Bounds for the eigenvalues of the continuous algebraic Riccati equation
NASA Astrophysics Data System (ADS)
Liu, Jianzhou; Zhang, Juan
2011-10-01
By using singular value decomposition and majorisation inequalities, we propose new upper and lower bounds for summations of eigenvalues (including the trace) of the solution of the continuous algebraic Riccati equation. These bounds improve and extend some of the previous results. Finally, we give corresponding numerical examples to illustrate the effectiveness of our results.
Neural network architecture for solving the algebraic matrix Riccati equation
NASA Astrophysics Data System (ADS)
Ham, Fredric M.; Collins, Emmanuel G.
1996-03-01
This paper presents a neurocomputing approach for solving the algebraic matrix Riccati equation. This approach is able to utilize a good initial condition to reduce the computation time in comparison to standard methods for solving the Riccati equation. The repeated solutions of closely related Riccati equations appears in homotopy algorithms to solve certain problems in fixed-architecture control. Hence, the new approach has the potential to significantly speed-up these algorithms. It also has potential applications in adaptive control. The structured neural network architecture is trained using error backpropagation based on a steepest-descent learning rule. An example is given which illustrates the advantage of utilizing a good initial condition (i.e., initial setting of the neural network synaptic weight matrix) in the structured neural network.
Algebraic Riccati equations in zero-sum differential games
NASA Technical Reports Server (NTRS)
Johnson, T. L.; Chao, A.
1974-01-01
The procedure for finding the closed-loop Nash equilibrium solution of two-player zero-sum linear time-invariant differential games with quadratic performance criteria and classical information pattern may be reduced in most cases to the solution of an algebraic Riccati equation. Based on the results obtained by Willems, necessary and sufficient conditions for existence of solutions to these equations are derived, and explicit conditions for a scalar example are given.
Numerical Solvers for Generalized Algebraic Riccati Equations
NASA Astrophysics Data System (ADS)
Ivanov, I. G.; Rusinova, R. I.
2009-10-01
We consider a new type nonlinear matrix equation. We investigate the existence a positive definite solution and two iterative methods for computing this solution. The first method is the classical Newton procedure and the second is a new Stein iteration. In this paper it is proved that a new Stein iteration has convergence properties to those of the Newton iteration.
A multilayer recurrent neural network for solving continuous-time algebraic Riccati equations.
Wang, Jun; Wu, Guang
1998-07-01
A multilayer recurrent neural network is proposed for solving continuous-time algebraic matrix Riccati equations in real time. The proposed recurrent neural network consists of four bidirectionally connected layers. Each layer consists of an array of neurons. The proposed recurrent neural network is shown to be capable of solving algebraic Riccati equations and synthesizing linear-quadratic control systems in real time. Analytical results on stability of the recurrent neural network and solvability of algebraic Riccati equations by use of the recurrent neural network are discussed. The operating characteristics of the recurrent neural network are also demonstrated through three illustrative examples.
A Numerical Algorithm for Finding Solution of Cross-Coupled Algebraic Riccati Equations
NASA Astrophysics Data System (ADS)
Mukaidani, Hiroaki; Yamamoto, Seiji; Yamamoto, Toru
In this letter, a computational approach for solving cross-coupled algebraic Riccati equations (CAREs) is investigated. The main purpose of this letter is to propose a new algorithm that combines Newton's method with a gradient-based iterative (GI) algorithm for solving CAREs. In particular, it is noteworthy that both a quadratic convergence under an appropriate initial condition and reduction in dimensions for matrix computation are both achieved. A numerical example is provided to demonstrate the efficiency of this proposed algorithm.
Upper solution bounds of the continuous coupled algebraic Riccati matrix equation
NASA Astrophysics Data System (ADS)
Liu, Jianzhou; Zhang, Juan
2011-04-01
In this article, by using some matrix identities, we construct the equivalent form of the continuous coupled algebraic Riccati equation (CCARE). Further, with the aid of the eigenvalue inequalities of matrix's product, by solving the linear inequalities utilising the properties of M-matrix and its inverse matrix, new upper matrix bounds for the solutions of the CCARE are established, which improve and extend some of the recent results. Finally, a corresponding numerical example is proposed to illustrate the effectiveness of the derived results.
NASA Astrophysics Data System (ADS)
Ivanov, I. G.; Netov, N. C.; Bogdanova, B. C.
2015-10-01
This paper addresses the problem of solving a generalized algebraic Riccati equation with an indefinite sign of its quadratic term. We extend the approach introduced by Lanzon, Feng, Anderson and Rotkowitz (2008) for solving similar Riccati equations. We numerically investigate two types of iterative methods for computing the stabilizing solution. The first type of iterative methods constructs two matrix sequences, where the sum of them converges to the stabilizing solution. The second type of methods defines one matrix sequence which converges to the stabilizing solution. Computer realizations of the presented methods are numerically tested and compared on the test of family examples. Based on the experiments some conclusions are derived.
NASA Technical Reports Server (NTRS)
Ito, Kazufumi
1987-01-01
The linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces is considered. The optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence pi sup N of finite dimensional approximations of the solution to ARE. A sufficient condition that shows pi sup N converges strongly to pi is obtained. Under this condition, a formula is derived which can be used to obtain a rate of convergence of pi sup N to pi. The results of the Galerkin approximation is demonstrated and applied for parabolic systems and the averaging approximation for hereditary differential systems.
Novel insights on the stabilising solution to the continuous-time algebraic Riccati equation
NASA Astrophysics Data System (ADS)
Rojas, A. J.
2014-11-01
In the present paper we present a closed-form solution, as a function of the closed-loop poles, for the continuous-time algebraic Riccati equations (CAREs) related to single-input single-output systems with non-repeated poles. The proposed solution trades the standard numerical algorithm approach for one based on a spectral factorisation argument, offering potential insight into any control technique based on a CARE and its solution. As an example, we present the equivalence of two fairly recent control over network results. Furthermore we apply the proposed result to the formula for the optimal regulator gain matrix k (or equivalently the Luenberger's observer gain l) and present an example. Finally, we conclude by discussing the possible extension of the proposed closed-form solution to the repeated eigenvalues case and to the case when the CARE is related to multiple-input multiple-output systems.
NASA Astrophysics Data System (ADS)
Liu, Jianzhou; Zhang, Juan
2011-08-01
In this article, applying the properties of M-matrix and non-negative matrix, utilising eigenvalue inequalities of matrix's sum and product, we firstly develop new upper and lower matrix bounds of the solution for discrete coupled algebraic Riccati equation (DCARE). Secondly, we discuss the solution existence uniqueness condition of the DCARE using the developed upper and lower matrix bounds and a fixed point theorem. Thirdly, a new fixed iterative algorithm of the solution for the DCARE is shown. Finally, the corresponding numerical examples are given to illustrate the effectiveness of the developed results.
New application to Riccati equation
NASA Astrophysics Data System (ADS)
Taogetusang; Sirendaoerji; Li, Shu-Min
2010-08-01
To seek new infinite sequence of exact solutions to nonlinear evolution equations, this paper gives the formula of nonlinear superposition of the solutions and Bäcklund transformation of Riccati equation. Based on the tanh-function expansion method and homogenous balance method, new infinite sequence of exact solutions to Zakharov-Kuznetsov equation, Karamoto-Sivashinsky equation and the set of (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov equations are obtained with the aid of symbolic computation system Mathematica. The method is of significance to construct infinite sequence exact solutions to other nonlinear evolution equations.
Lie theoretic aspects of the Riccati equation
NASA Technical Reports Server (NTRS)
Hermann, R.; Martin, C.
1977-01-01
Various features of the application of Lie theory to matrix Riccati equations, of basic importance in control and system theories, are discussed. Particular consideration is given to centralizer foliation, the Cartan decomposition, matrix Riccati equations as Lie systems on Grassmanians, local analysis near a zero point of a vector field, linearization in homogeneous space, the tangent bundle in terms of partitioned matrices, and stability properties of fixed points of Riccati vector fields.
NASA Astrophysics Data System (ADS)
Moraes Rêgo, Patrícia Helena; Viana da Fonseca Neto, João; Ferreira, Ernesto M.
2015-08-01
The main focus of this article is to present a proposal to solve, via UDUT factorisation, the convergence and numerical stability problems that are related to the covariance matrix ill-conditioning of the recursive least squares (RLS) approach for online approximations of the algebraic Riccati equation (ARE) solution associated with the discrete linear quadratic regulator (DLQR) problem formulated in the actor-critic reinforcement learning and approximate dynamic programming context. The parameterisations of the Bellman equation, utility function and dynamic system as well as the algebra of Kronecker product assemble a framework for the solution of the DLQR problem. The condition number and the positivity parameter of the covariance matrix are associated with statistical metrics for evaluating the approximation performance of the ARE solution via RLS-based estimators. The performance of RLS approximators is also evaluated in terms of consistence and polarisation when associated with reinforcement learning methods. The used methodology contemplates realisations of online designs for DLQR controllers that is evaluated in a multivariable dynamic system model.
NASA Astrophysics Data System (ADS)
Gardas, Bartłomiej
2011-04-01
In this paper we revisit the problem of decoherence by applying the block operator matrices analysis. The Riccati algebraic equation associated with the Hamiltonian describing the process of decoherence is studied. We prove that if the environment responsible for decoherence is invariant with respect to the antilinear transformation then the antilinear operator solves the Riccati equation in question. We also argue that this solution leads to neither a linear nor an antilinear operator similarity matrix. Therefore, we cannot use the standard procedure for solving a linear differential equation (e.g., Schrödinger equation). Furthermore, the explicit solution of the Riccati equation is found for the case where the environmental operators commute with each other. We discuss the connection between our results and the standard description of decoherence (one that uses the Kraus representation). We show that the reduced dynamics we obtain does not have the Kraus representation if the initial correlations between the system and its environment are present. However, for any initial state of the system (even when the correlations occur) reduced dynamics can be written in a manageable way.
Integrability of Lie Systems Through Riccati Equations
NASA Astrophysics Data System (ADS)
Cariñena, José F.; de Lucas, Javier
Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method to construct integrability conditions for Riccati equations following these approaches. This approach provides us with a unified geometrical viewpoint that allows us to analyse some previous works on the topic and explain new properties. Moreover, this new approach can be straightforwardly generalised to describe integrability conditions for any Lie system. Finally, we show the usefulness of our treatment in order to study the problem of the linearisability of Riccati equations.
NASA Astrophysics Data System (ADS)
Nishiyama, Seiya; da Providência, João
2015-02-01
In this paper we present the induced representation of SO(2N) canonical transformation group and introduce (SO(2N))/(U(N)) coset variables. We give a derivation of the time-dependent Hartree-Bogoliubov (TDHB) equation on the Kähler coset space (G)/(H) = (SO(2N))/(U(N)) from the Euler-Lagrange equation of motion for the coset variables. The TDHB wave function represents the TD behavior of Bose condensate of fermion pairs. It is a good approximation for the ground state of the fermion system with a pairing interaction, producing the spontaneous Bose condensation. To describe the classical motion on the coset manifold, we start from the local equation of motion. This equation becomes a Riccati-type equation. After giving a simple two-level model and a solution for a coset variable, we can get successfully a general solution of time-dependent Riccati-Hartree-Bogoliubov equation for the coset variables. We obtain the Harish-Chandra decomposition for the SO(2N) matrix based on the nonlinear Möbius transformation together with the geodesic flow on the manifold.
On Hilbert-Schmidt norm convergence of Galerkin approximation for operator Riccati equations
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1988-01-01
An abstract approximation framework for the solution of operator algebraic Riccati equations is developed. The approach taken is based on a formulation of the Riccati equation as an abstract nonlinear operator equation on the space of Hilbert-Schmidt operators. Hilbert-Schmidt norm convergence of solutions to generic finite dimensional Galerkin approximations to the Riccati equation to the solution of the original infinite dimensional problem is argued. The application of the general theory is illustrated via an operator Riccati equation arising in the linear-quadratic design of an optimal feedback control law for a 1-D heat/diffusion equation. Numerical results demonstrating the convergence of the associated Hilbert-Schmidt kernels are included.
Riccati equations for holographic 2-point functions
NASA Astrophysics Data System (ADS)
Papadimitriou, Ioannis; Taliotis, Anastasios
2014-04-01
Any second order homogeneous linear ordinary differential equation can be transformed into a first order non-linear Riccati equation. We argue that the Riccati form of the linearized fluctuation equations that determine the holographic 2-point functions simplifies considerably the numerical computation of such 2-point functions and of the corresponding transport coefficients by computing directly the response functions, eliminating the arbitrary source from the start. Moreover, it provides a neat criterion for the infrared regularity of the fluctuations. In particular, it is shown that the infrared regularity conditions for scalar and tensor fluctuations coincide, and hence they are either both regular or both singular. We demonstrate our numerical recipe based on the Riccati equations by computing the holographic 2-point functions for the stress tensor and a scalar operator in a number of asymptotically anti de Sitter backgrounds of bottom up scalar-gravity models. Analytical results are obtained for the 2-point function of the transverse traceless part of the stress tensor in two confining geometries, including a geometry that belongs to the class of IHQCD. We find that in this background the spin-2 spectrum is linear and, as expected, the position space 2-point function decays exponentially at large distances at a rate proportional to the confinement scale.
Riccati equation and the problem of decoherence
NASA Astrophysics Data System (ADS)
Gardas, Bartłomiej
2010-06-01
The block operator matrix theory is used to investigate the problem of a single qubit. We establish a connection between the Riccati operator equation and the possibility of obtaining an exact reduced dynamics for the qubit in question. The model of the half spin particle in the rotating magnetic field coupling with the external environment is discussed. We show that the model defined in such a way can be reduced to a time independent problem.
NASA Astrophysics Data System (ADS)
Dragan, Vasile; Ivanov, Ivan
2011-04-01
In this article, the problem of the numerical computation of the stabilising solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H ∞ control problem for a class of stochastic systems affected by state-dependent and control-dependent white noise and subjected to Markovian jumping. The stabilising solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilising solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The proposed algorithm extends to this general framework the method proposed in Lanzon, Feng, Anderson, and Rotkowitz (Lanzon, A., Feng, Y., Anderson, B.D.O., and Rotkowitz, M. (2008), 'Computing the Positive Stabilizing Solution to Algebraic Riccati Equations with an Indefinite Quadratic Term Viaa Recursive Method,' IEEE Transactions on Automatic Control, 53, pp. 2280-2291). In the proof of the convergence of the proposed algorithm different concepts associated the generalised Lyapunov operators as stability, stabilisability and detectability are widely involved. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.
Notes on the Riccati operator equation in open quantum systems
NASA Astrophysics Data System (ADS)
Gardas, Bartłomiej; Puchała, Zbigniew
2012-01-01
A recent problem [B. Gardas, J. Math. Phys. 52, 042104 (2011)] concerning an antilinear solution of the Riccati equation is solved. We also exemplify that a simplification of the Riccati equation, even under reasonable assumptions, can lead to a not equivalent equation.
Riccati equations for bounded radiating systems
NASA Astrophysics Data System (ADS)
Maharaj, S. D.; Tiwari, A. K.; Mohanlal, R.; Narain, R.
2016-09-01
We systematically analyze the nonlinear partial differential equation that determines the behaviour of a bounded radiating spherical mass in general relativity. Four categories of solution are possible. These are identified in terms of restrictions on the gravitational potentials. One category of solution can be related to the horizon function transformation which was recently introduced. A Lie symmetry analysis of the resulting Riccati equation shows that several new classes of exact solutions are possible. The relationship between the horizon function, Euclidean star models, and other earlier investigations is clarified.
Wavelet operational matrix method for solving the Riccati differential equation
NASA Astrophysics Data System (ADS)
Li, Yuanlu; Sun, Ning; Zheng, Bochao; Wang, Qi; Zhang, Yingchao
2014-03-01
A Haar wavelet operational matrix method (HWOMM) was derived to solve the Riccati differential equations. As a result, the computation of the nonlinear term was simplified by using the Block pulse function to expand the Haar wavelet one. The proposed method can be used to solve not only the classical Riccati differential equations but also the fractional ones. The capability and the simplicity of the proposed method was demonstrated by some examples and comparison with other methods.
a Multiple Riccati Equations Rational-Exponent Method and its Application to Whitham-Broer Equation
NASA Astrophysics Data System (ADS)
Liu, Qing; Wang, Zi-Hua; Jia, Dong-Li
2013-03-01
According to two dependent solutions to a generalized Riccati equation together with the equation itself, a multiple Riccati equations rational-exponent method is proposed and applied to Whitham-Broer-Kaup equation. It shows that this method is a more concise and efficient approach and can uniformly derive many types of combined solutions to nonlinear partial differential equations.
A neuro approach to solve fuzzy Riccati differential equations
Shahrir, Mohammad Shazri; Kumaresan, N. Kamali, M. Z. M.; Ratnavelu, Kurunathan
2015-10-22
There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.
A neuro approach to solve fuzzy Riccati differential equations
NASA Astrophysics Data System (ADS)
Shahrir, Mohammad Shazri; Kumaresan, N.; Kamali, M. Z. M.; Ratnavelu, Kurunathan
2015-10-01
There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.
Algebraic methods for the solution of some linear matrix equations
NASA Technical Reports Server (NTRS)
Djaferis, T. E.; Mitter, S. K.
1979-01-01
The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.
Liouvillian propagators, Riccati equation and differential Galois theory
NASA Astrophysics Data System (ADS)
Acosta-Humánez, Primitivo; Suazo, Erwin
2013-11-01
In this paper a Galoisian approach to building propagators through Riccati equations is presented. The main result corresponds to the relationship between the Galois integrability of the linear Schrödinger equation and the virtual solvability of the differential Galois group of its associated characteristic equation. As the main application of this approach we solve Ince’s differential equation through the Hamiltonian algebrization procedure and the Kovacic algorithm to find the propagator for a generalized harmonic oscillator. This propagator has applications which describe the process of degenerate parametric amplification in quantum optics and light propagation in a nonlinear anisotropic waveguide. Toy models of propagators inspired by integrable Riccati equations and integrable characteristic equations are also presented.
Relations between nonlinear Riccati equations and other equations in fundamental physics
NASA Astrophysics Data System (ADS)
Schuch, Dieter
2014-10-01
Many phenomena in the observable macroscopic world obey nonlinear evolution equations while the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. Linearizing nonlinear dynamics would destroy the fundamental property of this theory, however, it can be shown that quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown that the information about the dynamics of quantum systems with analytical solutions can not only be obtainable from the time-dependent Schrödinger equation but equally-well from a complex Riccati equation. Comparison with supersymmetric quantum mechanics shows that even additional information can be obtained from the nonlinear formulation. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation for any potential. Extension of the Riccati formulation to include irreversible dissipative effects is straightforward. Via (real and complex) Riccati equations, other fields of physics can also be treated within the same formalism, e.g., statistical thermodynamics, nonlinear dynamical systems like those obeying a logistic equation as well as wave equations in classical optics, Bose- Einstein condensates and cosmological models. Finally, the link to abstract "quantizations" such as the Pythagorean triples and Riccati equations connected with trigonometric and hyperbolic functions will be shown.
Computational complexities and storage requirements of some Riccati equation solvers
NASA Technical Reports Server (NTRS)
Utku, Senol; Garba, John A.; Ramesh, A. V.
1989-01-01
The linear optimal control problem of an nth-order time-invariant dynamic system with a quadratic performance functional is usually solved by the Hamilton-Jacobi approach. This leads to the solution of the differential matrix Riccati equation with a terminal condition. The bulk of the computation for the optimal control problem is related to the solution of this equation. There are various algorithms in the literature for solving the matrix Riccati equation. However, computational complexities and storage requirements as a function of numbers of state variables, control variables, and sensors are not available for all these algorithms. In this work, the computational complexities and storage requirements for some of these algorithms are given. These expressions show the immensity of the computational requirements of the algorithms in solving the Riccati equation for large-order systems such as the control of highly flexible space structures. The expressions are also needed to compute the speedup and efficiency of any implementation of these algorithms on concurrent machines.
A Riccati equation based approach to isotropic scalar field cosmologies
NASA Astrophysics Data System (ADS)
Harko, Tiberiu; Lobo, Francisco S. N.; Mak, M. K.
2014-05-01
Gravitationally coupled scalar fields ϕ, distinguished by the choice of an effective self-interaction potential V(ϕ), simulating a temporarily nonvanishing cosmological term, can generate both inflation and late time acceleration. In scalar field cosmological models the evolution of the Hubble function is determined, in terms of the interaction potential, by a Riccati type equation. In the present work, we investigate scalar field cosmological models that can be obtained as solutions of the Riccati evolution equation for the Hubble function. Four exact integrability cases of the field equations are presented, representing classes of general solutions of the Riccati evolution equation. The solutions correspond to cosmological models in which the Hubble function is proportional to the scalar field potential plus a linearly decreasing function of time, models with the time variation of the scalar field potential proportional to the potential minus its square, models in which the potential is the sum of an arbitrary function and the square of the function integral, and models in which the potential is the sum of an arbitrary function and the derivative of its square root, respectively. The cosmological properties of all models are investigated in detail, and it is shown that they can describe the inflationary or the late accelerating phase in the evolution of the universe.
The number of polynomial solutions of polynomial Riccati equations
NASA Astrophysics Data System (ADS)
Gasull, Armengol; Torregrosa, Joan; Zhang, Xiang
2016-11-01
Consider real or complex polynomial Riccati differential equations a (x) y ˙ =b0 (x) +b1 (x) y +b2 (x)y2 with all the involved functions being polynomials of degree at most η. We prove that the maximum number of polynomial solutions is η + 1 (resp. 2) when η ≥ 1 (resp. η = 0) and that these bounds are sharp. For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most η ≥ 1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2η (resp. 3) when η ≥ 2 (resp. η = 1) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain.
Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation
NASA Astrophysics Data System (ADS)
Naher, Hasibun; Abdullah, Farah Aini; Mohyud-Din, Syed Tauseef
2013-05-01
In this article, the generalized Riccati equation mapping together with the basic (G'/G)-expansion method is implemented which is advance mathematical tool to investigate nonlinear partial differential equations. Moreover, the auxiliary equation G'(ϕ) = h + f G(ϕ) + g G2(ϕ) is used with arbitrary constant coefficients and called the generalized Riccati equation. By applying this method, we have constructed abundant traveling wave solutions in a uniform way for the Sawada-Kotera equation. The obtained solutions of this equation have vital and noteworthy explanations for some practical physical phenomena.
Newton's laws of motion in the form of a Riccati equation.
Nowakowski, Marek; Rosu, Haret C
2002-04-01
We discuss two applications of a Riccati equation to Newton's laws of motion. The first one is the motion of a particle under the influence of a power law central potential V(r)=kr(epsilon). For zero total energy we show that the equation of motion can be cast in the Riccati form. We briefly show here an analogy to barotropic Friedmann-Robertson-Lemaitre cosmology where the expansion of the universe can be also shown to obey a Riccati equation. A second application in classical mechanics, where again the Riccati equation appears naturally, are problems involving quadratic friction. We use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problems.
NASA Astrophysics Data System (ADS)
Feng, Qing-Hua
2014-08-01
In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space-time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained.
An Exact Mapping from Navier-Stokes Equation to Schr"odinger Equation via Riccati Equation
NASA Astrophysics Data System (ADS)
Christianto, Vic; Smarandache, Florentin
2010-03-01
In the present article we argue that it is possible to write down Schr"odinger representation of Navier-Stokes equation via Riccati equation. The proposed approach, while differs appreciably from other method such as what is proposed by R. M. Kiehn, has an advantage, i.e. it enables us extend further to quaternionic and biquaternionic version of Navier-Stokes equation, for instance via Kravchenko's and Gibbon's route. Further observation is of course recommended in order to refute or verify this proposition.
Generalized creation and annihilation operators via complex nonlinear Riccati equations
NASA Astrophysics Data System (ADS)
Schuch, Dieter; Castaños, Octavio; Rosas-Ortiz, Oscar
2013-06-01
Based on Gaussian wave packet solutions of the time-dependent Schrödinger equation, a generalization of the conventional creation and annihilation operators and the corresponding coherent states can be obtained. This generalization includes systems where also the width of the coherent states is time-dependent as they occur for harmonic oscillators with time-dependent frequency or systems in contact with a dissipative environment. The key point is the replacement of the frequency ω0 that occurs in the usual definition of the creation/annihilation operator by a complex time-dependent function that fulfils a nonlinear Riccati equation. This equation and its solutions depend on the system under consideration and on the (complex) initial conditions. Formal similarities also exist with supersymmetric quantum mechanics. The generalized creation and annihilation operators also allow to construct exact analytic solutions of the free motion Schrödinger equation in terms of Hermite polynomials with time-dependent variable.
The Rational Solutions to a Generalized Riccati Equation and Their Application
NASA Astrophysics Data System (ADS)
Liu, Qing; Shen, Shi-Yao; Wang, Zi-Hua
2013-02-01
Based on the rational solutions to a generalized Riccati equation, a new method which is called as rational function method is proposed. We apply this method to solve the coupled mKdV equations and derive a set of rational solutions. This method is also available for seeking their solutions to other NLPDEs. It shows that the rational function method is more universal and powerful than the auxiliary Riccati equation method.
Riccati equation for simulation of leads in quantum transport
NASA Astrophysics Data System (ADS)
Bravi, M.; Farchioni, R.; Grosso, G.; Pastori Parravicini, G.
2014-10-01
We present a theoretical procedure with numerical demonstration of a workable and efficient method to evaluate the surface Green's function of semi-infinite leads connected to a device. Such a problem always occurs in quantum transport calculations but also in the study of surfaces and heterojunctions. We show here that these semi-infinite leads can be properly described by real-energy Green's functions obtained analytically by a smart solution of the Riccati matrix equation. The performance of our method is demonstrated in the case of a multichain two-dimensional electron-gas system, composed of a central ribbon connected to two semi-infinite leads, pierced by two opposite magnetic fields.
NASA Technical Reports Server (NTRS)
Garzia, M. R.; Loparo, K. A.; Martin, C. F.
1982-01-01
This paper looks at the structure of the solution of a matrix Riccati differential equation under a predefined group of transformations. The group of transformations used is an expanded form of the feedback group. It is shown that this group of transformations is a subgroup of the symplectic group. The orbits of the Riccati differential equation under the action of this group are studied and it is seen how these techniques apply to a decentralized optimal control problem.
NASA Astrophysics Data System (ADS)
Schuch, Dieter
2014-04-01
Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. I claim that linearity in quantum mechanics is not as essential as it apparently seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated and compared with similar space-dependent Riccati equations in supersymmetric quantum mechanics. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, like statistical thermodynamics and cosmology.
Error Estimates for Approximate Solutions of the Riccati Equation with Real or Complex Potentials
NASA Astrophysics Data System (ADS)
Finster, Felix; Smoller, Joel
2010-09-01
A method is presented for obtaining rigorous error estimates for approximate solutions of the Riccati equation, with real or complex potentials. Our main tool is to derive invariant region estimates for complex solutions of the Riccati equation. We explain the general strategy for applying these estimates and illustrate the method in typical examples, where the approximate solutions are obtained by gluing together WKB and Airy solutions of corresponding one-dimensional Schrödinger equations. Our method is motivated by, and has applications to, the analysis of linear wave equations in the geometry of a rotating black hole.
Hopf algebras and Dyson-Schwinger equations
NASA Astrophysics Data System (ADS)
Weinzierl, Stefan
2016-06-01
In this paper I discuss Hopf algebras and Dyson-Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.
Matrix Riccati equation formulation for radiative transfer in a plane-parallel geometry
NASA Astrophysics Data System (ADS)
Chang, Hung-Wen; Wu, Tso-Lun
1997-01-01
In this paper, we formulate the radiative transfer problem as an initial value problem via a pair of nonlinear matrix differential equations (matrix Riccati equations or MREs) which describe the reflection ( R) and transmission ( T) matrices of the specific intensities in a plane-parallel geometry. One first computes R and T matrices of some small but finite layer thickness (equivalent optical thickness 0959-7174/7/1/009/img1) and then repetitively applies the doubling method to the reflection and transmission matrices 0959-7174/7/1/009/img2 and 0959-7174/7/1/009/img3 until reaching the desired layer thickness. The initial matrices 0959-7174/7/1/009/img4 and 0959-7174/7/1/009/img5 can be computed quite accurately by either of the following methods: multiple-order, multiple-scattering approximation, iterative method or fourth-order Runge - Kutta techniques. In addition, the reflection coefficient matrix of a semi-infinite medium satisfies an algebraic matrix equation which can be solved repetitively by a matrix method. MREs offer an alternative way of solving plane-parallel radiative transport problems. This method requires only elementary matrix operations (addition, multiplication and inversion). For vector and/or beam-wave radiative transfer problems, large matrices are required to describe the physics adequately, and the MRE method provides a significant reduction in computer memory and computation time.
Semi analytical solution of second order fuzzy Riccati equation by homotopy perturbation method
NASA Astrophysics Data System (ADS)
Jameel, A. F.; Ismail, Ahmad Izani Md
2014-07-01
In this work, the Homotopy Perturbation Method (HPM) is formulated to find a semi-analytical solution of the Fuzzy Initial Value Problem (FIVP) involving nonlinear second order Riccati equation. This method is based upon homotopy perturbation theory. This method allows for the solution of the differential equation to be calculated in the form of an infinite series in which the components can be easily calculated. The effectiveness of the algorithm is demonstrated by solving nonlinear second order fuzzy Riccati equation. The results indicate that the method is very effective and simple to apply.
NASA Astrophysics Data System (ADS)
Lu, Bin
2012-06-01
In this Letter, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Bäcklund transformation of fractional Riccati equation are employed for constructing the exact solutions of nonlinear fractional partial differential equations. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations.
First integrals for time-dependent higher-order Riccati equations by nonholonomic transformation
NASA Astrophysics Data System (ADS)
Guha, Partha; Ghose Choudhury, A.; Khanra, Barun
2011-08-01
We exploit the notion of nonholonomic transformations to deduce a time-dependent first integral for a (generalized) second-order nonautonomous Riccati differential equation. It is further shown that the method can also be used to compute the first integrals of a particular class of third-order time-dependent ordinary differential equations and is therefore quite robust.
New conditions for obtaining the exact solutions of the general Riccati equation.
Bougoffa, Lazhar
2014-01-01
We propose a direct method for solving the general Riccati equation y' = f(x) + g(x)y + h(x)y(2). We first reduce it into an equivalent equation, and then we formulate the relations between the coefficients functions f(x), g(x), and h(x) of the equation to obtain an equivalent separable equation from which the previous equation can be solved in closed form. Several examples are presented to demonstrate the efficiency of this method.
SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations
NASA Astrophysics Data System (ADS)
Gusev, Sergei V.; Shiriaev, Anton S.; Freidovich, Leonid B.
2016-07-01
Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.
NASA Astrophysics Data System (ADS)
Kengne, Emmanuel; Saydé, Michel; Ben Hamouda, Fathi; Lakhssassi, Ahmed
2013-11-01
Analytical entire traveling wave solutions to the 1+1 density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method are presented in this paper. This equation can be regarded as an extension case of the Fisher-Kolmogoroff equation, which is used for studying insect and animal dispersal with growth dynamics. The analytical solutions are then used to investigate the effect of equation parameters on the population distribution.
NASA Astrophysics Data System (ADS)
Bota, C.; Cǎruntu, B.; Bundǎu, O.
2013-10-01
In this paper we applied the Squared Remainder Minimization Method (SRMM) to find analytic approximate polynomial solutions for Riccati differential equations. Two examples are included to demonstrated the validity and applicability of the method. The results are compared to those obtained by other methods.
Solving Absolute Value Equations Algebraically and Geometrically
ERIC Educational Resources Information Center
Shiyuan, Wei
2005-01-01
The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.
NASA Astrophysics Data System (ADS)
Schuch, Dieter
2015-06-01
It is shown that a nonlinear reformulation of time-dependent and time-independent quantum mechanics in terms of Riccati equations not only provides additional information about the physical system, but also allows for formal comparison with other nonlinear theories. This is demonstrated for the nonlinear Burgers and Korteweg-de Vries equations with soliton solutions. As Riccati equations can be linearized to corresponding Schrödinger equations, this also applies to the Riccati equations that can be obtained by integrating the nonlinear soliton equations, resulting in a time-independent Schrödinger equation with Rosen-Morse potential and its supersymmetric partner. Because both soliton equations lead to the same Riccati equation, relations between the Burgers and Korteweg-de Vries equations can be established. Finally, a connection with the inverse scattering method is mentioned.
State-Dependent Riccati Equation Regulation of Systems with State and Control Nonlinearities
NASA Technical Reports Server (NTRS)
Beeler, Scott C.; Cox, David E. (Technical Monitor)
2004-01-01
The state-dependent Riccati equations (SDRE) is the basis of a technique for suboptimal feedback control of a nonlinear quadratic regulator (NQR) problem. It is an extension of the Riccati equation used for feedback control of linear problems, with the addition of nonlinearities in the state dynamics of the system resulting in a state-dependent gain matrix as the solution of the equation. In this paper several variations on the SDRE-based method will be considered for the feedback control problem with control nonlinearities. The control nonlinearities may result in complications in the numerical implementation of the control, which the different versions of the SDRE method must try to overcome. The control methods will be applied to three test problems and their resulting performance analyzed.
Direct solution of the backscatter equation for ABS instruments through a Riccati reformulation
NASA Astrophysics Data System (ADS)
Hamilton, L. J.; Hall, M. V.
2012-09-01
Measurements of suspended sediment concentration profiles in aquatic environments may be made by remote sensing acoustic backscatter (ABS) instruments. These typically operate at MHz frequencies and narrow beamwidths (1-2°). The backscatter equation for these instruments is not in a closed form. It may be solved by iteration. A direct general solution has previously been presented by recasting into the form of a Bernoulli equation [Thosteson, E.D., Hanes, D.M., 1998. Journal of the Acoustical Society of America 104 (2), 820-830 (Pt. 1)]. It is shown that the direct solution is also enabled by recasting as a Riccati equation. The direct solution obtained by the Riccati formulation has the same form as given previously, but its derivation clarifies inconsistencies in previous work which have disguised its potential usefulness.
New Conditions for Obtaining the Exact Solutions of the General Riccati Equation
Bougoffa, Lazhar
2014-01-01
We propose a direct method for solving the general Riccati equation y′ = f(x) + g(x)y + h(x)y2. We first reduce it into an equivalent equation, and then we formulate the relations between the coefficients functions f(x), g(x), and h(x) of the equation to obtain an equivalent separable equation from which the previous equation can be solved in closed form. Several examples are presented to demonstrate the efficiency of this method. PMID:25202724
NASA Astrophysics Data System (ADS)
Xu, Ling; Cheng, Xuan; Dai, Chao-Qing
2015-12-01
Although the mapping method based on Riccati equation was proposed to obtain variable separation solutions many years ago, two important problems have not been studied: i) the equivalence of variable separation solutions by means of the mapping method based on Riccati equation with the radical sign combined ansatz; and ii) lack of physical meanings for some localized structures constructed by variable separation solutions. In this paper, we re-study the (2+1)-dimensional Boiti-Leon-Pempinelli equation via the mapping method based on Riccati equation and prove that nine types of variable separation solutions are actually equivalent to each other. Moreover, we also re-study localized structures constructed by variable separation solutions. Results indicate that some localized structures reported in the literature are lacking real values due to the appearance of the divergent and un-physical phenomenon for the initial field. Therefore, we must be careful with the initial field to avoid the appearance of some un-physical or even divergent structures in it when we construct localized structures for the potential field.
Some Remarks on the Riccati Equation Expansion Method for Variable Separation of Nonlinear Models
NASA Astrophysics Data System (ADS)
Zhang, Yu-Peng; Dai, Chao-Qing
2015-10-01
Based on the Riccati equation expansion method, 11 kinds of variable separation solutions with different forms of (2+1)-dimensional modified Korteweg-de Vries equation are obtained. The following two remarks on the Riccati equation expansion method for variable separation are made: (i) a remark on the equivalence of different solutions constructed by the Riccati equation expansion method. From analysis, we find that these seemly independent solutions with different forms actually depend on each other, and they can transform from one to another via some relations. We should avoid arbitrarily asserting so-called "new" solutions; (ii) a remark on the construction of localised excitation based on variable separation solutions. For two or multi-component systems, we must be careful with excitation structures constructed by all components for the same model lest the appearance of some un-physical structures. We hope that these results are helpful to deeply study exact solutions of nonlinear models in physical, engineering and biophysical contexts.
NASA Astrophysics Data System (ADS)
Feng, Qing-Hua
2013-05-01
In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann—Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.
NASA Astrophysics Data System (ADS)
Huang, Qing; Wang, Li-Zhen; Zuo, Su-Li
2016-02-01
In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie's modified Riemann-Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada-Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed. Supported by the National Natural Science Foundation of China under Grant Nos. 11101332, 11201371, 11371293 and the Natural Science Foundation of Shaanxi Province under Grant No. 2015JM1037
Nonlocal Symmetry of the Lax Equation Related to Riccati-Type Pseudopotential
NASA Astrophysics Data System (ADS)
Wang, Yun-Hu; Chen, Yong; Xin, Xiang-Peng
2012-12-01
We investigate the Lax equation that can be employed to describe motions of long waves in shallow water under gravity. A nonlocal symmetry of this equation is given and used to find exact solutions and derive lower integrable models from higher ones. It is interesting that this nonlocal symmetry links with its corresponding Riccati-type pseudopotential. By introducing suitable and simple auxiliary dependent variables, the nonlocal symmetry is localized and used to generate new solutions from trivial solutions. Meanwhile, this equation is reduced to an ordinary differential equation by means of this nonlocal symmetry and some local symmetries.
Islam, Md Shafiqul; Khan, Kamruzzaman; Akbar, M Ali; Mastroberardino, Antonio
2014-10-01
The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin-Bona-Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.
Islam, Md. Shafiqul; Khan, Kamruzzaman; Akbar, M. Ali; Mastroberardino, Antonio
2014-01-01
The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering. PMID:26064530
Integration-free interval doubling for Riccati equation solutions
NASA Technical Reports Server (NTRS)
Bierman, G. J.; Sidhu, G. S.
1976-01-01
Various algorithms are given for the case of constant coefficients. The algorithms are based on two ideas: first, relate the Re solution with general initial conditions to anchored RE solutions; and second, when the coefficients are constant the anchored solutions have a basic shift-invariance property. These ideas are used to construct an integration free superlinearly convergent iterative solution to the algebraic RE. The algorithm, arranged in square-root form, is thought to be numerically stable and competitive with other methods of solving the algebraic RE.
On a Equation in Finite Algebraically Structures
ERIC Educational Resources Information Center
Valcan, Dumitru
2013-01-01
Solving equations in finite algebraically structures (semigroups with identity, groups, rings or fields) many times is not easy. Even the professionals can have trouble in such cases. Therefore, in this paper we proposed to solve in the various finite groups or fields, a binomial equation of the form (1). We specify that this equation has been…
NASA Astrophysics Data System (ADS)
Guha, Partha; Ghose Choudhury, A.; Khanra, Barun
2012-11-01
We introduce a new transformation (nonlocal) to find the general solutions of some equations belonging to the third and fourth-order time dependent Riccati class of equations. These are in turn related to the Chazy polynomial class and the time dependent F-XVI Bureau symbol PI equations respectively.
Lie algebras and linear differential equations.
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Systems and Methods for Parameter Dependent Riccati Equation Approaches to Adaptive Control
NASA Technical Reports Server (NTRS)
Kim, Kilsoo (Inventor); Yucelen, Tansel (Inventor); Calise, Anthony J. (Inventor)
2015-01-01
Systems and methods for adaptive control are disclosed. The systems and methods can control uncertain dynamic systems. The control system can comprise a controller that employs a parameter dependent Riccati equation. The controller can produce a response that causes the state of the system to remain bounded. The control system can control both minimum phase and non-minimum phase systems. The control system can augment an existing, non-adaptive control design without modifying the gains employed in that design. The control system can also avoid the use of high gains in both the observer design and the adaptive control law.
Time-evolution of quantum systems via a complex nonlinear Riccati equation. II. Dissipative systems
NASA Astrophysics Data System (ADS)
Cruz, Hans; Schuch, Dieter; Castaños, Octavio; Rosas-Ortiz, Oscar
2016-10-01
In our former contribution (Cruz et al., 2015), we have shown the sensitivity to the choice of initial conditions in the evolution of Gaussian wave packets via the nonlinear Riccati equation. The formalism developed in the previous work is extended to effective approaches for the description of dissipative quantum systems. By means of simple examples we show the effects of the environment on the quantum uncertainties, correlation function, quantum energy contribution and tunnelling currents. We prove that the environmental parameter γ is strongly related with the sensitivity to the choice of initial conditions.
High tip angle approximation based on a modified Bloch-Riccati equation.
Boulant, Nicolas; Hoult, David I
2012-02-01
When designing a radio-frequency pulse to produce a desired dependence of magnetization on frequency or position, the small flip angle approximation is often used as a first step, and a Fourier relation between pulse and transverse magnetization is then invoked. However, common intuition often leads to linear scaling of the resulting pulse so as to produce a larger flip angle than the approximation warrants--with surprisingly good results. Starting from a modified version of the Bloch-Riccati equation, a differential equation in the flip angle itself, rather than in magnetization, is derived. As this equation has a substantial linear component that is an instance of Fourier's equation, the intuitive approach is seen to be justified. Examples of the accuracy of this higher tip angle approximation are given for both constant- and variable-phase pulses.
NASA Astrophysics Data System (ADS)
Vitanov, Nikolay K.
2010-08-01
We search for traveling-wave solutions of the class of equations ∑p=1N1αp{∂pQ}/{∂tp}+∑q=1N2βq{∂qQ}/{∂xq}+∑m=1MμmQm=0 where αp,βq and μm are parameters. We obtain such solutions by the method of simplest equation for the cases when the simplest equation is the the equation of Bernoulli or the equation of Riccati. We modify the methodology of the simplest equation of Kudryashov as follows. Kudryashov uses the first step of the test for Painleve property in order to determine the size of the solution of the studied PDE. We divide the studied PDEs in two parts: part A which contains the derivatives, and part B which contains the rest of the equation. The application of the ansatz or the extended ansatz of Kudryashov, transforms part A and part B to two polynomials. We balance the highest powers of the polynomials for the parts A and B and, thus obtain a balance equation which depends on the kind of the used simplest equation. The balance equations are connected to nonlinear algebraic systems of relationships among the parameters of the equations and the parameters of the solution. On the basis of these systems, we obtain numerous solutions of the studied class of equations.
NASA Astrophysics Data System (ADS)
Castaños, Octavio; Schuch, Dieter; Rosas-Ortiz, Oscar
2013-02-01
Based on the Gaussian wave packet solution for the harmonic oscillator and the corresponding creation and annihilation operators, a generalization is presented that also applies for wave packets with time-dependent width as they occur for systems with different initial conditions, time-dependent frequency or in contact with a dissipative environment. In all these cases, the corresponding coherent states, position and momentum uncertainties and quantum mechanical energy contributions can be obtained in the same form if the creation and annihilation operators are expressed in terms of a complex variable that fulfils a nonlinear Riccati equation which determines the time-evolution of the wave packet width. The solutions of this Riccati equation depend on the physical system under consideration and on the (complex) initial conditions and have close formal similarities with general superpotentials leading to isospectral potentials in supersymmetric quantum mechanics. The definition of the generalized creation and annihilation operator is also in agreement with a factorization of the operator corresponding to the Ermakov invariant that exists in all cases considered.
Optical systolic solutions of linear algebraic equations
NASA Technical Reports Server (NTRS)
Neuman, C. P.; Casasent, D.
1984-01-01
The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.
Cruz, Hans; Schuch, Dieter; Castaños, Octavio; Rosas-Ortiz, Oscar
2015-09-15
The sensitivity of the evolution of quantum uncertainties to the choice of the initial conditions is shown via a complex nonlinear Riccati equation leading to a reformulation of quantum dynamics. This sensitivity is demonstrated for systems with exact analytic solutions with the form of Gaussian wave packets. In particular, one-dimensional conservative systems with at most quadratic Hamiltonians are studied.
On the stability of a differential Riccati equation for continuous-discrete observers
NASA Astrophysics Data System (ADS)
Boizot, Nicolas; Busvelle, Eric
2016-02-01
This article deals with the stability properties of a continuous-discrete Riccati differential equation. The main motivation comes from the theory of Kalman filters for continuous-time nonlinear systems with sampled measurements, where this type of equation often arises. Stability is to be understood in the following sense: under appropriate hypotheses, the solution matrix is always symmetric positive definite and bounded from above and below. No uniformity is expected from the sampling procedure, only an upper bound on the elapsed time between consecutive samples is needed. The exposition consists of two parts. First, the stability properties are proven and second, by means of three examples, it is shown how the article's main result applies.
NASA Astrophysics Data System (ADS)
Yang, Yun-Qing; Wang, Yun-Hu; Li, Xin; Cheng, Xue-Ping
2014-03-01
We extend the method of constructing Bäcklund transformations for integrable equations through Riccati equations to the nonisospectral and the variable-coefficient equations. By taking nonisospectral and generalized variable-coefficient Korteweg—de Vries (KdV) equations as examples, their Bäcklund transformations are obtained under a more generalized constrain condition. In addition, the Lax pairs and infinite numbers of conservation laws of these equations are given. Especially, some classical equations such as the cylindrical KdV equation are just the special cases of the constrain condition.
On the generalization of statistical thermodynamic functions by a Riccati differential equation.
NASA Astrophysics Data System (ADS)
Peña, J. J.; Rubio-Ponce, A.; Morales, J.
2016-08-01
In this work, we propose a non-linear differential equation of Riccati-type, where the standard partition function Z(T) is taken as its particular solution leading to their generalization Zg(T); from there, other related statistical thermodynamic functions are generalized. As an useful application of our proposal, other thermodynamic functions, namely, the internal energy, heat capacity, Helmholtz free energy and entropy, associated to the model of the ideal monatomic gas in D-dimensions are generalized. According to our results, thermodynamic properties derived from the standard partition functions by means of ordinary statistical mechanics are incomplete. In fact, although asymptotically with the increasing of temperature the generalized statistical thermodynamic functions reduce to the standard ones, these contain an extra term which is dominant at very low temperature indicating that standard findings should be corrected.
Nonlinear closed loop optimal control: a modified state-dependent Riccati equation.
Rafee Nekoo, S
2013-03-01
The state-dependent Riccati equation (SDRE), as a controller, has been introduced and implemented since the 90s. In this article, the other aspects of this controller are declared which shows the capability of this technique. First, a general case which has control nonlinearities and time varying weighting matrix Q is solved with three approaches: exact solution (ES), online control update (OCU) and power series approximation (PSA). The proposed PSA in this paper is able to deal with time varying or state-dependent Q in nonlinear systems. As a result of having the solution of nonlinear systems with complex Q containing constraints, using OCU and proposed PSA, a method is introduced to prevent the collision of an end-effector of a robot and an obstacle which shows the adaptability of the SDRE controller. Two examples to support the idea are presented and conferred. Supplementing constraints to the SDRE via matrix Q, this approach is named a modified SDRE.
Laurent phenomenon algebras and the discrete BKP equation
NASA Astrophysics Data System (ADS)
Okubo, Naoto
2016-09-01
We construct the Laurent phenomenon algebras the cluster variables of which satisfy the discrete BKP equation, the discrete Sawada-Kotera equation and other difference equations obtained by its reduction. These Laurent phenomenon algebras are constructed from seeds with a generalization of mutation-period property. We show that a reduction of a seed corresponds to a reduction of a difference equation.
Laurent phenomenon algebras and the discrete BKP equation
NASA Astrophysics Data System (ADS)
Okubo, Naoto
2016-09-01
We construct the Laurent phenomenon algebras the cluster variables of which satisfy the discrete BKP equation, the discrete Sawada–Kotera equation and other difference equations obtained by its reduction. These Laurent phenomenon algebras are constructed from seeds with a generalization of mutation-period property. We show that a reduction of a seed corresponds to a reduction of a difference equation.
NASA Astrophysics Data System (ADS)
Zhang, Wei; Su, Housheng; Wang, Hongwei; Han, Zhengzhi
2012-12-01
This paper aims to design full-order and reduced-order observers for one-sided Lipschitz nonlinear systems. The system under consideration is an extension of its known Lipschitz counterpart and possesses inherent advantages with respect to conservativeness. For such system, we first develop a novel Riccati equation approach to design a full-order observer, for which rigorous mathematical analysis is performed. Consequently, we show that the conditions under which a full-order observer exists also guarantee the existence of a reduced-order observer. A design method for the reduced-order observer that is dependent on the solution of the Riccati equation is then presented. The proposed conditions are easily and numerically tractable via standard numerical software. Furthermore, it is theoretically proven that the obtained conditions are less conservative than some existing ones in recent literature. The effectiveness of the proposed observers is illustrated via a simulative example.
Algebraic approach to solve tt dilepton equations
Sonnenschein, Lars
2005-11-01
The set of nonlinear equations describing the standard model kinematics of the top quark antiquark production system in the dilepton decay channel has at most a fourfold ambiguity due to two not fully reconstructed neutrinos. Its most precise solution is of major importance for measurements of top quark properties like the top quark mass and tt spin correlations. Simple algebraic operations allow one to transform the nonlinear equations into a system of two polynomial equations with two unknowns. These two polynomials of multidegree eight can in turn be analytically reduced to one polynomial with one unknown by means of resultants. The obtained univariate polynomial is of degree 16. The number of its real solutions is determined analytically by means of Sturm's theorem, which is as well used to isolate each real solution into a unique pairwise disjoint interval. The solutions are polished by seeking the sign change of the polynomial in a given interval through binary bracketing.
Numerical solutions of matrix Riccati equations for radiative transfer in a plane-parallel geometry
NASA Astrophysics Data System (ADS)
Chang, Hung-Wen; Wu, Tso-Lun
1997-01-01
In this paper, we conduct numerical experiments with matrix Riccati equations (MREs) which describe the reflection ( R) and transmission ( T) matrices of the specific intensities in a layer containing randomly distributed scattering particles. The theoretical formulation of MREs is discussed in our previous paper where we show that R and T for a thick layer can be efficiently computed by successively doubling R and T matrices for a thin layer (with small optical thickness 0959-7174/7/1/010/img1). We can compute 0959-7174/7/1/010/img2 and 0959-7174/7/1/010/img3 very accurately using either a fourth-order Runge - Kutta scheme or the fourth-order iterative solution. The differences between these results and those computed by the eigenmode expansion technique (EMET) are very small (< 0.1%). Although the MRE formulation cannot be extended to handle the inhomogeneous term (source term) in the differential equation, we show that the force term can be reformulated as an equivalent boundary condition which is consistent with MRE methods. MRE methods offer an alternative way of solving plane-parallel radiative transport problems. For large problems that do not fit into computer memory, the MRE method provides a significant reduction in computer memory and computational time.
On the properties of a variant of the Riccati system of equations
NASA Astrophysics Data System (ADS)
Sarkar, Amartya; Guha, Partha; Ghose-Choudhury, Anindya; Bhattacharjee, J. K.; Mallik, A. K.; Leach, P. G. L.
2012-10-01
A variant of the generalized Riccati system of equations, \\ddot{x} + \\alpha \\dot{x} x^{2n+1} + x^{4n+3} = 0, is considered. It is shown that for α = 2n + 3 the system admits a bilagrangian description and the dynamics has a node at the origin, whereas for α much smaller than a critical value the dynamics is periodic, the origin being a centre. It is found that the solution changes from being periodic to aperiodic at a critical point, \\alpha _c = 2\\sqrt{2(n+1)}, which is independent of the initial conditions. This behaviour is explained by finding a scaling argument via which the phase trajectories corresponding to different initial conditions collapse onto a single universal orbit. Numerical evidence for the transition is shown. Further, using a perturbative renormalization group argument, it is conjectured that the oscillator, \\ddot{x} + (2n+3)\\dot{x}x^{2n+1}+x^{4n+3} +\\omega ^2 x = 0, exhibits isochronous oscillations. The correctness of the conjecture is established numerically.
NASA Astrophysics Data System (ADS)
Park, Han-Earl; Kim, Young-Rok
2016-01-01
A relative navigation method for autonomous formation flying using the state-dependent Riccati equation filter (SDREF) is presented. In the SDREF, nonlinear relative dynamics, including J2 perturbation, are parameterized into a state-dependent coefficient (SDC) form without any loss of nonlinearity. The relative navigation algorithm is established based on the carrier-phase differential GPS (CDGPS) and single-frequency GPS data, in which the SDREF is used as a nonlinear estimator. To evaluate the SDREF performance, two different extended Kalman filters (EKFR1 and EKFR2) are introduced. The dynamic models of all the filters are based on relative motion including J2 perturbation. However, the SDREF and the EKFR1 use linear state propagation, whereas EKFR2 employs nonlinear state propagation. The navigation simulation is performed for each filter using live GPS signals simulated by a GPS signal generator, and the result is analyzed in terms of estimation accuracy and computational load. As a result, the SDREF provides a relative navigation solution with 3-D RMS accuracies of 6.0 mm and 0.153 mm/s for position and velocity, respectively, for a separation of 50 km with a computation time of approximately 34 s. The simulation results demonstrate that the SDREF estimates the relative states as rapidly as the EKFR1 and as accurately as the EKFR2, which means that the developed SDREF combines the strong points of EKFR1 and EKFR2 and overcomes their disadvantages.
Korayem, M H; Nekoo, S R
2015-07-01
This work studies an optimal control problem using the state-dependent Riccati equation (SDRE) in differential form to track for time-varying systems with state and control nonlinearities. The trajectory tracking structure provides two nonlinear differential equations: the state-dependent differential Riccati equation (SDDRE) and the feed-forward differential equation. The independence of the governing equations and stability of the controller are proven along the trajectory using the Lyapunov approach. Backward integration (BI) is capable of solving the equations as a numerical solution; however, the forward solution methods require the closed-form solution to fulfill the task. A closed-form solution is introduced for SDDRE, but the feed-forward differential equation has not yet been obtained. Different ways of solving the problem are expressed and analyzed. These include BI, closed-form solution with corrective assumption, approximate solution, and forward integration. Application of the tracking problem is investigated to control robotic manipulators possessing rigid or flexible joints. The intention is to release a general program for automatic implementation of an SDDRE controller for any manipulator that obeys the Denavit-Hartenberg (D-H) principle when only D-H parameters are received as input data.
NASA Astrophysics Data System (ADS)
Lu, Bin
2012-12-01
In this paper, a effective method for searching infinite sequence periodic and solitary wave solutions to nonlinear partial differential equations (NLPDEs) is proposed. A simple transformation technique and division theorem are used to reduce some class of NLPDEs to the Riccati equation, and then the infinite sequence periodic and solitary wave solutions of some class of NLPDEs are constructed by using Bäcklund transformation of Riccati equation and nonlinear superposition principle. As illustrative examples, we obtain the infinite sequence travelling-wave solutions of the three special equations, respectively.
NASA Astrophysics Data System (ADS)
Schuch, Dieter
2012-08-01
Quantum mechanics is essentially described in terms of complex quantities like wave functions. The interesting point is that phase and amplitude of the complex wave function are not independent of each other, but coupled by some kind of conservation law. This coupling exists in time-independent quantum mechanics and has a counterpart in its time-dependent form. It can be traced back to a reformulation of quantum mechanics in terms of nonlinear real Ermakov equations or equivalent complex nonlinear Riccati equations, where the quadratic term in the latter equation explains the origin of the phase-amplitude coupling. Since realistic physical systems are always in contact with some kind of environment this aspect is also taken into account. In this context, different approaches for describing open quantum systems, particularly effective ones, are discussed and compared. Certain kinds of nonlinear modifications of the Schrödinger equation are discussed as well as their interrelations and their relations to linear approaches via non-unitary transformations. The modifications of the aforementioned Ermakov and Riccati equations when environmental effects are included can be determined in the time-dependent case. From formal similarities conclusions can be drawn how the equations of time-independent quantum mechanics can be modified to also incluce the enviromental aspects.
Korayem, M H; Nekoo, S R
2015-01-01
This article investigates finite-time optimal and suboptimal controls for time-varying systems with state and control nonlinearities. The state-dependent Riccati equation (SDRE) controller was the main framework. A finite-time constraint imposed on the equation changes it to a differential equation, known as the state-dependent differential Riccati equation (SDDRE) and this equation was applied to the problem reported in this study that provides general formulation and stability analysis. The following four solution methods were developed for solving the SDDRE; backward integration, state transition matrix (STM) and the Lyapunov based method. In the Lyapunov approach, both positive and negative definite solutions to related SDRE were used to provide suboptimal gain for the SDDRE. Finite-time suboptimal control is applied for robotic manipulator, as finite-time constraint strongly decreases state error and operation time. General state-dependent coefficient (SDC) parameterizations for rigid and flexible joint arms (prismatic or revolute joints) are introduced. By including nonlinear control inputs in the formulation, the actuator׳s limits can be inserted directly to the state-space equation of a manipulator. A finite-time SDRE was implemented on a 6R manipulator both in theory and experimentally. And a reduced 3R arm was modeled and tested as a flexible joint robot (FJR). Evaluations of load carrying capacity and operation time were investigated to assess the capability of this approach, both of which showed significant improvement.
A New Reynolds Stress Algebraic Equation Model
NASA Technical Reports Server (NTRS)
Shih, Tsan-Hsing; Zhu, Jiang; Lumley, John L.
1994-01-01
A general turbulent constitutive relation is directly applied to propose a new Reynolds stress algebraic equation model. In the development of this model, the constraints based on rapid distortion theory and realizability (i.e. the positivity of the normal Reynolds stresses and the Schwarz' inequality between turbulent velocity correlations) are imposed. Model coefficients are calibrated using well-studied basic flows such as homogeneous shear flow and the surface flow in the inertial sublayer. The performance of this model is then tested in complex turbulent flows including the separated flow over a backward-facing step and the flow in a confined jet. The calculation results are encouraging and point to the success of the present model in modeling turbulent flows with complex geometries.
Numerical solution of integral-algebraic equations for multistep methods
NASA Astrophysics Data System (ADS)
Budnikova, O. S.; Bulatov, M. V.
2012-05-01
Systems of Volterra linear integral equations with identically singular matrices in the principal part (called integral-algebraic equations) are examined. Multistep methods for the numerical solution of a selected class of such systems are proposed and justified.
Algebraic Flux Correction II. Compressible Euler Equations
NASA Astrophysics Data System (ADS)
Kuzmin, Dmitri; Möller, Matthias
Algebraic flux correction schemes of TVD and FCT type are extended to systems of hyperbolic conservation laws. The group finite element formulation is employed for the treatment of the compressible Euler equations. An efficient algorithm is proposed for the edge-by-edge matrix assembly. A generalization of Roe's approximate Riemann solver is derived by rendering all off-diagonal matrix blocks positive semi-definite. Another usable low-order method is constructed by adding scalar artificial viscosity proportional to the spectral radius of the cumulative Roe matrix. The limiting of antidiffusive fluxes is performed using a transformation to the characteristic variables or a suitable synchronization of correction factors for the conservative ones. The outer defect correction loop is equipped with a block-diagonal preconditioner so as to decouple the discretized Euler equations and solve them in a segregated fashion. As an alternative, a strongly coupled solution strategy (global BiCGSTAB method with a block-Gauß-Seidel preconditioner) is introduced for applications which call for the use of large time steps. Various algorithmic aspects including the implementation of characteristic boundary conditions are addressed. Simulation results are presented for inviscid flows in a wide range of Mach numbers.
Kaernbach, C; König, P; Schillen, T
1987-02-01
Recent experimental observations of otoacoustic emissions suggest the existence of spontaneous emitters of sound on the basilar membrane. These tend to send off waves not only in the normal direction of propagation. It is therefore significant to study the environmental conditions such an emitter finds inside the cochlea. The impedance relations seen by these emitters are described by the Riccati equation for an inhomogeneous transmission line. The results reported in this paper differ considerably for forward and backward excitation. This reflects the quite different behavior of the cochlea pertaining to waves traveling forward and backward. Because of reflections, backward waves cannot be treated with the Liouville-Green approximation.
Fault Detection in Differential Algebraic Equations
NASA Astrophysics Data System (ADS)
Scott, Jason Roderick
Fault detection and identification (FDI) is important in almost all real systems. Fault detection is the supervision of technical processes aimed at detecting undesired or unpermitted states (faults) and taking appropriate actions to avoid dangerous situations, or to ensure efficiency in a system. This dissertation develops and extends fault detection techniques for systems modeled by differential algebraic equations (DAEs). First, a passive, observer-based approach is developed and linear filters are constructed to identify faults by filtering residual information. The method presented here uses the least squares completion to compute an ordinary differential equation (ODE) that contains the solution of the DAE and applies the observer directly to this ODE. While observers have been applied to ODE models for the purpose of fault detection in the past, the use of observers on completions of DAEs is a new idea. Moreover, the resulting residuals are modified requiring additional analysis. Robustness with respect to disturbances is also addressed by a novel frequency filtering technique. Active detection, as opposed to passive detection where outputs are passively monitored, allows the injection of an auxiliary control signal to test the system. These algorithms compute an auxiliary input signal guaranteeing fault detection, assuming bounded noise. In the second part of this dissertation, a novel active detection approach for DAE models is developed by taking linear transformations of the DAEs and solving a bi-layer optimization problem. An efficient real-time detection algorithm is also provided, as is the extension to model uncertainty. The existence of a class of problems where the algorithm breaks down is revealed and an alternative algorithm that finds a nearly minimal auxiliary signal is presented. Finally, asynchronous signal design, that is, applying the test signal on a different interval than the observation window, is explored and discussed.
Dual number coefficient octonion algebra, field equations and conservation laws
NASA Astrophysics Data System (ADS)
Chanyal, B. C.; Chanyal, S. K.
2016-08-01
Starting with octonion algebra, we develop the dual number coefficient octonion (DNCO) algebra having sixteen components. DNCO forms of generalized potential, field and current equations are discussed in consistent manner. We have made an attempt to write the DNCO form of generalized Dirac-Maxwell's equations in presence of electric and magnetic charges (dyons). Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for dyons in terms of DNCO algebra. Further, we discuss the DNCO form of linear momentum conservation law for dyons.
Exact solution of some linear matrix equations using algebraic methods
NASA Technical Reports Server (NTRS)
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
Effective mass Schrödinger equation and nonlinear algebras
NASA Astrophysics Data System (ADS)
Roy, B.; Roy, P.
2005-06-01
Using supersymmetry we obtain solutions of Schrödinger equation with a position dependent effective mass exhibiting a harmonic oscillator like spectrum. We also discuss the underlying nonlinear algebraic symmetry of the problem.
Comparative study of homotopy continuation methods for nonlinear algebraic equations
NASA Astrophysics Data System (ADS)
Nor, Hafizudin Mohamad; Ismail, Ahmad Izani Md.; Majid, Ahmad Abd.
2014-07-01
We compare some recent homotopy continuation methods to see which method has greater applicability and greater accuracy. We test the methods on systems of nonlinear algebraic equations. The results obtained indicate the superior accuracy of Newton Homotopy Continuation Method (NHCM).
NASA Astrophysics Data System (ADS)
Ibáñez, Javier; Hernández, Vicente
2011-03-01
Differential Matrix Riccati Equations (DMREs) appear in several branches of science such as applied physics and engineering. For example, these equations play a fundamental role in control theory, optimal control, filtering and estimation, decoupling and order reduction, etc. In this paper a new method based on a theorem proved in this paper is described for solving DMREs by a piecewise-linearized approach. This method is applied for developing two block-oriented algorithms based on diagonal Padé approximants. MATLAB versions of the above algorithms are developed, comparing, under equal conditions, accuracy and computational costs with other piecewise-linearized algorithms implemented by the authors. Experimental results show the advantages of solving stiff or non-stiff DMREs by the implemented algorithms.
Algebraic approximations for transcendental equations with applications in nanophysics
NASA Astrophysics Data System (ADS)
Barsan, Victor
2015-09-01
Using algebraic approximations of trigonometric or hyperbolic functions, a class of transcendental equations can be transformed in tractable, algebraic equations. Studying transcendental equations this way gives the eigenvalues of Sturm-Liouville problems associated to wave equation, mainly to Schroedinger equation; these algebraic approximations provide approximate analytical expressions for the energy of electrons and phonons in quantum wells, quantum dots (QDs) and quantum wires, in the frame of one-particle models of such systems. The advantage of this approach, compared to the numerical calculations, is that the final result preserves the functional dependence on the physical parameters of the problem. The errors of this method, situated between some few percentages and ?, are carefully analysed. Several applications, for quantum wells, QDs and quantum wires, are presented.
NASA Astrophysics Data System (ADS)
Balagović, Martina
2015-03-01
We show that, under Drinfeld's degeneration (Proceedings of the International Congress of Mathematicians. American Mathematical Society, Providence, pp 798-820, 1987) of quantum loop algebras to Yangians, the trigonometric dynamical difference equations [Etingof and Varchenko (Adv Math 167:74-127, 2002)] for the quantum affine algebra degenerate to the trigonometric Casimir differential equations [Toledano Laredo (J Algebra 329:286-327, 2011)] for Yangians.
Stability of Linear Equations--Algebraic Approach
ERIC Educational Resources Information Center
Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.
2012-01-01
This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…
Exact solution of some linear matrix equations using algebraic methods
NASA Technical Reports Server (NTRS)
Djaferis, T. E.; Mitter, S. K.
1979-01-01
Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.
Zeta functional equation on Jordan algebras of type II
NASA Astrophysics Data System (ADS)
Kayoya, J. B.
2005-02-01
Using the Jordan algebras methods, specially the properties of Peirce decomposition and the Frobenius transformation, we compute the coefficients of the zeta functional equation, in the case of Jordan algebras of type II. As particular cases of our result, we can cite the case of studied by Gelbart [Mem. Amer. Math. Soc. 108 (1971)] and Godement and Jacquet [Zeta functions of simple algebras, Lecture Notes in Math., vol. 260, Springer-Verlag, Berlin, 1972], and the case of studied by Muro [Adv. Stud. Pure Math. 15 (1989) 429]. Let us also mention, that recently, Bopp and Rubenthaler have obtained a more general result on the zeta functional equation by using methods based on the algebraic properties of regular graded algebras which are in one-to-one correspondence with simple Jordan algebras [Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces, IRMA, Strasbourg, 2003]. The method used in this paper is a direct application of specific properties of Jordan algebras of type II.
ERIC Educational Resources Information Center
Falcon, Raymond
2009-01-01
Teachers use action research in order to improve their teaching and student learning. This action research will analyze students' algebraic reasoning in finding values of variables in systems of equations pictorially and algebraically. This research will look at students solving linear systems of equations without knowing the algebraic algorithms.…
Allidina, A.Y.; Malinowski, K.; Singh, M.G.
1982-12-01
The possibilities were explored for enhancing parallelism in the simulation of systems described by algebraic equations, ordinary differential equations and partial differential equations. These techniques, using multiprocessors, were developed to speed up simulations, e.g. for nuclear accidents. Issues involved in their design included suitable approximations to bring the problem into a numerically manageable form and a numerical procedure to perform the computations necessary to solve the problem accurately. Parallel processing techniques used as simulation procedures, and a design of a simulation scheme and simulation procedure employing parallel computer facilities, were both considered.
The exotic conformal Galilei algebra and nonlinear partial differential equations
NASA Astrophysics Data System (ADS)
Cherniha, Roman; Henkel, Malte
2010-09-01
The conformal Galilei algebra (CGA) and the exotic conformal Galilei algebra (ECGA) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the CGA but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under CGA. It is further shown that there are systems of non-linear PDEs admitting ECGA with the realisation obtained very recently in [D. Martelli and Y. Tachikawa, arXiv:0903.5184v2 [hep-th] (2009)]. Moreover, wide classes of non-linear systems, invariant under two different 10-dimensional subalgebras of ECGA are explicitly constructed and an example with possible physical interpretation is presented.
Reconstruction of symmetric Dirac-Maxwell equations using nonassociative algebra
NASA Astrophysics Data System (ADS)
Kalauni, Pushpa; Barata, J. C. A.
2015-01-01
In the presence of sources, the usual Maxwell equations are neither symmetric nor invariant with respect to the duality transformation between electric and magnetic fields. Dirac proposed the existence of magnetic monopoles for symmetrizing the Maxwell equations. In the present work, we obtain the fully symmetric Dirac-Maxwell's equations (i.e. with electric and magnetic charges and currents) as a single equation by using 4 × 4 matrix presentation of fields and derivative operators. This matrix representation has been derived with the help of the algebraic properties of quaternions and octonions. Such description gives a compact representation of electric and magnetic counterparts of the field in a single equation.
He's iteration formulation for solving nonlinear algebraic equations
NASA Astrophysics Data System (ADS)
Qian, W.-X.; Ye, Y.-H.; Chen, J.; Mo, L.-F.
2008-02-01
Newton iteration method is sensitive to initial guess and its slope. To overcome the shortcoming, He's iteration method is used to solve nonlinear algebraic equations where Newton iteration method becomes invalid. Some examples are given, showing that the method is effective.
Numerical solution to systems of delay integrodifferential algebraic equations
NASA Astrophysics Data System (ADS)
Dmitriev, S. S.; Kuznetsov, E. B.
2008-03-01
The numerical solution of the initial value problem for a system of delay integrodifferential algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which is the arc length along the integral curve of the problem. The efficiency of the transformation is demonstrated using test examples.
NASA Astrophysics Data System (ADS)
Jung, Junoh; Park, Sang-Young; Kim, Sung-Woo; Eun, Young Ho; Chang, Young-Keun
2013-02-01
A nonlinear control technique pertaining to attitude synchronization problems is presented for formation flying spacecraft by utilizing the State-Dependent Riccati Equation (SDRE) technique. An attitude controller consisting of relative control and absolute control is designed using a reaction wheel assembly for regulator and tracking problems. To achieve effective relative control, the selective state-dependent connectivity is also adopted. The global asymptotic stability of the controller is confirmed using the Lyapunov theorem and is verified by Monte-Carlo simulations. An air-bearing-based Hardware-In-the-Loop Simulator (HILS) is also developed to validate the proposed control laws in real-time environments. The SDRE controller is discretized for implementation of a real-time processor in the HILS. The pointing errors are about 0.2° in the numerical simulations and about 1° in the HILS simulations, and experimental simulations confirm the effectiveness of the control algorithm for attitude synchronization in a spacecraft formation flying mission. Consequently, experiments using the HILS in a real-time environment can appropriately perform spacecraft attitude synchronization algorithms for formation flying spacecraft.
A Realizable Reynolds Stress Algebraic Equation Model
NASA Technical Reports Server (NTRS)
Shih, Tsan-Hsing; Zhu, Jiang; Lumley, John L.
1993-01-01
The invariance theory in continuum mechanics is applied to analyze Reynolds stresses in high Reynolds number turbulent flows. The analysis leads to a turbulent constitutive relation that relates the Reynolds stresses to the mean velocity gradients in a more general form in which the classical isotropic eddy viscosity model is just the linear approximation of the general form. On the basis of realizability analysis, a set of model coefficients are obtained which are functions of the time scale ratios of the turbulence to the mean strain rate and the mean rotation rate. The coefficients will ensure the positivity of each component of the mean rotation rate. These coefficients will ensure the positivity of each component of the turbulent kinetic energy - realizability that most existing turbulence models fail to satisfy. Separated flows over backward-facing step configurations are taken as applications. The calculations are performed with a conservative finite-volume method. Grid-independent and numerical diffusion-free solutions are obtained by using differencing schemes of second-order accuracy on sufficiently fine grids. The calculated results are compared in detail with the experimental data for both mean and turbulent quantities. The comparison shows that the present proposal significantly improves the predictive capability of K-epsilon based two equation models. In addition, the proposed model is able to simulate rotational homogeneous shear flows with large rotation rates which all conventional eddy viscosity models fail to simulate.
NASA Astrophysics Data System (ADS)
Matsuno, Yoshimasa
2004-02-01
The multisoliton solution of the Benjamin-Ono equation is derived from the system of nonlinear algebraic equations. This finding is unexpected from the scheme of the inverse scattering transform method, which constructs the multisoliton solution through the system of linear algebraic equations. The anlaysis developed here is also applied to the rational multisoliton solution of the Kadomtsev-Petviashvili equation.
NASA Astrophysics Data System (ADS)
Dragan, Vasile; Morozan, Toader; Stoica, Adrian-Mihail
2010-04-01
In this article an iterative method to compute the maximal solution and the stabilising solution, respectively, of a wide class of discrete-time nonlinear equations on the linear space of symmetric matrices is proposed. The class of discrete-time nonlinear equations under consideration contains, as special cases, different types of discrete-time Riccati equations involved in various control problems for discrete-time stochastic systems. This article may be viewed as an addendum of the work of Dragan and Morozan (Dragan, V. and Morozan, T. (2009), 'A Class of Discrete Time Generalized Riccati Equations', Journal of Difference Equations and Applications, first published on 11 December 2009 (iFirst), doi: 10.1080/10236190802389381) where necessary and sufficient conditions for the existence of the maximal solution and stabilising solution of this kind of discrete-time nonlinear equations are given. The aim of this article is to provide a procedure for numerical computation of the maximal solution and the stabilising solution, respectively, simpler than the method based on the Newton-Kantorovich algorithm.
Numerical integration of systems of delay differential-algebraic equations
NASA Astrophysics Data System (ADS)
Kuznetsov, E. B.; Mikryukov, V. N.
2007-01-01
The numerical solution of the initial value problem for a system of delay differential-algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which ensures the best condition for the corresponding system of continuation equations. The best argument is the arc length along the integral curve of the problem. Algorithms and programs based on the continuous and discrete continuation methods are developed for the numerical integration of this problem. The efficiency of the suggested transformation is demonstrated using test examples.
ERIC Educational Resources Information Center
Novotna, Jarmila; Hoch, Maureen
2008-01-01
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…
Solving the generalized Langevin equation with the algebraically correlated noise
NASA Astrophysics Data System (ADS)
Srokowski, T.; Płoszajczak, M.
1998-04-01
We solve the Langevin equation with the memory kernel. The stochastic force possesses algebraic correlations, proportional to 1/t. The velocity autocorrelation function and related quantities characterizing transport properties are calculated with the assumption that the system is in thermal equilibrium. Stochastic trajectories are simulated numerically, using the kangaroo process as a noise generator. Results of this simulation resemble Lévy walks with divergent moments of the velocity distribution. We consider motion of a Brownian particle, both without any external potential and in the harmonic oscillator field, in particular the escape from a potential well. The results are compared with memory-free calculations for the Brownian particle.
NASA Astrophysics Data System (ADS)
Tala-Tebue, E.; Tsobgni-Fozap, D. C.; Kenfack-Jiotsa, A.; Kofane, T. C.
2014-06-01
Using the Jacobi elliptic functions and the alternative ( G'/ G-expansion method including the generalized Riccati equation, we derive exact soliton solutions for a discrete nonlinear electrical transmission line in (2+1) dimension. More precisely, these methods are general as they lead us to diverse solutions that have not been previously obtained for the nonlinear electrical transmission lines. This study seeks to show that it is not often necessary to transform the equation of the network into a well-known differential equation before finding its solutions. The solutions obtained by the current methods are generalized periodic solutions of nonlinear equations. The shape of solutions can be well controlled by adjusting the parameters of the network. These exact solutions may have significant applications in telecommunication systems where solitons are used to codify or for the transmission of data.
Riccati generalization of self-similar solutions of nonautonomous Gross-Pitaevskii equation
NASA Astrophysics Data System (ADS)
Panigrahi, P. K.; Gupta, Rama; Goyal, Amit; Kumar, C. N.
2013-07-01
We present a systematic analytical approach to construct a family of self-similar waves, related through a free parameter, in quasi one-dimension Gross-Pitaevskii equation with time-varying parameters. This approach enables us to control the dynamics of dark and bright similaritons, and first- and second- order self-similar rogue waves in Bose-Einstein condensate through the modulation of time dependent trapping potential. The analysis is done for the sech2- type time-varying quadratic trapping potential for two different choices of linear potential.
NASA Astrophysics Data System (ADS)
Saker, S. H.; O'Regan, Donal
2011-01-01
In this paper, we establish some new sufficient conditions for oscillation of the second-order neutral functional dynamic equation (p(t)([y(t)+r(t)y(τ(t))]Δ)γ)Δ+f(t,y(θ(t))=0,t∈[t0,∞)T, on a time scale T, where |f(t,u)|⩾q(t)|uγ|, r, p and q are real valued rd-continuous positive functions defined on T, γ⩾1 is the quotient of odd positive integers. Our results improve existence results in the literature in the sense that our results do not require pΔ(t)⩾0, and ∫t0∞θγ(s)q(s)[1-r(θ(s))]γΔs=∞. Some examples are given to illustrate the main results.
ERIC Educational Resources Information Center
Wasserman, Nicholas H.
2014-01-01
Algebraic structures are a necessary aspect of algebraic thinking for K-12 students and teachers. An approach for introducing the algebraic structure of groups and fields through the arithmetic properties required for solving simple equations is summarized; the collective (not individual) importance of these axioms as a foundation for algebraic…
On Traveling Waves in Lattices: The Case of Riccati Lattices
NASA Astrophysics Data System (ADS)
Dimitrova, Zlatinka
2012-09-01
The method of simplest equation is applied for analysis of a class of lattices described by differential-difference equations that admit traveling-wave solutions constructed on the basis of the solution of the Riccati equation. We denote such lattices as Riccati lattices. We search for Riccati lattices within two classes of lattices: generalized Lotka-Volterra lattices and generalized Holling lattices. We show that from the class of generalized Lotka-Volterra lattices only the Wadati lattice belongs to the class of Riccati lattices. Opposite to this many lattices from the Holling class are Riccati lattices. We construct exact traveling wave solutions on the basis of the solution of Riccati equation for three members of the class of generalized Holling lattices.
NASA Astrophysics Data System (ADS)
Pogrebkov, A. K.
2016-06-01
We show that the non-Abelian Hirota difference equation is directly related to a commutator identity on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra lead to a linear difference equation. We develop a special dressing procedure that results in an integrable non-Abelian Hirota difference equation and propose two regular reduction procedures that lead to a set of known equations, Abelian or non-Abelian, and also to some new integrable equations.
Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme
NASA Astrophysics Data System (ADS)
Mazzocco, Marta
2016-09-01
In this paper we produce seven new algebras as confluences of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877–912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory.
Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme
NASA Astrophysics Data System (ADS)
Mazzocco, Marta
2016-09-01
In this paper we produce seven new algebras as confluences of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877-912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory.
Supporting Students' Understanding of Linear Equations with One Variable Using Algebra Tiles
ERIC Educational Resources Information Center
Saraswati, Sari; Putri, Ratu Ilma Indra; Somakim
2016-01-01
This research aimed to describe how algebra tiles can support students' understanding of linear equations with one variable. This article is a part of a larger research on learning design of linear equations with one variable using algebra tiles combined with balancing method. Therefore, it will merely discuss one activity focused on how students…
NASA Astrophysics Data System (ADS)
Volkov, V. V.; Erokhin, V. I.
2010-04-01
The properties of a mathematical programming problem that arises in finding a stable (in the sense of Tikhonov) solution to a system of linear algebraic equations with an approximately given augmented coefficient matrix are examined. Conditions are obtained that determine whether this problem can be reduced to the minimization of a smoothing functional or to the minimal matrix correction of the underlying system of linear algebraic equations. A method for constructing (exact or approximately given) model systems of linear algebraic equations with known Tikhonov solutions is described. Sharp lower bounds are derived for the maximal error in the solution of an approximately given system of linear algebraic equations under finite perturbations of its coefficient matrix. Numerical examples are given.
The coquaternion algebra and complex partial differential equations
NASA Astrophysics Data System (ADS)
Dimiev, Stancho; Konstantinov, Mihail; Todorov, Vladimir
2009-11-01
In this paper we consider the problem of differentiation of coquaternionic functions. Let us recall that coquaternions are elements of an associative non-commutative real algebra with zero divisor, introduced by James Cockle (1849) under the name of split-quaternions or coquaternions. Developing two type complex representations for Cockle algebra (complex and paracomplex ones) we present the problem in a non-commutative form of the δ¯-type holomorphy. We prove that corresponding differentiable coquaternionic functions, smooth and analytic, satisfy PDE of complex, and respectively of real variables. Applications for coquaternionic polynomials are sketched.
Block method of Runge Kutta type for solving differential algebraic equation
NASA Astrophysics Data System (ADS)
Wen, Khoo Kai; Majid, Zanariah Abdul; Senu, Norazak
2015-10-01
In this paper, a self-starting block method of Runge Kutta type is proposed to solve semi-explicit index-1 differential algebraic equation (DAE). Semi-explicit DAE consists of a system of ordinary differential equations with algebraic constraints. This method will compute the solutions of DAE at two points simultaneously in a block by block steps using constant step size. The DAE is a stiff equation, therefore the Newton iteration is needed during the implementation. Numerical examples are given in order to illustrate the efficiency of the block method when solving the DAE.
NASA Astrophysics Data System (ADS)
Hartwig, J. T.; Stokman, J. V.
2013-02-01
We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schrödinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations as functions of the momenta. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with delta-function interactions is indicated.
Hartwig, J. T.; Stokman, J. V.
2013-02-15
We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schroedinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schroedinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations as functions of the momenta. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with delta-function interactions is indicated.
Numerical solutions of linear differential-algebraic equation systems via Hartley series
NASA Astrophysics Data System (ADS)
Ünal, Emrah; Yalçın, Numan; ćelik, Ercan
2014-08-01
In this paper, Hartley series are presented first. Then, the operational matrix of integration together with the product and coefficient matrices are presented. They are used to transform linear differential equation systems to a set of linear algebraic equations. Finally, numerical examples are given.
Newton equation for canonical, Lie-algebraic, and quadratic deformation of classical space
Daszkiewicz, Marcin; Walczyk, Cezary J.
2008-05-15
The Newton equation describing particle motion in a constant external field force on canonical, Lie-algebraic, and quadratic space-time is investigated. We show that for canonical deformation of space-time the dynamical effects are absent, while in the case of Lie-algebraic noncommutativity, when spatial coordinates commute to the time variable, the additional acceleration of the particle is generated. We also indicate that in the case of spatial coordinates commuting in a Lie-algebraic way, as well as for quadratic deformation, there appear additional velocity and position-dependent forces.
Cognitive Load in Algebra: Element Interactivity in Solving Equations
ERIC Educational Resources Information Center
Ngu, Bing Hiong; Chung, Siu Fung; Yeung, Alexander Seeshing
2015-01-01
Central to equation solving is the maintenance of equivalence on both sides of the equation. However, when the process involves an interaction of multiple elements, solving an equation can impose a high cognitive load. The balance method requires operations on both sides of the equation, whereas the inverse method involves operations on one side…
Numerical Treatment of Differential-Algebraic Equations with Index 2
NASA Astrophysics Data System (ADS)
Attili, Basem S.
2007-09-01
We will consider index-2 differential algebraic systems. Since they are usually harder to solve, we will show how to reduce the index 2 problem to index 1 DAE which becomes easier to solve numerically. For the numerical treatment, we will treat the resulting index-1 DAE using power series solutions coupled with pade' approximation for better convergence results. Numerical examples will be presented also.
Solving the Langevin equation with stochastic algebraically correlated noise
NASA Astrophysics Data System (ADS)
Płoszajczak, M.; Srokowski, T.
1997-05-01
The long time tail in the velocity and force autocorrelation function has been found recently in molecular dynamics simulations of peripheral collisions of ions. Simulation of those slowly decaying correlations in the stochastic transport theory requires the development of new methods of generating stochastic force of arbitrarily long correlation times. In this paper we propose a Markovian process, the multidimensional kangaroo process, which permits the description of various algebraically correlated stochastic processes.
NASA Technical Reports Server (NTRS)
Byrnes, C. I.
1980-01-01
It is noted that recent work by Kamen (1979) on the stability of half-plane digital filters shows that the problem of the existence of a feedback law also arises for other Banach algebras in applications. This situation calls for a realization theory and stabilizability criteria for systems defined over Banach for Frechet algebra A. Such a theory is developed here, with special emphasis placed on the construction of finitely generated realizations, the existence of coprime factorizations for T(s) defined over A, and the solvability of the quadratic optimal control problem and the associated algebraic Riccati equation over A.
Convergence of Galerkin Solutions for Linear Differential Algebraic Equations in Hilbert Spaces
NASA Astrophysics Data System (ADS)
Matthes, Michael; Tischendorf, Caren
2010-09-01
The simulation of complex systems describing different physical effects becomes more and more of interest in various applications. Examples are couplings describing interactions between circuits and semiconductor devices, circuits and electromagnetic fields, fluids and structures. The modeling of such complex processes [1, 2, 3, 4, 7, 8] often leads to coupled systems that are composed of ordinary differential equations (ODEs), differential-algebraic equations (DAEs) and partial differential equations (PDEs). Such coupled systems can be regarded in the general framework of abstract differential-algebraic equations. Here, we discuss a Galerkin approach for handling linear abstract differential-algebraic equations with monotone operators. It is shown to provide solutions that converge to the unique solution of the abstract differential-algebraic system. Furthermore, the solution is proved to depend continuously on the data. The most interesting point of the Galerkin approach is the choice of basis functions. They have to be chosen in proper subspaces in order to guarantee that the solution satisfies the non-dynamic constraints. In contrast to other approaches as e.g. [5, 6], this approach allows time dependent operators but needs monotonicity.
A numerical method for solving partial differential algebraic equations
NASA Astrophysics Data System (ADS)
Diep, Nguyen Khac; Chistyakov, V. F.
2013-06-01
Linear systems of partial differential equations with constant coefficient matrices are considered. The matrices multiplying the derivatives of the sought vector function are assumed to be singular. The structure of solutions to such systems is examined. The numerical solution of initialboundary value problems for such equations by applying implicit difference schemes is discussed.
Alternative Representations for Algebraic Problem Solving: When Are Graphs Better than Equations?
ERIC Educational Resources Information Center
Mielicki, Marta K.; Wiley, Jennifer
2016-01-01
Successful algebraic problem solving entails adaptability of solution methods using different representations. Prior research has suggested that students are more likely to prefer symbolic solution methods (equations) over graphical ones, even when graphical methods should be more efficient. However, this research has not tested how representation…
On a modification of minimal iteration methods for solving systems of linear algebraic equations
NASA Astrophysics Data System (ADS)
Yukhno, L. F.
2010-04-01
Modifications of certain minimal iteration methods for solving systems of linear algebraic equations are proposed and examined. The modified methods are shown to be superior to the original versions with respect to the round-off error accumulation, which makes them applicable to solving ill-conditioned problems. Numerical results demonstrating the efficiency of the proposed modifications are given.
Flipping an Algebra Classroom: Analyzing, Modeling, and Solving Systems of Linear Equations
ERIC Educational Resources Information Center
Kirvan, Rebecca; Rakes, Christopher R.; Zamora, Regie
2015-01-01
The present study investigated whether flipping an algebra classroom led to a stronger focus on conceptual understanding and improved learning of systems of linear equations for 54 seventh- and eighth-grade students using teacher journal data and district-mandated unit exam items. Multivariate analysis of covariance was used to compare scores on…
NASA Astrophysics Data System (ADS)
Shcheglova, A. A.
2009-09-01
Linear control differential algebraic equations are considered. The issue of minimum dimension of the control vector necessitated for complete controllability of the system on any closed interval from the domain of definition is investigated. The problem is analyzed in connection with the time invariant systems having regular matrix pencils and also systems with real-analytic or smooth coefficients, which possess some structural forms.
ERIC Educational Resources Information Center
Foley, Greg
2011-01-01
Continuous feed and bleed ultrafiltration, modeled with the gel polarization model for the limiting flux, is shown to provide a rich source of non-linear algebraic equations that can be readily solved using numerical and graphical techniques familiar to undergraduate students. We present a variety of numerical problems in the design, analysis, and…
ERIC Educational Resources Information Center
Gasyna, Zbigniew L.
2008-01-01
Computational experiment is proposed in which a linear algebra method is applied to the solution of the Schrodinger equation for a diatomic oscillator. Calculations of the vibration-rotation spectrum for the HCl molecule are presented and the results show excellent agreement with experimental data. (Contains 1 table and 1 figure.)
Sun, Leping
2016-01-01
This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2-delay differential algebraic equations. We obtain two sufficient conditions under which the methods are stable and asymptotically stable. At last, examples show that our methods are true. PMID:27441132
New solutions of reflection equation derived from type B BMW algebras
NASA Astrophysics Data System (ADS)
Häring-Oldenburg, Reinhard
1996-09-01
We use B-type knot theory to find new solutions of Sklyanin's reflection equation in a systematic way. This generalizes the well known Baxterization of Birman - Wenzl algebras and should describe integrable systems which are restricted to a half plane.
ERIC Educational Resources Information Center
Tonisson, Eno
2015-01-01
Sometimes Computer Algebra Systems (CAS) offer an answer that is somewhat different from the answer that is probably expected by the student or teacher. These (somewhat unexpected) answers could serve as a catalyst for rich mathematical discussion. In this study, over 120 equations from school mathematics were solved using 8 different CAS. Many…
Using Spreadsheets to Make Algebra More Accessible--Part 2: Solutions to Equations
ERIC Educational Resources Information Center
Green, John
2009-01-01
This article is the second in a series of two papers which suggest some practical, spreadsheet-based ideas for helping students to make appropriate connections between particular algebraic concepts. Solving equations has traditionally been taught as a pen-and-paper process. Spreadsheets, such as that of Excel, provide a contemporary, and powerful…
Translation of Algebraic Equations and Its Relation to Formal Operational Reasoning.
ERIC Educational Resources Information Center
Niaz, Mansoor
A large proportion of college students majoring in science are unable to translate even simple sentences into algebraic equations. Given the following sentence, "There are six times as many students (S) as professors (P) at this university," 37% of 150 freshmen engineering students in a study conducted in 1981 by Clement, Lockhead, and Monk wrote…
ERIC Educational Resources Information Center
Maat, Siti Mistima; Zakaria, Effandi
2011-01-01
Ordinary differential equations (ODEs) are one of the important topics in engineering mathematics that lead to the understanding of technical concepts among students. This study was conducted to explore the students' understanding of ODEs when they solve ODE questions using a traditional method as well as a computer algebraic system, particularly…
On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Tam, Honwah
2016-03-01
In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz–Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), and Hong Kong Research Grant Council under Grant No. HKBU202512, as well as the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016
On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Tam, Honwah
2016-03-01
In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz-Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), and Hong Kong Research Grant Council under Grant No. HKBU202512, as well as the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016
The change of the brain activation patterns as children learn algebra equation solving
Qin, Yulin; Carter, Cameron S.; Silk, Eli M.; Stenger, V. Andrew; Fissell, Kate; Goode, Adam; Anderson, John R.
2004-01-01
In a brain imaging study of children learning algebra, it is shown that the same regions are active in children solving equations as are active in experienced adults solving equations. As with adults, practice in symbol manipulation produces a reduced activation in prefrontal cortex area. However, unlike adults, practice seems also to produce a decrease in a parietal area that is holding an image of the equation. This finding suggests that adolescents' brain responses are more plastic and change more with practice. These results are integrated in a cognitive model that predicts both the behavioral and brain imaging results. PMID:15064407
A fifth order implicit method for the numerical solution of differential-algebraic equations
NASA Astrophysics Data System (ADS)
Skvortsov, L. M.
2015-06-01
An implicit two-step Runge-Kutta method of fifth order is proposed for the numerical solution of differential and differential-algebraic equations. The location of nodes in this method makes it possible to estimate the values of higher derivatives at the initial and terminal points of an integration step. Consequently, the proposed method can be regarded as a finite-difference analog of the Obrechkoff method. Numerical results, some of which are presented in this paper, show that our method preserves its order while solving stiff equations and equations of indices two and three. This is the main advantage of the proposed method as compared with the available ones.
Tracking children's mental states while solving algebra equations.
Anderson, John R; Betts, Shawn; Ferris, Jennifer L; Fincham, Jon M
2012-11-01
Behavioral and function magnetic resonance imagery (fMRI) data were combined to infer the mental states of students as they interacted with an intelligent tutoring system. Sixteen children interacted with a computer tutor for solving linear equations over a six-day period (days 0-5), with days 1 and 5 occurring in an fMRI scanner. Hidden Markov model algorithms combined a model of student behavior with multi-voxel imaging pattern data to predict the mental states of students. We separately assessed the algorithms' ability to predict which step in a problem-solving sequence was performed and whether the step was performed correctly. For day 1, the data patterns of other students were used to predict the mental states of a target student. These predictions were improved on day 5 by adding information about the target student's behavioral and imaging data from day 1. Successful tracking of mental states depended on using the combination of a behavioral model and multi-voxel pattern analysis, illustrating the effectiveness of an integrated approach to tracking the cognition of individuals in real time as they perform complex tasks.
NASA Astrophysics Data System (ADS)
Nordtvedt, K.
2015-11-01
A local system of bodies in General Relativity whose exterior metric field asymptotically approaches the Minkowski metric effaces any effects of the matter distribution exterior to its Minkowski boundary condition. To enforce to all orders this property of gravity which appears to hold in nature, a method using linear algebraic scaling equations is developed which generates by an iterative process an N-body Lagrangian expansion for gravity's motion-independent potentials which fulfills exterior effacement along with needed metric potential expansions. Then additional properties of gravity - interior effacement and Lorentz time dilation and spatial contraction - produce additional iterative, linear algebraic equations for obtaining the full non-linear and motion-dependent N-body gravity Lagrangian potentials as well.
Determining the multiplicity of a root of a nonlinear algebraic equation
NASA Astrophysics Data System (ADS)
Kalitkin, N. N.; Poshivailo, I. P.
2008-07-01
Newton’s method is most frequently used to find the roots of a nonlinear algebraic equation. The convergence domain of Newton’s method can be expanded by applying a generalization known as the continuous analogue of Newton’s method. For the classical and generalized Newton methods, an effective root-finding technique is proposed that simultaneously determines root multiplicity. Roots of high multiplicity (up to 10) can be calculated with a small error. The technique is illustrated using numerical examples.
NASA Astrophysics Data System (ADS)
Yukhno, L. F.
2008-12-01
For an overdetermined system of linear algebraic equations, systems obtained by introducing independent random errors into the original right-hand side are examined. Under certain assumptions on how these random variables are distributed, a practical stopping criterion is proposed for an iterative process that minimizes the sum of the squares of the residuals for the above systems. Numerical results demonstrating the efficiency of this criterion for some ill-conditioned problems are presented.
Numerical solution of differential-algebraic equations using the spline collocation-variation method
NASA Astrophysics Data System (ADS)
Bulatov, M. V.; Rakhvalov, N. P.; Solovarova, L. S.
2013-03-01
Numerical methods for solving initial value problems for differential-algebraic equations are proposed. The approximate solution is represented as a continuous vector spline whose coefficients are found using the collocation conditions stated for a subgrid with the number of collocation points less than the degree of the spline and the minimality condition for the norm of this spline in the corresponding spaces. Numerical results for some model problems are presented.
Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras
NASA Astrophysics Data System (ADS)
Grahovski, Georgi G.; Mikhailov, Alexander V.
2013-12-01
Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initial-boundary problems are formulated.
NASA Astrophysics Data System (ADS)
Man, Yiu-Kwong
2010-10-01
In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle-Singer method for computing first integrals of differential equations. Some examples on using this approach are provided.
Middle school students' reading comprehension of mathematical texts and algebraic equations
NASA Astrophysics Data System (ADS)
Duru, Adem; Koklu, Onder
2011-06-01
In this study, middle school students' abilities to translate mathematical texts into algebraic representations and vice versa were investigated. In addition, students' difficulties in making such translations and the potential sources for these difficulties were also explored. Both qualitative and quantitative methods were used to collect data for this study: questionnaire and clinical interviews. The questionnaire consisted of two general types of items: (1) selected-response (multiple-choice) items for which the respondent selects from multiple options and (2) open-ended items for which the respondent constructs a response. In order to further investigate the students' strategies while they were translating the given mathematical texts to algebraic equations and vice versa, five randomly chosen (n = 5) students were interviewed. Data were collected in the 2007-2008 school year from 185 middle-school students in five teachers' classrooms in three different schools in the city of Adıyaman, Turkey. After the analysis of data, it was found that students who participated in this study had difficulties in translating the mathematical texts into algebraic equations by using symbols. It was also observed that these students had difficulties in translating the symbolic representations into mathematical texts because of their weak reading comprehension. In addition, finding of this research revealed that students' difficulties in translating the given mathematical texts into symbolic representations or vice versa come from different sources.
NASA Astrophysics Data System (ADS)
Latini, D.; Ragnisco, O.; Ballesteros, A.; Enciso, A.; Herranz, F. J.; Riglioni, D.
2016-01-01
In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. Here we will follow the same path to tackle the Darboux III system, another maximally superintegrable system, which can indeed be viewed as a natural deformation of the isotropic harmonic oscillator where the flat Euclidean space is again replaced by another space of non-constant curvature.
Addendum to ‘Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model’
NASA Astrophysics Data System (ADS)
Li, Zi-Min; Batchelor, Murray T.
2016-09-01
In our recent paper Li and Batchelor (2015 J. Phys. A: Math. Theor. 48 454005) we obtained exceptional points in the eigenspectrum of the generalized Rabi model in terms of a set of algebraic equations. We also gave a proof for the number of roots of the constraint polynomials defining these exceptional solutions as a function of the system parameters and discussed the number of crossing points in the eigenspectrum. This approach however, only covered a subset of all exceptional points in the eigenspectrum. In this addendum, we clarify the distinction between the exceptional parts of the eigenspectrum for this model and discuss the subset of exceptional points not determined in our paper.
Benhammouda, Brahim
2016-01-01
Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems. PMID:27330880
Numerical solution of differential algebraic equations (DAEs) by mix-multistep method
NASA Astrophysics Data System (ADS)
Rahim, Yong Faezah; Suleiman, Mohamed; Ibrahim, Zarina Bibi
2014-06-01
Differential Algebraic Equations (DAEs) are regarded as stiff Ordinary Differential Equations (ODEs). Therefore they are solved using implicit method such as Backward Differentiation Formula (BDF) type of methods which require the use of Newton iteration which need much computational effort. However, not all of the ODEs in DAE system are stiff. In this paper, we describe a new technique for solving DAE, where the ODEs are treated as non-stiff at the start of the integration and putting the non-stiff ODEs into stiff subsystem should instability occurs. Adams type of method is used to solve the non-stiff part and BDF method for solving the stiff part. This strategy is shown to be competitive in terms of computational effort and accuracy. Numerical experiments are presented to validate its efficiency.
Data fitting in partial differential algebraic equations: some academic and industrial applications
NASA Astrophysics Data System (ADS)
Schittkowski, K.
2004-02-01
The paper introduces a numerical method to estimate parameters in systems of one-dimensional partial differential algebraic equations. Proceeding from given experimental data, i.e., observation times and measurements, the minimum least-squares distance of measured data from a fitting criterion is computed, which depends on the solution of the dynamical system. We present a typical black box approach that is easily implemented proceeding from some standard numerical analysis tools. Main emphasis of the paper is to present a couple of practical applications from industry and academia, to give an impression on the complexity of real-life systems of partial differential equations. The domains of application are pharmaceutics, geology, mechanical engineering, chemical engineering, food engineering, and electrical engineering.
Nonlocal symmetries of Riccati and Abel chains and their similarity reductions
NASA Astrophysics Data System (ADS)
Bruzon, M. S.; Gandarias, M. L.; Senthilvelan, M.
2012-02-01
We study nonlocal symmetries and their similarity reductions of Riccati and Abel chains. Our results show that all the equations in Riccati chain share the same form of nonlocal symmetry. The similarity reduced Nth order ordinary differential equation (ODE), N = 2, 3, 4, …, in this chain yields (N - 1)th order ODE in the same chain. All the equations in the Abel chain also share the same form of nonlocal symmetry (which is different from the one that exist in Riccati chain) but the similarity reduced Nth order ODE, N = 2, 3, 4, …, in the Abel chain always ends at the (N - 1)th order ODE in the Riccati chain. We describe the method of finding general solution of all the equations that appear in these chains from the nonlocal symmetry.
NASA Astrophysics Data System (ADS)
Fukuchi, Tsugio
2014-06-01
The finite difference method (FDM) based on Cartesian coordinate systems can be applied to numerical analyses over any complex domain. A complex domain is usually taken to mean that the geometry of an immersed body in a fluid is complex; here, it means simply an analytical domain of arbitrary configuration. In such an approach, we do not need to treat the outer and inner boundaries differently in numerical calculations; both are treated in the same way. Using a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. Discretization of the space differential in the FDM is usually derived using the Taylor series expansion; however, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences. In using the polynomial interpolations, we can numerically solve the Poisson equation freely over any complex domain. Only a particular type of partial differential equation, Poisson's equations, is treated; however, the arguments put forward have wider generality in numerical calculations using the FDM.
Mixed superposition rules and the Riccati hierarchy
NASA Astrophysics Data System (ADS)
Grabowski, Janusz; de Lucas, Javier
Mixed superposition rules, i.e., functions describing the general solution of a system of first-order differential equations in terms of a generic family of particular solutions of first-order systems and some constants, are studied. The main achievement is a generalization of the celebrated Lie-Scheffers Theorem, characterizing systems admitting a mixed superposition rule. This somehow unexpected result says that such systems are exactly Lie systems, i.e., they admit a standard superposition rule. This provides a new and powerful tool for finding Lie systems, which is applied here to studying the Riccati hierarchy and to retrieving some known results in a more efficient and simpler way.
NASA Astrophysics Data System (ADS)
Muthuvalu, Mohana Sundaram; Aruchunan, Elayaraja; Koh, Wei Sin; Akhir, Mohd Kamalrulzaman Md; Sulaiman, Jumat; Karim, Samsul Ariffin Abdul
2014-07-01
In this paper, the application of the Accelerated Over-Relaxation (AOR) iterative method is extended to solve first order composite closed Newton-Cotes quadrature (1-CCNC) algebraic equations arising from second kind linear Fredholm integral equations. The formulation and implementation of the method are also discussed. In addition, numerical results by solving several test problems are included and compared with the conventional iterative methods.
Mizuno, Yuta; Arasaki, Yasuki; Takatsuka, Kazuo
2016-01-14
A complicated yet interesting induced photon emission can take place by a nonadiabatic intramolecular electron transfer system like LiF under an intense CW laser [Y. Arasaki, S. Scheit, and K. Takatsuka, J. Chem. Phys. 138, 161103 (2013)]. Behind this phenomena, the crossing point between two potential energy curves of covalent and ionic natures in diabatic representation is forced to oscillate, since only the ionic potential curve is shifted significantly up and down repeatedly (called the Dynamical Stark effect). The wavepacket pumped initially to the excited covalent potential curve frequently encounters such a dynamically moving crossing point and thereby undergoes very complicated dynamics including wavepacket bifurcation and deformation. Intramolecular electron transfer thus driven by the coupling between nonadiabatic state-mixing and laser fields induces irregular photon emission. Here in this report we discuss the complicated spectral features of this kind of photon emission induced by infrared laser. In the low frequency domain, the photon emission is much more involved than those of ultraviolet/visible driving fields, since many field-dressed states are created on the ionic potential, which have their own classical turning points and crossing points with the covalent counterpart. To analyze the physics behind the phenomena, we develop a perturbation theoretic approach to the Riccati equation that is transformed from coupled first-order linear differential equations with periodic coefficients, which are supposed to produce the so-called Floquet states. We give mathematical expressions for the Floquet energies, frequencies, and intensities of the photon emission spectra, and the cutoff energy of their harmonic generation. Agreement between these approximate quantities and those estimated with full quantum calculations is found to be excellent. Furthermore, the present analysis provides with notions to facilitate deeper understanding for the physical and
NASA Astrophysics Data System (ADS)
Mizuno, Yuta; Arasaki, Yasuki; Takatsuka, Kazuo
2016-01-01
A complicated yet interesting induced photon emission can take place by a nonadiabatic intramolecular electron transfer system like LiF under an intense CW laser [Y. Arasaki, S. Scheit, and K. Takatsuka, J. Chem. Phys. 138, 161103 (2013)]. Behind this phenomena, the crossing point between two potential energy curves of covalent and ionic natures in diabatic representation is forced to oscillate, since only the ionic potential curve is shifted significantly up and down repeatedly (called the Dynamical Stark effect). The wavepacket pumped initially to the excited covalent potential curve frequently encounters such a dynamically moving crossing point and thereby undergoes very complicated dynamics including wavepacket bifurcation and deformation. Intramolecular electron transfer thus driven by the coupling between nonadiabatic state-mixing and laser fields induces irregular photon emission. Here in this report we discuss the complicated spectral features of this kind of photon emission induced by infrared laser. In the low frequency domain, the photon emission is much more involved than those of ultraviolet/visible driving fields, since many field-dressed states are created on the ionic potential, which have their own classical turning points and crossing points with the covalent counterpart. To analyze the physics behind the phenomena, we develop a perturbation theoretic approach to the Riccati equation that is transformed from coupled first-order linear differential equations with periodic coefficients, which are supposed to produce the so-called Floquet states. We give mathematical expressions for the Floquet energies, frequencies, and intensities of the photon emission spectra, and the cutoff energy of their harmonic generation. Agreement between these approximate quantities and those estimated with full quantum calculations is found to be excellent. Furthermore, the present analysis provides with notions to facilitate deeper understanding for the physical and
Mizuno, Yuta; Arasaki, Yasuki; Takatsuka, Kazuo
2016-01-14
A complicated yet interesting induced photon emission can take place by a nonadiabatic intramolecular electron transfer system like LiF under an intense CW laser [Y. Arasaki, S. Scheit, and K. Takatsuka, J. Chem. Phys. 138, 161103 (2013)]. Behind this phenomena, the crossing point between two potential energy curves of covalent and ionic natures in diabatic representation is forced to oscillate, since only the ionic potential curve is shifted significantly up and down repeatedly (called the Dynamical Stark effect). The wavepacket pumped initially to the excited covalent potential curve frequently encounters such a dynamically moving crossing point and thereby undergoes very complicated dynamics including wavepacket bifurcation and deformation. Intramolecular electron transfer thus driven by the coupling between nonadiabatic state-mixing and laser fields induces irregular photon emission. Here in this report we discuss the complicated spectral features of this kind of photon emission induced by infrared laser. In the low frequency domain, the photon emission is much more involved than those of ultraviolet/visible driving fields, since many field-dressed states are created on the ionic potential, which have their own classical turning points and crossing points with the covalent counterpart. To analyze the physics behind the phenomena, we develop a perturbation theoretic approach to the Riccati equation that is transformed from coupled first-order linear differential equations with periodic coefficients, which are supposed to produce the so-called Floquet states. We give mathematical expressions for the Floquet energies, frequencies, and intensities of the photon emission spectra, and the cutoff energy of their harmonic generation. Agreement between these approximate quantities and those estimated with full quantum calculations is found to be excellent. Furthermore, the present analysis provides with notions to facilitate deeper understanding for the physical and
Activities for Students: Biology as a Source for Algebra Equations--The Heart
ERIC Educational Resources Information Center
Horak, Virginia M.
2005-01-01
The high school course that integrated first year algebra with an introductory environmental biology/anatomy and physiology course, in order to solve algebra problems is discussed. Lessons and activities for the course were taken by identifying the areas where mathematics and biology content intervenes may help students understand biology concepts…
Balancing the Equation: Do Course Variations in Algebra 1 Provide Equal Student Outcomes?
ERIC Educational Resources Information Center
Kenfield, Danielle M.
2013-01-01
Historically, algebra has served as a gatekeeper that divides students into academic programs with varying opportunities to learn and controls access to higher education and career opportunities. Successful completion of Algebra 1 demonstrates mathematical proficiency and allows access to a sequential and progressive path of advanced study that…
NASA Astrophysics Data System (ADS)
Yukhno, L. F.
2007-12-01
The use of modifications of certain well-known methods of the conjugate direction type for solving systems of linear algebraic equations with rectangular matrices is examined. The modified methods are shown to be superior to the original versions with respect to the round-off accumulation; the advantage is especially large for ill-conditioned matrices. Examples are given of the efficient use of the modified methods for solving certain fairly large ill-conditioned problems.
NASA Astrophysics Data System (ADS)
Yukhno, L. F.
2007-11-01
A modification of certain well-known methods of the conjugate direction type is proposed and examined. The modified methods are more stable with respect to the accumulation of round-off errors. Moreover, these methods are applicable for solving ill-conditioned systems of linear algebraic equations that, in particular, arise as approximations of ill-posed problems. Numerical results illustrating the advantages of the proposed modification are presented.
NASA Astrophysics Data System (ADS)
Akbari, M. R.; Ganji, D. D.; Ahmadi, A. R.; Kachapi, Sayyid H. Hashemi
2014-03-01
In the current paper, a simplified model of Tower Cranes has been presented in order to investigate and analyze the nonlinear differential equation governing on the presented system in three different cases by Algebraic Method (AGM). Comparisons have been made between AGM and Numerical Solution, and these results have been indicated that this approach is very efficient and easy so it can be applied for other nonlinear equations. It is citable that there are some valuable advantages in this way of solving differential equations and also the answer of various sets of complicated differential equations can be achieved in this manner which in the other methods, so far, they have not had acceptable solutions. The simplification of the solution procedure in Algebraic Method and its application for solving a wide variety of differential equations not only in Vibrations but also in different fields of study such as fluid mechanics, chemical engineering, etc. make AGM be a powerful and useful role model for researchers in order to solve complicated nonlinear differential equations.
Shifted Riccati Procedure: Application to Conformal Barotropic FRW Cosmologies
NASA Astrophysics Data System (ADS)
Rosu, Haret C.; Khmelnytskaya, Kira V.
2011-02-01
In the case of barotropic FRW cosmologies, the Hubble parameter in conformal time is the solution of a simple Riccati equation of constant coefficients. We consider these cosmologies in the framework of nonrelativistic supersymmetry that has been effective in the area of supersymmetric quantum mechanics. Recalling that Faraoni [Amer. J. Phys. 67 (1999), 732-734] showed how to reduce the barotropic FRW system of differential equations to simple harmonic oscillator differential equations, we set the latter equations in the supersymmetric approach and divide their solutions into two classes of 'bosonic' (nonsingular) and 'fermionic' (singular) cosmological zero-mode solutions. The fermionic equations can be considered as representing cosmologies of Stephani type, i.e., inhomogeneous and curvature-changing in the conformal time. We next apply the so-called shifted Riccati procedure by introducing a constant additive parameter, denoted by S, in the common Riccati solution of these supersymmetric partner cosmologies. This leads to barotropic Stephani cosmologies with periodic singularities in their spatial curvature indices that we call U and V cosmologies, the first being of bosonic type and the latter of fermionic type. We solve completely these cyclic singular cosmologies at the level of their zero modes showing that an acceptable shift parameter should be purely imaginary, which in turn introduces a parity-time (PT) property of the partner curvature indices.
The consistent Riccati expansion and new interaction solution for a Boussinesq-type coupled system
NASA Astrophysics Data System (ADS)
Ruan, Shao-Qing; Yu, Wei-Feng; Yu, Jun; Yu, Guo-Xiang
2015-06-01
Starting from the Davey-Stewartson equation, a Boussinesq-type coupled equation system is obtained by using a variable separation approach. For the Boussinesq-type coupled equation system, its consistent Riccati expansion (CRE) solvability is studied with the help of a Riccati equation. It is significant that the soliton-cnoidal wave interaction solution, expressed explicitly by Jacobi elliptic functions and the third type of incomplete elliptic integral, of the system is also given. Project supported by the National Natural Science Foundation of China (Grant No. 11275129).
Bethe subalgebras in affine Birman-Murakami-Wenzl algebras and flat connections for q-KZ equations
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Kirillov, A. N.; Tarasov, V. O.
2016-05-01
Commutative sets of Jucys-Murphy elements for affine braid groups of {A}(1),{B}(1),{C}(1),{D}(1) types were defined. Construction of R-matrix representations of the affine braid group of type {C}(1) and its distinguished commutative subgroup generated by the {C}(1)-type Jucys-Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik-Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the {C}(1)-type Jucys-Murphy elements. We specify our general construction to the case of the Birman-Murakami-Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl-Cherednik elements {Y}\\prime {{s}} in the double affine Hecke algebra of type A. Dedicated to Professor Rodney Baxter on the occasion of his 75th Birthday.
Bethe subalgebras in affine Birman–Murakami–Wenzl algebras and flat connections for q-KZ equations
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Kirillov, A. N.; Tarasov, V. O.
2016-05-01
Commutative sets of Jucys–Murphy elements for affine braid groups of {A}(1),{B}(1),{C}(1),{D}(1) types were defined. Construction of R-matrix representations of the affine braid group of type {C}(1) and its distinguished commutative subgroup generated by the {C}(1)-type Jucys–Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik–Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the {C}(1)-type Jucys–Murphy elements. We specify our general construction to the case of the Birman–Murakami–Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl–Cherednik elements {Y}\\prime {{s}} in the double affine Hecke algebra of type A. Dedicated to Professor Rodney Baxter on the occasion of his 75th Birthday.
Nonlinear dynamics of DNA - Riccati generalized solitary wave solutions
NASA Astrophysics Data System (ADS)
Alka, W.; Goyal, Amit; Nagaraja Kumar, C.
2011-01-01
We study the nonlinear dynamics of DNA, for longitudinal and transverse motions, in the framework of the microscopic model of Peyrard and Bishop. The coupled nonlinear partial differential equations for dynamics of DNA model, which consists of two long elastic homogeneous strands connected with each other by an elastic membrane, have been solved for solitary wave solution which is further generalized using Riccati parameterized factorization method.
NASA Astrophysics Data System (ADS)
Gladwin Pradeep, R.; Chandrasekar, V. K.; Mohanasubha, R.; Senthilvelan, M.; Lakshmanan, M.
2016-07-01
We identify contact transformations which linearize the given equations in the Riccati and Abel chains of nonlinear scalar and coupled ordinary differential equations to the same order. The identified contact transformations are not of Cole-Hopf type and are new to the literature. The linearization of Abel chain of equations is also demonstrated explicitly for the first time. The contact transformations can be utilized to derive dynamical symmetries of the associated nonlinear ODEs. The wider applicability of identifying this type of contact transformations and the method of deriving dynamical symmetries by using them is illustrated through two dimensional generalizations of the Riccati and Abel chains as well.
Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients
Kalchev, D
2012-04-02
This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-{rho}AMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and the computed solution used to evaluate some functional(s) of interest that then determine if the coefficient sample is acceptable or not. The MCMC process is hence computationally intensive and requires the solvers used to be efficient and fast. This fact that at every step of MCMC the resulting linear system changes, makes an already existing solver built for the old problem perhaps not as efficient for the problem corresponding to the new sampled coefficient. This motivates the main goal of our study, namely, to adapt an already existing solver to handle the problem (with changed coefficient) with the objective to achieve this goal to be faster and more efficient than building a completely new solver from scratch. Our approach utilizes the local element matrices (for the problem with changed coefficients) to build local problems associated with constructed by the method agglomerated elements (a set of subdomains that cover the given computational domain). We solve a generalized eigenproblem for each set in a subspace spanned by the previous local coarse space (used for the old solver) and a vector, component of the error, that the old solver cannot handle. A portion of the spectrum of these local eigen-problems (corresponding to eigenvalues close to zero) form the
NASA Astrophysics Data System (ADS)
Akhunov, R. R.; Gazizov, T. R.; Kuksenko, S. P.
2016-08-01
The mean time needed to solve a series of systems of linear algebraic equations (SLAEs) as a function of the number of SLAEs is investigated. It is proved that this function has an extremum point. An algorithm for adaptively determining the time when the preconditioner matrix should be recalculated when a series of SLAEs is solved is developed. A numerical experiment with multiply solving a series of SLAEs using the proposed algorithm for computing 100 capacitance matrices with two different structures—microstrip when its thickness varies and a modal filter as the gap between the conductors varies—is carried out. The speedups turned out to be close to the optimal ones.
Middle School Students' Reading Comprehension of Mathematical Texts and Algebraic Equations
ERIC Educational Resources Information Center
Duru, Adem; Koklu, Onder
2011-01-01
In this study, middle school students' abilities to translate mathematical texts into algebraic representations and vice versa were investigated. In addition, students' difficulties in making such translations and the potential sources for these difficulties were also explored. Both qualitative and quantitative methods were used to collect data…
Yu, Zhang; Zhang, Yufeng
2009-01-30
Three semi-direct sum Lie algebras are constructed, which is an efficient and new way to obtain discrete integrable couplings. As its applications, three discrete integrable couplings associated with the modified KdV lattice equation are worked out. The approach can be used to produce other discrete integrable couplings of the discrete hierarchies of solition equations. PMID:20119478
NASA Astrophysics Data System (ADS)
Levin, A. M.; Olshanetsky, M. A.; Zotov, A. V.
2016-08-01
We construct twisted Calogero-Moser systems with spins as Hitchin systems derived from the Higgs bundles over elliptic curves, where the transition operators are defined by arbitrary finite-order automorphisms of the underlying Lie algebras. We thus obtain a spin generalization of the twisted D'Hoker-Phong and Bordner-Corrigan-Sasaki-Takasaki systems. In addition, we construct the corresponding twisted classical dynamical r-matrices and the Knizhnik-Zamolodchikov-Bernard equations related to the automorphisms of Lie algebras.
Generalized Flip-Flop Input Equations Based on a Four-Valued Boolean Algebra
NASA Technical Reports Server (NTRS)
Tucker, Jerry H.; Tapia, Moiez A.
1996-01-01
A procedure is developed for obtaining generalized flip-flop input equations, and a concise method is presented for representing these equations. The procedure is based on solving a four-valued characteristic equation of the flip-flop, and can encompass flip-flops that are too complex to approach intuitively. The technique is presented using Karnaugh maps, but could easily be implemented in software.
Algebraic Equations and Inequalities: Issues for Research and Teaching. Research Forum
ERIC Educational Resources Information Center
Bazzini, Luciana; Tsamir, Pessia
2004-01-01
The presentations address a variety of difficulties occurring in students' solutions of equations and inequalities, and suggest different reasons for these difficulties. When analyzing students' performances, [BB] and [TTT] mention students' tendencies to make irrelevant connections between equations and inequalities as a problematic phenomenon.…
NASA Astrophysics Data System (ADS)
Krasnikov, S. D.; Kuznetsov, E. B.
2016-09-01
Numerical continuation of solution through certain singular points of the curve of the set of solutions to a system of nonlinear algebraic or transcendental equations with a parameter is considered. Bifurcation points of codimension two and three are investigated. Algorithms and computer programs are developed that implement the procedure of discrete parametric continuation of the solution and find all branches at simple bifurcation points of codimension two and three. Corresponding theorems are proved, and each algorithm is rigorously justified. A novel algorithm for the estimation of errors of tangential vectors at simple bifurcation points of a finite codimension m is proposed. The operation of the computer programs is demonstrated by test examples, which allows one to estimate their efficiency and confirm the theoretical results.
ERIC Educational Resources Information Center
Samuel, Koji; Mulenga, H. M.; Angel, Mukuka
2016-01-01
This paper investigates the challenges faced by secondary school teachers and pupils in the teaching and learning of algebraic linear equations. The study involved 80 grade 11 pupils and 15 teachers of mathematics, drawn from 4 selected secondary schools in Mufulira district, Zambia in Central Africa. A descriptive survey method was employed to…
ERIC Educational Resources Information Center
Fuchs, Lynn S.; Zumeta, Rebecca O.; Schumacher, Robin Finelli; Powell, Sarah R.; Seethaler, Pamela M.; Hamlett, Carol L.; Fuchs, Douglas
2010-01-01
The purpose of this study was to assess the effects of schema-broadening instruction (SBI) on second graders' word-problem-solving skills and their ability to represent the structure of word problems using algebraic equations. Teachers (n = 18) were randomly assigned to conventional word-problem instruction or SBI word-problem instruction, which…
Profiles of Algebraic Competence
ERIC Educational Resources Information Center
Humberstone, J.; Reeve, R.A.
2008-01-01
The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map…
Computer subroutine ISUDS accurately solves large system of simultaneous linear algebraic equations
NASA Technical Reports Server (NTRS)
Collier, G.
1967-01-01
Computer program, an Iterative Scheme Using a Direct Solution, obtains double precision accuracy using a single-precision coefficient matrix. ISUDS solves a system of equations written in matrix form as AX equals B, where A is a square non-singular coefficient matrix, X is a vector, and B is a vector.
ERIC Educational Resources Information Center
Ngu, Bing Hiong; Yeung, Alexander Seeshing
2013-01-01
Text editing directs students' attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three…
Chandrasekhar equations for infinite dimensional systems
NASA Technical Reports Server (NTRS)
Ito, K.; Powers, R. K.
1985-01-01
Chandrasekhar equations are derived for linear time invariant systems defined on Hilbert spaces using a functional analytic technique. An important consequence of this is that the solution to the evolutional Riccati equation is strongly differentiable in time and one can define a strong solution of the Riccati differential equation. A detailed discussion on the linear quadratic optimal control problem for hereditary differential systems is also included.
Algebra for Gifted Third Graders.
ERIC Educational Resources Information Center
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
Numerical algebraic geometry and algebraic kinematics
NASA Astrophysics Data System (ADS)
Wampler, Charles W.; Sommese, Andrew J.
In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.
Zhuk, Sergiy
2013-10-15
In this paper we present Kalman duality principle for a class of linear Differential-Algebraic Equations (DAE) with arbitrary index and time-varying coefficients. We apply it to an ill-posed minimax control problem with DAE constraint and derive a corresponding dual control problem. It turns out that the dual problem is ill-posed as well and so classical optimality conditions are not applicable in the general case. We construct a minimizing sequence u-circumflex{sub {epsilon}} for the dual problem applying Tikhonov method. Finally we represent u-circumflex{sub {epsilon}} in the feedback form using Riccati equation on a subspace which corresponds to the differential part of the DAE.
NASA Astrophysics Data System (ADS)
Shabana, Ahmed A.; Hussein, Bassam A.
2009-11-01
In this paper, a two-loop implicit sparse matrix numerical integration (TLISMNI) procedure for the solution of constrained rigid and flexible multibody system differential and algebraic equations is proposed. The proposed method ensures that the kinematic constraint equations are satisfied at the position, velocity and acceleration levels. In this method, a sparse Lagrangian augmented form of the equations of motion that ensures that the constraints are satisfied at the acceleration level is first used to solve for all the accelerations and Lagrange multipliers. The independent coordinates and velocities are then identified and integrated using HTT or Newmark formulas, expressed in this paper in terms of the independent accelerations only. The constraint equations at the position level are then used within an iterative Newton-Raphson procedure to determine the dependent coordinates. The dependent velocities are determined by solving a linear system of algebraic equations. In order to effectively exploit efficient sparse matrix techniques and have minimum storage requirements, a two-loop iterative method is proposed. Equally important, the proposed method avoids the use of numerical differentiation which is commonly associated with the use of implicit integration methods in multibody system algorithms. Numerical examples are presented in order to demonstrate the use of the new integration procedure.
Colored Quantum Algebra and Its Bethe State
NASA Astrophysics Data System (ADS)
Wang, Jin-Zheng; Jia, Xiao-Yu; Wang, Shi-Kun
2014-12-01
We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.
NASA Technical Reports Server (NTRS)
Ruge, J. W.; Stueben, K.
1987-01-01
The state of the art in algebraic multgrid (AMG) methods is discussed. The interaction between the relaxation process and the coarse grid correction necessary for proper behavior of the solution probes is discussed in detail. Sufficient conditions on relaxation and interpolation for the convergence of the V-cycle are given. The relaxation used in AMG, what smoothing means in an algebraic setting, and how it relates to the existing theory are considered. Some properties of the coarse grid operator are discussed, and results on the convergence of two-level and multilevel convergence are given. Details of an algorithm particularly studied for problems obtained by discretizing a single elliptic, second order partial differential equation are given. Results of experiments with such problems using both finite difference and finite element discretizations are presented.
NASA Astrophysics Data System (ADS)
Kipps, Mark R.
1994-03-01
The modeling of power systems has been primarily driven by the commercial power utility industry. These models usually involve the assumption that system bus voltage and frequency are constant. However, in applications such as shipboard power systems this infinite bus assumption is not valid. This thesis investigates the modeling of a synchronous generator and various loads in a modular fashion on a finite bus. The simulation presented allows the interconnection of multiple state-space models via a bus voltage model. The major difficulty encountered in building a model which computes bus voltage at each time step is that bus voltage is a function of current and current derivative terms. Bus voltage is also an input to the state equations which produce the current and current derivatives. This creates an algebraic loop which is a form of implicit differential equation. A routine has been developed by Linda Petzold of Lawrence Livermore Laboratory for solving these types of equations. The routine, called Differential Algebraic System Solver (DASSL), has been implemented in a pre-release version of the software Advanced Continuous Simulation Language (ACSL) and has been made available to the Naval Postgraduate School on a trial basis. An isolated power system is modeled using this software and the DASSL routine. The system response to several dynamic situations is studied and the results are presented.
Teaching Algebra without Algebra
ERIC Educational Resources Information Center
Kalman, Richard S.
2008-01-01
Algebra is, among other things, a shorthand way to express quantitative reasoning. This article illustrates ways for the classroom teacher to convert algebraic solutions to verbal problems into conversational solutions that can be understood by students in the lower grades. Three reasonably typical verbal problems that either appeared as or…
Second-Order Algebraic Theories
NASA Astrophysics Data System (ADS)
Fiore, Marcelo; Mahmoud, Ola
Fiore and Hur [10] recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding.
Chudnovsky, D. V.
1978-01-01
For systems of nonlinear equations having the form [Ln - (∂/∂t), Lm - (∂/∂y)] = 0 the class of meromorphic solutions obtained from the linear equations [Formula: see text] is presented. PMID:16592559
NASA Astrophysics Data System (ADS)
Rosu, H. C.; Khmelnytskaya, K. V.
2011-09-01
We determine the kind of parametric oscillators that are generated in the usual factorization procedure of second-order linear differential equations when one introduces a constant shift of the Riccati solution of the classical harmonic oscillator. The mathematical results show that some of these oscillators could be of physical nature. We give the solutions of the obtained second-order differential equations and the values of the shift parameter providing strictly periodic and antiperiodic solutions. We also notice that this simple problem presents parity-time (PT) symmetry. Possible applications are mentioned.
NASA Astrophysics Data System (ADS)
Hiley, B. J.
In this chapter, we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. By using this algebra, we show that it contains both the Weyl-von Neumann and the Moyal quantum algebras. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave-function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.
NASA Astrophysics Data System (ADS)
Smirnov, Andrey
2010-08-01
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl( N;?)-case is discussed.
NASA Astrophysics Data System (ADS)
Smirnov, Andrey
2010-08-01
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.
Covariant deformed oscillator algebras
NASA Technical Reports Server (NTRS)
Quesne, Christiane
1995-01-01
The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.
Cornish-Bowden, A
1976-01-01
Methods of deriving rate equations that rely on repetition of terms for identification of redundant or invalid terms give incorrect results if used with mechanisms in which some rate constants appear more than once. PMID:999635
Characteristic Numbers of Matrix Lie Algebras
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Fan, En-Gui
2008-04-01
A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.
The Riccati transfer matrix method. [for computerized structural analysis
NASA Technical Reports Server (NTRS)
Horner, G. C.; Pilkey, W. D.
1977-01-01
The Riccati transfer matrix method is a new technique for analyzing structural members. This new technique makes use of an existing large catalog of transfer matrices for various structural members such as rotating shafts. The numerical instability encountered when calculating high resonant frequencies, static response of a flexible member on a stiff foundation, or the response of a long member by the transfer matrix method is eliminated by the Riccati transfer matrix method. The computational time and storage requirements of the Riccati transfer matrix method are about half the values for the transfer matrix method. A rotating shaft analysis demonstrates the numerical accuracy of the method.
Riccati parameterized self-similar waves in tapered graded-index waveguides
NASA Astrophysics Data System (ADS)
Goyal, Amit; Gupta, Rama; Loomba, Shally; Kumar, C. N.
2012-10-01
We present a large family of self-similar waves by tailoring the tapering function, through Riccati parameter, in a tapered graded-index nonlinear waveguide amplifier. We show the existence of bright similaritons, self-similar Akhmediev breathers and self-similar rogue waves for generalized nonlinear Schrödinger equation with constant dispersion and nonlinearity, and a distributed gain. We illustrate the procedure to amplify the intensity of self-similar waves using isospectral Hamiltonian approach. This approach provides a handle to find analytically a wide class of tapering function and thus enabling one to control the self-similar wave structure and dynamical behavior.
ERIC Educational Resources Information Center
Pavelle, Richard; And Others
1981-01-01
Describes the nature and use of computer algebra and its applications to various physical sciences. Includes diagrams illustrating, among others, a computer algebra system and flow chart of operation of the Euclidean algorithm. (SK)
A Riccati type PDE for light-front higher helicity vertices
NASA Astrophysics Data System (ADS)
Bengtsson, Anders K. H.
2014-09-01
This paper is based on a curious observation about an equation related to the tracelessness constraints of higher spin gauge fields. A similar equation also occurs in the theory of continuous spin representations of the Poincaré group. Expressed in an oscillator basis for the higher spin fields, the equation becomes a non-linear partial differential operator of the Riccati type acting on the vertex functions. The consequences of the equation for the cubic vertex is investigated in the light-front formulation of higher spin theory. The vertex is fixed by the PDE up to a set of terms that can be considered as boundary data for the PDE. These terms can serve as off-shell quantum corrections.
Algebraic integrability: a survey.
Vanhaecke, Pol
2008-03-28
We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems. PMID:17588863
ERIC Educational Resources Information Center
Buerman, Margaret
2007-01-01
Finding real-world examples for middle school algebra classes can be difficult but not impossible. As we strive to accomplish teaching our students how to solve and graph equations, we neglect to teach the big ideas of algebra. One of those big ideas is functions. This article gives three examples of functions that are found in Arches National…
Adaptive Algebraic Multigrid Methods
Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J
2004-04-09
Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.
NASA Astrophysics Data System (ADS)
Carlen, E. A.; Carvalho, M. C.; Orlandi, E.
1999-06-01
This is the first of two papers devoted to the study of a nonlocal evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider subcritical temperatures, for which there are two local equilibria, and begin the proof of a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibria; i.e., the fronts. We shall show in the second paper that an initial perturbation v 0 of a front that is sufficiently small in L 2 norm, and sufficiently localized that ∫ x 2 v 0( x)2 dx<∞, yields a solution that relaxes to another front, selected by a conservation law, in the L 1 norm at an algebraic rate that we explicitly estimate. There we also obtain rates for the relaxation in the L 2 norm and the rate of decrease of the excess free energy. Here we prove a number of estimates essential for this result. Moreover, the estimates proved here suffice to establish the main result in an important special case.
Invertible linear transformations and the Lie algebras
NASA Astrophysics Data System (ADS)
Zhang, Yufeng; Tam, Honwah; Guo, Fukui
2008-07-01
With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity.
ERIC Educational Resources Information Center
Benjamin, Carl; And Others
Presented are student performance objectives, a student progress chart, and assignment sheets with objective and diagnostic measures for the stated performance objectives in College Algebra II. Topics covered include: differencing and complements; real numbers; factoring; fractions; linear equations; exponents and radicals; complex numbers,…
ERIC Educational Resources Information Center
Schaufele, Christopher; Zumoff, Nancy
Earth Algebra is an entry level college algebra course that incorporates the spirit of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics at the college level. The context of the course places mathematics at the center of one of the major current concerns of the world. Through…
ERIC Educational Resources Information Center
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
NASA Astrophysics Data System (ADS)
Liu, Yu-Ping; Gao, Yi-Tian; Wei, Guang-Mei
2012-02-01
The inhomogeneous nonlinear Schrödinger-type (NLS) model from certain plasmas and optical fibers is investigated with symbolic computation. An improved Γ-Riccati Bäcklund transformation (Γ-R BT) is presented, which can generate successively a hierarchy of solutions through algebraic manipulations. Based on the improved Γ-R BT, the Darboux transformation is obtained, the analytic one/two-soliton-like solutions are presented, and the physical characteristics of the influences of the coefficient parameters on the propagation of the soliton pulses are discussed graphically.
Generalized Galilean algebras and Newtonian gravity
NASA Astrophysics Data System (ADS)
González, N.; Rubio, G.; Salgado, P.; Salgado, S.
2016-04-01
The non-relativistic versions of the generalized Poincaré algebras and generalized AdS-Lorentz algebras are obtained. These non-relativistic algebras are called, generalized Galilean algebras of type I and type II and denoted by GBn and GLn respectively. Using a generalized Inönü-Wigner contraction procedure we find that the generalized Galilean algebras of type I can be obtained from the generalized Galilean algebras type II. The S-expansion procedure allows us to find the GB5 algebra from the Newton Hooke algebra with central extension. The procedure developed in Ref. [1] allows us to show that the nonrelativistic limit of the five dimensional Einstein-Chern-Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.
Formal scattering theory by an algebraic approach
NASA Astrophysics Data System (ADS)
Alhassid, Y.; Levine, R. D.
1985-02-01
Formal scattering theory is recast in a Lie-algebraic form. The central result is an algebraic Lippmann-Schwinger equation for the wave operator from which an algebraic form of the Born series (containing only linked terms) is obtained. When a finite Lie algebra is sufficient, The Mo/ller wave operator, on the energy shell, can be solved for explicitly as an element of the corresponding group. The method is illustrated for the separable potential whose relevant algebra is found to be U(1,1).
Elementary Algebra Connections to Precalculus
ERIC Educational Resources Information Center
Lopez-Boada, Roberto; Daire, Sandra Arguelles
2013-01-01
This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…
Rota-Baxter operators on Witt and Virasoro algebras
NASA Astrophysics Data System (ADS)
Gao, Xu; Liu, Ming; Bai, Chengming; Jing, Naihuan
2016-10-01
The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.
Direct and inverse relationships between Riccati systems coupled with multiplicative terms.
Navickas, Z; Vilkas, R; Telksnys, T; Ragulskis, M
2016-01-01
An analytical and computational framework for the derivation of solitary solutions to biological systems describing the cooperation and competition of species and expressed by the system of Riccati equations coupled with multiplicative terms is presented in this paper. It is demonstrated that relationships between these solitary solutions can be either direct or inverse. Thus, an infinitesimal perturbation of one population would lead to an infinitesimal change in the other population - if only both solitary solutions are coupled with the direct relationship. But, in general, that is not true if solitary solutions are coupled with the inverse relationship - an infinitesimal perturbation of one population may result into a non-infinitesimal change in the other population. Necessary and sufficient conditions for the existence of solitary solutions are derived in the space of the system's parameters and initial conditions. PMID:27159649
Riccati parameterized self-similar waves in two-dimensional graded-index waveguide
NASA Astrophysics Data System (ADS)
Kumar De, Kanchan; Goyal, Amit; Raju, Thokala Soloman; Kumar, C. N.; Panigrahi, Prasanta K.
2015-04-01
An analytical method based on gauge-similarity transformation technique has been employed for mapping a (2+1)- dimensional variable coefficient coupled nonlinear Schrödinger equations (vc-CNLSE) with dispersion, nonlinearity and gain to standard NLSE. Under certain functional relations we construct a large family of self-similar waves in the form of bright similaritons, Akhmediev breathers and rogue waves. We report the effect of dispersion on the intensity of the solitary waves. Further, we illustrate the procedure to amplify the intensity of self-similar waves using isospectral Hamiltonian approach. This approach provides an efficient mechanism to generate analytically a wide class of tapering profiles and widths by exploiting the Riccati parameter. Equivalently, it enables one to control efficiently the self-similar wave structures and hence their evolution.
Direct and inverse relationships between Riccati systems coupled with multiplicative terms.
Navickas, Z; Vilkas, R; Telksnys, T; Ragulskis, M
2016-01-01
An analytical and computational framework for the derivation of solitary solutions to biological systems describing the cooperation and competition of species and expressed by the system of Riccati equations coupled with multiplicative terms is presented in this paper. It is demonstrated that relationships between these solitary solutions can be either direct or inverse. Thus, an infinitesimal perturbation of one population would lead to an infinitesimal change in the other population - if only both solitary solutions are coupled with the direct relationship. But, in general, that is not true if solitary solutions are coupled with the inverse relationship - an infinitesimal perturbation of one population may result into a non-infinitesimal change in the other population. Necessary and sufficient conditions for the existence of solitary solutions are derived in the space of the system's parameters and initial conditions.
ERIC Educational Resources Information Center
Chazan, Daniel; Sela, Hagit; Herbst, Patricio
2012-01-01
We illustrate a method, which is modeled on "breaching experiments," for studying tacit norms that govern classroom interaction around particular mathematical content. Specifically, this study explores norms that govern teachers' expectations for the doing of word problems in school algebra. Teacher study groups discussed representations of…
A Riccati approach for constrained linear quadratic optimal control
NASA Astrophysics Data System (ADS)
Sideris, Athanasios; Rodriguez, Luis A.
2011-02-01
An active-set method is proposed for solving linear quadratic optimal control problems subject to general linear inequality path constraints including mixed state-control and state-only constraints. A Riccati-based approach is developed for efficiently solving the equality constrained optimal control subproblems generated during the procedure. The solution of each subproblem requires computations that scale linearly with the horizon length. The algorithm is illustrated with numerical examples.
Static friction, differential algebraic systems and numerical stability
NASA Astrophysics Data System (ADS)
Chen, Jian; Schinner, Alexander; Matuttis, Hans-Georg
We show how Differential Algebraic Systems (Ordinary Differential Equations with algebraic constraints) in mechanics are affected by stability issues and we implement Lubich's projection method to reduce the error to practically zero. Then, we explain how the "numerically exact" implementation for static friction by Differential Algebraic Systems can be stabilized. We conclude by comparing the corresponding steps in the "Contact mechanics" introduced by Moreau.
Exploring Algebraic Misconceptions with Technology
ERIC Educational Resources Information Center
Sakow, Matthew; Karaman, Ruveyda
2015-01-01
Many students struggle with algebra, from simplifying expressions to solving systems of equations. Students also have misconceptions about the meaning of variables. In response to the question "Can x + y + z ever equal x + p + z?" during a student interview, the student claimed, "Never . . . because p has to have a different value…
Kinds of Knowledge in Algebra.
ERIC Educational Resources Information Center
Lewis, Clayton
Solving equations in elementary algebra requires knowledge of the permitted operations, and knowledge of what operation to use at a given point in the solution process. While just these kinds of knowledge would be adequate for an ideal solver, human solvers appear to need and use other kinds of knowledge. First, many errors seem to indicate that…
Math for All Learners: Algebra.
ERIC Educational Resources Information Center
Meader, Pam; Storer, Judy
This book consists of a series of activities aimed at providing a problem solving, hands-on approach so that students can experience concepts in algebra. Topics include ratio and proportion, patterns and formulas, integers, polynomials, linear equations, graphs, and probability. The activities come in the form of reproducible blackline masters…
Algebraic Systems and Pushdown Automata
NASA Astrophysics Data System (ADS)
Petre, Ion; Salomaa, Arto
We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.
ERIC Educational Resources Information Center
Blakley, G. R.
1982-01-01
Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)
NASA Astrophysics Data System (ADS)
Vaninsky, Alexander
2011-04-01
This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos - satisfying an axiom sin2 + cos2 = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two different interpretations of the TF are discussed with many others potentially possible. The main objective of this article is to introduce a broader view of trigonometry that can serve as motivation for mathematics students and teachers to study and teach abstract algebraic structures.
Titration Calculations with Computer Algebra Software
ERIC Educational Resources Information Center
Lachance, Russ; Biaglow, Andrew
2012-01-01
This article examines the symbolic algebraic solution of the titration equations for a diprotic acid, as obtained using "Mathematica," "Maple," and "Mathcad." The equilibrium and conservation equations are solved symbolically by the programs to eliminate the approximations that normally would be performed by the student. Of the three programs,…
Algebraic operator approach to gas kinetic models
NASA Astrophysics Data System (ADS)
Il'ichov, L. V.
1997-02-01
Some general properties of the linear Boltzmann kinetic equation are used to present it in the form ∂ tϕ = - Â†Âϕ with the operators ÂandÂ† possessing some nontrivial algebraic properties. When applied to the Keilson-Storer kinetic model, this method gives an example of quantum ( q-deformed) Lie algebra. This approach provides also a natural generalization of the “kangaroo model”.
Quantization of Algebraic Reduction
Sniatycki, Jeodrzej
2007-11-14
For a Poisson algebra obtained by algebraic reduction of symmetries of a quantizable system we develop an analogue of geometric quantization based on the quantization structure of the original system.
Shifted genus expanded W ∞ algebra and shifted Hurwitz numbers
NASA Astrophysics Data System (ADS)
Zheng, Quan
2016-05-01
We construct the shifted genus expanded W ∞ algebra, which is isomorphic to the central subalgebra A ∞ of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra Λ* defined by Okounkov and Olshanskii. As an application, we get some differential equations for the generating functions of the shifted Hurwitz numbers; thus, we can express the generating functions in terms of the shifted genus expanded cut-and-join operators.
Learning Algebra in a Computer Algebra Environment
ERIC Educational Resources Information Center
Drijvers, Paul
2004-01-01
This article summarises a doctoral thesis entitled "Learning algebra in a computer algebra environment, design research on the understanding of the concept of parameter" (Drijvers, 2003). It describes the research questions, the theoretical framework, the methodology and the results of the study. The focus of the study is on the understanding of…
NASA Technical Reports Server (NTRS)
Iachello, Franco
1995-01-01
An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.
Orientation in operator algebras
Alfsen, Erik M.; Shultz, Frederic W.
1998-01-01
A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics. PMID:9618457
Developing Thinking in Algebra
ERIC Educational Resources Information Center
Mason, John; Graham, Alan; Johnson-Wilder, Sue
2005-01-01
This book is for people with an interest in algebra whether as a learner, or as a teacher, or perhaps as both. It is concerned with the "big ideas" of algebra and what it is to understand the process of thinking algebraically. The book has been structured according to a number of pedagogic principles that are exposed and discussed along the way,…
Connecting Arithmetic to Algebra
ERIC Educational Resources Information Center
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
NASA Astrophysics Data System (ADS)
Yao, Ruo-Xia; Wang, Wei; Chen, Ting-Hua
2014-11-01
Motivated by the widely used ansätz method and starting from the modified Riemann—Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.
Computer algebra and transport theory.
Warsa, J. S.
2004-01-01
Modern symbolic algebra computer software augments and complements more traditional approaches to transport theory applications in several ways. The first area is in the development and enhancement of numerical solution methods for solving the Boltzmann transport equation. Typically, special purpose computer codes are designed and written to solve specific transport problems in particular ways. Different aspects of the code are often written from scratch and the pitfalls of developing complex computer codes are numerous and well known. Software such as MAPLE and MATLAB can be used to prototype, analyze, verify and determine the suitability of numerical solution methods before a full-scale transport application is written. Once it is written, the relevant pieces of the full-scale code can be verified using the same tools I that were developed for prototyping. Another area is in the analysis of numerical solution methods or the calculation of theoretical results that might otherwise be difficult or intractable. Algebraic manipulations are done easily and without error and the software also provides a framework for any additional numerical calculations that might be needed to complete the analysis. We will discuss several applications in which we have extensively used MAPLE and MATLAB in our work. All of them involve numerical solutions of the S{sub N} transport equation. These applications encompass both of the two main areas in which we have found computer algebra software essential.
Computer algebra and operators
NASA Technical Reports Server (NTRS)
Fateman, Richard; Grossman, Robert
1989-01-01
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
Discrete Minimal Surface Algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Hoppe, Jens
2010-05-01
We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
2003-06-03
The ALGEBRA II program allows the user to manipulate data from a finite element analysis before it is plotted by evaluating algebraic expressions. The equation variables are dependent on the input database variable names. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database which can be read by plot programs. Code is written in a portable form as possible. Fortran code is written in ANSI Standard FORTRAN-77. Machine-specific routines are limited in number and are grouped together to minimize the time required to adapt them to a new system. SEACAS codes has been ported to several Unix systems.
2003-06-03
The ALGEBRA II program allows the user to manipulate data from a finite element analysis before it is plotted by evaluating algebraic expressions. The equation variables are dependent on the input database variable names. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database which can be read by plot programs. Code is written in a portable form as possible. Fortran codemore » is written in ANSI Standard FORTRAN-77. Machine-specific routines are limited in number and are grouped together to minimize the time required to adapt them to a new system. SEACAS codes has been ported to several Unix systems.« less
ERIC Educational Resources Information Center
Lin, Cheng-Yao; Kuo, Yu-Chun; Ko, Yi-Yin
2015-01-01
The purpose of this study was to investigate elementary pre-service teachers' content knowledge in algebra (Linear Equation, Quadratic Equation, Functions, System Equations and Polynomials) as well as their technological pedagogical content knowledge (TPACK) in teaching algebra. Participants were 79 undergraduate pre-service teachers who were…
Octonic Massive Field Equations
NASA Astrophysics Data System (ADS)
Demir, Süleyman; Kekeç, Seray
2016-07-01
In the present paper we propose the octonic form of massive field equations based on the analogy with electromagnetism and linear gravity. Using the advantages of octon algebra the Maxwell-Dirac-Proca equations have been reformulated in compact and elegant way. The energy-momentum relations for massive field are discussed.
Multidimensional integrable systems and deformations of Lie algebra homomorphisms
Dunajski, Maciej; Grant, James D. E.; Strachan, Ian A. B.
2007-09-15
We use deformations of Lie algebra homomorphisms to construct deformations of dispersionless integrable systems arising as symmetry reductions of anti-self-dual Yang-Mills equations with a gauge group Diff(S{sup 1})
A Richer Understanding of Algebra
ERIC Educational Resources Information Center
Foy, Michelle
2008-01-01
Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number, pre-algebra, and algebra…
Invariant algebraic surfaces for a virus dynamics
NASA Astrophysics Data System (ADS)
Valls, Claudia
2015-08-01
In this paper, we provide a complete classification of the invariant algebraic surfaces and of the rational first integrals for a well-known virus system. In the proofs, we use the weight-homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.
Transformation of time dependence to linear algebra
NASA Astrophysics Data System (ADS)
Menšík, Miroslav
2005-10-01
Reduced density matrix and memory function in the Nakajima-Zwanzig equation are expanded in properly chosen basis of special functions. This trick completely transforms time dependence to linear algebra. Then, the master equation for memory function is constructed and expanded in the same basis functions. For the model of a simple harmonic oscillator it is shown that this trick introduces infinite partial summation of the memory function in the system-bath interaction.
Connecting Algebra and Chemistry.
ERIC Educational Resources Information Center
O'Connor, Sean
2003-01-01
Correlates high school chemistry curriculum with high school algebra curriculum and makes the case for an integrated approach to mathematics and science instruction. Focuses on process integration. (DDR)
Noncommutative Pfaffians associated with the orthogonal algebra
Artamonov, Dmitrii V; Golubeva, Valentina A
2012-12-31
Commutators of Pfaffians associated with the orthogonal algebra are found in skew-symmetric and root realizations of o{sub N}. A generating function of Pfaffians is proved to satisfy the reflection equation. A relation between Pfaffians in skew-symmetric and root realizations of o{sub N} is established. Using these results we construct an integrable equation of Knizhnik-Zamolodchikov type using the Capelli central elements in U(o{sub N}), which are sums of squares of the considered Pfaffians. A classical limit of the obtained Knizhnik-Zamolodchikov type equation turns out to be a very specific system of equations of isomonodromic deformations. Bibliography: 18 titles.
Difference equations and some of their solutions
NASA Astrophysics Data System (ADS)
Atakishiyev, Natig M.
1996-04-01
Some methods for solving difference equations are discussed. As particular examples we consider in detail the difference equations corresponding to the Kravchuk functions, q-harmonic oscillator wavefunctions, and the Clebsch-Gordan coefficients for the quantum algebra suq(2).
Successfully Transitioning to Linear Equations
ERIC Educational Resources Information Center
Colton, Connie; Smith, Wendy M.
2014-01-01
The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this…
Algebraic Reasoning through Patterns
ERIC Educational Resources Information Center
Rivera, F. D.; Becker, Joanne Rossi
2009-01-01
This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…
ERIC Educational Resources Information Center
Merlin, Ethan M.
2013-01-01
This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…
Unified derivation of exact solutions to the relativistic Coulomb problem: Lie algebraic approach
NASA Astrophysics Data System (ADS)
Panahi, H.; Baradaran, M.; Savadi, A.
2015-10-01
Exact algebraic solutions of the D-dimensional Dirac and Klein-Gordon equations for the Coulomb potential are obtained in a unified treatment. It is shown that two cases are reducible to the same basic equation, which can be solved exactly. Using the Lie algebraic approach, the general exact solutions of the problem are obtained within the framework of representation theory of the sl(2) Lie algebra.
Computational algebraic geometry of epidemic models
NASA Astrophysics Data System (ADS)
Rodríguez Vega, Martín.
2014-06-01
Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.
Lie algebra extensions of current algebras on S3
NASA Astrophysics Data System (ADS)
Kori, Tosiaki; Imai, Yuto
2015-06-01
An affine Kac-Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac-Moody algebras give it for two-dimensional conformal field theory.
Leibniz algebras associated with representations of filiform Lie algebras
NASA Astrophysics Data System (ADS)
Ayupov, Sh. A.; Camacho, L. M.; Khudoyberdiyev, A. Kh.; Omirov, B. A.
2015-12-01
In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra nn,1. We introduce a Fock module for the algebra nn,1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra nn,1 with condition that the ideal I is a Fock nn,1-module, where I is the ideal generated by squares of elements from L. We also consider Leibniz algebras with corresponding Lie algebra nn,1 and such that the action I ×nn,1 → I gives rise to a minimal faithful representation of nn,1. The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4.
Maxwell Equations and the Redundant Gauge Degree of Freedom
ERIC Educational Resources Information Center
Wong, Chun Wa
2009-01-01
On transformation to the Fourier space (k,[omega]), the partial differential Maxwell equations simplify to algebraic equations, and the Helmholtz theorem of vector calculus reduces to vector algebraic projections. Maxwell equations and their solutions can then be separated readily into longitudinal and transverse components relative to the…
Operator product expansion algebra
Holland, Jan; Hollands, Stefan
2013-07-15
We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean φ{sup 4}-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of hep-th/1105.3375, that the 3-point OPE,
Algebraic Semantics for Narrative
ERIC Educational Resources Information Center
Kahn, E.
1974-01-01
This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)
Ten-Year-Old Students Solving Linear Equations
ERIC Educational Resources Information Center
Brizuela, Barbara; Schliemann, Analucia
2004-01-01
In this article, the authors seek to re-conceptualize the perspective regarding students' difficulties with algebra. While acknowledging that students "do" have difficulties when learning algebra, they also argue that the generally espoused criteria for algebra as the ability to work with the syntactical rules for solving equations is…
Aprepro - Algebraic Preprocessor
2005-08-01
Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.
Geometric Algebra for Physicists
NASA Astrophysics Data System (ADS)
Doran, Chris; Lasenby, Anthony
2007-11-01
Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.
Description of DASSL: a differential/algebraic system solver
Petzold, L.R.
1982-09-01
This paper describes a new code DASSL, for the numerical solution of implicit systems of differential/algebraic equations. These equations are written in the form F(t,y,y') = 0, and they can include systems which are substantially more complex than standard form ODE systems y' = f(t,y). Differential/algebraic equations occur in several diverse applications in the physical world. We outline the algorithms and strategies used in DASSL, and explain some of the features of the code. In addition, we outline briefly what needs to be done to solve a problem using DASSL.
Parallel Multigrid Equation Solver
2001-09-07
Prometheus is a fully parallel multigrid equation solver for matrices that arise in unstructured grid finite element applications. It includes a geometric and an algebraic multigrid method and has solved problems of up to 76 mullion degrees of feedom, problems in linear elasticity on the ASCI blue pacific and ASCI red machines.
Algebra 2r, Mathematics (Experimental): 5216.23.
ERIC Educational Resources Information Center
Ellis, June
The third in a series of six guidebooks on minimum course content for second-year algebra, this booklet covers relations, functions, and solving and graphing linear equations, linear inequalities, systems of equations, and systems of inequalities. Overall course goals are specified, a course outline is provided, performance objectives are listed,…
Symmetric linear systems - An application of algebraic systems theory
NASA Technical Reports Server (NTRS)
Hazewinkel, M.; Martin, C.
1983-01-01
Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.
Some remarks on representations of Yang-Mills algebras
NASA Astrophysics Data System (ADS)
Herscovich, Estanislao
2015-01-01
In this article, we present some new properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra 𝔶𝔪(n) on n generators, for n ≥ 2m. We derive from this that any semisimple Lie algebra and even any affine Kac-Moody algebra is a quotient of 𝔶𝔪(n) for n ≥ 4. Combining this with previous results on representations of Yang-Mills algebras given in [Herscovich and Solotar, Ann. Math. 173(2), 1043-1080 (2011)], one may obtain solutions to the Yang-Mills equations by differential operators acting on sections of twisted vector bundles on the affine space of dimension n ≥ 4 associated to representations of any semisimple Lie algebra. We also show that this quotient property does not hold for n = 3, since any morphism of Lie algebras from 𝔶𝔪(3) to 𝔰𝔩(2, k) has in fact solvable image.
The algebraic criteria for the stability of control systems
NASA Technical Reports Server (NTRS)
Cremer, H.; Effertz, F. H.
1986-01-01
This paper critically examines the standard algebraic criteria for the stability of linear control systems and their proofs, reveals important previously unnoticed connections, and presents new representations. Algebraic stability criteria have also acquired significance for stability studies of non-linear differential equation systems by the Krylov-Bogoljubov-Magnus Method, and allow realization conditions to be determined for classes of broken rational functions as frequency characteristics of electrical network.
MODEL IDENTIFICATION AND COMPUTER ALGEBRA
Bollen, Kenneth A.; Bauldry, Shawn
2011-01-01
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods. PMID:21769158
MODEL IDENTIFICATION AND COMPUTER ALGEBRA.
Bollen, Kenneth A; Bauldry, Shawn
2010-10-01
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods.
Numerical linear algebra for reconstruction inverse problems
NASA Astrophysics Data System (ADS)
Nachaoui, Abdeljalil
2004-01-01
Our goal in this paper is to discuss various issues we have encountered in trying to find and implement efficient solvers for a boundary integral equation (BIE) formulation of an iterative method for solving a reconstruction problem. We survey some methods from numerical linear algebra, which are relevant for the solution of this class of inverse problems. We motivate the use of our constructing algorithm, discuss its implementation and mention the use of preconditioned Krylov methods.
NASA Astrophysics Data System (ADS)
Rosu, Haret C.; Mancas, Stefan C.; Chen, Pisin
2014-04-01
In the context of supersymmetric quantum mechanics, we define a potential through a particular Riccati solution of the composition form (F∘f)(x)=F(f(x) and obtain a generalized Mielnik construction of one-parameter isospectral potentials when we use the general Riccati solution. Some examples for special cases of F and f are given to illustrate the method. An interesting result is obtained in the case of a parametric double well potential generated by this method, for which it is shown that the parameter of the potential controls the heights of the localization probability in the two wells, and for certain values of the parameter the height of the localization probability can be higher in the smaller well.
ERIC Educational Resources Information Center
Actuarial Foundation, 2013
2013-01-01
"Solving the Unknown with Algebra" is a new math program aligned with the National Council of Teachers of Mathematics (NCTM) standards and designed to help students practice pre-algebra skills including using formulas, solving for unknowns, and manipulating equations. Developed by The Actuarial Foundation with Scholastic, this program provides…
Solving stochastic epidemiological models using computer algebra
NASA Astrophysics Data System (ADS)
Hincapie, Doracelly; Ospina, Juan
2011-06-01
Mathematical modeling in Epidemiology is an important tool to understand the ways under which the diseases are transmitted and controlled. The mathematical modeling can be implemented via deterministic or stochastic models. Deterministic models are based on short systems of non-linear ordinary differential equations and the stochastic models are based on very large systems of linear differential equations. Deterministic models admit complete, rigorous and automatic analysis of stability both local and global from which is possible to derive the algebraic expressions for the basic reproductive number and the corresponding epidemic thresholds using computer algebra software. Stochastic models are more difficult to treat and the analysis of their properties requires complicated considerations in statistical mathematics. In this work we propose to use computer algebra software with the aim to solve epidemic stochastic models such as the SIR model and the carrier-borne model. Specifically we use Maple to solve these stochastic models in the case of small groups and we obtain results that do not appear in standard textbooks or in the books updated on stochastic models in epidemiology. From our results we derive expressions which coincide with those obtained in the classical texts using advanced procedures in mathematical statistics. Our algorithms can be extended for other stochastic models in epidemiology and this shows the power of computer algebra software not only for analysis of deterministic models but also for the analysis of stochastic models. We also perform numerical simulations with our algebraic results and we made estimations for the basic parameters as the basic reproductive rate and the stochastic threshold theorem. We claim that our algorithms and results are important tools to control the diseases in a globalized world.
Local Algebras of Differential Operators
NASA Astrophysics Data System (ADS)
Church, P. T.; Timourian, J. G.
2002-05-01
There is an increasing literature devoted to the study of boundary value problems using singularity theory. The resulting differential operators are typically Fredholm with index 0, defined on infinite-dimensional spaces, and they have often led to folds, cusps, and even higher-order Morin singularities. In this paper we develop some of the local algebras of germs of such differential Fredholm operators, extending the theory of the finite-dimensional case. We apply this work to nonlinear elliptic boundary value problems: in particular, we make further progress on a question proposed and initially studied by Ruf [1999, J. Differential Equations 151, 111-133]. We also make comments on several problems raised by others.
Spatial Operator Algebra for multibody system dynamics
NASA Technical Reports Server (NTRS)
Rodriguez, G.; Jain, A.; Kreutz-Delgado, K.
1992-01-01
The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.
Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction
ERIC Educational Resources Information Center
Wasserman, Nicholas H.
2016-01-01
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…
A computational algorithm for a disturbance rejection problem in structural acoustics
Hendrickson, E.
1994-12-31
We consider a disturbance rejection problem for an optimal control problem arising in structural acoustics. The solution to the control problem with unbounded control action involves the solution to an algebraic Riccati equation as well as an associated linear, nonhomogeneous differential equation. The goal is to construct a numerical algorithm based on the FEM for the computations of solutions to associated matrix algebraic Riccati equations. The Riccati-based feedback control will be used to minimize the effect of the disturbance for a given performance index.
Development of abstract mathematical reasoning: the case of algebra
Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja
2014-01-01
Algebra typically represents the students’ first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students’ ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16–17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15–16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students’ transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition. PMID:25228874
Development of abstract mathematical reasoning: the case of algebra.
Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja
2014-01-01
Algebra typically represents the students' first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students' ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16-17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15-16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students' transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition.
Computer Program For Linear Algebra
NASA Technical Reports Server (NTRS)
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
Revisiting Newtonian and Non-Newtonian Fluid Mechanics Using Computer Algebra
ERIC Educational Resources Information Center
Knight, D. G.
2006-01-01
This article illustrates how a computer algebra system, such as Maple[R], can assist in the study of theoretical fluid mechanics, for both Newtonian and non-Newtonian fluids. The continuity equation, the stress equations of motion, the Navier-Stokes equations, and various constitutive equations are treated, using a full, but straightforward,…
Pseudo Algebraically Closed Extensions
NASA Astrophysics Data System (ADS)
Bary-Soroker, Lior
2009-07-01
This PhD deals with the notion of pseudo algebraically closed (PAC) extensions of fields. It develops a group-theoretic machinery, based on a generalization of embedding problems, to study these extensions. Perhaps the main result is that although there are many PAC extensions, the Galois closure of a proper PAC extension is separably closed. The dissertation also contains the following subjects. The group theoretical counterpart of pseudo algebraically closed extensions, the so-called projective pairs. Applications to seemingly unrelated subjects, e.g., an analog of Dirichlet's theorem about primes in arithmetic progression for polynomial rings in one variable over infinite fields.
Renormalization group flows and continual Lie algebras
NASA Astrophysics Data System (ADS)
Bakas, Ioannis
2003-08-01
We study the renormalization group flows of two-dimensional metrics in sigma models using the one-loop beta functions, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates. In this algebraic setting, the logarithm of the world-sheet length scale, t, is interpreted as Dynkin parameter on the root system of a novel continual Lie algebra, denoted by Script G(d/dt;1), with anti-symmetric Cartan kernel K(t,t') = delta'(t-t'); as such, it coincides with the Cartan matrix of the superalgebra sl(N|N+1) in the large-N limit. The resulting Toda field equation is a non-linear generalization of the heat equation, which is integrable in target space and shares the same dissipative properties in time, t. We provide the general solution of the renormalization group flows in terms of free fields, via Bäcklund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultra-violet limit by gluing together two copies of Witten's two-dimensional black hole in the asymptotic region. We also provide some new solutions that describe the renormalization group flow of negatively curved spaces in different patches, which look like a cane in the infra-red region. Finally, we revisit the transition of a flat cone C/Zn to the plane, as another special solution, and note that tachyon condensation in closed string theory exhibits a hidden relation to the infinite dimensional algebra Script G(d/dt;1) in the regime of gravity. Its exponential growth holds the key for the construction of conserved currents and their systematic interpretation in string theory, but they still remain unknown.
Assessing Elementary Algebra with STACK
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2007-01-01
This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…
Stochastic regulator theory for a class of abstract wave equations
NASA Technical Reports Server (NTRS)
Balakrishnan, A. V.
1991-01-01
A class of steady-state stochastic regulator problems for abstract wave equations in a Hilbert space - of relevance to the problem of feedback control of large space structures using co-located controls/sensors - is studied. Both the control operator, as well as the observation operator, are finite-dimensional. As a result, the usual condition of exponential stabilizability invoked for existence of solutions to the steady-state Riccati equations is not valid. Fortunately, for the problems considered it turns out that strong stabilizability suffices. In particular, a closed form expression is obtained for the minimal (asymptotic) performance criterion as the control effort is allowed to grow without bound.
Classical mechanics on noncommutative space with Lie-algebraic structure
Miao Yangang; Wang Xudong; Yu Shaojie
2011-08-15
Highlights: > Suggest a useful method to look for new Lie-algebraic noncommutative spaces. > Find out two new Lie-algebraic noncommutative spaces. > Derive Newton and Hamilton equations that present unimaginable extra forces. > Analyse the source of unimaginable extra forces from space noncummutativity. > Provide various intriguing classical trajectories. - Abstract: We investigate the kinetics of a nonrelativistic particle interacting with a constant external force on a Lie-algebraic noncommutative space. The structure constants of a Lie algebra, also called noncommutative parameters, are constrained in general due to some algebraic properties, such as the antisymmetry and Jacobi identity. Through solving the constraint equations the structure constants satisfy, we obtain two new sorts of algebraic structures, each of which corresponds to one type of noncommutative spaces. Based on such types of noncommutative spaces as the starting point, we analyze the classical motion of the particle interacting with a constant external force by means of the Hamiltonian formalism on a Poisson manifold. Our results not only include that of a recent work as our special cases, but also provide new trajectories of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable tx-dot-,(xx-dot)-, and (xx-double dot)-dependence besides with the usual t-, x-, and x-dot-dependence, originating from a variety of noncommutativity between different spatial coordinates and between spatial coordinates and momenta as well, deform greatly the particle's ordinary trajectories we are quite familiar with on the Euclidean (commutative) space.
Computer algebra methods in the study of nonlinear differential systems
NASA Astrophysics Data System (ADS)
Irtegov, V. D.; Titorenko, T. N.
2013-06-01
Some issues concerning computer algebra methods as applied to the qualitative analysis of differential equations with first integrals are discussed. The problems of finding stationary sets and analyzing their stability and bifurcations are considered. Special attention is given to algorithms for finding and analyzing peculiar stationary sets. It is shown that computer algebra tools, combined with qualitative analysis methods for differential equations, make it possible not only to enhance the computational efficiency of classical algorithms, but also to implement new approaches to the solution of well-known problems and, in this way, to obtain new results.
The impact of fraction magnitude knowledge on algebra performance and learning.
Booth, Julie L; Newton, Kristie J; Twiss-Garrity, Laura K
2014-02-01
Knowledge of fractions is thought to be crucial for success with algebra, but empirical evidence supporting this conjecture is just beginning to emerge. In the current study, Algebra 1 students completed magnitude estimation tasks on three scales (0-1 [fractions], 0-1,000,000, and 0-62,571) just before beginning their unit on equation solving. Results indicated that fraction magnitude knowledge, and not whole number knowledge, was especially related to students' pretest knowledge of equation solving and encoding of equation features. Pretest fraction knowledge was also predictive of students' improvement in equation solving and equation encoding skills. Students' placement of unit fractions (e.g., those with a numerator of 1) was not especially useful for predicting algebra performance and learning in this population. Placement of non-unit fractions was more predictive, suggesting that proportional reasoning skills might be an important link between fraction knowledge and learning algebra.
Thinking Visually about Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2015-01-01
Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…
Computer Algebra versus Manipulation
ERIC Educational Resources Information Center
Zand, Hossein; Crowe, David
2004-01-01
In the UK there is increasing concern about the lack of skill in algebraic manipulation that is evident in students entering mathematics courses at university level. In this note we discuss how the computer can be used to ameliorate some of the problems. We take as an example the calculations needed in three dimensional vector analysis in polar…
ERIC Educational Resources Information Center
Glick, David
1995-01-01
Presents a technique that helps students concentrate more on the science and less on the mechanics of algebra while dealing with introductory physics formulas. Allows the teacher to do complex problems at a lower level and not be too concerned about the mathematical abilities of the students. (JRH)
ERIC Educational Resources Information Center
Nwabueze, Kenneth K.
2004-01-01
The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the…
A spatial operator algebra for manipulator modeling and control
NASA Technical Reports Server (NTRS)
Rodriguez, G.; Kreutz, K.; Milman, M.
1988-01-01
A powerful new spatial operator algebra for modeling, control, and trajectory design of manipulators is discussed along with its implementation in the Ada programming language. Applications of this algebra to robotics include an operator representation of the manipulator Jacobian matrix; the robot dynamical equations formulated in terms of the spatial algebra, showing the complete equivalence between the recursive Newton-Euler formulations to robot dynamics; the operator factorization and inversion of the manipulator mass matrix which immediately results in O(N) recursive forward dynamics algorithms; the joint accelerations of a manipulator due to a tip contact force; the recursive computation of the equivalent mass matrix as seen at the tip of a manipulator; and recursive forward dynamics of a closed chain system. Finally, additional applications and current research involving the use of the spatial operator algebra are discussed in general terms.
Constitutive relations in optics in terms of geometric algebra
NASA Astrophysics Data System (ADS)
Dargys, A.
2015-11-01
To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p-form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium.
Symbolic and algebraic computation
Not Available
1990-01-01
This book contains subjects under the following headings: Foundations of symbolic computation; Computational logics; systems Algorithms on polynormal; Integrative and differential equations; and Differential equations.
ERIC Educational Resources Information Center
Usman, Ahmed Ibrahim
2015-01-01
Knowledge and understanding of mathematical operations serves as a pre-reequisite for the successful translation of algebraic word problems. This study explored pre-service teachers' ability to recognize mathematical operations as well as use of those capabilities in constructing algebraic expressions, equations, and their solutions. The outcome…
ERIC Educational Resources Information Center
McNeil, Nicole M.; Rittle-Johnson, Bethany; Hattikudur, Shanta; Petersen, Lori A.
2010-01-01
This study examined if solving arithmetic problems hinders undergraduates' accuracy on algebra problems. The hypothesis was that solving arithmetic problems would hinder accuracy because it activates an operational view of equations, even in educated adults who have years of experience with algebra. In three experiments, undergraduates (N = 184)…
Algebraic Bethe ansatz for the Temperley-Lieb spin-1 chain
NASA Astrophysics Data System (ADS)
Nepomechie, Rafael I.; Pimenta, Rodrigo A.
2016-09-01
We use the algebraic Bethe ansatz to obtain the eigenvalues and eigenvectors of the spin-1 Temperley-Lieb open quantum chain with "free" boundary conditions. We exploit the associated reflection algebra in order to prove the off-shell equation satisfied by the Bethe vectors.
Capitalizing on Basic Brain Processes in Developmental Algebra--Part 3
ERIC Educational Resources Information Center
Laughbaum, Edward D.
2011-01-01
In Part Three, the author reviews the basic ideas presented in Parts One and Two while arguing why the traditional equation-solving developmental algebra curricula is not a good choice for implementing neural response strategies presented in the first two parts. He continues by showing that the developmental algebra student audience is simply…
Algebraic connectivity and graph robustness.
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T.
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
LAPACK: Linear algebra software for supercomputers
Bischof, C.H.
1991-01-01
This paper presents an overview of the LAPACK library, a portable, public-domain library to solve the most common linear algebra problems. This library provides a uniformly designed set of subroutines for solving systems of simultaneous linear equations, least-squares problems, and eigenvalue problems for dense and banded matrices. We elaborate on the design methodologies incorporated to make the LAPACK codes efficient on today's high-performance architectures. In particular, we discuss the use of block algorithms and the reliance on the Basic Linear Algebra Subprograms. We present performance results that show the suitability of the LAPACK approach for vector uniprocessors and shared-memory multiprocessors. We also address some issues that have to be dealt with in tuning LAPACK for specific architectures. Lastly, we present results that show that the LAPACK software can be adapted with little effort to distributed-memory environments, and we discuss future efforts resulting from this project. 31 refs., 10 figs., 2 tabs.
NASA Astrophysics Data System (ADS)
Dankova, T. S.; Rosensteel, G.
1998-10-01
Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of coadjoint orbits for the groups U(n) and O(2n). The general theory of mean fields is formulated for an arbitrary Lie algebra L of fermion operators. The moment map provides the correspondence between the Hilbert space of microscopic wave functions and the dual space L^* of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The mean field theories for the Elliott SU(3) and symplectic Sp(3,R) algebras are constructed explicitly in the coadjoint orbit framework.
ERIC Educational Resources Information Center
Beigie, Darin
2014-01-01
Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…
Lee, Jaehoon; Wilczek, Frank
2013-11-27
Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumps and an anisotropy in one part.
NASA Technical Reports Server (NTRS)
Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.
1999-01-01
This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras
Yu, Zhang; Zhang, Yufeng
2009-01-01
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings. PMID:20084092
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras.
Yu, Zhang; Zhang, Yufeng
2009-01-15
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra smu(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra smu(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras smu(6) and E is used to directly construct integrable couplings.
On the cohomology of Leibniz conformal algebras
NASA Astrophysics Data System (ADS)
Zhang, Jiao
2015-04-01
We construct a new cohomology complex of Leibniz conformal algebras with coefficients in a representation instead of a module. The low-dimensional cohomology groups of this complex are computed. Meanwhile, we construct a Leibniz algebra from a Leibniz conformal algebra and prove that the category of Leibniz conformal algebras is equivalent to the category of equivalence classes of formal distribution Leibniz algebras.
Assessing Algebraic Solving Ability: A Theoretical Framework
ERIC Educational Resources Information Center
Lian, Lim Hooi; Yew, Wun Thiam
2012-01-01
Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…
Some remarks on unilateral matrix equations
Cerchiai, Bianca L.; Zumino, Bruno
2001-02-01
We briefly review the results of our paper LBNL-46775: We study certain solutions of left-unilateral matrix equations. These are algebraic equations where the coefficients and the unknown are square matrices of the same order, or, more abstractly, elements of an associative, but possibly noncommutative algebra, and all coefficients are on the left. Recently such equations have appeared in a discussion of generalized Born-Infeld theories. In particular, two equations, their perturbative solutions and the relation between them are studied, applying a unified approach based on the generalized Bezout theorem for matrix polynomials.
Computer Algebra Systems and Theorems on Real Roots of Polynomials
ERIC Educational Resources Information Center
Aidoo, Anthony Y.; Manthey, Joseph L.; Ward, Kim Y.
2010-01-01
A computer algebra system is used to derive a theorem on the existence of roots of a quadratic equation on any bounded real interval. This is extended to a cubic polynomial. We discuss how students could be led to derive and prove these theorems. (Contains 1 figure.)
First Course in Algebra, Student's Text, Part II, Unit 10.
ERIC Educational Resources Information Center
Allen, Frank B.; And Others
Unit 10 in the SMSG's secondary school mathematics series is a student text covering the following topics in Algebra I: factors and exponents, radicals, polynomial and rational expressions, truth sets of open sentences, graphs of open sentences in two variables, systems of equations and inequalities, quadratic polynomials, and functions. (DT)
Writing to Promote and Assess Conceptual Understanding in College Algebra
ERIC Educational Resources Information Center
Gay, A. Susan; Peterson, Ingrid
2014-01-01
Concept-focused quiz questions required College Algebra students to write about their understanding. The questions can be viewed in three broad categories: a focus on sense-making, a focus on describing a mathematical object such as a graph or an equation, and a focus on understanding vocabulary. Student responses from 10 classes were analyzed.…
Fixing Ganache: Another Real-Life Use for Algebra
ERIC Educational Resources Information Center
Kalman, Adam M.
2011-01-01
This article presents a real-world application of proportional reasoning and equation solving. The author describes how students adjust ingredient amounts in a recipe for chocolate ganache. Using this real-world scenario provided students an opportunity to solve a difficult and nonstandard algebra problem, a lot of practice with fractions, a…
Geometric and Algebraic Approaches in the Concept of Complex Numbers
ERIC Educational Resources Information Center
Panaoura, A.; Elia, I.; Gagatsis, A.; Giatilis, G.-P.
2006-01-01
This study explores pupils' performance and processes in tasks involving equations and inequalities of complex numbers requiring conversions from a geometric representation to an algebraic representation and conversions in the reverse direction, and also in complex numbers problem solving. Data were collected from 95 pupils of the final grade from…
Learning Activity Package, Algebra 124, LAPs 46-55.
ERIC Educational Resources Information Center
Holland, Bill
A series of 10 teacher-prepared Learning Activity Packages (LAPs) in advanced algebra and trigonometry, these units cover absolute value, inequalities, exponents, radicals, and complex numbers; functions; higher degree equations and the derivative; the trigonometric functions; graphs and applications of the trigonometric functions; sequences and…
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s. PMID:26806075
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s.
Algebraic dichotomies with an application to the stability of Riemann solutions of conservation laws
NASA Astrophysics Data System (ADS)
Lin, Xiao-Biao
Recently, there has been some interest on the stability of waves where the functions involved grow or decay at an algebraic rate |. In this paper we define the so-called algebraic dichotomy that may aid in treating such problems. We discuss the basic properties of the algebraic dichotomy, methods of detecting it, and calculating the power of the weight function. We present several examples: (1) The Bessel equation. (2) The n-degree Fisher type equation. (3) Hyperbolic conservation laws in similarity coordinates. (4) A system of conservation laws with a Dafermos type viscous regularization. We show that the linearized system generates an analytic semigroup in the space of algebraic decay functions. This example motivates our work on algebraic dichotomies.
Learning Activity Package, Algebra 93-94, LAPs 12-22.
ERIC Educational Resources Information Center
Evans, Diane
A set of 11 teacher-prepared Learning Activity Packages (LAPs) in beginning algebra, these units cover sets, properties of operations, operations over real numbers, open expressions, solution sets of equations and inequalities, equations and inequalities with two variables, solution sets of equations with two variables, exponents, factoring and…
On Lie systems and Kummer-Schwarz equations
NASA Astrophysics Data System (ADS)
de Lucas, J.; Sardón, C.
2013-03-01
A Lie system is a system of first-order differential equations admitting a superposition rule, i.e., a map that expresses its general solution in terms of a generic family of particular solutions and certain constants. In this work, we use the geometric theory of Lie systems to prove that the explicit integration of second- and third-order Kummer-Schwarz equations is equivalent to obtaining a particular solution of a Lie system on SL(2,{R}). This same result can be extended to Riccati, Milne-Pinney, and to the here defined generalised Kummer-Schwarz equations, which include several types of Kummer-Schwarz equations as particular cases. We demonstrate that all the above-mentioned equations related to the same Lie system on SL(2,{R}) can be integrated simultaneously, which retrieves and generalizes in a unified and simpler manner previous results appearing in the literature. As a byproduct, we recover various properties of the Schwarzian derivative.
ERIC Educational Resources Information Center
Alibali, Martha W.; Kao, Yvonne S.; Brown, Alayna N.; Nathan, Mitchell J.; Stephens, Ana C.
2009-01-01
This study investigated middle school students' conceptual understanding of algebraic equations. Participants in the study--257 sixth- and seventh-grade students--were asked to solve one set of algebraic equations and to generate story problems corresponding with another set of equations. Structural aspects of the equations, including the number…
Multifractal vector fields and stochastic Clifford algebra
NASA Astrophysics Data System (ADS)
Schertzer, Daniel; Tchiguirinskaia, Ioulia
2015-12-01
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.
Multifractal vector fields and stochastic Clifford algebra
Schertzer, Daniel Tchiguirinskaia, Ioulia
2015-12-15
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.
Multifractal vector fields and stochastic Clifford algebra.
Schertzer, Daniel; Tchiguirinskaia, Ioulia
2015-12-01
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality. PMID:26723166
Multifractal vector fields and stochastic Clifford algebra.
Schertzer, Daniel; Tchiguirinskaia, Ioulia
2015-12-01
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.
2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW*-algebras
NASA Astrophysics Data System (ADS)
Ayupov, Shavkat; Kudaybergenov, Karimbergen
2016-03-01
The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M 2n (A), n ≥ 2, is a derivation. We apply this result to AW*-algebras and show that any 2-local derivation on an arbitrary AW*-algebra is a derivation.
Structured adaptive grid generation using algebraic methods
NASA Technical Reports Server (NTRS)
Yang, Jiann-Cherng; Soni, Bharat K.; Roger, R. P.; Chan, Stephen C.
1993-01-01
The accuracy of the numerical algorithm depends not only on the formal order of approximation but also on the distribution of grid points in the computational domain. Grid adaptation is a procedure which allows optimal grid redistribution as the solution progresses. It offers the prospect of accurate flow field simulations without the use of an excessively timely, computationally expensive, grid. Grid adaptive schemes are divided into two basic categories: differential and algebraic. The differential method is based on a variational approach where a function which contains a measure of grid smoothness, orthogonality and volume variation is minimized by using a variational principle. This approach provided a solid mathematical basis for the adaptive method, but the Euler-Lagrange equations must be solved in addition to the original governing equations. On the other hand, the algebraic method requires much less computational effort, but the grid may not be smooth. The algebraic techniques are based on devising an algorithm where the grid movement is governed by estimates of the local error in the numerical solution. This is achieved by requiring the points in the large error regions to attract other points and points in the low error region to repel other points. The development of a fast, efficient, and robust algebraic adaptive algorithm for structured flow simulation applications is presented. This development is accomplished in a three step process. The first step is to define an adaptive weighting mesh (distribution mesh) on the basis of the equidistribution law applied to the flow field solution. The second, and probably the most crucial step, is to redistribute grid points in the computational domain according to the aforementioned weighting mesh. The third and the last step is to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentration
Plethystic algebras and vector symmetric functions.
Rota, G C; Stein, J A
1994-01-01
An isomorphism is established between the plethystic Hopf algebra Pleth(Super[L]) and the algebra of vector symmetric functions. The Hall inner product of symmetric function theory is extended to the Hopf algebra Pleth(Super[L]). PMID:11607504
Algebra and Algebraic Thinking in School Math: 70th YB
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, 2008
2008-01-01
Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…
Abstract Algebra to Secondary School Algebra: Building Bridges
ERIC Educational Resources Information Center
Christy, Donna; Sparks, Rebecca
2015-01-01
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
Handheld Computer Algebra Systems in the Pre-Algebra Classroom
ERIC Educational Resources Information Center
Gantz, Linda Ann Galofaro
2010-01-01
This mixed method analysis sought to investigate several aspects of student learning in pre-algebra through the use of computer algebra systems (CAS) as opposed to non-CAS learning. This research was broken into two main parts, one which compared results from both the experimental group (instruction using CAS, N = 18) and the control group…
NASA Technical Reports Server (NTRS)
Collins, Emmanuel G., Jr.; Richter, Stephen
1990-01-01
One well known deficiency of LQG compensators is that they do not guarantee any measure of robustness. This deficiency is especially highlighted when considering control design for complex systems such as flexible structures. There has thus been a need to generalize LQG theory to incorporate robustness constraints. Here we describe the maximum entropy approach to robust control design for flexible structures, a generalization of LQG theory, pioneered by Hyland, which has proved useful in practice. The design equations consist of a set of coupled Riccati and Lyapunov equations. A homotopy algorithm that is used to solve these design equations is presented.
Statecharts Via Process Algebra
NASA Technical Reports Server (NTRS)
Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance
1999-01-01
Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumpsmore » and an anisotropy in one part.« less
Double Precision Differential/Algebraic Sensitivity Analysis Code
1995-06-02
DDASAC solves nonlinear initial-value problems involving stiff implicit systems of ordinary differential and algebraic equations. Purely algebraic nonlinear systems can also be solved, given an initial guess within the region of attraction of a solution. Options include automatic reconciliation of inconsistent initial states and derivatives, automatic initial step selection, direct concurrent parametric sensitivity analysis, and stopping at a prescribed value of any user-defined functional of the current solution vector. Local error control (in the max-normmore » or the 2-norm) is provided for the state vector and can include the sensitivities on request.« less
Linear Algebra and Image Processing
ERIC Educational Resources Information Center
Allali, Mohamed
2010-01-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)
Linear algebra and image processing
NASA Astrophysics Data System (ADS)
Allali, Mohamed
2010-09-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty.
A Programmed Course in Algebra.
ERIC Educational Resources Information Center
Mewborn, Ancel C.; Hively, Wells II
This programed textbook consists of short sections of text interspersed with questions designed to aid the student in understanding the material. The course is designed to increase the student's understanding of some of the basic ideas of algebra. Some general experience and manipulative skill with respect to high school algebra is assumed.…
ERIC Educational Resources Information Center
1997
Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…
Gamow functionals on operator algebras
NASA Astrophysics Data System (ADS)
Castagnino, M.; Gadella, M.; Betán, R. Id; Laura, R.
2001-11-01
We obtain the precise form of two Gamow functionals representing the exponentially decaying part of a quantum resonance and its mirror image that grows exponentially, as a linear, positive and continuous functional on an algebra containing observables. These functionals do not admit normalization and, with an appropriate choice of the algebra, are time reversal of each other.
Online Algebraic Tools for Teaching
ERIC Educational Resources Information Center
Kurz, Terri L.
2011-01-01
Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)
Patterns to Develop Algebraic Reasoning
ERIC Educational Resources Information Center
Stump, Sheryl L.
2011-01-01
What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…
ERIC Educational Resources Information Center
Instructional Objectives Exchange, Los Angeles, CA.
A complete set of behavioral objectives for first-year algebra taught in any of grades 8 through 12 is presented. Three to six sample test items and answers are provided for each objective. Objectives were determined by surveying the most used secondary school algebra textbooks. Fourteen major categories are included: (1) whole numbers--operations…
Elementary maps on nest algebras
NASA Astrophysics Data System (ADS)
Li, Pengtong
2006-08-01
Let , be algebras and let , be maps. An elementary map of is an ordered pair (M,M*) such that for all , . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.
Condensing Algebra for Technical Mathematics.
ERIC Educational Resources Information Center
Greenfield, Donald R.
Twenty Algebra-Packets (A-PAKS) were developed by the investigator for technical education students at the community college level. Each packet contained a statement of rationale, learning objectives, performance activities, performance test, and performance test answer key. The A-PAKS condensed the usual sixteen weeks of algebra into a six-week…
Thermodynamics. [algebraic structure
NASA Technical Reports Server (NTRS)
Zeleznik, F. J.
1976-01-01
The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory's theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not present.
Partial Differential Algebraic Sensitivity Analysis Code
1995-05-15
PDASAC solves stiff, nonlinear initial-boundary-value in a timelike dimension t and a space dimension x. Plane, circular cylindrical or spherical boundaries can be handled. Mixed-order systems of partial differential and algebraic equations can be analyzed with members of order or 0 or 1 in t, 0,1 or 2 in x. Parametric sensitivities of the calculated states are compted simultaneously on request, via the Jacobian of the state equations. Initial and boundary conditions are efficiently reconciled.more » Local error control (in the max-norm or the 2-norm) is provided for the state vector and can include the parametric sensitivites if desired.« less
Numerical stability in problems of linear algebra.
NASA Technical Reports Server (NTRS)
Babuska, I.
1972-01-01
Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.
Abstract numeric relations and the visual structure of algebra.
Landy, David; Brookes, David; Smout, Ryan
2014-09-01
Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition, it has often been assumed that skilled users of these formalisms treat situations in terms of semantic properties encoded in an abstract syntax that governs the use of notation without particular regard to the details of the physical structure of the equation itself (Anderson, 2005; Hegarty, Mayer, & Monk, 1995). We explore how the notational structure of verbal descriptions or algebraic equations (e.g., the spatial proximity of certain words or the visual alignment of numbers and symbols in an equation) plays a role in the process of interpreting or constructing symbolic equations. We propose in particular that construction processes involve an alignment of notational structures across representation systems, biasing reasoners toward the selection of formal notations that maintain the visuospatial structure of source representations. For example, in the statement "There are 5 elephants for every 3 rhinoceroses," the spatial proximity of 5 and elephants and 3 and rhinoceroses will bias reasoners to write the incorrect expression 5E = 3R, because that expression maintains the spatial relationships encoded in the source representation. In 3 experiments, participants constructed equations with given structure, based on story problems with a variety of phrasings. We demonstrate how the notational alignment approach accounts naturally for a variety of previously reported phenomena in equation construction and successfully predicts error patterns that are not accounted for by prior explanations, such as the left to right transcription heuristic.
ERIC Educational Resources Information Center
Gonzalez-Vega, Laureano
1999-01-01
Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)
Hierarchies of nonlinear integrable equations and their symmetries in 2 + 1 dimensions
NASA Astrophysics Data System (ADS)
Cheng, Yi
1990-11-01
For a given nonlinear integrable equation in 2 + 1 dimensions, an approach is described to construct the hierarchies of equations and relevant Lie algebraic properties. The commutability and noncommutability of equations of the flow, their symmetries and mastersymmetries are then derived as direct results of these algebraic properties. The details for the modified Kadomtsev-Petviashvilli equation are shown as an example and the main results for the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Katera-Sawada equation are given.
NASA Astrophysics Data System (ADS)
Cardona, Carlos; Gomez, Humberto
2016-06-01
Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a mathbb{C}{P}^2 space. We show that for the simplest integrand, namely the n - gon, our proposal indeed reproduces the expected result. By using the recently formulated Λ-algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.
NASA Astrophysics Data System (ADS)
Doikou, Anastasia; Ioannidou, Theodora
2011-04-01
A non-compact version of the Weyl equation is proposed, based on the infinite dimensional spin zero representation of the mathfrak{s}{mathfrak{l}_2} algebra. Solutions of the aforementioned equation are obtained in terms of the Kummer functions. In this context, we discuss the ADHMN approach in order to construct the corresponding non-compact BPS monopoles.
Quantum algebra of N superspace
Hatcher, Nicolas; Restuccia, A.; Stephany, J.
2007-08-15
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.
Constraint algebra in bigravity
Soloviev, V. O.
2015-07-15
The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.
Constraint algebra in bigravity
NASA Astrophysics Data System (ADS)
Soloviev, V. O.
2015-07-01
The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.
Chen, J.; Safro, I.
2011-01-01
Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.
ERIC Educational Resources Information Center
Powell, Sarah R.; Fuchs, Lynn S.
2014-01-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 second-grade students, we administered: (1) measures of calculations and…
ERIC Educational Resources Information Center
Ross, Amanda; Willson, Victor
2012-01-01
This study examined the effects of types of representations, constructivist teaching approaches, and student engagement on middle school algebra students' procedural knowledge and conceptual understanding. Data gathered from 16 video lessons and algebra pretest/posttests were used to run three multilevel structural equation models. Symbolic…
Readiness and Preparation for Beginning Algebra.
ERIC Educational Resources Information Center
Rotman, Jack W.
Drawing from experience at Lansing Community College (LCC), this paper discusses how to best prepare students for success in a beginning algebra course. First, an overview is presented of LCC's developmental math sequence, which includes Basic Arithmetic (MTH 008), Pre-Algebra (MTH 009), Beginning Algebra (MTH 012), and Intermediate Algebra (MTH…
Two-parameter twisted quantum affine algebras
NASA Astrophysics Data System (ADS)
Jing, Naihuan; Zhang, Honglian
2016-09-01
We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.
TBGG- INTERACTIVE ALGEBRAIC GRID GENERATION
NASA Technical Reports Server (NTRS)
Smith, R. E.
1994-01-01
TBGG, Two-Boundary Grid Generation, applies an interactive algebraic grid generation technique in two dimensions. The program incorporates mathematical equations that relate the computational domain to the physical domain. TBGG has application to a variety of problems using finite difference techniques, such as computational fluid dynamics. Examples include the creation of a C-type grid about an airfoil and a nozzle configuration in which no left or right boundaries are specified. The underlying two-boundary technique of grid generation is based on Hermite cubic interpolation between two fixed, nonintersecting boundaries. The boundaries are defined by two ordered sets of points, referred to as the top and bottom. Left and right side boundaries may also be specified, and call upon linear blending functions to conform interior interpolation to the side boundaries. Spacing between physical grid coordinates is determined as a function of boundary data and uniformly spaced computational coordinates. Control functions relating computational coordinates to parametric intermediate variables that affect the distance between grid points are embedded in the interpolation formulas. A versatile control function technique with smooth cubic spline functions is also presented. The TBGG program is written in FORTRAN 77. It works best in an interactive graphics environment where computational displays and user responses are quickly exchanged. The program has been implemented on a CDC Cyber 170 series computer using NOS 2.4 operating system, with a central memory requirement of 151,700 (octal) 60 bit words. TBGG requires a Tektronix 4015 terminal and the DI-3000 Graphics Library of Precision Visuals, Inc. TBGG was developed in 1986.
A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
NASA Astrophysics Data System (ADS)
Cariñena, José F.; Guha, Partha; de Lucas, Javier
2013-03-01
A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geometric way. This allows us to study the transformation of these equations into simpler canonical forms, which solves a gap in the previous literature, and other relevant differential equations, which leads to derive new constants of motion for families of second-order Gambier equations. Additionally, we describe general solutions of certain second-order Gambier equations in terms of particular solutions of Riccati equations, linear systems, and t-dependent frequency harmonic oscillators.
Cartooning in Algebra and Calculus
ERIC Educational Resources Information Center
Moseley, L. Jeneva
2014-01-01
This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.
NASA Technical Reports Server (NTRS)
Klumpp, A. R.; Lawson, C. L.
1988-01-01
Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.
GCD, LCM, and Boolean Algebra?
ERIC Educational Resources Information Center
Cohen, Martin P.; Juraschek, William A.
1976-01-01
This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)
Hopf algebras and topological recursion
NASA Astrophysics Data System (ADS)
Esteves, João N.
2015-11-01
We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293-309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347-452).
Sjaardema, G.; Gilkey, A.; Smith, M.; Forsythe, C.
2005-04-11
The ALGEBRA program allows the user to manipulate data from a finite element analysis before it is plotted. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database. The ALGEBRA program evaluates user-supplied functions of the data and writes the results to an output EXODUS II database that can be read by plot programs.
Algebraic Rossby Solitary Waves Excited by Non-Stationary External Source
NASA Astrophysics Data System (ADS)
Yang, Hong-Wei; Yin, Bao-Shu; Dong, Huan-He; Shi, Yun-Long
2012-09-01
The paper deals with the effects of non-stationary external source forcing and dissipation on algebraic Rossby solitary waves. From quasi-geostrophic potential vorticity equation, basing on the multiple-scale method, an inhomogeneous Korteweg-de Vries—Benjamin—Ono—Burgers (KdV-B-O-Burgers) equation is obtained. This equation has not been previously derived for Rossby waves. By analysis and calculation, four conservation laws associated with the above equation are first obtained. With the help of pseudo-spectral method, the waterfall plots are obtained and the evolutional characters of algebraic Rossby solitary waves are studied. The results show that non-stationary external source and dissipation have great effect on the generation and evolution of algebraic solitary Rossby waves.
Hunt, H. B.; Marathe, M. V.; Stearns, R. E.
2001-01-01
We demonstrate how the concepts of algebraic representability and strongly-local reductions developed here and in [HSM00] can be used to characterize the computational complexity/efficient approximability of a number of basic problems and their variants, on various abstract algebraic structures F. These problems include the following: (1) A1gebra:Determine the solvability, unique solvability, number of solutions, etc., of a system of equations on F. Determine the equivalence of two formulas or straight-line programs on F. 2. 0ptimization:Let {epsilon} > 0. (a) Determine the maximum number of simultaneously satisfiable equations in a system of equations on F; or approximate this number within a multiplicative factor of n{sup {epsilon}}. (b) Determine the maximum value of an objective function subject to satisfiable algebraically expressed constraints on F; or approximate this maximum value within a multiplicative factor of n{sup {epsilon}}. (c) Given a formula or straight-line program, find a minimum size equivalent formula or straightline program; or find an equivalent formula or straight-line program of size {le} f (minimum). Both finite and infinite algebraic structures are considered. These finite structures include all finite nondegenerate lattices and all finite rings or semi-rings with a nonzero element idempotent under multiplication (e.g. all non-degenerate finite unitary rings or semi-rings); and these infinite structures include the natural numbers, integers, real numbers, various algebras on these structures, all ordered rings, many cancellative semi-rings, and all infinite lattices with two elements a,b such that a is covered by b. Our results significantly extend a number of results by Ladner [La89], Condon, et. al. [CF+93], Khanna, et.al [KSW97], Cr951 and Zuckerman [Zu93] on the complexity and approximbaility of combinatorial problems.
Super-Poincarè algebras, space-times, and supergravities. II
NASA Astrophysics Data System (ADS)
Santi, A.; Spiro, A.
2012-03-01
The presentation of supergravity theories of our previous paper "Super-Poincarè algebras, space-times, and supergravities. I" is re-formulated in the language of Berezin-Leites-Kostant theory of supermanifolds. It is also shown that the equations of Cremmer, Julia, and Scherk's theory of 11D-supergravity are equivalent to manifestly covariant equations on a supermanifold.
Traveling kinks in cubic nonlinear Ginzburg-Landau equations.
Rosu, H C; Cornejo-Pérez, O; Ojeda-May, P
2012-03-01
Nonlinear cubic Euler-Lagrange equations of motion in the traveling variable are usually derived from Ginzburg-Landau free energy functionals frequently encountered in several fields of physics. Many authors considered in the past damped versions of such equations, with the damping term added by hand simulating the friction due to the environment. It is known that even in this damped case kink solutions can exist. By means of a factorization method, we provide analytic formulas for several possible kink solutions of such equations of motion in the undriven and constant field driven cases, including the recently introduced Riccati parameter kinks, which were not considered previously in such a context. The latter parameter controls the delay of the switching stage of the kinks. The delay is caused by antikink components that are introduced in the structure of the solution through this parameter. PMID:22587214
A perturbative solution to metadynamics ordinary differential equation
NASA Astrophysics Data System (ADS)
Tiwary, Pratyush; Dama, James F.; Parrinello, Michele
2015-12-01
Metadynamics is a popular enhanced sampling scheme wherein by periodic application of a repulsive bias, one can surmount high free energy barriers and explore complex landscapes. Recently, metadynamics was shown to be mathematically well founded, in the sense that the biasing procedure is guaranteed to converge to the true free energy surface in the long time limit irrespective of the precise choice of biasing parameters. A differential equation governing the post-transient convergence behavior of metadynamics was also derived. In this short communication, we revisit this differential equation, expressing it in a convenient and elegant Riccati-like form. A perturbative solution scheme is then developed for solving this differential equation, which is valid for any generic biasing kernel. The solution clearly demonstrates the robustness of metadynamics to choice of biasing parameters and gives further confidence in the widely used method.
Quadratic algebra for superintegrable monopole system in a Taub-NUT space
NASA Astrophysics Data System (ADS)
Hoque, Md Fazlul; Marquette, Ian; Zhang, Yao-Zhong
2016-09-01
We introduce a Hartmann system in the generalized Taub-NUT space with Abelian monopole interaction. This quantum system includes well known Kaluza-Klein monopole and MIC-Zwanziger monopole as special cases. It is shown that the corresponding Schrödinger equation of the Hamiltonian is separable in both spherical and parabolic coordinates. We obtain the integrals of motion of this superintegrable model and construct the quadratic algebra and Casimir operator. This algebra can be realized in terms of a deformed oscillator algebra and has finite dimensional unitary representations (unirreps) which provide energy spectra of the system. This result coincides with the physical spectra obtained from the separation of variables.
Integrable maps from Galois differential algebras, Borel transforms and number sequences
NASA Astrophysics Data System (ADS)
Tempesta, Piergiulio
A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation, in particular symmetries and integrability (see Tempesta, 2010 [40]). Our approach is based on the properties of a suitable Galois differential algebra, that we shall call a Rota algebra. A formulation of the procedure in terms of category theory is proposed. In order to render the lattice dynamics confined, a Borel regularization is also adopted. As a byproduct of the theory, a connection between number sequences and integrability is discussed.
Geometric Approaches to Quadratic Equations from Other Times and Places.
ERIC Educational Resources Information Center
Allaire, Patricia R.; Bradley, Robert E.
2001-01-01
Focuses on geometric solutions of quadratic problems. Presents a collection of geometric techniques from ancient Babylonia, classical Greece, medieval Arabia, and early modern Europe to enhance the quadratic equation portion of an algebra course. (KHR)
Iterative methods for elliptic finite element equations on general meshes
NASA Technical Reports Server (NTRS)
Nicolaides, R. A.; Choudhury, Shenaz
1986-01-01
Iterative methods for arbitrary mesh discretizations of elliptic partial differential equations are surveyed. The methods discussed are preconditioned conjugate gradients, algebraic multigrid, deflated conjugate gradients, an element-by-element techniques, and domain decomposition. Computational results are included.
Phase shift of interacting algebraic solitary waves in a two-layer fluid system
Matsuno, Y. )
1994-09-05
The interaction of interfacial solitary waves of algebraic type is investigated on the basis of a higher-order Benjamin-Ono equation. By developing a multisoliton perturbation theory, we show analytically that the overtaking collision between two solitary waves exhibits the phase shift but the amplitudes are not altered after interaction. The prediction of the phase shift that takes place between algebraic solitary waves is the first example reported in the literature.
Relativistic algebraic spinors and quantum motions in phase space
Holland, P.R.
1986-08-01
Following suggestions of Schonberg and Bohm, we study the tensorial phase space representation of the Dirac and Feynman-Gell-Mann equations in terms of the complex Dirac algebra C/sub 4/, a Jordan-Wigner algebra G/sub 4/, and Wigner transformations. To do this we solve the problem of the conditions under which elements in C/sub 4/ generate minimal ideals, and extend this to G/sub 4/. This yields the linear theory of Dirac spin spaces and tensor representations of Dirac spinors, and the spin-1/2 wave equations are represented through fermionic state vectors in a higher space as a set of interconnected tensor relations.
Universal vertex-IRF transformation for quantum affine algebras
Buffenoir, E.; Roche, Ph.; Terras, V.
2012-10-15
We construct a universal solution of the generalized coboundary equation in the case of quantum affine algebras, which is an extension of our previous work to U{sub q}(A{sub r}{sup (1)}). This universal solution has a simple Gauss decomposition which is constructed using Sevostyanov's characters of twisted quantum Borel algebras. We show that in the evaluation representations it gives a vertex-face transformation between a vertex type solution and a face type solution of the quantum dynamical Yang-Baxter equation. In particular, in the evaluation representation of U{sub q}(A{sub 1}{sup (1)}), it gives Baxter's well-known transformation between the 8-vertex model and the interaction-round-faces (IRF) height model.
NASA Astrophysics Data System (ADS)
Dumansky, Alexander M.; Tairova, Lyudmila P.
2008-09-01
A method for the construction of hereditary constitutive equation is proposed for the laminate on the basis of hereditary constitutive equations of a layer. The method is developed from the assumption that in the directions of axes of orthotropy the layer follows elastic behavior, and obeys hereditary constitutive equations under shear. The constitutive equations of the laminate are constructed on the basis of classical laminate theory and algebra of resolvent operators. Effective matrix algorithm and relationships of operator algebra are used to derive visco-elastic stiffness and compliance of the laminate. The example of construction of hereditary constitutive equations of cross-ply carbon fiber-reinforced plastic is presented.
Lie algebraic methods for particle tracking calculations
Douglas, D.R.; Dragt, A.J.
1983-08-01
A study of the nonlinear stability of an accelerator or storage ring lattice typically includes particle tracking simulations. Such simulations trace rays through linear and nonlinear lattice elements by numerically evaluating linear matrix or impulsive nonlinear transformations. Using the mathematical tools of Lie groups and algebras, one may construct a formalism which makes explicit use of Hamilton's equations and which allows the description of groups of linear and nonlinear lattice elements by a single transformation. Such a transformation will be exactly canonical and will describe finite length linear and nonlinear elements through third (octupole) order. It is presently possible to include effects such as fringing fields and potentially possible to extend the formalism to include nonlinearities of higher order, multipole errors, and magnet misalignments. We outline this Lie algebraic formalism and its use in particle tracking calculations. A computer code, MARYLIE, has been constructed on the basis of this formalism. We describe the use of this program for tracking and provide examples of its application. 6 references, 3 figures.
NASA Astrophysics Data System (ADS)
Masoero, Davide; Raimondo, Andrea; Valeri, Daniele
2016-09-01
We assess the ODE/IM correspondence for the quantum g -KdV model, for a non-simply laced Lie algebra g. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra g^{(1)} , and constructing the relevant {Ψ} -system among subdominant solutions. We then use the {Ψ} -system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum g -KdV model. We also consider generalized Airy functions for twisted Kac-Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.
Symmetries of stochastic differential equations: A geometric approach
NASA Astrophysics Data System (ADS)
De Vecchi, Francesco C.; Morando, Paola; Ugolini, Stefania
2016-06-01
A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an algebra of strong symmetries for a modified SDE is proved under suitable regularity assumptions. This general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional Brownian motion.
Catmull-Rom Curve Fitting and Interpolation Equations
ERIC Educational Resources Information Center
Jerome, Lawrence
2010-01-01
Computer graphics and animation experts have been using the Catmull-Rom smooth curve interpolation equations since 1974, but the vector and matrix equations can be derived and simplified using basic algebra, resulting in a simple set of linear equations with constant coefficients. A variety of uses of Catmull-Rom interpolation are demonstrated,…
Solving Differential Equations in R: Package deSolve
In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines appr...
Equations of gas dynamics admitting an infinite number of symmetries
Meshkov, A.G.; Mikhalyaev, B.B.
1988-02-01
All the equations of state for which the equations of one-dimensional gas dynamics have an infinite Lie-Baecklund algebra are found. In all these cases, the gas-dynamic equations can either be integrated directly or represented in Lax form. A method for constructing an infinite set of conservation laws is indicated.
BRST charges for finite nonlinear algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.
2010-07-01
Some ingredients of the BRST construction for quantum Lie algebras are applied to a wider class of quadratic algebras of constraints. We build the BRST charge for a quantum Lie algebra with three generators and ghost-anti-ghosts commuting with constraints. We consider a one-parametric family of quadratic algebras with three generators and show that the BRST charge acquires the conventional form after a redefinition of ghosts. The modified ghosts form a quadratic algebra. The family possesses a nonlinear involution, which implies the existence of two independent BRST charges for each algebra in the family. These BRST charges anticommute and form a double BRST complex.
Some Remarks on Kite Pseudo Effect Algebras
NASA Astrophysics Data System (ADS)
Dvurečenskij, Anatolij; Holland, W. Charles
2014-05-01
Recently a new family of pseudo effect algebras, called kite pseudo effect algebras, was introduced. Such an algebra starts with a po-group G, a set I and with two bijections λ, ρ: I→ I. Using a clever construction on the ordinal sum of ( G +) I and ( G -) I , we can define a pseudo effect algebra which can be non-commutative even if G is an Abelian po-group. In the paper we give a characterization of subdirect product of subdirectly irreducible kite pseudo effect algebras, and we show that every kite pseudo effect algebra is an interval in a unital po-loop.
Computational algebraic topology-based video restoration
NASA Astrophysics Data System (ADS)
Rochel, Alban; Ziou, Djemel; Auclair-Fortier, Marie-Flavie
2005-03-01
This paper presents a scheme for video denoising by diffusion of gray levels, based on the Computational Algebraic Topology (CAT) image model. The diffusion approach is similar to the one used to denoise static images. Rather than using the heat transfer partial differential equation, discretizing it and solving it by a purely mathematical process, the CAT approach considers the global expression of the heat transfer and decomposes it into elementary physical laws. Some of these laws describe conservative relations, leading to error-free expressions, whereas others depend on metric quantities and require approximation. This scheme allows for a physical interpretation for each step of the resolution process. We propose a nonlinear and an anisotropic diffusion algorithms based on the extension to video of an existing 2D algorithm thanks to the flexibility of the topological support. Finally it is validated with experimental results.
NASA Astrophysics Data System (ADS)
Imai, Kenji
2014-02-01
In this paper, a new n-dimensional homogeneous Lotka-Volterra (HLV) equation, which possesses a Lie symmetry, is derived by the extension from a three-dimensional HLV equation. Its integrability is shown from the viewpoint of Lie symmetries. Furthermore, we derive dynamical systems of higher order, which possess the Lie symmetry, using the algebraic structure of this HLV equation.
Constructing a parasupersymmetric Virasoro algebra
NASA Astrophysics Data System (ADS)
Kuwata, S.
2011-03-01
We construct a para SUSY Virasoro algebra by generalizing the ordinary fermion in SUSY Virasoro algebra (Ramond or Neveu-Schwarz algebra) to the parafermion. First, we obtain a polynomial relation (PR) between different-mode parafermion fi's by generalizing the corresponding single-mode PR to such that is invariant under the unitary transformation of fi (Green's condition). Differently from a usual context, where the Green's condition is imposed only on the defining relation of fi (degree three with respect to fi and fi†), we impose it on any degree of PR. For the case of order-two parafermion (the simplest case of para SUSY), we calculate a PR between the parasupercharge G0, the bosonic hamiltonian LB0 and parafermionic one LF0, although it is difficult to obtain a PR between G0 and the total hamiltonian L0 (= LB0 + LF0). Finally, we construct a para SUSY Virasoro algebra by generalizing L0 to the Ln's such that form a Virasoro algebra.
ERIC Educational Resources Information Center
Gale, David; And Others
Four units make up the contents of this document. The first examines applications of finite mathematics to business and economies. The user is expected to learn the method of optimization in optimal assignment problems. The second module presents applications of difference equations to economics and social sciences, and shows how to: 1) interpret…
1992-05-04
DOE-MACSYMA (Project MAC''s SYmbolic MAnipulation system) is a large computer programming system written in LISP. With DOE-MACSYMA the user can differentiate, integrate, take limits, solve systems of linear or polynomial equations, factor polynomials, expand functions in Laurent or Taylor series, solve differential equations (using direct or transform methods), compute Poisson series, plot curves, and manipulate matrices and tensors. A language similar to ALGOL-60 permits users to write their own programs for transforming symbolic expressions. Franz Lisp OPUS 38 provides the environment for the Encore, Celerity, and DEC VAX11 UNIX,SUN(OPUS) versions under UNIX and the Alliant version under Concentrix. Kyoto Common Lisp (KCL) provides the environment for the SUN(KCL),Convex, and IBM PC under UNIX and Data General under AOS/VS.
1992-05-04
DOE-MACSYMA (Project MAC''s SYmbolic MAnipulation system) is a large computer programming system written in LISP. With DOE-MACSYMA the user can differentiate, integrate, take limits, solve systems of linear or polynomial equations, factor polynomials, expand functions in Laurent or Taylor series, solve differential equations (using direct or transform methods), compute Poisson series, plot curves, and manipulate matrices and tensors. A language similar to ALGOL-60 permits users to write their own programs for transforming symbolic expressions. Franzmore » Lisp OPUS 38 provides the environment for the Encore, Celerity, and DEC VAX11 UNIX,SUN(OPUS) versions under UNIX and the Alliant version under Concentrix. Kyoto Common Lisp (KCL) provides the environment for the SUN(KCL),Convex, and IBM PC under UNIX and Data General under AOS/VS.« less
Spatial operator algebra framework for multibody system dynamics
NASA Technical Reports Server (NTRS)
Rodriguez, G.; Jain, Abhinandan; Kreutz, K.
1989-01-01
The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.
Algebraic multigrid methods applied to problems in computational structural mechanics
NASA Technical Reports Server (NTRS)
Mccormick, Steve; Ruge, John
1989-01-01
The development of algebraic multigrid (AMG) methods and their application to certain problems in structural mechanics are described with emphasis on two- and three-dimensional linear elasticity equations and the 'jacket problems' (three-dimensional beam structures). Various possible extensions of AMG are also described. The basic idea of AMG is to develop the discretization sequence based on the target matrix and not the differential equation. Therefore, the matrix is analyzed for certain dependencies that permit the proper construction of coarser matrices and attendant transfer operators. In this manner, AMG appears to be adaptable to structural analysis applications.
A Metric Conceptual Space Algebra
NASA Astrophysics Data System (ADS)
Adams, Benjamin; Raubal, Martin
The modeling of concepts from a cognitive perspective is important for designing spatial information systems that interoperate with human users. Concept representations that are built using geometric and topological conceptual space structures are well suited for semantic similarity and concept combination operations. In addition, concepts that are more closely grounded in the physical world, such as many spatial concepts, have a natural fit with the geometric structure of conceptual spaces. Despite these apparent advantages, conceptual spaces are underutilized because existing formalizations of conceptual space theory have focused on individual aspects of the theory rather than the creation of a comprehensive algebra. In this paper we present a metric conceptual space algebra that is designed to facilitate the creation of conceptual space knowledge bases and inferencing systems. Conceptual regions are represented as convex polytopes and context is built in as a fundamental element. We demonstrate the applicability of the algebra to spatial information systems with a proof-of-concept application.
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-07-01
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.
Moving frames and prolongation algebras
NASA Technical Reports Server (NTRS)
Estabrook, F. B.
1982-01-01
Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.
NASA Astrophysics Data System (ADS)
Khalilov, E. H.
2016-07-01
The surface integral equation for a spatial mixed boundary value problem for the Helmholtz equation is considered. At a set of chosen points, the equation is replaced with a system of algebraic equations, and the existence and uniqueness of the solution of this system is established. The convergence of the solutions of this system to the exact solution of the integral equation is proven, and the convergence rate of the method is determined.
Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers
Nataf, J.M.; Winkelmann, F.
1992-09-01
We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK`s symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of these methods to solving the partial differential equations for two-dimensional heat flow is illustrated.
Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers
Nataf, J.M.; Winkelmann, F.
1992-09-01
We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK's symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of these methods to solving the partial differential equations for two-dimensional heat flow is illustrated.
Motivating Activities that Lead to Algebra
ERIC Educational Resources Information Center
Menon, Ramakrishnan
2004-01-01
Four activities consisting of puzzles are introduced, which help students to recognize the strength of algebraic generalizations. They also assist them to comprehend algebraic concepts, and enable them to develop their individual puzzles and games.
Large chiral diffeomorphisms on Riemann surfaces and W-algebras
Bandelloni, G.; Lazzarini, S.
2006-10-15
The diffeomorphism action lifted on truncated (chiral) Taylor expansion of a complex scalar field over a Riemann surface is presented in the paper under the name of large diffeomorphisms. After an heuristic approach, we show how a linear truncation in the Taylor expansion can generate an algebra of symmetry characterized by some structure functions. Such a linear truncation is explicitly realized by introducing the notion of Forsyth frame over the Riemann surface with the help of a conformally covariant algebraic differential equation. The large chiral diffeomorphism action is then implemented through a Becchi-Rouet-Stora (BRS) formulation (for a given order of truncation) leading to a more algebraic setup. In this context the ghost fields behave as holomorphically covariant jets. Subsequently, the link with the so-called W-algebras is made explicit once the ghost parameters are turned from jets into tensorial ghost ones. We give a general solution with the help of the structure functions pertaining to all the possible truncations lower or equal to the given order. This provides another contribution to the relationship between Korteweg-de Vries (KdV) flows and W-diffeomorphims.
Scalable Parallel Algebraic Multigrid Solvers
Bank, R; Lu, S; Tong, C; Vassilevski, P
2005-03-23
The authors propose a parallel algebraic multilevel algorithm (AMG), which has the novel feature that the subproblem residing in each processor is defined over the entire partition domain, although the vast majority of unknowns for each subproblem are associated with the partition owned by the corresponding processor. This feature ensures that a global coarse description of the problem is contained within each of the subproblems. The advantages of this approach are that interprocessor communication is minimized in the solution process while an optimal order of convergence rate is preserved; and the speed of local subproblem solvers can be maximized using the best existing sequential algebraic solvers.
Discrimination in a General Algebraic Setting.
Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras.
Discrimination in a General Algebraic Setting
Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras. PMID:26171421
Spatial-Operator Algebra For Robotic Manipulators
NASA Technical Reports Server (NTRS)
Rodriguez, Guillermo; Kreutz, Kenneth K.; Milman, Mark H.
1991-01-01
Report discusses spatial-operator algebra developed in recent studies of mathematical modeling, control, and design of trajectories of robotic manipulators. Provides succinct representation of mathematically complicated interactions among multiple joints and links of manipulator, thereby relieving analyst of most of tedium of detailed algebraic manipulations. Presents analytical formulation of spatial-operator algebra, describes some specific applications, summarizes current research, and discusses implementation of spatial-operator algebra in the Ada programming language.
Twining characters and orbit Lie algebras
Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert; Schweigert, Christoph
1996-12-05
We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.
New family of Maxwell like algebras
NASA Astrophysics Data System (ADS)
Concha, P. K.; Durka, R.; Merino, N.; Rodríguez, E. K.
2016-08-01
We introduce an alternative way of closing Maxwell like algebras. We show, through a suitable change of basis, that resulting algebras are given by the direct sums of the AdS and the Maxwell algebras already known in the literature. Casting the result into the S-expansion method framework ensures the straightaway construction of the gravity theories based on a found enlargement.
Unifying the Algebra for All Movement
ERIC Educational Resources Information Center
Eddy, Colleen M.; Quebec Fuentes, Sarah; Ward, Elizabeth K.; Parker, Yolanda A.; Cooper, Sandi; Jasper, William A.; Mallam, Winifred A.; Sorto, M. Alejandra; Wilkerson, Trena L.
2015-01-01
There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the "Algebra for All" movement. In light of this…
Build an Early Foundation for Algebra Success
ERIC Educational Resources Information Center
Knuth, Eric; Stephens, Ana; Blanton, Maria; Gardiner, Angela
2016-01-01
Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What's needed is a consistent, algebra-infused mathematics curriculum all…
Difficulties in Initial Algebra Learning in Indonesia
ERIC Educational Resources Information Center
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-01-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was…
A Balancing Act: Making Sense of Algebra
ERIC Educational Resources Information Center
Gavin, M. Katherine; Sheffield, Linda Jensen
2015-01-01
For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…
Algebra? A Gate! A Barrier! A Mystery!
ERIC Educational Resources Information Center
Mathematics Educatio Dialogues, 2000
2000-01-01
This issue of Mathematics Education Dialogues focuses on the nature and the role of algebra in the K-14 curriculum. Articles on this theme include: (1) "Algebra For All? Why?" (Nel Noddings); (2) "Algebra For All: It's a Matter of Equity, Expectations, and Effectiveness" (Dorothy S. Strong and Nell B. Cobb); (3) "Don't Delay: Build and Talk about…
Computer Algebra Systems, Pedagogy, and Epistemology
ERIC Educational Resources Information Center
Bosse, Michael J.; Nandakumar, N. R.
2004-01-01
The advent of powerful Computer Algebra Systems (CAS) continues to dramatically affect curricula, pedagogy, and epistemology in secondary and college algebra classrooms. However, epistemological and pedagogical research regarding the role and effectiveness of CAS in the learning of algebra lags behind. This paper investigates concerns regarding…
Teaching Strategies to Improve Algebra Learning
ERIC Educational Resources Information Center
Zbiek, Rose Mary; Larson, Matthew R.
2015-01-01
Improving student learning is the primary goal of every teacher of algebra. Teachers seek strategies to help all students learn important algebra content and develop mathematical practices. The new Institute of Education Sciences[IES] practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students"…
Leapfrog variants of iterative methods for linear algebra equations
NASA Technical Reports Server (NTRS)
Saylor, Paul E.
1988-01-01
Two iterative methods are considered, Richardson's method and a general second order method. For both methods, a variant of the method is derived for which only even numbered iterates are computed. The variant is called a leapfrog method. Comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large. In the case of Richardson's method, it is possible to express the final iterate in terms of only the initial approximation, a variant of the iteration called the grand-leap method. In the case of the grand-leap variant, a set of parameters is required. An algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for Gaussian quadrature. General algorithms to implement the leapfrog and grand-leap methods are presented. Algorithms for the important special case of the Chebyshev method are also given.
Powell, Sarah R; Fuchs, Lynn S
2014-08-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2(nd)- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty.
Powell, Sarah R; Fuchs, Lynn S
2014-08-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2(nd)- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty. PMID:25309044
Lie algebras for time evolution with applications from chaos studies to spintronics
NASA Astrophysics Data System (ADS)
Wendler, Tim G.; Berrondo, Manuel; Beus, Ty; Sayer, Ryan T.; van Huele, Jean-Francois S.
2012-10-01
We illustrate the power of Lie algebras in computing the time evolution of quantum systems with time-dependent physical parameters. By factorizing the quantum mechanical time evolution operator and using the linear independence of the Lie algebra generators, we reduce the operator equations to systems of coupled ordinary differential equations of scalar functions applicable to a variety of dynamical systems. We use the results to explore the possibility of detecting chaos in quantum nonlinear oscillators based on criteria from classical chaos studies and to follow spin currents in time-dependent spin-orbit coupled media.
Finding the Axis of Revolution of an Algebraic Surface of Revolution.
Alcazar, Juan G; Goldman, Ron
2016-09-01
We present an algorithm for extracting the axis of revolution from the implicit equation of an algebraic surface of revolution based on three distinct computational methods: factoring the highest order form into quadrics, contracting the tensor of the highest order form, and using univariate resultants and gcds. We compare and contrast the advantages and disadvantages of each of these three techniques and we derive conditions under which each technique is most appropriate. In addition, we provide several necessary conditions for an implicit algebraic equation to represent a surface of revolution.
The Clifford algebra of physical space and Dirac theory
NASA Astrophysics Data System (ADS)
Vaz, Jayme, Jr.
2016-09-01
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein–Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.
The Clifford algebra of physical space and Dirac theory
NASA Astrophysics Data System (ADS)
Vaz, Jayme, Jr.
2016-09-01
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.
Nonlocal Symmetry and Interaction Solutions of a Generalized Kadomtsev—Petviashvili Equation
NASA Astrophysics Data System (ADS)
Huang, Li-Li; Chen, Yong; Ma, Zheng-Yi
2016-08-01
A generalized Kadomtsev—Petviashvili equation is studied by nonlocal symmetry method and consistent Riccati expansion (CRE) method in this paper. Applying the truncated Painlevé analysis to the generalized Kadomtsev—Petviashvili equation, some Bäcklund transformations (BTs) including auto-BT and non-auto-BT are obtained. The auto-BT leads to a nonlocal symmetry which corresponds to the residual of the truncated Painlevé expansion. Then the nonlocal symmetry is localized to the corresponding nonlocal group by introducing two new variables. Further, by applying the Lie point symmetry method to the prolonged system, a new type of finite symmetry transformation is derived. In addition, the generalized Kadomtsev—Petviashvili equation is proved consistent Riccati expansion (CRE) solvable. As a result, the soliton-cnoidal wave interaction solutions of the equation are explicitly given, which are difficult to be found by other traditional methods. Moreover, figures are given out to show the properties of the explicit analytic interaction solutions. Supported by the Global Change Research Program of China under Grant No. 2015CB953904, National Natural Science Foundation of under Grant Nos. 11275072 and 11435005, Doctoral Program of Higher Education of China under Grant No. 20120076110024, the Network Information Physics Calculation of Basic Research Innovation Research Group of China under Grant No. 61321064, and Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things under Grant No. ZF1213, and Zhejiang Provincial Natural Science Foundation of China under Grant No. LY14A010005
Entropy algebras and Birkhoff factorization
NASA Astrophysics Data System (ADS)
Marcolli, Matilde; Tedeschi, Nicolas
2015-11-01
We develop notions of Rota-Baxter structures and associated Birkhoff factorizations, in the context of min-plus semirings and their thermodynamic deformations, including deformations arising from quantum information measures such as the von Neumann entropy. We consider examples related to Manin's renormalization and computation program, to Markov random fields and to counting functions and zeta functions of algebraic varieties.
Algebraic Activities Aid Discovery Lessons
ERIC Educational Resources Information Center
Wallace-Gomez, Patricia
2013-01-01
After a unit on the rules for positive and negative numbers and the order of operations for evaluating algebraic expressions, many students believe that they understand these principles well enough, but they really do not. They clearly need more practice, but not more of the same kind of drill. Wallace-Gomez provides three graphing activities that…
ERIC Educational Resources Information Center
Bosse, Michael J.; Ries, Heather; Chandler, Kayla
2012-01-01
Secondary school mathematics teachers often need to answer the "Why do we do that?" question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students' conceptual knowledge of secondary mathematics. This…
Dimension independence in exterior algebra.
Hawrylycz, M
1995-01-01
The identities between homogeneous expressions in rank 1 vectors and rank n - 1 covectors in a Grassmann-Cayley algebra of rank n, in which one set occurs multilinearly, are shown to represent a set of dimension-independent identities. The theorem yields an infinite set of nontrivial geometric identities from a given identity. PMID:11607520
ERIC Educational Resources Information Center
Oishi, Lindsay
2011-01-01
"Solve for x." While many people first encountered this enigmatic instruction in high school, the last 20 years have seen a strong push to get students to take algebra in eighth grade or even before. Today, concerns about the economy highlight a familiar worry: American eighth-graders trailed their peers in five Asian countries on the 2007 TIMSS…
Weaving Geometry and Algebra Together
ERIC Educational Resources Information Center
Cetner, Michelle
2015-01-01
When thinking about student reasoning and sense making, teachers must consider the nature of tasks given to students along with how to plan to use the tasks in the classroom. Students should be presented with tasks in a way that encourages them to draw connections between algebraic and geometric concepts. This article focuses on the idea that it…
Algebraic methods in system theory
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Willems, J. C.; Willsky, A. S.
1975-01-01
Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.
Algebra from Chips and Chopsticks
ERIC Educational Resources Information Center
Yun, Jeong Oak; Flores, Alfinio
2012-01-01
Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…
Celestial mechanics with geometric algebra
NASA Technical Reports Server (NTRS)
Hestenes, D.
1983-01-01
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.
Algebra for All. Research Brief
ERIC Educational Resources Information Center
Bleyaert, Barbara
2009-01-01
The call for "algebra for all" is not a recent phenomenon. Concerns about the inadequacy of math (and science) preparation in America's high schools have been a steady drumbeat since the 1957 launch of Sputnik; a call for raising standards and the number of math (and science) courses required for graduation has been a part of countless national…
Adventures in Flipping College Algebra
ERIC Educational Resources Information Center
Van Sickle, Jenna
2015-01-01
This paper outlines the experience of a university professor who implemented flipped learning in two sections of college algebra courses for two semesters. It details how the courses were flipped, what technology was used, advantages, challenges, and results. It explains what students do outside of class, what they do inside class, and discusses…
ERIC Educational Resources Information Center
Deakin, Michael A. B.
1974-01-01
Euler's famous formula, e to the (i, pi) power equals -1, is developed by a purely algebraic method that avoids the use of both trigonometry and calculus. A heuristic outline is given followed by the rigorous theory. Pedagogical considerations for classroom presentation are suggested. (LS)
Inequalities, Assessment and Computer Algebra
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2015-01-01
The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in…
Algebra, Home Mortgages, and Recessions
ERIC Educational Resources Information Center
Mariner, Jean A. Miller; Miller, Richard A.
2009-01-01
The current financial crisis and recession in the United States present an opportunity to discuss relevant applications of some topics in typical first-and second-year algebra and precalculus courses. Real-world applications of percent change, exponential functions, and sums of finite geometric sequences can help students understand the problems…
Math Sense: Algebra and Geometry.
ERIC Educational Resources Information Center
Howett, Jerry
This book is designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four strands: (1) skill lessons…
NASA Astrophysics Data System (ADS)
Kuipers, J.
2012-06-01
New features of the symbolic algebra package Form 4 are discussed. Most importantly, these features include polynomial factorization and polynomial gcd computation. Examples of their use are shown. One of them is an exact version of Mincer which gives answers in terms of rational polynomials and 5 master integrals.
An algebraic criterion for the onset of chaos in nonlinear dynamic systems
NASA Technical Reports Server (NTRS)
Unal, A.; Tobak, M.
1987-01-01
The correspondence between iterated integrals and a noncommutative algebra is used to recast the given dynamical system from the time domain to the Laplace-Borel transform domain. It is then shown that the following algebraic criterion has to be satisfied for the outset of chaos: the limit (as tau approaches infinity and x sub 0 approaches infinity) of ((sigma(k=0) (tau sup k) / (k* x sub 0 sup k)) G II G = 0, where G is the generating power series of the trajectories, the symbol II is the shuffle product (le melange) of the noncommutative algebra, x sub 0 is a noncommutative variable, and tau is the correlation parameter. In the given equation, symbolic forms for both G and II can be obtained by use of one of the currently available symbolic languages such as PLI, REDUCE, and MACSYMA. Hence, the criterion is a computer-algebraic one.
Array algebra estimation in signal processing
NASA Astrophysics Data System (ADS)
Rauhala, U. A.
A general theory of linear estimators called array algebra estimation is interpreted in some terms of multidimensional digital signal processing, mathematical statistics, and numerical analysis. The theory has emerged during the past decade from the new field of a unified vector, matrix and tensor algebra called array algebra. The broad concepts of array algebra and its estimation theory cover several modern computerized sciences and technologies converting their established notations and terminology into one common language. Some concepts of digital signal processing are adopted into this language after a review of the principles of array algebra estimation and its predecessors in mathematical surveying sciences.
NASA Astrophysics Data System (ADS)
Hu, Xiao-Rui; Chen, Yong
2015-09-01
For the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the BKK system in the sense of having a consistent Riccati expansion (CRE) is investigated. The interaction solutions between soliton and cnoidal periodic wave are explicitly studied. Project supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ13A010014) and the National Natural Science Foundation of China (Grant Nos. 11326164, 11401528, 11435005, and 11375090).
Filiform Lie algebras of order 3
Navarro, R. M.
2014-04-15
The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.
Atomic effect algebras with compression bases
Caragheorgheopol, Dan; Tkadlec, Josef
2011-01-15
Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.
NASA Astrophysics Data System (ADS)
Essén, Hanno; Nordmark, Arne B.
2016-09-01
The canonical Poisson bracket algebra of four-dimensional relativistic mechanics is used to derive the equation of motion for a charged particle, with the Lorentz force, and the homogeneous Maxwell equations.
NASA Astrophysics Data System (ADS)
Essén, Hanno; Nordmark, Arne B.
2016-09-01
The canonical Poisson bracket algebra of four-dimensional relativistic mechanics is used to derive the equation of motion for a charged particle, with the Lorentz force, and the homogeneous Maxwell equations.
Generalized quantum statistics and Lie (super)algebras
NASA Astrophysics Data System (ADS)
Stoilova, N. I.
2016-03-01
Generalized quantum statistics, such as paraboson and parafermion statistics, are characterized by triple relations which are related to Lie (super)algebras of type B. The correspondence of the Fock spaces of parabosons, parafermions as well as the Fock space of a system of parafermions and parabosons to irreducible representations of (super)algebras of type B will be pointed out. Example of generalized quantum statistics connected to the basic classical Lie superalgebra B(1|1) ≡ osp(3|2) with interesting physical properties, such as noncommutative coordinates, will be given. Therefore the article focuses on the question, addressed already in 1950 by Wigner: do the equation of motion determine the quantum mechanical commutation relation?
A Unified Introduction to Ordinary Differential Equations
ERIC Educational Resources Information Center
Lutzer, Carl V.
2006-01-01
This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)
Synthesizing Strategies Creatively: Solving Linear Equations
ERIC Educational Resources Information Center
Ponce, Gregorio A.; Tuba, Imre
2015-01-01
New strategies can ignite teachers' imagination to create new lessons or adapt lessons created by others. In this article, the authors present the experience of an algebra teacher and his students solving linear and literal equations and explain how the use of ideas found in past NCTM journals helped bring this lesson to life. The…
A new model for algebraic Rossby solitary waves in rotation fluid and its solution
NASA Astrophysics Data System (ADS)
Chen, Yao-Deng; Yang, Hong-Wei; Gao, Yu-Fang; Yin, Bao-Shu; Feng, Xing-Ru
2015-09-01
A generalized Boussinesq equation that includes the dissipation effect is derived to describe a kind of algebraic Rossby solitary waves in a rotating fluid by employing perturbation expansions and stretching transformations of time and space. Using this equation, the conservation laws of algebraic Rossby solitary waves are discussed. It is found that the mass, the momentum, the energy, and the velocity of center of gravity of the algebraic solitary waves are conserved in the propagation process. Finally, the analytical solution of the equation is generated. Based on the analytical solution, the properties of the algebraic solitary waves and the dissipation effect are discussed. The results point out that, similar to classic solitary waves, the dissipation can cause the amplitude and the speed of solitary waves to decrease; however, unlike classic solitary waves, the algebraic solitary waves can split during propagation and the decrease of the detuning parameter can accelerate the occurrence of the solitary waves fission phenomenon. Project supported by the Shandong Provincial Key Laboratory of Marine Ecology and Environment and Disaster Prevention and Mitigation Project, China (Grant No. 2012010), the National Natural Science Foundation of China (Grant Nos. 41205082 and 41476019), the Special Funds for Theoretical Physics of the National Natural Science Foundation of China (Grant No. 11447205), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), China.
A Z{sub 3} generalization of Pauli's principle, quark algebra and the Lorentz invariance
Kerner, Richard
2012-09-24
The fundamental difference between bosons and fermions is that they obey two alternative representations of the Z{sub 2} group, resulting in symmetric or anti-symmetric binary commutation relations. Our aim is to explore possibilities offered by ternary Z{sub 3} generalization commutation relations. This leads to cubic and ternary algebras which are a direct generalization of usual commutation relations, with Z{sub 3}-grading replacing the usual Z{sub 2}-grading. Properties and structure of such algebras are discussed, with special interest in a low-dimensional one, with two generators. Invariant cubic forms on such algebras are introduced, and it is shown how the SL(2,C) group arises naturally as the symmetry group preserving these forms. In the case of lowest dimension, with only two generators, it is shown how the cubic combinations of elements of the same Z{sub 3} grade behave like Lorentz spinors, while binary products of elements of this algebra with an element of the conjugate algebra behave like Lorentz vectors. The wave equation generalizing the Dirac operator to the Z{sub 3}-graded case is introduced, whose diagonalization leads to a third-order equation. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry obeyed by quark states.
Simple derivation of the Lindblad equation
NASA Astrophysics Data System (ADS)
Pearle, Philip
2012-07-01
The Lindblad equation is an evolution equation for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is Hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad equation are given. The derivation of the Lindblad equation presented here is ‘simple’ in that all it uses is the expression of a Hermitian matrix in terms of its orthonormal eigenvectors and real eigenvalues. Thus, it is appropriate for students who have learned the algebra of quantum theory. Where helpful, arguments are first given in a two-dimensional Hilbert space.
Analytical solution of tt dilepton equations
Sonnenschein, Lars
2006-03-01
The top quark antiquark production system in the dilepton decay channel is described by a set of equations which is nonlinear in the unknown neutrino momenta. Its most precise and least time consuming solution is of major importance for measurements of top quark properties like the top quark mass and tt spin correlations. The initial system of equations can be transformed into two polynomial equations with two unknowns by means of elementary algebraic operations. These two polynomials of multidegree two can be reduced to one univariate polynomial of degree four by means of resultants. The obtained quartic equation is solved analytically.
Linear algebra algorithms for divisors on an algebraic curve
NASA Astrophysics Data System (ADS)
Khuri-Makdisi, Kamal
We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and matrices of size $O(g^2)\\times O(g)$. When the base field $k$ is finite, or if $C$ has a rational point over $k$, these give algorithms for working on the Jacobian of $C$ that require $O(g^4)$ field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to work with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Hess' 1999 Ph.D. thesis, which works with function fields as extensions of $k[x]$. However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor), superelliptic curves (Galbraith, Paulus, and Smart), and $C_{ab}$ curves (Harasawa and Suzuki); in all those cases, one can attain a complexity of $O(g^2)$.
Using trees to compute approximate solutions to ordinary differential equations exactly
NASA Technical Reports Server (NTRS)
Grossman, Robert
1991-01-01
Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.
Alternative algebraic approaches in quantum chemistry
Mezey, Paul G.
2015-01-22
Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed.
The algebras of large N matrix mechanics
Halpern, M.B.; Schwartz, C.
1999-09-16
Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.
Solution Methods for Certain Evolution Equations
NASA Astrophysics Data System (ADS)
Vega-Guzman, Jose Manuel
Solution methods for certain linear and nonlinear evolution equations are presented in this dissertation. Emphasis is placed mainly on the analytical treatment of nonautonomous differential equations, which are challenging to solve despite the existent numerical and symbolic computational software programs available. Ideas from the transformation theory are adopted allowing one to solve the problems under consideration from a non-traditional perspective. First, the Cauchy initial value problem is considered for a class of nonautonomous and inhomogeneous linear diffusion-type equation on the entire real line. Explicit transformations are used to reduce the equations under study to their corresponding standard forms emphasizing on natural relations with certain Riccati(and/or Ermakov)-type systems. These relations give solvability results for the Cauchy problem of the parabolic equation considered. The superposition principle allows to solve formally this problem from an unconventional point of view. An eigenfunction expansion approach is also considered for this general evolution equation. Examples considered to corroborate the efficacy of the proposed solution methods include the Fokker-Planck equation, the Black-Scholes model and the one-factor Gaussian Hull-White model. The results obtained in the first part are used to solve the Cauchy initial value problem for certain inhomogeneous Burgers-type equation. The connection between linear (the Diffusion-type) and nonlinear (Burgers-type) parabolic equations is stress in order to establish a strong commutative relation. Traveling wave solutions of a nonautonomous Burgers equation are also investigated. Finally, it is constructed explicitly the minimum-uncertainty squeezed states for quantum harmonic oscillators. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. It is shown that the product of the variances attains the required minimum value
BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1994-01-01
The Basic Linear Algebra Subprogram (BLAS) library is a collection of FORTRAN callable routines for employing standard techniques in performing the basic operations of numerical linear algebra. The BLAS library was developed to provide a portable and efficient source of basic operations for designers of programs involving linear algebraic computations. The subprograms available in the library cover the operations of dot product, multiplication of a scalar and a vector, vector plus a scalar times a vector, Givens transformation, modified Givens transformation, copy, swap, Euclidean norm, sum of magnitudes, and location of the largest magnitude element. Since these subprograms are to be used in an ANSI FORTRAN context, the cases of single precision, double precision, and complex data are provided for. All of the subprograms have been thoroughly tested and produce consistent results even when transported from machine to machine. BLAS contains Assembler versions and FORTRAN test code for any of the following compilers: Lahey F77L, Microsoft FORTRAN, or IBM Professional FORTRAN. It requires the Microsoft Macro Assembler and a math co-processor. The PC implementation allows individual arrays of over 64K. The BLAS library was developed in 1979. The PC version was made available in 1986 and updated in 1988.
Introduction to Image Algebra Ada
NASA Astrophysics Data System (ADS)
Wilson, Joseph N.
1991-07-01
Image Algebra Ada (IAA) is a superset of the Ada programming language designed to support use of the Air Force Armament Laboratory's image algebra in the development of computer vision application programs. The IAA language differs from other computer vision languages is several respects. It is machine independent, and an IAA translator has been implemented in the military standard Ada language. Its image operands and operations can be used to program a range of both low- and high-level vision algorithms. This paper provides an overview of the image algebra constructs supported in IAA and describes the embodiment of these constructs in the IAA extension of Ada. Examples showing the use of IAA for a range of computer vision tasks are given. The design of IAA as a superset of Ada and the implementation of the initial translator in Ada represent critical choices. The authors discuss the reasoning behind these choices as well as the benefits and drawbacks associated with them. Implementation strategies associated with the use of Ada as an implementation language for IAA are also discussed. While one can look on IAA as a program design language (PDL) for specifying Ada programs, it is useful to consider IAA as a separate language superset of Ada. This admits the possibility of directly translating IAA for implementation on special purpose architectures. This paper explores strategies for porting IAA to various architectures and notes the critical language and implementation features for porting to different architectures.
Modular properties of characters of the W3 algebra
NASA Astrophysics Data System (ADS)
Iles, Nicholas J.; Watts, Gérard M. T.
2016-01-01
In a previous work, exact formulae and differential equations were found for traces of powers of the zero mode in the W 3 algebra. In this paper we investigate their modular properties, in particular we find the exact result for the modular transformations of traces of W 0 n for n = 1 , 2 , 3, solving exactly the problem studied approximately by Gaberdiel, Hartman and Jin. We also find modular differential equations satisfied by traces with a single W 0 inserted, and relate them to differential equations studied by Mathur et al. We find that, remarkably, these all seem to be related to weight 0 modular forms with expansions with non-negative integer coefficients.
Polarization ellipse and Stokes parameters in geometric algebra.
Santos, Adler G; Sugon, Quirino M; McNamara, Daniel J
2012-01-01
In this paper, we use geometric algebra to describe the polarization ellipse and Stokes parameters. We show that a solution to Maxwell's equation is a product of a complex basis vector in Jackson and a linear combination of plane wave functions. We convert both the amplitudes and the wave function arguments from complex scalars to complex vectors. This conversion allows us to separate the electric field vector and the imaginary magnetic field vector, because exponentials of imaginary scalars convert vectors to imaginary vectors and vice versa, while exponentials of imaginary vectors only rotate the vector or imaginary vector they are multiplied to. We convert this expression for polarized light into two other representations: the Cartesian representation and the rotated ellipse representation. We compute the conversion relations among the representation parameters and their corresponding Stokes parameters. And finally, we propose a set of geometric relations between the electric and magnetic fields that satisfy an equation similar to the Poincaré sphere equation.
A Unified Approach to Teaching Quadratic and Cubic Equations.
ERIC Educational Resources Information Center
Ward, A. J. B.
2003-01-01
Presents a simple method for teaching the algebraic solution of cubic equations via completion of the cube. Shows that this method is readily accepted by students already familiar with completion of the square as a method for quadratic equations. (Author/KHR)
XXZ-type Bethe ansatz equations and quasi-polynomials
NASA Astrophysics Data System (ADS)
Li, Jian Rong; Tarasov, Vitaly
2013-01-01
We study solutions of the Bethe ansatz equation for the XXZ-type integrable model associated with the Lie algebra fraktur sfraktur lN. We give a correspondence between solutions of the Bethe ansatz equations and collections of quasi-polynomials. This extends the results of E. Mukhin and A. Varchenko for the XXX-type model and the trigonometric Gaudin model.
Algebra: A Challenge at the Crossroads of Policy and Practice
ERIC Educational Resources Information Center
Stein, Mary Kay; Kaufman, Julia Heath; Sherman, Milan; Hillen, Amy F.
2011-01-01
The authors review what is known about early and universal algebra, including who is getting access to algebra and student outcomes associated with algebra course taking in general and specifically with universal algebra policies. The findings indicate that increasing numbers of students, some of whom are underprepared, are taking algebra earlier.…
Loop equations and KDV hierarchy in 2-D quantum gravity
Fucito, F. ); Martellini, M. )
1992-04-20
In this paper a derivation of the loop equation for two-dimensional quantum gravity from the KdV equations and the string equation of the one-matrix model is given. The loop equation was found to be equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the virasoro algebra. Starting form the equations expressing these constraints. The authors are able to rederive the equations of the KdV hierarchy using the vertex operator construction of the A{sup (I)}{sub I} infinite dimensional twisted Kac-Moody algebra. From these considerations it follows that the solutions of the string equation of the one-matrix model are given by a subset of the solutions of the KdV hierarchy.
Software package and API in MATLAB for working with fuzzy algebras
NASA Astrophysics Data System (ADS)
Zahariev, Zlatko
2009-11-01
New software package for fuzzy calculus is presented in this paper. Most important feature of this package is solving fuzzy linear systems of equations and inequalities in fuzzy algebras. Together with this, some other functionality is implemented. An example is also given.
Mathematics for High School, First Course in Algebra, Part 3. Preliminary Edition.
ERIC Educational Resources Information Center
Allen, Frank B.; And Others
This is the third part of a three-part SMSG algebra text for high school students. Chapter titles include: Truth Sets of Open Sentences; Graphs of Open Sentences in Two Variables; Systems of Equations and Inequalities; Quadratic Polynomials; and Functions. (MK)
NASA Technical Reports Server (NTRS)
Rostand, Philippe
1988-01-01
The incorporation of algebraic turbulence models in a solver for the 2-D compressible Navier-Stokes equations using triangular grids is described. A practical way to use the Cebeci Smith model, and to modify it in separated regions is proposed. The ability of the model to predict high speed, perfect gas boundary layers is investigated from a numerical point of view.
Cognitive Tutor[R] Algebra I. What Works Clearinghouse Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2007
2007-01-01
"Cognitive Tutor[R] Algebra I," a full year course, delivers instruction in single variable data, simplifying linear expressions, mathematical modeling, solving systems with linear equations, problem solving using proportional reasoning, and powers and exponents. Students work at their own pace to develop problem-solving skills. The duration of…