Advances in Mathematical Physics The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . All rights reserved. A Numberable Set of Exact Solutions for the Macroscopic Approach to Extended Thermodynamics of Polyatomic Gases with Many Moments Thu, 25 Aug 2016 12:09:40 +0000 A new model for Polyatomic Gases with an arbitrary but fixed number of moments has been recently proposed and investigated in the framework of Extended Thermodynamics; the arbitrariness of the number of moments is linked to a number and the resulting model is called an -Model. This model has been elaborated in order to take into account the entropy principle, the Galilean relativity principle, and some symmetry conditions. It has been proved that the solution for all these conditions exists, but it has not been written explicitly because hard notation is necessary; it has only been shown how the theory is self-generating in the sense that if we know the closure of the -Model, then we will be able to find that of -Model. Up to now only a single particular solution has been found in this regard. Instead of this, we find here a numberable set of exact solutions which hold for every fixed number . Maria Cristina Carrisi, Rita Enoh Tchame, Marcel Obounou, and Sebastiano Pennisi Copyright © 2016 Maria Cristina Carrisi et al. All rights reserved. From Boole to Leggett-Garg: Epistemology of Bell-Type Inequalities Thu, 25 Aug 2016 11:22:12 +0000 In 1862, George Boole derived an inequality for variables that represents a demarcation line between possible and impossible experience. This inequality forms an important milestone in the epistemology of probability theory and probability measures. In 1985 Leggett and Garg derived a physics related inequality, mathematically identical to Boole’s, that according to them represents a demarcation between macroscopic realism and quantum mechanics. We show that a wide gulf separates the “sense impressions” and corresponding data, as well as the postulates of macroscopic realism, from the mathematical abstractions that are used to derive the inequality of Leggett and Garg. If the gulf can be bridged, one may indeed derive the said inequality, which is then clearly a demarcation between possible and impossible experience: it cannot be violated and is not violated by quantum theory. This implies that the Leggett-Garg inequality does not mean that the SQUID flux is not there when nobody looks, as Leggett and Garg suggest, but instead that the probability measures may not be what Leggett and Garg have assumed them to be, when no data can be secured that directly relate to them. We show that similar considerations apply to other quantum interpretation-puzzles such as two-slit experiments. Karl Hess, Hans De Raedt, and Kristel Michielsen Copyright © 2016 Karl Hess et al. All rights reserved. Two-Dimensional Nonlinear Propagation of Ion Acoustic Waves through KPB and KP Equations in Weakly Relativistic Plasmas Wed, 24 Aug 2016 08:11:15 +0000 Two-dimensional three-component plasma system consisting of nonextensive electrons, positrons, and relativistic thermal ions is considered. The well-known Kadomtsev-Petviashvili-Burgers and Kadomtsev-Petviashvili equations are derived to study the basic characteristics of small but finite amplitude ion acoustic waves of the plasmas by using the reductive perturbation method. The influences of positron concentration, electron-positron and ion-electron temperature ratios, strength of electron and positrons nonextensivity, and relativistic streaming factor on the propagation of ion acoustic waves in the plasmas are investigated. It is revealed that the electrostatic compressive and rarefactive ion acoustic waves are obtained for superthermal electrons and positrons, but only compressive ion acoustic waves are found and the potential profiles become steeper in case of subthermal positrons and electrons. M. G. Hafez, M. R. Talukder, and M. Hossain Ali Copyright © 2016 M. G. Hafez et al. All rights reserved. Higher Spin Symmetries of the Free Schrödinger Equation Sun, 21 Aug 2016 11:18:39 +0000 It is shown that the Schrödinger symmetry algebra of a free particle in spatial dimensions can be embedded into a representation of the higher spin algebra. The latter spans an infinite dimensional algebra of higher-order symmetry generators of the free Schrödinger equation. An explicit representation of the maximal finite dimensional subalgebra of the higher spin algebra is given in terms of nonrelativistic generators. We show also how to convert Vasiliev’s equations into an explicit nonrelativistic covariant form, such that they might apply to nonrelativistic systems. Our procedure reveals that the space of solutions of the Schrödinger equation can be regarded also as a supersymmetric module. Mauricio Valenzuela Copyright © 2016 Mauricio Valenzuela. All rights reserved. A Mixed Finite Element Formulation for the Conservative Fractional Diffusion Equations Tue, 16 Aug 2016 07:35:49 +0000 We consider a boundary-value problem of one-side conservative elliptic equation involving Riemann-Liouville fractional integral. The appearance of the singular term in the solution leads to lower regularity of the solution of the equation, so to the lower order convergence rate for the numerical solution. In this paper, by the dividing of equation, we drop the lower regularity term in the solution successfully and get a new fractional elliptic equation which has full regularity. We present a theoretical framework of mixed finite element approximation to the new fractional elliptic equation and derive the error estimates for unknown function, its derivative, and fractional-order flux. Some numerical results are illustrated to confirm the optimal error estimates. Suxiang Yang and Huanzhen Chen Copyright © 2016 Suxiang Yang and Huanzhen Chen. All rights reserved. Strong Isomorphism in Eisert-Wilkens-Lewenstein Type Quantum Games Sun, 14 Aug 2016 07:12:15 +0000 The aim of this paper is to bring together the notions of quantum game and game isomorphism. The work is intended as an attempt to introduce a new criterion for quantum game schemes. The generally accepted requirement forces a quantum scheme to generate the classical game in a particular case. Now, given a quantum game scheme and two isomorphic classical games, we additionally require the resulting quantum games to be isomorphic as well. We are concerned with the Eisert-Wilkens-Lewenstein quantum game scheme and the strong isomorphism between games in strategic form. Piotr Frąckiewicz Copyright © 2016 Piotr Frąckiewicz. All rights reserved. Formal First Integrals of General Dynamical Systems Tue, 09 Aug 2016 12:18:33 +0000 The goal of this paper is trying to make a complete study on the integrability for general analytic nonlinear systems by first integrals. We will firstly give an exhaustive discussion on analytic planar systems. Then a class of higher dimensional systems with invariant manifolds will be considered; we will develop several criteria for existence of formal integrals and give some applications to illustrate our results at last. Jia Jiao, Wenlei Li, and Qingjian Zhou Copyright © 2016 Jia Jiao et al. All rights reserved. Theory of Nonlocal Point Transformations in General Relativity Mon, 08 Aug 2016 07:10:28 +0000 A discussion of the functional setting customarily adopted in General Relativity (GR) is proposed. This is based on the introduction of the notion of nonlocal point transformations (NLPTs). While allowing the extension of the traditional concept of GR-reference frame, NLPTs are important because they permit the explicit determination of the map between intrinsically different and generally curved space-times expressed in arbitrary coordinate systems. For this purpose in the paper the mathematical foundations of NLPT-theory are laid down and basic physical implications are considered. In particular, explicit applications of the theory are proposed, which concern a solution to the so-called Einstein teleparallel problem in the framework of NLPT-theory; the determination of the tensor transformation laws holding for the acceleration 4-tensor with respect to the group of NLPTs and the identification of NLPT-acceleration effects, namely, the relationship established via general NLPT between particle 4-acceleration tensors existing in different curved space-times; the construction of the nonlocal transformation law connecting different diagonal metric tensors solution to the Einstein field equations; and the diagonalization of nondiagonal metric tensors. Massimo Tessarotto and Claudio Cremaschini Copyright © 2016 Massimo Tessarotto and Claudio Cremaschini. All rights reserved. Vector Solitons of a Coupled Schrödinger System with Variable Coefficients Thu, 04 Aug 2016 07:59:34 +0000 We show the existence of waveforms of finite-energy (vector solitons) for a coupled nonlinear Schrödinger system with inhomogeneous coefficients. Furthermore, some of these solutions are approximated using a Newton-type iteration, combined with a collocation-spectral strategy to discretize the corresponding soliton equations. Some numerical simulations concerned with analysis of a collision of two oncoming vector solitons of the system are also performed. Juan Carlos Muñoz Grajales Copyright © 2016 Juan Carlos Muñoz Grajales. All rights reserved. Explicit Solutions of the Boundary Value Problems for an Ellipse with Double Porosity Sun, 31 Jul 2016 12:53:50 +0000 The basic two-dimensional boundary value problems of the fully coupled linear equilibrium theory of elasticity for solids with double porosity structure are reduced to the solvability of two types of a problem. The first one is the BVPs for the equations of classical elasticity of isotropic bodies, and the other is the BVPs for the equations of pore and fissure fluid pressures. The solutions of these equations are presented by means of elementary (harmonic, metaharmonic, and biharmonic) functions. On the basis of the gained results, we constructed an explicit solution of some basic BVPs for an ellipse in the form of absolutely uniformly convergent series. Lamara Bitsadze and Natela Zirakashvili Copyright © 2016 Lamara Bitsadze and Natela Zirakashvili. All rights reserved. Electromagnetic Scattering at the Waveguide Step between Equilateral Triangular Waveguides Sun, 31 Jul 2016 11:47:18 +0000 The analysis of the electromagnetic scattering at discontinuities between equilateral triangular waveguides is studied. The complete electromagnetic solution is derived using analytical closed form expressions for the mode spectrum of the equilateral waveguide. The mathematical formulation of the electromagnetic scattering problem is based on the quasi-analytical Mode-Matching method. This method benefits from the electromagnetic field division into symmetries as well as from the plane wave formulation presented for the expressions involved. The unification of the surface integrals used in the method thanks to the plane wave formulation is revealed, leading to expressions that are very well suited for its implementation in an electromagnetic analysis and design code. The obtained results for some cases of interest (building blocks for microwave components for communication systems) are verified using other numerical methods included in a commercial software package, showing the potential of the presented approach based on quasi-analytic expressions. Ana Morán-López, Juan Córcoles, Jorge A. Ruiz-Cruz, José R. Montejo-Garai, and Jesús M. Rebollar Copyright © 2016 Ana Morán-López et al. All rights reserved. Elastic Equilibrium of Porous Cosserat Media with Double Porosity Sun, 31 Jul 2016 10:06:36 +0000 The static equilibrium of porous elastic materials with double porosity is considered in the case of an elastic Cosserat medium. The corresponding three-dimensional system of differential equations is derived. Detailed consideration is given to the case of plane deformation. A two-dimensional system of equations of plane deformation is written in the complex form and its general solution is represented by means of three analytic functions of a complex variable and two solutions of Helmholtz equations. The constructed general solution enables one to solve analytically a sufficiently wide class of plane boundary value problems of the elastic equilibrium of porous Cosserat media with double porosity. A concrete boundary value problem for a concentric ring is solved. Roman Janjgava Copyright © 2016 Roman Janjgava. All rights reserved. Thermal Stability Investigation in a Reactive Sphere of Combustible Material Sun, 31 Jul 2016 08:30:21 +0000 An investigation of thermal stability in a stockpile of combustible material is considered. The combustible material is any carbon containing material that can react with oxygen trapped in a stockpile due to exothermic chemical reaction. The complicated process is modelled in a sphere and one-dimensional energy equation is used to solve the problem. The semi-implicit finite difference method (FDM) is applied to tackle the nonlinear differential equation governing the problem. Graphical solutions are displayed to describe effects of embedded kinetic parameters on the temperature of the system. R. S. Lebelo Copyright © 2016 R. S. Lebelo. All rights reserved. Lax Triples for Integrable Surfaces in Three-Dimensional Space Thu, 28 Jul 2016 09:55:15 +0000 We study Lax triples (i.e., Lax representations consisting of three linear equations) associated with families of surfaces immersed in three-dimensional Euclidean space . We begin with a natural integrable deformation of the principal chiral model. Then, we show that all deformations linear in the spectral parameter are trivial unless we admit Lax representations in a larger space. We present an explicit example of triply orthogonal systems with Lax representation in the group . Finally, the obtained results are interpreted in the context of the soliton surfaces approach. Jan L. Cieśliński and Artur Kobus Copyright © 2016 Jan L. Cieśliński and Artur Kobus. All rights reserved. Drinfeld Realization of Quantum Twisted Affine Algebras via Braid Group Wed, 27 Jul 2016 10:54:42 +0000 The Drinfeld realization of quantum affine algebras has been tremendously useful since its discovery. Combining techniques of Beck and Nakajima with our previous approach, we give a complete and conceptual proof of the Drinfeld realization for the twisted quantum affine algebras using Lusztig’s braid group action. Naihuan Jing and Honglian Zhang Copyright © 2016 Naihuan Jing and Honglian Zhang. All rights reserved. New Applications of a Kind of Infinitesimal-Operator Lie Algebra Mon, 25 Jul 2016 11:30:43 +0000 Applying some reduced Lie algebras of Lie symmetry operators of a Lie transformation group, we obtain an invariant of a second-order differential equation which can be generated by a Euler-Lagrange formulism. A corresponding discrete equation approximating it is given as well. Finally, we make use of the Lie algebras to generate some new integrable systems including () and () dimensions. Honwah Tam, Yufeng Zhang, and Xiangzhi Zhang Copyright © 2016 Honwah Tam et al. All rights reserved. A Lipschitz Stability Estimate for the Inverse Source Problem and the Numerical Scheme Wed, 20 Jul 2016 07:12:40 +0000 We consider the inverse source problem for heat equation, where the source term has the form . We give a numerical algorithm to compute unknown source term . Also, we give a stability estimate in the case that is a piecewise constant function. Xianzheng Jia Copyright © 2016 Xianzheng Jia. All rights reserved. A General Scheme for Information Interception in the Ping-Pong Protocol Sun, 17 Jul 2016 13:24:51 +0000 The existence of undetectable eavesdropping of dense coded information has been already demonstrated by Pavičić for the quantum direct communication based on the ping-pong paradigm. However, (a) the explicit scheme of the circuit is only given and no design rules are provided; (b) the existence of losses is implicitly assumed; (c) the attack has been formulated against qubit based protocol only and it is not clear whether it can be adapted to higher dimensional systems. These deficiencies are removed in the presented contribution. A new generic eavesdropping scheme built on a firm theoretical background is proposed. In contrast to the previous approach, it does not refer to the properties of the vacuum state, so it is fully consistent with the absence of losses assumption. Moreover, the scheme applies to the communication paradigm based on signal particles of any dimensionality. It is also shown that some well known attacks are special cases of the proposed scheme. Piotr Zawadzki and Jarosław Adam Miszczak Copyright © 2016 Piotr Zawadzki and Jarosław Adam Miszczak. All rights reserved. Multidimensional Analysis of Plates with Irregularities Using Higher-Order Finite Elements Based on Lobatto Shape Functions Wed, 13 Jul 2016 06:49:50 +0000 Direct modeling and simulation of engineering problems with various irregularities are computationally very inefficient and in some cases impossible, even in these days of massively parallel computational systems. As a result, in recent times, a number of schemes have been put forward to tract such problems in a computationally efficient manner. Needless to say, such schemes are still going through evolutionary stages. This paper addresses direct solution based on the selective use of different dimensional models at different regions of the problem domain. For the multidimensional approach, a higher-order transition element is developed to connect the different element types where two- and three-dimensional laminated elements based on higher-order subparametric concept are considered. Modeling simplicity and calculation efficiency of the multidimensional approach are shown for the analysis of cantilever plates with stepped section and patch-repaired plates. Jae S. Ahn Copyright © 2016 Jae S. Ahn. All rights reserved. Enriched Meshfree Method for an Accurate Numerical Solution of the Motz Problem Sun, 10 Jul 2016 09:27:21 +0000 We present an enriched meshfree solution of the Motz problem. The Motz problem has been known as a benchmark problem to verify the efficiency of numerical methods in the presence of a jump boundary data singularity at a point, where an abrupt change occurs for the boundary condition. We propose a singular basis function enrichment technique in the context of partition of unity based meshfree method. We take the leading terms of the local series expansion at the point singularity and use them as enrichment functions for the local approximation space. As a result, we obtain highly accurate leading coefficients of the Motz problem that are comparable to the most accurate numerical solution. The proposed singular enrichment technique is highly effective in the case of the local series expansion of the solution being known. The enrichment technique that is used in this study can be applied to monotone singularities (of type with ) as well as oscillating singularities (of type ). It is the first attempt to apply singular meshfree enrichment technique to the Motz problem. Won-Tak Hong Copyright © 2016 Won-Tak Hong. All rights reserved. A Type of Multigrid Method Based on the Fixed-Shift Inverse Iteration for the Steklov Eigenvalue Problem Tue, 05 Jul 2016 08:08:22 +0000 For the Steklov eigenvalue problem, we establish a type of multigrid discretizations based on the fixed-shift inverse iteration and study in depth its a priori/a posteriori error estimates. In addition, we also propose an adaptive algorithm on the basis of the a posteriori error estimates. Finally, we present some numerical examples to validate the efficiency of our method. Feiyan Li and Hai Bi Copyright © 2016 Feiyan Li and Hai Bi. All rights reserved. Econophysics, Statistical Mechanics for Financial Applications, and Financial Mathematics Sun, 03 Jul 2016 09:56:50 +0000 Doojin Ryu and Kiseop Lee Copyright © 2016 Doojin Ryu and Kiseop Lee. All rights reserved. Application of Perturbation Theory to a Master Equation Thu, 30 Jun 2016 14:54:48 +0000 We develop a matrix perturbation method for the Lindblad master equation. The first- and second-order corrections are obtained and the method is generalized for higher orders. The perturbation method developed is applied to the problem of a lossy cavity filled with a Kerr medium; the second-order corrections are estimated and compared with the known exact analytic solution. The comparison is done by calculating the -function, the average number of photons, and the distance between density matrices. B. M. Villegas-Martínez, F. Soto-Eguibar, and H. M. Moya-Cessa Copyright © 2016 B. M. Villegas-Martínez et al. All rights reserved. Computing Coherence Vectors and Correlation Matrices with Application to Quantum Discord Quantification Wed, 29 Jun 2016 12:10:23 +0000 Coherence vectors and correlation matrices are important functions frequently used in physics. The numerical calculation of these functions directly from their definitions, which involves Kronecker products and matrix multiplications, may seem to be a reasonable option. Notwithstanding, as we demonstrate in this paper, some algebraic manipulations before programming can reduce considerably their computational complexity. Besides, we provide Fortran code to generate generalized Gell-Mann matrices and to compute the optimized and unoptimized versions of associated Bloch’s vectors and correlation matrix in the case of bipartite quantum systems. As a code test and application example, we consider the calculation of Hilbert-Schmidt quantum discords. Jonas Maziero Copyright © 2016 Jonas Maziero. All rights reserved. Homotopy Analysis Solution for Magnetohydrodynamic Squeezing Flow in Porous Medium Wed, 22 Jun 2016 12:24:47 +0000 The aim of the present work is to analyze the magnetohydrodynamic (MHD) squeezing flow through porous medium using homotopy analysis method (HAM). Fourth-order boundary value problem is modeled through stream function and transformation . Absolute residuals are used to check the efficiency and consistency of HAM. Other analytical techniques are compared with the present work. It is shown that results of good agreement can be obtained by choosing a suitable value of convergence control parameter in the valid region . The influence of different parameters on the flow is argued theoretically as well as graphically. Inayat Ullah, M. T. Rahim, Hamid Khan, and Mubashir Qayyum Copyright © 2016 Inayat Ullah et al. All rights reserved. Analytical Investigation of Magnetohydrodynamic Flow over a Nonlinear Porous Stretching Sheet Thu, 16 Jun 2016 06:35:17 +0000 We investigated the magnetohydrodynamic (MHD) boundary layer flow over a nonlinear porous stretching sheet with the help of semianalytical method known as optimal homotopy asymptotic method (OHAM). The effects of different parameters on fluid flow are investigated and discussed. The obtained results are compared with numerical Runge-Kutta-Fehlberg fourth-fifth-order method. It is found that the OHAM solution agrees well with numerical as well as published data for different assigned values of parameters; this thus indicates the feasibility of the proposed method (OHAM). Fazle Mabood and Nopparat Pochai Copyright © 2016 Fazle Mabood and Nopparat Pochai. All rights reserved. Interactions of Delta Shock Waves for Zero-Pressure Gas Dynamics with Energy Conservation Law Wed, 15 Jun 2016 15:54:04 +0000 We study the interactions of delta shock waves and vacuum states for the system of conservation laws of mass, momentum, and energy in zero-pressure gas dynamics. The Riemann problems with initial data of three piecewise constant states are solved case by case, and four different configurations of Riemann solutions are constructed. Furthermore, the numerical simulations completely coinciding with theoretical analysis are shown. Wei Cai and Yanyan Zhang Copyright © 2016 Wei Cai and Yanyan Zhang. All rights reserved. An Operational Matrix Technique for Solving Variable Order Fractional Differential-Integral Equation Based on the Second Kind of Chebyshev Polynomials Wed, 15 Jun 2016 12:17:56 +0000 An operational matrix technique is proposed to solve variable order fractional differential-integral equation based on the second kind of Chebyshev polynomials in this paper. The differential operational matrix and integral operational matrix are derived based on the second kind of Chebyshev polynomials. Using two types of operational matrixes, the original equation is transformed into the arithmetic product of several dependent matrixes, which can be viewed as an algebraic system after adopting the collocation points. Further, numerical solution of original equation is obtained by solving the algebraic system. Finally, several examples show that the numerical algorithm is computationally efficient. Jianping Liu, Xia Li, and Limeng Wu Copyright © 2016 Jianping Liu et al. All rights reserved. CRE Solvability, Exact Soliton-Cnoidal Wave Interaction Solutions, and Nonlocal Symmetry for the Modified Boussinesq Equation Wed, 15 Jun 2016 06:33:43 +0000 It is proved that the modified Boussinesq equation is consistent Riccati expansion (CRE) solvable; two types of special soliton-cnoidal wave interaction solution of the equation are explicitly given, which is difficult to be found by other traditional methods. Moreover, the nonlocal symmetry related to the consistent tanh expansion (CTE) and the residual symmetry from the truncated Painlevé expansion, as well as the relationship between them, are obtained. The residual symmetry is localized after embedding the original system in an enlarged one. The symmetry group transformation of the enlarged system is derived by applying the Lie point symmetry approach. Wenguang Cheng and Biao Li Copyright © 2016 Wenguang Cheng and Biao Li. All rights reserved. A Subdivision Based Iterative Collocation Algorithm for Nonlinear Third-Order Boundary Value Problems Thu, 09 Jun 2016 09:29:21 +0000 We construct an algorithm for the numerical solution of nonlinear third-order boundary value problems. This algorithm is based on eight-point binary subdivision scheme. Proposed algorithm is stable and convergent and gives more accurate results than fourth-degree B-spline algorithm. Syeda Tehmina Ejaz and Ghulam Mustafa Copyright © 2016 Syeda Tehmina Ejaz and Ghulam Mustafa. All rights reserved.