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Sample records for algebraic equations dae

  1. 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.

  2. Sensitivity Analysis of Differential-Algebraic Equations and Partial Differential Equations

    SciTech Connect

    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.

  3. Computing spacetime curvature via differential-algebraic equations

    SciTech Connect

    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.

  4. A novel technique to solve nonlinear higher-index Hessenberg differential-algebraic equations by Adomian decomposition method.

    PubMed

    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

  5. 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…

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. 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.

  11. 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.

  12. 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…

  13. 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.

  14. 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.

  15. 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.

  16. 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.

  17. 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.

  18. 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.

  19. 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.

  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.

  1. 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.

  2. 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…

  3. 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.

  4. 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.

  5. 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.

  6. Algebraic Reasoning in the Middle Grades: A View of Student Strategies in Pictorial and Algebraic System of Equations

    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.…

  7. Note on parallel processing techniques for algebraic equations, ordinary differential equations and partial differential equations

    SciTech Connect

    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.

  8. New developments in the numerical solution of differential/algebraic systems

    SciTech Connect

    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.

  9. 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.

  10. 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.

  11. Solving Differential Equations in R: Package deSolve

    EPA Science Inventory

    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...

  12. 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.

  13. How Structure Sense for Algebraic Expressions or Equations Is Related to Structure Sense for Abstract Algebra

    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…

  14. 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.

  15. Bilinear equations and q-discrete Painlevé equations satisfied by variables and coefficients in cluster algebras

    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.

  16. Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations

    SciTech Connect

    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.

  17. Introducing Algebraic Structures through Solving Equations: Vertical Content Knowledge for K-12 Mathematics Teachers

    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…

  18. Algebraic Equations: Can Middle-School Students Meaningfully Translate from Words to Mathematical Symbols?

    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,…

  19. Commutator identities on associative algebras, the non-Abelian Hirota difference equation and its reductions

    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.

  20. 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.

  1. Solution of the Dirac equation with some superintegrable potentials by the quadratic algebra approach

    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.

  2. 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…

  3. 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.

  4. 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…

  5. Extended trigonometric Cherednik algebras and nonstationary Schroedinger equations with delta-potentials

    SciTech Connect

    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.

  6. 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…

  7. Kalman Duality Principle for a Class of Ill-Posed Minimax Control Problems with Linear Differential-Algebraic Constraints

    SciTech Connect

    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.

  8. 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.

  9. Differences between Expected Answers and the Answers Given by Computer Algebra Systems to School Equations

    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…

  10. Exploring Students' Understanding of Ordinary Differential Equations Using Computer Algebraic System (CAS)

    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…

  11. 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…

  12. 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…

  13. 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…

  14. Solution of the Schrodinger Equation for a Diatomic Oscillator Using Linear Algebra: An Undergraduate Computational Experiment

    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.)

  15. 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…

  16. 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.

  17. Solution of Algebraic Equations in the Analysis, Design, and Optimization of Continuous Ultrafiltration

    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…

  18. 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

  19. 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

  20. 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.

  1. Tracking children's mental states while solving algebra equations.

    PubMed

    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

  2. Obtaining General Relativity's N-body non-linear Lagrangian from iterative, linear algebraic scaling equations

    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.

  3. Modeling scalar dissipation and scalar variance in large eddy simulation: Algebraic and transport equation closures

    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.

  4. 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.

  5. Crossing the Cognitive Gap between Arithmetic and Algebra: Operating on the Unknown in the Context of Equations.

    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)

  6. Hierarchies of finite-dimensional Lax equations with a spectral parameter on a Riemann surface and semisimple Lie algebras

    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.

  7. FAST TRACK COMMUNICATION: On the Liouvillian solution of second-order linear differential equations and algebraic invariant curves

    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.

  8. Modeling boyciana-fish-human interaction with partial differential algebraic equations.

    PubMed

    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

  9. 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.

  10. The moser type reduction of integrable Riccati differential equations and its Lie-algebraic structure

    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.

  11. 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.

  12. Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces

    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.

  13. Hirota equations associated with simply laced affine Lie algebras: The cuspidal class E6 of {{e}}_6^{(1)}

    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.

  14. 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.

  15. Quadratic Algebra Approach to the Dirac Equation with Spin and Pseudospin Symmetry for the 4D Harmonic Oscillator and U(1) Monopole

    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.

  16. 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…

  17. 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…

  18. Young Students Learning Formal Algebraic Notation and Solving Linear Equations: Are Commonly Experienced Difficulties Avoidable?

    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"…

  19. 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…

  20. Analyzing the nonlinear vibrational wave differential equation for the simplified model of Tower Cranes by Algebraic Method

    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.

  1. A functional realization of 𝔰𝔩(3, ℝ) providing minimal Vessiot-Guldberg-Lie algebras of nonlinear second-order ordinary differential equations as proper subalgebras

    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.

  2. 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.

  3. Numerical solvers to the stabilizing solution of perturbed algebraic Riccati equations in LQ zero-sum games

    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.

  4. 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.

  5. Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

    SciTech Connect

    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

  6. 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…

  7. Symmetry Lie algebras and properties of linear ordinary differential equations with maximal dimension

    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.

  8. Three semi-direct sum Lie algebras and three discrete integrable couplings associated with the modified KdV lattice equation.

    PubMed

    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

  9. A Wave Equation including Leptons and Quarks for the Standard Model of Quantum Physics in Clifford Algebra

    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.

  10. Stiff DAE integrator with sensitivity analysis capabilities

    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.

  11. 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.

  12. 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.…

  13. 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.

  14. The Effects of Schema-Broadening Instruction on Second Graders' Word-Problem Performance and Their Ability to Represent Word Problems with Algebraic Equations: A Randomized Control Study

    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…

  15. 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…

  16. Algebra Word Problem Solving Approaches in a Chemistry Context: Equation Worked Examples versus Text Editing

    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…

  17. 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.

  18. 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.

  19. A constrained regularization method for inverting data represented by linear algebraic or integral equations

    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.

  20. 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.

  1. Exponential integrators for the incompressible Navier-Stokes equations.

    SciTech Connect

    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

  2. Application of reduced order modeling techniques to problems in heat conduction, isoelectric focusing and differential algebraic equations

    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.

  3. 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…

  4. A modular approach to modeling an isolated power system on a finite voltage bus using a differential algebraic equation solving routine

    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.

  5. Convergence of the standard RLS method and UDUT factorisation of covariance matrix for solving the algebraic Riccati equation of the DLQR via heuristic approximate dynamic programming

    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.

  6. The Algebraic Way

    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.

  7. 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.

  8. Quantitative determination of the steady-state kinetics of multienzyme reactions using the algebraic rate equations for the component single-enzyme reactions.

    PubMed Central

    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

  9. 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)

  10. 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.

  11. Algebraic integrability: a survey.

    PubMed

    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

  12. The Algebra of the Arches

    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…

  13. 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.

  14. Modeling the nonlinear hysteretic response in DAE experiments of Berea sandstone: A case-study

    SciTech Connect

    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.

  15. 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.

  16. 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.

  17. Adaptive Algebraic Multigrid Methods

    SciTech Connect

    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.

  18. 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…

  19. Algebraic Rate of Decay for the Excess Free Energy and Stability of Fronts for a Nonlocal Phase Kinetics Equation with a Conservation Law. I

    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.

  20. 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.

  1. Earth Algebra.

    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…

  2. Kiddie Algebra

    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.…

  3. 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.

  4. 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.

  5. 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…

  6. Is the Role of Equations in the Doing of Word Problems in School Algebra Changing? Initial Indications from Teacher Study Groups

    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…

  7. 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°).

  8. 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.

  9. 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…

  10. 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.

  11. 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)

  12. 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…

  13. 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…

  14. Adaptive Algebraic Smoothers

    SciTech Connect

    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.

  15. 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.].

  16. 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.

  17. 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.

  18. Algebraic vs physical N = 6 3-algebras

    SciTech Connect

    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.

  19. Chemical Equation Balancing.

    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)

  20. $\\frac{{\\rm SO}(2N)}{U(N)}$ Riccati-Hartree-Bogoliubov equation based on the SO(2N) Lie algebra of the fermion operators

    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.

  1. 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.

  2. 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

  3. Historical Topics in Algebra.

    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…

  4. Deformations of 3-algebras

    SciTech Connect

    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.

  5. Injectable in situ forming depot systems: PEG-DAE as novel solvent for improved PLGA storage stability.

    PubMed

    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

  6. Quantum cluster algebras and quantum nilpotent algebras

    PubMed Central

    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

  7. 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,…

  8. Deforming the Maxwell-Sim algebra

    SciTech Connect

    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).

  9. Filtering Algebraic Multigrid and Adaptive Strategies

    SciTech Connect

    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.

  10. 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 = - †Âϕ with the operators Âand† 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”.

  11. Single axioms for Boolean algebra.

    SciTech Connect

    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.

  12. 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'.

  13. 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…

  14. Numerical solution of nonlinear algebraic equations in stiff ODE solving (1986--89)---Quasi-Newton updating for large scale nonlinear systems (1989--90)

    SciTech Connect

    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.

  15. Algebraic theory of molecules

    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.

  16. Orientation in operator algebras

    PubMed Central

    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

  17. 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,…

  18. 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…

  19. 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…

  20. Ternary Virasoro - Witt algebra.

    SciTech Connect

    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.

  1. GNSS algebraic structures

    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

  2. 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.

  3. Computer algebra and transport theory.

    SciTech Connect

    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.

  4. 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.

  5. 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.

  6. ALGEBRA IIVer 1.22

    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

  7. 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…

  8. 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.

  9. Pre-Service Teachers' Perceptions and Beliefs of Technological Pedagogical Content Knowledge on Algebra

    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…

  10. Directional Auger electron spectroscopy (DAES) and directional elastic peak electron spectroscopy (DEPES) in the investigation of the crystalline structure of surface layers: the Ag/Cu(111) interface

    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.

  11. A Study for the Restoration of Hong Dae-Yong Honsangui - Focusing on the structure and operating mechanism

    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.

  12. 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.

  13. 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.

  14. 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.

  15. 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…

  16. 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.

  17. 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…

  18. 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…

  19. 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…

  20. Short single axioms for boolean algebra.

    SciTech Connect

    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.

  1. 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)

  2. Noncommutative Pfaffians associated with the orthogonal algebra

    SciTech Connect

    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.

  3. Algebraic Flux Correction II

    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.

  4. Teaching Structure in Algebra

    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…

  5. Ready, Set, Algebra?

    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…

  6. Linear-Algebra Programs

    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.

  7. Catching Up on Algebra

    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…

  8. 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…

  9. 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.

  10. 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.

  11. 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.

  12. 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.

  13. 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)

  14. 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)

  15. Aprepro - Algebraic Preprocessor

    2005-08-01

    Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.

  16. 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.

  17. Contractions and deformations of quasiclassical Lie algebras preserving a nondegenerate quadratic Casimir operator

    SciTech Connect

    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.

  18. 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.

  19. 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…

  20. Description of DASSL: a differential/algebraic system solver

    SciTech Connect

    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.

  1. Dynamics of gelling liquids: algebraic relaxation.

    PubMed

    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

  2. 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.

  3. 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.

  4. 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.

  5. MODEL IDENTIFICATION AND COMPUTER ALGEBRA.

    PubMed

    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

  6. 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.

  7. 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.

  8. Parallel Multigrid Equation Solver

    2001-09-07

    Prometheus is a fully parallel multigrid equation solver for matrices that arise in unstructured grid finite element applications. It includes a geometric and an algebraic multigrid method and has solved problems of up to 76 mullion degrees of feedom, problems in linear elasticity on the ASCI blue pacific and ASCI red machines.

  9. 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.

  10. 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.

  11. Solving the Unknown with Algebra: Poster/Teaching Guide for Pre-Algebra Students. Expect the Unexpected with Math[R

    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…

  12. On Griess Algebras

    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.

  13. 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…

  14. 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.

  15. 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.

  16. 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.

  17. Algebraic geometric codes

    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.

  18. Extended conformal algebras

    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.

  19. 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,…

  20. 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…

  1. 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…

  2. 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.

  3. 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.

  4. Algebraic Artful Aids.

    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)

  5. From Arithmetic to Algebra

    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.…

  6. 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…

  7. The Power of Algebra.

    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…

  8. 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,…

  9. Pre-Algebra.

    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…

  10. Computers in Abstract Algebra

    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…

  11. On modified Weyl Heisenberg algebras, noncommutativity, matrix-valued Planck constant and QM in Clifford spaces

    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.

  12. XML algebras for data mining

    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.

  13. 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.

  14. 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.

  15. 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.

  16. 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.

  17. Algebraic connectivity and graph robustness.

    SciTech Connect

    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.

  18. 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.

  19. 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.

  20. Secondary School Pre-Service Mathematics Teachers' Content Knowledge of Algebraic Word Problem in Nigeria

    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…

  1. The Effect of Instruction with Graphing Calculators on How General Mathematics Students Naturalistically Solve Algebraic Problems.

    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…

  2. 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…

  3. 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.

  4. Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras

    SciTech Connect

    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.

  5. Algebraic Mean Field Theory

    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.

  6. The Algebra Artist

    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®,…

  7. Algebraic Multigrid Benchmark

    SciTech Connect

    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.

  8. 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.

  9. The tensor hierarchy algebra

    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.

  10. Priority in Process Algebras

    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.

  11. The tensor hierarchy algebra

    SciTech Connect

    Palmkvist, Jakob

    2014-01-15

    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.

  12. Compactly Generated de Morgan Lattices, Basic Algebras and Effect Algebras

    NASA Astrophysics Data System (ADS)

    Paseka, Jan; Riečanová, Zdenka

    2010-12-01

    We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.

  13. Locally finite dimensional Lie algebras

    NASA Astrophysics Data System (ADS)

    Hennig, Johanna

    We prove that in a locally finite dimensional Lie algebra L, any maximal, locally solvable subalgebra is the stabilizer of a maximal, generalized flag in an integrable, faithful module over L. Then we prove two structure theorems for simple, locally finite dimensional Lie algebras over an algebraically closed field of characteristic p which give sufficient conditions for the algebras to be of the form [K(R, *), K( R, *)] / (Z(R) ∩ [ K(R, *), K(R, *)]) for a simple, locally finite dimensional associative algebra R with involution *. Lastly, we explore the noncommutative geometry of locally simple representations of the diagonal locally finite Lie algebras sl(ninfinity), o( ninfinity), and sp(n infinity).

  14. Duncan F. Gregory, William Walton and the development of British algebra: 'algebraical geometry', 'geometrical algebra', abstraction.

    PubMed

    Verburgt, Lukas M

    2016-01-01

    This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s. PMID:26806075

  15. On the cohomology of Leibniz conformal algebras

    NASA Astrophysics Data System (ADS)

    Zhang, Jiao

    2015-04-01

    We construct a new cohomology complex of Leibniz conformal algebras with coefficients in a representation instead of a module. The low-dimensional cohomology groups of this complex are computed. Meanwhile, we construct a Leibniz algebra from a Leibniz conformal algebra and prove that the category of Leibniz conformal algebras is equivalent to the category of equivalence classes of formal distribution Leibniz algebras.

  16. Assessing Algebraic Solving Ability: A Theoretical Framework

    ERIC Educational Resources Information Center

    Lian, Lim Hooi; Yew, Wun Thiam

    2012-01-01

    Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…

  17. Computer Algebra Systems and Theorems on Real Roots of Polynomials

    ERIC Educational Resources Information Center

    Aidoo, Anthony Y.; Manthey, Joseph L.; Ward, Kim Y.

    2010-01-01

    A computer algebra system is used to derive a theorem on the existence of roots of a quadratic equation on any bounded real interval. This is extended to a cubic polynomial. We discuss how students could be led to derive and prove these theorems. (Contains 1 figure.)

  18. Writing to Promote and Assess Conceptual Understanding in College Algebra

    ERIC Educational Resources Information Center

    Gay, A. Susan; Peterson, Ingrid

    2014-01-01

    Concept-focused quiz questions required College Algebra students to write about their understanding. The questions can be viewed in three broad categories: a focus on sense-making, a focus on describing a mathematical object such as a graph or an equation, and a focus on understanding vocabulary. Student responses from 10 classes were analyzed.…

  19. Fixing Ganache: Another Real-Life Use for Algebra

    ERIC Educational Resources Information Center

    Kalman, Adam M.

    2011-01-01

    This article presents a real-world application of proportional reasoning and equation solving. The author describes how students adjust ingredient amounts in a recipe for chocolate ganache. Using this real-world scenario provided students an opportunity to solve a difficult and nonstandard algebra problem, a lot of practice with fractions, a…

  20. First Course in Algebra, Student's Text, Part II, Unit 10.

    ERIC Educational Resources Information Center

    Allen, Frank B.; And Others

    Unit 10 in the SMSG's secondary school mathematics series is a student text covering the following topics in Algebra I: factors and exponents, radicals, polynomial and rational expressions, truth sets of open sentences, graphs of open sentences in two variables, systems of equations and inequalities, quadratic polynomials, and functions. (DT)

  1. Geometric and Algebraic Approaches in the Concept of Complex Numbers

    ERIC Educational Resources Information Center

    Panaoura, A.; Elia, I.; Gagatsis, A.; Giatilis, G.-P.

    2006-01-01

    This study explores pupils' performance and processes in tasks involving equations and inequalities of complex numbers requiring conversions from a geometric representation to an algebraic representation and conversions in the reverse direction, and also in complex numbers problem solving. Data were collected from 95 pupils of the final grade from…

  2. Learning Activity Package, Algebra 124, LAPs 46-55.

    ERIC Educational Resources Information Center

    Holland, Bill

    A series of 10 teacher-prepared Learning Activity Packages (LAPs) in advanced algebra and trigonometry, these units cover absolute value, inequalities, exponents, radicals, and complex numbers; functions; higher degree equations and the derivative; the trigonometric functions; graphs and applications of the trigonometric functions; sequences and…

  3. Infinite-dimensional Lie algebras, classical r-matrices, and Lax operators: Two approaches

    NASA Astrophysics Data System (ADS)

    Skrypnyk, T.

    2013-10-01

    For each finite-dimensional simple Lie algebra {g}, starting from a general {g}⊗ {g}-valued solutions r(u, v) of the generalized classical Yang-Baxter equation, we construct infinite-dimensional Lie algebras widetilde{{g}}-_r of {g}-valued meromorphic functions. We outline two ways of embedding of the Lie algebra widetilde{{g}}-_r into a larger Lie algebra with Kostant-Adler-Symmes decomposition. The first of them is an embedding of widetilde{{g}}-_r into Lie algebra widetilde{{g}}(u^{-1},u)) of formal Laurent power series. The second is an embedding of widetilde{{g}}-_r as a quasigraded Lie subalgebra into a quasigraded Lie algebra widetilde{{g}}_r: widetilde{{g}}_r=widetilde{{g}}-_r+widetilde{{g}}+_r, such that the Kostant-Adler-Symmes decomposition is consistent with a chosen quasigrading. We construct dual spaces widetilde{{g}}^*_r, (widetilde{{g}}^{± }_r)^* and explicit form of the Lax operators L(u), L±(u) as elements of these spaces. We develop a theory of integrable finite-dimensional hamiltonian systems and soliton hierarchies based on Lie algebras widetilde{{g}}_r, widetilde{{g}}^{± }_r. We consider examples of such systems and soliton equations and obtain the most general form of integrable tops, Kirchhoff-type integrable systems, and integrable Landau-Lifshitz-type equations corresponding to the Lie algebra {g}.

  4. Semi-direct sums of Lie algebras and continuous integrable couplings

    NASA Astrophysics Data System (ADS)

    Ma, Wen-Xiu; Xu, Xi-Xiang; Zhang, Yufeng

    2006-02-01

    A relation between semi-direct sums of Lie algebras and integrable couplings of continuous soliton equations is presented, and correspondingly, a feasible way to construct integrable couplings is furnished. A direct application to the AKNS spectral problem leads to a novel hierarchy of integrable couplings of the AKNS hierarchy of soliton equations. It is also indicated that the study of integrable couplings using semi-direct sums of Lie algebras is an important step towards complete classification of integrable systems.

  5. Higher level twisted Zhu algebras

    SciTech Connect

    Ekeren, Jethro van

    2011-05-15

    The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper, we consider the general setup of a vertex algebra V, graded by {Gamma}/Z for some subgroup {Gamma} of R containing Z, and with a Hamiltonian operator H having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level p Zhu algebras Zhu{sub p,{Gamma}}(V), and we prove the following theorems: For each p, there is a bijection between the irreducible Zhu{sub p,{Gamma}}(V)-modules and the irreducible {Gamma}-twisted positive energy V-modules, and V is ({Gamma}, H)-rational if and only if all its Zhu algebras Zhu{sub p,{Gamma}}(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for H. We provide an explicit description of the level p Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra Vir{sup c} and the universal affine Kac-Moody vertex algebra V{sup k}(g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.

  6. Algebraic dichotomies with an application to the stability of Riemann solutions of conservation laws

    NASA Astrophysics Data System (ADS)

    Lin, Xiao-Biao

    Recently, there has been some interest on the stability of waves where the functions involved grow or decay at an algebraic rate |. In this paper we define the so-called algebraic dichotomy that may aid in treating such problems. We discuss the basic properties of the algebraic dichotomy, methods of detecting it, and calculating the power of the weight function. We present several examples: (1) The Bessel equation. (2) The n-degree Fisher type equation. (3) Hyperbolic conservation laws in similarity coordinates. (4) A system of conservation laws with a Dafermos type viscous regularization. We show that the linearized system generates an analytic semigroup in the space of algebraic decay functions. This example motivates our work on algebraic dichotomies.

  7. Dual spaces of differential Lie algebras

    SciTech Connect

    Kupershmidt, B.A.

    1982-01-01

    We present a mathematical scheme which serves as an infinite-dimensional generalization of Poisson structures on dual spaces of finite-dimensional Lie algebras, which are well known and widely used in classical mechanics. These structures have recently appeared in the theory of Lax equations, long waves in hydrodynamics, and various other physical models: compressible hydrodynamics, magnetohydrodynamics, multifluid plasmas, elasticity, superfluid /sup 4/He and /sup 3/He-A, Ginzburg-Landau theory of superconductors, and classical chromohydrodynamics (the generalization of plasma physics to Yang-Mills interactions).

  8. Multifractal vector fields and stochastic Clifford algebra

    NASA Astrophysics Data System (ADS)

    Schertzer, Daniel; Tchiguirinskaia, Ioulia

    2015-12-01

    In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.

  9. Multifractal vector fields and stochastic Clifford algebra

    SciTech Connect

    Schertzer, Daniel Tchiguirinskaia, Ioulia

    2015-12-15

    In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.

  10. Multifractal vector fields and stochastic Clifford algebra.

    PubMed

    Schertzer, Daniel; Tchiguirinskaia, Ioulia

    2015-12-01

    In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality. PMID:26723166

  11. Learning Activity Package, Algebra 93-94, LAPs 12-22.

    ERIC Educational Resources Information Center

    Evans, Diane

    A set of 11 teacher-prepared Learning Activity Packages (LAPs) in beginning algebra, these units cover sets, properties of operations, operations over real numbers, open expressions, solution sets of equations and inequalities, equations and inequalities with two variables, solution sets of equations with two variables, exponents, factoring and…

  12. Dirac Field, Gravity, Inertial Effects, and Computer Algebra

    NASA Astrophysics Data System (ADS)

    Vulcanov, Dumitru N.; Cotăescu, Ion I.

    The article presents some new results obtained for the non-relativistic approximation of the Dirac equation in a non-inertial reference frame — rotated and accelerated — and in Schwarzschild gravitational field. These results are obtained with new routines of algebraic programming in REDUCE + EXCALC language for the Dirac equation in a non-inertial reference frame and after three successive Foldy-Wouthuysen transformations.

  13. Some remarks on unilateral matrix equations

    SciTech Connect

    Cerchiai, Bianca L.; Zumino, Bruno

    2001-02-01

    We briefly review the results of our paper LBNL-46775: We study certain solutions of left-unilateral matrix equations. These are algebraic equations where the coefficients and the unknown are square matrices of the same order, or, more abstractly, elements of an associative, but possibly noncommutative algebra, and all coefficients are on the left. Recently such equations have appeared in a discussion of generalized Born-Infeld theories. In particular, two equations, their perturbative solutions and the relation between them are studied, applying a unified approach based on the generalized Bezout theorem for matrix polynomials.

  14. Handheld Computer Algebra Systems in the Pre-Algebra Classroom

    ERIC Educational Resources Information Center

    Gantz, Linda Ann Galofaro

    2010-01-01

    This mixed method analysis sought to investigate several aspects of student learning in pre-algebra through the use of computer algebra systems (CAS) as opposed to non-CAS learning. This research was broken into two main parts, one which compared results from both the experimental group (instruction using CAS, N = 18) and the control group…

  15. Abstract Algebra to Secondary School Algebra: Building Bridges

    ERIC Educational Resources Information Center

    Christy, Donna; Sparks, Rebecca

    2015-01-01

    The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…

  16. Algebra and Algebraic Thinking in School Math: 70th YB

    ERIC Educational Resources Information Center

    National Council of Teachers of Mathematics, 2008

    2008-01-01

    Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…

  17. Geometric and algebraic multigrid techniques for fluid dynamics problems on unstructured grids

    NASA Astrophysics Data System (ADS)

    Volkov, K. N.; Emel'yanov, V. N.; Teterina, I. V.

    2016-02-01

    Issues concerning the implementation and practical application of geometric and algebraic multigrid techniques for solving systems of difference equations generated by the finite volume discretization of the Euler and Navier-Stokes equations on unstructured grids are studied. The construction of prolongation and interpolation operators, as well as grid levels of various resolutions, is discussed. The results of the application of geometric and algebraic multigrid techniques for the simulation of inviscid and viscous compressible fluid flows over an airfoil are compared. Numerical results show that geometric methods ensure faster convergence and weakly depend on the method parameters, while the efficiency of algebraic methods considerably depends on the input parameters.

  18. Structured adaptive grid generation using algebraic methods

    NASA Technical Reports Server (NTRS)

    Yang, Jiann-Cherng; Soni, Bharat K.; Roger, R. P.; Chan, Stephen C.

    1993-01-01

    The accuracy of the numerical algorithm depends not only on the formal order of approximation but also on the distribution of grid points in the computational domain. Grid adaptation is a procedure which allows optimal grid redistribution as the solution progresses. It offers the prospect of accurate flow field simulations without the use of an excessively timely, computationally expensive, grid. Grid adaptive schemes are divided into two basic categories: differential and algebraic. The differential method is based on a variational approach where a function which contains a measure of grid smoothness, orthogonality and volume variation is minimized by using a variational principle. This approach provided a solid mathematical basis for the adaptive method, but the Euler-Lagrange equations must be solved in addition to the original governing equations. On the other hand, the algebraic method requires much less computational effort, but the grid may not be smooth. The algebraic techniques are based on devising an algorithm where the grid movement is governed by estimates of the local error in the numerical solution. This is achieved by requiring the points in the large error regions to attract other points and points in the low error region to repel other points. The development of a fast, efficient, and robust algebraic adaptive algorithm for structured flow simulation applications is presented. This development is accomplished in a three step process. The first step is to define an adaptive weighting mesh (distribution mesh) on the basis of the equidistribution law applied to the flow field solution. The second, and probably the most crucial step, is to redistribute grid points in the computational domain according to the aforementioned weighting mesh. The third and the last step is to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentration

  19. Statecharts Via Process Algebra

    NASA Technical Reports Server (NTRS)

    Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance

    1999-01-01

    Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics

  20. Algebraic Multigrid Benchmark

    2013-05-06

    AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumpsmore » and an anisotropy in one part.« less

  1. Middle School Students' Conceptual Understanding of Equations: Evidence From Writing Story Problems. WCER Working Paper No. 2009-3

    ERIC Educational Resources Information Center

    Alibali, Martha W.; Kao, Yvonne S.; Brown, Alayna N.; Nathan, Mitchell J.; Stephens, Ana C.

    2009-01-01

    This study investigated middle school students' conceptual understanding of algebraic equations. Participants in the study--257 sixth- and seventh-grade students--were asked to solve one set of algebraic equations and to generate story problems corresponding with another set of equations. Structural aspects of the equations, including the number…

  2. Integrability of the Wong Equations in the Class of Linear Integrals of Motion

    NASA Astrophysics Data System (ADS)

    Magazev, A. A.

    2016-04-01

    The Wong equations, which describe the motion of a classical charged particle with isospin in an external gauge field, are considered. The structure of the Lie algebra of the linear integrals of motion of these equations is investigated. An algebraic condition for integrability of the Wong equations is formulated. Some examples are considered.

  3. The Algebra of Complex Numbers.

    ERIC Educational Resources Information Center

    LePage, Wilbur R.

    This programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex number…

  4. Algebraic Squares: Complete and Incomplete.

    ERIC Educational Resources Information Center

    Gardella, Francis J.

    2000-01-01

    Illustrates ways of using algebra tiles to give students a visual model of competing squares that appear in algebra as well as in higher mathematics. Such visual representations give substance to the symbolic manipulation and give students who do not learn symbolically a way of understanding the underlying concepts of completing the square. (KHR)

  5. Online Algebraic Tools for Teaching

    ERIC Educational Resources Information Center

    Kurz, Terri L.

    2011-01-01

    Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)

  6. Condensing Algebra for Technical Mathematics.

    ERIC Educational Resources Information Center

    Greenfield, Donald R.

    Twenty Algebra-Packets (A-PAKS) were developed by the investigator for technical education students at the community college level. Each packet contained a statement of rationale, learning objectives, performance activities, performance test, and performance test answer key. The A-PAKS condensed the usual sixteen weeks of algebra into a six-week…

  7. Algebraic Thinking in Adult Education

    ERIC Educational Resources Information Center

    Manly, Myrna; Ginsburg, Lynda

    2010-01-01

    In adult education, algebraic thinking can be a sense-making tool that introduces coherence among mathematical concepts for those who previously have had trouble learning math. Further, a modeling approach to algebra connects mathematics and the real world, demonstrating the usefulness of math to those who have seen it as just an academic…

  8. Linear Algebra and Image Processing

    ERIC Educational Resources Information Center

    Allali, Mohamed

    2010-01-01

    We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)

  9. Algebra: Grades 8-12.

    ERIC Educational Resources Information Center

    Instructional Objectives Exchange, Los Angeles, CA.

    A complete set of behavioral objectives for first-year algebra taught in any of grades 8 through 12 is presented. Three to six sample test items and answers are provided for each objective. Objectives were determined by surveying the most used secondary school algebra textbooks. Fourteen major categories are included: (1) whole numbers--operations…

  10. Exploring Algebraic Patterns through Literature.

    ERIC Educational Resources Information Center

    Austin, Richard A.; Thompson, Denisse R.

    1997-01-01

    Presents methods for using literature to develop algebraic thinking in an environment that connects algebra to various situations. Activities are based on the book "Anno's Magic Seeds" with additional resources listed. Students express a constant function, exponential function, and a recursive function in their own words as well as writing about…

  11. Learning Algebra from Worked Examples

    ERIC Educational Resources Information Center

    Lange, Karin E.; Booth, Julie L.; Newton, Kristie J.

    2014-01-01

    For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is…

  12. Thermodynamics. [algebraic structure

    NASA Technical Reports Server (NTRS)

    Zeleznik, F. J.

    1976-01-01

    The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory's theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not present.

  13. Clifford algebras and Hestenes spinors

    NASA Astrophysics Data System (ADS)

    Lounesto, Pertti

    1993-09-01

    This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within the real Clifford algebra Cl 1,3 ≃ M2 (H). Hestenes invented first in 1966 his ideal spinorsφ in Cl_{1,3 _2}^1 (1 - γ _{03} ) and later 1967/75 he recognized the importance of his operator spinors ψ ∈ Cl{/1,3 + } ≃ M2 (C). This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are given for a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decomposition C ⊗ Cl 1,3 and not in C ⊗ M4 (R)=M4 (C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor. Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case ψ†γ0ψ=0, ψ†γ0γ0123ψ=0. The null case is thoroughly scrutinized for the first time with a new concept called boomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation. Projection operators P±, Σ± are resettled among their new relatives in End(Cl 1,3 ). Finally, a new mapping, called tilt, is introduced to enable a transition from Cl 1,3 to the (graded) opposite algebra Cl 3,1 without resorting to complex numbers, that is, not using a replacement γμ → iγμ.

  14. Double Precision Differential/Algebraic Sensitivity Analysis Code

    1995-06-02

    DDASAC solves nonlinear initial-value problems involving stiff implicit systems of ordinary differential and algebraic equations. Purely algebraic nonlinear systems can also be solved, given an initial guess within the region of attraction of a solution. Options include automatic reconciliation of inconsistent initial states and derivatives, automatic initial step selection, direct concurrent parametric sensitivity analysis, and stopping at a prescribed value of any user-defined functional of the current solution vector. Local error control (in the max-normmore » or the 2-norm) is provided for the state vector and can include the sensitivities on request.« less

  15. Invariants of triangular Lie algebras

    NASA Astrophysics Data System (ADS)

    Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman

    2007-07-01

    Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of Boyko et al (2006 J. Phys. A: Math. Gen.39 5749 (Preprint math-ph/0602046)), developed further in Boyko et al (2007 J. Phys. A: Math. Theor.40 113 (Preprint math-ph/0606045)), is used to determine the invariants. A conjecture of Tremblay and Winternitz (2001 J. Phys. A: Math. Gen.34 9085), concerning the number of independent invariants and their form, is corroborated.

  16. Recursion operators, conservation laws, and integrability conditions for difference equations

    NASA Astrophysics Data System (ADS)

    Mikhailov, A. V.; Wang, Jing Ping; Xenitidis, P.

    2011-04-01

    We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations.

  17. Regular perturbation theory of relativistic corrections: II. Algebraic approximation

    NASA Astrophysics Data System (ADS)

    Rutkowski, A.; Kozłowski, R.; Rutkowska, D.

    2001-01-01

    A four-component equivalent of the Schrödinger equation, describing both the nonrelativistic electron and the nonrelativistic positron, is introduced. The difference between this equation and the Dirac equation is treated as a perturbation. The relevant perturbation equations and formulas for corrections to the energy are derived. Owing to the semibounded character of the Schrödinger Hamiltonian of the unperturbed equation the variational perturbation method is formulated. The Hylleraas functionals become then either upper or lower bounds to the respective exact corrections to the energy. In order to demonstrate the usefulness of this approach to the problem of the variational optimization of nonlinear parameters, the perturbation corrections to wave functions for the of hydrogenlike atoms have been approximated in terms of exponential basis functions. The Dirac equation in this algebraic approximation is solved iteratively starting with the solution of the Schrödinger equation.

  18. Partial Differential Algebraic Sensitivity Analysis Code

    1995-05-15

    PDASAC solves stiff, nonlinear initial-boundary-value in a timelike dimension t and a space dimension x. Plane, circular cylindrical or spherical boundaries can be handled. Mixed-order systems of partial differential and algebraic equations can be analyzed with members of order or 0 or 1 in t, 0,1 or 2 in x. Parametric sensitivities of the calculated states are compted simultaneously on request, via the Jacobian of the state equations. Initial and boundary conditions are efficiently reconciled.more » Local error control (in the max-norm or the 2-norm) is provided for the state vector and can include the parametric sensitivites if desired.« less

  19. Numerical stability in problems of linear algebra.

    NASA Technical Reports Server (NTRS)

    Babuska, I.

    1972-01-01

    Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.

  20. Method of lines solution of Richards` equation

    SciTech Connect

    Kelley, C.T.; Miller, C.T.; Tocci, M.D.

    1996-12-31

    We consider the method of lines solution of Richard`s equation, which models flow through porous media, as an example of a situation in which the method can give incorrect results because of premature termination of the nonlinear corrector iteration. This premature termination arises when the solution has a sharp moving front and the Jacobian is ill-conditioned. While this problem can be solved by tightening the tolerances provided to the ODE or DAE solver used for the temporal integration, it is more efficient to modify the termination criteria of the nonlinear solver and/or recompute the Jacobian more frequently. In this paper we continue previous work on this topic by analyzing the modifications in more detail and giving a strategy on how the modifications can be turned on and off in response to changes in the character of the solution.

  1. Using Linear Algebra to Introduce Computer Algebra, Numerical Analysis, Data Structures and Algorithms (and To Teach Linear Algebra, Too).

    ERIC Educational Resources Information Center

    Gonzalez-Vega, Laureano

    1999-01-01

    Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)

  2. Ternary Z3 -graded generalization of Heisenberg's algebra

    NASA Astrophysics Data System (ADS)

    Kerner, Richard

    2015-04-01

    We investigate a ternary, Z3-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, j = e 2πi/3, one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra. A cubic analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator. A set of eigenstates in coordinate representation is constructed in terms of functions satisfying linear differential equation of third order.

  3. Abstract numeric relations and the visual structure of algebra.

    PubMed

    Landy, David; Brookes, David; Smout, Ryan

    2014-09-01

    Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition, it has often been assumed that skilled users of these formalisms treat situations in terms of semantic properties encoded in an abstract syntax that governs the use of notation without particular regard to the details of the physical structure of the equation itself (Anderson, 2005; Hegarty, Mayer, & Monk, 1995). We explore how the notational structure of verbal descriptions or algebraic equations (e.g., the spatial proximity of certain words or the visual alignment of numbers and symbols in an equation) plays a role in the process of interpreting or constructing symbolic equations. We propose in particular that construction processes involve an alignment of notational structures across representation systems, biasing reasoners toward the selection of formal notations that maintain the visuospatial structure of source representations. For example, in the statement "There are 5 elephants for every 3 rhinoceroses," the spatial proximity of 5 and elephants and 3 and rhinoceroses will bias reasoners to write the incorrect expression 5E = 3R, because that expression maintains the spatial relationships encoded in the source representation. In 3 experiments, participants constructed equations with given structure, based on story problems with a variety of phrasings. We demonstrate how the notational alignment approach accounts naturally for a variety of previously reported phenomena in equation construction and successfully predicts error patterns that are not accounted for by prior explanations, such as the left to right transcription heuristic. PMID:24820674

  4. Algebraic distance on graphs.

    SciTech Connect

    Chen, J.; Safro, I.

    2011-01-01

    Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.

  5. Constraint algebra in bigravity

    SciTech Connect

    Soloviev, V. O.

    2015-07-15

    The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.

  6. Quantum algebra of N superspace

    SciTech Connect

    Hatcher, Nicolas; Restuccia, A.; Stephany, J.

    2007-08-15

    We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.

  7. Using Homemade Algebra Tiles To Develop Algebra and Prealgebra Concepts.

    ERIC Educational Resources Information Center

    Leitze, Annette Ricks; Kitt, Nancy A.

    2000-01-01

    Describes how to use homemade tiles, sketches, and the box method to reach a broader group of students for successful algebra learning. Provides a list of concepts appropriate for such an approach. (KHR)

  8. Distance geometry and geometric algebra

    NASA Astrophysics Data System (ADS)

    Dress, Andreas W. M.; Havel, Timothy F.

    1993-10-01

    As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18)

  9. Loop Virasoro Lie conformal algebra

    SciTech Connect

    Wu, Henan Chen, Qiufan; Yue, Xiaoqing

    2014-01-15

    The Lie conformal algebra of loop Virasoro algebra, denoted by CW, is introduced in this paper. Explicitly, CW is a Lie conformal algebra with C[∂]-basis (L{sub i} | i∈Z) and λ-brackets [L{sub i} {sub λ} L{sub j}] = (−∂−2λ)L{sub i+j}. Then conformal derivations of CW are determined. Finally, rank one conformal modules and Z-graded free intermediate series modules over CW are classified.

  10. Time-reversal and the Bessel equation

    NASA Astrophysics Data System (ADS)

    Alfinito, Eleonora; Vitiello, Giuseppe

    2015-07-01

    The system of two damped/amplified oscillator equations is of widespread interest in the study of many physical problems and phenomena, from inflationary models of the Universe to thermal field theories, in condensed matter physics as well in high energy physics, and also in neuroscience. In this report we review the equivalence, in a suitable parametrization, between such a system of equations and the Bessel equations. In this connection, we discuss the breakdown of loop-antiloop symmetry, its relation with time-reversal symmetry and the mechanism of group contraction. Euclidean algebras such as e(2) and e(3) are also discussed in relation with Virasoro-like algebra.

  11. Hierarchies of nonlinear integrable equations and their symmetries in 2 + 1 dimensions

    NASA Astrophysics Data System (ADS)

    Cheng, Yi

    1990-11-01

    For a given nonlinear integrable equation in 2 + 1 dimensions, an approach is described to construct the hierarchies of equations and relevant Lie algebraic properties. The commutability and noncommutability of equations of the flow, their symmetries and mastersymmetries are then derived as direct results of these algebraic properties. The details for the modified Kadomtsev-Petviashvilli equation are shown as an example and the main results for the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Katera-Sawada equation are given.

  12. Elliptic scattering equations

    NASA Astrophysics Data System (ADS)

    Cardona, Carlos; Gomez, Humberto

    2016-06-01

    Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a mathbb{C}{P}^2 space. We show that for the simplest integrand, namely the n - gon, our proposal indeed reproduces the expected result. By using the recently formulated Λ-algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.

  13. Does Early Algebraic Reasoning Differ as a Function of Students' Difficulty with Calculations versus Word Problems?

    ERIC Educational Resources Information Center

    Powell, Sarah R.; Fuchs, Lynn S.

    2014-01-01

    According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 second-grade students, we administered: (1) measures of calculations and…

  14. The Effects of Representations, Constructivist Approaches, and Engagement on Middle School Students' Algebraic Procedure and Conceptual Understanding

    ERIC Educational Resources Information Center

    Ross, Amanda; Willson, Victor

    2012-01-01

    This study examined the effects of types of representations, constructivist teaching approaches, and student engagement on middle school algebra students' procedural knowledge and conceptual understanding. Data gathered from 16 video lessons and algebra pretest/posttests were used to run three multilevel structural equation models. Symbolic…

  15. The non-compact Weyl equation

    NASA Astrophysics Data System (ADS)

    Doikou, Anastasia; Ioannidou, Theodora

    2011-04-01

    A non-compact version of the Weyl equation is proposed, based on the infinite dimensional spin zero representation of the mathfrak{s}{mathfrak{l}_2} algebra. Solutions of the aforementioned equation are obtained in terms of the Kummer functions. In this context, we discuss the ADHMN approach in order to construct the corresponding non-compact BPS monopoles.

  16. TBGG- INTERACTIVE ALGEBRAIC GRID GENERATION

    NASA Technical Reports Server (NTRS)

    Smith, R. E.

    1994-01-01

    TBGG, Two-Boundary Grid Generation, applies an interactive algebraic grid generation technique in two dimensions. The program incorporates mathematical equations that relate the computational domain to the physical domain. TBGG has application to a variety of problems using finite difference techniques, such as computational fluid dynamics. Examples include the creation of a C-type grid about an airfoil and a nozzle configuration in which no left or right boundaries are specified. The underlying two-boundary technique of grid generation is based on Hermite cubic interpolation between two fixed, nonintersecting boundaries. The boundaries are defined by two ordered sets of points, referred to as the top and bottom. Left and right side boundaries may also be specified, and call upon linear blending functions to conform interior interpolation to the side boundaries. Spacing between physical grid coordinates is determined as a function of boundary data and uniformly spaced computational coordinates. Control functions relating computational coordinates to parametric intermediate variables that affect the distance between grid points are embedded in the interpolation formulas. A versatile control function technique with smooth cubic spline functions is also presented. The TBGG program is written in FORTRAN 77. It works best in an interactive graphics environment where computational displays and user responses are quickly exchanged. The program has been implemented on a CDC Cyber 170 series computer using NOS 2.4 operating system, with a central memory requirement of 151,700 (octal) 60 bit words. TBGG requires a Tektronix 4015 terminal and the DI-3000 Graphics Library of Precision Visuals, Inc. TBGG was developed in 1986.

  17. Deformations of Poisson brackets and extensions of Lie algebras of contact vector fields

    NASA Astrophysics Data System (ADS)

    Ovsienko, V.; Roger, C.

    1992-12-01

    CONTENTSIntroduction § 1. Main theoremsChapter I. Algebra § 2. Moyal deformations of the Poisson bracket and *-product on \\mathbb R^{2n} § 3. Algebraic construction § 4. Central extensions § 5. ExamplesChapter II. Deformations of the Poisson bracket and *-product on an arbitrary symplectic manifold § 6. Formal deformations: definitions § 7. Graded Lie algebras as a means of describing deformations § 8. Cohomology computations and their consequences § 9. Existence of a *-productChapter III. Extensions of the Lie algebra of contact vector fields on an arbitrary contact manifold §10. Lagrange bracket §11. Extensions and modules of tensor fieldsAppendix 1. Extensions of the Lie algebra of differential operatorsAppendix 2. Examples of equations of Korteweg-de Vries typeReferences

  18. Sequential products on effect algebras

    NASA Astrophysics Data System (ADS)

    Gudder, Stan; Greechie, Richard

    2002-02-01

    A sequential effect algebra (SEA) is an effect algebra on which a sequential product with natural properties is defined. The properties of sequential products on Hilbert space effect algebras are discussed. For a general SEA, relationships between sequential independence, coexistence and compatibility are given. It is shown that the sharp elements of a SEA form an orthomodular poset. The sequential center of a SEA is discussed and a characterization of when the sequential center is isomorphic to a fuzzy set system is presented. It is shown that the existence, of a sequential product is a strong restriction that eliminates many effect algebras from being SEA's. For example, there are no finite nonboolean SEA's, A measure of sharpness called the sharpness index is studied. The existence of horizontal sums of SEA's is characterized and examples of horizontal sums and tensor products are presented.

  19. Curvature calculations with spacetime algebra

    SciTech Connect

    Hestenes, D.

    1986-06-01

    A new method for calculating the curvature tensor is developed and applied to the Scharzschild case. The method employs Clifford algebra and has definite advantages over conventional methods using differential forms or tensor analysis.

  20. GCD, LCM, and Boolean Algebra?

    ERIC Educational Resources Information Center

    Cohen, Martin P.; Juraschek, William A.

    1976-01-01

    This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)

  1. Cartooning in Algebra and Calculus

    ERIC Educational Resources Information Center

    Moseley, L. Jeneva

    2014-01-01

    This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.

  2. Ada Linear-Algebra Program

    NASA Technical Reports Server (NTRS)

    Klumpp, A. R.; Lawson, C. L.

    1988-01-01

    Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.

  3. Semiclassical states on Lie algebras

    SciTech Connect

    Tsobanjan, Artur

    2015-03-15

    The effective technique for analyzing representation-independent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finite-dimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following the methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.

  4. Numerical solution of nonlinear algebraic equations in stiff ODE solving (1986--89)---Quasi-Newton updating for large scale nonlinear systems (1989--90). Final report, 1986--1990

    SciTech Connect

    Walker, H.F.

    1990-12-31

    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.

  5. Conservation laws, differential identities, and constraints of partial differential equations

    NASA Astrophysics Data System (ADS)

    Zharinov, V. V.

    2015-11-01

    We consider specific cohomological properties such as low-dimensional conservation laws and differential identities of systems of partial differential equations (PDEs). We show that such properties are inherent to complex systems such as evolution systems with constraints. The mathematical tools used here are the algebraic analysis of PDEs and cohomologies over differential algebras and modules.

  6. Hopf algebras and topological recursion

    NASA Astrophysics Data System (ADS)

    Esteves, João N.

    2015-11-01

    We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293-309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347-452).

  7. ALGEBRA v.1.27

    2005-04-11

    The ALGEBRA program allows the user to manipulate data from a finite element analysis before it is plotted. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database. The ALGEBRA program evaluates user-supplied functions of the data and writes the results to an output EXODUS II database that can be read by plot programs.

  8. Strongly-local reductions and the complexity/efficient approximability of algebra and optimization on abstract algebraic structures

    SciTech Connect

    Hunt, H. B.; Marathe, M. V.; Stearns, R. E.

    2001-01-01

    We demonstrate how the concepts of algebraic representability and strongly-local reductions developed here and in [HSM00] can be used to characterize the computational complexity/efficient approximability of a number of basic problems and their variants, on various abstract algebraic structures F. These problems include the following: (1) A1gebra:Determine the solvability, unique solvability, number of solutions, etc., of a system of equations on F. Determine the equivalence of two formulas or straight-line programs on F. 2. 0ptimization:Let {epsilon} > 0. (a) Determine the maximum number of simultaneously satisfiable equations in a system of equations on F; or approximate this number within a multiplicative factor of n{sup {epsilon}}. (b) Determine the maximum value of an objective function subject to satisfiable algebraically expressed constraints on F; or approximate this maximum value within a multiplicative factor of n{sup {epsilon}}. (c) Given a formula or straight-line program, find a minimum size equivalent formula or straightline program; or find an equivalent formula or straight-line program of size {le} f (minimum). Both finite and infinite algebraic structures are considered. These finite structures include all finite nondegenerate lattices and all finite rings or semi-rings with a nonzero element idempotent under multiplication (e.g. all non-degenerate finite unitary rings or semi-rings); and these infinite structures include the natural numbers, integers, real numbers, various algebras on these structures, all ordered rings, many cancellative semi-rings, and all infinite lattices with two elements a,b such that a is covered by b. Our results significantly extend a number of results by Ladner [La89], Condon, et. al. [CF+93], Khanna, et.al [KSW97], Cr951 and Zuckerman [Zu93] on the complexity and approximbaility of combinatorial problems.

  9. Learning Activity Package, Algebra 103-104, LAPs 23-33.

    ERIC Educational Resources Information Center

    Evans, Diane

    This set of 11 teacher-prepared Learning Activity Packages (LAPs) in intermediate algebra covers number systems; exponents and radicals; polynomials and factoring; rational expressions; coordinate geometry; relations, functions, and inequalities; quadratic equations and inequalities; Quadratic functions; systems of equations and inequalities;…

  10. Nonnumeric Computer Applications to Algebra, Trigonometry and Calculus.

    ERIC Educational Resources Information Center

    Stoutemyer, David R.

    1983-01-01

    Described are computer program packages requiring little or no knowledge of computer programing for college algebra, calculus, and abstract algebra. Widely available computer algebra systems are listed. (MNS)

  11. Relativistic algebraic spinors and quantum motions in phase space

    SciTech Connect

    Holland, P.R.

    1986-08-01

    Following suggestions of Schonberg and Bohm, we study the tensorial phase space representation of the Dirac and Feynman-Gell-Mann equations in terms of the complex Dirac algebra C/sub 4/, a Jordan-Wigner algebra G/sub 4/, and Wigner transformations. To do this we solve the problem of the conditions under which elements in C/sub 4/ generate minimal ideals, and extend this to G/sub 4/. This yields the linear theory of Dirac spin spaces and tensor representations of Dirac spinors, and the spin-1/2 wave equations are represented through fermionic state vectors in a higher space as a set of interconnected tensor relations.

  12. Integrable maps from Galois differential algebras, Borel transforms and number sequences

    NASA Astrophysics Data System (ADS)

    Tempesta, Piergiulio

    A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation, in particular symmetries and integrability (see Tempesta, 2010 [40]). Our approach is based on the properties of a suitable Galois differential algebra, that we shall call a Rota algebra. A formulation of the procedure in terms of category theory is proposed. In order to render the lattice dynamics confined, a Borel regularization is also adopted. As a byproduct of the theory, a connection between number sequences and integrability is discussed.

  13. Realization theory and quadratic optimal controllers for systems defined over Banach and Frechet algebras

    NASA Technical Reports Server (NTRS)

    Byrnes, C. I.

    1980-01-01

    It is noted that recent work by Kamen (1979) on the stability of half-plane digital filters shows that the problem of the existence of a feedback law also arises for other Banach algebras in applications. This situation calls for a realization theory and stabilizability criteria for systems defined over Banach for Frechet algebra A. Such a theory is developed here, with special emphasis placed on the construction of finitely generated realizations, the existence of coprime factorizations for T(s) defined over A, and the solvability of the quadratic optimal control problem and the associated algebraic Riccati equation over A.

  14. Virasoro algebra in the KN algebra; Bosonic string with fermionic ghosts on Riemann surfaces

    SciTech Connect

    Koibuchi, H. )

    1991-10-10

    In this paper the bosonic string model with fermionic ghosts is considered in the framework of the KN algebra. The authors' attentions are paid to representations of KN algebra and a Clifford algebra of the ghosts. The authors show that a Virasoro-like algebra is obtained from KN algebra when KN algebra has certain antilinear anti-involution, and that it is isomorphic to the usual Virasoro algebra. The authors show that there is an expected relation between a central charge of this Virasoro-like algebra and an anomaly of the combined system.

  15. The topology of Liouville foliation for the Sokolov integrable case on the Lie algebra so(4)

    SciTech Connect

    Haghighatdoost, Gorbanali; Oshemkov, Andrey A

    2009-06-30

    Several new integrable cases for Euler's equations on some six-dimensional Lie algebras were found by Sokolov in 2004. In this paper we study topological properties of one of these integrable cases on the Lie algebra so(4). In particular, for the system under consideration the bifurcation diagrams of the momentum mapping are constructed and all Fomenko invariants are calculated. Thereby, the classification of isoenergy surfaces for this system up to the rough Liouville equivalence is obtained. Bibliography: 9 titles.

  16. Spectral properties of sums of Hermitian matrices and algebraic geometry

    NASA Astrophysics Data System (ADS)

    Chau Huu-Tai, P.; Van Isacker, P.

    2016-04-01

    It is shown that all the eigenvectors of a sum of Hermitian matrices belong to the same algebraic variety. A polynomial system characterizing this variety is given and a set of nonlinear equations is derived which allows the construction of the variety. Moreover, in some specific cases, explicit expressions for the eigenvectors and eigenvalues can be obtained. Explicit solutions of selected models are also derived.

  17. Iterative methods for elliptic finite element equations on general meshes

    NASA Technical Reports Server (NTRS)

    Nicolaides, R. A.; Choudhury, Shenaz

    1986-01-01

    Iterative methods for arbitrary mesh discretizations of elliptic partial differential equations are surveyed. The methods discussed are preconditioned conjugate gradients, algebraic multigrid, deflated conjugate gradients, an element-by-element techniques, and domain decomposition. Computational results are included.

  18. COMMENT: Comment on `Dirac theory in spacetime algebra'

    NASA Astrophysics Data System (ADS)

    Baylis, William E.

    2002-06-01

    In contrast to formulations of the Dirac theory by Hestenes and by the present author, the formulation recently presented by Joyce (Joyce W P 2001 J. Phys. A: Math. Gen. 34 1991-2005) is equivalent to the usual Dirac equation only in the case of vanishing mass. For nonzero mass, solutions to Joyce's equation can be solutions either of the Dirac equation in the Hestenes form or of the same equation with the sign of the mass reversed, and in general they are mixtures of the two possibilities. Because of this relationship, Joyce obtains twice as many linearly independent plane-wave solutions for a given momentum eigenvalue as exist in the conventional theory. A misconception about the symmetry of the Hestenes equation and the geometric significance of the algebraic spinors is also briefly discussed.

  19. Switched electrical networks and bilinear equations

    NASA Technical Reports Server (NTRS)

    Wood, J. R.

    1974-01-01

    An investigation is conducted concerning the state equations which arise in the description of power processing systems. Attention is given to the role played by Lie groups and Lie algebras in the characterization of the dynamical features of the systems. The bilinear equations used for the representation of the network characteristics are discussed along with the nature of the solutions for the equations. The application of the described approaches is illustrated with the aid of a number of network examples.

  20. Algebraic geometry realization of quantum Hall soliton

    NASA Astrophysics Data System (ADS)

    Abounasr, R.; Ait Ben Haddou, M.; El Rhalami, A.; Saidi, E. H.

    2005-02-01

    Using the Iqbal-Netzike-Vafa dictionary giving the correspondence between the H2 homology of del Pezzo surfaces and p-branes, we develop a way to approach the system of brane bounds in M-theory on S1. We first review the structure of 10-dimensional quantum Hall soliton (QHS) from the view of M-theory on S1. Then, we show how the D0 dissolution in D2-brane is realized in M-theory language and derive the p-brane constraint equations used to define appropriately the QHS. Finally, we build an algebraic geometry realization of the QHS in type IIA superstring and show how to get its type IIB dual. Other aspects are also discussed.

  1. Ternary generalization of Heisenberg's algebra

    NASA Astrophysics Data System (ADS)

    Kerner, Richard

    2015-06-01

    A concise study of ternary and cubic algebras with Z3 grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, S3, and its abelian subgroup Z3 may play an important role in quantum physics. We show then how most of important algebras with Z2 grading can be generalized with ternary composition laws combined with a Z3 grading. We investigate in particular a ternary, Z3-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, , one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra. An analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator, and some properties of its eigenfunctions are briefly discussed.

  2. Beyond Dirac - a Unified Algebra

    NASA Astrophysics Data System (ADS)

    Lundberg, Wayne R.

    2001-10-01

    A introductory insight will be shared regarding a 'separation of variables' approach to understanding the relationship between QCD and the origins of cosmological and particle mass. The discussion will then build upon work presented at DFP 2000, focussing on the formal basis for using 3x3x3 matrix algebra as it underlies and extends Dirac notation. A set of restrictions are established which break the multiple symmetries of the 3x3x3 matrix algebra, yielding Standard Model QCD objects and interactions. It will be shown that the 3x3x3 matrix representation unifies the algebra of strong and weak (and by extension, electromagnetic) interactions. A direct correspondence to string theoretic objects is established by considering the string to be partitioned in thirds. Rubik's cube is used as a graphical means of handling algebraic manipulation of 3x3x3 algebra. Further, its potential utility for advancing pedagogical methods through active engagement is discussed. A simulated classroom exercize will be conducted.

  3. Computer Algebra System

    1992-05-04

    DOE-MACSYMA (Project MAC''s SYmbolic MAnipulation system) is a large computer programming system written in LISP. With DOE-MACSYMA the user can differentiate, integrate, take limits, solve systems of linear or polynomial equations, factor polynomials, expand functions in Laurent or Taylor series, solve differential equations (using direct or transform methods), compute Poisson series, plot curves, and manipulate matrices and tensors. A language similar to ALGOL-60 permits users to write their own programs for transforming symbolic expressions. Franzmore » Lisp OPUS 38 provides the environment for the Encore, Celerity, and DEC VAX11 UNIX,SUN(OPUS) versions under UNIX and the Alliant version under Concentrix. Kyoto Common Lisp (KCL) provides the environment for the SUN(KCL),Convex, and IBM PC under UNIX and Data General under AOS/VS.« less

  4. Symmetries of stochastic differential equations: A geometric approach

    NASA Astrophysics Data System (ADS)

    De Vecchi, Francesco C.; Morando, Paola; Ugolini, Stefania

    2016-06-01

    A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an algebra of strong symmetries for a modified SDE is proved under suitable regularity assumptions. This general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional Brownian motion.

  5. Optimal Assignment Problem Applications of Finite Mathematics to Business and Economics. [and] Difference Equations with Applications. Applications of Difference Equations to Economics and Social Sciences. [and] Selected Applications of Mathematics to Finance and Investment. Applications of Elementary Algebra to Finance. [and] Force of Interest. Applications of Calculus to Finance. UMAP Units 317, 322, 381, 382.

    ERIC Educational Resources Information Center

    Gale, David; And Others

    Four units make up the contents of this document. The first examines applications of finite mathematics to business and economies. The user is expected to learn the method of optimization in optimal assignment problems. The second module presents applications of difference equations to economics and social sciences, and shows how to: 1) interpret…

  6. Finite-difference methods for solving loaded parabolic equations

    NASA Astrophysics Data System (ADS)

    Abdullayev, V. M.; Aida-zade, K. R.

    2016-01-01

    Loaded partial differential equations are solved numerically. For illustrative purposes, a boundary value problem for a parabolic equation with various point loads is considered. By applying difference approximations, the problems are reduced to systems of algebraic equations of special structure, which are solved using a parametric representation involving solutions of auxiliary linear systems with tridiagonal matrices. Numerical results are presented and analyzed.

  7. Catmull-Rom Curve Fitting and Interpolation Equations

    ERIC Educational Resources Information Center

    Jerome, Lawrence

    2010-01-01

    Computer graphics and animation experts have been using the Catmull-Rom smooth curve interpolation equations since 1974, but the vector and matrix equations can be derived and simplified using basic algebra, resulting in a simple set of linear equations with constant coefficients. A variety of uses of Catmull-Rom interpolation are demonstrated,…

  8. Algebraic Lattices in QFT Renormalization

    NASA Astrophysics Data System (ADS)

    Borinsky, Michael

    2016-04-01

    The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.

  9. Algebraic Lattices in QFT Renormalization

    NASA Astrophysics Data System (ADS)

    Borinsky, Michael

    2016-07-01

    The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.

  10. Moving frames and prolongation algebras

    NASA Technical Reports Server (NTRS)

    Estabrook, F. B.

    1982-01-01

    Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.

  11. Spatial operator algebra framework for multibody system dynamics

    NASA Technical Reports Server (NTRS)

    Rodriguez, G.; Jain, Abhinandan; Kreutz, K.

    1989-01-01

    The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.

  12. Integrability in three dimensions: Algebraic Bethe ansatz for anyonic models

    NASA Astrophysics Data System (ADS)

    Khachatryan, Sh.; Ferraz, A.; Klümper, A.; Sedrakyan, A.

    2015-10-01

    We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2 + 1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang-Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.

  13. Algebraic multigrid methods applied to problems in computational structural mechanics

    NASA Technical Reports Server (NTRS)

    Mccormick, Steve; Ruge, John

    1989-01-01

    The development of algebraic multigrid (AMG) methods and their application to certain problems in structural mechanics are described with emphasis on two- and three-dimensional linear elasticity equations and the 'jacket problems' (three-dimensional beam structures). Various possible extensions of AMG are also described. The basic idea of AMG is to develop the discretization sequence based on the target matrix and not the differential equation. Therefore, the matrix is analyzed for certain dependencies that permit the proper construction of coarser matrices and attendant transfer operators. In this manner, AMG appears to be adaptable to structural analysis applications.

  14. Homogeneous Lotka-Volterra Equation Possessing a Lie Symmetry: Extension to n-Dimensional Equation and Integrability

    NASA Astrophysics Data System (ADS)

    Imai, Kenji

    2014-02-01

    In this paper, a new n-dimensional homogeneous Lotka-Volterra (HLV) equation, which possesses a Lie symmetry, is derived by the extension from a three-dimensional HLV equation. Its integrability is shown from the viewpoint of Lie symmetries. Furthermore, we derive dynamical systems of higher order, which possess the Lie symmetry, using the algebraic structure of this HLV equation.

  15. Motivating Activities that Lead to Algebra

    ERIC Educational Resources Information Center

    Menon, Ramakrishnan

    2004-01-01

    Four activities consisting of puzzles are introduced, which help students to recognize the strength of algebraic generalizations. They also assist them to comprehend algebraic concepts, and enable them to develop their individual puzzles and games.

  16. Scalable Parallel Algebraic Multigrid Solvers

    SciTech Connect

    Bank, R; Lu, S; Tong, C; Vassilevski, P

    2005-03-23

    The authors propose a parallel algebraic multilevel algorithm (AMG), which has the novel feature that the subproblem residing in each processor is defined over the entire partition domain, although the vast majority of unknowns for each subproblem are associated with the partition owned by the corresponding processor. This feature ensures that a global coarse description of the problem is contained within each of the subproblems. The advantages of this approach are that interprocessor communication is minimized in the solution process while an optimal order of convergence rate is preserved; and the speed of local subproblem solvers can be maximized using the best existing sequential algebraic solvers.

  17. Computational triadic algebras of signs

    SciTech Connect

    Zadrozny, W.

    1996-12-31

    We present a finite model of Peirce`s ten classes of signs. We briefly describe Peirce`s taxonomy of signs; we prove that any finite collection of signs can be extended to a finite algebra of signs in which all interpretants are themselves being interpreted; and we argue that Peirce`s ten classes of signs can be defined using constraints on algebras of signs. The paper opens the possibility of defining multimodal cognitive agents using Peirce`s classes of signs, and is a first step towards building a computational logic of signs based on Peirce`s taxonomies.

  18. Large chiral diffeomorphisms on Riemann surfaces and W-algebras

    SciTech Connect

    Bandelloni, G.; Lazzarini, S.

    2006-10-15

    The diffeomorphism action lifted on truncated (chiral) Taylor expansion of a complex scalar field over a Riemann surface is presented in the paper under the name of large diffeomorphisms. After an heuristic approach, we show how a linear truncation in the Taylor expansion can generate an algebra of symmetry characterized by some structure functions. Such a linear truncation is explicitly realized by introducing the notion of Forsyth frame over the Riemann surface with the help of a conformally covariant algebraic differential equation. The large chiral diffeomorphism action is then implemented through a Becchi-Rouet-Stora (BRS) formulation (for a given order of truncation) leading to a more algebraic setup. In this context the ghost fields behave as holomorphically covariant jets. Subsequently, the link with the so-called W-algebras is made explicit once the ghost parameters are turned from jets into tensorial ghost ones. We give a general solution with the help of the structure functions pertaining to all the possible truncations lower or equal to the given order. This provides another contribution to the relationship between Korteweg-de Vries (KdV) flows and W-diffeomorphims.

  19. Making Algebra Work: Instructional Strategies that Deepen Student Understanding, within and between Algebraic Representations

    ERIC Educational Resources Information Center

    Star, Jon R.; Rittle-Johnson, Bethany

    2009-01-01

    Competence in algebra is increasingly recognized as a critical milestone in students' middle and high school years. The transition from arithmetic to algebra is a notoriously difficult one, and improvements in algebra instruction are greatly needed (National Research Council, 2001). Algebra historically has represented students' first sustained…

  20. Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers

    SciTech Connect

    Nataf, J.M.; Winkelmann, F.

    1992-09-01

    We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK's symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of these methods to solving the partial differential equations for two-dimensional heat flow is illustrated.

  1. Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers

    SciTech Connect

    Nataf, J.M.; Winkelmann, F.

    1992-09-01

    We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK`s symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of these methods to solving the partial differential equations for two-dimensional heat flow is illustrated.

  2. Spatial-Operator Algebra For Robotic Manipulators

    NASA Technical Reports Server (NTRS)

    Rodriguez, Guillermo; Kreutz, Kenneth K.; Milman, Mark H.

    1991-01-01

    Report discusses spatial-operator algebra developed in recent studies of mathematical modeling, control, and design of trajectories of robotic manipulators. Provides succinct representation of mathematically complicated interactions among multiple joints and links of manipulator, thereby relieving analyst of most of tedium of detailed algebraic manipulations. Presents analytical formulation of spatial-operator algebra, describes some specific applications, summarizes current research, and discusses implementation of spatial-operator algebra in the Ada programming language.

  3. The weak Hopf algebras related to generalized Kac-Moody algebra

    SciTech Connect

    Wu Zhixiang

    2006-06-15

    We define a kind of quantized enveloping algebra of a generalized Kac-Moody algebra G by adding a generator J satisfying J{sup m}=J{sup m-1} for some integer m. We denote this algebra by wU{sub q}{sup {tau}}(G). This algebra is a weak Hopf algebra if and only if m=2. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usually quantum envelope algebra U{sub q}(G) of a generalized Kac-Moody algebra G.

  4. Algebra? A Gate! A Barrier! A Mystery!

    ERIC Educational Resources Information Center

    Mathematics Educatio Dialogues, 2000

    2000-01-01

    This issue of Mathematics Education Dialogues focuses on the nature and the role of algebra in the K-14 curriculum. Articles on this theme include: (1) "Algebra For All? Why?" (Nel Noddings); (2) "Algebra For All: It's a Matter of Equity, Expectations, and Effectiveness" (Dorothy S. Strong and Nell B. Cobb); (3) "Don't Delay: Build and Talk about…

  5. Graphing Calculator Use in Algebra Teaching

    ERIC Educational Resources Information Center

    Dewey, Brenda L.; Singletary, Ted J.; Kinzel, Margaret T.

    2009-01-01

    This study examines graphing calculator technology availability, characteristics of teachers who use it, teacher attitudes, and how use reflects changes to algebra curriculum and instructional practices. Algebra I and Algebra II teachers in 75 high school and junior high/middle schools in a diverse region of a northwestern state were surveyed.…

  6. New family of Maxwell like algebras

    NASA Astrophysics Data System (ADS)

    Concha, P. K.; Durka, R.; Merino, N.; Rodríguez, E. K.

    2016-08-01

    We introduce an alternative way of closing Maxwell like algebras. We show, through a suitable change of basis, that resulting algebras are given by the direct sums of the AdS and the Maxwell algebras already known in the literature. Casting the result into the S-expansion method framework ensures the straightaway construction of the gravity theories based on a found enlargement.

  7. Build an Early Foundation for Algebra Success

    ERIC Educational Resources Information Center

    Knuth, Eric; Stephens, Ana; Blanton, Maria; Gardiner, Angela

    2016-01-01

    Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What's needed is a consistent, algebra-infused mathematics curriculum all…

  8. A Balancing Act: Making Sense of Algebra

    ERIC Educational Resources Information Center

    Gavin, M. Katherine; Sheffield, Linda Jensen

    2015-01-01

    For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…

  9. Difficulties in Initial Algebra Learning in Indonesia

    ERIC Educational Resources Information Center

    Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja

    2014-01-01

    Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was…

  10. Teaching Strategies to Improve Algebra Learning

    ERIC Educational Resources Information Center

    Zbiek, Rose Mary; Larson, Matthew R.

    2015-01-01

    Improving student learning is the primary goal of every teacher of algebra. Teachers seek strategies to help all students learn important algebra content and develop mathematical practices. The new Institute of Education Sciences[IES] practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students"…

  11. Lessons for Algebraic Thinking. Grades K-2.

    ERIC Educational Resources Information Center

    von Rotz, Leyani; Burns, Marilyn

    Algebra is one of the top priorities of mathematics instruction for the elementary and middle grades. This book is designed to help K-2 teachers meet the challenge of making algebra an integral part of their mathematics instruction and realize both what to teach and how to teach central algebraic concepts. Classroom-tested lessons help teachers…

  12. Unifying the Algebra for All Movement

    ERIC Educational Resources Information Center

    Eddy, Colleen M.; Quebec Fuentes, Sarah; Ward, Elizabeth K.; Parker, Yolanda A.; Cooper, Sandi; Jasper, William A.; Mallam, Winifred A.; Sorto, M. Alejandra; Wilkerson, Trena L.

    2015-01-01

    There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the "Algebra for All" movement. In light of this…

  13. Justification of the collocation method for the integral equation for a mixed boundary value problem for the Helmholtz equation

    NASA Astrophysics Data System (ADS)

    Khalilov, E. H.

    2016-07-01

    The surface integral equation for a spatial mixed boundary value problem for the Helmholtz equation is considered. At a set of chosen points, the equation is replaced with a system of algebraic equations, and the existence and uniqueness of the solution of this system is established. The convergence of the solutions of this system to the exact solution of the integral equation is proven, and the convergence rate of the method is determined.

  14. Leapfrog variants of iterative methods for linear algebra equations

    NASA Technical Reports Server (NTRS)

    Saylor, Paul E.

    1988-01-01

    Two iterative methods are considered, Richardson's method and a general second order method. For both methods, a variant of the method is derived for which only even numbered iterates are computed. The variant is called a leapfrog method. Comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large. In the case of Richardson's method, it is possible to express the final iterate in terms of only the initial approximation, a variant of the iteration called the grand-leap method. In the case of the grand-leap variant, a set of parameters is required. An algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for Gaussian quadrature. General algorithms to implement the leapfrog and grand-leap methods are presented. Algorithms for the important special case of the Chebyshev method are also given.

  15. Weaving Geometry and Algebra Together

    ERIC Educational Resources Information Center

    Cetner, Michelle

    2015-01-01

    When thinking about student reasoning and sense making, teachers must consider the nature of tasks given to students along with how to plan to use the tasks in the classroom. Students should be presented with tasks in a way that encourages them to draw connections between algebraic and geometric concepts. This article focuses on the idea that it…

  16. Inequalities, Assessment and Computer Algebra

    ERIC Educational Resources Information Center

    Sangwin, Christopher J.

    2015-01-01

    The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in…

  17. Putting the Modern in Algebra

    ERIC Educational Resources Information Center

    Bosse, Michael J.; Ries, Heather; Chandler, Kayla

    2012-01-01

    Secondary school mathematics teachers often need to answer the "Why do we do that?" question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students' conceptual knowledge of secondary mathematics. This…

  18. Implementing Change in College Algebra

    ERIC Educational Resources Information Center

    Haver, William E.

    2007-01-01

    In this paper, departments are urged to consider implementing the type of changes proposed in Beyond Crossroads in College Algebra. The author of this paper is chair of the Curriculum Renewal Across the First Two Years (CRAFTY) Committee of the Mathematical Association of America. The committee has members from two-year colleges, four-year…

  19. Algebraic Activities Aid Discovery Lessons

    ERIC Educational Resources Information Center

    Wallace-Gomez, Patricia

    2013-01-01

    After a unit on the rules for positive and negative numbers and the order of operations for evaluating algebraic expressions, many students believe that they understand these principles well enough, but they really do not. They clearly need more practice, but not more of the same kind of drill. Wallace-Gomez provides three graphing activities that…

  20. Fuzzy-algebra uncertainty assessment

    SciTech Connect

    Cooper, J.A.; Cooper, D.K.

    1994-12-01

    A significant number of analytical problems (for example, abnormal-environment safety analysis) depend on data that are partly or mostly subjective. Since fuzzy algebra depends on subjective operands, we have been investigating its applicability to these forms of assessment, particularly for portraying uncertainty in the results of PRA (probabilistic risk analysis) and in risk-analysis-aided decision-making. Since analysis results can be a major contributor to a safety-measure decision process, risk management depends on relating uncertainty to only known (not assumed) information. The uncertainties due to abnormal environments are even more challenging than those in normal-environment safety assessments; and therefore require an even more judicious approach. Fuzzy algebra matches these requirements well. One of the most useful aspects of this work is that we have shown the potential for significant differences (especially in perceived margin relative to a decision threshold) between fuzzy assessment and probabilistic assessment based on subtle factors inherent in the choice of probability distribution models. We have also shown the relation of fuzzy algebra assessment to ``bounds`` analysis, as well as a description of how analyses can migrate from bounds analysis to fuzzy-algebra analysis, and to probabilistic analysis as information about the process to be analyzed is obtained. Instructive examples are used to illustrate the points.

  1. Entropy algebras and Birkhoff factorization

    NASA Astrophysics Data System (ADS)

    Marcolli, Matilde; Tedeschi, Nicolas

    2015-11-01

    We develop notions of Rota-Baxter structures and associated Birkhoff factorizations, in the context of min-plus semirings and their thermodynamic deformations, including deformations arising from quantum information measures such as the von Neumann entropy. We consider examples related to Manin's renormalization and computation program, to Markov random fields and to counting functions and zeta functions of algebraic varieties.

  2. Algebra for All. Research Brief

    ERIC Educational Resources Information Center

    Bleyaert, Barbara

    2009-01-01

    The call for "algebra for all" is not a recent phenomenon. Concerns about the inadequacy of math (and science) preparation in America's high schools have been a steady drumbeat since the 1957 launch of Sputnik; a call for raising standards and the number of math (and science) courses required for graduation has been a part of countless national…

  3. A New Age for Algebra

    ERIC Educational Resources Information Center

    Oishi, Lindsay

    2011-01-01

    "Solve for x." While many people first encountered this enigmatic instruction in high school, the last 20 years have seen a strong push to get students to take algebra in eighth grade or even before. Today, concerns about the economy highlight a familiar worry: American eighth-graders trailed their peers in five Asian countries on the 2007 TIMSS…

  4. An Introduction to Algebraic Multigrid

    SciTech Connect

    Falgout, R D

    2006-04-25

    Algebraic multigrid (AMG) solves linear systems based on multigrid principles, but in a way that only depends on the coefficients in the underlying matrix. The author begins with a basic introduction to AMG methods, and then describes some more recent advances and theoretical developments

  5. Adventures in Flipping College Algebra

    ERIC Educational Resources Information Center

    Van Sickle, Jenna

    2015-01-01

    This paper outlines the experience of a university professor who implemented flipped learning in two sections of college algebra courses for two semesters. It details how the courses were flipped, what technology was used, advantages, challenges, and results. It explains what students do outside of class, what they do inside class, and discusses…

  6. Algebra, Home Mortgages, and Recessions

    ERIC Educational Resources Information Center

    Mariner, Jean A. Miller; Miller, Richard A.

    2009-01-01

    The current financial crisis and recession in the United States present an opportunity to discuss relevant applications of some topics in typical first-and second-year algebra and precalculus courses. Real-world applications of percent change, exponential functions, and sums of finite geometric sequences can help students understand the problems…

  7. Algebra from Chips and Chopsticks

    ERIC Educational Resources Information Center

    Yun, Jeong Oak; Flores, Alfinio

    2012-01-01

    Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…

  8. Celestial mechanics with geometric algebra

    NASA Technical Reports Server (NTRS)

    Hestenes, D.

    1983-01-01

    Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.

  9. Algebraic methods in system theory

    NASA Technical Reports Server (NTRS)

    Brockett, R. W.; Willems, J. C.; Willsky, A. S.

    1975-01-01

    Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.

  10. Principals + Algebra (- Fear) = Instructional Leadership

    ERIC Educational Resources Information Center

    Carver, Cynthia L.

    2010-01-01

    Recent state legislation in Michigan mandates that all graduating seniors successfully pass algebra I and II. Numerous initiatives have been enacted to help mathematics teachers meet this challenge, yet school principals have had little preparation for the necessary curricular and instructional changes. To address this unmet need, university-based…

  11. Experts Question California's Algebra Edict

    ERIC Educational Resources Information Center

    Cavanagh, Sean

    2008-01-01

    Business leaders from important sectors of the American economy have been urging schools to set higher standards in math and science--and California officials, in mandating that 8th graders be tested in introductory algebra, have responded with one of the highest such standards in the land. Still, many California educators and school…

  12. The Clifford algebra of physical space and Dirac theory

    NASA Astrophysics Data System (ADS)

    Vaz, Jayme, Jr.

    2016-09-01

    The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein–Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.

  13. Invariance properties of a general bond-pricing equation

    NASA Astrophysics Data System (ADS)

    Sinkala, W.; Leach, P. G. L.; O'Hara, J. G.

    We perform the group classification of a bond-pricing partial differential equation of mathematical finance to discover the combinations of arbitrary parameters that allow the partial differential equation to admit a nontrivial symmetry Lie algebra. As a result of the group classification we propose "natural" values for the arbitrary parameters in the partial differential equation, some of which validate the choices of parameters in such classical models as that of Vasicek and Cox-Ingersoll-Ross. For each set of these natural parameter values we compute the admitted Lie point symmetries, identify the corresponding symmetry Lie algebra and solve the partial differential equation.

  14. Partially-massless higher-spin algebras and their finite-dimensional truncations

    NASA Astrophysics Data System (ADS)

    Joung, Euihun; Mkrtchyan, Karapet

    2016-01-01

    The global symmetry algebras of partially-massless (PM) higher-spin (HS) fields in (A)dS d+1 are studied. The algebras involving PM generators up to depth 2 ( ℓ - 1) are defined as the maximal symmetries of free conformal scalar field with 2 ℓ order wave equation in d dimensions. We review the construction of these algebras by quotienting certain ideals in the universal enveloping algebra of ( A) dS d+1 isometries. We discuss another description in terms of Howe duality and derive the formula for computing trace in these algebras. This enables us to explicitly calculate the bilinear form for this one-parameter family of algebras. In particular, the bilinear form shows the appearance of additional ideal for any non-negative integer values of ℓ - d/2 , which coincides with the annihilator of the one-row ℓ-box Young diagram representation of s{o}_{d+2} . Hence, the corresponding finite-dimensional coset algebra spanned by massless and PM generators is equivalent to the symmetries of this representation.

  15. Finding the Axis of Revolution of an Algebraic Surface of Revolution.

    PubMed

    Alcazar, Juan G; Goldman, Ron

    2016-09-01

    We present an algorithm for extracting the axis of revolution from the implicit equation of an algebraic surface of revolution based on three distinct computational methods: factoring the highest order form into quadrics, contracting the tensor of the highest order form, and using univariate resultants and gcds. We compare and contrast the advantages and disadvantages of each of these three techniques and we derive conditions under which each technique is most appropriate. In addition, we provide several necessary conditions for an implicit algebraic equation to represent a surface of revolution. PMID:26561460

  16. Does Early Algebraic Reasoning Differ as a Function of Students' Difficulty with Calculations versus Word Problems?

    PubMed

    Powell, Sarah R; Fuchs, Lynn S

    2014-08-01

    According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2(nd)- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty. PMID:25309044

  17. Individual differences in algebraic cognition: Relation to the approximate number and semantic memory systems.

    PubMed

    Geary, David C; Hoard, Mary K; Nugent, Lara; Rouder, Jeffrey N

    2015-12-01

    The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 ninth graders (92 girls) while controlling for parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x,y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation but not to schema memory. Frequency of fact retrieval errors was related to schema memory but not to coordinate plane or expression evaluation accuracy. The results suggest that the ANS may contribute to or be influenced by spatial-numerical and numerical-only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest that different brain and cognitive systems are engaged during the learning of different components of algebraic competence while controlling for demographic and domain general abilities. PMID:26255604

  18. The Exocenter of a Generalized Effect Algebra

    NASA Astrophysics Data System (ADS)

    Foulis, David J.; Pulmannová, Sylvia

    2011-12-01

    Elements of the exocenter of a generalized effect algebra (GEA) correspond to decompositions of the GEA as a direct sum and thus the exocenter is a generalization to GEAs of the center of an effect algebra. The exocenter of a GEA is shown to be a boolean algebra, and the notion of a hull mapping for an effect algebra is generalized to a hull system for a GEA. We study Dedekind orthocompleteness of GEAs and extend to GEAs the notion of a centrally orthocomplete effect algebra.

  19. Array algebra estimation in signal processing

    NASA Astrophysics Data System (ADS)

    Rauhala, U. A.

    A general theory of linear estimators called array algebra estimation is interpreted in some terms of multidimensional digital signal processing, mathematical statistics, and numerical analysis. The theory has emerged during the past decade from the new field of a unified vector, matrix and tensor algebra called array algebra. The broad concepts of array algebra and its estimation theory cover several modern computerized sciences and technologies converting their established notations and terminology into one common language. Some concepts of digital signal processing are adopted into this language after a review of the principles of array algebra estimation and its predecessors in mathematical surveying sciences.

  20. Filiform Lie algebras of order 3

    SciTech Connect

    Navarro, R. M.

    2014-04-15

    The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.

  1. Atomic effect algebras with compression bases

    SciTech Connect

    Caragheorgheopol, Dan; Tkadlec, Josef

    2011-01-15

    Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.

  2. Atomic effect algebras with compression bases

    NASA Astrophysics Data System (ADS)

    Caragheorgheopol, Dan; Tkadlec, Josef

    2011-01-01

    Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.

  3. The sixth Painlevé transcendent and uniformization of algebraic curves

    NASA Astrophysics Data System (ADS)

    Brezhnev, Yurii V.

    2016-02-01

    We exhibit a remarkable connection between sixth equation of Painlevé list and infinite families of explicitly uniformizable algebraic curves. Fuchsian equations, congruences for group transformations, differential calculus of functions and differentials on corresponding Riemann surfaces, Abelian integrals, analytic connections (generalizations of Chazy's equations), and other attributes of uniformization can be obtained for these curves. As byproducts of the theory, we establish relations between Picard-Hitchin's curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous differential equation which Apéry used to prove the irrationality of Riemann's ζ (3).

  4. Generalized quantum statistics and Lie (super)algebras

    NASA Astrophysics Data System (ADS)

    Stoilova, N. I.

    2016-03-01

    Generalized quantum statistics, such as paraboson and parafermion statistics, are characterized by triple relations which are related to Lie (super)algebras of type B. The correspondence of the Fock spaces of parabosons, parafermions as well as the Fock space of a system of parafermions and parabosons to irreducible representations of (super)algebras of type B will be pointed out. Example of generalized quantum statistics connected to the basic classical Lie superalgebra B(1|1) ≡ osp(3|2) with interesting physical properties, such as noncommutative coordinates, will be given. Therefore the article focuses on the question, addressed already in 1950 by Wigner: do the equation of motion determine the quantum mechanical commutation relation?

  5. Matrix De Rham Complex and Quantum A-infinity algebras

    NASA Astrophysics Data System (ADS)

    Barannikov, S.

    2014-04-01

    I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A ∞-algebras, introduced in Barannikov (Modular operads and non-commutative Batalin-Vilkovisky geometry. IMRN, vol. 2007, rnm075. Max Planck Institute for Mathematics 2006-48, 2007), is represented via de Rham differential acting on the supermatrix spaces related with Bernstein-Leites simple associative algebras with odd trace q( N), and gl( N| N). I also show that the matrix Lagrangians from Barannikov (Noncommutative Batalin-Vilkovisky geometry and matrix integrals. Isaac Newton Institute for Mathematical Sciences, Cambridge University, 2006) are represented by equivariantly closed differential forms.

  6. The linear algebraic method for electron-molecule collisions

    SciTech Connect

    Collins, L.A.; Schneider, B.I.

    1995-09-01

    In order to find numerical solutions to many problems in physics, chemistry and engineering it is necessary to place the equations of motion (classical or quantal) of the variables of dynamical interest on a discrete mesh. The formulation of scattering theory in quantum mechanics is no exception and leads to partial differential or integral equations which may only be solved on digital computers. Typical approaches introduce a numerical grid or basis set expansion of the scattering wavefunction in order to reduce `the problem to the solution of a set of algebraic equations. Often it is more convenient to deal with the scattering matrix or phase amplitude rather than the wavefunction but the essential features of the numerics are unchanged. In this section we will formulate the Linear Algebraic Method (LAM) for electron-atom/molecule scattering for a simple, one-dimensional radial potential. This will illustrate the basic approach and enable the uninitiated reader to follow the subsequent discussion of the general, multi-channel, electron-molecule formulation without undue difficulty. We begin by writing the Schroedinger equation for the s-wave scattering of a structureless particle by a short-range, local potential.

  7. A Z{sub 3} generalization of Pauli's principle, quark algebra and the Lorentz invariance

    SciTech Connect

    Kerner, Richard

    2012-09-24

    The fundamental difference between bosons and fermions is that they obey two alternative representations of the Z{sub 2} group, resulting in symmetric or anti-symmetric binary commutation relations. Our aim is to explore possibilities offered by ternary Z{sub 3} generalization commutation relations. This leads to cubic and ternary algebras which are a direct generalization of usual commutation relations, with Z{sub 3}-grading replacing the usual Z{sub 2}-grading. Properties and structure of such algebras are discussed, with special interest in a low-dimensional one, with two generators. Invariant cubic forms on such algebras are introduced, and it is shown how the SL(2,C) group arises naturally as the symmetry group preserving these forms. In the case of lowest dimension, with only two generators, it is shown how the cubic combinations of elements of the same Z{sub 3} grade behave like Lorentz spinors, while binary products of elements of this algebra with an element of the conjugate algebra behave like Lorentz vectors. The wave equation generalizing the Dirac operator to the Z{sub 3}-graded case is introduced, whose diagonalization leads to a third-order equation. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry obeyed by quark states.

  8. A new model for algebraic Rossby solitary waves in rotation fluid and its solution

    NASA Astrophysics Data System (ADS)

    Chen, Yao-Deng; Yang, Hong-Wei; Gao, Yu-Fang; Yin, Bao-Shu; Feng, Xing-Ru

    2015-09-01

    A generalized Boussinesq equation that includes the dissipation effect is derived to describe a kind of algebraic Rossby solitary waves in a rotating fluid by employing perturbation expansions and stretching transformations of time and space. Using this equation, the conservation laws of algebraic Rossby solitary waves are discussed. It is found that the mass, the momentum, the energy, and the velocity of center of gravity of the algebraic solitary waves are conserved in the propagation process. Finally, the analytical solution of the equation is generated. Based on the analytical solution, the properties of the algebraic solitary waves and the dissipation effect are discussed. The results point out that, similar to classic solitary waves, the dissipation can cause the amplitude and the speed of solitary waves to decrease; however, unlike classic solitary waves, the algebraic solitary waves can split during propagation and the decrease of the detuning parameter can accelerate the occurrence of the solitary waves fission phenomenon. Project supported by the Shandong Provincial Key Laboratory of Marine Ecology and Environment and Disaster Prevention and Mitigation Project, China (Grant No. 2012010), the National Natural Science Foundation of China (Grant Nos. 41205082 and 41476019), the Special Funds for Theoretical Physics of the National Natural Science Foundation of China (Grant No. 11447205), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), China.

  9. Procedural Embodiment and Magic in Linear Equations

    ERIC Educational Resources Information Center

    de Lima, Rosana Nogueira; Tall, David

    2008-01-01

    How do students think about algebra? Here we consider a theoretical framework which builds from natural human functioning in terms of embodiment--perceiving the world, acting on it and reflecting on the effect of the actions--to shift to the use of symbolism to solve linear equations. In the main, the students involved in this study do not…

  10. A Unified Introduction to Ordinary Differential Equations

    ERIC Educational Resources Information Center

    Lutzer, Carl V.

    2006-01-01

    This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)

  11. Synthesizing Strategies Creatively: Solving Linear Equations

    ERIC Educational Resources Information Center

    Ponce, Gregorio A.; Tuba, Imre

    2015-01-01

    New strategies can ignite teachers' imagination to create new lessons or adapt lessons created by others. In this article, the authors present the experience of an algebra teacher and his students solving linear and literal equations and explain how the use of ideas found in past NCTM journals helped bring this lesson to life. The…

  12. Relativistic version of the Feynman–Dyson–Hughes derivation of the Lorentz force law and Maxwell's homogeneous equations

    NASA Astrophysics Data System (ADS)

    Essén, Hanno; Nordmark, Arne B.

    2016-09-01

    The canonical Poisson bracket algebra of four-dimensional relativistic mechanics is used to derive the equation of motion for a charged particle, with the Lorentz force, and the homogeneous Maxwell equations.

  13. Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type

    NASA Astrophysics Data System (ADS)

    Khongsap, Ta; Wang, Weiqiang

    2009-01-01

    We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.

  14. The algebras of large N matrix mechanics

    SciTech Connect

    Halpern, M.B.; Schwartz, C.

    1999-09-16

    Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.

  15. Alternative algebraic approaches in quantum chemistry

    SciTech Connect

    Mezey, Paul G.

    2015-01-22

    Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed.

  16. Elliptic Quantum Groups E_{τ ,η } ({s}{l}_2 ) and Quasi-Hopf Algebras

    NASA Astrophysics Data System (ADS)

    Enriquez, B.; Felder, G.

    1998-08-01

    We construct an algebra morphism from the elliptic quantum group E_{τ ,η } ({s}{l}_2 ) to a certain elliptic version of the "quantum loop groups in higher genus" studied by V. Rubtsov and the first author. This provides an embedding of E_{τ ,η } ({s}{l}_2 ) in an algebra "with central extension". In particular we construct L ±-operators obeying a dynamical version of the Reshetikhin-:Semenov-Tian-Shansky relations. To do that, we construct the factorization of a certain twist of the quantum loop algebra, that automatically satisfies the "twisted cocycle equation" of O. Babelon, D. Bernard and E. Billey, and therefore provides a solution of the dynamical Yang-Baxter equation.

  17. BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS

    NASA Technical Reports Server (NTRS)

    Krogh, F. T.

    1994-01-01

    The Basic Linear Algebra Subprogram (BLAS) library is a collection of FORTRAN callable routines for employing standard techniques in performing the basic operations of numerical linear algebra. The BLAS library was developed to provide a portable and efficient source of basic operations for designers of programs involving linear algebraic computations. The subprograms available in the library cover the operations of dot product, multiplication of a scalar and a vector, vector plus a scalar times a vector, Givens transformation, modified Givens transformation, copy, swap, Euclidean norm, sum of magnitudes, and location of the largest magnitude element. Since these subprograms are to be used in an ANSI FORTRAN context, the cases of single precision, double precision, and complex data are provided for. All of the subprograms have been thoroughly tested and produce consistent results even when transported from machine to machine. BLAS contains Assembler versions and FORTRAN test code for any of the following compilers: Lahey F77L, Microsoft FORTRAN, or IBM Professional FORTRAN. It requires the Microsoft Macro Assembler and a math co-processor. The PC implementation allows individual arrays of over 64K. The BLAS library was developed in 1979. The PC version was made available in 1986 and updated in 1988.

  18. Equivalence groupoids of classes of linear ordinary differential equations and their group classification

    NASA Astrophysics Data System (ADS)

    Boyko, Vyacheslav M.; Popovych, Roman O.; Shapoval, Nataliya M.

    2015-06-01

    Admissible point transformations of classes of rth order linear ordinary differential equations (in particular, the whole class of such equations and its subclasses of equations in the rational form, the Laguerre-Forsyth form, the first and second Arnold forms) are exhaustively described. Using these results, the group classification of such equations is carried out within the algebraic approach in three different ways.

  19. Modular properties of characters of the W3 algebra

    NASA Astrophysics Data System (ADS)

    Iles, Nicholas J.; Watts, Gérard M. T.

    2016-01-01

    In a previous work, exact formulae and differential equations were found for traces of powers of the zero mode in the W 3 algebra. In this paper we investigate their modular properties, in particular we find the exact result for the modular transformations of traces of W 0 n for n = 1 , 2 , 3, solving exactly the problem studied approximately by Gaberdiel, Hartman and Jin. We also find modular differential equations satisfied by traces with a single W 0 inserted, and relate them to differential equations studied by Mathur et al. We find that, remarkably, these all seem to be related to weight 0 modular forms with expansions with non-negative integer coefficients.

  20. Dual isomonodromic deformations and moment maps to loop algebras

    NASA Astrophysics Data System (ADS)

    Harnad, J.

    1994-12-01

    The Hamiltonian structure of the monodromy preserving deformation equations of Jimbo et al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to “dual” pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painlevé transcendents P V and VI .

  1. Algebraic reconstruction techniques for spectral reconstruction in diffuse optical tomography

    SciTech Connect

    Brendel, Bernhard; Ziegler, Ronny; Nielsen, Tim

    2008-12-01

    Reconstruction in diffuse optical tomography (DOT) necessitates solving the diffusion equation, which is nonlinear with respect to the parameters that have to be reconstructed. Currently applied solving methods are based on the linearization of the equation. For spectral three-dimensional reconstruction, the emerging equation system is too large for direct inversion, but the application of iterative methods is feasible. Computational effort and speed of convergence of these iterative methods are crucial since they determine the computation time of the reconstruction. In this paper, the iterative methods algebraic reconstruction technique (ART) and conjugated gradients (CGs) as well as a new modified ART method are investigated for spectral DOT reconstruction. The aim of the modified ART scheme is to speed up the convergence by considering the specific conditions of spectral reconstruction. As a result, it converges much faster to favorable results than conventional ART and CG methods.

  2. Using trees to compute approximate solutions to ordinary differential equations exactly

    NASA Technical Reports Server (NTRS)

    Grossman, Robert

    1991-01-01

    Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.

  3. Maximal Abelian subalgebras of pseudoeuclidean real Lie algebras and their application in physics

    NASA Astrophysics Data System (ADS)

    Thomova, Zora

    1998-12-01

    We construct the conjugacy classes of maximal abelian subalgebras (MASAs) of the real pseudoeuclidean Lie algebras e(p, q) under the conjugation by the corresponding pseudoeuclidean Lie groups E(p, q). The algebra e( p, q) is a semi-direct sum of the pseudoorthogonal algebra o(p, q) and the abelian ideal of translations T(p + q). We use this particular structure to construct first the splitting MASAs, which are themselves direct sums of subalgebras of o(p, q) and T(p + q). Splitting MASAs give rise to the nonsplitting MASAs of e(p, q). The results for q = 0, 1 and 2 are entirely explicit. MASAs of e(p, 0) and e( p, 1) are used to construct conformally nonequivalent coordinate systems in which the wave equation and Hamilton-Jacobi equations allow the separation of variables. As an application of subgroup classification we perform symmetry reduction for two nonlinear partial differential equations. The method of symmetry reduction is used to obtain analytical solutions of the Landau-Lifshitz and a nonlinear diffusion equations. The symmetry group is found for both equations and all two-dimensional subgroups are classified. These are used to reduce both equations to ordinary differential equations, which are solved in terms of elliptic functions.

  4. (Fuzzy) Ideals of BN-Algebras

    PubMed Central

    Walendziak, Andrzej

    2015-01-01

    The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained. PMID:26125050

  5. Lax operator algebras and integrable systems

    NASA Astrophysics Data System (ADS)

    Sheinman, O. K.

    2016-02-01

    A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero-Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac-Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann-Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. Bibliography: 51 titles.

  6. Algebra: A Challenge at the Crossroads of Policy and Practice

    ERIC Educational Resources Information Center

    Stein, Mary Kay; Kaufman, Julia Heath; Sherman, Milan; Hillen, Amy F.

    2011-01-01

    The authors review what is known about early and universal algebra, including who is getting access to algebra and student outcomes associated with algebra course taking in general and specifically with universal algebra policies. The findings indicate that increasing numbers of students, some of whom are underprepared, are taking algebra earlier.…

  7. Coverings of topological semi-abelian algebras

    NASA Astrophysics Data System (ADS)

    Mucuk, Osman; Demir, Serap

    2016-08-01

    In this work, we study on a category of topological semi-abelian algebras which are topological models of given an algebraic theory T whose category of models is semi-abelian; and investigate some results on the coverings of topological models of such theories yielding semi-abelian categories. We also consider the internal groupoid structure in the semi-abelian category of T-algebras, and give a criteria for the lifting of internal groupoid structure to the covering groupoids.

  8. Finite-dimensional simple graded algebras

    SciTech Connect

    Bahturin, Yu A; Zaicev, M V; Sehgal, S K

    2008-08-31

    Let R be a finite-dimensional algebra over an algebraically closed field F graded by an arbitrary group G. In the paper it is proved that if the characteristic of F is zero or does not divide the order of any finite subgroup of G, then R is graded simple if and only if it is isomorphic to a matrix algebra over a finite-dimensional graded skew field. Bibliography: 24 titles.

  9. Software package and API in MATLAB for working with fuzzy algebras

    NASA Astrophysics Data System (ADS)

    Zahariev, Zlatko

    2009-11-01

    New software package for fuzzy calculus is presented in this paper. Most important feature of this package is solving fuzzy linear systems of equations and inequalities in fuzzy algebras. Together with this, some other functionality is implemented. An example is also given.

  10. Computing the Moore-Penrose Inverse of a Matrix with a Computer Algebra System

    ERIC Educational Resources Information Center

    Schmidt, Karsten

    2008-01-01

    In this paper "Derive" functions are provided for the computation of the Moore-Penrose inverse of a matrix, as well as for solving systems of linear equations by means of the Moore-Penrose inverse. Making it possible to compute the Moore-Penrose inverse easily with one of the most commonly used Computer Algebra Systems--and to have the blueprint…

  11. Cauchy problem and Green's functions for first order differential operators and algebraic quantization

    SciTech Connect

    Muehlhoff, Rainer

    2011-02-15

    Existence and uniqueness of advanced and retarded fundamental solutions (Green's functions) and of global solutions to the Cauchy problem is proved for a general class of first order linear differential operators on vector bundles over globally hyperbolic Lorentzian manifolds. This is a core ingredient to CAR-/CCR-algebraic constructions of quantum field theories on curved spacetimes, particularly for higher spin field equations.

  12. How Our Methods of Writing Algebra Have Evolved: A Thread through History

    ERIC Educational Resources Information Center

    Oliver, Jack

    2007-01-01

    Today's generation does not always have its clever ways of writing algebraic equations and expressions. This paper attempts to trace how this system has developed since the dawn of civilization. The author looks at a few snapshots taken at distinct times to illustrate this progress. (Contains 4 tables.)

  13. Cognitive Tutor[R] Algebra I. What Works Clearinghouse Intervention Report

    ERIC Educational Resources Information Center

    What Works Clearinghouse, 2007

    2007-01-01

    "Cognitive Tutor[R] Algebra I," a full year course, delivers instruction in single variable data, simplifying linear expressions, mathematical modeling, solving systems with linear equations, problem solving using proportional reasoning, and powers and exponents. Students work at their own pace to develop problem-solving skills. The duration of…

  14. Algebraic Reasoning: Professor Arbegla Introduces Variables and Functions. GEMS Teacher's Guide for Grades 3-5.

    ERIC Educational Resources Information Center

    Kopp, Jaine; Bergman, Lincoln

    This teacher guide helps build a solid foundation in algebra for students in grades 3-5 in which students gain essential understanding of properties of numbers, variables, functions, equations, and formulas. Throughout the problem solving activities, students use computational skills and gain a deeper understanding of the number system. Students…

  15. Algebraic Bethe Ansatz for Open XXX Model with Triangular Boundary Matrices

    NASA Astrophysics Data System (ADS)

    Belliard, Samuel; Crampé, Nicolas; Ragoucy, Eric

    2013-05-01

    We consider an open XXX spin chain with two general boundary matrices whose entries obey a relation, which is equivalent to the possibility to put simultaneously the two matrices in a upper-triangular form. We construct Bethe vectors by means of a generalized algebraic Bethe ansatz. As usual, the method uses Bethe equations and provides transfer matrix eigenvalues.

  16. Exploring Interactive and Dynamic Simulations Using a Computer Algebra System in an Advanced Placement Chemistry Course

    ERIC Educational Resources Information Center

    Matsumoto, Paul S.

    2014-01-01

    The article describes the use of Mathematica, a computer algebra system (CAS), in a high school chemistry course. Mathematica was used to generate a graph, where a slider controls the value of parameter(s) in the equation; thus, students can visualize the effect of the parameter(s) on the behavior of the system. Also, Mathematica can show the…

  17. Algebraic turbulence models for the computation of two-dimensional high speed flows using unstructured grids

    NASA Technical Reports Server (NTRS)

    Rostand, Philippe

    1988-01-01

    The incorporation of algebraic turbulence models in a solver for the 2-D compressible Navier-Stokes equations using triangular grids is described. A practical way to use the Cebeci Smith model, and to modify it in separated regions is proposed. The ability of the model to predict high speed, perfect gas boundary layers is investigated from a numerical point of view.

  18. Algebra and Problem-Solving in Down Syndrome: A Study with 15 Teenagers

    ERIC Educational Resources Information Center

    Martinez, Elisabetta Monari; Pellegrini, Katia

    2010-01-01

    There is a common opinion that mathematics is difficult for persons with Down syndrome, because of a weakness in numeracy and in abstract thinking. Since 1996, some single case studies have suggested that new opportunities in mathematics are possible for these students: some of them learned algebra and also learned to use equations in…

  19. A Functional Perspective on the Teaching of Algebra: Current Challenges and the Contribution of Technology

    ERIC Educational Resources Information Center

    Lagrange, Jean-Baptiste

    2014-01-01

    From the early nineties, most reformed curricula at upper secondary level choose to give functions a major position and a priority over rational expressions and equations of traditional algebra. The goal of this paper is to introduce key challenges resulting from this choice and to discuss the contribution that software environments associating…

  20. Algebraic logic of concepts and its machine implementation in the algebras of deontic and axiological notions

    NASA Astrophysics Data System (ADS)

    Manerowska, Anna; Nieznański, Edward; Mulawka, Jan

    2013-10-01

    Our aim is to present the algebra of concepts in two formal languages. First, after introducing a primary relation between concepts, which is subsumption, we shall specify in a language that uses quantifiers, the Boolean algebra of general concepts. Next, we shall note down the same algebra in simplified non-quantifying language, in order to use it as basis for two specific implementations, i.e. to create the Boolean algebras of deontic concepts and axiological concepts.

  1. XXZ-type Bethe ansatz equations and quasi-polynomials

    NASA Astrophysics Data System (ADS)

    Li, Jian Rong; Tarasov, Vitaly

    2013-01-01

    We study solutions of the Bethe ansatz equation for the XXZ-type integrable model associated with the Lie algebra fraktur sfraktur lN. We give a correspondence between solutions of the Bethe ansatz equations and collections of quasi-polynomials. This extends the results of E. Mukhin and A. Varchenko for the XXX-type model and the trigonometric Gaudin model.

  2. A Unified Approach to Teaching Quadratic and Cubic Equations.

    ERIC Educational Resources Information Center

    Ward, A. J. B.

    2003-01-01

    Presents a simple method for teaching the algebraic solution of cubic equations via completion of the cube. Shows that this method is readily accepted by students already familiar with completion of the square as a method for quadratic equations. (Author/KHR)

  3. Representations of Super Yang-Mills Algebras

    NASA Astrophysics Data System (ADS)

    Herscovich, Estanislao

    2013-06-01

    We study in this article the representation theory of a family of super algebras, called the super Yang-Mills algebras, by exploiting the Kirillov orbit method à la Dixmier for nilpotent super Lie algebras. These super algebras are an extension of the so-called Yang-Mills algebras, introduced by A. Connes and M. Dubois-Violette in (Lett Math Phys 61(2):149-158, 2002), and in fact they appear as a "background independent" formulation of supersymmetric gauge theory considered in physics, in a similar way as Yang-Mills algebras do the same for the usual gauge theory. Our main result states that, under certain hypotheses, all Clifford-Weyl super algebras {{Cliff}q(k) ⊗ Ap(k)}, for p ≥ 3, or p = 2 and q ≥ 2, appear as a quotient of all super Yang-Mills algebras, for n ≥ 3 and s ≥ 1. This provides thus a family of representations of the super Yang-Mills algebras.

  4. Difficulties in initial algebra learning in Indonesia

    NASA Astrophysics Data System (ADS)

    Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja

    2014-12-01

    Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students' difficulties in algebra. In order to do so, a literature study was carried out on students' difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematization, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganize the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed.

  5. Multicloning and Multibroadcasting in Operator Algebras

    NASA Astrophysics Data System (ADS)

    Kaniowski, Krzysztof; Lubnauer, Katarzyna; Łuczak, Andrzej

    2015-12-01

    We investigate multicloning and multibroadcasting in the general operator algebra framework in arbitrary dimension, generalizing thus results obtained in this framework for simple cloning and broadcasting.

  6. On Realization of Generalized Effect Algebras

    NASA Astrophysics Data System (ADS)

    Paseka, Jan

    2012-12-01

    A well-known fact is that there is a finite orthomodular lattice with an order determining set of states which is not representable in the standard quantum logic, the lattice L(H) of all closed subspaces of a separable complex Hilbert space. We show that a generalized effect algebra is representable in the operator generalized effect algebra G(H) of effects of a complex Hilbert space H iff it has an order determining set of generalized states. This extends the corresponding results for effect algebras of Riečanová and Zajac. Further, any operator generalized effect algebra G(H) possesses an order determining set of generalized states.

  7. Literal algebra for satellite dynamics. [perturbation analysis

    NASA Technical Reports Server (NTRS)

    Gaposchkin, E. M.

    1975-01-01

    A description of the rather general class of operations available is given and the operations are related to problems in satellite dynamics. The implementation of an algebra processor is discussed. The four main categories of symbol processors are related to list processing, string manipulation, symbol manipulation, and formula manipulation. Fundamental required operations for an algebra processor are considered. It is pointed out that algebra programs have been used for a number of problems in celestial mechanics with great success. The advantage of computer algebra is its accuracy and speed.

  8. Banach Algebras Associated to Lax Pairs

    NASA Astrophysics Data System (ADS)

    Glazebrook, James F.

    2015-04-01

    Lax pairs featuring in the theory of integrable systems are known to be constructed from a commutative algebra of formal pseudodifferential operators known as the Burchnall- Chaundy algebra. Such pairs induce the well known KP flows on a restricted infinite-dimensional Grassmannian. The latter can be exhibited as a Banach homogeneous space constructed from a Banach *-algebra. It is shown that this commutative algebra of operators generating Lax pairs can be associated with a commutative C*-subalgebra in the C*-norm completion of the *-algebra. In relationship to the Bose-Fermi correspondence and the theory of vertex operators, this C*-algebra has an association with the CAR algebra of operators as represented on Fermionic Fock space by the Gelfand-Naimark-Segal construction. Instrumental is the Plücker embedding of the restricted Grassmannian into the projective space of the associated Hilbert space. The related Baker and tau-functions provide a connection between these two C*-algebras, following which their respective state spaces and Jordan-Lie-Banach algebras structures can be compared.

  9. Type-Decomposition of an Effect Algebra

    NASA Astrophysics Data System (ADS)

    Foulis, David J.; Pulmannová, Sylvia

    2010-10-01

    Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras. We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice.

  10. A Structure of BCI-Algebras

    NASA Astrophysics Data System (ADS)

    Chajda, Ivan

    2014-10-01

    Commutative BCI-algebras can be considered as semilattices whose sections are equipped with certain involutions. A similar view can be applied to commutative BCK-algebras. However, for general BCK-algebras a certain construction was settled by the author and J. Kühr (Miskolc Math. Notes 8:11-21, 2007) showing that they can be considered as structures essentially weaker than semilattices but still with certain involutions in sections. The aim of this paper is to involve a similar approach for BCI-algebras.

  11. Loop equations and KDV hierarchy in 2-D quantum gravity

    SciTech Connect

    Fucito, F. ); Martellini, M. )

    1992-04-20

    In this paper a derivation of the loop equation for two-dimensional quantum gravity from the KdV equations and the string equation of the one-matrix model is given. The loop equation was found to be equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the virasoro algebra. Starting form the equations expressing these constraints. The authors are able to rederive the equations of the KdV hierarchy using the vertex operator construction of the A{sup (I)}{sub I} infinite dimensional twisted Kac-Moody algebra. From these considerations it follows that the solutions of the string equation of the one-matrix model are given by a subset of the solutions of the KdV hierarchy.

  12. Analysis on singular spaces: Lie manifolds and operator algebras

    NASA Astrophysics Data System (ADS)

    Nistor, Victor

    2016-07-01

    We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference Noncommutative geometry and applications, Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds-called "Lie manifolds" -that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here-work that spans over close to two decades-was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.

  13. Prospective Teachers' Views on the Use of Calculators with Computer Algebra System in Algebra Instruction

    ERIC Educational Resources Information Center

    Ozgun-Koca, S. Ash

    2010-01-01

    Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This…

  14. Results of Using Algebra Tiles as Meaningful Representations of Algebra Concepts.

    ERIC Educational Resources Information Center

    Sharp, Janet M.

    Mathematical meanings can be developed when individuals construct translations between algebra symbol systems and physical systems that represent one another. Previous research studies indicated (1) few high school students connect whole number manipulations to algebraic manipulations and (2) students who encounter algebraic ideas through…

  15. Some C∗-algebras which are coronas of non-C∗-Banach algebras

    NASA Astrophysics Data System (ADS)

    Voiculescu, Dan-Virgil

    2016-07-01

    We present results and motivating problems in the study of commutants of hermitian n-tuples of Hilbert space operators modulo normed ideals. In particular, the C∗-algebras which arise in this context as coronas of non-C∗-Banach algebras, the connections with normed ideal perturbations of operators, the hyponormal operators and the bidual Banach algebras one encounters are discussed.

  16. Leibniz algebras associated with some finite-dimensional representation of Diamond Lie algebra

    NASA Astrophysics Data System (ADS)

    Camacho, Luisa M.; Ladra, Manuel; Karimjanov, Iqboljon A.; Omirov, Bakhrom A.

    2016-03-01

    In this paper we classify Leibniz algebras whose associated Lie algebra is four-dimensional Diamond Lie algebra 𝕯 and the ideal generated by squares of elements is represented by one of the finite-dimensional indecomposable D-modules Un 1, Un 2 or Wn 1 or Wn 2.

  17. Algebraic Bethe ansatz for Q-operators: the Heisenberg spin chain

    NASA Astrophysics Data System (ADS)

    Frassek, Rouven

    2015-07-01

    We diagonalize Q-operators for rational homogeneous {sl}(2)-invariant Heisenberg spin chains using the algebraic Bethe ansatz. After deriving the fundamental commutation relations relevant for this case from the Yang-Baxter equation we demonstrate that the Q-operators act diagonally on the Bethe vectors if the Bethe equations are satisfied. In this way we provide a direct proof that the eigenvalues of the Q-operators studied here are given by Baxter's Q-functions.

  18. Primordial fluctuations from deformed quantum algebras

    SciTech Connect

    Day, Andrew C.; Brown, Iain A.; Seahra, Sanjeev S. E-mail: ibrown@astro.uio.no

    2014-03-01

    We study the implications of deformed quantum algebras for the generation of primordial perturbations from slow-roll inflation. Specifically, we assume that the quantum commutator of the inflaton's amplitude and momentum in Fourier space gets modified at energies above some threshold M{sub *}. We show that when the commutator is modified to be a function of the momentum only, the problem of solving for the post-inflationary spectrum of fluctuations is formally equivalent to solving a one-dimensional Schr and quot;odinger equation with a time dependent potential. Depending on the class of modification, we find results either close to or significantly different from nearly scale invariant spectra. For the former case, the power spectrum is characterized by step-like behaviour at some pivot scale, where the magnitude of the jump is O(H{sup 2}/M{sub *}{sup 2}). (H is the inflationary Hubble parameter.) We use our calculated power spectra to generate predictions for the cosmic microwave background and baryon acoustic oscillations, hence demonstrating that certain types of deformations are incompatible with current observations.

  19. Primordial fluctuations from deformed quantum algebras

    NASA Astrophysics Data System (ADS)

    Day, Andrew C.; Brown, Iain A.; Seahra, Sanjeev S.

    2014-03-01

    We study the implications of deformed quantum algebras for the generation of primordial perturbations from slow-roll inflation. Specifically, we assume that the quantum commutator of the inflaton's amplitude and momentum in Fourier space gets modified at energies above some threshold Mstar. We show that when the commutator is modified to be a function of the momentum only, the problem of solving for the post-inflationary spectrum of fluctuations is formally equivalent to solving a one-dimensional Schr"odinger equation with a time dependent potential. Depending on the class of modification, we find results either close to or significantly different from nearly scale invariant spectra. For the former case, the power spectrum is characterized by step-like behaviour at some pivot scale, where the magnitude of the jump is Script O(H2/Mstar2). (H is the inflationary Hubble parameter.) We use our calculated power spectra to generate predictions for the cosmic microwave background and baryon acoustic oscillations, hence demonstrating that certain types of deformations are incompatible with current observations.

  20. Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients.

    PubMed

    Boyko, Vyacheslav M; Popovych, Roman O; Shapoval, Nataliya M

    2013-01-01

    Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach. PMID:23564972

  1. A Classroom Note on: An Alternative Method for Solving Linear Equations

    ERIC Educational Resources Information Center

    Klikovac, Ida; Riedinger, Michael

    2011-01-01

    The method of "Double False Position" is an arithmetic approach to solving linear equations that pre-dates current algebraic methods by more than 3,000 years. The method applies to problems that, in algebraic notation, would be expressed as y = L(x), where L(x) is a linear function of x. Double False Position works by evaluating the described…

  2. The Algebra of Lexical Semantics

    NASA Astrophysics Data System (ADS)

    Kornai, András

    The current generative theory of the lexicon relies primarily on tools from formal language theory and mathematical logic. Here we describe how a different formal apparatus, taken from algebra and automata theory, resolves many of the known problems with the generative lexicon. We develop a finite state theory of word meaning based on machines in the sense of Eilenberg [11], a formalism capable of describing discrepancies between syntactic type (lexical category) and semantic type (number of arguments). This mechanism is compared both to the standard linguistic approaches and to the formalisms developed in AI/KR.

  3. Full-deautonomisation of a lattice equation

    NASA Astrophysics Data System (ADS)

    Willox, R.; Mase, T.; Ramani, A.; Grammaticos, B.

    2016-07-01

    In this letter we report on the unexpected possibility of applying the full-deautonomisation approach we recently proposed for predicting the algebraic entropy of second-order birational mappings, to discrete lattice equations. Moreover, we show, on two examples, that the full-deautonomisation technique can in fact also be successfully applied to reductions of these lattice equations to mappings with orders higher than 2. In particular, we apply this technique to a recently discovered lattice equation that has confined singularities while being nonintegrable, and we show that our approach accurately predicts this nonintegrable character. Finally, we demonstrate how our method can even be used to predict the algebraic entropy for some nonconfining higher order mappings.

  4. Non-partial wave treatment of reactive and non-reactive scattering Coupled integral equation formalism.

    NASA Technical Reports Server (NTRS)

    Hayes, E. F.; Kouri, D. J.

    1971-01-01

    Coupled integral equations are derived for the full scattering amplitudes for both reactive and nonreactive channels. The equations do not involve any partial wave expansion and are obtained using channel operators for reactive and nonreactive collisions. These coupled integral equations are similar in nature to equations derived for purely nonreactive collisions of structureless particles. Using numerical quadrature techniques, these equations may be reduced to simultaneous algebraic equations which may then be solved.

  5. Strengthening Effect Algebras in a Logical Perspective: Heyting-Wajsberg Algebras

    NASA Astrophysics Data System (ADS)

    Konig, Martinvaldo

    2014-10-01

    Heyting effect algebras are lattice-ordered pseudoboolean effect algebras endowed with a pseudocomplementation that maps on the center (i.e. Boolean elements). They are the algebraic counterpart of an extension of both Łukasiewicz many-valued logic and intuitionistic logic. We show that Heyting effect algebras are termwise equivalent to Heyting-Wajsberg algebras where the two different logical implications are defined as primitive operators. We prove this logic to be decidable, to be strongly complete and to have the deduction-detachment theorem.

  6. Automorphisms and Derivations of the Insertion-Elimination Algebra and Related Graded Lie Algebras

    NASA Astrophysics Data System (ADS)

    Ondrus, Matthew; Wiesner, Emilie

    2016-07-01

    This paper addresses several structural aspects of the insertion-elimination algebra {mathfrak{g}}, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of {mathfrak{g}}, the automorphism group of {mathfrak{g}}, the derivation Lie algebra of {mathfrak{g}}, and a generating set. Several results are stated in terms of Lie algebras admitting a triangular decomposition and can be used to reproduce results for the generalized Virasoro algebras.

  7. Realizations of conformal current-type Lie algebras

    SciTech Connect

    Pei Yufeng; Bai Chengming

    2010-05-15

    In this paper we obtain the realizations of some infinite-dimensional Lie algebras, named 'conformal current-type Lie algebras', in terms of a two-dimensional Novikov algebra and its deformations. Furthermore, Ovsienko and Roger's loop cotangent Virasoro algebra, which can be regarded as a nice generalization of the Virasoro algebra with two space variables, is naturally realized as an affinization of the tensor product of a deformation of the two-dimensional Novikov algebra and the Laurent polynomial algebra. These realizations shed new light on various aspects of the structure and representation theory of the corresponding infinite-dimensional Lie algebras.

  8. Is Algebra Really Difficult for All Students?

    ERIC Educational Resources Information Center

    Egodawatte, Gunawardena

    2009-01-01

    Research studies have shown that students encounter difficulties in transitioning from arithmetic to algebra. Errors made by high school students were analyzed for patterns and their causes. The origins of errors were: intuitive assumptions, failure to understand the syntax of algebra, analogies with other familiar symbol systems such as the…

  9. Some Applications of Algebraic System Solving

    ERIC Educational Resources Information Center

    Roanes-Lozano, Eugenio

    2011-01-01

    Technology and, in particular, computer algebra systems, allows us to change both the way we teach mathematics and the mathematical curriculum. Curiously enough, unlike what happens with linear system solving, algebraic system solving is not widely known. The aim of this paper is to show that, although the theory lying behind the "exact solve"…

  10. A Technology-Intensive Approach to Algebra.

    ERIC Educational Resources Information Center

    Heid, M. Kathleen; Zbiek, Rose Mary

    1995-01-01

    Computer-Intensive Algebra (CIA) focuses on the use of technology to help develop a rich understanding of fundamental algebraic concepts in real-world settings using computing tools for easy access to numerical, graphical, and symbolic representations of mathematical ideas. (MKR)

  11. An Inquiry-Based Linear Algebra Class

    ERIC Educational Resources Information Center

    Wang, Haohao; Posey, Lisa

    2011-01-01

    Linear algebra is a standard undergraduate mathematics course. This paper presents an overview of the design and implementation of an inquiry-based teaching material for the linear algebra course which emphasizes discovery learning, analytical thinking and individual creativity. The inquiry-based teaching material is designed to fit the needs of a…

  12. Algebra in the Early Years? Yes!

    ERIC Educational Resources Information Center

    Taylor-Cox, Jennifer

    2003-01-01

    Suggests ways early years educators can begin teaching young children to think algebraically and prepare them for success in algebra. Focuses on ways to promote mathematical patterns, mathematical situations and structures, models of quantitative relationship, and change. Describes how first-graders used real object representations to better…

  13. Algebraic Thinking: A Problem Solving Approach

    ERIC Educational Resources Information Center

    Windsor, Will

    2010-01-01

    Algebraic thinking is a crucial and fundamental element of mathematical thinking and reasoning. It initially involves recognising patterns and general mathematical relationships among numbers, objects and geometric shapes. This paper will highlight how the ability to think algebraically might support a deeper and more useful knowledge, not only of…

  14. New directions in algebraic dynamical systems

    NASA Astrophysics Data System (ADS)

    Schmidt, Klaus; Verbitskiy, Evgeny

    2011-02-01

    The logarithmic Mahler measure of certain multivariate polynomials occurs frequently as the entropy or the free energy of solvable lattice models (especially dimer models). It is also known that the entropy of an algebraic dynamical system is the logarithmic Mahler measure of the defining polynomial. The connection between the lattice models and the algebraic dynamical systems is still rather mysterious.

  15. Cartan calculus on quantum Lie algebras

    SciTech Connect

    Schupp, P.; Watts, P.; Zumino, B.

    1993-12-09

    A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions au into one big algebra, the ``Cartan Calculus.``

  16. Low Performers Found Unready to Take Algebra

    ERIC Educational Resources Information Center

    Cavanagh, Sean

    2008-01-01

    As state and school leaders across the country push to have more students take algebra in 8th grade, a new study argues that middle schoolers struggling the most in math are being enrolled in that course despite being woefully unprepared. "The Misplaced Math Student: Lost in Eighth Grade Algebra," scheduled for release by the Brookings Institution…

  17. Calif. Laws Shift Gears on Algebra, Textbooks

    ERIC Educational Resources Information Center

    Robelen, Erik W.

    2012-01-01

    New laws in California have set the state on a course for some potentially significant changes to the curriculum, including a measure that revisits the matter of teaching Algebra 1 in 8th grade and another that revamps the state's textbook-adoption process and hands districts greater leeway in choosing instructional materials. The algebra-related…

  18. Success in Algebra among Community College Students

    ERIC Educational Resources Information Center

    Reyes, Czarina

    2010-01-01

    College algebra is a required course for most majors, but is viewed by many as a gatekeeper course for degree completion by students. With almost half a million students taking college algebra each year, faculty are experimenting with new course lengths of time that might result in higher success, completion, and retention rates for college…

  19. Using the Internet To Investigate Algebra.

    ERIC Educational Resources Information Center

    Sherwood, Walter

    The lesson plans in this book engage students by using a tool they enjoy--the Internet--to explore key concepts in algebra. Working either individually or in groups, students learn to approach algebra from a problem solving perspective. Each lesson shows learners how to use the Internet as a resource for gathering facts, data, and other…

  20. Algebraic Geodesics on Three-Dimensional Quadrics

    NASA Astrophysics Data System (ADS)

    Kai, Yue

    2015-12-01

    By Hamilton-Jacobi method, we study the problem of algebraic geodesics on the third-order surface. By the implicit function theorem, we proved the existences of the real geodesics which are the intersections of two algebraic surfaces, and we also give some numerical examples.

  1. Algebraic Formulas for Areas between Curves.

    ERIC Educational Resources Information Center

    Gabai, Hyman

    1982-01-01

    Korean secondary school students preparing for college learn about a simple algebraic formula for area bounded by a parabola and line. The approach does not seem well-known among American students. It is noted that, while the formula derivations rely on integration, algebra students could use the formulas without proofs. (MP)

  2. Classical and quantum Kummer shape algebras

    NASA Astrophysics Data System (ADS)

    Odzijewicz, A.; Wawreniuk, E.

    2016-07-01

    We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras, here called Kummer shape algebras. The resolution of identity for a wide class of reproducing kernels is found. A number of examples, illustrating this theory, are also presented.

  3. Young Mathematicians at Work: Constructing Algebra

    ERIC Educational Resources Information Center

    Fosnot, Catherine Twomey; Jacob, Bill

    2010-01-01

    This book provides a landscape of learning that helps teachers recognize, support, and celebrate their students' capacity to structure their worlds algebraically. It identifies the models, contexts, and landmarks that facilitate algebraic thinking in young students and provides insightful and practical methods for teachers, math supervisors, and…

  4. Focus on Fractions to Scaffold Algebra

    ERIC Educational Resources Information Center

    Ooten, Cheryl Thomas

    2013-01-01

    Beginning algebra is a gatekeeper course into the pipeline to higher mathematics courses required for respected professions in engineering, science, statistics, mathematics, education, and technology. Beginning algebra can also be a perfect storm if the necessary foundational skills are not within a student's grasp. What skills ensure beginning…

  5. Fourier theory and C∗-algebras

    NASA Astrophysics Data System (ADS)

    Bédos, Erik; Conti, Roberto

    2016-07-01

    We discuss a number of results concerning the Fourier series of elements in reduced twisted group C∗-algebras of discrete groups, and, more generally, in reduced crossed products associated to twisted actions of discrete groups on unital C∗-algebras. A major part of the article gives a review of our previous work on this topic, but some new results are also included.

  6. Situated Learning in an Abstract Algebra Classroom

    ERIC Educational Resources Information Center

    Ticknor, Cindy S.

    2012-01-01

    Advisory committees of mathematics consider abstract algebra as an essential component of the mathematical preparation of secondary teachers, yet preservice teachers find it challenging to connect the topics addressed in this advanced course with the high school algebra they must someday teach. This study analyzed the mathematical content…

  7. Teaching Algebra to Students with Learning Disabilities

    ERIC Educational Resources Information Center

    Impecoven-Lind, Linda S.; Foegen, Anne

    2010-01-01

    Algebra is a gateway to expanded opportunities, but it often poses difficulty for students with learning disabilities. Consequently, it is essential to identify evidence-based instructional strategies for these students. The authors begin by identifying three areas of algebra difficulty experienced by students with disabilities: cognitive…

  8. Arithmetic and Cognitive Contributions to Algebra

    ERIC Educational Resources Information Center

    Cirino, Paul T.; Tolar, Tammy D.; Fuchs, Lynn S.

    2013-01-01

    Algebra is a prerequisite for access to STEM careers and occupational success (NMAP, 2008a), yet algebra is difficult for students through high school (US DOE, 2008). Growth in children's conceptual and procedural arithmetical knowledge is reciprocal, although conceptual knowledge has more impact on procedural knowledge than the reverse…

  9. Just Say Yes to Early Algebra!

    ERIC Educational Resources Information Center

    Stephens, Ana; Blanton, Maria; Knuth, Eric; Isler, Isil; Gardiner, Angela Murphy

    2015-01-01

    Mathematics educators have argued for some time that elementary school students are capable of engaging in algebraic thinking and should be provided with rich opportunities to do so. Recent initiatives like the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) have taken up this call by reiterating the place of early algebra in…

  10. Fusion rule algebras from graph theory

    NASA Astrophysics Data System (ADS)

    Caselle, M.; Ponzano, G.

    1989-06-01

    We describe a new class of fusion algebras related to graph theory which bear intriguing connections with group algebras. The structure constants and the matrix S, which diagonalizes the fusion rules, are explicitly computed in terms of SU(2) coupling coefficients.

  11. NINTH YEAR MATHEMATICS. COURSE I, ALGEBRA.

    ERIC Educational Resources Information Center

    New York State Education Dept., Albany.

    THIS GUIDE OUTLINES THE MINIMUM MATERIAL FOR WHICH STUDENTS OF NINTH YEAR MATHEMATICS - COURSE 1 - ALGEBRA WERE HELD RESPONSIBLE ON THE REGENTS EXAMINATIONS BEGINNING IN JUNE, 1966. THE REPORT ALSO PRESENTS THE SCOPE AND CONTENT OF THE ALGEBRA COURSE AND POSSIBLE SUGGESTIONS FOR TEACHING THE MATERIAL TO STUDENTS. (RP)

  12. Modern Algebra, Mathematics: 5293.36.

    ERIC Educational Resources Information Center

    Edwards, Raymond J.

    This guidebook covers Boolean algebra, matrices, linear transformations of the plane, characteristic values, vectors, and algebraic structures. Overall course goals and performance objectives for each unit are specified; sequencing of units and various time schedules are suggested. A sample pretest and posttest are given, and an annotated list of…

  13. Algebra: How Is It for You?

    ERIC Educational Resources Information Center

    Rickard, Caroline

    2008-01-01

    Shortly after starting work for the University of Chichester in the School of Teacher Education, the author was planning a session relating to algebra and found herself inspired by an article in MT182: "Algebraic Infants" by Andrews and Sayers (2003). Based on the making of families of "Multilink" animals, Andrews and Sayers (2003) claim that…

  14. Teaching Modeling and Axiomatization with Boolean Algebra.

    ERIC Educational Resources Information Center

    De Villiers, Michael D.

    1987-01-01

    Presented is an alternative approach to the traditional teaching of Boolean algebra for secondary school mathematics. The main aim of the approach is to use Boolean algebra to teach pupils such mathematical processes as modeling and axiomatization. A course using the approach is described. (RH)

  15. Loop realizations of quantum affine algebras

    SciTech Connect

    Cautis, Sabin; Licata, Anthony

    2012-12-15

    We give a simplified description of quantum affine algebras in their loop presentation. This description is related to Drinfeld's new realization via halves of vertex operators. We also define an idempotent version of the quantum affine algebra which is suitable for categorification.

  16. Performance of Renormalization Group Algebraic Turbulence Model on Boundary Layer Transition Simulation

    NASA Technical Reports Server (NTRS)

    Ahn, Kyung H.

    1994-01-01

    The RNG-based algebraic turbulence model, with a new method of solving the cubic equation and applying new length scales, is introduced. An analysis is made of the RNG length scale which was previously reported and the resulting eddy viscosity is compared with those from other algebraic turbulence models. Subsequently, a new length scale is introduced which actually uses the two previous RNG length scales in a systematic way to improve the model performance. The performance of the present RNG model is demonstrated by simulating the boundary layer flow over a flat plate and the flow over an airfoil.

  17. Algebraic Apect of Helicities in Hadron Physics

    NASA Astrophysics Data System (ADS)

    An, Murat; Ji, Chueng

    2015-04-01

    We examined the relation of polarization vectors and spinors of (1 , 0) ⊕(0 , 1) representation of Lorentz group in Clifford algebra Cl1 , 3 , their relation with standard algebra, and properties of these spinors. Cl1 , 3 consists of different grades:e.g. the first and the second grades represent (1 / 2 , 1 / 2) and (1 , 0) ⊕(0 , 1) representation of spin groups respectively with 4 and 6 components. However, these Clifford numbers are not the helicity eigenstates and thus we transform them into combinations of helicity eigenstates by expressing them as spherical harmonics. We relate the spin-one polarization vectors and (1 , 0) ⊕(0 , 1) spinors under one simple transformation with the spin operators. We also link our work with Winnberg's work of a superfield of a spinors of Clifford algebra by giving a physical meaning to Grassmann variables and discuss how Grassman algebra is linked with Clifford algebra.

  18. Classification of commutator algebras leading to the new type of closed Baker-Campbell-Hausdorff formulas

    NASA Astrophysics Data System (ADS)

    Matone, Marco

    2015-11-01

    We show that there are {\\it 13 types} of commutator algebras leading to the new closed forms of the Baker-Campbell-Hausdorff (BCH) formula $$\\exp(X)\\exp(Y)\\exp(Z)=\\exp({AX+BZ+CY+DI}) \\ , $$ derived in arXiv:1502.06589, JHEP {\\bf 1505} (2015) 113. This includes, as a particular case, $\\exp(X) \\exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$. The algorithm exploits the associativity of the BCH formula and is based on the decomposition $\\exp(X)\\exp(Y)\\exp(Z)=\\exp(X)\\exp({\\alpha Y}) \\exp({(1-\\alpha) Y}) \\exp(Z)$, with $\\alpha$ fixed in such a way that it reduces to $\\exp({\\tilde X})\\exp({\\tilde Y})$, with $\\tilde X$ and $\\tilde Y$ satisfying the Van-Brunt and Visser condition $[\\tilde X,\\tilde Y]=\\tilde u\\tilde X+\\tilde v\\tilde Y+\\tilde cI$. It turns out that $e^\\alpha$ satisfies, in the generic case, an algebraic equation whose exponents depend on the parameters defining the commutator algebra. In nine {\\it types} of commutator algebras, such an equation leads to rational solutions for $\\alpha$. We find all the equations that characterize the solution of the above decomposition problem by combining it with the Jacobi identity.

  19. AN ADA LINEAR ALGEBRA PACKAGE MODELED AFTER HAL/S

    NASA Technical Reports Server (NTRS)

    Klumpp, A. R.

    1994-01-01

    This package extends the Ada programming language to include linear algebra capabilities similar to those of the HAL/S programming language. The package is designed for avionics applications such as Space Station flight software. In addition to the HAL/S built-in functions, the package incorporates the quaternion functions used in the Shuttle and Galileo projects, and routines from LINPAK that solve systems of equations involving general square matrices. Language conventions in this package follow those of HAL/S to the maximum extent practical and minimize the effort required for writing new avionics software and translating existent software into Ada. Valid numeric types in this package include scalar, vector, matrix, and quaternion declarations. (Quaternions are fourcomponent vectors used in representing motion between two coordinate frames). Single precision and double precision floating point arithmetic is available in addition to the standard double precision integer manipulation. Infix operators are used instead of function calls to define dot products, cross products, quaternion products, and mixed scalar-vector, scalar-matrix, and vector-matrix products. The package contains two generic programs: one for floating point, and one for integer. The actual component type is passed as a formal parameter to the generic linear algebra package. The procedures for solving systems of linear equations defined by general matrices include GEFA, GECO, GESL, and GIDI. The HAL/S functions include ABVAL, UNIT, TRACE, DET, INVERSE, TRANSPOSE, GET, PUT, FETCH, PLACE, and IDENTITY. This package is written in Ada (Version 1.2) for batch execution and is machine independent. The linear algebra software depends on nothing outside the Ada language except for a call to a square root function for floating point scalars (such as SQRT in the DEC VAX MATHLIB library). This program was developed in 1989, and is a copyrighted work with all copyright vested in NASA.

  20. Scattering equations and global duality of residues

    NASA Astrophysics Data System (ADS)

    Søgaard, Mads; Zhang, Yang

    2016-05-01

    We examine the polynomial form of the scattering equations by means of computational algebraic geometry. The scattering equations are the backbone of the Cachazo-He-Yuan (CHY) representation of the S-matrix. We explain how the Bezoutian matrix facilitates the calculation of amplitudes in the CHY formalism, without explicitly solving the scattering equations or summing over the individual residues. Since for n -particle scattering the size of the Bezoutian matrix grows only as (n -3 )×(n -3 ), our algorithm is very efficient for analytic and numeric amplitude computations.