forcing Term by the Generalized tanh-coth Method and the -Expansion Method Tue, 03 Jun 2014 10:57:04 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/948072/ An application of the generalized tanh-coth method and the -expansion method to search for exact solutions of nonlinear partial differential equations is analyzed. These methods are used for the KdV equation with forcing term. The generalized tanh-coth method and the -expansion method were used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. It is shown that the generalized tanh-coth method and the -expansion method, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear problems. Jalil Manafian and Monireh Nasrollahpour Copyright © 2014 Jalil Manafian and Monireh Nasrollahpour. All rights reserved. Sufficiency and Duality in Nonsmooth Multiobjective Programming Problem under Generalized Univex Functions Thu, 22 May 2014 12:17:48 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/904640/ We consider a nonsmooth multiobjective programming problem where the functions involved are nondifferentiable. The class of univex functions is generalized to a far wider class of --V-type I univex functions. Then, through various nontrivial examples, we illustrate that the class introduced is new and extends several known classes existing in the literature. Based upon these generalized functions, Karush-Kuhn-Tucker type sufficient optimality conditions are established. Further, we derive weak, strong, converse, and strict converse duality theorems for Mond-Weir type multiobjective dual program. Pallavi Kharbanda, Divya Agarwal, and Deepa Sinha Copyright © 2014 Pallavi Kharbanda et al. All rights reserved. Gravity Modulation Effects of Hydromagnetic Elastico-Viscous Fluid Flow past a Porous Plate in Slip Flow Regime Wed, 21 May 2014 07:29:08 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/492906/ The two-dimensional hydromagnetic free convective flow of elastico-viscous fluid (Walters liquid Model ) with simultaneous heat and mass transfer past an infinite vertical porous plate under the influence of gravity modulation effects has been analysed. Generalized Navier’s boundary condition has been used to study the characteristics of slip flow regime. Fluctuating characteristics of temperature and concentration are considered in the neighbourhood of the surface having periodic suction. The governing equations of fluid motion are solved analytically by using perturbation technique. Various fluid flow characteristics (velocity profile, viscous drag, etc.) are analyzed graphically for various values of flow parameters involved in the solution. A special emphasis is given on the gravity modulation effects on both Newtonian and non-Newtonian fluids. Debasish Dey Copyright © 2014 Debasish Dey. All rights reserved. Application of Galerkin Method to Kirchhoff Plates Stochastic Bending Problem Thu, 15 May 2014 12:38:41 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/604368/ In this paper, the Galerkin method is used to obtain approximate solutions for Kirchhoff plates stochastic bending problem with uncertainty over plates flexural rigidity coefficient. The uncertainty in the rigidity coefficient is represented by means of parameterized stochastic processes. A theorem of Lax-Milgram type, about existence and uniqueness of the theoretical solutions, is presented and used in selection of the approximate solution space. The Wiener-Askey scheme of generalized polynomials chaos (gPC) is used to model the stochastic behavior of the displacement solutions. The performance of the approximate Galerkin solution scheme developed herein is evaluated by comparing first and second order moments of the approximate solution with the same moments evaluated from Monte Carlo simulation. Rapid convergence of approximate Galerkin's solution to the first and second order moments is observed, for the problems studied herein. Results also show that using the developed Galerkin's scheme one gets adequate estimates for accrued probability function to a random variable generated by the stochastic process of displacement. Cláudio Roberto Ávila da Silva Júnior, Milton Kist, and Marcelo Borges dos Santos Copyright © 2014 Cláudio Roberto Ávila da Silva Júnior et al. All rights reserved. Iterative and Algebraic Algorithms for the Computation of the Steady State Kalman Filter Gain Sun, 04 May 2014 13:35:51 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/417623/ The Kalman filter gain arises in linear estimation and is associated with linear systems. The gain is a matrix through which the estimation and the prediction of the state as well as the corresponding estimation and prediction error covariance matrices are computed. For time invariant and asymptotically stable systems, there exists a steady state value of the Kalman filter gain. The steady state Kalman filter gain is usually derived via the steady state prediction error covariance by first solving the corresponding Riccati equation. In this paper, we present iterative per-step and doubling algorithms as well as an algebraic algorithm for the steady state Kalman filter gain computation. These algorithms hold under conditions concerning the system parameters. The advantage of these algorithms is the autonomous computation of the steady state Kalman filter gain. Nicholas Assimakis and Maria Adam Copyright © 2014 Nicholas Assimakis and Maria Adam. All rights reserved. Wolfe Type Second Order Nondifferentiable Symmetric Duality in Multiobjective Programming over Cone with Generalized (K, F)-Convexity Thu, 24 Apr 2014 06:38:46 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/687917/ A new class of second order (K, F) pseudoconvex function is introduced with example. A pair of Wolfe type second order nondifferentiable symmetric dual programs over arbitrary cones with square root term is formulated. The duality results are established under second order (K, F) pseudoconvexity assumption. Also a Wolfe type second order minimax mixed integer programming problem is formulated and the symmetric duality results are established under second order (K, F) pseudoconvexity assumption. A. K. Tripathy Copyright © 2014 A. K. Tripathy. All rights reserved. Mathematical Model of Stock Prices via a Fractional Brownian Motion Model with Adaptive Parameters Mon, 07 Apr 2014 12:32:48 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/791418/ The paper presents a mathematical model of stock prices using a fractional Brownian motion model with adaptive parameters (FBMAP). The accuracy index of the proposed model is compared with the Brownian motion model with adaptive parameters (BMAP). The parameters in both models are adapted at any time. The ADVANC Info Service Public Company Limited (ADVANC) and Land and Houses Public Company Limited (LH) closed prices are concerned in the paper. The Brownian motion model with adaptive parameters (BMAP) and fractional Brownian motion model with adaptive parameters (FBMAP) are applied to identify ADVANC and LH closed prices. The simulation results show that the FBMAP is more suitable for forecasting the ADVANC and LH closed price than the BMAP. Tidarut Areerak Copyright © 2014 Tidarut Areerak. All rights reserved. Dynamic Behavior of a One-Dimensional Wave Equation with Memory and Viscous Damping Tue, 01 Apr 2014 14:22:46 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/984098/ We study the dynamic behavior of a one-dimensional wave equation with both exponential polynomial kernel memory and viscous damping under the Dirichlet boundary condition. By introducing some new variables, the time-variant system is changed into a time-invariant one. The detailed spectral analysis is presented. It is shown that all eigenvalues of the system approach a line that is parallel to the imaginary axis. The residual and continuous spectral sets are shown to be empty. The main result is the spectrum-determined growth condition that is one of the most difficult problems for infinite-dimensional systems. Consequently, an exponential stability is concluded. Jing Wang Copyright © 2014 Jing Wang. All rights reserved. Stability Criteria for Uncertain Discrete-Time Systems under the Influence of Saturation Nonlinearities and Time-Varying Delay Tue, 01 Apr 2014 11:48:57 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/861759/ The problem of global asymptotic stability of a class of uncertain discrete-time systems in the presence of saturation nonlinearities and interval-like time-varying delay in the state is considered. The uncertainties associated with the system parameters are assumed to be deterministic and normbounded. The objective of the paper is to propose stability criteria having considerably smaller numerical complexity. Two new delay-dependent stability criteria are derived by estimating the forward difference of the Lyapunov functional using the concept of reciprocal convexity and method of scale inequality, respectively. The presented criteria are compared with a previously reported criterion. A numerical example is provided to illustrate the effectiveness of the presented criteria. Siva Kumar Tadepalli, V. Krishna Rao Kandanvli, and Haranath Kar Copyright © 2014 Siva Kumar Tadepalli et al. All rights reserved. A Hybrid Feature Selection Method Based on Rough Conditional Mutual Information and Naive Bayesian Classifier Sun, 30 Mar 2014 14:06:57 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/382738/ We introduced a novel hybrid feature selection method based on rough conditional mutual information and Naive Bayesian classifier. Conditional mutual information is an important metric in feature selection, but it is hard to compute. We introduce a new measure called rough conditional mutual information which is based on rough sets; it is shown that the new measure can substitute Shannon’s conditional mutual information. Thus rough conditional mutual information can also be used to filter the irrelevant and redundant features. Subsequently, to reduce the feature and improve classification accuracy, a wrapper approach based on naive Bayesian classifier is used to search the optimal feature subset in the space of a candidate feature subset which is selected by filter model. Finally, the proposed algorithms are tested on several UCI datasets compared with other classical feature selection methods. The results show that our approach obtains not only high classification accuracy, but also the least number of selected features. Zilin Zeng, Hongjun Zhang, Rui Zhang, and Youliang Zhang Copyright © 2014 Zilin Zeng et al. All rights reserved. Robust Eye Localization by Combining Classification and Regression Methods Sun, 30 Mar 2014 07:42:31 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/804291/ Eye localization is an important part in face recognition system, because its precision closely affects the performance of the system. In this paper we analyze the limitations of classification and regression methods and propose a robust and accurate eye localization method combining these two methods. The classification method in eye localization is robust, but its precision is not so high, while the regression method is sensitive to the initial position, but in case the initial position is near to the eye position, it can converge to the eye position accurately. Experiments on BioID and LFW databases show that the proposed method gives very good results on both low and high quality images. Pak Il Nam, Ri Song Jin, and Peter Peer Copyright © 2014 Pak Il Nam et al. All rights reserved. Asymptotic Stability Analysis and Optimality Algorithm for Uncertain Neutral Systems with Saturation Thu, 27 Mar 2014 12:44:46 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/805798/ The certain and uncertain neutral systems with time-delay and saturating actuator are considered in this paper. In order to analyse and optimize the system, auxiliary functions are presented based on additive decomposition approach and the relationship among them is discussed. As the novel stability criterion, two sufficient conditions are obtained for asymptotic stability of the neutral systems. Furthermore, the paper gives the stability analysis algorithm and optimality algorithm to optimize the result. Finally, from the two-stage dissolution tank of solid caustic soda in a chemical plant, three numerical examples are implemented to show the effectiveness of the proposed method. Xinghua Liu Copyright © 2014 Xinghua Liu. All rights reserved. On Second-Order Differential Equations with Nonsmooth Second Member Mon, 24 Mar 2014 08:40:14 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/305718/ In an abstract framework, we consider the following initial value problem: u′′ + μAu + F(u)u = f  in  (0,T), , where is a positive function and f a nonsmooth function. Given u0, u1, and f we determine in order to have a solution u of the previous equation. We analyze two cases of . In our approach, we use the Theory of Linear Operators in Hilbert Spaces, the compactness Aubin-Lions Theorem, and an argument of Fixed Point. One of our two results provides an answer in a certain sense to an open question formulated by Lions in (1981, Page 284). M. Milla Miranda, A. T. Lourêdo, and L. A. Medeiros Copyright © 2014 M. Milla Miranda et al. All rights reserved. Parameters Estimation of the Rakhmatov and Vrudhula Model from the Optimization Method Search in Improved Network Sun, 23 Mar 2014 13:13:32 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/958538/ This paper presents a proposition of a new optimization method called Search in Improved Network, which is an extension of the method Search in Modified Network, to calculate the empirical parameters of the Rakhmatov and Vrudhula model for predicting batteries lifetime used in mobile devices. The new method is evaluated according to the following methodology. At first empirical parameters are computed considering the optimization methods Search in Improved Network, Search in Modified Network, and Least Squares, as well as the experimental data obtained from a testbed, considering a Lhition-Ion battery, model BL5F, used in the Nokia N95 cell phone. In a second moment, the Rakhmatov and Vrudhula model is validated for each set of parameters obtained, and the simulated data from the model are compared with a set of experimental data. From simulations results a comparative analysis is performed and it is found that by the application of the method Search in Improved Network in the parameters estimation of the Rakhmatov and Vrudhula model it is possible to obtain an easy and intuitive implementation, improving the results obtained in the model accuracy, as well as preserving the runtime. B. F. Silva, P. S. Sausen, and A. Sausen Copyright © 2014 B. F. Silva et al. All rights reserved. Sliding Mode Control for the Synchronous Generator Thu, 20 Mar 2014 12:23:03 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/256504/ Based on the Lyapunov stability theorem and sliding mode control technique, a design of the nonlinear controller is proposed for the dual-excited and steam-valving control of the synchronous generators with matched and mismatched perturbations in this paper. By using some constant gains designed in the sliding surface function, the perturbations in the power system can be suppressed, and the property of asymptotical stability of the rotor angle and the voltage can be achieved at the same time. Yaote Chang and Chih-Chin Wen Copyright © 2014 Yaote Chang and Chih-Chin Wen. All rights reserved. A Note on the Adaptive Estimation of a Multiplicative Separable Regression Function Thu, 20 Mar 2014 11:27:48 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/271303/ We investigate the estimation of a multiplicative separable regression function from a bidimensional nonparametric regression model with random design. We present a general estimator for this problem and study its mean integrated squared error (MISE) properties. A wavelet version of this estimator is developed. In some situations, we prove that it attains the standard unidimensional rate of convergence under the MISE over Besov balls. Christophe Chesneau Copyright © 2014 Christophe Chesneau. All rights reserved. Selecting the Best of Portfolio Using OWA Operator Weights in Cross Efficiency-Evaluation Wed, 19 Mar 2014 13:46:09 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/978314/ The present study is an attempt toward evaluating the performance of portfolios and asset selection using cross-efficiency evaluation. Cross-efficiency evaluation is an effective way of ranking decision making units (DMUs) in data envelopment analysis (DEA). The most widely used approach is to evaluate the efficiencies in each row or column in the cross-efficiency matrix with equal weights into an average cross-efficiency score for each DMU and consider it as the overall performance measurement of the DMU. This paper focuses on the evaluation process of the efficiencies in the cross-efficiency matrix and proposes the use of ordered weighted averaging (OWA) operator weights for cross-efficiency evaluation. The OWA operator weights are generated by the minimax disparity approach and allow the decision maker (DM) or investor to select the best assets that are characterized by an orness degree. The problem consists of choosing an optimal set of assets in order to minimize the risk and maximize return. This method is illustrated by application in mutual funds and weights are obtained via OWA operator for making the best portfolio. The finding could be used for constructing the best portfolio in stock companies, in various finance organization, and public and private sector companies. Masoud Sanei and Shokoofeh Banihashemi Copyright © 2014 Masoud Sanei and Shokoofeh Banihashemi. All rights reserved. Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems Wed, 19 Mar 2014 11:56:51 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/472371/ We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method. Bin Wang, Yuangui Zhou, Jianyi Xue, and Delan Zhu Copyright © 2014 Bin Wang et al. All rights reserved. Three-Dimensional Modeling of Tsunami Generation and Propagation under the Effect of Stochastic Seismic Fault Source Model in Linearized Shallow-Water Wave Theory Wed, 19 Mar 2014 09:03:47 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/874230/ Tsunami generation and propagation caused by stochastic seismic fault driven by two Gaussian white noises in the - and -directions are investigated. This model is used to study the tsunami amplitude amplification under the effect of the noise intensities, spreading uplift length and rise times of the three-dimensional stochastic fault source model. Tsunami waveforms within the frame of the linearized shallow-water theory for constant water depth are analyzed analytically by transform methods (Laplace in time and Fourier in space). The amplification of tsunami amplitudes builds up progressively as time increases during the generation process due to wave focusing while the maximum wave amplitude decreases with time during the propagation process due to the geometric spreading and also due to dispersion. The maximum amplitude amplification is proportional to the propagation length of the stochastic source model and inversely proportional to the water depth. The increase of the normalized noise intensities on the bottom topography leads to an increase in oscillations and amplitude in the free surface elevation. We derived and analyzed the mean and variance of the random tsunami waves as a function of the propagated uplift length, noise intensities, and the average depth of the ocean along the generation and propagation path. Allam A. Allam, M. A. Omar, and Khaled T. Ramadan Copyright © 2014 Allam A. Allam et al. All rights reserved. Improved Liu-Type Estimator of Parameters in Two Seemingly Unrelated Regressions Sun, 16 Mar 2014 12:46:59 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/679835/ We consider the parameter estimation in two seemingly unrelated regression systems. To overcome the multicollinearity, we propose a Liu-type estimator in seemingly unrelated regression systems. The superiority of the new estimator over the classic estimator in the mean square error is discussed and we also discuss the admissibility of the Liu-type estimator. Jibo Wu Copyright © 2014 Jibo Wu. All rights reserved. Some Remarks on the Mathieu Series Thu, 13 Mar 2014 11:20:40 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/985782/ The object of this note is to present new expressions for the classical Mathieu series in terms of hyperbolic functions. The derivation is based on elementary arguments concerning the integral representation of the series. The results are used afterwards to prove, among others, a new relationship between the Mathieu series and its alternating companion. A recursion formula for the Mathieu series is also presented. As a byproduct, some closed-form evaluations of integrals involving hyperbolic functions are inferred. Robert Frontczak Copyright © 2014 Robert Frontczak. All rights reserved. Improved Qrginv Algorithm for Computing Moore-Penrose Inverse Matrices Wed, 12 Mar 2014 09:59:42 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/641706/ Katsikis et al. presented a computational method in order to calculate the Moore-Penrose inverse of an arbitrary matrix (including singular and rectangular) (2011). In this paper, an improved version of this method is presented for computing the pseudo inverse of an real matrix A with rank . Numerical experiments show that the resulting pseudoinverse matrix is reasonably accurate and its computation time is significantly less than that obtained by Katsikis et al. Alireza Ataei Copyright © 2014 Alireza Ataei. All rights reserved. Strong Pullback Attractors for Nonautonomous Suspension Bridge Equations Sun, 09 Mar 2014 09:45:55 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/253532/ We prove the existence of a pullback -attractor in for the nonautonomous suspension bridge equations. Wenchao Ju and Xuan Wang Copyright © 2014 Wenchao Ju and Xuan Wang. All rights reserved. A Modified Approach to the New Solutions of Generalized mKdV Equation Using -Expansion Wed, 05 Mar 2014 15:43:55 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/856396/ The modified -expansion method is applied for finding new solutions of the generalized mKdV equation. By taking an appropriate transformation, the generalized mKdV equation is solved in different cases and hyperbolic, trigonometric, and rational function solutions are obtained. Yu Zhou and Ying Wang Copyright © 2014 Yu Zhou and Ying Wang. All rights reserved. On the Dynamics of an Oil Price Model Wed, 05 Mar 2014 08:42:38 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/375856/ We address the dynamic of an energy price model in a more general approach; to our knowledge, in previous works, different conditions are imposed on the energy price function at a particular value in order to characterize stability and unstability for real or complex eigenvalues. Nonexistence of periodic orbits is shown by means of Dulac’s criterion; we also exhibit some pictures of solutions. Finally we modify this model and apply it to some particular cases in Venezuela economy. Teodoro Lara Copyright © 2014 Teodoro Lara. All rights reserved. On Optimal Control Problem for Backward Stochastic Doubly Systems Tue, 04 Mar 2014 09:03:53 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/903912/ We are going to study an approach of optimal control problems where the state equation is a backward doubly stochastic differential equation, and the set of strict (classical) controls need not be convex and the diffusion coefficient and the generator coefficient depend on the terms control. The main result is necessary conditions as well as a sufficient condition for optimality in the form of a relaxed maximum principle. Adel Chala Copyright © 2014 Adel Chala. All rights reserved. The Applications of Cardinal Trigonometric Splines in Solving Nonlinear Integral Equations Tue, 04 Mar 2014 08:44:18 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/213909/ The cardinal trigonometric splines on small compact supports are employed to solve integral equations. The unknown function is expressed as a linear combination of cardinal trigonometric splines functions. Then a simple system of equations on the coefficients is deducted. When solving the Volterra integral equations, the system is triangular, so it is relatively straight forward to solve the nonlinear system of the coefficients and a good approximation of the original solution is obtained. The sufficient condition for the existence of the solution is discussed and the convergence rate is investigated. Jin Xie, Xiaoyan Liu, and Lixiang Xu Copyright © 2014 Jin Xie et al. All rights reserved. Generalizing Krawtchouk Polynomials Using Hadamard Matrices Tue, 04 Mar 2014 00:00:00 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/498135/ We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged in a square matrix; in particular, the case where this matrix is a Hadamard matrix is considered. Orthogonality relations and recurrence relations are established, and coefficients for the expansion of any polynomial in terms of m-polynomials are obtained. We conclude this paper by an implementation of m-polynomials and some of the results obtained for them in Mathematica. Peter S. Chami, Bernd Sing, and Norris Sookoo Copyright © 2014 Peter S. Chami et al. All rights reserved. Some Regularity Criteria for the 3D Boussinesq Equations in the Class Sun, 23 Feb 2014 08:44:45 +0000 http://www.hindawi.com/journals/isrn.applied.mathematics/2014/564758/ We consider the three-dimensional Boussinesq equations and obtain some regularity criteria via the velocity gradient (or the vorticity, or the deformation tensor) and the temperature gradient. Zujin Zhang Copyright © 2014 Zujin Zhang. All rights reserved.