Advances in Mathematical Physics The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. A Simpler GMRES Method for Oscillatory Integrals with Irregular Oscillations Thu, 02 Jul 2015 11:17:37 +0000 A simpler GMRES method for computing oscillatory integral is presented. Theoretical analysis shows that this method is mathematically equivalent to the GMRES method proposed by Olver (2009). Moreover, the simpler GMRES does not require upper Hessenberg matrix factorization, which leads to much simpler program and requires less work. Numerical experiments are conducted to illustrate the performance of the new method and show that in some cases the simpler GMRES method could achieve higher accuracy than GMRES. Qinghua Wu and Meiying Xiang Copyright © 2015 Qinghua Wu and Meiying Xiang. All rights reserved. Existence of Exponential -Stability Nonconstant Equilibrium of Markovian Jumping Nonlinear Diffusion Equations via Ekeland Variational Principle Thu, 25 Jun 2015 07:09:37 +0000 The authors obtained a delay-dependent exponential -stability criterion for a class of Markovian jumping nonlinear diffusion equations by employing the Lyapunov stability theory and some variational methods. As far as we know, it is the first time to apply Ekeland variational principle to obtain the existence of exponential stability equilibrium of -Laplacian dynamic system so that some methods used in this paper are different from those methods of many previous related literatures. In addition, the obtained existence criterion is only involved in the activation functions so that the criterion is simpler and easier than other existence criteria to be verified in practical application. Moreover, a numerical example shows the effectiveness of the proposed methods owing to the large allowable variation range of time-delay. Ruofeng Rao and Shouming Zhong Copyright © 2015 Ruofeng Rao and Shouming Zhong. All rights reserved. Centroids of Lie Supertriple Systems Thu, 18 Jun 2015 08:40:20 +0000 We derive certain structural results concerning centroids of Lie supertriple systems. Centroids of the tensor product of a Lie supertriple system and a unital commutative associative algebra are studied. Furthermore, the centroid of a tensor product of a simple Lie supertriple system and a polynomial ring is partly determined. Jianrong Peng, Liangyun Chen, and Bing Sun Copyright © 2015 Jianrong Peng et al. All rights reserved. The Breather-Like and Rational Solutions for the Integrable Kadomtsev-Petviashvili-Based System Tue, 16 Jun 2015 07:16:10 +0000 The integrable Kadomtsev-Petviashvili-based system is studied. The breather-like (a pulsating mode) and rational solutions are presented applying Hirota bilinear method and Taylor series. The intricate structures of the rational solitary wave solution are discussed mathematically and graphically. The existence conditions of three different solitary wave solution structure for the short-wave field are given by the theory of extreme value analysis. By controlling the wave number of the background plane wave we may control the the behavior of rational solitary wave. However, the shape of the rational solitary wave solution for the real long-wave field is not affected as the wave number is varied. Chuanjian Wang, Zhengde Dai, and Changfu Liu Copyright © 2015 Chuanjian Wang et al. All rights reserved. Energy Distribution of a Regular Black Hole Solution in Einstein-Nonlinear Electrodynamics Thu, 11 Jun 2015 12:26:00 +0000 A study about the energy momentum of a new four-dimensional spherically symmetric, static and charged, regular black hole solution developed in the context of general relativity coupled to nonlinear electrodynamics is presented. Asymptotically, this new black hole solution behaves as the Reissner-Nordström solution only for the particular value , where is a positive integer parameter appearing in the mass function of the solution. The calculations are performed by use of the Einstein, Landau-Lifshitz, Weinberg, and Møller energy momentum complexes. In all the aforementioned prescriptions, the expressions for the energy of the gravitating system considered depend on the mass of the black hole, its charge , a positive integer α, and the radial coordinate r. In all these pseudotensorial prescriptions, the momenta are found to vanish, while the Landau-Lifshitz and Weinberg prescriptions give the same result for the energy distribution. In addition, the limiting behavior of the energy for the cases , , and is studied. The special case and is also examined. We conclude that the Einstein and Møller energy momentum complexes can be considered as the most reliable tools for the study of the energy momentum localization of a gravitating system. I. Radinschi, F. Rahaman, Th. Grammenos, A. Spanou, and Sayeedul Islam Copyright © 2015 I. Radinschi et al. All rights reserved. Comparative Solution of Nonlinear Quintic Cubic Oscillator Using Modified Homotopy Perturbation Method Wed, 03 Jun 2015 14:00:12 +0000 We use modified homotopy perturbation method to find out the solution of nonlinear cubic quintic equation. Besides this method solution of the problem with the following methods is discussed, Energy Balance Method and He’s Frequency Formulation method and then compare the results with each other and Global Error Method. The results show that these three methods are effective as global error method for nonlinear cubic quintic oscillator equation with multiple nonlinear terms and have different effects on the solution. In particular, the homotopy perturbation solution is quite surprising. A cubic quintic nonlinear oscillator is used as an example to compare the results. Muhammad Suleman and Qingbiao Wu Copyright © 2015 Muhammad Suleman and Qingbiao Wu. All rights reserved. Geometry of the Solutions of Localized Induction Equation in the Pseudo-Galilean Space Sun, 31 May 2015 16:45:26 +0000 We study the surfaces corresponding to solutions of the localized induction equation in the pseudo-Galilean space . We classify such surfaces with null curvature and characterize some special curves on these surfaces in . Muhittin Evren Aydin, Adela Mihai, Alper Osman Ogrenmis, and Mahmut Ergut Copyright © 2015 Muhittin Evren Aydin et al. All rights reserved. Adaptive Synchronization of Chaotic Systems considering Performance Parameters of Operational Amplifiers Sun, 31 May 2015 06:26:42 +0000 This paper addresses an adaptive control approach for synchronizing two chaotic oscillators with saturated nonlinear function series as nonlinear functions. Mathematical models to characterize the behavior of the transmitter and receiver circuit were derived, including in the latter the adaptive control and taking into account, for both chaotic oscillators, the most influential performance parameters associated with operational amplifiers. Asymptotic stability of the full synchronization system is studied by using Lyapunov direct method. Theoretical derivations and related results are experimentally validated through implementations from commercially available devices. Finally, the full synchronization system can easily be reproducible at a low cost. Sergio Ruíz-Hernández, Eduardo Ortega-Torres, Carlos Sánchez-López, Miguel Angel Carrasco-Aguilar, Rocio Ochoa-Montiel, Rocio Ilhuicatzi-Roldán, and Marco Antonio Taneco-Hernández Copyright © 2015 Sergio Ruíz-Hernández et al. All rights reserved. Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States Wed, 27 May 2015 09:59:44 +0000 Previously, a formula, incorporating a hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments of density-matrices () and their partial transposes (), was applied with to the generalized two-qubit separability probability question. The formula can, furthermore, be viewed, as we note here, as an averaging over “induced measures in the space of mixed quantum states.” The associated induced-measure separability probabilities () are found—via a high-precision density approximation procedure—to assume interesting, relatively simple rational values in the two-re[al]bit (), (standard) two-qubit (), and two-quater[nionic]bit () cases. We deduce rather simple companion (rebit, qubit, quaterbit, …) formulas that successfully reproduce the rational values assumed for general  . These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula. Paul B. Slater and Charles F. Dunkl Copyright © 2015 Paul B. Slater and Charles F. Dunkl. All rights reserved. Coupling Influence on Signal Readout of a Dual-Parameter LC Resonant System Mon, 25 May 2015 11:59:08 +0000 Dual-parameter inductive-capacitive (LC) resonant sensor is gradually becoming the measurement trend in complex harsh environments; however, the coupling between inductors greatly affects the readout signal, which becomes very difficult to resolve by means of simple mathematical tools. By changing the values of specific variables in a MATLAB code, the influence of coupling between coils on the readout signal is analyzed. Our preliminary conclusions underline that changing the coupling to antenna greatly affects the readout signal, but it simultaneously influences the other signal. When , it is better to broaden the difference between the two coupling coefficients and . On the other side, when is smaller than , it is better to decrease the coupling between sensor inductors , in order to obtain two readout signals averaged in strength. Finally, a test system including a discrete capacitor soldered to a printed circuit board (PCB) based planar spiral coil is built, and the readout signals under different relative inductors positions are analyzed. All experimental results are in good agreement with the results of the MATLAB simulation. Jijun Xiong, Tanyong Wei, Tao Luo, Qiulin Tan, Chenyang Xue, Jun Liu, and Wendong Zhang Copyright © 2015 Jijun Xiong et al. All rights reserved. Comparison of 3D Adaptive Remeshing Strategies for Finite Element Simulations of Electromagnetic Heating of Gold Nanoparticles Sun, 24 May 2015 11:54:32 +0000 The optical properties of metallic nanoparticles are well known, but the study of their thermal behavior is in its infancy. However the local heating of surrounding medium, induced by illuminated nanostructures, opens the way to new sensors and devices. Consequently the accurate calculation of the electromagnetically induced heating of nanostructures is of interest. The proposed multiphysics problem cannot be directly solved with the classical refinement method of Comsol Multiphysics and a 3D adaptive remeshing process based on an a posteriori error estimator is used. In this paper the efficiency of three remeshing strategies for solving the multiphysics problem is compared. The first strategy uses independent remeshing for each physical quantity to reach a given accuracy. The second strategy only controls the accuracy on temperature. The third strategy uses a linear combination of the two normalized targets (the electric field intensity and the temperature). The analysis of the performance of each strategy is based on the convergence of the remeshing process in terms of number of elements. The efficiency of each strategy is also characterized by the number of computation iterations, the number of elements, the CPU time, and the RAM required to achieve a given target accuracy. Fadhil Mezghani, Dominique Barchiesi, Abel Cherouat, Thomas Grosges, and Houman Borouchaki Copyright © 2015 Fadhil Mezghani et al. All rights reserved. Exact Solutions for Some Fractional Differential Equations Wed, 20 May 2015 11:31:02 +0000 The extended Jacobi elliptic function expansion method is used for solving fractional differential equations in the sense of Jumarie’s modified Riemann-Liouville derivative. By means of this approach, a few fractional differential equations are successfully solved. As a result, some new Jacobi elliptic function solutions including solitary wave solutions and trigonometric function solutions are established. The proposed method can also be applied to other fractional differential equations. Abdullah Sonmezoglu Copyright © 2015 Abdullah Sonmezoglu. All rights reserved. Analytical Models for Gravitating Radiating Systems Mon, 11 May 2015 12:39:46 +0000 We analyse the gravitational behaviour of a relativistic heat conducting fluid in a shear-free spherically symmetric spacetime. We show that the isotropy of pressure is a consistency condition which realises a second order nonlinear ordinary differential equation with variable coefficients in the gravitational potentials. Several new classes of solutions are found to the governing equation by imposing various forms on one of the potentials. Interestingly, a complex transformation leads to an exact solution with only real metric functions. All solutions are written in terms of elementary functions. We demonstrate graphically that the fluid pressure, energy density, and heat flux are well behaved for the model, and the model is consistent with a core-envelope framework. B. P. Brassel, S. D. Maharaj, and G. Govender Copyright © 2015 B. P. Brassel et al. All rights reserved. Integrodifferential Equations of the Vector Problem of Electromagnetic Wave Diffraction by a System of Nonintersecting Screens and Inhomogeneous Bodies Wed, 06 May 2015 07:11:42 +0000 The vector problem of electromagnetic wave diffraction by a system of bodies and infinitely thin screens is considered in a quasi-classical formulation. The solution is sought in the classical sense but is defined not in the entire space but rather everywhere except for the screen edges. The original boundary value problem for Maxwell’s equations system is reduced to a system of integrodifferential equations in the regions occupied by the bodies and on the screen surfaces. The integrodifferential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator. Y. G. Smirnov and A. A. Tsupak Copyright © 2015 Y. G. Smirnov and A. A. Tsupak. All rights reserved. Study on Impeller Fracture Model Based on Vibration Characteristics and Fractal Theory Tue, 05 May 2015 12:27:44 +0000 During the operation of centrifugal compressor, failure easily occurs in the presence of complicated external forces. The failure process characterizes with strong nonlinearity, and hence it is difficult to be described by conventional methods. In this paper, firstly, the cracks in different positions are described using crack fractal theory. The basic failure modes of the impeller are summarized. Secondly, a three-dimensional finite element model of the impeller is constructed. Then the von Mises stress under the centrifugal force is calculated, and the corresponding impeller failure process is simulated by “element life and death technology” in ANSYS. Finally, the impeller failure mechanism is analyzed. It can be found that the static stress is not the main cause of the impeller failure, and the dynamic characteristics of the impeller are not perfect because of the pitch vibration modes which appeared in the investigated frequency range. Meanwhile, the natural frequency of the impeller also cannot avoid the frequency of the excitation force. Xiaolong Zhang, Ruishan Yuan, and Yonghui Xie Copyright © 2015 Xiaolong Zhang et al. All rights reserved. Existence of Center for Planar Differential Systems with Impulsive Perturbations Tue, 05 May 2015 07:35:26 +0000 We present a method that uses successor functions in ordinary differential systems to address the “center-focus” problem of a class of planar systems that have an impulsive perturbation. By deriving solution formulae for impulsive systems, several interesting criteria for distinguishing between the center and the focus of linear and nonlinear planar systems with state-dependent impulsions are established. The conditions describing the stability of the focus of the considered models are also given. The computing methods presented here are more convenient for determining the center of impulsive systems than those in the literature. Numerical examples are given to show the effectiveness of the theoretical results. Dengguo Xu Copyright © 2015 Dengguo Xu. All rights reserved. ()-Dimensional mKdV Hierarchy and Chirp Effect of Rossby Solitary Waves Tue, 28 Apr 2015 07:36:32 +0000 By constructing a kind of generalized Lie algebra, based on generalized Tu scheme, a new ()-dimensional mKdV hierarchy is derived which popularizes the results of ()-dimensional integrable system. Furthermore, the ()-dimensional mKdV equation can be applied to describe the propagation of the Rossby solitary waves in the plane of ocean and atmosphere, which is different from the ()-dimensional mKdV equation. By virtue of Riccati equation, some solutions of ()-dimensional mKdV equation are obtained. With the help of solitary wave solutions, similar to the fiber soliton communication, the chirp effect of Rossby solitary waves is discussed and some conclusions are given. Chunlei Wang, Yong Zhang, Baoshu Yin, and Xiaoen Zhang Copyright © 2015 Chunlei Wang et al. All rights reserved. Corrigendum to “Convergence Analysis of Legendre Pseudospectral Scheme for Solving Nonlinear Systems of Volterra Integral Equations” Mon, 27 Apr 2015 09:33:35 +0000 Emran Tohidi, O. R. Navid Samadi, and S. Shateyi Copyright © 2015 Emran Tohidi et al. All rights reserved. Mannheim Curves in Nonflat 3-Dimensional Space Forms Mon, 20 Apr 2015 11:46:03 +0000 We consider the Mannheim curves in nonflat 3-dimensional space forms (Riemannian or Lorentzian) and we give the concept of Mannheim curves. In addition, we investigate the properties of nonnull Mannheim curves and their partner curves. We come to the conclusion that a necessary and sufficient condition is that a linear relationship with constant coefficients will exist between the curvature and the torsion of the given original curves. In the case of null curve, we reveal that there are no null Mannheim curves in the 3-dimensional de Sitter space. Wenjing Zhao, Donghe Pei, and Xinyu Cao Copyright © 2015 Wenjing Zhao et al. All rights reserved. Jordan Isomorphisms on Nest Subalgebras Sun, 19 Apr 2015 07:59:35 +0000 This paper is devoted to the study of Jordan isomorphisms on nest subalgebras of factor von Neumann algebras. It is shown that every Jordan isomorphism between the two nest subalgebras and is either an isomorphism or an anti-isomorphism. Aili Yang Copyright © 2015 Aili Yang. All rights reserved. The Nonlinear Hydroelastic Response of a Semi-Infinite Elastic Plate Floating on a Fluid due to Incident Progressive Waves Wed, 15 Apr 2015 12:51:48 +0000 The nonlinear hydroelastic response of very large floating structures (VLFSs) or an ice sheet floating on the surface of deep water, idealized as a semi-infinite thin elastic plate, is investigated analytically in the case of nonlinear incident waves. Assuming that the fluid is inviscid and incompressible and the motion is irrotational, we consider incident progressive waves with a given angular frequency within the framework of potential flow theory. With the aid of the homotopy analysis method (HAM), the convergent analytical series solutions are derived by solving the simultaneous equations in which we apply a convergence-control parameter to obtain convergent solutions with relatively few terms. The clear calculation results are represented to show nonlinear wave-plate interaction. The effects of different physical parameters, including incident wave amplitude, Young’s modulus, the thickness and density of the plate on the wave scattering, and the hydroelastic response of the floating plate, are considered. We find that the variations of the plate stiffness, thickness, and density greatly change amount of wave energy which is reflected into the open water region and is transmitted into the plate-covered region. Further, the hydroelastic response of the plate also can be affected by the amplitude of incident wave. Ping Wang Copyright © 2015 Ping Wang. All rights reserved. A Meshless Method Based on the Fundamental Solution and Radial Basis Function for Solving an Inverse Heat Conduction Problem Tue, 14 Apr 2015 15:36:15 +0000 We propose a new meshless method to solve a backward inverse heat conduction problem. The numerical scheme, based on the fundamental solution of the heat equation and radial basis functions (RBFs), is used to obtain a numerical solution. Since the coefficients matrix is ill-conditioned, the Tikhonov regularization (TR) method is employed to solve the resulted system of linear equations. Also, the generalized cross-validation (GCV) criterion is applied to choose a regularization parameter. A test problem demonstrates the stability, accuracy, and efficiency of the proposed method. Muhammad Arghand and Majid Amirfakhrian Copyright © 2015 Muhammad Arghand and Majid Amirfakhrian. All rights reserved. Generalized Wavelet Fisher’s Information of Signals Wed, 08 Apr 2015 13:52:55 +0000 This paper defines the generalized wavelet Fisher information of parameter . This information measure is obtained by generalizing the time-domain definition of Fisher’s information of Furuichi to the wavelet domain and allows to quantify smoothness and correlation, among other signals characteristics. Closed-form expressions of generalized wavelet Fisher information for signals are determined and a detailed discussion of their properties, characteristics and their relationship with wavelet -Fisher information are given. Information planes of signals Fisher information are obtained and, based on these, potential applications are highlighted. Finally, generalized wavelet Fisher information is applied to the problem of detecting and locating weak structural breaks in stationary signals, particularly for fractional Gaussian noise series. It is shown that by using a joint Fisher/F-Statistic procedure, significant improvements in time and accuracy are achieved in comparison with the sole application of the F-statistic. Julio Ramírez-Pacheco, Homero Toral-Cruz, Luis Rizo-Domínguez, and Joaquin Cortez-Gonzalez Copyright © 2015 Julio Ramírez-Pacheco et al. All rights reserved. Generalized Taylor Series Method for Solving Nonlinear Fractional Differential Equations with Modified Riemann-Liouville Derivative Wed, 08 Apr 2015 13:27:04 +0000 We propose an efficient analytic method for solving nonlinear differential equations of fractional order. The fractional derivative is defined in the sense of modified Riemann-Liouville derivative. A new technique for calculating the generalized Taylor series coefficients (also known as “generalized differential transforms,” GDTs) of nonlinear functions and a new approach of the generalized Taylor series method (GTSM) are presented. This new method offers a simple algorithm for computing GDTs of nonlinear functions and avoids massive computational work that usually arises in the standard method. Several illustrative examples are demonstrated to show effectiveness of the proposed method. Süleyman Öğrekçi Copyright © 2015 Süleyman Öğrekçi. All rights reserved. Influence of Hierarchic Structure on the Moisture Permeability of Biomimic Woven Fabric Using Fractal Derivative Method Wed, 08 Apr 2015 12:00:16 +0000 The relationship between the unique internal structure of biomimic woven fabric and its moisture management property is investigated using fractal derivative method. The biomimic fabric exhibits a fractal hierarchic inner structure, and its fractal hierarchy can be further extended by fleece finishing treatment on both surfaces of the fabric. Fractal derivative analysis indicates that the fuzzy biomimic fabric with a higher hierarchic construction after fleece finishing performs better in moisture permeability, and the result was proved by experimental tests. Jie Fan, Na Zhu, Li-li Wang, Zhi Liu, Chang-Yuan Wang, and Yong Liu Copyright © 2015 Jie Fan et al. All rights reserved. The Global Symmetry Group of Quantum Spectral Beams and Geometric Phase Factors Tue, 07 Apr 2015 10:15:56 +0000 We propose a cohomological modelling schema of quantum state spaces and their connectivity structures in relation to the formulation of global geometric phase phenomena. In the course of this schema, we introduce the notion of Hermitian differential line sheaves or unitary rays and classify their gauge equivalence classes in terms of a global differential invariant given by the de Rham cohomology class of the curvature. Furthermore, we formulate and interpret physically the curvature recognition integrality theorem for unitary rays. Using this recognition theorem, we define the notion of a quantum spectral beam and show that it has an affine space structure with structure group given by the characters of the fundamental group. Elias Zafiris Copyright © 2015 Elias Zafiris. All rights reserved. Unsteady Flows of a Generalized Fractional Burgers’ Fluid between Two Side Walls Perpendicular to a Plate Sun, 05 Apr 2015 14:21:22 +0000 The unsteady flows of a generalized fractional Burgers’ fluid between two side walls perpendicular to a plate are studied for the case of Rayleigh-Stokes’ first and second problems. Exact solutions of the velocity fields are derived in terms of the generalized Mittag-Leffler function by using the double Fourier transform and discrete Laplace transform of sequential fractional derivatives. The solution for Rayleigh-Stokes’ first problem is represented as the sum of the Newtonian solutions and the non-Newtonian contributions, based on which the solution for Rayleigh-Stokes’ second problem is constructed by the Duhamel’s principle. The solutions for generalized second-grade fluid, generalized Maxwell fluid, and generalized Oldroyd-B fluid performing the same motions appear as limiting cases of the present solutions. Furthermore, the influences of fractional parameters and material parameters on the unsteady flows are discussed by graphical illustrations. Jianhong Kang, Yingke Liu, and Tongqiang Xia Copyright © 2015 Jianhong Kang et al. All rights reserved. Existence of Electrically Charged Structures with Regular Center in Nonlinear Electrodynamics Minimally Coupled to Gravity Thu, 26 Mar 2015 13:13:53 +0000 We address the question of correct description of Lagrange dynamics for regular electrically charged structures in nonlinear electrodynamics coupled to gravity. Regular spherically symmetric configuration satisfying the weak energy condition has obligatory de Sitter center in which the electric field vanishes while the energy density of electromagnetic vacuum achieves its maximal value. The Maxwell weak field limit as requires vanishing electric field at infinity. A field invariant evolves between two minus zero in the center and at infinity which makes a Lagrangian with nonequal asymptotic limits inevitably branching. We formulate the appropriate nonuniform variational problem including the proper boundary conditions and present the example of the spherically symmetric Lagrangian describing electrically charged structure with the regular center. Irina Dymnikova, Evgeny Galaktionov, and Eduard Tropp Copyright © 2015 Irina Dymnikova et al. All rights reserved. Approximate Analytical Solutions of the Fractional-Order Brusselator System Using the Polynomial Least Squares Method Tue, 24 Mar 2015 11:37:48 +0000 The paper presents a new method, called the Polynomial Least Squares Method (PLSM). PLSM allows us to compute approximate analytical solutions for the Brusselator system, which is a fractional-order system of nonlinear differential equations. Constantin Bota and Bogdan Căruntu Copyright © 2015 Constantin Bota and Bogdan Căruntu. All rights reserved. Power Load Prediction Based on Fractal Theory Thu, 19 Mar 2015 13:17:10 +0000 The basic theories of load forecasting on the power system are summarized. Fractal theory, which is a new algorithm applied to load forecasting, is introduced. Based on the fractal dimension and fractal interpolation function theories, the correlation algorithms are applied to the model of short-term load forecasting. According to the process of load forecasting, the steps of every process are designed, including load data preprocessing, similar day selecting, short-term load forecasting, and load curve drawing. The attractor is obtained using an improved deterministic algorithm based on the fractal interpolation function, a day’s load is predicted by three days’ historical loads, the maximum relative error is within 3.7%, and the average relative error is within 1.6%. The experimental result shows the accuracy of this prediction method, which has a certain application reference value in the field of short-term load prediction. Liang Jian-Kai, Carlo Cattani, and Song Wan-Qing Copyright © 2015 Liang Jian-Kai et al. All rights reserved.