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.
Sensitivity Analysis of Differential-Algebraic Equations and Partial Differential Equations
Petzold, L; Cao, Y; Li, S; Serban, R
2005-08-09
Sensitivity analysis generates essential information for model development, design optimization, parameter estimation, optimal control, model reduction and experimental design. In this paper we describe the forward and adjoint methods for sensitivity analysis, and outline some of our recent work on theory, algorithms and software for sensitivity analysis of differential-algebraic equation (DAE) and time-dependent partial differential equation (PDE) systems.
Computing spacetime curvature via differential-algebraic equations
Ashby, S.F.; Lee, S.L.; Petzold, L.R.; Saylor, P.E.; Seidel, E.
1996-01-01
The equations that govern the behavior of physical systems can often solved numerically using a method of lines approach and differential-algebraic equation (DAE) solvers. For example, such an approach can be used to solve the Einstein field equations of general relativity, and thereby simulate significant astrophysical events. In this paper, we describe some preliminary work in which two model problems in general relativity are formulated, spatially discretized, and then numerically solved as a DAE. In particular, we seek to reproduce the solution to the spherically symmetric Schwarzschild spacetime. This is an important testbed calculation in numerical relativity since the solution is the steady-state for the collision of two (or more) non-rotating black holes. Moreover, analytic late-time properties of the Schwarzschild spacetime are well known and can be used the accuracy of the simulation.
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
Introducing DAE Systems in Undergraduate and Graduate Chemical Engineering Curriculum
ERIC Educational Resources Information Center
Mandela, Ravi Kumar; Sridhar, L. N.; Rengaswamy, Raghunathan
2010-01-01
Models play an important role in understanding chemical engineering systems. While differential equation models are taught in standard modeling and control courses, Differential Algebraic Equation (DAE) system models are not usually introduced. These models appear naturally in several chemical engineering problems. In this paper, the introduction…
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.
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.
DAEs and PDEs in elastic multibody systems
NASA Astrophysics Data System (ADS)
Simeon, B.
1998-12-01
Elastic multibody systems arise in the simulation of vehicles, robots, air- and spacecrafts. They feature a mixed structure with differential-algebraic equations (DAEs) governing the gross motion and partial differential equations (PDEs) describing the elastic deformation of particular bodies. We introduce a general modelling framework for this new application field and discuss numerical simulation techniques from several points of view. Due to different time scales, singular perturbation theory and model reduction play an important role. A slider crank mechanism with a 2D FE grid for the elastic connecting rod illustrates the techniques.
One-Equation Algebraic Model Of Turbulence
NASA Technical Reports Server (NTRS)
Baldwin, B. S.; Barth, T. J.
1993-01-01
One-equation model of turbulence based on standard equations of k-epsilon model of turbulence, where k is turbulent energy and e is rate of dissipation of k. Derivation of one-equation model motivated partly by inaccuracies of flows computed by some Navier-Stokes-equations-solving algorithms incorporating algebraic models of turbulence. Satisfies need to avoid having to determine algebraic length scales.
An algebraic approach to the scattering equations
NASA Astrophysics Data System (ADS)
Huang, Rijun; Rao, Junjie; Feng, Bo; He, Yang-Hui
2015-12-01
We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.
Trajectories of a DAE near a pseudo-equilibrium
NASA Astrophysics Data System (ADS)
Beardmore, R. E.; Laister, R.; Peplow, A.
2004-01-01
We consider a class of differential-algebraic equations (DAEs) defined by analytic nonlinearities and study its singular solutions. The main assumption used is that the linearization of the DAE represents a Kronecker index-2 matrix pencil and that the constraint manifold has a quadratic fold along its singularity. From these assumptions we obtain a normal form for the DAE where the presence of the singularity and its effects on the dynamics of the problem are made explicit in the form of a quasi-linear differential equation. Subsequently, two distinct types of singular points are identified through which there pass exactly two analytic solutions: pseudo-nodes and pseudo-saddles. We also demonstrate that a singular point called a pseudo-node supports an uncountable infinity of solutions which are not analytic in general. Moreover, akin to known results in the literature for DAEs with singular equilibria, a degenerate singularity is found through which there passes one analytic solution such that the singular point in question is contained within a quasi-invariant manifold of solutions. We call this type of singularity a pseudo-centre and it provides not only a manifold of solutions which intersects the singularity, but also a local flow on that manifold which solves the DAE.
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…
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.
The kinematic algebras from the scattering equations
NASA Astrophysics Data System (ADS)
Monteiro, Ricardo; O'Connell, Donal
2014-03-01
We study kinematic algebras associated to the recently proposed scattering equations, which arise in the description of the scattering of massless particles. In particular, we describe the role that these algebras play in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex which is associated to each solution of those equations. We also identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.
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.
The Dirac equation and Hestenes' geometric algebra
NASA Astrophysics Data System (ADS)
Hamilton, J. Dwayne
1984-06-01
Hestenes' geometric algebra and Dirac spinors are reviewed and united into a common mathematical formalism, a unification that establishes the Dirac equation as being manifestly covariant under the Lorentz group, and one that needs no matrix representation of the Dirac algebra. New and simple methods of amplitude or ``trace'' calculations are then described. A number of problems are then considered within the context of the new approach, such as relativistic spin projections, new and covariant C and T-transformations and spinors for massless and Majorana fields.
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.
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.
Affine Vertex Operator Algebras and Modular Linear Differential Equations
NASA Astrophysics Data System (ADS)
Arike, Yusuke; Kaneko, Masanobu; Nagatomo, Kiyokazu; Sakai, Yuichi
2016-05-01
In this paper, we list all affine vertex operator algebras of positive integral levels whose dimensions of spaces of characters are at most 5 and show that a basis of the space of characters of each affine vertex operator algebra in the list gives a fundamental system of solutions of a modular linear differential equation. Further, we determine the dimensions of the spaces of characters of affine vertex operator algebras whose numbers of inequivalent simple modules are not exceeding 20.
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.
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.
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.
Algebraic Approach to the Computation of the Defining Polynomial of the Algebraic Riccati Equation
NASA Astrophysics Data System (ADS)
Kitamoto, Takuya
The algebraic Riccati equation, which we denote by ’ARE’ in the rest of the paper, is one of the most important equations of the post modern control theory. It plays important role for solving H 2 and H ∞ optimal control problems.
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.
New developments in the numerical solution of differential/algebraic systems
Petzold, L.R.
1987-04-01
In this paper we survey some recent developments in the numerical solution of nonlinear differential/algebraic equation (DAE) systems of the form 0 = F(t,y,y'), where the initial values of y are known and par. deltaF/par. deltay' may be singular. These systems arise in the simulation of electrical networks, as well as in many other applications. DAE systems include standard form ODEs as a special case, but they also include problems which are in many ways quite different from ODEs. We examine the classification of DAE systems according to the degree of singularity of the system, and present some results on the analytical structure of these systems. We give convergence results for backward differentiation formulas applied to DAEs and examine some of the software issues involved in the numerical solution of DAEs. One-step methods are potentially advantageous for solving DAE systems with frequent discontinuities. However, recent results indicate that there is a reduction in the order of accuracy of many implicit Runge-Kutta methods even for simple DAE systems. We examine the current state of solving DAE systems by implicit Runge-Kutta methods. Finding a consistent set of initial conditions is often a problem for DAEs arising in applications. We explore some numerical methods for obtaining a consistent set of initial conditions. 21 refs.
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.
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.
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...
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.
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)
Okubo, Naoto
2015-08-01
We construct cluster algebras the variables and coefficients of which satisfy the discrete mKdV equation, the discrete Toda equation and other integrable bilinear equations, several of which lead to q-discrete Painlevé equations. These cluster algebras are obtained from quivers with an infinite number of vertices or with the mutation-period property. We will also show that a suitable transformation of quivers corresponds to a reduction of the difference equation.
Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations
Wackerbauer, R. ); Huebler, A. . Center for Complex Systems Research); Mayer-Kress, G. California Univ., Santa Cruz, CA . Dept. of Mathematics)
1991-07-25
The relation between the parameters of a differential equation and corresponding discrete maps are becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbation methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. A new iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODE's with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is polynomial and if the ODE has fixed point in the origin. Approximations of different orders respectively of the rest term are investigated for several nonlinear systems. 31 refs., 16 figs.
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…
ERIC Educational Resources Information Center
Capraro, Mary Margaret; Joffrion, Heather
2006-01-01
Using symbolic algebra to represent and solve linear equations is one of the expectations within the "Algebra" content standard for the 6-8-grade band in the National Council of Teachers of Mathematics (NCTM) "Principles and Standards for School Mathematics" (2000). Students' understanding of these concepts, even before a formal algebra course,…
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.
NASA Astrophysics Data System (ADS)
Aghaei, S.; Chenaghlou, A.
2014-02-01
The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E8 (case (3b)), S4 and S2, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.
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…
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.
The Algebra Solution to Mathematics Reform: Completing the Equation
ERIC Educational Resources Information Center
Spielhagen, Frances R.
2011-01-01
How can we increase mathematics achievement among all students? This book provides a straightforward explanation of how changing mathematics tracking policies to provide algebra instruction to all students by at least eighth grade can bring about changes in both student achievement and teacher performance. Spielhagen chronicles the success of a…
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.
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…
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.
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.
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…
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…
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…
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…
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…
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.)
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…
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
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…
On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Honwah, Tam
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
Soliton equations related to the affine Kac-Moody algebra D{4/(1)}
NASA Astrophysics Data System (ADS)
Gerdjikov, V. S.; Mladenov, D. M.; Stefanov, A. A.; Varbev, S. K.
2015-06-01
We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D {4/(1)} by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data.
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. PMID:21932262
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.
NASA Astrophysics Data System (ADS)
Knudsen, E.; Richardson, E. S.; Doran, E. M.; Pitsch, H.; Chen, J. H.
2012-05-01
Scalar dissipation rates and subfilter scalar variances are important modeling parameters in large eddy simulations (LES) of reacting flows. Currently available models capture the general behavior of these parameters, but these models do not always perform with the degree of accuracy that is needed for predictive LES. Here, two direct numerical simulations (DNS) are used to analyze LES dissipation rate and variance models, and to propose a new model for the dissipation rate that is based on a transport equation. The first DNS that is considered is a non-premixed auto-igniting C2H4 jet flame simulation originally performed by Yoo et al. [Proc. Combust. Inst. 33, 1619-1627 (2011)], 10.1016/j.proci.2010.06.147. A LES of this case is run using algebraic models for the dissipation rate and subfilter variance. It is shown that the algebraic models fail to adequately reproduce the DNS results. This motivates the introduction of a transport equation model for the LES dissipation rate. Closure of the equation is addressed by formulating a new adapted dynamic approach. This approach borrows dynamically computed information from LES quantities that, unlike the dissipation rate, do not reside on the smallest flow length scales. The adapted dynamic approach is analyzed by considering a second DNS of scalar mixing in homogeneous isotropic turbulence. Data from this second DNS are used to confirm that the adapted dynamic approach successfully closes the dissipation rate equation over a wide range of LES filter widths. The first reacting jet case is then returned to and used to test the LES transport equation models. The transport equation model for the dissipation rate is shown to be more accurate than its algebraic counterpoint, and the dissipation rate is eliminated as a source of error in the transported variance model.
Parabolic orbit determination. Comparison of the Olbers method and algebraic equations
NASA Astrophysics Data System (ADS)
Kuznetsov, V. B.
2016-05-01
In this paper, the Olbers method for the preliminary parabolic orbit determination (in the Lagrange-Subbotin modification) and the method based on systems of algebraic equations for two or three variables proposed by the author are compared. The maximum number of possible solutions is estimated. The problem of selection of the true solution from the set of solutions obtained both using additional equations and by the problem reduction to finding the objective function minimum is considered. The results of orbit determination of the comets 153P/Ikeya-Zhang and 2007 N3 Lulin are cited as examples.
ERIC Educational Resources Information Center
Linchevski, Liora; Herscovics, Nicolas
1996-01-01
Reports the results of a teaching experiment involving like terms and equations in algebra. Seventh-grade students (n=6) experienced difficulties in decomposing an additive term into a difference. (Author/MKR)
NASA Astrophysics Data System (ADS)
Sheinman, O. K.
2015-12-01
Based on ℤ-gradings of semisimple Lie algebras and invariant polynomials on them, we construct hierarchies of Lax equations with a spectral parameter on a Riemann surface and prove the commutativity of the corresponding flows.
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.
Modeling boyciana-fish-human interaction with partial differential algebraic equations.
Jiang, Yushan; Zhang, Qingling; Wang, Haiyan
2016-07-01
Under the influence of human population distribution, the boyciana-fish ecological system is considered. First, the system can be described as a nonlinear partial differential algebraic equations system (PDAEs) with Neumann boundary conditions and ratio-dependent functional response. Second, we examine the system's persistence properties: the loacl stabilities of positive steady states, the absorbtion region and the global stability. And the proposed approach is illustrated by numerical simulation. Finally, by using the realistic data collected in the past fourteen years, the PDAEs parameter optimization model is built to predict the boyciana population. PMID:27155570
Symmetries of the Schrödinger equation and algebra/superalgebra duality
NASA Astrophysics Data System (ADS)
Toppan, Francesco
2015-04-01
Some key features of the symmetries of the Schrodinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving first and second-order differential operators. It provides different viewpoints for the spectrum-generating subalgebras. The representation- dependent notion of on-shell symmetry is introduced. The difference in associating the time-derivative symmetry operator with either a root or a Cartan generator of the sl(2) subalgebra is discussed. In application to one-dimensional Lagrangian superconformal sigma-models it implies superconformal actions which are either supersymmetric or non-supersymmetric.
NASA Astrophysics Data System (ADS)
Napora, Jolanta
2000-10-01
A given Riccati equation, as is well known, can be naturally reduced to a system of nonlinear evolution equations on an infinite-dimensional functional manifold with Cauchy-Goursat initial data. We describe the Lie algebraic reduction procedure of nonlocal type for this infinite-dimensional dynamical system upon the set of critical points of an invariant Lagrangian functional. As one of our main results, we show that the reduced dynamical system generates the completely integrable Hamiltonian flow on this submanifold with respect to the canonical symplectic structure upon it. The above also makes it possible to find effectively its finite-dimensional Lax type representation via both the well known Moser type reduction procedure and the dual momentum mapping scheme on some matrix manifold.
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 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.
NASA Astrophysics Data System (ADS)
Dodd, R. K.
2014-02-01
In this paper we derive Hirota equations associated with the simply laced affine Lie algebras {{g}}^{(1)}, where {{g}} is one of the simply laced complex Lie algebras {{a}}_n, {{d}}_n, {{e}}_6, {{e}}_7 or {{e}}_8, defined by finite order automorphisms of {{g}} which we call Lepowsky automorphisms. In particular, we investigate the Hirota equations for Lepowsky automorphisms of {{e}}_6 defined by the cuspidal class E6 of the Weyl group W(E6) of {{e}}_6. We also investigate the relationship between the Lepowsky automorphisms of the simply laced complex Lie algebras {{g}} and the conjugate canonical automorphisms defined by Kac. This analysis is applied to identify the canonical automorphisms for the cuspidal class E6 of {{e}}_6.
Numerical prediction of kinetic model for enzymatic hydrolysis of cellulose using DAE-QMOM approach
NASA Astrophysics Data System (ADS)
Jamil, N. M.; Wang, Q.
2016-06-01
Bioethanol production from lignocellulosic biomass consists of three fundamental processes; pre-treatment, enzymatic hydrolysis, and fermentation. In enzymatic hydrolysis phase, the enzymes break the cellulose chains into sugar in the form of cellobiose or glucose. A currently proposed kinetic model for enzymatic hydrolysis of cellulose that uses population balance equation (PBE) mechanism was studied. The complexity of the model due to integrodifferential equations makes it difficult to find the analytical solution. Therefore, we solved the full model of PBE numerically by using DAE-QMOM approach. The computation was carried out using MATLAB software. The numerical results were compared to the asymptotic solution developed in the author's previous paper and the results of Griggs et al. Besides confirming the findings were consistent with those references, some significant characteristics were also captured. The PBE model for enzymatic hydrolysis process can be solved using DAE-QMOM method. Also, an improved understanding of the physical insights of the model was achieved.
NASA Astrophysics Data System (ADS)
Aghaei, S.; Chenaghlou, A.
2015-01-01
In this paper, we study the Dirac equation with spin and pseudospin symmetry by the quadratic algebra approach for the 4-dimensional harmonic oscillator. By realization of the quadratic algebras in the deformed oscillator algebra, we obtain the relativistic energy spectrum. Also, by regarding the generalized Kustaanheimo-Stiefel transformation, we obtain the relativistic energy spectrum for the charge-dyon system with the U(1) monopole.
From Function to Equations: Introduction of Algebraic Thinking to 13 Year-Old Students
ERIC Educational Resources Information Center
Farmaki, Vasiliki; Klaoudatos, Nikos; Verikios, Petros
2004-01-01
The various difficulties and cognitive obstacles that students face when they are introduced to algebra are well documented and described in the relevant bibliography. If it is possible, in order to avoid these difficulties, we have adopted the functional approach widening the meaning of algebraic thinking. In this paper, which is part of wider…
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…
ERIC Educational Resources Information Center
Hewitt, Dave
2012-01-01
This study looks at a mixed ability group of 21 Year 5 primary students (aged 9-10 years old) who had previously never had formal instruction using letters to stand for unknowns or variables in a mathematics context; nor had they been introduced to formal algebraic notation. Three lessons were taught using the computer software "Grid Algebra"…
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…
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.
NASA Astrophysics Data System (ADS)
Campoamor-Stursberg, R.
2016-06-01
A functional realization of the Lie algebra s l (" separators=" 3 , R) as a Vessiot-Guldberg-Lie algebra of second order differential equation (SODE) Lie systems is proposed. It is shown that a minimal Vessiot-Guldberg-Lie algebra L V G is obtained from proper subalgebras of s l (" separators=" 3 , R) for each of the SODE Lie systems of this type by particularization of one functional and two scalar parameters of the s l (" separators=" 3 , R) -realization. The relation between the various Vessiot-Guldberg-Lie algebras by means of a limiting process in the scalar parameters further allows to define a notion of contraction of SODE Lie systems.
A Dirac-Dunkl Equation on S 2 and the Bannai-Ito Algebra
NASA Astrophysics Data System (ADS)
De Bie, Hendrik; Genest, Vincent X.; Vinet, Luc
2016-05-01
The Dirac-Dunkl operator on the two-sphere associated to the Z23 reflection group is considered. Its symmetries are found and are shown to generate the Bannai-Ito algebra. Representations of the Bannai-Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac-Dunkl operator are obtained using a Cauchy-Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai-Ito algebra.
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.
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.
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
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…
NASA Astrophysics Data System (ADS)
Folly-Gbetoula, Mensah; Kara, A. H.
2015-04-01
Solutions of linear iterative equations and expressions for these solutions in terms of the parameters of the first-order source equation are obtained. Based on certain properties of iterative equations, finding the solutions is reduced to finding solutions of the second-order source equation. We have therefore found classes of solutions to the source equations by letting the parameters of the source equation be functions of a specific type such as monomials, functions of exponential and logarithmic type.
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)
Daviau, Claude; Bertrand, Jacques
A wave equation with mass term is studied for all particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks $u$ and $d$ with three states of color and antiquarks $\\overline{u}$ and $\\overline{d}$. This wave equation is form invariant under the $Cl_3^*$ group generalizing the relativistic invariance. It is gauge invariant under the $U(1)\\times SU(2) \\times SU(3)$ group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra $Cl_{1,5}$. All features of the standard model, charge conjugation, color, left waves, Lagrangian formalism, are linked to the geometry of this extended space-time.
Stiff DAE integrator with sensitivity analysis capabilities
Energy Science and Technology Software Center (ESTSC)
2007-11-26
IDAS is a general purpose (serial and parallel) solver for differential equation (ODE) systems with senstivity analysis capabilities. It provides both forward and adjoint sensitivity analysis options.
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.…
The Dirac equation in an external electromagnetic field: symmetry algebra and exact integration
NASA Astrophysics Data System (ADS)
Breev, A. I.; Shapovalov, A. V.
2016-01-01
Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are not obtained by separation of variables for several external electromagnetic fields. We have considered an example of crossed electric and magnetic fields of a special type for which the Dirac equation admits a nonlocal symmetry operator.
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…
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…
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.
Realizations of Galilei algebras
NASA Astrophysics Data System (ADS)
Nesterenko, Maryna; Pošta, Severin; Vaneeva, Olena
2016-03-01
All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations.
NASA Astrophysics Data System (ADS)
Provencher, Stephen W.
1982-09-01
CONTIN is a portable Fortran IV package for inverting noisy linear operator equations. These problems occur in the analysis of data from a wide variety experiments. They are generally ill-posed problems, which means that errors in an unregularized inversion are unbounded. Instead, CONTIN seeks the optimal solution by incorporating parsimony and any statistical prior knowledge into the regularizor and absolute prior knowledge into equallity and inequality constraints. This can be greatly increase the resolution and accuracyh of the solution. CONTIN is very flexible, consisting of a core of about 50 subprograms plus 13 small "USER" subprograms, which the user can easily modify to specify special-purpose constraints, regularizors, operator equations, simulations, statistical weighting, etc. Specjial collections of USER subprograms are available for photon correlation spectroscopy, multicomponent spectra, and Fourier-Bessel, Fourier and Laplace transforms. Numerically stable algorithms are used throughout CONTIN. A fairly precise definition of information content in terms of degrees of freedom is given. The regularization parameter can be automatically chosen on the basis of an F-test and confidence region. The interpretation of the latter and of error estimates based on the covariance matrix of the constrained regularized solution are discussed. The strategies, methods and options in CONTIN are outlined. The program itself is described in the following paper.
Evaluating numerical ODE/DAE methods, algorithms and software
NASA Astrophysics Data System (ADS)
Soderlind, Gustaf; Wang, Lina
2006-01-01
Until recently, the testing of ODE/DAE software has been limited to simple comparisons and benchmarking. The process of developing software from a mathematically specified method is complex: it entails constructing control structures and objectives, selecting iterative methods and termination criteria, choosing norms and many more decisions. Most software constructors have taken a heuristic approach to these design choices, and as a consequence two different implementations of the same method may show significant differences in performance. Yet it is common to try to deduce from software comparisons that one method is better than another. Such conclusions are not warranted, however, unless the testing is carried out under true ceteris paribus conditions. Moreover, testing is an empirical science and as such requires a formal test protocol; without it conclusions are questionable, invalid or even false.We argue that ODE/DAE software can be constructed and analyzed by proven, "standard" scientific techniques instead of heuristics. The goals are computational stability, reproducibility, and improved software quality. We also focus on different error criteria and norms, and discuss modifications to DASPK and RADAU5. Finally, some basic principles of a test protocol are outlined and applied to testing these codes on a variety of problems.
Exponential integrators for the incompressible Navier-Stokes equations.
Newman, Christopher K.
2004-07-01
We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size. Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step. Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L{sup 2} inner product. We examine this L{sup 2} projection and an H{sup 1} projection induced by the H{sup 1} semi-inner product. The H
NASA Astrophysics Data System (ADS)
Mathai, Pramod P.
the uncertainty in the parameters of the differential equations. There is a clear need to design better experiments for IEF without the current overhead of expensive chemicals and labor. We show how with a simpler modeling of the underlying chemistry, we can still achieve the accuracy that has been achieved in existing literature for modeling small ranges of pH (hydrogen ion concentration) in IEF, but with far less computational time. We investigate a further reduction of time by modeling the IEF problem using the Proper Orthogonal Decomposition (POD) technique and show why POD may not be sufficient due to the underlying constraints. The final problem that we address in this thesis addresses a certain class of dynamics with high stiffness - in particular, differential algebraic equations. With the help of simple examples, we show how the traditional POD procedure will fail to model certain high stiffness problems due to a particular behavior of the vector field which we will denote as twist. We further show how a novel augmentation to the traditional POD algorithm can model-reduce problems with twist in a computationally cheap manner without any additional data requirements.
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…
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.
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)
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.
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.
Stoner, C D
1993-01-01
Methods are given whereby the steady-state kinetic characteristics of multienzyme reactions consisting of individual single-enzyme reactions linked by freely diffusible intermediates can be determined quantitatively from the experimentally determined complete algebraic rate equations for the individual reactions. The approach is based on the fact that a valid steady-state rate equation for such a multienzyme reaction, in terms of the rate equations for the individual reactions, can be obtained simply from knowledge of the relative rates of the individual reactions when the multienzyme reaction is in the steady state. A number of model multienzyme reactions, which differ as to structural arrangement of the individual reactions, are examined by this approach. Simple mathematical methods which are applicable to most of these models are given for direct calculation of dependent variables. It is either pointed out or demonstrated with Mathematica that the rate equations for all of these models can be handled very easily with the aid of a personal computer equipped with appropriate equation-solving software. Since the approach permits evaluation of all dependent variables for any specific combination of values for the kinetic parameters and independent variables, numerical values for the flux control coefficients of the individual enzymes can be obtained by direct calculation for a wide variety of conditions and can be compared with those obtained according to the methods of Metabolic Control Analysis. Several such comparisons have been made and in all cases identical results were obtained. The intuitive notion that the individual enzymes of a multienzyme reaction would be equally rate limiting if the total amount of enzyme were being used with maximum efficiency is tested and shown to be incorrect. In the course of this test the flux control coefficient for the individual enzymes were found to be appropriate indicators of relative rate limitation or control by the
Computer Algebra Systems in Undergraduate Instruction.
ERIC Educational Resources Information Center
Small, Don; And Others
1986-01-01
Computer algebra systems (such as MACSYMA and muMath) can carry out many of the operations of calculus, linear algebra, and differential equations. Use of them with sketching graphs of rational functions and with other topics is discussed. (MNS)
Quantum computation using geometric algebra
NASA Astrophysics Data System (ADS)
Matzke, Douglas James
This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.
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…
Modeling the nonlinear hysteretic response in DAE experiments of Berea sandstone: A case-study
NASA Astrophysics Data System (ADS)
Pecorari, Claudio
2015-03-01
Dynamic acousto-elasticity (DAE) allows probing the instantaneous state of a material while the latter slowly and periodically is changed by an external, dynamic source. In DAE investigations of geo-materials, hysteresis of the material's modulus defect displays intriguing features which have not yet been interpreted in terms of any specific mechanism occurring at atomic or mesoscale. Here, experimental results on dry Berea sandstone, which is the rock type best investigated by means of a DAE technique, are analyzed in terms of three rheological models providing simplified representations of mechanisms involving dislocations interacting with point defects which are distributed along the dislocations' core or glide planes, and microcracks with finite stiffness in compression. Constitutive relations linking macroscopic strain and stress are derived. From the latter, the modulus defect associated to each mechanism is recovered. These models are employed to construct a composite one which is capable of reproducing several of the main features observed in the experimental data. The limitations of the present approach and, possibly, of the current implementation of DAE are discussed.
Modeling the nonlinear hysteretic response in DAE experiments of Berea sandstone: A case-study
Pecorari, Claudio
2015-03-31
Dynamic acousto-elasticity (DAE) allows probing the instantaneous state of a material while the latter slowly and periodically is changed by an external, dynamic source. In DAE investigations of geo-materials, hysteresis of the material's modulus defect displays intriguing features which have not yet been interpreted in terms of any specific mechanism occurring at atomic or mesoscale. Here, experimental results on dry Berea sandstone, which is the rock type best investigated by means of a DAE technique, are analyzed in terms of three rheological models providing simplified representations of mechanisms involving dislocations interacting with point defects which are distributed along the dislocations' core or glide planes, and microcracks with finite stiffness in compression. Constitutive relations linking macroscopic strain and stress are derived. From the latter, the modulus defect associated to each mechanism is recovered. These models are employed to construct a composite one which is capable of reproducing several of the main features observed in the experimental data. The limitations of the present approach and, possibly, of the current implementation of DAE are discussed.
Phase noise in oscillators as differential-algebraic systems with colored noise sources
NASA Astrophysics Data System (ADS)
Demir, Alper
2004-05-01
Oscillators are key components of many kinds of systems, particularly electronic and opto-electronic systems. Undesired perturbations, i.e. noise, in practical systems adversely affect the spectral and timing properties of the signals generated by oscillators resulting in phase noise and timing jitter, which are key performance limiting factors, being major contributors to bit-error-rate (BER) of RF and possibly optical communication systems, and creating synchronization problems in clocked and sampled-data electronic systems. In this paper, we review our work on the theory and numerical methods for nonlinear perturbation and noise analysis of oscillators described by a system of differential-algebraic equations (DAEs) with white and colored noise sources. The bulk of the work reviewed in this paper first appeared in [1], then in [2] and [3]. Prior to the work mentioned above, we developed a theory and numerical methods for nonlinear perturbation and noise analysis of oscillators described by a system of ordinary differential equations (ODEs) with white noise sources only [4, 5]. In this paper, we also discuss some open problems and issues in the modeling and analysis of phase noise both in free running oscillators and in phase/injection-locked ones.
Semigroups and computer algebra in algebraic structures
NASA Astrophysics Data System (ADS)
Bijev, G.
2012-11-01
Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.
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.
UCSMP Algebra. What Works Clearinghouse Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2007
2007-01-01
"University of Chicago School Mathematics Project (UCSMP) Algebra," designed to increase students' skills in algebra, is appropriate for students in grades 7-10, depending on the students' incoming knowledge. This one-year course highlights applications, uses statistics and geometry to develop the algebra of linear equations and inequalities, and…
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
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.…
Explicit travelling waves and invariant algebraic curves
NASA Astrophysics Data System (ADS)
Gasull, Armengol; Giacomini, Hector
2015-06-01
We introduce a precise definition of algebraic travelling wave solution of n-th order partial differential equations and prove that the only algebraic travelling waves solutions for the celebrated Fisher-Kolmogorov equation are the ones found in 1979 by Ablowitz and Zeppetella. This question is equivalent to study when an associated one-parameter family of planar ordinary differential systems has invariant algebraic curves.
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.
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…
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…
Analysis of observational records of Dae-gyupyo in Joseon Dynasty
NASA Astrophysics Data System (ADS)
Mihn, Byeong-Hee; Lee, Ki-Won; Kim, Sang-Hyuk; Ahn, Young Sook; Lee, Yong Sam
2012-09-01
It is known that Dae-gyupyo (the Large Noon Gnomon) and So-gyupyo (the Small Noon Gnomon) were constructed in the reign of King Sejong (1418--1450) of the Joseon Dynasty. Gyupyo is an astronomical instrument for measuring the length of the shadow cast by a celestial body at the meridian passage time; it consists of two basic parts: a measuring scale and a vertical column. According to the Veritable Records of King Sejong and of King Myeongjong (1545--1567), the column of Dae-gyupyo was 40 Cheok (˜ 8 m) in height from the measuring scale and had a cross-bar, like the Guibiao of Shoujing Guo of the Yuan Dynasty in China. In the latter Veritable Records, three observations of the Sun on the date of the winter solstice and two of the full Moon on the first month in a luni-solar calendar are also recorded. In particular, the observational record of Dae-gyupyo for the Sun on Dec. 12, 1563 is ˜ 1 m shorter than the previous two records. To explain this, we investigated two possibilities: the vertical column was inclined, and the cross-bar was lowered. The cross-bar was attached to the column by a supporting arm; that should be installed at an angle of ˜ 36.9° to the north on the basis of a geometric structure inferred from the records of Yuanshi (History of the Yuan Dynasty). We found that it was possible that the vertical column was inclined ˜ 7.7° to the south or the supporting arm was tilted ˜ 58.3° downward. We suggest that the arm was tilted by ˜ 95° (= 36.9° + 58.3°).
Stability of algebraically unstable dispersive flows
NASA Astrophysics Data System (ADS)
King, Kristina; Zaretzky, Paula; Weinstein, Steven; Cromer, Michael; Barlow, Nathaniel
2015-11-01
A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to a class of partial differential equations describing wave propagation in dispersive media. There are several morphological differences between algebraically growing disturbances and the exponentially growing wave packets inherent to classical linear stability analysis, and these are elucidated in this study.
The Structural Algebra Option: A Discussion Paper.
ERIC Educational Resources Information Center
Kirshner, David
The goal of this paper is to renew interest in the structural option to algebra instruction. Concern for the usual secondary school algebra curriculum related to simplifying expressions, solving equations, and rationalizing numerators and denominators is viewed from three pedagogical approaches: (1) structural approach, (2) empirical approach, and…
Role of division algebra in seven-dimensional gauge theory
NASA Astrophysics Data System (ADS)
Kalauni, Pushpa; Barata, J. C. A.
2015-03-01
The algebra of octonions 𝕆 forms the largest normed division algebra over the real numbers ℝ, complex numbers ℂ and quaternions ℍ. The usual three-dimensional vector product is given by quaternions, while octonions produce seven-dimensional vector product. Thus, octonionic algebra is closely related to the seven-dimensional algebra, therefore one can extend generalization of rotations in three dimensions to seven dimensions using octonions. An explicit algebraic description of octonions has been given to describe rotational transformation in seven-dimensional space. We have also constructed a gauge theory based on non-associative algebra to discuss Yang-Mills theory and field equation in seven-dimensional space.
Lima Beans, Paper Cups, and Algebra.
ERIC Educational Resources Information Center
Loewen, A. C.
1991-01-01
An activity in which students use manipulative materials to help solve simple algebraic equations using the operations of adding inverses, removing opposites, and sharing equally is presented. Directions, examples, the rationale, and cautions are included. (KR)
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…
Philip, Bobby; Chartier, Dr Timothy
2012-01-01
methods based on Local Sensitivity Analysis (LSA). The method can be used in the context of geometric and algebraic multigrid methods for constructing smoothers, and in the context of Krylov methods for constructing block preconditioners. It is suitable for both constant and variable coecient problems. Furthermore, the method can be applied to systems arising from both scalar and coupled system partial differential equations (PDEs), as well as linear systems that do not arise from PDEs. The simplicity of the method will allow it to be easily incorporated into existing multigrid and Krylov solvers while providing a powerful tool for adaptively constructing methods tuned to a problem.
Lie Triple Derivations of CSL Algebras
NASA Astrophysics Data System (ADS)
Yu, Weiyan; Zhang, Jianhua
2013-06-01
Let [InlineEquation not available: see fulltext.] be a commutative subspace lattice generated by finite many commuting independent nests on a complex separable Hilbert space [InlineEquation not available: see fulltext.] with [InlineEquation not available: see fulltext.], and [InlineEquation not available: see fulltext.] the associated CSL algebra. It is proved that every Lie triple derivation from [InlineEquation not available: see fulltext.] into any σ-weakly closed algebra [InlineEquation not available: see fulltext.] containing [InlineEquation not available: see fulltext.] is of the form X→ XT- TX+ h( X) I, where [InlineEquation not available: see fulltext.] and h is a linear mapping from [InlineEquation not available: see fulltext.] into ℂ such that h([[ A, B], C])=0 for all [InlineEquation not available: see fulltext.].
Algebraic method for finding equivalence groups
NASA Astrophysics Data System (ADS)
Bihlo, Alexander; Dos Santos Cardoso-Bihlo, Elsa; Popovych, Roman O.
2015-06-01
The algebraic method for computing the complete point symmetry group of a system of differential equations is extended to finding the complete equivalence group of a class of such systems. The extended method uses the knowledge of the corresponding equivalence algebra. Two versions of the method are presented, where the first involves the automorphism group of this algebra and the second is based on a list of its megaideals. We illustrate the megaideal-based version of the method with the computation of the complete equivalence group of a class of nonlinear wave equations with applications in nonlinear elasticity.
Twisted Quantum Toroidal Algebras
NASA Astrophysics Data System (ADS)
Jing, Naihuan; Liu, Rongjia
2014-09-01
We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.
Algebraic vs physical N = 6 3-algebras
Cantarini, Nicoletta; Kac, Victor G.
2014-01-15
In our previous paper, we classified linearly compact algebraic simple N = 6 3-algebras. In the present paper, we classify their “physical” counterparts, which actually appear in the N = 6 supersymmetric 3-dimensional Chern-Simons theories.
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)
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.
The Universal C*-Algebra of the Electromagnetic Field
NASA Astrophysics Data System (ADS)
Buchholz, Detlev; Ciolli, Fabio; Ruzzi, Giuseppe; Vasselli, Ezio
2016-02-01
A universal C*-algebra of the electromagnetic field is constructed. It is represented in any quantum field theory which incorporates electromagnetism and expresses basic features of the field such as Maxwell's equations, Poincaré covariance and Einstein causality. Moreover, topological properties of the field resulting from Maxwell's equations are encoded in the algebra, leading to commutation relations with values in its center. The representation theory of the algebra is discussed with focus on vacuum representations, fixing the dynamics of the field.
PREFACE: 5th DAE-BRNS Workshop on Hadron Physics (Hadron 2011)
NASA Astrophysics Data System (ADS)
Jyoti Roy, Bidyut; Chatterjee, A.; Kailas, S.
2012-07-01
The 5th DAE-BRNS Workshop on Hadron Physics was held at the Bhabha Atomic Research Centre (BARC), Mumbai from 31 October to 4 November 2011. This workshop series, supported by the Board of Research in Nuclear Sciences, Department of Atomic Energy (BRNS, DAE), Govt. of India, began ten years ago with the first one being held at BARC, Mumbai in October 2002. The second one was held at Puri in 2005, organized jointly by Institute of Physics, Bhubneswar and Saha Institute of Nuclear Physics, Kolkata. The 3rd and 4th ones took place, respectively, at Shantineketan in 2006, organized by Visva Bharati University, and at Aligarh in 2008, organized by Aligarh Muslim University, Aligarh. The aim of the present workshop was to bring together the experts and young researchers in the field of hadron physics (both experiment and theory) and to have in-depth discussions on the current research activities in this field. The format of the workshop was: a series of review lectures by various experts from India and abroad, the presentation of advanced research results by researchers in the field, and a review of major experimental programs being planned and pursued in major laboratories in the field of hadron physics, with the aim of providing a platform for the young participants for interaction with their peers. The upcoming international FAIR facility at GSI is a unique future facility for studies of hadron physics in the charm sector and hyper nuclear physics. The Indian hadron physics community is involved in this mega science project and is working with the PANDA collaboration on the development of detectors, simulation and software tools for the hadron physics programme with antiprotons at FAIR. A one-day discussion session was held at this workshop to discuss India-PANDA activities, the current collaboration status and the work plan. This volume presents the workshop proceedings consisting of lectures and seminars which were delivered during the workshop. We are thankful to
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, Inc., Reston, VA.
This is a reprint of the historical capsules dealing with algebra from the 31st Yearbook of NCTM,"Historical Topics for the Mathematics Classroom." Included are such themes as the change from a geometric to an algebraic solution of problems, the development of algebraic symbolism, the algebraic contributions of different countries, the origin and…
Figueroa-O'Farrill, Jose Miguel
2009-11-15
We phrase deformations of n-Leibniz algebras in terms of the cohomology theory of the associated Leibniz algebra. We do the same for n-Lie algebras and for the metric versions of n-Leibniz and n-Lie algebras. We place particular emphasis on the case of n=3 and explore the deformations of 3-algebras of relevance to three-dimensional superconformal Chern-Simons theories with matter.
Schoenhammer, K; Petersen, H; Guethlein, F; Goepferich, A
2009-04-17
Injectable in situ forming depots (ISFD) that contain a peptide or a protein within a polymeric solution comprise an attractive, but challenging application system. Beyond chemical compatibility, local tolerability and acute toxicity, an important factor for an ISFD is its storage stability as a liquid. In this study, poly(D,L-lactide-co-glycolide) (PLGA) degradation in the presence of poly(ethyleneglycol) (PEG) as biocompatible solvent was investigated as a function of storage temperature and water content. The PLGA molecular weight (Mw) was determined by gel permeation chromatography (GPC), and monitored by NMR during degradation. Rapid PLGA degradation of 75% at 25 degrees C storage temperature was shown to be the result of a transesterification using conventional PEG as solvent. A significant improvement with only 3% Mw loss was obtained by capping the PEG hydroxy- with an alkyl- endgroup to have poly(ethyleneglycol) dialkylether (PEG-DAE). The formation of PEG-PLGA block co-polymers was confirmed by NMR, only for PEG300. Reaction rate constants were used to compare PLGA degradation dissolved in conventional and alkylated PEGs. The degradation kinetics in PEG-DAE were almost completely insensitive to 1% additional water in the solution. The transesterification of the hydroxy endgroups of PEG with PLGA was the major degradation mechanism, even under hydrous conditions. The use of PEG-DAE for injectable polymeric solutions, showed PLGA stability under the chosen conditions for at least 2 months. Based on the results obtained here, PEG-DAE appears to be a promising excipient for PLGA-based, parenteral ISFD. PMID:19135512
Quantum cluster algebras and quantum nilpotent algebras
Goodearl, Kenneth R.; Yakimov, Milen T.
2014-01-01
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197
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,…
Deforming the Maxwell-Sim algebra
Gibbons, G. W.; Gomis, Joaquim; Pope, C. N.
2010-09-15
The Maxwell algebra is a noncentral extension of the Poincare algebra, in which the momentum generators no longer commute, but satisfy [P{sub {mu}},P{sub {nu}}]=Z{sub {mu}{nu}}. The charges Z{sub {mu}{nu}} commute with the momenta, and transform tensorially under the action of the angular momentum generators. If one constructs an action for a massive particle, invariant under these symmetries, one finds that it satisfies the equations of motion of a charged particle interacting with a constant electromagnetic field via the Lorentz force. In this paper, we explore the analogous constructions where one starts instead with the ISim subalgebra of Poincare, this being the symmetry algebra of very special relativity. It admits an analogous noncentral extension, and we find that a particle action invariant under this Maxwell-Sim algebra again describes a particle subject to the ordinary Lorentz force. One can also deform the ISim algebra to DISim{sub b}, where b is a nontrivial dimensionless parameter. We find that the motion described by an action invariant under the corresponding Maxwell-DISim algebra is that of a particle interacting via a Finslerian modification of the Lorentz force. In an appendix is it shown that the DISim{sub b} algebra is isomorphic to the extended Schroedinger algebra with its standard deformation parameter z, when b=(1/1-z).
Filtering Algebraic Multigrid and Adaptive Strategies
Nagel, A; Falgout, R D; Wittum, G
2006-01-31
Solving linear systems arising from systems of partial differential equations, multigrid and multilevel methods have proven optimal complexity and efficiency properties. Due to shortcomings of geometric approaches, algebraic multigrid methods have been developed. One example is the filtering algebraic multigrid method introduced by C. Wagner. This paper proposes a variant of Wagner's method with substantially improved robustness properties. The method is used in an adaptive, self-correcting framework and tested numerically.
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”.
Single axioms for Boolean algebra.
McCune, W.
2000-06-30
Explicit single axioms are presented for Boolean algebra in terms of (1) the Sheffer stroke; (2) disjunction and negation; (3) disjunction, conjunction, and negation; and (4) disjunction, conjunction, negation, 0, and 1. It was previously known that single axioms exist for these systems, but the procedures to generate them are exponential, producing huge equations. Automated deduction techniques were applied to find axioms of lengths 105, 131, 111, and 127, respectively, each with six variables.
Traditional vectors as an introduction to geometric algebra
NASA Astrophysics Data System (ADS)
Carroll, J. E.
2003-07-01
The 2002 Oersted Medal Lecture by David Hestenes concerns the many advantages for education in physics if geometric algebra were to replace standard vector algebra. However, such a change has difficulties for those who have been taught traditionally. A new way of introducing geometric algebra is presented here using a four-element array composed of traditional vector and scalar products. This leads to an explicit 4 × 4 matrix representation which contains key requirements for three-dimensional geometric algebra. The work can be extended to include Maxwell's equations where it is found that curl and divergence appear naturally together. However, to obtain an explicit representation of space-time algebra with the correct behaviour under Lorentz transformations, an 8 × 8 matrix representation has to be formed. This leads to a Dirac representation of Maxwell's equations showing that space-time algebra has hidden within its formalism the symmetry of 'parity, charge conjugation and time reversal'.
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…
Walker, H.F.
1990-01-01
During the 1986--1989 project period, two major areas of research developed into which most of the work fell: matrix-free'' methods for solving linear systems, by which we mean iterative methods that require only the action of the coefficient matrix on vectors and not the coefficient matrix itself, and Newton-like methods for underdetermined nonlinear systems. In the 1990 project period of the renewal grant, a third major area of research developed: inexact Newton and Newton iterative methods and their applications to large-scale nonlinear systems, especially those arising in discretized problems. An inexact Newton method is any method in which each step reduces the norm of the local linear model of the function of interest. A Newton iterative method is any implementation of Newton's method in which the linear systems that characterize Newton steps (the Newton equations'') are solved only approximately using an iterative linear solver. Newton iterative methods are properly considered special cases of inexact Newton methods. We describe the work in these areas and in other areas in this paper.
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…
Applied Algebra Curriculum Modules.
ERIC Educational Resources Information Center
Texas State Technical Coll., Marshall.
This collection of 11 applied algebra curriculum modules can be used independently as supplemental modules for an existing algebra curriculum. They represent diverse curriculum styles that should stimulate the teacher's creativity to adapt them to other algebra concepts. The selected topics have been determined to be those most needed by students…
Ternary Virasoro - Witt algebra.
Zachos, C.; Curtright, T.; Fairlie, D.; High Energy Physics; Univ. of Miami; Univ. of Durham
2008-01-01
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.
NASA Astrophysics Data System (ADS)
Lannes, A.; Teunissen, P. J. G.
2011-05-01
The first objective of this paper is to show that some basic concepts used in global navigation satellite systems (GNSS) are similar to those introduced in Fourier synthesis for handling some phase calibration problems. In experimental astronomy, the latter are at the heart of what is called `phase closure imaging.' In both cases, the analysis of the related structures appeals to the algebraic graph theory and the algebraic number theory. For example, the estimable functions of carrier-phase ambiguities, which were introduced in GNSS to correct some rank defects of the undifferenced equations, prove to be `closure-phase ambiguities:' the so-called `closure-delay' (CD) ambiguities. The notion of closure delay thus generalizes that of double difference (DD). The other estimable functional variables involved in the phase and code undifferenced equations are the receiver and satellite pseudo-clock biases. A related application, which corresponds to the second objective of this paper, concerns the definition of the clock information to be broadcasted to the network users for their precise point positioning (PPP). It is shown that this positioning can be achieved by simply having access to the satellite pseudo-clock biases. For simplicity, the study is restricted to relatively small networks. Concerning the phase for example, these biases then include five components: a frequency-dependent satellite-clock error, a tropospheric satellite delay, an ionospheric satellite delay, an initial satellite phase, and an integer satellite ambiguity. The form of the PPP equations to be solved by the network user is then similar to that of the traditional PPP equations. As soon as the CD ambiguities are fixed and validated, an operation which can be performed in real time via appropriate decorrelation techniques, estimates of these float biases can be immediately obtained. No other ambiguity is to be fixed. The satellite pseudo-clock biases can thus be obtained in real time. This is
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.
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.
Semigroups And Computer Algebra In Discrete Structures
NASA Astrophysics Data System (ADS)
Bijev, G.
2010-10-01
Some concepts in semigroup theory are interpreted in discrete structures such as finite lattices, binary relations, and finite semilattices. An algebraic approach to the pseudoinverse generalization problem in Boolean vector spaces is used. By analogy with the linear spaces in the linear algebra semilattice homomorphisms, isomorphisms, projections on Boolean vector spaces are defined and some properties of them are investigated in detail. Maps, corresponding to them in the linear algebra, are connected with matrices and their pseudouinverse. Important properties of these maps, which are essential for solving linear systems, remain the same in the Boolean vector spaces. Stochastic experiments using the maps defined and computer algebra methods have been made for solving linear equations Ax = b. The Hamming distance between b and the projection p(b) = Ax of b is equal or close to the least possible one, if the system has no solutions.
Energy Science and Technology Software Center (ESTSC)
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
Algebra. Instructor Edition. Post-Secondary Vocational-Technical Level.
ERIC Educational Resources Information Center
Eads, Freeman D.; Hinton, Barbara E.
This instructor's guide consists of materials for use in teaching a course in algebra designed for students enrolled in postsecondary vocational and technical education programs. Addressed in the eight individual units of the guide are the following topics: signed numbers, algebraic notation, equations and inequalities, solving algebraic…
Modular operads and the quantum open-closed homotopy algebra
NASA Astrophysics Data System (ADS)
Doubek, Martin; Jurčo, Branislav; Münster, Korbinian
2015-12-01
We verify that certain algebras appearing in string field theory are algebras over Feynman transform of modular operads which we describe explicitly. Equivalent description in terms of solutions of generalized BV master equations are explained from the operadic point of view.
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…
NASA Astrophysics Data System (ADS)
Mróz, S.; Nowicki, M.
1993-11-01
Dependence of the Auger signal (directional Auger electron spectroscopy — DAES) and the elastically scattered electron intensity (directional elastic peak electron spectroscopy — DEPES) on the direction of the primary electron beam ( E = 600-1500 eV) was measured using a retarding field analyser (LEED optics) for the Cu(111) face, both clean and covered with silver up to 12 ML. Well-developed maxima of DAES and DEPES signals appear when the primary beam is parallel to one of the close-packed rows of atoms in the sample surface layer, while the angular distribution of the emitted electrons is averaged over the large acceptance angle of the RFA and does not influence appreciably the DAES and DEPES profiles. From the positions of the maxima mentioned above the atomic structure of a few surface layers can be determined. The silver layer was found to be rotated for 60° with respect to the Cu(111) substrate.
NASA Astrophysics Data System (ADS)
Lee, Yong Sam; Kim, Sang Hyuk; Park, Je Hoon
2013-09-01
Honsangui (celestial globe) which is a water-hammering method astronomical clock is recorded in "Juhaesuyong" which is Volume VI of supplement from "Damheonseo", written by Hong Dae-Yong (1731~1783). We made out the conceptual design of Hong Dae-Yong's Honsangui through the study on its structure and working mechanism. Honsangui consist of three rings and two layers, the structure of rings which correspond to outer layer is similar to his own Tongcheonui (armillary sphere) which is a kind of armillary sphere. Honsang sphere which correspond to inner layer depicts constellations and milky way and two beads hang on it as Sun and Moon respectively for realize the celestial motion. Tongcheonui is operated by the pendulum power but Honsangui is operated by water-hammering method mechanism. This Honsangui's working mechanism is the traditional way of Joseon and it was simplified the working mechanism of Shui y'n i hsiang t'ai which is a representative astronomical clock of China. This record of Honsangui is the only historical record about the water-hammering method working mechanism of Joseon Era and it provide the study of water-hammering method mechanism with a vital clue.
Octonic Massive Field Equations
NASA Astrophysics Data System (ADS)
Demir, Süleyman; Kekeç, Seray
2016-03-01
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.
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.
Four Lie algebras associated with R6 and their applications
NASA Astrophysics Data System (ADS)
Zhang, Yufeng; Tam, Honwah
2010-09-01
The first part in the paper reads that a three-dimensional Lie algebra is first introduced, whose corresponding loop algebra is constructed, for which isospectral problems are established. By employing zero curvature equations, a modified Kaup-Newell (mKN) soliton hierarchy of evolution equations is obtained. The corresponding hereditary operator and Hamiltonian structure are worked out, respectively. Then two types of enlarging semisimple Lie algebras isomorphic to the linear space R6 are followed to construct, one of them is a complex Lie algebra. Their corresponding loop algebras are also given so that two types of new isospectral problems are introduced to generate two kinds of integrable couplings of the above mKN hierarchy. The hereditary operators, Hamiltonian structures of the hierarchies are produced again, respectively. The exact computing formulas of the constant γ appearing in the trace identity and the variational identity are derived under the semisimple algebras. The second part of this paper is devoted to constructing two kinds of Lie algebras by using product of complex vectors, which are also isomorphic to the linear space R6. Then we make use of the corresponding loop algebras to produce two integrable hierarchies along with bi-Hamiltonian structures. From various aspects, we give some ways for constructing Lie algebras which have extensive applications in generating integrable Hamiltonian systems.
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.
Learning Activity Package, Algebra-Trigonometry.
ERIC Educational Resources Information Center
Holland, Bill
A series of ten teacher-prepared Learning Activity Packages (LAPs) in advanced algebra and trigonometry, the units cover logic; absolute value, inequalities, exponents, and complex numbers; functions; higher degree equations and the derivative; the trigonometric function; graphs and applications of the trigonometric functions; sequences and…
Algebra 1Q, Mathematics: 5215.12.
ERIC Educational Resources Information Center
Hirigoyen, Hector
This is the second of the six guidebooks on minimum course content for first-year algebra; it includes the ordered field properties of the real number system, solution of linear equations and inequalities, verbal problems, exponents and operations with polynomials. Overall goals for the course are stated; performance objectives for each unit, a…
Algebra 1p, Mathematics: 5215.11.
ERIC Educational Resources Information Center
Strachan, Florence; Hirigoyen, Hector
This is the first of six guidebooks on minimum course content for first-year algebra; it introduces the language of sets, the fundamental operations and properties of the real number system, the use of variables, and the solution of simple linear equations and inequalities. Overall goals for the course are stated; then performance objectives, a…
Short single axioms for boolean algebra.
McCune, W.; Veroff, R.; Fitelson, B.; Harris, K.; Feist, A.; Wos, L.; Mathematics and Computer Science; Univ. of New Mexico; Univ. of Wisconsin at Madison; Duke Univ.
2002-01-01
We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke. Automated deduction techniques were used in several parts of the work.
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.
NASA Astrophysics Data System (ADS)
Kuzmin, Dmitri; Möller, Matthias; Gurris, Marcel
Flux limiting for hyperbolic systems requires a careful generalization of the design principles and algorithms introduced in the context of scalar conservation laws. In this chapter, we develop FCT-like algebraic flux correction schemes for the Euler equations of gas dynamics. In particular, we discuss the construction of artificial viscosity operators, the choice of variables to be limited, and the transformation of antidiffusive fluxes. An a posteriori control mechanism is implemented to make the limiter failsafe. The numerical treatment of initial and boundary conditions is discussed in some detail. The initialization is performed using an FCT-constrained L 2 projection. The characteristic boundary conditions are imposed in a weak sense, and an approximate Riemann solver is used to evaluate the fluxes on the boundary. We also present an unconditionally stable semi-implicit time-stepping scheme and an iterative solver for the fully discrete problem. The results of a numerical study indicate that the nonlinearity and non-differentiability of the flux limiter do not inhibit steady state convergence even in the case of strongly varying Mach numbers. Moreover, the convergence rates improve as the pseudo-time step is increased.
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…
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…
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
Cavanagh, Sean
2008-01-01
A popular humorist and avowed mathphobe once declared that in real life, there's no such thing as algebra. Kathie Wilson knows better. Most of the students in her 8th grade class will be thrust into algebra, the definitive course that heralds the beginning of high school mathematics, next school year. The problem: Many of them are about three…
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…
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.
Coreflections in Algebraic Quantum Logic
NASA Astrophysics Data System (ADS)
Jacobs, Bart; Mandemaker, Jorik
2012-07-01
Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.
Developing Algebraic Thinking.
ERIC Educational Resources Information Center
Alejandre, Suzanne
2002-01-01
Presents a teaching experience that resulted in students getting to a point of full understanding of the kinesthetic activity and the algebra behind it. Includes a lesson plan for a traffic jam activity. (KHR)
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)
Aprepro - Algebraic Preprocessor
Energy Science and Technology Software Center (ESTSC)
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.
Campoamor-Stursberg, R.
2008-05-15
By means of contractions of Lie algebras, we obtain new classes of indecomposable quasiclassical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise naturally from noncompact real simple algebras with nonsimple complexification, where we impose that a nondegenerate quadratic Casimir operator is preserved by the limiting process. We further consider the converse problem and obtain sufficient conditions on integrable cocycles of quasiclassical Lie algebras in order to preserve nondegenerate quadratic Casimir operators by the associated linear deformations.
DG Poisson algebra and its universal enveloping algebra
NASA Astrophysics Data System (ADS)
Lü, JiaFeng; Wang, XingTing; Zhuang, GuangBin
2016-05-01
In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.
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…
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.
Dynamics of gelling liquids: algebraic relaxation.
Srivastava, Sunita; Kumar, C N; Tankeshwar, K
2009-08-19
The sol-gel system which is known, experimentally, to exhibit a power law decay of stress autocorrelation function has been studied theoretically. A second-order nonlinear differential equation obtained from Mori's integro-differential equation is derived which provides the algebraic decay of a time correlation function. Involved parameters in the expression obtained are related to exact properties of the corresponding correlation function. The algebraic model has been applied to Lennard-Jones and sol-gel systems. The model shows the behaviour of viscosity as has been observed in computer simulation and theoretical studies. The expression obtained for the viscosity predicts a logarithmic divergence at a critical value of the parameter in agreement with the prediction of other theories. PMID:21828600
Twisting algebraically special solutions in five dimensions
NASA Astrophysics Data System (ADS)
Bernardi de Freitas, Gabriel; Godazgar, Mahdi; Reall, Harvey S.
2016-05-01
We determine the general form of the solutions of the five-dimensional vacuum Einstein equations with cosmological constant for which (i) the Weyl tensor is everywhere type II or more special in the null alignment classification of Coley et al, and (ii) the 3 × 3 matrix encoding the expansion, shear and twist of the aligned null direction has rank 2. The dependence of the solution on two coordinates is determined explicitly, so the Einstein equation reduces to PDEs in the three remaining coordinates, just as for four-dimensional (4d) algebraically special solutions. The solutions fall into several families. One of these consists of warped products of 4d algebraically special solutions. The others are new.
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.
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. PMID:21769158
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.
Prediction of Complex Aerodynamic Flows with Explicit Algebraic Stress Models
NASA Technical Reports Server (NTRS)
Abid, Ridha; Morrison, Joseph H.; Gatski, Thomas B.; Speziale, Charles G.
1996-01-01
An explicit algebraic stress equation, developed by Gatski and Speziale, is used in the framework of K-epsilon formulation to predict complex aerodynamic turbulent flows. The nonequilibrium effects are modeled through coefficients that depend nonlinearly on both rotational and irrotational strains. The proposed model was implemented in the ISAAC Navier-Stokes code. Comparisons with the experimental data are presented which clearly demonstrate that explicit algebraic stress models can predict the correct response to nonequilibrium flow.
Parallel Multigrid Equation Solver
Energy Science and Technology Software Center (ESTSC)
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.
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.
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.
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…
NASA Astrophysics Data System (ADS)
Roitman, Michael
2008-08-01
In this paper we prove that for any commutative (but in general non-associative) algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V0 Å V2 Å V3 Å ¼, such that dim V0 = 1 and V2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
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…
Sedeonic Equations of Massive Fields
NASA Astrophysics Data System (ADS)
Mironov, Sergey V.; Mironov, Victor L.
2015-01-01
Prior work on space-time sedeon algebra models relativistic quantum mechanical equation of motion with corresponding field equations, mediated by massive or massless spin-1 or spin-1/2 particles. In the massless spin-1 case, such exchange particles mediate fields in analogy to Maxwell's equations in Lorentz gauge. This paper demonstrates fundamental aspects of massive field's theory, such as gauge invariance, charge conservation, Poynting's theorem, potential of a stationary scalar point source, plane wave solution, and interaction between point sources. We briefly discuss some aspects of sedeonic algebra and their potential physical applications.
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.
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.
NASA Technical Reports Server (NTRS)
Shahshahani, M.
1991-01-01
The performance characteristics are discussed of certain algebraic geometric codes. Algebraic geometric codes have good minimum distance properties. On many channels they outperform other comparable block codes; therefore, one would expect them eventually to replace some of the block codes used in communications systems. It is suggested that it is unlikely that they will become useful substitutes for the Reed-Solomon codes used by the Deep Space Network in the near future. However, they may be applicable to systems where the signal to noise ratio is sufficiently high so that block codes would be more suitable than convolutional or concatenated codes.
NASA Astrophysics Data System (ADS)
Bouwknegt, Peter
1988-06-01
We investigate extensions of the Virasoro algebra by a single primary field of integer or halfinteger conformal dimension Δ. We argue that for vanishing structure constant CΔΔΔ, the extended conformal algebra can only be associative for a generic c-value if Δ=1/2, 1, 3/2, 2 or 3. For the other Δ<=5 we compute the finite set of allowed c-values and identify the rational solutions. The case CΔΔΔ≠0 is also briefly discussed. I would like to thank Kareljan Schoutens for discussions and Sander Bais for a careful reading of the manuscript.
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,…
Teaching Arithmetic and Algebraic Expressions
ERIC Educational Resources Information Center
Subramaniam, K.; Banerjee, Rakhi
2004-01-01
A teaching intervention study was conducted with sixth grade students to explore the interconnections between students' growing understanding of arithmetic expressions and beginning algebra. Three groups of students were chosen, with two groups receiving instruction in arithmetic and algebra, and one group in algebra without arithmetic. Students…
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…
Spinors in the hyperbolic algebra
NASA Astrophysics Data System (ADS)
Ulrych, S.
2006-01-01
The three-dimensional universal complex Clifford algebra Cbar3,0 is used to represent relativistic vectors in terms of paravectors. In analogy to the Hestenes spacetime approach spinors are introduced in an algebraic form. This removes the dependance on an explicit matrix representation of the algebra.
An extended Dirac equation in noncommutative spacetime
NASA Astrophysics Data System (ADS)
Mendes, R. Vilela
2016-05-01
Stabilizing, by deformation, the algebra of relativistic quantum mechanics a noncommutative spacetime geometry is obtained. The exterior algebra of this geometry leads to an extended massless Dirac equation which has both a massless and a large mass solution. The nature of the solutions is discussed as well as the effects of coupling the two solutions.
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
Ketterlin-Geller, Leanne R.; Jungjohann, Kathleen; Chard, David J.; Baker, Scott
2007-01-01
Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…
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
Boiteau, Denise; Stansfield, David
This document describes mathematical programs on the basic concepts of algebra produced by Louisiana Public Broadcasting. Programs included are: (1) "Inverse Operations"; (2) "The Order of Operations"; (3) "Basic Properties" (addition and multiplication of numbers and variables); (4) "The Positive and Negative Numbers"; and (5) "Using Positive…
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,…
ERIC Educational Resources Information Center
Kennedy, John
This text provides information and exercises on arithmetic topics which should be mastered before a student enrolls in an Elementary Algebra course. Section I describes the fundamental properties and relationships of whole numbers, focusing on basic operations, divisibility tests, exponents, order of operations, prime numbers, greatest common…
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…
NASA Astrophysics Data System (ADS)
Castro, Carlos
2006-11-01
A novel Weyl-Heisenberg algebra in Clifford spaces is constructed that is based on a matrix-valued {\\cal H}^{AB} extension of Planck's constant. As a result of this modified Weyl-Heisenberg algebra one will no longer be able to measure, simultaneously, the pairs of variables (x, px), (x, py), (x, pz), (y, px), ... with absolute precision. New Klein-Gordon and Dirac wave equations and dispersion relations in Clifford spaces are presented. The latter Dirac equation is a generalization of the Dirac-Lanczos-Barut-Hestenes equation. We display the explicit isomorphism between Yang's noncommutative spacetime algebra and the area-coordinates algebra associated with Clifford spaces. The former Yang's algebra involves noncommuting coordinates and momenta with a minimum Planck scale λ (ultraviolet cutoff) and a minimum momentum p = planck/R (maximal length R, infrared cutoff). The double-scaling limit of Yang's algebra λ → 0, R → ∞, in conjunction with the large n → ∞ limit, leads naturally to the area quantization condition λR = L2 = nλ2 (in Planck area units) given in terms of the discrete angular-momentum eigenvalues n. It is shown how modified Newtonian dynamics is also a consequence of Yang's algebra resulting from the modified Poisson brackets. Finally, another noncommutative algebra which differs from Yang's algebra and related to the minimal length uncertainty relations is presented. We conclude with a discussion of the implications of noncommutative QM and QFT's in Clifford spaces.
NASA Astrophysics Data System (ADS)
Zhang, Ming; Yao, JingTao
2004-04-01
The XML is a new standard for data representation and exchange on the Internet. There are studies on XML query languages as well as XML algebras in literature. However, attention has not been paid to research on XML algebras for data mining due to partially the fact that there is no widely accepted definition of XML mining tasks. This paper tries to examine the XML mining tasks and provide guidelines to design XML algebras for data mining. Some summarization and comparison have been done to existing XML algebras. We argue that by adding additional operators for mining tasks, XML algebras may work well for data mining with XML documents.
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.
Generalized Kaluza-Klein monopole, quadratic algebras and ladder operators
NASA Astrophysics Data System (ADS)
Marquette, Ian
2011-06-01
We present a generalized Kaluza-Klein monopole system. We solve this quantum superintegrable system on a Euclidean Taub Nut manifold using the separation of variables of the corresponding Schrödinger equation in spherical and parabolic coordinates. We present the integrals of motion of this system, the quadratic algebra generated by these integrals, the realization in terms of a deformed oscillator algebra using the Daskaloyannis construction and the energy spectrum. The structure constants and the Casimir operator are functions not only of the Hamiltonian but also of other two integrals commuting with all generators of the quadratic algebra and forming an Abelian subalgebra. We present another algebraic derivation of the energy spectrum of this system using the factorization method and ladder operators.
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.
On Dunkl angular momenta algebra
NASA Astrophysics Data System (ADS)
Feigin, Misha; Hakobyan, Tigran
2015-11-01
We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl( N ) version of the subalge-bra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.
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.
Integrability of Hamiltonian systems with algebraic potentials
NASA Astrophysics Data System (ADS)
Maciejewski, Andrzej J.; Przybylska, Maria
2016-01-01
Problem of integrability for Hamiltonian systems with potentials that are algebraic thus multivalued functions of coordinates is discussed. Introducing potential as a new variable the original Hamiltonian system on 2n dimensional phase space is extended to 2 n + 1 dimensional system with rational right-hand sides. For extended system its non-canonical degenerated Poisson structure of constant rank 2n and rational Hamiltonian is identified. For algebraic homogeneous potentials of non-zero rational homogeneity degree necessary integrability conditions are formulated. These conditions are deduced from an analysis of the differential Galois group of variational equations around particular solutions of a straight line type. Obtained integrability obstructions are applied to the class of monomial homogeneous potentials. Some integrable potentials satisfying these conditions are found.
ODE/IM correspondence and modified affine Toda field equations
NASA Astrophysics Data System (ADS)
Ito, Katsushi; Locke, Christopher
2014-08-01
We study the two-dimensional affine Toda field equations for affine Lie algebra gˆ modified by a conformal transformation and the associated linear equations. In the conformal limit, the associated linear problem reduces to a (pseudo-)differential equation. For classical affine Lie algebra gˆ, we obtain a (pseudo-)differential equation corresponding to the Bethe equations for the Langlands dual of the Lie algebra g, which were found by Dorey et al. in study of the ODE/IM correspondence.
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
Merriweather, Michelle; Tharp, Marcia L.
1999-01-01
Focuses on changes in attitude toward mathematics and calculator use and changes in how general mathematics students naturalistically solve algebraic problems. Uses a survey to determine whether a student is rule-based. Concludes that the rule-based students used an equation to solve the algebraic word problem whereas the non-rule-based students…
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…
Switched electrical networks and bilinear equations
NASA Technical Reports Server (NTRS)
Wood, J. R.
1974-01-01
State equations arising in the description of power processing systems are described. The role played by Lie groups and Lie algebras in characterizing the inherent dynamical features of these systems is outlined, and network examples are presented for illustration.
Marquette, Ian
2013-07-15
We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.
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®,…
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.
Vertex Algebras, Kac-Moody Algebras, and the Monster
NASA Astrophysics Data System (ADS)
Borcherds, Richard E.
1986-05-01
It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The ``Moonshine'' representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.
NASA Astrophysics Data System (ADS)
Palmkvist, Jakob
2014-01-01
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D - 2 - p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
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.