Advances in Mathematical Physics The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . All rights reserved. Integrable 2D Time-Irreversible Systems with a Cubic Second Integral Wed, 04 May 2016 09:05:41 +0000 We construct a very rare integrable 2D mechanical system which admits a complementary integral of motion cubic in the velocities in the presence of conservative potential and velocity-dependent (gyroscopic) forces. Special cases are given interpretation as a motion of a particle on a sphere endowed with a Riemannian metric, a particle in the Euclidean plane, and new generalizations of two cases of motion of a rigid body with a cubic integral, known by names of Goriachev-Chaplygin and Goriachev. H. M. Yehia and A. A. Elmandouh Copyright © 2016 H. M. Yehia and A. A. Elmandouh. All rights reserved. Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System Tue, 19 Apr 2016 14:36:11 +0000 The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions. Salim Medjber, Hacene Bekkar, Salah Menouar, and Jeong Ryeol Choi Copyright © 2016 Salim Medjber et al. All rights reserved. The Effect of Initial State Error for Nonlinear Systems with Delay via Iterative Learning Control Thu, 07 Apr 2016 14:08:47 +0000 An iterative learning control problem for nonlinear systems with delays is studied in detail in this paper. By introducing the λ-norm and being inspired by retarded Gronwall-like inequality, the novel sufficient conditions for robust convergence of the tracking error, whose initial states are not zero, with time delays are obtained. Finally, simulation example is given to illustrate the effectiveness of the proposed method. Zhang Qunli Copyright © 2016 Zhang Qunli. All rights reserved. A Soliton Hierarchy Associated with a Spectral Problem of 2nd Degree in a Spectral Parameter and Its Bi-Hamiltonian Structure Wed, 30 Mar 2016 13:18:31 +0000 Associated with , a new matrix spectral problem of 2nd degree in a spectral parameter is proposed and its corresponding soliton hierarchy is generated within the zero curvature formulation. Bi-Hamiltonian structures of the presented soliton hierarchy are furnished by using the trace identity, and thus, all presented equations possess infinitely commuting many symmetries and conservation laws, which implies their Liouville integrability. Yuqin Yao, Shoufeng Shen, and Wen-Xiu Ma Copyright © 2016 Yuqin Yao et al. All rights reserved. Firm Growth Function and Extended-Gibrat’s Property Tue, 22 Mar 2016 08:49:25 +0000 We analytically show that the logarithmic average sales of firms first follow power-law growth and subsequently follow exponential growth, if the growth-rate distributions of the sales obey the extended-Gibrat’s property and Gibrat’s law. Here, the extended-Gibrat’s property and Gibrat’s law are statistically observed in short-term data, which denote the dependence of the growth-rate distributions on the initial values. In the derivation, we analytically show that the parameter of the extended-Gibrat’s property is identical to the power-law growth exponent and that it also decides the parameter of the exponential growth. By employing around one million bits of exhaustive sales data of Japanese firms in the ORBIS database, we confirmed our analytic results. Atushi Ishikawa, Shouji Fujimoto, Takayuki Mizuno, and Tsutomu Watanabe Copyright © 2016 Atushi Ishikawa et al. All rights reserved. Comparing First-Order Microscopic and Macroscopic Crowd Models for an Increasing Number of Massive Agents Sun, 20 Mar 2016 10:14:42 +0000 A comparison between first-order microscopic and macroscopic differential models of crowd dynamics is established for an increasing number of pedestrians. The novelty is the fact of considering massive agents, namely, particles whose individual mass does not become infinitesimal when grows. This implies that the total mass of the system is not constant but grows with . The main result is that the two types of models approach one another in the limit , provided the strength and/or the domain of pedestrian interactions are properly modulated by at either scale. This is consistent with the idea that pedestrians may adapt their interpersonal attitudes according to the overall level of congestion. Alessandro Corbetta and Andrea Tosin Copyright © 2016 Alessandro Corbetta and Andrea Tosin. All rights reserved. On Pseudo-Petrov Symmetric Riemannian Manifolds Tue, 15 Mar 2016 12:22:51 +0000 The present paper deals with pseudo-Petrov symmetric Riemannian manifolds whose space-matter tensor satisfies a special condition. Firstly, basic results of pseudo-Petrov symmetric Riemannian manifolds are obtained. Then, pseudo-Petrov symmetric manifolds which are Einstein, quasi-Einstein, and locally decomposable are examined and some theorems involving these manifolds are proved. Finally, two examples proving the existence of pseudo-Petrov symmetric Riemannian manifolds are given. Sanjib Kumar Jana, Fusun Nurcan, Amit Kumar Debnath, and Joydeep Sengupta Copyright © 2016 Sanjib Kumar Jana et al. All rights reserved. The Ritz Method for Boundary Problems with Essential Conditions as Constraints Sun, 13 Mar 2016 12:22:04 +0000 We give an elementary derivation of an extension of the Ritz method to trial functions that do not satisfy essential boundary conditions. As in the Babuška-Brezzi approach boundary conditions are treated as variational constraints and Lagrange multipliers are used to remove them. However, we avoid the saddle point reformulation of the problem and therefore do not have to deal with the Babuška-Brezzi inf-sup condition. In higher dimensions boundary weights are used to approximate the boundary conditions, and the assumptions in our convergence proof are stated in terms of completeness of the trial functions and of the boundary weights. These assumptions are much more straightforward to verify than the Babuška-Brezzi condition. We also discuss limitations of the method and implementation issues that follow from our analysis and examine a number of examples, both analytic and numerical. Vojin Jovanovic and Sergiy Koshkin Copyright © 2016 Vojin Jovanovic and Sergiy Koshkin. All rights reserved. On Functions of Several Split-Quaternionic Variables Sun, 13 Mar 2016 07:43:23 +0000 Alesker studied a relation between the determinant of a quaternionic Hessian of a function and a specific complex volume form. In this note we show that similar relation holds for functions of several split-quaternionic variables and point to some relations with geometry. Gueo Grantcharov and Camilo Montoya Copyright © 2016 Gueo Grantcharov and Camilo Montoya. All rights reserved. A New Nonlinear Diffusion Equation Model for Noisy Image Segmentation Wed, 09 Mar 2016 13:44:46 +0000 Image segmentation and image denoising are two important and fundamental topics in the field of image processing. Geometric active contour model based on level set method can deal with the problem of image segmentation, but it does not consider the problem of image denoising. In this paper, a new diffusion equation model for noisy image segmentation is proposed by incorporating some classical diffusion equation denoising models into the segmental process. An assumption about the connection between the image intensity and level set function is given firstly. Some classical denoising models are employed to describe the evolution of level set function secondly. The final nonlinear diffusion equation model for noisy image segmentation is built thirdly. Then image segmentation and image denoising are combined in a united framework. The segmental results can be presented by level set function. Experimental results show that the new model has the advantage of noise resistance and is superior to traditional segmentation model. Bo Chen, Xiao-Hui Zhou, Li-Wei Zhang, Jie Wang, Wei-Qiang Zhang, and Chen Zhang Copyright © 2016 Bo Chen et al. All rights reserved. The Interval Slope Method for Long-Term Forecasting of Stock Price Trends Sun, 06 Mar 2016 14:19:07 +0000 A stock price is a typical but complex type of time series data. We used the effective prediction of long-term time series data to schedule an investment strategy and obtain higher profit. Due to economic, environmental, and other factors, it is very difficult to obtain a precise long-term stock price prediction. The exponentially segmented pattern (ESP) is introduced here and used to predict the fluctuation of different stock data over five future prediction intervals. The new feature of stock pricing during the subinterval, named the interval slope, can characterize fluctuations in stock price over specific periods. The cumulative distribution function (CDF) of MSE was compared to those of MMSE-BC and SVR. We concluded that the interval slope developed here can capture more complex dynamics of stock price trends. The mean stock price can then be predicted over specific time intervals relatively accurately, in which multiple mean values over time intervals are used to express the time series in the long term. In this way, the prediction of long-term stock price can be more precise and prevent the development of cumulative errors. Chun-xue Nie and Xue-bo Jin Copyright © 2016 Chun-xue Nie and Xue-bo Jin. All rights reserved. A Nonlinear Schrödinger Equation Resonating at an Essential Spectrum Wed, 02 Mar 2016 09:04:55 +0000 We consider the nonlinear Schrödinger equation . The potential function satisfies that the essential spectrum of the Schrödinger operator is and this Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. The nonlinearity satisfies the resonance type condition . Under some additional conditions on and , we prove that this equation has infinitely many solutions. Shaowei Chen and Haijun Zhou Copyright © 2016 Shaowei Chen and Haijun Zhou. All rights reserved. A Note on the Discrete Spectrum of Gaussian Wells (I): The Ground State Energy in One Dimension Mon, 29 Feb 2016 17:26:30 +0000 The ground state energy of is computed for small values of by means of an approximation of an integral operator in momentum space. Such an approximation leads to a transcendental equation for which is the root. G. Muchatibaya, S. Fassari, F. Rinaldi, and J. Mushanyu Copyright © 2016 G. Muchatibaya et al. All rights reserved. The Approximate Solution of Some Plane Boundary Value Problems of the Moment Theory of Elasticity Mon, 29 Feb 2016 17:23:44 +0000 We consider a two-dimensional system of differential equations of the moment theory of elasticity. The general solution of this system is represented by two arbitrary harmonic functions and solution of the Helmholtz equation. Based on the general solution, an algorithm of constructing approximate solutions of boundary value problems is developed. Using the proposed method, the approximate solutions of some problems on stress concentration on the contours of holes are constructed. The values of stress concentration coefficients obtained in the case of moment elasticity and for the classical elastic medium are compared. In the final part of the paper, we construct the approximate solution of a nonlocal problem whose exact solution is already known and compare our approximate solution with the exact one. Supposedly, the proposed method makes it possible to construct approximate solutions of quite a wide class of boundary value problems. Roman Janjgava Copyright © 2016 Roman Janjgava. All rights reserved. Variational Multiscale Element Free Galerkin Method Coupled with Low-Pass Filter for Burgers’ Equation with Small Diffusion Mon, 29 Feb 2016 12:58:29 +0000 Variational multiscale element free Galerkin (VMEFG) method is applied to Burgers’ equation. It can be found that, for the very small diffusivity coefficients, VMEFG method still suffers from instability in the presence of boundary or interior layers. In order to overcome this problem, the high order low-pass filter is used to smooth the solution. Three test examples with very small diffusion are presented and the solutions obtained are compared with exact solutions and some other numerical methods. The numerical results are found in which the VMEFG coupled with low-pass filter works very well for Burgers’ equation with very small diffusivity coefficients. Ping Zhang, Xiaohua Zhang, and Laizhong Song Copyright © 2016 Ping Zhang et al. All rights reserved. Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation Mon, 22 Feb 2016 06:26:32 +0000 A variety of methods for examining the properties and solutions of nonlinear evolution equations are explored by using the Vakhnenko equation (VE) as an example. The VE, which arises in modelling the propagation of high-frequency waves in a relaxing medium, has periodic and solitary traveling wave solutions some of which are loop-like in nature. The VE can be written in an alternative form, known as the Vakhnenko-Parkes equation (VPE), by a change of independent variables. The VPE has an -soliton solution which is discussed in detail. Individual solitons are hump-like in nature whereas the corresponding solution to the VE comprises -loop-like solitons. Aspects of the inverse scattering transform (IST) method, as applied originally to the KdV equation, are used to find one- and two-soliton solutions to the VPE even though the VPE’s spectral equation is third-order and not second-order. A Bäcklund transformation for the VPE is used to construct conservation laws. The standard IST method for third-order spectral problems is used to investigate solutions corresponding to bound states of the spectrum and to a continuous spectrum. This leads to -soliton solutions and -mode periodic solutions, respectively. Interactions between these types of solutions are investigated. V. O. Vakhnenko and E. J. Parkes Copyright © 2016 V. O. Vakhnenko and E. J. Parkes. All rights reserved. The Stability of Interbank Market Network: A Perspective on Contagion and Risk Sharing Thu, 18 Feb 2016 09:17:37 +0000 As an important part of the financial system, interbank market provides banks with liquidity and credit lending and also is the main channel for risk contagion. In this paper, we test the existence of systematic risk contagion within the Chinese interbank market. By building the networks of the Chinese interbank market for each year and using the measure of mutual information, we quantitatively detect the changes of interbank market networks and observe that the correlations between banks become increasingly tighter in recent years. With the bilateral risk exposure among Chinese listed commercial banks, we find that the possibility of systemic risk contagion in Chinese interbank market is fairly small. But of great concern on each individual bank, the matter is different. Our simulation shows that the failures of three special banks (i.e., Agricultural Bank of China and Bank of China and Industrial and Commercial Bank of China) most likely lead to systemic risk contagion. Furthermore, we test the antirisk ability of the Chinese interbank market from the perspective of risk sharing and discover that the interbank market is stable when the loss scale is lower than forty percent of banks’ total core capital. Chi Xie, Yang Liu, Gang-Jin Wang, and Yan Xu Copyright © 2016 Chi Xie et al. All rights reserved. Direct Scaling Analysis of Fermionic Multiparticle Correlated Anderson Models with Infinite-Range Interaction Thu, 18 Feb 2016 08:53:33 +0000 We adapt the method of direct scaling analysis developed earlier for single-particle Anderson models, to the fermionic multiparticle models with finite or infinite interaction on graphs. Combined with a recent eigenvalue concentration bound for multiparticle systems, the new method leads to a simpler proof of the multiparticle dynamical localization with optimal decay bounds in a natural distance in the multiparticle configuration space, for a large class of strongly mixing random external potentials. Earlier results required the random potential to be IID. Victor Chulaevsky Copyright © 2016 Victor Chulaevsky. All rights reserved. Time Decay for Nonlinear Dissipative Schrödinger Equations in Optical Fields Sun, 14 Feb 2016 11:24:25 +0000 We consider the initial value problem for the nonlinear dissipative Schrödinger equations with a gauge invariant nonlinearity of order for arbitrarily large initial data, where the lower bound is a positive root of for and for Our purpose is to extend the previous results for higher space dimensions concerning -time decay and to improve the lower bound of under the same dissipative condition on : and as in the previous works. Nakao Hayashi, Chunhua Li, and Pavel I. Naumkin Copyright © 2016 Nakao Hayashi et al. All rights reserved. Description of the Magnetic Field and Divergence of Multisolenoid Aharonov-Bohm Potential Tue, 09 Feb 2016 08:36:20 +0000 Explicit formulas for the magnetic field and divergence of multisolenoid Aharonov-Bohm potential are obtained; the mathematical essence of this potential is explained. It is shown that the magnetic field and divergence of this potential are very singular generalized functions concentrated at a finite number of thin solenoids. Deficiency index is found for the minimal operator generated by the Aharonov-Bohm differential expression. Araz R. Aliev, Elshad H. Eyvazov, Said F. M. Ibrahim, and Hassan A. Zedan Copyright © 2016 Araz R. Aliev et al. All rights reserved. Kubo Fluctuation Relations in the Generalized Elastic Model Sun, 31 Jan 2016 09:07:31 +0000 The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. In this paper we show that the Fractional Langevin Equation (FLE) is a suitable framework for the study of the tracer (probe) particle dynamics, when an external force acts only on a single point (tagged probe) belonging to the system. With the help of the Fox function formalism we study the scaling behaviour of the noise- and force-propagators for large and short times (distances). We show that the Kubo fluctuation relations are exactly fulfilled when a time periodic force is exerted on the tagged probe. Most importantly, by studying the large and low frequency behaviour of the complex mobility we illustrate surprising nontrivial physical scenarios. Our analysis shows that the system splits into two distinct regions whose size depends on the applied frequency, characterized by very different response to the periodic perturbation exerted, both in the phase shift and in the amplitude. Alessandro Taloni Copyright © 2016 Alessandro Taloni. All rights reserved. A Comparative Approach to the Solution of the Zabolotskaya-Khokhlov Equation by Iteration Methods Tue, 26 Jan 2016 13:58:04 +0000 We employed different iteration methods like Homotopy Analysis Method (HAM), Adomian Decomposition Method (ADM), and Variational Iteration Method (VIM) to find the approximate solution to the Zabolotskaya-Khokhlov (ZK) equation. Iteration methods are used to solve linear and nonlinear PDEs whose classical methods are either very complex or too limited to apply. A comparison study has been made to see which of these methods converges to the approximate solution rapidly. The result revealed that, amongst these methods, ADM is more effective and simpler tool in its nature which does not require any transformation or linearization. Saeed Ahmed and Muhammad Kalim Copyright © 2016 Saeed Ahmed and Muhammad Kalim. All rights reserved. Explicit Solution of Reinsurance-Investment Problem for an Insurer with Dynamic Income under Vasicek Model Tue, 26 Jan 2016 13:46:24 +0000 Unlike traditionally used reserves models, this paper focuses on a reserve process with dynamic income to study the reinsurance-investment problem for an insurer under Vasicek stochastic interest rate model. The insurer’s dynamic income is given by the remainder after a dynamic reward budget being subtracted from the insurer’s net premium which is calculated according to expected premium principle. Applying stochastic control technique, a Hamilton-Jacobi-Bellman equation is established and the explicit solution is obtained under the objective of maximizing the insurer’s power utility of terminal wealth. Some economic interpretations of the obtained results are explained in detail. In addition, numerical analysis and several graphics are given to illustrate our results more meticulous. De-Lei Sheng Copyright © 2016 De-Lei Sheng. All rights reserved. Tight --Frame and Its Novel Characterizations via Atomic Systems Tue, 26 Jan 2016 07:15:57 +0000 -frame is a generalization of -frame. We generalize the tight -frame to --frame via atomic systems. In this paper, the definition of tight --frame is put forward; equivalent characterizations and necessary conditions of tight --frame are given. In particular, the necessary and sufficient condition for tight --frame being tight -frame is obtained. Finally, by means of methods and techniques of frame theory, several properties of tight --frame are given. Yongdong Huang and Dingli Hua Copyright © 2016 Yongdong Huang and Dingli Hua. All rights reserved. Singularity Analysis for a Class of Porous Medium Equation with Time-Dependent Coefficients Tue, 19 Jan 2016 09:17:04 +0000 This paper concerns the singularity and global regularity for the porous medium equation with time-dependent coefficients under homogeneous Dirichlet boundary conditions. Firstly, some global regularity results are established. Furthermore, we investigate the blow-up solution to the boundary value problem. The upper and lower estimates to the lifespan of the singular solution are also obtained here. Anyin Xia, Xianxiang Pu, and Shan Li Copyright © 2016 Anyin Xia et al. All rights reserved. Finite Time Control for Fractional Order Nonlinear Hydroturbine Governing System via Frequency Distributed Model Sun, 17 Jan 2016 08:41:05 +0000 This paper studies the application of frequency distributed model for finite time control of a fractional order nonlinear hydroturbine governing system (HGS). Firstly, the mathematical model of HGS with external random disturbances is introduced. Secondly, a novel terminal sliding surface is proposed and its stability to origin is proved based on the frequency distributed model and Lyapunov stability theory. Furthermore, based on finite time stability and sliding mode control theory, a robust control law to ensure the occurrence of the sliding motion in a finite time is designed for stabilization of the fractional order HGS. Finally, simulation results show the effectiveness and robustness of the proposed scheme. Bin Wang, Lin Yin, Shaojie Wang, Shirui Miao, Tantan Du, and Chao Zuo Copyright © 2016 Bin Wang et al. All rights reserved. Inverse Uniqueness in Interior Transmission Problem and Its Eigenvalue Tunneling in Simple Domain Tue, 12 Jan 2016 09:25:16 +0000 We study inverse uniqueness with a knowledge of spectral data of an interior transmission problem in a penetrable simple domain. We expand the solution in a series of one-dimensional problems in the far-fields. We define an ODE by restricting the PDE along a fixed scattered direction. Accordingly, we obtain a Sturm-Liouville problem for each scattered direction. There exists the correspondence between the ODE spectrum and the PDE spectrum. We deduce the inverse uniqueness on the index of refraction from the discussion on the uniqueness anglewise of the Strum-Liouville problem. Lung-Hui Chen Copyright © 2016 Lung-Hui Chen. All rights reserved. Upper Bounds on the Degeneracy of the Ground State in Quantum Field Models Wed, 06 Jan 2016 14:05:08 +0000 Axiomatic abstract formulations are presented to derive upper bounds on the degeneracy of the ground state in quantum field models including massless ones. In particular, given is a sufficient condition under which the degeneracy of the ground state of the perturbed Hamiltonian is less than or equal to the degeneracy of the ground state of the unperturbed one. Applications of the abstract theory to models in quantum field theory are outlined. Asao Arai and Daiju Funakawa Copyright © 2016 Asao Arai and Daiju Funakawa. All rights reserved. Seiberg-Witten Like Equations on Pseudo-Riemannian Manifolds with Structure Wed, 06 Jan 2016 13:28:13 +0000 We consider 7-dimensional pseudo-Riemannian manifolds with structure group . On such manifolds, the space of 2-forms splits orthogonally into components . We define self-duality of a 2-form by considering the part as the bundle of self-dual 2-forms. We express the spinor bundle and the Dirac operator and write down Seiberg-Witten like equations on such manifolds. Finally we get explicit forms of these equations on and give some solutions. Nülifer Özdemir and Nedim Deǧirmenci Copyright © 2016 Nülifer Özdemir and Nedim Deǧirmenci. All rights reserved. MHD Flow due to the Nonlinear Stretching of a Porous Sheet Tue, 05 Jan 2016 14:13:45 +0000 The MHD flow due to the nonlinear stretching of a porous sheet is investigated. A closed form solution is obtained when the stretching rate is inversely proportional to the distance from the origin. Otherwise a uniformly valid asymptotic expansion, for large magnetic interaction number , is developed. It coincides with a homotopy perturbation expansion for the problem. The asymptotic/homotopy perturbation expansion gives results in excellent agreement with accurate numerical results, for large as well as small values of . For large , the expansion, being asymptotic, needs a small number of terms, regardless of the mass transfer rate or the degree of nonlinearity. For small , the expansion is a homotopy perturbation one. It needs considerably increasing number of terms with higher injection rates and/or with stretching rates approaching the inverse proportionality. It may even fail. Tarek M. A. El-Mistikawy Copyright © 2016 Tarek M. A. El-Mistikawy. All rights reserved.