Numerical solution of differential algebraic equations (DAEs) by mix-multistep method
NASA Astrophysics Data System (ADS)
Rahim, Yong Faezah; Suleiman, Mohamed; Ibrahim, Zarina Bibi
2014-06-01
Differential Algebraic Equations (DAEs) are regarded as stiff Ordinary Differential Equations (ODEs). Therefore they are solved using implicit method such as Backward Differentiation Formula (BDF) type of methods which require the use of Newton iteration which need much computational effort. However, not all of the ODEs in DAE system are stiff. In this paper, we describe a new technique for solving DAE, where the ODEs are treated as non-stiff at the start of the integration and putting the non-stiff ODEs into stiff subsystem should instability occurs. Adams type of method is used to solve the non-stiff part and BDF method for solving the stiff part. This strategy is shown to be competitive in terms of computational effort and accuracy. Numerical experiments are presented to validate its efficiency.
Block method of Runge Kutta type for solving differential algebraic equation
NASA Astrophysics Data System (ADS)
Wen, Khoo Kai; Majid, Zanariah Abdul; Senu, Norazak
2015-10-01
In this paper, a self-starting block method of Runge Kutta type is proposed to solve semi-explicit index-1 differential algebraic equation (DAE). Semi-explicit DAE consists of a system of ordinary differential equations with algebraic constraints. This method will compute the solutions of DAE at two points simultaneously in a block by block steps using constant step size. The DAE is a stiff equation, therefore the Newton iteration is needed during the implementation. Numerical examples are given in order to illustrate the efficiency of the block method when solving the DAE.
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.
Benhammouda, Brahim
2016-01-01
Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems. PMID:27330880
Introducing DAE Systems in Undergraduate and Graduate Chemical Engineering Curriculum
ERIC Educational Resources Information Center
Mandela, Ravi Kumar; Sridhar, L. N.; Rengaswamy, Raghunathan
2010-01-01
Models play an important role in understanding chemical engineering systems. While differential equation models are taught in standard modeling and control courses, Differential Algebraic Equation (DAE) system models are not usually introduced. These models appear naturally in several chemical engineering problems. In this paper, the introduction…
Hopf algebras and Dyson-Schwinger equations
NASA Astrophysics Data System (ADS)
Weinzierl, Stefan
2016-06-01
In this paper I discuss Hopf algebras and Dyson-Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.
Numerical Treatment of Differential-Algebraic Equations with Index 2
NASA Astrophysics Data System (ADS)
Attili, Basem S.
2007-09-01
We will consider index-2 differential algebraic systems. Since they are usually harder to solve, we will show how to reduce the index 2 problem to index 1 DAE which becomes easier to solve numerically. For the numerical treatment, we will treat the resulting index-1 DAE using power series solutions coupled with pade' approximation for better convergence results. Numerical examples will be presented also.
Solving Absolute Value Equations Algebraically and Geometrically
ERIC Educational Resources Information Center
Shiyuan, Wei
2005-01-01
The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.
DAEs and PDEs in elastic multibody systems
NASA Astrophysics Data System (ADS)
Simeon, B.
1998-12-01
Elastic multibody systems arise in the simulation of vehicles, robots, air- and spacecrafts. They feature a mixed structure with differential-algebraic equations (DAEs) governing the gross motion and partial differential equations (PDEs) describing the elastic deformation of particular bodies. We introduce a general modelling framework for this new application field and discuss numerical simulation techniques from several points of view. Due to different time scales, singular perturbation theory and model reduction play an important role. A slider crank mechanism with a 2D FE grid for the elastic connecting rod illustrates the techniques.
On a Equation in Finite Algebraically Structures
ERIC Educational Resources Information Center
Valcan, Dumitru
2013-01-01
Solving equations in finite algebraically structures (semigroups with identity, groups, rings or fields) many times is not easy. Even the professionals can have trouble in such cases. Therefore, in this paper we proposed to solve in the various finite groups or fields, a binomial equation of the form (1). We specify that this equation has been…
Lie algebras and linear differential equations.
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Convergence of Galerkin Solutions for Linear Differential Algebraic Equations in Hilbert Spaces
NASA Astrophysics Data System (ADS)
Matthes, Michael; Tischendorf, Caren
2010-09-01
The simulation of complex systems describing different physical effects becomes more and more of interest in various applications. Examples are couplings describing interactions between circuits and semiconductor devices, circuits and electromagnetic fields, fluids and structures. The modeling of such complex processes [1, 2, 3, 4, 7, 8] often leads to coupled systems that are composed of ordinary differential equations (ODEs), differential-algebraic equations (DAEs) and partial differential equations (PDEs). Such coupled systems can be regarded in the general framework of abstract differential-algebraic equations. Here, we discuss a Galerkin approach for handling linear abstract differential-algebraic equations with monotone operators. It is shown to provide solutions that converge to the unique solution of the abstract differential-algebraic system. Furthermore, the solution is proved to depend continuously on the data. The most interesting point of the Galerkin approach is the choice of basis functions. They have to be chosen in proper subspaces in order to guarantee that the solution satisfies the non-dynamic constraints. In contrast to other approaches as e.g. [5, 6], this approach allows time dependent operators but needs monotonicity.
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.
Laurent phenomenon algebras and the discrete BKP equation
NASA Astrophysics Data System (ADS)
Okubo, Naoto
2016-09-01
We construct the Laurent phenomenon algebras the cluster variables of which satisfy the discrete BKP equation, the discrete Sawada-Kotera equation and other difference equations obtained by its reduction. These Laurent phenomenon algebras are constructed from seeds with a generalization of mutation-period property. We show that a reduction of a seed corresponds to a reduction of a difference equation.
Laurent phenomenon algebras and the discrete BKP equation
NASA Astrophysics Data System (ADS)
Okubo, Naoto
2016-09-01
We construct the Laurent phenomenon algebras the cluster variables of which satisfy the discrete BKP equation, the discrete Sawada–Kotera equation and other difference equations obtained by its reduction. These Laurent phenomenon algebras are constructed from seeds with a generalization of mutation-period property. We show that a reduction of a seed corresponds to a reduction of a difference equation.
Algebraic approach to solve tt dilepton equations
Sonnenschein, Lars
2005-11-01
The set of nonlinear equations describing the standard model kinematics of the top quark antiquark production system in the dilepton decay channel has at most a fourfold ambiguity due to two not fully reconstructed neutrinos. Its most precise solution is of major importance for measurements of top quark properties like the top quark mass and tt spin correlations. Simple algebraic operations allow one to transform the nonlinear equations into a system of two polynomial equations with two unknowns. These two polynomials of multidegree eight can in turn be analytically reduced to one polynomial with one unknown by means of resultants. The obtained univariate polynomial is of degree 16. The number of its real solutions is determined analytically by means of Sturm's theorem, which is as well used to isolate each real solution into a unique pairwise disjoint interval. The solutions are polished by seeking the sign change of the polynomial in a given interval through binary bracketing.
A New Reynolds Stress Algebraic Equation Model
NASA Technical Reports Server (NTRS)
Shih, Tsan-Hsing; Zhu, Jiang; Lumley, John L.
1994-01-01
A general turbulent constitutive relation is directly applied to propose a new Reynolds stress algebraic equation model. In the development of this model, the constraints based on rapid distortion theory and realizability (i.e. the positivity of the normal Reynolds stresses and the Schwarz' inequality between turbulent velocity correlations) are imposed. Model coefficients are calibrated using well-studied basic flows such as homogeneous shear flow and the surface flow in the inertial sublayer. The performance of this model is then tested in complex turbulent flows including the separated flow over a backward-facing step and the flow in a confined jet. The calculation results are encouraging and point to the success of the present model in modeling turbulent flows with complex geometries.
Numerical solution of integral-algebraic equations for multistep methods
NASA Astrophysics Data System (ADS)
Budnikova, O. S.; Bulatov, M. V.
2012-05-01
Systems of Volterra linear integral equations with identically singular matrices in the principal part (called integral-algebraic equations) are examined. Multistep methods for the numerical solution of a selected class of such systems are proposed and justified.
Algebraic Flux Correction II. Compressible Euler Equations
NASA Astrophysics Data System (ADS)
Kuzmin, Dmitri; Möller, Matthias
Algebraic flux correction schemes of TVD and FCT type are extended to systems of hyperbolic conservation laws. The group finite element formulation is employed for the treatment of the compressible Euler equations. An efficient algorithm is proposed for the edge-by-edge matrix assembly. A generalization of Roe's approximate Riemann solver is derived by rendering all off-diagonal matrix blocks positive semi-definite. Another usable low-order method is constructed by adding scalar artificial viscosity proportional to the spectral radius of the cumulative Roe matrix. The limiting of antidiffusive fluxes is performed using a transformation to the characteristic variables or a suitable synchronization of correction factors for the conservative ones. The outer defect correction loop is equipped with a block-diagonal preconditioner so as to decouple the discretized Euler equations and solve them in a segregated fashion. As an alternative, a strongly coupled solution strategy (global BiCGSTAB method with a block-Gauß-Seidel preconditioner) is introduced for applications which call for the use of large time steps. Various algorithmic aspects including the implementation of characteristic boundary conditions are addressed. Simulation results are presented for inviscid flows in a wide range of Mach numbers.
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.
Dual number coefficient octonion algebra, field equations and conservation laws
NASA Astrophysics Data System (ADS)
Chanyal, B. C.; Chanyal, S. K.
2016-08-01
Starting with octonion algebra, we develop the dual number coefficient octonion (DNCO) algebra having sixteen components. DNCO forms of generalized potential, field and current equations are discussed in consistent manner. We have made an attempt to write the DNCO form of generalized Dirac-Maxwell's equations in presence of electric and magnetic charges (dyons). Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for dyons in terms of DNCO algebra. Further, we discuss the DNCO form of linear momentum conservation law for dyons.
Exact solution of some linear matrix equations using algebraic methods
NASA Technical Reports Server (NTRS)
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
Effective mass Schrödinger equation and nonlinear algebras
NASA Astrophysics Data System (ADS)
Roy, B.; Roy, P.
2005-06-01
Using supersymmetry we obtain solutions of Schrödinger equation with a position dependent effective mass exhibiting a harmonic oscillator like spectrum. We also discuss the underlying nonlinear algebraic symmetry of the problem.
Comparative study of homotopy continuation methods for nonlinear algebraic equations
NASA Astrophysics Data System (ADS)
Nor, Hafizudin Mohamad; Ismail, Ahmad Izani Md.; Majid, Ahmad Abd.
2014-07-01
We compare some recent homotopy continuation methods to see which method has greater applicability and greater accuracy. We test the methods on systems of nonlinear algebraic equations. The results obtained indicate the superior accuracy of Newton Homotopy Continuation Method (NHCM).
Algebraic approximations for transcendental equations with applications in nanophysics
NASA Astrophysics Data System (ADS)
Barsan, Victor
2015-09-01
Using algebraic approximations of trigonometric or hyperbolic functions, a class of transcendental equations can be transformed in tractable, algebraic equations. Studying transcendental equations this way gives the eigenvalues of Sturm-Liouville problems associated to wave equation, mainly to Schroedinger equation; these algebraic approximations provide approximate analytical expressions for the energy of electrons and phonons in quantum wells, quantum dots (QDs) and quantum wires, in the frame of one-particle models of such systems. The advantage of this approach, compared to the numerical calculations, is that the final result preserves the functional dependence on the physical parameters of the problem. The errors of this method, situated between some few percentages and ?, are carefully analysed. Several applications, for quantum wells, QDs and quantum wires, are presented.
NASA Astrophysics Data System (ADS)
Balagović, Martina
2015-03-01
We show that, under Drinfeld's degeneration (Proceedings of the International Congress of Mathematicians. American Mathematical Society, Providence, pp 798-820, 1987) of quantum loop algebras to Yangians, the trigonometric dynamical difference equations [Etingof and Varchenko (Adv Math 167:74-127, 2002)] for the quantum affine algebra degenerate to the trigonometric Casimir differential equations [Toledano Laredo (J Algebra 329:286-327, 2011)] for Yangians.
Stability of Linear Equations--Algebraic Approach
ERIC Educational Resources Information Center
Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.
2012-01-01
This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…
Exact solution of some linear matrix equations using algebraic methods
NASA Technical Reports Server (NTRS)
Djaferis, T. E.; Mitter, S. K.
1979-01-01
Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.
Zeta functional equation on Jordan algebras of type II
NASA Astrophysics Data System (ADS)
Kayoya, J. B.
2005-02-01
Using the Jordan algebras methods, specially the properties of Peirce decomposition and the Frobenius transformation, we compute the coefficients of the zeta functional equation, in the case of Jordan algebras of type II. As particular cases of our result, we can cite the case of studied by Gelbart [Mem. Amer. Math. Soc. 108 (1971)] and Godement and Jacquet [Zeta functions of simple algebras, Lecture Notes in Math., vol. 260, Springer-Verlag, Berlin, 1972], and the case of studied by Muro [Adv. Stud. Pure Math. 15 (1989) 429]. Let us also mention, that recently, Bopp and Rubenthaler have obtained a more general result on the zeta functional equation by using methods based on the algebraic properties of regular graded algebras which are in one-to-one correspondence with simple Jordan algebras [Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces, IRMA, Strasbourg, 2003]. The method used in this paper is a direct application of specific properties of Jordan algebras of type II.
ERIC Educational Resources Information Center
Falcon, Raymond
2009-01-01
Teachers use action research in order to improve their teaching and student learning. This action research will analyze students' algebraic reasoning in finding values of variables in systems of equations pictorially and algebraically. This research will look at students solving linear systems of equations without knowing the algebraic algorithms.…
Allidina, A.Y.; Malinowski, K.; Singh, M.G.
1982-12-01
The possibilities were explored for enhancing parallelism in the simulation of systems described by algebraic equations, ordinary differential equations and partial differential equations. These techniques, using multiprocessors, were developed to speed up simulations, e.g. for nuclear accidents. Issues involved in their design included suitable approximations to bring the problem into a numerically manageable form and a numerical procedure to perform the computations necessary to solve the problem accurately. Parallel processing techniques used as simulation procedures, and a design of a simulation scheme and simulation procedure employing parallel computer facilities, were both considered.
The exotic conformal Galilei algebra and nonlinear partial differential equations
NASA Astrophysics Data System (ADS)
Cherniha, Roman; Henkel, Malte
2010-09-01
The conformal Galilei algebra (CGA) and the exotic conformal Galilei algebra (ECGA) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the CGA but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under CGA. It is further shown that there are systems of non-linear PDEs admitting ECGA with the realisation obtained very recently in [D. Martelli and Y. Tachikawa, arXiv:0903.5184v2 [hep-th] (2009)]. Moreover, wide classes of non-linear systems, invariant under two different 10-dimensional subalgebras of ECGA are explicitly constructed and an example with possible physical interpretation is presented.
Reconstruction of symmetric Dirac-Maxwell equations using nonassociative algebra
NASA Astrophysics Data System (ADS)
Kalauni, Pushpa; Barata, J. C. A.
2015-01-01
In the presence of sources, the usual Maxwell equations are neither symmetric nor invariant with respect to the duality transformation between electric and magnetic fields. Dirac proposed the existence of magnetic monopoles for symmetrizing the Maxwell equations. In the present work, we obtain the fully symmetric Dirac-Maxwell's equations (i.e. with electric and magnetic charges and currents) as a single equation by using 4 × 4 matrix presentation of fields and derivative operators. This matrix representation has been derived with the help of the algebraic properties of quaternions and octonions. Such description gives a compact representation of electric and magnetic counterparts of the field in a single equation.
He's iteration formulation for solving nonlinear algebraic equations
NASA Astrophysics Data System (ADS)
Qian, W.-X.; Ye, Y.-H.; Chen, J.; Mo, L.-F.
2008-02-01
Newton iteration method is sensitive to initial guess and its slope. To overcome the shortcoming, He's iteration method is used to solve nonlinear algebraic equations where Newton iteration method becomes invalid. Some examples are given, showing that the method is effective.
Numerical solution to systems of delay integrodifferential algebraic equations
NASA Astrophysics Data System (ADS)
Dmitriev, S. S.; Kuznetsov, E. B.
2008-03-01
The numerical solution of the initial value problem for a system of delay integrodifferential algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which is the arc length along the integral curve of the problem. The efficiency of the transformation is demonstrated using test examples.
Bounds for the eigenvalues of the continuous algebraic Riccati equation
NASA Astrophysics Data System (ADS)
Liu, Jianzhou; Zhang, Juan
2011-10-01
By using singular value decomposition and majorisation inequalities, we propose new upper and lower bounds for summations of eigenvalues (including the trace) of the solution of the continuous algebraic Riccati equation. These bounds improve and extend some of the previous results. Finally, we give corresponding numerical examples to illustrate the effectiveness of our results.
Algebraic Riccati equations in zero-sum differential games
NASA Technical Reports Server (NTRS)
Johnson, T. L.; Chao, A.
1974-01-01
The procedure for finding the closed-loop Nash equilibrium solution of two-player zero-sum linear time-invariant differential games with quadratic performance criteria and classical information pattern may be reduced in most cases to the solution of an algebraic Riccati equation. Based on the results obtained by Willems, necessary and sufficient conditions for existence of solutions to these equations are derived, and explicit conditions for a scalar example are given.
Numerical Solvers for Generalized Algebraic Riccati Equations
NASA Astrophysics Data System (ADS)
Ivanov, I. G.; Rusinova, R. I.
2009-10-01
We consider a new type nonlinear matrix equation. We investigate the existence a positive definite solution and two iterative methods for computing this solution. The first method is the classical Newton procedure and the second is a new Stein iteration. In this paper it is proved that a new Stein iteration has convergence properties to those of the Newton iteration.
Solving Differential Equations in R: Package deSolve
In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines appr...
A Realizable Reynolds Stress Algebraic Equation Model
NASA Technical Reports Server (NTRS)
Shih, Tsan-Hsing; Zhu, Jiang; Lumley, John L.
1993-01-01
The invariance theory in continuum mechanics is applied to analyze Reynolds stresses in high Reynolds number turbulent flows. The analysis leads to a turbulent constitutive relation that relates the Reynolds stresses to the mean velocity gradients in a more general form in which the classical isotropic eddy viscosity model is just the linear approximation of the general form. On the basis of realizability analysis, a set of model coefficients are obtained which are functions of the time scale ratios of the turbulence to the mean strain rate and the mean rotation rate. The coefficients will ensure the positivity of each component of the mean rotation rate. These coefficients will ensure the positivity of each component of the turbulent kinetic energy - realizability that most existing turbulence models fail to satisfy. Separated flows over backward-facing step configurations are taken as applications. The calculations are performed with a conservative finite-volume method. Grid-independent and numerical diffusion-free solutions are obtained by using differencing schemes of second-order accuracy on sufficiently fine grids. The calculated results are compared in detail with the experimental data for both mean and turbulent quantities. The comparison shows that the present proposal significantly improves the predictive capability of K-epsilon based two equation models. In addition, the proposed model is able to simulate rotational homogeneous shear flows with large rotation rates which all conventional eddy viscosity models fail to simulate.
NASA Astrophysics Data System (ADS)
Matsuno, Yoshimasa
2004-02-01
The multisoliton solution of the Benjamin-Ono equation is derived from the system of nonlinear algebraic equations. This finding is unexpected from the scheme of the inverse scattering transform method, which constructs the multisoliton solution through the system of linear algebraic equations. The anlaysis developed here is also applied to the rational multisoliton solution of the Kadomtsev-Petviashvili equation.
Numerical integration of systems of delay differential-algebraic equations
NASA Astrophysics Data System (ADS)
Kuznetsov, E. B.; Mikryukov, V. N.
2007-01-01
The numerical solution of the initial value problem for a system of delay differential-algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which ensures the best condition for the corresponding system of continuation equations. The best argument is the arc length along the integral curve of the problem. Algorithms and programs based on the continuous and discrete continuation methods are developed for the numerical integration of this problem. The efficiency of the suggested transformation is demonstrated using test examples.
ERIC Educational Resources Information Center
Novotna, Jarmila; Hoch, Maureen
2008-01-01
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…
Neural network architecture for solving the algebraic matrix Riccati equation
NASA Astrophysics Data System (ADS)
Ham, Fredric M.; Collins, Emmanuel G.
1996-03-01
This paper presents a neurocomputing approach for solving the algebraic matrix Riccati equation. This approach is able to utilize a good initial condition to reduce the computation time in comparison to standard methods for solving the Riccati equation. The repeated solutions of closely related Riccati equations appears in homotopy algorithms to solve certain problems in fixed-architecture control. Hence, the new approach has the potential to significantly speed-up these algorithms. It also has potential applications in adaptive control. The structured neural network architecture is trained using error backpropagation based on a steepest-descent learning rule. An example is given which illustrates the advantage of utilizing a good initial condition (i.e., initial setting of the neural network synaptic weight matrix) in the structured neural network.
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.
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…
NASA Astrophysics Data System (ADS)
Pogrebkov, A. K.
2016-06-01
We show that the non-Abelian Hirota difference equation is directly related to a commutator identity on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra lead to a linear difference equation. We develop a special dressing procedure that results in an integrable non-Abelian Hirota difference equation and propose two regular reduction procedures that lead to a set of known equations, Abelian or non-Abelian, and also to some new integrable equations.
Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme
NASA Astrophysics Data System (ADS)
Mazzocco, Marta
2016-09-01
In this paper we produce seven new algebras as confluences of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877–912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory.
Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme
NASA Astrophysics Data System (ADS)
Mazzocco, Marta
2016-09-01
In this paper we produce seven new algebras as confluences of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877-912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory.
Supporting Students' Understanding of Linear Equations with One Variable Using Algebra Tiles
ERIC Educational Resources Information Center
Saraswati, Sari; Putri, Ratu Ilma Indra; Somakim
2016-01-01
This research aimed to describe how algebra tiles can support students' understanding of linear equations with one variable. This article is a part of a larger research on learning design of linear equations with one variable using algebra tiles combined with balancing method. Therefore, it will merely discuss one activity focused on how students…
A multilayer recurrent neural network for solving continuous-time algebraic Riccati equations.
Wang, Jun; Wu, Guang
1998-07-01
A multilayer recurrent neural network is proposed for solving continuous-time algebraic matrix Riccati equations in real time. The proposed recurrent neural network consists of four bidirectionally connected layers. Each layer consists of an array of neurons. The proposed recurrent neural network is shown to be capable of solving algebraic Riccati equations and synthesizing linear-quadratic control systems in real time. Analytical results on stability of the recurrent neural network and solvability of algebraic Riccati equations by use of the recurrent neural network are discussed. The operating characteristics of the recurrent neural network are also demonstrated through three illustrative examples.
NASA Astrophysics Data System (ADS)
Volkov, V. V.; Erokhin, V. I.
2010-04-01
The properties of a mathematical programming problem that arises in finding a stable (in the sense of Tikhonov) solution to a system of linear algebraic equations with an approximately given augmented coefficient matrix are examined. Conditions are obtained that determine whether this problem can be reduced to the minimization of a smoothing functional or to the minimal matrix correction of the underlying system of linear algebraic equations. A method for constructing (exact or approximately given) model systems of linear algebraic equations with known Tikhonov solutions is described. Sharp lower bounds are derived for the maximal error in the solution of an approximately given system of linear algebraic equations under finite perturbations of its coefficient matrix. Numerical examples are given.
The coquaternion algebra and complex partial differential equations
NASA Astrophysics Data System (ADS)
Dimiev, Stancho; Konstantinov, Mihail; Todorov, Vladimir
2009-11-01
In this paper we consider the problem of differentiation of coquaternionic functions. Let us recall that coquaternions are elements of an associative non-commutative real algebra with zero divisor, introduced by James Cockle (1849) under the name of split-quaternions or coquaternions. Developing two type complex representations for Cockle algebra (complex and paracomplex ones) we present the problem in a non-commutative form of the δ¯-type holomorphy. We prove that corresponding differentiable coquaternionic functions, smooth and analytic, satisfy PDE of complex, and respectively of real variables. Applications for coquaternionic polynomials are sketched.
NASA Astrophysics Data System (ADS)
Hartwig, J. T.; Stokman, J. V.
2013-02-01
We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schrödinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations as functions of the momenta. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with delta-function interactions is indicated.
Hartwig, J. T.; Stokman, J. V.
2013-02-15
We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schroedinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schroedinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations as functions of the momenta. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with delta-function interactions is indicated.
Numerical solutions of linear differential-algebraic equation systems via Hartley series
NASA Astrophysics Data System (ADS)
Ünal, Emrah; Yalçın, Numan; ćelik, Ercan
2014-08-01
In this paper, Hartley series are presented first. Then, the operational matrix of integration together with the product and coefficient matrices are presented. They are used to transform linear differential equation systems to a set of linear algebraic equations. Finally, numerical examples are given.
Newton equation for canonical, Lie-algebraic, and quadratic deformation of classical space
Daszkiewicz, Marcin; Walczyk, Cezary J.
2008-05-15
The Newton equation describing particle motion in a constant external field force on canonical, Lie-algebraic, and quadratic space-time is investigated. We show that for canonical deformation of space-time the dynamical effects are absent, while in the case of Lie-algebraic noncommutativity, when spatial coordinates commute to the time variable, the additional acceleration of the particle is generated. We also indicate that in the case of spatial coordinates commuting in a Lie-algebraic way, as well as for quadratic deformation, there appear additional velocity and position-dependent forces.
Cognitive Load in Algebra: Element Interactivity in Solving Equations
ERIC Educational Resources Information Center
Ngu, Bing Hiong; Chung, Siu Fung; Yeung, Alexander Seeshing
2015-01-01
Central to equation solving is the maintenance of equivalence on both sides of the equation. However, when the process involves an interaction of multiple elements, solving an equation can impose a high cognitive load. The balance method requires operations on both sides of the equation, whereas the inverse method involves operations on one side…
Zhuk, Sergiy
2013-10-15
In this paper we present Kalman duality principle for a class of linear Differential-Algebraic Equations (DAE) with arbitrary index and time-varying coefficients. We apply it to an ill-posed minimax control problem with DAE constraint and derive a corresponding dual control problem. It turns out that the dual problem is ill-posed as well and so classical optimality conditions are not applicable in the general case. We construct a minimizing sequence u-circumflex{sub {epsilon}} for the dual problem applying Tikhonov method. Finally we represent u-circumflex{sub {epsilon}} in the feedback form using Riccati equation on a subspace which corresponds to the differential part of the DAE.
Solving the Langevin equation with stochastic algebraically correlated noise
NASA Astrophysics Data System (ADS)
Płoszajczak, M.; Srokowski, T.
1997-05-01
The long time tail in the velocity and force autocorrelation function has been found recently in molecular dynamics simulations of peripheral collisions of ions. Simulation of those slowly decaying correlations in the stochastic transport theory requires the development of new methods of generating stochastic force of arbitrarily long correlation times. In this paper we propose a Markovian process, the multidimensional kangaroo process, which permits the description of various algebraically correlated stochastic processes.
A numerical method for solving partial differential algebraic equations
NASA Astrophysics Data System (ADS)
Diep, Nguyen Khac; Chistyakov, V. F.
2013-06-01
Linear systems of partial differential equations with constant coefficient matrices are considered. The matrices multiplying the derivatives of the sought vector function are assumed to be singular. The structure of solutions to such systems is examined. The numerical solution of initialboundary value problems for such equations by applying implicit difference schemes is discussed.
Alternative Representations for Algebraic Problem Solving: When Are Graphs Better than Equations?
ERIC Educational Resources Information Center
Mielicki, Marta K.; Wiley, Jennifer
2016-01-01
Successful algebraic problem solving entails adaptability of solution methods using different representations. Prior research has suggested that students are more likely to prefer symbolic solution methods (equations) over graphical ones, even when graphical methods should be more efficient. However, this research has not tested how representation…
On a modification of minimal iteration methods for solving systems of linear algebraic equations
NASA Astrophysics Data System (ADS)
Yukhno, L. F.
2010-04-01
Modifications of certain minimal iteration methods for solving systems of linear algebraic equations are proposed and examined. The modified methods are shown to be superior to the original versions with respect to the round-off error accumulation, which makes them applicable to solving ill-conditioned problems. Numerical results demonstrating the efficiency of the proposed modifications are given.
Flipping an Algebra Classroom: Analyzing, Modeling, and Solving Systems of Linear Equations
ERIC Educational Resources Information Center
Kirvan, Rebecca; Rakes, Christopher R.; Zamora, Regie
2015-01-01
The present study investigated whether flipping an algebra classroom led to a stronger focus on conceptual understanding and improved learning of systems of linear equations for 54 seventh- and eighth-grade students using teacher journal data and district-mandated unit exam items. Multivariate analysis of covariance was used to compare scores on…
NASA Astrophysics Data System (ADS)
Shcheglova, A. A.
2009-09-01
Linear control differential algebraic equations are considered. The issue of minimum dimension of the control vector necessitated for complete controllability of the system on any closed interval from the domain of definition is investigated. The problem is analyzed in connection with the time invariant systems having regular matrix pencils and also systems with real-analytic or smooth coefficients, which possess some structural forms.
ERIC Educational Resources Information Center
Foley, Greg
2011-01-01
Continuous feed and bleed ultrafiltration, modeled with the gel polarization model for the limiting flux, is shown to provide a rich source of non-linear algebraic equations that can be readily solved using numerical and graphical techniques familiar to undergraduate students. We present a variety of numerical problems in the design, analysis, and…
ERIC Educational Resources Information Center
Gasyna, Zbigniew L.
2008-01-01
Computational experiment is proposed in which a linear algebra method is applied to the solution of the Schrodinger equation for a diatomic oscillator. Calculations of the vibration-rotation spectrum for the HCl molecule are presented and the results show excellent agreement with experimental data. (Contains 1 table and 1 figure.)
Sun, Leping
2016-01-01
This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2-delay differential algebraic equations. We obtain two sufficient conditions under which the methods are stable and asymptotically stable. At last, examples show that our methods are true. PMID:27441132
New solutions of reflection equation derived from type B BMW algebras
NASA Astrophysics Data System (ADS)
Häring-Oldenburg, Reinhard
1996-09-01
We use B-type knot theory to find new solutions of Sklyanin's reflection equation in a systematic way. This generalizes the well known Baxterization of Birman - Wenzl algebras and should describe integrable systems which are restricted to a half plane.
ERIC Educational Resources Information Center
Tonisson, Eno
2015-01-01
Sometimes Computer Algebra Systems (CAS) offer an answer that is somewhat different from the answer that is probably expected by the student or teacher. These (somewhat unexpected) answers could serve as a catalyst for rich mathematical discussion. In this study, over 120 equations from school mathematics were solved using 8 different CAS. Many…
Using Spreadsheets to Make Algebra More Accessible--Part 2: Solutions to Equations
ERIC Educational Resources Information Center
Green, John
2009-01-01
This article is the second in a series of two papers which suggest some practical, spreadsheet-based ideas for helping students to make appropriate connections between particular algebraic concepts. Solving equations has traditionally been taught as a pen-and-paper process. Spreadsheets, such as that of Excel, provide a contemporary, and powerful…
Translation of Algebraic Equations and Its Relation to Formal Operational Reasoning.
ERIC Educational Resources Information Center
Niaz, Mansoor
A large proportion of college students majoring in science are unable to translate even simple sentences into algebraic equations. Given the following sentence, "There are six times as many students (S) as professors (P) at this university," 37% of 150 freshmen engineering students in a study conducted in 1981 by Clement, Lockhead, and Monk wrote…
ERIC Educational Resources Information Center
Maat, Siti Mistima; Zakaria, Effandi
2011-01-01
Ordinary differential equations (ODEs) are one of the important topics in engineering mathematics that lead to the understanding of technical concepts among students. This study was conducted to explore the students' understanding of ODEs when they solve ODE questions using a traditional method as well as a computer algebraic system, particularly…
On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Tam, Honwah
2016-03-01
In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz–Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), and Hong Kong Research Grant Council under Grant No. HKBU202512, as well as the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016
On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Tam, Honwah
2016-03-01
In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz-Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), and Hong Kong Research Grant Council under Grant No. HKBU202512, as well as the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016
The change of the brain activation patterns as children learn algebra equation solving
Qin, Yulin; Carter, Cameron S.; Silk, Eli M.; Stenger, V. Andrew; Fissell, Kate; Goode, Adam; Anderson, John R.
2004-01-01
In a brain imaging study of children learning algebra, it is shown that the same regions are active in children solving equations as are active in experienced adults solving equations. As with adults, practice in symbol manipulation produces a reduced activation in prefrontal cortex area. However, unlike adults, practice seems also to produce a decrease in a parietal area that is holding an image of the equation. This finding suggests that adolescents' brain responses are more plastic and change more with practice. These results are integrated in a cognitive model that predicts both the behavioral and brain imaging results. PMID:15064407
A fifth order implicit method for the numerical solution of differential-algebraic equations
NASA Astrophysics Data System (ADS)
Skvortsov, L. M.
2015-06-01
An implicit two-step Runge-Kutta method of fifth order is proposed for the numerical solution of differential and differential-algebraic equations. The location of nodes in this method makes it possible to estimate the values of higher derivatives at the initial and terminal points of an integration step. Consequently, the proposed method can be regarded as a finite-difference analog of the Obrechkoff method. Numerical results, some of which are presented in this paper, show that our method preserves its order while solving stiff equations and equations of indices two and three. This is the main advantage of the proposed method as compared with the available ones.
Tracking children's mental states while solving algebra equations.
Anderson, John R; Betts, Shawn; Ferris, Jennifer L; Fincham, Jon M
2012-11-01
Behavioral and function magnetic resonance imagery (fMRI) data were combined to infer the mental states of students as they interacted with an intelligent tutoring system. Sixteen children interacted with a computer tutor for solving linear equations over a six-day period (days 0-5), with days 1 and 5 occurring in an fMRI scanner. Hidden Markov model algorithms combined a model of student behavior with multi-voxel imaging pattern data to predict the mental states of students. We separately assessed the algorithms' ability to predict which step in a problem-solving sequence was performed and whether the step was performed correctly. For day 1, the data patterns of other students were used to predict the mental states of a target student. These predictions were improved on day 5 by adding information about the target student's behavioral and imaging data from day 1. Successful tracking of mental states depended on using the combination of a behavioral model and multi-voxel pattern analysis, illustrating the effectiveness of an integrated approach to tracking the cognition of individuals in real time as they perform complex tasks.
NASA Astrophysics Data System (ADS)
Nordtvedt, K.
2015-11-01
A local system of bodies in General Relativity whose exterior metric field asymptotically approaches the Minkowski metric effaces any effects of the matter distribution exterior to its Minkowski boundary condition. To enforce to all orders this property of gravity which appears to hold in nature, a method using linear algebraic scaling equations is developed which generates by an iterative process an N-body Lagrangian expansion for gravity's motion-independent potentials which fulfills exterior effacement along with needed metric potential expansions. Then additional properties of gravity - interior effacement and Lorentz time dilation and spatial contraction - produce additional iterative, linear algebraic equations for obtaining the full non-linear and motion-dependent N-body gravity Lagrangian potentials as well.
Determining the multiplicity of a root of a nonlinear algebraic equation
NASA Astrophysics Data System (ADS)
Kalitkin, N. N.; Poshivailo, I. P.
2008-07-01
Newton’s method is most frequently used to find the roots of a nonlinear algebraic equation. The convergence domain of Newton’s method can be expanded by applying a generalization known as the continuous analogue of Newton’s method. For the classical and generalized Newton methods, an effective root-finding technique is proposed that simultaneously determines root multiplicity. Roots of high multiplicity (up to 10) can be calculated with a small error. The technique is illustrated using numerical examples.
NASA Astrophysics Data System (ADS)
Yukhno, L. F.
2008-12-01
For an overdetermined system of linear algebraic equations, systems obtained by introducing independent random errors into the original right-hand side are examined. Under certain assumptions on how these random variables are distributed, a practical stopping criterion is proposed for an iterative process that minimizes the sum of the squares of the residuals for the above systems. Numerical results demonstrating the efficiency of this criterion for some ill-conditioned problems are presented.
A Numerical Algorithm for Finding Solution of Cross-Coupled Algebraic Riccati Equations
NASA Astrophysics Data System (ADS)
Mukaidani, Hiroaki; Yamamoto, Seiji; Yamamoto, Toru
In this letter, a computational approach for solving cross-coupled algebraic Riccati equations (CAREs) is investigated. The main purpose of this letter is to propose a new algorithm that combines Newton's method with a gradient-based iterative (GI) algorithm for solving CAREs. In particular, it is noteworthy that both a quadratic convergence under an appropriate initial condition and reduction in dimensions for matrix computation are both achieved. A numerical example is provided to demonstrate the efficiency of this proposed algorithm.
Numerical solution of differential-algebraic equations using the spline collocation-variation method
NASA Astrophysics Data System (ADS)
Bulatov, M. V.; Rakhvalov, N. P.; Solovarova, L. S.
2013-03-01
Numerical methods for solving initial value problems for differential-algebraic equations are proposed. The approximate solution is represented as a continuous vector spline whose coefficients are found using the collocation conditions stated for a subgrid with the number of collocation points less than the degree of the spline and the minimality condition for the norm of this spline in the corresponding spaces. Numerical results for some model problems are presented.
Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras
NASA Astrophysics Data System (ADS)
Grahovski, Georgi G.; Mikhailov, Alexander V.
2013-12-01
Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initial-boundary problems are formulated.
A New Robust Solver for Saturated-Unsaturated Richards' Equation
NASA Astrophysics Data System (ADS)
Barajas-Solano, D. A.; Tartakovsky, D. M.
2012-12-01
We present a novel approach for the numerical integration of the saturated-unsaturated Richards' equation, a degenerate parabolic partial differential equation that models flow in porous media. The method is based on the mixed (pore pressure-water content) form of RE, written as a set of differential algebraic equations (DAEs) of index-1 for the fully saturated case and index-2 for the partially saturated case. A DAE-based approach allows us to overcome the numerical challenges posed by the degenerate nature of the Richards' equation. The resulting set of DAEs is solved using the stiffly-accurate, single-step, 3-stage implicit Runge-Kutta method Radau IIA, chosen for its favorable accuracy and stability properties, and its ease of implementation. For each time step a nonlinear system of equations on the intermediate Runge-Kutta states of the pore pressure is solved, written so to ensure that the next step pore pressure and water content correspond to one another correctly. The implementation of our approach compares favorably to state-of-the-art DAE-based solvers in both one- and two-dimensional simulations. These solvers use multi-step backward difference formulas together with a pressure-based form of Richards' equation. To the best of our knowledge, our method is the first instance of a successful DAE-based solver that uses the mixed form of Richards' equation. We consider this a promising line of research, with future work to be done on the use of globally convergent methods for the solution of the occurring nonlinear systems of equations.
NASA Astrophysics Data System (ADS)
Man, Yiu-Kwong
2010-10-01
In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle-Singer method for computing first integrals of differential equations. Some examples on using this approach are provided.
Middle school students' reading comprehension of mathematical texts and algebraic equations
NASA Astrophysics Data System (ADS)
Duru, Adem; Koklu, Onder
2011-06-01
In this study, middle school students' abilities to translate mathematical texts into algebraic representations and vice versa were investigated. In addition, students' difficulties in making such translations and the potential sources for these difficulties were also explored. Both qualitative and quantitative methods were used to collect data for this study: questionnaire and clinical interviews. The questionnaire consisted of two general types of items: (1) selected-response (multiple-choice) items for which the respondent selects from multiple options and (2) open-ended items for which the respondent constructs a response. In order to further investigate the students' strategies while they were translating the given mathematical texts to algebraic equations and vice versa, five randomly chosen (n = 5) students were interviewed. Data were collected in the 2007-2008 school year from 185 middle-school students in five teachers' classrooms in three different schools in the city of Adıyaman, Turkey. After the analysis of data, it was found that students who participated in this study had difficulties in translating the mathematical texts into algebraic equations by using symbols. It was also observed that these students had difficulties in translating the symbolic representations into mathematical texts because of their weak reading comprehension. In addition, finding of this research revealed that students' difficulties in translating the given mathematical texts into symbolic representations or vice versa come from different sources.
NASA Astrophysics Data System (ADS)
Latini, D.; Ragnisco, O.; Ballesteros, A.; Enciso, A.; Herranz, F. J.; Riglioni, D.
2016-01-01
In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. Here we will follow the same path to tackle the Darboux III system, another maximally superintegrable system, which can indeed be viewed as a natural deformation of the isotropic harmonic oscillator where the flat Euclidean space is again replaced by another space of non-constant curvature.
Addendum to ‘Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model’
NASA Astrophysics Data System (ADS)
Li, Zi-Min; Batchelor, Murray T.
2016-09-01
In our recent paper Li and Batchelor (2015 J. Phys. A: Math. Theor. 48 454005) we obtained exceptional points in the eigenspectrum of the generalized Rabi model in terms of a set of algebraic equations. We also gave a proof for the number of roots of the constraint polynomials defining these exceptional solutions as a function of the system parameters and discussed the number of crossing points in the eigenspectrum. This approach however, only covered a subset of all exceptional points in the eigenspectrum. In this addendum, we clarify the distinction between the exceptional parts of the eigenspectrum for this model and discuss the subset of exceptional points not determined in our paper.
Upper solution bounds of the continuous coupled algebraic Riccati matrix equation
NASA Astrophysics Data System (ADS)
Liu, Jianzhou; Zhang, Juan
2011-04-01
In this article, by using some matrix identities, we construct the equivalent form of the continuous coupled algebraic Riccati equation (CCARE). Further, with the aid of the eigenvalue inequalities of matrix's product, by solving the linear inequalities utilising the properties of M-matrix and its inverse matrix, new upper matrix bounds for the solutions of the CCARE are established, which improve and extend some of the recent results. Finally, a corresponding numerical example is proposed to illustrate the effectiveness of the derived results.
Data fitting in partial differential algebraic equations: some academic and industrial applications
NASA Astrophysics Data System (ADS)
Schittkowski, K.
2004-02-01
The paper introduces a numerical method to estimate parameters in systems of one-dimensional partial differential algebraic equations. Proceeding from given experimental data, i.e., observation times and measurements, the minimum least-squares distance of measured data from a fitting criterion is computed, which depends on the solution of the dynamical system. We present a typical black box approach that is easily implemented proceeding from some standard numerical analysis tools. Main emphasis of the paper is to present a couple of practical applications from industry and academia, to give an impression on the complexity of real-life systems of partial differential equations. The domains of application are pharmaceutics, geology, mechanical engineering, chemical engineering, food engineering, and electrical engineering.
NASA Technical Reports Server (NTRS)
Ito, Kazufumi
1987-01-01
The linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces is considered. The optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence pi sup N of finite dimensional approximations of the solution to ARE. A sufficient condition that shows pi sup N converges strongly to pi is obtained. Under this condition, a formula is derived which can be used to obtain a rate of convergence of pi sup N to pi. The results of the Galerkin approximation is demonstrated and applied for parabolic systems and the averaging approximation for hereditary differential systems.
Novel insights on the stabilising solution to the continuous-time algebraic Riccati equation
NASA Astrophysics Data System (ADS)
Rojas, A. J.
2014-11-01
In the present paper we present a closed-form solution, as a function of the closed-loop poles, for the continuous-time algebraic Riccati equations (CAREs) related to single-input single-output systems with non-repeated poles. The proposed solution trades the standard numerical algorithm approach for one based on a spectral factorisation argument, offering potential insight into any control technique based on a CARE and its solution. As an example, we present the equivalence of two fairly recent control over network results. Furthermore we apply the proposed result to the formula for the optimal regulator gain matrix k (or equivalently the Luenberger's observer gain l) and present an example. Finally, we conclude by discussing the possible extension of the proposed closed-form solution to the repeated eigenvalues case and to the case when the CARE is related to multiple-input multiple-output systems.
NASA Astrophysics Data System (ADS)
Fukuchi, Tsugio
2014-06-01
The finite difference method (FDM) based on Cartesian coordinate systems can be applied to numerical analyses over any complex domain. A complex domain is usually taken to mean that the geometry of an immersed body in a fluid is complex; here, it means simply an analytical domain of arbitrary configuration. In such an approach, we do not need to treat the outer and inner boundaries differently in numerical calculations; both are treated in the same way. Using a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. Discretization of the space differential in the FDM is usually derived using the Taylor series expansion; however, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences. In using the polynomial interpolations, we can numerically solve the Poisson equation freely over any complex domain. Only a particular type of partial differential equation, Poisson's equations, is treated; however, the arguments put forward have wider generality in numerical calculations using the FDM.
Numerical prediction of kinetic model for enzymatic hydrolysis of cellulose using DAE-QMOM approach
NASA Astrophysics Data System (ADS)
Jamil, N. M.; Wang, Q.
2016-06-01
Bioethanol production from lignocellulosic biomass consists of three fundamental processes; pre-treatment, enzymatic hydrolysis, and fermentation. In enzymatic hydrolysis phase, the enzymes break the cellulose chains into sugar in the form of cellobiose or glucose. A currently proposed kinetic model for enzymatic hydrolysis of cellulose that uses population balance equation (PBE) mechanism was studied. The complexity of the model due to integrodifferential equations makes it difficult to find the analytical solution. Therefore, we solved the full model of PBE numerically by using DAE-QMOM approach. The computation was carried out using MATLAB software. The numerical results were compared to the asymptotic solution developed in the author's previous paper and the results of Griggs et al. Besides confirming the findings were consistent with those references, some significant characteristics were also captured. The PBE model for enzymatic hydrolysis process can be solved using DAE-QMOM method. Also, an improved understanding of the physical insights of the model was achieved.
NASA Astrophysics Data System (ADS)
Muthuvalu, Mohana Sundaram; Aruchunan, Elayaraja; Koh, Wei Sin; Akhir, Mohd Kamalrulzaman Md; Sulaiman, Jumat; Karim, Samsul Ariffin Abdul
2014-07-01
In this paper, the application of the Accelerated Over-Relaxation (AOR) iterative method is extended to solve first order composite closed Newton-Cotes quadrature (1-CCNC) algebraic equations arising from second kind linear Fredholm integral equations. The formulation and implementation of the method are also discussed. In addition, numerical results by solving several test problems are included and compared with the conventional iterative methods.
Activities for Students: Biology as a Source for Algebra Equations--The Heart
ERIC Educational Resources Information Center
Horak, Virginia M.
2005-01-01
The high school course that integrated first year algebra with an introductory environmental biology/anatomy and physiology course, in order to solve algebra problems is discussed. Lessons and activities for the course were taken by identifying the areas where mathematics and biology content intervenes may help students understand biology concepts…
Balancing the Equation: Do Course Variations in Algebra 1 Provide Equal Student Outcomes?
ERIC Educational Resources Information Center
Kenfield, Danielle M.
2013-01-01
Historically, algebra has served as a gatekeeper that divides students into academic programs with varying opportunities to learn and controls access to higher education and career opportunities. Successful completion of Algebra 1 demonstrates mathematical proficiency and allows access to a sequential and progressive path of advanced study that…
NASA Astrophysics Data System (ADS)
Yukhno, L. F.
2007-12-01
The use of modifications of certain well-known methods of the conjugate direction type for solving systems of linear algebraic equations with rectangular matrices is examined. The modified methods are shown to be superior to the original versions with respect to the round-off accumulation; the advantage is especially large for ill-conditioned matrices. Examples are given of the efficient use of the modified methods for solving certain fairly large ill-conditioned problems.
NASA Astrophysics Data System (ADS)
Yukhno, L. F.
2007-11-01
A modification of certain well-known methods of the conjugate direction type is proposed and examined. The modified methods are more stable with respect to the accumulation of round-off errors. Moreover, these methods are applicable for solving ill-conditioned systems of linear algebraic equations that, in particular, arise as approximations of ill-posed problems. Numerical results illustrating the advantages of the proposed modification are presented.
NASA Astrophysics Data System (ADS)
Akbari, M. R.; Ganji, D. D.; Ahmadi, A. R.; Kachapi, Sayyid H. Hashemi
2014-03-01
In the current paper, a simplified model of Tower Cranes has been presented in order to investigate and analyze the nonlinear differential equation governing on the presented system in three different cases by Algebraic Method (AGM). Comparisons have been made between AGM and Numerical Solution, and these results have been indicated that this approach is very efficient and easy so it can be applied for other nonlinear equations. It is citable that there are some valuable advantages in this way of solving differential equations and also the answer of various sets of complicated differential equations can be achieved in this manner which in the other methods, so far, they have not had acceptable solutions. The simplification of the solution procedure in Algebraic Method and its application for solving a wide variety of differential equations not only in Vibrations but also in different fields of study such as fluid mechanics, chemical engineering, etc. make AGM be a powerful and useful role model for researchers in order to solve complicated nonlinear differential equations.
NASA Astrophysics Data System (ADS)
Ivanov, I. G.; Netov, N. C.; Bogdanova, B. C.
2015-10-01
This paper addresses the problem of solving a generalized algebraic Riccati equation with an indefinite sign of its quadratic term. We extend the approach introduced by Lanzon, Feng, Anderson and Rotkowitz (2008) for solving similar Riccati equations. We numerically investigate two types of iterative methods for computing the stabilizing solution. The first type of iterative methods constructs two matrix sequences, where the sum of them converges to the stabilizing solution. The second type of methods defines one matrix sequence which converges to the stabilizing solution. Computer realizations of the presented methods are numerically tested and compared on the test of family examples. Based on the experiments some conclusions are derived.
Bethe subalgebras in affine Birman-Murakami-Wenzl algebras and flat connections for q-KZ equations
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Kirillov, A. N.; Tarasov, V. O.
2016-05-01
Commutative sets of Jucys-Murphy elements for affine braid groups of {A}(1),{B}(1),{C}(1),{D}(1) types were defined. Construction of R-matrix representations of the affine braid group of type {C}(1) and its distinguished commutative subgroup generated by the {C}(1)-type Jucys-Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik-Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the {C}(1)-type Jucys-Murphy elements. We specify our general construction to the case of the Birman-Murakami-Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl-Cherednik elements {Y}\\prime {{s}} in the double affine Hecke algebra of type A. Dedicated to Professor Rodney Baxter on the occasion of his 75th Birthday.
Bethe subalgebras in affine Birman–Murakami–Wenzl algebras and flat connections for q-KZ equations
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Kirillov, A. N.; Tarasov, V. O.
2016-05-01
Commutative sets of Jucys–Murphy elements for affine braid groups of {A}(1),{B}(1),{C}(1),{D}(1) types were defined. Construction of R-matrix representations of the affine braid group of type {C}(1) and its distinguished commutative subgroup generated by the {C}(1)-type Jucys–Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik–Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the {C}(1)-type Jucys–Murphy elements. We specify our general construction to the case of the Birman–Murakami–Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl–Cherednik elements {Y}\\prime {{s}} in the double affine Hecke algebra of type A. Dedicated to Professor Rodney Baxter on the occasion of his 75th Birthday.
Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients
Kalchev, D
2012-04-02
This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-{rho}AMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and the computed solution used to evaluate some functional(s) of interest that then determine if the coefficient sample is acceptable or not. The MCMC process is hence computationally intensive and requires the solvers used to be efficient and fast. This fact that at every step of MCMC the resulting linear system changes, makes an already existing solver built for the old problem perhaps not as efficient for the problem corresponding to the new sampled coefficient. This motivates the main goal of our study, namely, to adapt an already existing solver to handle the problem (with changed coefficient) with the objective to achieve this goal to be faster and more efficient than building a completely new solver from scratch. Our approach utilizes the local element matrices (for the problem with changed coefficients) to build local problems associated with constructed by the method agglomerated elements (a set of subdomains that cover the given computational domain). We solve a generalized eigenproblem for each set in a subspace spanned by the previous local coarse space (used for the old solver) and a vector, component of the error, that the old solver cannot handle. A portion of the spectrum of these local eigen-problems (corresponding to eigenvalues close to zero) form the
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.
NASA Astrophysics Data System (ADS)
Liu, Jianzhou; Zhang, Juan
2011-08-01
In this article, applying the properties of M-matrix and non-negative matrix, utilising eigenvalue inequalities of matrix's sum and product, we firstly develop new upper and lower matrix bounds of the solution for discrete coupled algebraic Riccati equation (DCARE). Secondly, we discuss the solution existence uniqueness condition of the DCARE using the developed upper and lower matrix bounds and a fixed point theorem. Thirdly, a new fixed iterative algorithm of the solution for the DCARE is shown. Finally, the corresponding numerical examples are given to illustrate the effectiveness of the developed results.
NASA Astrophysics Data System (ADS)
Akhunov, R. R.; Gazizov, T. R.; Kuksenko, S. P.
2016-08-01
The mean time needed to solve a series of systems of linear algebraic equations (SLAEs) as a function of the number of SLAEs is investigated. It is proved that this function has an extremum point. An algorithm for adaptively determining the time when the preconditioner matrix should be recalculated when a series of SLAEs is solved is developed. A numerical experiment with multiply solving a series of SLAEs using the proposed algorithm for computing 100 capacitance matrices with two different structures—microstrip when its thickness varies and a modal filter as the gap between the conductors varies—is carried out. The speedups turned out to be close to the optimal ones.
Middle School Students' Reading Comprehension of Mathematical Texts and Algebraic Equations
ERIC Educational Resources Information Center
Duru, Adem; Koklu, Onder
2011-01-01
In this study, middle school students' abilities to translate mathematical texts into algebraic representations and vice versa were investigated. In addition, students' difficulties in making such translations and the potential sources for these difficulties were also explored. Both qualitative and quantitative methods were used to collect data…
Yu, Zhang; Zhang, Yufeng
2009-01-30
Three semi-direct sum Lie algebras are constructed, which is an efficient and new way to obtain discrete integrable couplings. As its applications, three discrete integrable couplings associated with the modified KdV lattice equation are worked out. The approach can be used to produce other discrete integrable couplings of the discrete hierarchies of solition equations. PMID:20119478
NASA Astrophysics Data System (ADS)
Levin, A. M.; Olshanetsky, M. A.; Zotov, A. V.
2016-08-01
We construct twisted Calogero-Moser systems with spins as Hitchin systems derived from the Higgs bundles over elliptic curves, where the transition operators are defined by arbitrary finite-order automorphisms of the underlying Lie algebras. We thus obtain a spin generalization of the twisted D'Hoker-Phong and Bordner-Corrigan-Sasaki-Takasaki systems. In addition, we construct the corresponding twisted classical dynamical r-matrices and the Knizhnik-Zamolodchikov-Bernard equations related to the automorphisms of Lie algebras.
Petzold, L.R.; Rosen, J.B.
1997-12-30
Differential-algebraic equations arise in a wide variety of engineering and scientific problems. Relatively little work has been done regarding sensitivity analysis and model reduction for this class of problems. Efficient methods for sensitivity analysis are required in model development and as an intermediate step in design optimization of engineering processes. Reduced order models are needed for modelling complex physical phenomena like turbulent reacting flows, where it is not feasible to use a fully-detailed model. The objective of this work has been to develop numerical methods and software for sensitivity analysis and model reduction of nonlinear differential-algebraic systems, including large-scale systems. In collaboration with Peter Brown and Alan Hindmarsh of LLNL, the authors developed an algorithm for finding consistent initial conditions for several widely occurring classes of differential-algebraic equations (DAEs). The new algorithm is much more robust than the previous algorithm. It is also very easy to use, having been designed to require almost no information about the differential equation, Jacobian matrix, etc. in addition to what is already needed to take the subsequent time steps. The new algorithm has been implemented in a version of the software for solution of large-scale DAEs, DASPK, which has been made available on the internet. The new methods and software have been used to solve a Tokamak edge plasma problem at LLNL which could not be solved with the previous methods and software because of difficulties in finding consistent initial conditions. The capability of finding consistent initial values is also needed for the sensitivity and optimization efforts described in this paper.
Generalized Flip-Flop Input Equations Based on a Four-Valued Boolean Algebra
NASA Technical Reports Server (NTRS)
Tucker, Jerry H.; Tapia, Moiez A.
1996-01-01
A procedure is developed for obtaining generalized flip-flop input equations, and a concise method is presented for representing these equations. The procedure is based on solving a four-valued characteristic equation of the flip-flop, and can encompass flip-flops that are too complex to approach intuitively. The technique is presented using Karnaugh maps, but could easily be implemented in software.
Algebraic Equations and Inequalities: Issues for Research and Teaching. Research Forum
ERIC Educational Resources Information Center
Bazzini, Luciana; Tsamir, Pessia
2004-01-01
The presentations address a variety of difficulties occurring in students' solutions of equations and inequalities, and suggest different reasons for these difficulties. When analyzing students' performances, [BB] and [TTT] mention students' tendencies to make irrelevant connections between equations and inequalities as a problematic phenomenon.…
NASA Astrophysics Data System (ADS)
Krasnikov, S. D.; Kuznetsov, E. B.
2016-09-01
Numerical continuation of solution through certain singular points of the curve of the set of solutions to a system of nonlinear algebraic or transcendental equations with a parameter is considered. Bifurcation points of codimension two and three are investigated. Algorithms and computer programs are developed that implement the procedure of discrete parametric continuation of the solution and find all branches at simple bifurcation points of codimension two and three. Corresponding theorems are proved, and each algorithm is rigorously justified. A novel algorithm for the estimation of errors of tangential vectors at simple bifurcation points of a finite codimension m is proposed. The operation of the computer programs is demonstrated by test examples, which allows one to estimate their efficiency and confirm the theoretical results.
ERIC Educational Resources Information Center
Samuel, Koji; Mulenga, H. M.; Angel, Mukuka
2016-01-01
This paper investigates the challenges faced by secondary school teachers and pupils in the teaching and learning of algebraic linear equations. The study involved 80 grade 11 pupils and 15 teachers of mathematics, drawn from 4 selected secondary schools in Mufulira district, Zambia in Central Africa. A descriptive survey method was employed to…
ERIC Educational Resources Information Center
Fuchs, Lynn S.; Zumeta, Rebecca O.; Schumacher, Robin Finelli; Powell, Sarah R.; Seethaler, Pamela M.; Hamlett, Carol L.; Fuchs, Douglas
2010-01-01
The purpose of this study was to assess the effects of schema-broadening instruction (SBI) on second graders' word-problem-solving skills and their ability to represent the structure of word problems using algebraic equations. Teachers (n = 18) were randomly assigned to conventional word-problem instruction or SBI word-problem instruction, which…
Profiles of Algebraic Competence
ERIC Educational Resources Information Center
Humberstone, J.; Reeve, R.A.
2008-01-01
The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map…
Computer subroutine ISUDS accurately solves large system of simultaneous linear algebraic equations
NASA Technical Reports Server (NTRS)
Collier, G.
1967-01-01
Computer program, an Iterative Scheme Using a Direct Solution, obtains double precision accuracy using a single-precision coefficient matrix. ISUDS solves a system of equations written in matrix form as AX equals B, where A is a square non-singular coefficient matrix, X is a vector, and B is a vector.
ERIC Educational Resources Information Center
Ngu, Bing Hiong; Yeung, Alexander Seeshing
2013-01-01
Text editing directs students' attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three…
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.
Algebra for Gifted Third Graders.
ERIC Educational Resources Information Center
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
Exponential integrators for the incompressible Navier-Stokes equations.
Newman, Christopher K.
2004-07-01
We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size. Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step. Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L{sup 2} inner product. We examine this L{sup 2} projection and an H{sup 1} projection induced by the H{sup 1} semi-inner product. The H
Numerical algebraic geometry and algebraic kinematics
NASA Astrophysics Data System (ADS)
Wampler, Charles W.; Sommese, Andrew J.
In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.
NASA Astrophysics Data System (ADS)
Shabana, Ahmed A.; Hussein, Bassam A.
2009-11-01
In this paper, a two-loop implicit sparse matrix numerical integration (TLISMNI) procedure for the solution of constrained rigid and flexible multibody system differential and algebraic equations is proposed. The proposed method ensures that the kinematic constraint equations are satisfied at the position, velocity and acceleration levels. In this method, a sparse Lagrangian augmented form of the equations of motion that ensures that the constraints are satisfied at the acceleration level is first used to solve for all the accelerations and Lagrange multipliers. The independent coordinates and velocities are then identified and integrated using HTT or Newmark formulas, expressed in this paper in terms of the independent accelerations only. The constraint equations at the position level are then used within an iterative Newton-Raphson procedure to determine the dependent coordinates. The dependent velocities are determined by solving a linear system of algebraic equations. In order to effectively exploit efficient sparse matrix techniques and have minimum storage requirements, a two-loop iterative method is proposed. Equally important, the proposed method avoids the use of numerical differentiation which is commonly associated with the use of implicit integration methods in multibody system algorithms. Numerical examples are presented in order to demonstrate the use of the new integration procedure.
Colored Quantum Algebra and Its Bethe State
NASA Astrophysics Data System (ADS)
Wang, Jin-Zheng; Jia, Xiao-Yu; Wang, Shi-Kun
2014-12-01
We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.
NASA Technical Reports Server (NTRS)
Ruge, J. W.; Stueben, K.
1987-01-01
The state of the art in algebraic multgrid (AMG) methods is discussed. The interaction between the relaxation process and the coarse grid correction necessary for proper behavior of the solution probes is discussed in detail. Sufficient conditions on relaxation and interpolation for the convergence of the V-cycle are given. The relaxation used in AMG, what smoothing means in an algebraic setting, and how it relates to the existing theory are considered. Some properties of the coarse grid operator are discussed, and results on the convergence of two-level and multilevel convergence are given. Details of an algorithm particularly studied for problems obtained by discretizing a single elliptic, second order partial differential equation are given. Results of experiments with such problems using both finite difference and finite element discretizations are presented.
NASA Astrophysics Data System (ADS)
Kipps, Mark R.
1994-03-01
The modeling of power systems has been primarily driven by the commercial power utility industry. These models usually involve the assumption that system bus voltage and frequency are constant. However, in applications such as shipboard power systems this infinite bus assumption is not valid. This thesis investigates the modeling of a synchronous generator and various loads in a modular fashion on a finite bus. The simulation presented allows the interconnection of multiple state-space models via a bus voltage model. The major difficulty encountered in building a model which computes bus voltage at each time step is that bus voltage is a function of current and current derivative terms. Bus voltage is also an input to the state equations which produce the current and current derivatives. This creates an algebraic loop which is a form of implicit differential equation. A routine has been developed by Linda Petzold of Lawrence Livermore Laboratory for solving these types of equations. The routine, called Differential Algebraic System Solver (DASSL), has been implemented in a pre-release version of the software Advanced Continuous Simulation Language (ACSL) and has been made available to the Naval Postgraduate School on a trial basis. An isolated power system is modeled using this software and the DASSL routine. The system response to several dynamic situations is studied and the results are presented.
Teaching Algebra without Algebra
ERIC Educational Resources Information Center
Kalman, Richard S.
2008-01-01
Algebra is, among other things, a shorthand way to express quantitative reasoning. This article illustrates ways for the classroom teacher to convert algebraic solutions to verbal problems into conversational solutions that can be understood by students in the lower grades. Three reasonably typical verbal problems that either appeared as or…
Rashid, Naim U.; Sun, Wei; Ibrahim, Joseph G.
2014-01-01
In DAE (DNA After Enrichment)-seq experiments, genomic regions related with certain biological processes are enriched/isolated by an assay and are then sequenced on a high-throughput sequencing platform to determine their genomic positions. Statistical analysis of DAE-seq data aims to detect genomic regions with significant aggregations of isolated DNA fragments (“enriched regions”) versus all the other regions (“background”). However, many confounding factors may influence DAE-seq signals. In addition, the signals in adjacent genomic regions may exhibit strong correlations, which invalidate the independence assumption employed by many existing methods. To mitigate these issues, we develop a novel Autoregressive Hidden Markov Model (AR-HMM) to account for covariates effects and violations of the independence assumption. We demonstrate that our AR-HMM leads to improved performance in identifying enriched regions in both simulated and real datasets, especially in those in epigenetic datasets with broader regions of DAE-seq signal enrichment. We also introduce a variable selection procedure in the context of the HMM/AR-HMM where the observations are not independent and the mean value of each state-specific emission distribution is modeled by some covariates. We study the theoretical properties of this variable selection procedure and demonstrate its efficacy in simulated and real DAE-seq data. In summary, we develop several practical approaches for DAE-seq data analysis that are also applicable to more general problems in statistics. PMID:24678134
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.
Second-Order Algebraic Theories
NASA Astrophysics Data System (ADS)
Fiore, Marcelo; Mahmoud, Ola
Fiore and Hur [10] recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding.
Chudnovsky, D. V.
1978-01-01
For systems of nonlinear equations having the form [Ln - (∂/∂t), Lm - (∂/∂y)] = 0 the class of meromorphic solutions obtained from the linear equations [Formula: see text] is presented. PMID:16592559
Shimizu, H; Okubo, M; Nakamoto, A; Enomoto, M; Kojima, N
2006-12-11
Intercalation of an organic photochromic molecule into layered magnetic systems may provide multifunctional properties such as photomagnetism. To build up a photosensitive multifunctional magnet, an organic-inorganic hybrid system coupled with a photochromic diarylethene anion, 2,2'-dimethyl-3,3'-(perfluorocyclopentene-1,2-diyl)bis(benzo[b]thiophene-6-sulfonate) (DAE), and cobalt LDHs (layered double hydroxides), Co4(OH)7(DAE)0.5.3H2O, was synthesized by the anion exchange reaction between Co2(OH)3(CH3COO).H2O and DAE. In the dark and under UV-irradiated (313 nm) conditions, Co4(OH)7(DAE)0.5.3H2O with open and closed forms of DAE were obtained, respectively. The magnetic susceptibility measurements elucidated ferromagnetic intra- and interlayer interactions and Curie temperatures of TC = 9 and 20 K for cobalt LDHs with the open and closed forms of DAE, respectively. The enhancement of the Curie temperature from 9 to 20 K by substitution of the open form of DAE with the closed form of DAE as an intercalated molecule is attributed to the delocalization of the pi-electrons in the closed form of DAE, which enhances the interlayer magnetic interaction. The enhancement of the interlayer magnetic interaction induced by the delocalization of pi-electrons in intercalated molecules is strongly supported by the fact that the Curie temperature (26.0 K) of cobalt LDHs with (E,E)-2,4-hexadienedioate having a conjugated pi-electron system is enormously higher than that (7.0 K) of the cobalt LDHs with hexanedioate. By UV irradiation at 313 nm, Co4(OH)7(DAE)0.5.3H2O shows the photoisomerization of DAE from the open form to the closed one in the solid state, which leads to the enhancement of Curie temperature.
NASA Astrophysics Data System (ADS)
Hiley, B. J.
In this chapter, we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. By using this algebra, we show that it contains both the Weyl-von Neumann and the Moyal quantum algebras. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave-function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.
NASA Astrophysics Data System (ADS)
Smirnov, Andrey
2010-08-01
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl( N;?)-case is discussed.
NASA Astrophysics Data System (ADS)
Smirnov, Andrey
2010-08-01
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.
Covariant deformed oscillator algebras
NASA Technical Reports Server (NTRS)
Quesne, Christiane
1995-01-01
The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.
Cornish-Bowden, A
1976-01-01
Methods of deriving rate equations that rely on repetition of terms for identification of redundant or invalid terms give incorrect results if used with mechanisms in which some rate constants appear more than once. PMID:999635
Characteristic Numbers of Matrix Lie Algebras
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Fan, En-Gui
2008-04-01
A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.
ERIC Educational Resources Information Center
Pavelle, Richard; And Others
1981-01-01
Describes the nature and use of computer algebra and its applications to various physical sciences. Includes diagrams illustrating, among others, a computer algebra system and flow chart of operation of the Euclidean algorithm. (SK)
Modeling the nonlinear hysteretic response in DAE experiments of Berea sandstone: A case-study
NASA Astrophysics Data System (ADS)
Pecorari, Claudio
2015-03-01
Dynamic acousto-elasticity (DAE) allows probing the instantaneous state of a material while the latter slowly and periodically is changed by an external, dynamic source. In DAE investigations of geo-materials, hysteresis of the material's modulus defect displays intriguing features which have not yet been interpreted in terms of any specific mechanism occurring at atomic or mesoscale. Here, experimental results on dry Berea sandstone, which is the rock type best investigated by means of a DAE technique, are analyzed in terms of three rheological models providing simplified representations of mechanisms involving dislocations interacting with point defects which are distributed along the dislocations' core or glide planes, and microcracks with finite stiffness in compression. Constitutive relations linking macroscopic strain and stress are derived. From the latter, the modulus defect associated to each mechanism is recovered. These models are employed to construct a composite one which is capable of reproducing several of the main features observed in the experimental data. The limitations of the present approach and, possibly, of the current implementation of DAE are discussed.
Modeling the nonlinear hysteretic response in DAE experiments of Berea sandstone: A case-study
Pecorari, Claudio
2015-03-31
Dynamic acousto-elasticity (DAE) allows probing the instantaneous state of a material while the latter slowly and periodically is changed by an external, dynamic source. In DAE investigations of geo-materials, hysteresis of the material's modulus defect displays intriguing features which have not yet been interpreted in terms of any specific mechanism occurring at atomic or mesoscale. Here, experimental results on dry Berea sandstone, which is the rock type best investigated by means of a DAE technique, are analyzed in terms of three rheological models providing simplified representations of mechanisms involving dislocations interacting with point defects which are distributed along the dislocations' core or glide planes, and microcracks with finite stiffness in compression. Constitutive relations linking macroscopic strain and stress are derived. From the latter, the modulus defect associated to each mechanism is recovered. These models are employed to construct a composite one which is capable of reproducing several of the main features observed in the experimental data. The limitations of the present approach and, possibly, of the current implementation of DAE are discussed.
Algebraic integrability: a survey.
Vanhaecke, Pol
2008-03-28
We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems. PMID:17588863
ERIC Educational Resources Information Center
Buerman, Margaret
2007-01-01
Finding real-world examples for middle school algebra classes can be difficult but not impossible. As we strive to accomplish teaching our students how to solve and graph equations, we neglect to teach the big ideas of algebra. One of those big ideas is functions. This article gives three examples of functions that are found in Arches National…
Adaptive Algebraic Multigrid Methods
Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J
2004-04-09
Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.
NASA Astrophysics Data System (ADS)
Carlen, E. A.; Carvalho, M. C.; Orlandi, E.
1999-06-01
This is the first of two papers devoted to the study of a nonlocal evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider subcritical temperatures, for which there are two local equilibria, and begin the proof of a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibria; i.e., the fronts. We shall show in the second paper that an initial perturbation v 0 of a front that is sufficiently small in L 2 norm, and sufficiently localized that ∫ x 2 v 0( x)2 dx<∞, yields a solution that relaxes to another front, selected by a conservation law, in the L 1 norm at an algebraic rate that we explicitly estimate. There we also obtain rates for the relaxation in the L 2 norm and the rate of decrease of the excess free energy. Here we prove a number of estimates essential for this result. Moreover, the estimates proved here suffice to establish the main result in an important special case.
Invertible linear transformations and the Lie algebras
NASA Astrophysics Data System (ADS)
Zhang, Yufeng; Tam, Honwah; Guo, Fukui
2008-07-01
With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity.
ERIC Educational Resources Information Center
Benjamin, Carl; And Others
Presented are student performance objectives, a student progress chart, and assignment sheets with objective and diagnostic measures for the stated performance objectives in College Algebra II. Topics covered include: differencing and complements; real numbers; factoring; fractions; linear equations; exponents and radicals; complex numbers,…
ERIC Educational Resources Information Center
Schaufele, Christopher; Zumoff, Nancy
Earth Algebra is an entry level college algebra course that incorporates the spirit of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics at the college level. The context of the course places mathematics at the center of one of the major current concerns of the world. Through…
ERIC Educational Resources Information Center
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
Generalized Galilean algebras and Newtonian gravity
NASA Astrophysics Data System (ADS)
González, N.; Rubio, G.; Salgado, P.; Salgado, S.
2016-04-01
The non-relativistic versions of the generalized Poincaré algebras and generalized AdS-Lorentz algebras are obtained. These non-relativistic algebras are called, generalized Galilean algebras of type I and type II and denoted by GBn and GLn respectively. Using a generalized Inönü-Wigner contraction procedure we find that the generalized Galilean algebras of type I can be obtained from the generalized Galilean algebras type II. The S-expansion procedure allows us to find the GB5 algebra from the Newton Hooke algebra with central extension. The procedure developed in Ref. [1] allows us to show that the nonrelativistic limit of the five dimensional Einstein-Chern-Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.
Formal scattering theory by an algebraic approach
NASA Astrophysics Data System (ADS)
Alhassid, Y.; Levine, R. D.
1985-02-01
Formal scattering theory is recast in a Lie-algebraic form. The central result is an algebraic Lippmann-Schwinger equation for the wave operator from which an algebraic form of the Born series (containing only linked terms) is obtained. When a finite Lie algebra is sufficient, The Mo/ller wave operator, on the energy shell, can be solved for explicitly as an element of the corresponding group. The method is illustrated for the separable potential whose relevant algebra is found to be U(1,1).
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°).
Elementary Algebra Connections to Precalculus
ERIC Educational Resources Information Center
Lopez-Boada, Roberto; Daire, Sandra Arguelles
2013-01-01
This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…
Rota-Baxter operators on Witt and Virasoro algebras
NASA Astrophysics Data System (ADS)
Gao, Xu; Liu, Ming; Bai, Chengming; Jing, Naihuan
2016-10-01
The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.
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…
Static friction, differential algebraic systems and numerical stability
NASA Astrophysics Data System (ADS)
Chen, Jian; Schinner, Alexander; Matuttis, Hans-Georg
We show how Differential Algebraic Systems (Ordinary Differential Equations with algebraic constraints) in mechanics are affected by stability issues and we implement Lubich's projection method to reduce the error to practically zero. Then, we explain how the "numerically exact" implementation for static friction by Differential Algebraic Systems can be stabilized. We conclude by comparing the corresponding steps in the "Contact mechanics" introduced by Moreau.
Exploring Algebraic Misconceptions with Technology
ERIC Educational Resources Information Center
Sakow, Matthew; Karaman, Ruveyda
2015-01-01
Many students struggle with algebra, from simplifying expressions to solving systems of equations. Students also have misconceptions about the meaning of variables. In response to the question "Can x + y + z ever equal x + p + z?" during a student interview, the student claimed, "Never . . . because p has to have a different value…
Kinds of Knowledge in Algebra.
ERIC Educational Resources Information Center
Lewis, Clayton
Solving equations in elementary algebra requires knowledge of the permitted operations, and knowledge of what operation to use at a given point in the solution process. While just these kinds of knowledge would be adequate for an ideal solver, human solvers appear to need and use other kinds of knowledge. First, many errors seem to indicate that…
Math for All Learners: Algebra.
ERIC Educational Resources Information Center
Meader, Pam; Storer, Judy
This book consists of a series of activities aimed at providing a problem solving, hands-on approach so that students can experience concepts in algebra. Topics include ratio and proportion, patterns and formulas, integers, polynomials, linear equations, graphs, and probability. The activities come in the form of reproducible blackline masters…
Algebraic Systems and Pushdown Automata
NASA Astrophysics Data System (ADS)
Petre, Ion; Salomaa, Arto
We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.
ERIC Educational Resources Information Center
Blakley, G. R.
1982-01-01
Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)
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
NASA Astrophysics Data System (ADS)
Vaninsky, Alexander
2011-04-01
This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos - satisfying an axiom sin2 + cos2 = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two different interpretations of the TF are discussed with many others potentially possible. The main objective of this article is to introduce a broader view of trigonometry that can serve as motivation for mathematics students and teachers to study and teach abstract algebraic structures.
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.
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
Titration Calculations with Computer Algebra Software
ERIC Educational Resources Information Center
Lachance, Russ; Biaglow, Andrew
2012-01-01
This article examines the symbolic algebraic solution of the titration equations for a diprotic acid, as obtained using "Mathematica," "Maple," and "Mathcad." The equilibrium and conservation equations are solved symbolically by the programs to eliminate the approximations that normally would be performed by the student. Of the three programs,…
Algebraic operator approach to gas kinetic models
NASA Astrophysics Data System (ADS)
Il'ichov, L. V.
1997-02-01
Some general properties of the linear Boltzmann kinetic equation are used to present it in the form ∂ tϕ = - Â†Âϕ with the operators ÂandÂ† possessing some nontrivial algebraic properties. When applied to the Keilson-Storer kinetic model, this method gives an example of quantum ( q-deformed) Lie algebra. This approach provides also a natural generalization of the “kangaroo model”.
Quantization of Algebraic Reduction
Sniatycki, Jeodrzej
2007-11-14
For a Poisson algebra obtained by algebraic reduction of symmetries of a quantizable system we develop an analogue of geometric quantization based on the quantization structure of the original system.
Shifted genus expanded W ∞ algebra and shifted Hurwitz numbers
NASA Astrophysics Data System (ADS)
Zheng, Quan
2016-05-01
We construct the shifted genus expanded W ∞ algebra, which is isomorphic to the central subalgebra A ∞ of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra Λ* defined by Okounkov and Olshanskii. As an application, we get some differential equations for the generating functions of the shifted Hurwitz numbers; thus, we can express the generating functions in terms of the shifted genus expanded cut-and-join operators.
Learning Algebra in a Computer Algebra Environment
ERIC Educational Resources Information Center
Drijvers, Paul
2004-01-01
This article summarises a doctoral thesis entitled "Learning algebra in a computer algebra environment, design research on the understanding of the concept of parameter" (Drijvers, 2003). It describes the research questions, the theoretical framework, the methodology and the results of the study. The focus of the study is on the understanding of…
NASA Technical Reports Server (NTRS)
Iachello, Franco
1995-01-01
An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.
Orientation in operator algebras
Alfsen, Erik M.; Shultz, Frederic W.
1998-01-01
A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics. PMID:9618457
Developing Thinking in Algebra
ERIC Educational Resources Information Center
Mason, John; Graham, Alan; Johnson-Wilder, Sue
2005-01-01
This book is for people with an interest in algebra whether as a learner, or as a teacher, or perhaps as both. It is concerned with the "big ideas" of algebra and what it is to understand the process of thinking algebraically. The book has been structured according to a number of pedagogic principles that are exposed and discussed along the way,…
Connecting Arithmetic to Algebra
ERIC Educational Resources Information Center
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
NASA Astrophysics Data System (ADS)
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.
Computer algebra and transport theory.
Warsa, J. S.
2004-01-01
Modern symbolic algebra computer software augments and complements more traditional approaches to transport theory applications in several ways. The first area is in the development and enhancement of numerical solution methods for solving the Boltzmann transport equation. Typically, special purpose computer codes are designed and written to solve specific transport problems in particular ways. Different aspects of the code are often written from scratch and the pitfalls of developing complex computer codes are numerous and well known. Software such as MAPLE and MATLAB can be used to prototype, analyze, verify and determine the suitability of numerical solution methods before a full-scale transport application is written. Once it is written, the relevant pieces of the full-scale code can be verified using the same tools I that were developed for prototyping. Another area is in the analysis of numerical solution methods or the calculation of theoretical results that might otherwise be difficult or intractable. Algebraic manipulations are done easily and without error and the software also provides a framework for any additional numerical calculations that might be needed to complete the analysis. We will discuss several applications in which we have extensively used MAPLE and MATLAB in our work. All of them involve numerical solutions of the S{sub N} transport equation. These applications encompass both of the two main areas in which we have found computer algebra software essential.
Computer algebra and operators
NASA Technical Reports Server (NTRS)
Fateman, Richard; Grossman, Robert
1989-01-01
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
Discrete Minimal Surface Algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Hoppe, Jens
2010-05-01
We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
2003-06-03
The ALGEBRA II program allows the user to manipulate data from a finite element analysis before it is plotted by evaluating algebraic expressions. The equation variables are dependent on the input database variable names. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database which can be read by plot programs. Code is written in a portable form as possible. Fortran code is written in ANSI Standard FORTRAN-77. Machine-specific routines are limited in number and are grouped together to minimize the time required to adapt them to a new system. SEACAS codes has been ported to several Unix systems.
2003-06-03
The ALGEBRA II program allows the user to manipulate data from a finite element analysis before it is plotted by evaluating algebraic expressions. The equation variables are dependent on the input database variable names. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database which can be read by plot programs. Code is written in a portable form as possible. Fortran codemore » is written in ANSI Standard FORTRAN-77. Machine-specific routines are limited in number and are grouped together to minimize the time required to adapt them to a new system. SEACAS codes has been ported to several Unix systems.« less
ERIC Educational Resources Information Center
Lin, Cheng-Yao; Kuo, Yu-Chun; Ko, Yi-Yin
2015-01-01
The purpose of this study was to investigate elementary pre-service teachers' content knowledge in algebra (Linear Equation, Quadratic Equation, Functions, System Equations and Polynomials) as well as their technological pedagogical content knowledge (TPACK) in teaching algebra. Participants were 79 undergraduate pre-service teachers who were…
Octonic Massive Field Equations
NASA Astrophysics Data System (ADS)
Demir, Süleyman; Kekeç, Seray
2016-07-01
In the present paper we propose the octonic form of massive field equations based on the analogy with electromagnetism and linear gravity. Using the advantages of octon algebra the Maxwell-Dirac-Proca equations have been reformulated in compact and elegant way. The energy-momentum relations for massive field are discussed.
Multidimensional integrable systems and deformations of Lie algebra homomorphisms
Dunajski, Maciej; Grant, James D. E.; Strachan, Ian A. B.
2007-09-15
We use deformations of Lie algebra homomorphisms to construct deformations of dispersionless integrable systems arising as symmetry reductions of anti-self-dual Yang-Mills equations with a gauge group Diff(S{sup 1})
A Richer Understanding of Algebra
ERIC Educational Resources Information Center
Foy, Michelle
2008-01-01
Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number, pre-algebra, and algebra…
Invariant algebraic surfaces for a virus dynamics
NASA Astrophysics Data System (ADS)
Valls, Claudia
2015-08-01
In this paper, we provide a complete classification of the invariant algebraic surfaces and of the rational first integrals for a well-known virus system. In the proofs, we use the weight-homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.
Transformation of time dependence to linear algebra
NASA Astrophysics Data System (ADS)
Menšík, Miroslav
2005-10-01
Reduced density matrix and memory function in the Nakajima-Zwanzig equation are expanded in properly chosen basis of special functions. This trick completely transforms time dependence to linear algebra. Then, the master equation for memory function is constructed and expanded in the same basis functions. For the model of a simple harmonic oscillator it is shown that this trick introduces infinite partial summation of the memory function in the system-bath interaction.
Connecting Algebra and Chemistry.
ERIC Educational Resources Information Center
O'Connor, Sean
2003-01-01
Correlates high school chemistry curriculum with high school algebra curriculum and makes the case for an integrated approach to mathematics and science instruction. Focuses on process integration. (DDR)
Noncommutative Pfaffians associated with the orthogonal algebra
Artamonov, Dmitrii V; Golubeva, Valentina A
2012-12-31
Commutators of Pfaffians associated with the orthogonal algebra are found in skew-symmetric and root realizations of o{sub N}. A generating function of Pfaffians is proved to satisfy the reflection equation. A relation between Pfaffians in skew-symmetric and root realizations of o{sub N} is established. Using these results we construct an integrable equation of Knizhnik-Zamolodchikov type using the Capelli central elements in U(o{sub N}), which are sums of squares of the considered Pfaffians. A classical limit of the obtained Knizhnik-Zamolodchikov type equation turns out to be a very specific system of equations of isomonodromic deformations. Bibliography: 18 titles.
Difference equations and some of their solutions
NASA Astrophysics Data System (ADS)
Atakishiyev, Natig M.
1996-04-01
Some methods for solving difference equations are discussed. As particular examples we consider in detail the difference equations corresponding to the Kravchuk functions, q-harmonic oscillator wavefunctions, and the Clebsch-Gordan coefficients for the quantum algebra suq(2).
Successfully Transitioning to Linear Equations
ERIC Educational Resources Information Center
Colton, Connie; Smith, Wendy M.
2014-01-01
The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this…
Algebraic Reasoning through Patterns
ERIC Educational Resources Information Center
Rivera, F. D.; Becker, Joanne Rossi
2009-01-01
This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…
ERIC Educational Resources Information Center
Merlin, Ethan M.
2013-01-01
This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…
Unified derivation of exact solutions to the relativistic Coulomb problem: Lie algebraic approach
NASA Astrophysics Data System (ADS)
Panahi, H.; Baradaran, M.; Savadi, A.
2015-10-01
Exact algebraic solutions of the D-dimensional Dirac and Klein-Gordon equations for the Coulomb potential are obtained in a unified treatment. It is shown that two cases are reducible to the same basic equation, which can be solved exactly. Using the Lie algebraic approach, the general exact solutions of the problem are obtained within the framework of representation theory of the sl(2) Lie algebra.
Computational algebraic geometry of epidemic models
NASA Astrophysics Data System (ADS)
Rodríguez Vega, Martín.
2014-06-01
Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.
Lie algebra extensions of current algebras on S3
NASA Astrophysics Data System (ADS)
Kori, Tosiaki; Imai, Yuto
2015-06-01
An affine Kac-Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac-Moody algebras give it for two-dimensional conformal field theory.
Leibniz algebras associated with representations of filiform Lie algebras
NASA Astrophysics Data System (ADS)
Ayupov, Sh. A.; Camacho, L. M.; Khudoyberdiyev, A. Kh.; Omirov, B. A.
2015-12-01
In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra nn,1. We introduce a Fock module for the algebra nn,1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra nn,1 with condition that the ideal I is a Fock nn,1-module, where I is the ideal generated by squares of elements from L. We also consider Leibniz algebras with corresponding Lie algebra nn,1 and such that the action I ×nn,1 → I gives rise to a minimal faithful representation of nn,1. The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4.
Maxwell Equations and the Redundant Gauge Degree of Freedom
ERIC Educational Resources Information Center
Wong, Chun Wa
2009-01-01
On transformation to the Fourier space (k,[omega]), the partial differential Maxwell equations simplify to algebraic equations, and the Helmholtz theorem of vector calculus reduces to vector algebraic projections. Maxwell equations and their solutions can then be separated readily into longitudinal and transverse components relative to the…
Operator product expansion algebra
Holland, Jan; Hollands, Stefan
2013-07-15
We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean φ{sup 4}-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of hep-th/1105.3375, that the 3-point OPE,
Algebraic Semantics for Narrative
ERIC Educational Resources Information Center
Kahn, E.
1974-01-01
This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)
Ten-Year-Old Students Solving Linear Equations
ERIC Educational Resources Information Center
Brizuela, Barbara; Schliemann, Analucia
2004-01-01
In this article, the authors seek to re-conceptualize the perspective regarding students' difficulties with algebra. While acknowledging that students "do" have difficulties when learning algebra, they also argue that the generally espoused criteria for algebra as the ability to work with the syntactical rules for solving equations is…
Aprepro - Algebraic Preprocessor
2005-08-01
Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.
Geometric Algebra for Physicists
NASA Astrophysics Data System (ADS)
Doran, Chris; Lasenby, Anthony
2007-11-01
Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.
Description of DASSL: a differential/algebraic system solver
Petzold, L.R.
1982-09-01
This paper describes a new code DASSL, for the numerical solution of implicit systems of differential/algebraic equations. These equations are written in the form F(t,y,y') = 0, and they can include systems which are substantially more complex than standard form ODE systems y' = f(t,y). Differential/algebraic equations occur in several diverse applications in the physical world. We outline the algorithms and strategies used in DASSL, and explain some of the features of the code. In addition, we outline briefly what needs to be done to solve a problem using DASSL.
Parallel Multigrid Equation Solver
2001-09-07
Prometheus is a fully parallel multigrid equation solver for matrices that arise in unstructured grid finite element applications. It includes a geometric and an algebraic multigrid method and has solved problems of up to 76 mullion degrees of feedom, problems in linear elasticity on the ASCI blue pacific and ASCI red machines.
Algebra 2r, Mathematics (Experimental): 5216.23.
ERIC Educational Resources Information Center
Ellis, June
The third in a series of six guidebooks on minimum course content for second-year algebra, this booklet covers relations, functions, and solving and graphing linear equations, linear inequalities, systems of equations, and systems of inequalities. Overall course goals are specified, a course outline is provided, performance objectives are listed,…
Symmetric linear systems - An application of algebraic systems theory
NASA Technical Reports Server (NTRS)
Hazewinkel, M.; Martin, C.
1983-01-01
Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.
Some remarks on representations of Yang-Mills algebras
NASA Astrophysics Data System (ADS)
Herscovich, Estanislao
2015-01-01
In this article, we present some new properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra 𝔶𝔪(n) on n generators, for n ≥ 2m. We derive from this that any semisimple Lie algebra and even any affine Kac-Moody algebra is a quotient of 𝔶𝔪(n) for n ≥ 4. Combining this with previous results on representations of Yang-Mills algebras given in [Herscovich and Solotar, Ann. Math. 173(2), 1043-1080 (2011)], one may obtain solutions to the Yang-Mills equations by differential operators acting on sections of twisted vector bundles on the affine space of dimension n ≥ 4 associated to representations of any semisimple Lie algebra. We also show that this quotient property does not hold for n = 3, since any morphism of Lie algebras from 𝔶𝔪(3) to 𝔰𝔩(2, k) has in fact solvable image.
The algebraic criteria for the stability of control systems
NASA Technical Reports Server (NTRS)
Cremer, H.; Effertz, F. H.
1986-01-01
This paper critically examines the standard algebraic criteria for the stability of linear control systems and their proofs, reveals important previously unnoticed connections, and presents new representations. Algebraic stability criteria have also acquired significance for stability studies of non-linear differential equation systems by the Krylov-Bogoljubov-Magnus Method, and allow realization conditions to be determined for classes of broken rational functions as frequency characteristics of electrical network.
MODEL IDENTIFICATION AND COMPUTER ALGEBRA
Bollen, Kenneth A.; Bauldry, Shawn
2011-01-01
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods. PMID:21769158
MODEL IDENTIFICATION AND COMPUTER ALGEBRA.
Bollen, Kenneth A; Bauldry, Shawn
2010-10-01
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods.
Numerical linear algebra for reconstruction inverse problems
NASA Astrophysics Data System (ADS)
Nachaoui, Abdeljalil
2004-01-01
Our goal in this paper is to discuss various issues we have encountered in trying to find and implement efficient solvers for a boundary integral equation (BIE) formulation of an iterative method for solving a reconstruction problem. We survey some methods from numerical linear algebra, which are relevant for the solution of this class of inverse problems. We motivate the use of our constructing algorithm, discuss its implementation and mention the use of preconditioned Krylov methods.
ERIC Educational Resources Information Center
Actuarial Foundation, 2013
2013-01-01
"Solving the Unknown with Algebra" is a new math program aligned with the National Council of Teachers of Mathematics (NCTM) standards and designed to help students practice pre-algebra skills including using formulas, solving for unknowns, and manipulating equations. Developed by The Actuarial Foundation with Scholastic, this program provides…
Solving stochastic epidemiological models using computer algebra
NASA Astrophysics Data System (ADS)
Hincapie, Doracelly; Ospina, Juan
2011-06-01
Mathematical modeling in Epidemiology is an important tool to understand the ways under which the diseases are transmitted and controlled. The mathematical modeling can be implemented via deterministic or stochastic models. Deterministic models are based on short systems of non-linear ordinary differential equations and the stochastic models are based on very large systems of linear differential equations. Deterministic models admit complete, rigorous and automatic analysis of stability both local and global from which is possible to derive the algebraic expressions for the basic reproductive number and the corresponding epidemic thresholds using computer algebra software. Stochastic models are more difficult to treat and the analysis of their properties requires complicated considerations in statistical mathematics. In this work we propose to use computer algebra software with the aim to solve epidemic stochastic models such as the SIR model and the carrier-borne model. Specifically we use Maple to solve these stochastic models in the case of small groups and we obtain results that do not appear in standard textbooks or in the books updated on stochastic models in epidemiology. From our results we derive expressions which coincide with those obtained in the classical texts using advanced procedures in mathematical statistics. Our algorithms can be extended for other stochastic models in epidemiology and this shows the power of computer algebra software not only for analysis of deterministic models but also for the analysis of stochastic models. We also perform numerical simulations with our algebraic results and we made estimations for the basic parameters as the basic reproductive rate and the stochastic threshold theorem. We claim that our algorithms and results are important tools to control the diseases in a globalized world.
Local Algebras of Differential Operators
NASA Astrophysics Data System (ADS)
Church, P. T.; Timourian, J. G.
2002-05-01
There is an increasing literature devoted to the study of boundary value problems using singularity theory. The resulting differential operators are typically Fredholm with index 0, defined on infinite-dimensional spaces, and they have often led to folds, cusps, and even higher-order Morin singularities. In this paper we develop some of the local algebras of germs of such differential Fredholm operators, extending the theory of the finite-dimensional case. We apply this work to nonlinear elliptic boundary value problems: in particular, we make further progress on a question proposed and initially studied by Ruf [1999, J. Differential Equations 151, 111-133]. We also make comments on several problems raised by others.
Spatial Operator Algebra for multibody system dynamics
NASA Technical Reports Server (NTRS)
Rodriguez, G.; Jain, A.; Kreutz-Delgado, K.
1992-01-01
The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.
Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction
ERIC Educational Resources Information Center
Wasserman, Nicholas H.
2016-01-01
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…
Development of abstract mathematical reasoning: the case of algebra
Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja
2014-01-01
Algebra typically represents the students’ first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students’ ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16–17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15–16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students’ transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition. PMID:25228874
Development of abstract mathematical reasoning: the case of algebra.
Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja
2014-01-01
Algebra typically represents the students' first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students' ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16-17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15-16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students' transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition.
Computer Program For Linear Algebra
NASA Technical Reports Server (NTRS)
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
Revisiting Newtonian and Non-Newtonian Fluid Mechanics Using Computer Algebra
ERIC Educational Resources Information Center
Knight, D. G.
2006-01-01
This article illustrates how a computer algebra system, such as Maple[R], can assist in the study of theoretical fluid mechanics, for both Newtonian and non-Newtonian fluids. The continuity equation, the stress equations of motion, the Navier-Stokes equations, and various constitutive equations are treated, using a full, but straightforward,…
Pseudo Algebraically Closed Extensions
NASA Astrophysics Data System (ADS)
Bary-Soroker, Lior
2009-07-01
This PhD deals with the notion of pseudo algebraically closed (PAC) extensions of fields. It develops a group-theoretic machinery, based on a generalization of embedding problems, to study these extensions. Perhaps the main result is that although there are many PAC extensions, the Galois closure of a proper PAC extension is separably closed. The dissertation also contains the following subjects. The group theoretical counterpart of pseudo algebraically closed extensions, the so-called projective pairs. Applications to seemingly unrelated subjects, e.g., an analog of Dirichlet's theorem about primes in arithmetic progression for polynomial rings in one variable over infinite fields.
Renormalization group flows and continual Lie algebras
NASA Astrophysics Data System (ADS)
Bakas, Ioannis
2003-08-01
We study the renormalization group flows of two-dimensional metrics in sigma models using the one-loop beta functions, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates. In this algebraic setting, the logarithm of the world-sheet length scale, t, is interpreted as Dynkin parameter on the root system of a novel continual Lie algebra, denoted by Script G(d/dt;1), with anti-symmetric Cartan kernel K(t,t') = delta'(t-t'); as such, it coincides with the Cartan matrix of the superalgebra sl(N|N+1) in the large-N limit. The resulting Toda field equation is a non-linear generalization of the heat equation, which is integrable in target space and shares the same dissipative properties in time, t. We provide the general solution of the renormalization group flows in terms of free fields, via Bäcklund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultra-violet limit by gluing together two copies of Witten's two-dimensional black hole in the asymptotic region. We also provide some new solutions that describe the renormalization group flow of negatively curved spaces in different patches, which look like a cane in the infra-red region. Finally, we revisit the transition of a flat cone C/Zn to the plane, as another special solution, and note that tachyon condensation in closed string theory exhibits a hidden relation to the infinite dimensional algebra Script G(d/dt;1) in the regime of gravity. Its exponential growth holds the key for the construction of conserved currents and their systematic interpretation in string theory, but they still remain unknown.
Assessing Elementary Algebra with STACK
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2007-01-01
This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…
Classical mechanics on noncommutative space with Lie-algebraic structure
Miao Yangang; Wang Xudong; Yu Shaojie
2011-08-15
Highlights: > Suggest a useful method to look for new Lie-algebraic noncommutative spaces. > Find out two new Lie-algebraic noncommutative spaces. > Derive Newton and Hamilton equations that present unimaginable extra forces. > Analyse the source of unimaginable extra forces from space noncummutativity. > Provide various intriguing classical trajectories. - Abstract: We investigate the kinetics of a nonrelativistic particle interacting with a constant external force on a Lie-algebraic noncommutative space. The structure constants of a Lie algebra, also called noncommutative parameters, are constrained in general due to some algebraic properties, such as the antisymmetry and Jacobi identity. Through solving the constraint equations the structure constants satisfy, we obtain two new sorts of algebraic structures, each of which corresponds to one type of noncommutative spaces. Based on such types of noncommutative spaces as the starting point, we analyze the classical motion of the particle interacting with a constant external force by means of the Hamiltonian formalism on a Poisson manifold. Our results not only include that of a recent work as our special cases, but also provide new trajectories of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable tx-dot-,(xx-dot)-, and (xx-double dot)-dependence besides with the usual t-, x-, and x-dot-dependence, originating from a variety of noncommutativity between different spatial coordinates and between spatial coordinates and momenta as well, deform greatly the particle's ordinary trajectories we are quite familiar with on the Euclidean (commutative) space.
Computer algebra methods in the study of nonlinear differential systems
NASA Astrophysics Data System (ADS)
Irtegov, V. D.; Titorenko, T. N.
2013-06-01
Some issues concerning computer algebra methods as applied to the qualitative analysis of differential equations with first integrals are discussed. The problems of finding stationary sets and analyzing their stability and bifurcations are considered. Special attention is given to algorithms for finding and analyzing peculiar stationary sets. It is shown that computer algebra tools, combined with qualitative analysis methods for differential equations, make it possible not only to enhance the computational efficiency of classical algorithms, but also to implement new approaches to the solution of well-known problems and, in this way, to obtain new results.
The impact of fraction magnitude knowledge on algebra performance and learning.
Booth, Julie L; Newton, Kristie J; Twiss-Garrity, Laura K
2014-02-01
Knowledge of fractions is thought to be crucial for success with algebra, but empirical evidence supporting this conjecture is just beginning to emerge. In the current study, Algebra 1 students completed magnitude estimation tasks on three scales (0-1 [fractions], 0-1,000,000, and 0-62,571) just before beginning their unit on equation solving. Results indicated that fraction magnitude knowledge, and not whole number knowledge, was especially related to students' pretest knowledge of equation solving and encoding of equation features. Pretest fraction knowledge was also predictive of students' improvement in equation solving and equation encoding skills. Students' placement of unit fractions (e.g., those with a numerator of 1) was not especially useful for predicting algebra performance and learning in this population. Placement of non-unit fractions was more predictive, suggesting that proportional reasoning skills might be an important link between fraction knowledge and learning algebra.
Thinking Visually about Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2015-01-01
Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…
Computer Algebra versus Manipulation
ERIC Educational Resources Information Center
Zand, Hossein; Crowe, David
2004-01-01
In the UK there is increasing concern about the lack of skill in algebraic manipulation that is evident in students entering mathematics courses at university level. In this note we discuss how the computer can be used to ameliorate some of the problems. We take as an example the calculations needed in three dimensional vector analysis in polar…
ERIC Educational Resources Information Center
Glick, David
1995-01-01
Presents a technique that helps students concentrate more on the science and less on the mechanics of algebra while dealing with introductory physics formulas. Allows the teacher to do complex problems at a lower level and not be too concerned about the mathematical abilities of the students. (JRH)
ERIC Educational Resources Information Center
Nwabueze, Kenneth K.
2004-01-01
The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the…
A spatial operator algebra for manipulator modeling and control
NASA Technical Reports Server (NTRS)
Rodriguez, G.; Kreutz, K.; Milman, M.
1988-01-01
A powerful new spatial operator algebra for modeling, control, and trajectory design of manipulators is discussed along with its implementation in the Ada programming language. Applications of this algebra to robotics include an operator representation of the manipulator Jacobian matrix; the robot dynamical equations formulated in terms of the spatial algebra, showing the complete equivalence between the recursive Newton-Euler formulations to robot dynamics; the operator factorization and inversion of the manipulator mass matrix which immediately results in O(N) recursive forward dynamics algorithms; the joint accelerations of a manipulator due to a tip contact force; the recursive computation of the equivalent mass matrix as seen at the tip of a manipulator; and recursive forward dynamics of a closed chain system. Finally, additional applications and current research involving the use of the spatial operator algebra are discussed in general terms.
Constitutive relations in optics in terms of geometric algebra
NASA Astrophysics Data System (ADS)
Dargys, A.
2015-11-01
To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p-form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium.
Symbolic and algebraic computation
Not Available
1990-01-01
This book contains subjects under the following headings: Foundations of symbolic computation; Computational logics; systems Algorithms on polynormal; Integrative and differential equations; and Differential equations.
ERIC Educational Resources Information Center
Usman, Ahmed Ibrahim
2015-01-01
Knowledge and understanding of mathematical operations serves as a pre-reequisite for the successful translation of algebraic word problems. This study explored pre-service teachers' ability to recognize mathematical operations as well as use of those capabilities in constructing algebraic expressions, equations, and their solutions. The outcome…
ERIC Educational Resources Information Center
McNeil, Nicole M.; Rittle-Johnson, Bethany; Hattikudur, Shanta; Petersen, Lori A.
2010-01-01
This study examined if solving arithmetic problems hinders undergraduates' accuracy on algebra problems. The hypothesis was that solving arithmetic problems would hinder accuracy because it activates an operational view of equations, even in educated adults who have years of experience with algebra. In three experiments, undergraduates (N = 184)…
Algebraic Bethe ansatz for the Temperley-Lieb spin-1 chain
NASA Astrophysics Data System (ADS)
Nepomechie, Rafael I.; Pimenta, Rodrigo A.
2016-09-01
We use the algebraic Bethe ansatz to obtain the eigenvalues and eigenvectors of the spin-1 Temperley-Lieb open quantum chain with "free" boundary conditions. We exploit the associated reflection algebra in order to prove the off-shell equation satisfied by the Bethe vectors.
Capitalizing on Basic Brain Processes in Developmental Algebra--Part 3
ERIC Educational Resources Information Center
Laughbaum, Edward D.
2011-01-01
In Part Three, the author reviews the basic ideas presented in Parts One and Two while arguing why the traditional equation-solving developmental algebra curricula is not a good choice for implementing neural response strategies presented in the first two parts. He continues by showing that the developmental algebra student audience is simply…
Algebraic connectivity and graph robustness.
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T.
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
LAPACK: Linear algebra software for supercomputers
Bischof, C.H.
1991-01-01
This paper presents an overview of the LAPACK library, a portable, public-domain library to solve the most common linear algebra problems. This library provides a uniformly designed set of subroutines for solving systems of simultaneous linear equations, least-squares problems, and eigenvalue problems for dense and banded matrices. We elaborate on the design methodologies incorporated to make the LAPACK codes efficient on today's high-performance architectures. In particular, we discuss the use of block algorithms and the reliance on the Basic Linear Algebra Subprograms. We present performance results that show the suitability of the LAPACK approach for vector uniprocessors and shared-memory multiprocessors. We also address some issues that have to be dealt with in tuning LAPACK for specific architectures. Lastly, we present results that show that the LAPACK software can be adapted with little effort to distributed-memory environments, and we discuss future efforts resulting from this project. 31 refs., 10 figs., 2 tabs.
NASA Astrophysics Data System (ADS)
Dankova, T. S.; Rosensteel, G.
1998-10-01
Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of coadjoint orbits for the groups U(n) and O(2n). The general theory of mean fields is formulated for an arbitrary Lie algebra L of fermion operators. The moment map provides the correspondence between the Hilbert space of microscopic wave functions and the dual space L^* of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The mean field theories for the Elliott SU(3) and symplectic Sp(3,R) algebras are constructed explicitly in the coadjoint orbit framework.
ERIC Educational Resources Information Center
Beigie, Darin
2014-01-01
Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…
Lee, Jaehoon; Wilczek, Frank
2013-11-27
Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumps and an anisotropy in one part.
NASA Technical Reports Server (NTRS)
Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.
1999-01-01
This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras
Yu, Zhang; Zhang, Yufeng
2009-01-01
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings. PMID:20084092
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras.
Yu, Zhang; Zhang, Yufeng
2009-01-15
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra smu(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra smu(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras smu(6) and E is used to directly construct integrable couplings.
On the cohomology of Leibniz conformal algebras
NASA Astrophysics Data System (ADS)
Zhang, Jiao
2015-04-01
We construct a new cohomology complex of Leibniz conformal algebras with coefficients in a representation instead of a module. The low-dimensional cohomology groups of this complex are computed. Meanwhile, we construct a Leibniz algebra from a Leibniz conformal algebra and prove that the category of Leibniz conformal algebras is equivalent to the category of equivalence classes of formal distribution Leibniz algebras.
Assessing Algebraic Solving Ability: A Theoretical Framework
ERIC Educational Resources Information Center
Lian, Lim Hooi; Yew, Wun Thiam
2012-01-01
Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…
Some remarks on unilateral matrix equations
Cerchiai, Bianca L.; Zumino, Bruno
2001-02-01
We briefly review the results of our paper LBNL-46775: We study certain solutions of left-unilateral matrix equations. These are algebraic equations where the coefficients and the unknown are square matrices of the same order, or, more abstractly, elements of an associative, but possibly noncommutative algebra, and all coefficients are on the left. Recently such equations have appeared in a discussion of generalized Born-Infeld theories. In particular, two equations, their perturbative solutions and the relation between them are studied, applying a unified approach based on the generalized Bezout theorem for matrix polynomials.
Computer Algebra Systems and Theorems on Real Roots of Polynomials
ERIC Educational Resources Information Center
Aidoo, Anthony Y.; Manthey, Joseph L.; Ward, Kim Y.
2010-01-01
A computer algebra system is used to derive a theorem on the existence of roots of a quadratic equation on any bounded real interval. This is extended to a cubic polynomial. We discuss how students could be led to derive and prove these theorems. (Contains 1 figure.)
First Course in Algebra, Student's Text, Part II, Unit 10.
ERIC Educational Resources Information Center
Allen, Frank B.; And Others
Unit 10 in the SMSG's secondary school mathematics series is a student text covering the following topics in Algebra I: factors and exponents, radicals, polynomial and rational expressions, truth sets of open sentences, graphs of open sentences in two variables, systems of equations and inequalities, quadratic polynomials, and functions. (DT)
Writing to Promote and Assess Conceptual Understanding in College Algebra
ERIC Educational Resources Information Center
Gay, A. Susan; Peterson, Ingrid
2014-01-01
Concept-focused quiz questions required College Algebra students to write about their understanding. The questions can be viewed in three broad categories: a focus on sense-making, a focus on describing a mathematical object such as a graph or an equation, and a focus on understanding vocabulary. Student responses from 10 classes were analyzed.…
Fixing Ganache: Another Real-Life Use for Algebra
ERIC Educational Resources Information Center
Kalman, Adam M.
2011-01-01
This article presents a real-world application of proportional reasoning and equation solving. The author describes how students adjust ingredient amounts in a recipe for chocolate ganache. Using this real-world scenario provided students an opportunity to solve a difficult and nonstandard algebra problem, a lot of practice with fractions, a…
Geometric and Algebraic Approaches in the Concept of Complex Numbers
ERIC Educational Resources Information Center
Panaoura, A.; Elia, I.; Gagatsis, A.; Giatilis, G.-P.
2006-01-01
This study explores pupils' performance and processes in tasks involving equations and inequalities of complex numbers requiring conversions from a geometric representation to an algebraic representation and conversions in the reverse direction, and also in complex numbers problem solving. Data were collected from 95 pupils of the final grade from…
Learning Activity Package, Algebra 124, LAPs 46-55.
ERIC Educational Resources Information Center
Holland, Bill
A series of 10 teacher-prepared Learning Activity Packages (LAPs) in advanced algebra and trigonometry, these units cover absolute value, inequalities, exponents, radicals, and complex numbers; functions; higher degree equations and the derivative; the trigonometric functions; graphs and applications of the trigonometric functions; sequences and…
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s. PMID:26806075
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s.
Algebraic dichotomies with an application to the stability of Riemann solutions of conservation laws
NASA Astrophysics Data System (ADS)
Lin, Xiao-Biao
Recently, there has been some interest on the stability of waves where the functions involved grow or decay at an algebraic rate |. In this paper we define the so-called algebraic dichotomy that may aid in treating such problems. We discuss the basic properties of the algebraic dichotomy, methods of detecting it, and calculating the power of the weight function. We present several examples: (1) The Bessel equation. (2) The n-degree Fisher type equation. (3) Hyperbolic conservation laws in similarity coordinates. (4) A system of conservation laws with a Dafermos type viscous regularization. We show that the linearized system generates an analytic semigroup in the space of algebraic decay functions. This example motivates our work on algebraic dichotomies.
Learning Activity Package, Algebra 93-94, LAPs 12-22.
ERIC Educational Resources Information Center
Evans, Diane
A set of 11 teacher-prepared Learning Activity Packages (LAPs) in beginning algebra, these units cover sets, properties of operations, operations over real numbers, open expressions, solution sets of equations and inequalities, equations and inequalities with two variables, solution sets of equations with two variables, exponents, factoring and…
ERIC Educational Resources Information Center
Alibali, Martha W.; Kao, Yvonne S.; Brown, Alayna N.; Nathan, Mitchell J.; Stephens, Ana C.
2009-01-01
This study investigated middle school students' conceptual understanding of algebraic equations. Participants in the study--257 sixth- and seventh-grade students--were asked to solve one set of algebraic equations and to generate story problems corresponding with another set of equations. Structural aspects of the equations, including the number…
Multifractal vector fields and stochastic Clifford algebra
NASA Astrophysics Data System (ADS)
Schertzer, Daniel; Tchiguirinskaia, Ioulia
2015-12-01
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.
Multifractal vector fields and stochastic Clifford algebra
Schertzer, Daniel Tchiguirinskaia, Ioulia
2015-12-15
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.
Multifractal vector fields and stochastic Clifford algebra.
Schertzer, Daniel; Tchiguirinskaia, Ioulia
2015-12-01
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality. PMID:26723166
Multifractal vector fields and stochastic Clifford algebra.
Schertzer, Daniel; Tchiguirinskaia, Ioulia
2015-12-01
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.
2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW*-algebras
NASA Astrophysics Data System (ADS)
Ayupov, Shavkat; Kudaybergenov, Karimbergen
2016-03-01
The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M 2n (A), n ≥ 2, is a derivation. We apply this result to AW*-algebras and show that any 2-local derivation on an arbitrary AW*-algebra is a derivation.
Structured adaptive grid generation using algebraic methods
NASA Technical Reports Server (NTRS)
Yang, Jiann-Cherng; Soni, Bharat K.; Roger, R. P.; Chan, Stephen C.
1993-01-01
The accuracy of the numerical algorithm depends not only on the formal order of approximation but also on the distribution of grid points in the computational domain. Grid adaptation is a procedure which allows optimal grid redistribution as the solution progresses. It offers the prospect of accurate flow field simulations without the use of an excessively timely, computationally expensive, grid. Grid adaptive schemes are divided into two basic categories: differential and algebraic. The differential method is based on a variational approach where a function which contains a measure of grid smoothness, orthogonality and volume variation is minimized by using a variational principle. This approach provided a solid mathematical basis for the adaptive method, but the Euler-Lagrange equations must be solved in addition to the original governing equations. On the other hand, the algebraic method requires much less computational effort, but the grid may not be smooth. The algebraic techniques are based on devising an algorithm where the grid movement is governed by estimates of the local error in the numerical solution. This is achieved by requiring the points in the large error regions to attract other points and points in the low error region to repel other points. The development of a fast, efficient, and robust algebraic adaptive algorithm for structured flow simulation applications is presented. This development is accomplished in a three step process. The first step is to define an adaptive weighting mesh (distribution mesh) on the basis of the equidistribution law applied to the flow field solution. The second, and probably the most crucial step, is to redistribute grid points in the computational domain according to the aforementioned weighting mesh. The third and the last step is to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentration
Plethystic algebras and vector symmetric functions.
Rota, G C; Stein, J A
1994-01-01
An isomorphism is established between the plethystic Hopf algebra Pleth(Super[L]) and the algebra of vector symmetric functions. The Hall inner product of symmetric function theory is extended to the Hopf algebra Pleth(Super[L]). PMID:11607504
Algebra and Algebraic Thinking in School Math: 70th YB
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, 2008
2008-01-01
Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…
Abstract Algebra to Secondary School Algebra: Building Bridges
ERIC Educational Resources Information Center
Christy, Donna; Sparks, Rebecca
2015-01-01
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
Handheld Computer Algebra Systems in the Pre-Algebra Classroom
ERIC Educational Resources Information Center
Gantz, Linda Ann Galofaro
2010-01-01
This mixed method analysis sought to investigate several aspects of student learning in pre-algebra through the use of computer algebra systems (CAS) as opposed to non-CAS learning. This research was broken into two main parts, one which compared results from both the experimental group (instruction using CAS, N = 18) and the control group…
Statecharts Via Process Algebra
NASA Technical Reports Server (NTRS)
Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance
1999-01-01
Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumpsmore » and an anisotropy in one part.« less
Double Precision Differential/Algebraic Sensitivity Analysis Code
1995-06-02
DDASAC solves nonlinear initial-value problems involving stiff implicit systems of ordinary differential and algebraic equations. Purely algebraic nonlinear systems can also be solved, given an initial guess within the region of attraction of a solution. Options include automatic reconciliation of inconsistent initial states and derivatives, automatic initial step selection, direct concurrent parametric sensitivity analysis, and stopping at a prescribed value of any user-defined functional of the current solution vector. Local error control (in the max-normmore » or the 2-norm) is provided for the state vector and can include the sensitivities on request.« less
Linear Algebra and Image Processing
ERIC Educational Resources Information Center
Allali, Mohamed
2010-01-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)
Linear algebra and image processing
NASA Astrophysics Data System (ADS)
Allali, Mohamed
2010-09-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty.
A Programmed Course in Algebra.
ERIC Educational Resources Information Center
Mewborn, Ancel C.; Hively, Wells II
This programed textbook consists of short sections of text interspersed with questions designed to aid the student in understanding the material. The course is designed to increase the student's understanding of some of the basic ideas of algebra. Some general experience and manipulative skill with respect to high school algebra is assumed.…
ERIC Educational Resources Information Center
1997
Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…
Gamow functionals on operator algebras
NASA Astrophysics Data System (ADS)
Castagnino, M.; Gadella, M.; Betán, R. Id; Laura, R.
2001-11-01
We obtain the precise form of two Gamow functionals representing the exponentially decaying part of a quantum resonance and its mirror image that grows exponentially, as a linear, positive and continuous functional on an algebra containing observables. These functionals do not admit normalization and, with an appropriate choice of the algebra, are time reversal of each other.
Online Algebraic Tools for Teaching
ERIC Educational Resources Information Center
Kurz, Terri L.
2011-01-01
Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)
Patterns to Develop Algebraic Reasoning
ERIC Educational Resources Information Center
Stump, Sheryl L.
2011-01-01
What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…
ERIC Educational Resources Information Center
Instructional Objectives Exchange, Los Angeles, CA.
A complete set of behavioral objectives for first-year algebra taught in any of grades 8 through 12 is presented. Three to six sample test items and answers are provided for each objective. Objectives were determined by surveying the most used secondary school algebra textbooks. Fourteen major categories are included: (1) whole numbers--operations…
Elementary maps on nest algebras
NASA Astrophysics Data System (ADS)
Li, Pengtong
2006-08-01
Let , be algebras and let , be maps. An elementary map of is an ordered pair (M,M*) such that for all , . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.
Condensing Algebra for Technical Mathematics.
ERIC Educational Resources Information Center
Greenfield, Donald R.
Twenty Algebra-Packets (A-PAKS) were developed by the investigator for technical education students at the community college level. Each packet contained a statement of rationale, learning objectives, performance activities, performance test, and performance test answer key. The A-PAKS condensed the usual sixteen weeks of algebra into a six-week…
Thermodynamics. [algebraic structure
NASA Technical Reports Server (NTRS)
Zeleznik, F. J.
1976-01-01
The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory's theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not present.
Partial Differential Algebraic Sensitivity Analysis Code
1995-05-15
PDASAC solves stiff, nonlinear initial-boundary-value in a timelike dimension t and a space dimension x. Plane, circular cylindrical or spherical boundaries can be handled. Mixed-order systems of partial differential and algebraic equations can be analyzed with members of order or 0 or 1 in t, 0,1 or 2 in x. Parametric sensitivities of the calculated states are compted simultaneously on request, via the Jacobian of the state equations. Initial and boundary conditions are efficiently reconciled.more » Local error control (in the max-norm or the 2-norm) is provided for the state vector and can include the parametric sensitivites if desired.« less
Numerical stability in problems of linear algebra.
NASA Technical Reports Server (NTRS)
Babuska, I.
1972-01-01
Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.
Method of lines solution of Richards` equation
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.
Abstract numeric relations and the visual structure of algebra.
Landy, David; Brookes, David; Smout, Ryan
2014-09-01
Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition, it has often been assumed that skilled users of these formalisms treat situations in terms of semantic properties encoded in an abstract syntax that governs the use of notation without particular regard to the details of the physical structure of the equation itself (Anderson, 2005; Hegarty, Mayer, & Monk, 1995). We explore how the notational structure of verbal descriptions or algebraic equations (e.g., the spatial proximity of certain words or the visual alignment of numbers and symbols in an equation) plays a role in the process of interpreting or constructing symbolic equations. We propose in particular that construction processes involve an alignment of notational structures across representation systems, biasing reasoners toward the selection of formal notations that maintain the visuospatial structure of source representations. For example, in the statement "There are 5 elephants for every 3 rhinoceroses," the spatial proximity of 5 and elephants and 3 and rhinoceroses will bias reasoners to write the incorrect expression 5E = 3R, because that expression maintains the spatial relationships encoded in the source representation. In 3 experiments, participants constructed equations with given structure, based on story problems with a variety of phrasings. We demonstrate how the notational alignment approach accounts naturally for a variety of previously reported phenomena in equation construction and successfully predicts error patterns that are not accounted for by prior explanations, such as the left to right transcription heuristic.
ERIC Educational Resources Information Center
Gonzalez-Vega, Laureano
1999-01-01
Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)
Hierarchies of nonlinear integrable equations and their symmetries in 2 + 1 dimensions
NASA Astrophysics Data System (ADS)
Cheng, Yi
1990-11-01
For a given nonlinear integrable equation in 2 + 1 dimensions, an approach is described to construct the hierarchies of equations and relevant Lie algebraic properties. The commutability and noncommutability of equations of the flow, their symmetries and mastersymmetries are then derived as direct results of these algebraic properties. The details for the modified Kadomtsev-Petviashvilli equation are shown as an example and the main results for the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Katera-Sawada equation are given.
NASA Astrophysics Data System (ADS)
Cardona, Carlos; Gomez, Humberto
2016-06-01
Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a mathbb{C}{P}^2 space. We show that for the simplest integrand, namely the n - gon, our proposal indeed reproduces the expected result. By using the recently formulated Λ-algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.
NASA Astrophysics Data System (ADS)
Doikou, Anastasia; Ioannidou, Theodora
2011-04-01
A non-compact version of the Weyl equation is proposed, based on the infinite dimensional spin zero representation of the mathfrak{s}{mathfrak{l}_2} algebra. Solutions of the aforementioned equation are obtained in terms of the Kummer functions. In this context, we discuss the ADHMN approach in order to construct the corresponding non-compact BPS monopoles.
Quantum algebra of N superspace
Hatcher, Nicolas; Restuccia, A.; Stephany, J.
2007-08-15
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.
Constraint algebra in bigravity
Soloviev, V. O.
2015-07-15
The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.
Constraint algebra in bigravity
NASA Astrophysics Data System (ADS)
Soloviev, V. O.
2015-07-01
The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.
Chen, J.; Safro, I.
2011-01-01
Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.
ERIC Educational Resources Information Center
Powell, Sarah R.; Fuchs, Lynn S.
2014-01-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 second-grade students, we administered: (1) measures of calculations and…
ERIC Educational Resources Information Center
Ross, Amanda; Willson, Victor
2012-01-01
This study examined the effects of types of representations, constructivist teaching approaches, and student engagement on middle school algebra students' procedural knowledge and conceptual understanding. Data gathered from 16 video lessons and algebra pretest/posttests were used to run three multilevel structural equation models. Symbolic…
Readiness and Preparation for Beginning Algebra.
ERIC Educational Resources Information Center
Rotman, Jack W.
Drawing from experience at Lansing Community College (LCC), this paper discusses how to best prepare students for success in a beginning algebra course. First, an overview is presented of LCC's developmental math sequence, which includes Basic Arithmetic (MTH 008), Pre-Algebra (MTH 009), Beginning Algebra (MTH 012), and Intermediate Algebra (MTH…
Two-parameter twisted quantum affine algebras
NASA Astrophysics Data System (ADS)
Jing, Naihuan; Zhang, Honglian
2016-09-01
We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.
TBGG- INTERACTIVE ALGEBRAIC GRID GENERATION
NASA Technical Reports Server (NTRS)
Smith, R. E.
1994-01-01
TBGG, Two-Boundary Grid Generation, applies an interactive algebraic grid generation technique in two dimensions. The program incorporates mathematical equations that relate the computational domain to the physical domain. TBGG has application to a variety of problems using finite difference techniques, such as computational fluid dynamics. Examples include the creation of a C-type grid about an airfoil and a nozzle configuration in which no left or right boundaries are specified. The underlying two-boundary technique of grid generation is based on Hermite cubic interpolation between two fixed, nonintersecting boundaries. The boundaries are defined by two ordered sets of points, referred to as the top and bottom. Left and right side boundaries may also be specified, and call upon linear blending functions to conform interior interpolation to the side boundaries. Spacing between physical grid coordinates is determined as a function of boundary data and uniformly spaced computational coordinates. Control functions relating computational coordinates to parametric intermediate variables that affect the distance between grid points are embedded in the interpolation formulas. A versatile control function technique with smooth cubic spline functions is also presented. The TBGG program is written in FORTRAN 77. It works best in an interactive graphics environment where computational displays and user responses are quickly exchanged. The program has been implemented on a CDC Cyber 170 series computer using NOS 2.4 operating system, with a central memory requirement of 151,700 (octal) 60 bit words. TBGG requires a Tektronix 4015 terminal and the DI-3000 Graphics Library of Precision Visuals, Inc. TBGG was developed in 1986.
Cartooning in Algebra and Calculus
ERIC Educational Resources Information Center
Moseley, L. Jeneva
2014-01-01
This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.
NASA Technical Reports Server (NTRS)
Klumpp, A. R.; Lawson, C. L.
1988-01-01
Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.
GCD, LCM, and Boolean Algebra?
ERIC Educational Resources Information Center
Cohen, Martin P.; Juraschek, William A.
1976-01-01
This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)
Hopf algebras and topological recursion
NASA Astrophysics Data System (ADS)
Esteves, João N.
2015-11-01
We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293-309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347-452).
Sjaardema, G.; Gilkey, A.; Smith, M.; Forsythe, C.
2005-04-11
The ALGEBRA program allows the user to manipulate data from a finite element analysis before it is plotted. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database. The ALGEBRA program evaluates user-supplied functions of the data and writes the results to an output EXODUS II database that can be read by plot programs.
Algebraic Rossby Solitary Waves Excited by Non-Stationary External Source
NASA Astrophysics Data System (ADS)
Yang, Hong-Wei; Yin, Bao-Shu; Dong, Huan-He; Shi, Yun-Long
2012-09-01
The paper deals with the effects of non-stationary external source forcing and dissipation on algebraic Rossby solitary waves. From quasi-geostrophic potential vorticity equation, basing on the multiple-scale method, an inhomogeneous Korteweg-de Vries—Benjamin—Ono—Burgers (KdV-B-O-Burgers) equation is obtained. This equation has not been previously derived for Rossby waves. By analysis and calculation, four conservation laws associated with the above equation are first obtained. With the help of pseudo-spectral method, the waterfall plots are obtained and the evolutional characters of algebraic Rossby solitary waves are studied. The results show that non-stationary external source and dissipation have great effect on the generation and evolution of algebraic solitary Rossby waves.
Hunt, H. B.; Marathe, M. V.; Stearns, R. E.
2001-01-01
We demonstrate how the concepts of algebraic representability and strongly-local reductions developed here and in [HSM00] can be used to characterize the computational complexity/efficient approximability of a number of basic problems and their variants, on various abstract algebraic structures F. These problems include the following: (1) A1gebra:Determine the solvability, unique solvability, number of solutions, etc., of a system of equations on F. Determine the equivalence of two formulas or straight-line programs on F. 2. 0ptimization:Let {epsilon} > 0. (a) Determine the maximum number of simultaneously satisfiable equations in a system of equations on F; or approximate this number within a multiplicative factor of n{sup {epsilon}}. (b) Determine the maximum value of an objective function subject to satisfiable algebraically expressed constraints on F; or approximate this maximum value within a multiplicative factor of n{sup {epsilon}}. (c) Given a formula or straight-line program, find a minimum size equivalent formula or straightline program; or find an equivalent formula or straight-line program of size {le} f (minimum). Both finite and infinite algebraic structures are considered. These finite structures include all finite nondegenerate lattices and all finite rings or semi-rings with a nonzero element idempotent under multiplication (e.g. all non-degenerate finite unitary rings or semi-rings); and these infinite structures include the natural numbers, integers, real numbers, various algebras on these structures, all ordered rings, many cancellative semi-rings, and all infinite lattices with two elements a,b such that a is covered by b. Our results significantly extend a number of results by Ladner [La89], Condon, et. al. [CF+93], Khanna, et.al [KSW97], Cr951 and Zuckerman [Zu93] on the complexity and approximbaility of combinatorial problems.
Super-Poincarè algebras, space-times, and supergravities. II
NASA Astrophysics Data System (ADS)
Santi, A.; Spiro, A.
2012-03-01
The presentation of supergravity theories of our previous paper "Super-Poincarè algebras, space-times, and supergravities. I" is re-formulated in the language of Berezin-Leites-Kostant theory of supermanifolds. It is also shown that the equations of Cremmer, Julia, and Scherk's theory of 11D-supergravity are equivalent to manifestly covariant equations on a supermanifold.
Quadratic algebra for superintegrable monopole system in a Taub-NUT space
NASA Astrophysics Data System (ADS)
Hoque, Md Fazlul; Marquette, Ian; Zhang, Yao-Zhong
2016-09-01
We introduce a Hartmann system in the generalized Taub-NUT space with Abelian monopole interaction. This quantum system includes well known Kaluza-Klein monopole and MIC-Zwanziger monopole as special cases. It is shown that the corresponding Schrödinger equation of the Hamiltonian is separable in both spherical and parabolic coordinates. We obtain the integrals of motion of this superintegrable model and construct the quadratic algebra and Casimir operator. This algebra can be realized in terms of a deformed oscillator algebra and has finite dimensional unitary representations (unirreps) which provide energy spectra of the system. This result coincides with the physical spectra obtained from the separation of variables.
Integrable maps from Galois differential algebras, Borel transforms and number sequences
NASA Astrophysics Data System (ADS)
Tempesta, Piergiulio
A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation, in particular symmetries and integrability (see Tempesta, 2010 [40]). Our approach is based on the properties of a suitable Galois differential algebra, that we shall call a Rota algebra. A formulation of the procedure in terms of category theory is proposed. In order to render the lattice dynamics confined, a Borel regularization is also adopted. As a byproduct of the theory, a connection between number sequences and integrability is discussed.
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.
Geometric Approaches to Quadratic Equations from Other Times and Places.
ERIC Educational Resources Information Center
Allaire, Patricia R.; Bradley, Robert E.
2001-01-01
Focuses on geometric solutions of quadratic problems. Presents a collection of geometric techniques from ancient Babylonia, classical Greece, medieval Arabia, and early modern Europe to enhance the quadratic equation portion of an algebra course. (KHR)
Iterative methods for elliptic finite element equations on general meshes
NASA Technical Reports Server (NTRS)
Nicolaides, R. A.; Choudhury, Shenaz
1986-01-01
Iterative methods for arbitrary mesh discretizations of elliptic partial differential equations are surveyed. The methods discussed are preconditioned conjugate gradients, algebraic multigrid, deflated conjugate gradients, an element-by-element techniques, and domain decomposition. Computational results are included.
Phase shift of interacting algebraic solitary waves in a two-layer fluid system
Matsuno, Y. )
1994-09-05
The interaction of interfacial solitary waves of algebraic type is investigated on the basis of a higher-order Benjamin-Ono equation. By developing a multisoliton perturbation theory, we show analytically that the overtaking collision between two solitary waves exhibits the phase shift but the amplitudes are not altered after interaction. The prediction of the phase shift that takes place between algebraic solitary waves is the first example reported in the literature.
Relativistic algebraic spinors and quantum motions in phase space
Holland, P.R.
1986-08-01
Following suggestions of Schonberg and Bohm, we study the tensorial phase space representation of the Dirac and Feynman-Gell-Mann equations in terms of the complex Dirac algebra C/sub 4/, a Jordan-Wigner algebra G/sub 4/, and Wigner transformations. To do this we solve the problem of the conditions under which elements in C/sub 4/ generate minimal ideals, and extend this to G/sub 4/. This yields the linear theory of Dirac spin spaces and tensor representations of Dirac spinors, and the spin-1/2 wave equations are represented through fermionic state vectors in a higher space as a set of interconnected tensor relations.
Universal vertex-IRF transformation for quantum affine algebras
Buffenoir, E.; Roche, Ph.; Terras, V.
2012-10-15
We construct a universal solution of the generalized coboundary equation in the case of quantum affine algebras, which is an extension of our previous work to U{sub q}(A{sub r}{sup (1)}). This universal solution has a simple Gauss decomposition which is constructed using Sevostyanov's characters of twisted quantum Borel algebras. We show that in the evaluation representations it gives a vertex-face transformation between a vertex type solution and a face type solution of the quantum dynamical Yang-Baxter equation. In particular, in the evaluation representation of U{sub q}(A{sub 1}{sup (1)}), it gives Baxter's well-known transformation between the 8-vertex model and the interaction-round-faces (IRF) height model.
NASA Astrophysics Data System (ADS)
Dumansky, Alexander M.; Tairova, Lyudmila P.
2008-09-01
A method for the construction of hereditary constitutive equation is proposed for the laminate on the basis of hereditary constitutive equations of a layer. The method is developed from the assumption that in the directions of axes of orthotropy the layer follows elastic behavior, and obeys hereditary constitutive equations under shear. The constitutive equations of the laminate are constructed on the basis of classical laminate theory and algebra of resolvent operators. Effective matrix algorithm and relationships of operator algebra are used to derive visco-elastic stiffness and compliance of the laminate. The example of construction of hereditary constitutive equations of cross-ply carbon fiber-reinforced plastic is presented.
Lie algebraic methods for particle tracking calculations
Douglas, D.R.; Dragt, A.J.
1983-08-01
A study of the nonlinear stability of an accelerator or storage ring lattice typically includes particle tracking simulations. Such simulations trace rays through linear and nonlinear lattice elements by numerically evaluating linear matrix or impulsive nonlinear transformations. Using the mathematical tools of Lie groups and algebras, one may construct a formalism which makes explicit use of Hamilton's equations and which allows the description of groups of linear and nonlinear lattice elements by a single transformation. Such a transformation will be exactly canonical and will describe finite length linear and nonlinear elements through third (octupole) order. It is presently possible to include effects such as fringing fields and potentially possible to extend the formalism to include nonlinearities of higher order, multipole errors, and magnet misalignments. We outline this Lie algebraic formalism and its use in particle tracking calculations. A computer code, MARYLIE, has been constructed on the basis of this formalism. We describe the use of this program for tracking and provide examples of its application. 6 references, 3 figures.
NASA Astrophysics Data System (ADS)
Masoero, Davide; Raimondo, Andrea; Valeri, Daniele
2016-09-01
We assess the ODE/IM correspondence for the quantum g -KdV model, for a non-simply laced Lie algebra g. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra g^{(1)} , and constructing the relevant {Ψ} -system among subdominant solutions. We then use the {Ψ} -system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum g -KdV model. We also consider generalized Airy functions for twisted Kac-Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.
Symmetries of stochastic differential equations: A geometric approach
NASA Astrophysics Data System (ADS)
De Vecchi, Francesco C.; Morando, Paola; Ugolini, Stefania
2016-06-01
A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an algebra of strong symmetries for a modified SDE is proved under suitable regularity assumptions. This general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional Brownian motion.
Catmull-Rom Curve Fitting and Interpolation Equations
ERIC Educational Resources Information Center
Jerome, Lawrence
2010-01-01
Computer graphics and animation experts have been using the Catmull-Rom smooth curve interpolation equations since 1974, but the vector and matrix equations can be derived and simplified using basic algebra, resulting in a simple set of linear equations with constant coefficients. A variety of uses of Catmull-Rom interpolation are demonstrated,…
Equations of gas dynamics admitting an infinite number of symmetries
Meshkov, A.G.; Mikhalyaev, B.B.
1988-02-01
All the equations of state for which the equations of one-dimensional gas dynamics have an infinite Lie-Baecklund algebra are found. In all these cases, the gas-dynamic equations can either be integrated directly or represented in Lax form. A method for constructing an infinite set of conservation laws is indicated.
BRST charges for finite nonlinear algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.
2010-07-01
Some ingredients of the BRST construction for quantum Lie algebras are applied to a wider class of quadratic algebras of constraints. We build the BRST charge for a quantum Lie algebra with three generators and ghost-anti-ghosts commuting with constraints. We consider a one-parametric family of quadratic algebras with three generators and show that the BRST charge acquires the conventional form after a redefinition of ghosts. The modified ghosts form a quadratic algebra. The family possesses a nonlinear involution, which implies the existence of two independent BRST charges for each algebra in the family. These BRST charges anticommute and form a double BRST complex.
Some Remarks on Kite Pseudo Effect Algebras
NASA Astrophysics Data System (ADS)
Dvurečenskij, Anatolij; Holland, W. Charles
2014-05-01
Recently a new family of pseudo effect algebras, called kite pseudo effect algebras, was introduced. Such an algebra starts with a po-group G, a set I and with two bijections λ, ρ: I→ I. Using a clever construction on the ordinal sum of ( G +) I and ( G -) I , we can define a pseudo effect algebra which can be non-commutative even if G is an Abelian po-group. In the paper we give a characterization of subdirect product of subdirectly irreducible kite pseudo effect algebras, and we show that every kite pseudo effect algebra is an interval in a unital po-loop.
Computational algebraic topology-based video restoration
NASA Astrophysics Data System (ADS)
Rochel, Alban; Ziou, Djemel; Auclair-Fortier, Marie-Flavie
2005-03-01
This paper presents a scheme for video denoising by diffusion of gray levels, based on the Computational Algebraic Topology (CAT) image model. The diffusion approach is similar to the one used to denoise static images. Rather than using the heat transfer partial differential equation, discretizing it and solving it by a purely mathematical process, the CAT approach considers the global expression of the heat transfer and decomposes it into elementary physical laws. Some of these laws describe conservative relations, leading to error-free expressions, whereas others depend on metric quantities and require approximation. This scheme allows for a physical interpretation for each step of the resolution process. We propose a nonlinear and an anisotropic diffusion algorithms based on the extension to video of an existing 2D algorithm thanks to the flexibility of the topological support. Finally it is validated with experimental results.
NASA Astrophysics Data System (ADS)
Imai, Kenji
2014-02-01
In this paper, a new n-dimensional homogeneous Lotka-Volterra (HLV) equation, which possesses a Lie symmetry, is derived by the extension from a three-dimensional HLV equation. Its integrability is shown from the viewpoint of Lie symmetries. Furthermore, we derive dynamical systems of higher order, which possess the Lie symmetry, using the algebraic structure of this HLV equation.
Constructing a parasupersymmetric Virasoro algebra
NASA Astrophysics Data System (ADS)
Kuwata, S.
2011-03-01
We construct a para SUSY Virasoro algebra by generalizing the ordinary fermion in SUSY Virasoro algebra (Ramond or Neveu-Schwarz algebra) to the parafermion. First, we obtain a polynomial relation (PR) between different-mode parafermion fi's by generalizing the corresponding single-mode PR to such that is invariant under the unitary transformation of fi (Green's condition). Differently from a usual context, where the Green's condition is imposed only on the defining relation of fi (degree three with respect to fi and fi†), we impose it on any degree of PR. For the case of order-two parafermion (the simplest case of para SUSY), we calculate a PR between the parasupercharge G0, the bosonic hamiltonian LB0 and parafermionic one LF0, although it is difficult to obtain a PR between G0 and the total hamiltonian L0 (= LB0 + LF0). Finally, we construct a para SUSY Virasoro algebra by generalizing L0 to the Ln's such that form a Virasoro algebra.
ERIC Educational Resources Information Center
Gale, David; And Others
Four units make up the contents of this document. The first examines applications of finite mathematics to business and economies. The user is expected to learn the method of optimization in optimal assignment problems. The second module presents applications of difference equations to economics and social sciences, and shows how to: 1) interpret…
1992-05-04
DOE-MACSYMA (Project MAC''s SYmbolic MAnipulation system) is a large computer programming system written in LISP. With DOE-MACSYMA the user can differentiate, integrate, take limits, solve systems of linear or polynomial equations, factor polynomials, expand functions in Laurent or Taylor series, solve differential equations (using direct or transform methods), compute Poisson series, plot curves, and manipulate matrices and tensors. A language similar to ALGOL-60 permits users to write their own programs for transforming symbolic expressions. Franz Lisp OPUS 38 provides the environment for the Encore, Celerity, and DEC VAX11 UNIX,SUN(OPUS) versions under UNIX and the Alliant version under Concentrix. Kyoto Common Lisp (KCL) provides the environment for the SUN(KCL),Convex, and IBM PC under UNIX and Data General under AOS/VS.
1992-05-04
DOE-MACSYMA (Project MAC''s SYmbolic MAnipulation system) is a large computer programming system written in LISP. With DOE-MACSYMA the user can differentiate, integrate, take limits, solve systems of linear or polynomial equations, factor polynomials, expand functions in Laurent or Taylor series, solve differential equations (using direct or transform methods), compute Poisson series, plot curves, and manipulate matrices and tensors. A language similar to ALGOL-60 permits users to write their own programs for transforming symbolic expressions. Franzmore » Lisp OPUS 38 provides the environment for the Encore, Celerity, and DEC VAX11 UNIX,SUN(OPUS) versions under UNIX and the Alliant version under Concentrix. Kyoto Common Lisp (KCL) provides the environment for the SUN(KCL),Convex, and IBM PC under UNIX and Data General under AOS/VS.« less
Spatial operator algebra framework for multibody system dynamics
NASA Technical Reports Server (NTRS)
Rodriguez, G.; Jain, Abhinandan; Kreutz, K.
1989-01-01
The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.
Algebraic multigrid methods applied to problems in computational structural mechanics
NASA Technical Reports Server (NTRS)
Mccormick, Steve; Ruge, John
1989-01-01
The development of algebraic multigrid (AMG) methods and their application to certain problems in structural mechanics are described with emphasis on two- and three-dimensional linear elasticity equations and the 'jacket problems' (three-dimensional beam structures). Various possible extensions of AMG are also described. The basic idea of AMG is to develop the discretization sequence based on the target matrix and not the differential equation. Therefore, the matrix is analyzed for certain dependencies that permit the proper construction of coarser matrices and attendant transfer operators. In this manner, AMG appears to be adaptable to structural analysis applications.
A Metric Conceptual Space Algebra
NASA Astrophysics Data System (ADS)
Adams, Benjamin; Raubal, Martin
The modeling of concepts from a cognitive perspective is important for designing spatial information systems that interoperate with human users. Concept representations that are built using geometric and topological conceptual space structures are well suited for semantic similarity and concept combination operations. In addition, concepts that are more closely grounded in the physical world, such as many spatial concepts, have a natural fit with the geometric structure of conceptual spaces. Despite these apparent advantages, conceptual spaces are underutilized because existing formalizations of conceptual space theory have focused on individual aspects of the theory rather than the creation of a comprehensive algebra. In this paper we present a metric conceptual space algebra that is designed to facilitate the creation of conceptual space knowledge bases and inferencing systems. Conceptual regions are represented as convex polytopes and context is built in as a fundamental element. We demonstrate the applicability of the algebra to spatial information systems with a proof-of-concept application.
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-07-01
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.
Moving frames and prolongation algebras
NASA Technical Reports Server (NTRS)
Estabrook, F. B.
1982-01-01
Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.
NASA Astrophysics Data System (ADS)
Khalilov, E. H.
2016-07-01
The surface integral equation for a spatial mixed boundary value problem for the Helmholtz equation is considered. At a set of chosen points, the equation is replaced with a system of algebraic equations, and the existence and uniqueness of the solution of this system is established. The convergence of the solutions of this system to the exact solution of the integral equation is proven, and the convergence rate of the method is determined.
Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers
Nataf, J.M.; Winkelmann, F.
1992-09-01
We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK`s symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of these methods to solving the partial differential equations for two-dimensional heat flow is illustrated.
Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers
Nataf, J.M.; Winkelmann, F.
1992-09-01
We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK's symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of these methods to solving the partial differential equations for two-dimensional heat flow is illustrated.
Motivating Activities that Lead to Algebra
ERIC Educational Resources Information Center
Menon, Ramakrishnan
2004-01-01
Four activities consisting of puzzles are introduced, which help students to recognize the strength of algebraic generalizations. They also assist them to comprehend algebraic concepts, and enable them to develop their individual puzzles and games.
Large chiral diffeomorphisms on Riemann surfaces and W-algebras
Bandelloni, G.; Lazzarini, S.
2006-10-15
The diffeomorphism action lifted on truncated (chiral) Taylor expansion of a complex scalar field over a Riemann surface is presented in the paper under the name of large diffeomorphisms. After an heuristic approach, we show how a linear truncation in the Taylor expansion can generate an algebra of symmetry characterized by some structure functions. Such a linear truncation is explicitly realized by introducing the notion of Forsyth frame over the Riemann surface with the help of a conformally covariant algebraic differential equation. The large chiral diffeomorphism action is then implemented through a Becchi-Rouet-Stora (BRS) formulation (for a given order of truncation) leading to a more algebraic setup. In this context the ghost fields behave as holomorphically covariant jets. Subsequently, the link with the so-called W-algebras is made explicit once the ghost parameters are turned from jets into tensorial ghost ones. We give a general solution with the help of the structure functions pertaining to all the possible truncations lower or equal to the given order. This provides another contribution to the relationship between Korteweg-de Vries (KdV) flows and W-diffeomorphims.
Scalable Parallel Algebraic Multigrid Solvers
Bank, R; Lu, S; Tong, C; Vassilevski, P
2005-03-23
The authors propose a parallel algebraic multilevel algorithm (AMG), which has the novel feature that the subproblem residing in each processor is defined over the entire partition domain, although the vast majority of unknowns for each subproblem are associated with the partition owned by the corresponding processor. This feature ensures that a global coarse description of the problem is contained within each of the subproblems. The advantages of this approach are that interprocessor communication is minimized in the solution process while an optimal order of convergence rate is preserved; and the speed of local subproblem solvers can be maximized using the best existing sequential algebraic solvers.
Discrimination in a General Algebraic Setting.
Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras.
Discrimination in a General Algebraic Setting
Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras. PMID:26171421
Spatial-Operator Algebra For Robotic Manipulators
NASA Technical Reports Server (NTRS)
Rodriguez, Guillermo; Kreutz, Kenneth K.; Milman, Mark H.
1991-01-01
Report discusses spatial-operator algebra developed in recent studies of mathematical modeling, control, and design of trajectories of robotic manipulators. Provides succinct representation of mathematically complicated interactions among multiple joints and links of manipulator, thereby relieving analyst of most of tedium of detailed algebraic manipulations. Presents analytical formulation of spatial-operator algebra, describes some specific applications, summarizes current research, and discusses implementation of spatial-operator algebra in the Ada programming language.
Twining characters and orbit Lie algebras
Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert; Schweigert, Christoph
1996-12-05
We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.
New family of Maxwell like algebras
NASA Astrophysics Data System (ADS)
Concha, P. K.; Durka, R.; Merino, N.; Rodríguez, E. K.
2016-08-01
We introduce an alternative way of closing Maxwell like algebras. We show, through a suitable change of basis, that resulting algebras are given by the direct sums of the AdS and the Maxwell algebras already known in the literature. Casting the result into the S-expansion method framework ensures the straightaway construction of the gravity theories based on a found enlargement.
Unifying the Algebra for All Movement
ERIC Educational Resources Information Center
Eddy, Colleen M.; Quebec Fuentes, Sarah; Ward, Elizabeth K.; Parker, Yolanda A.; Cooper, Sandi; Jasper, William A.; Mallam, Winifred A.; Sorto, M. Alejandra; Wilkerson, Trena L.
2015-01-01
There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the "Algebra for All" movement. In light of this…
Build an Early Foundation for Algebra Success
ERIC Educational Resources Information Center
Knuth, Eric; Stephens, Ana; Blanton, Maria; Gardiner, Angela
2016-01-01
Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What's needed is a consistent, algebra-infused mathematics curriculum all…
Difficulties in Initial Algebra Learning in Indonesia
ERIC Educational Resources Information Center
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-01-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was…
A Balancing Act: Making Sense of Algebra
ERIC Educational Resources Information Center
Gavin, M. Katherine; Sheffield, Linda Jensen
2015-01-01
For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…
Algebra? A Gate! A Barrier! A Mystery!
ERIC Educational Resources Information Center
Mathematics Educatio Dialogues, 2000
2000-01-01
This issue of Mathematics Education Dialogues focuses on the nature and the role of algebra in the K-14 curriculum. Articles on this theme include: (1) "Algebra For All? Why?" (Nel Noddings); (2) "Algebra For All: It's a Matter of Equity, Expectations, and Effectiveness" (Dorothy S. Strong and Nell B. Cobb); (3) "Don't Delay: Build and Talk about…
Computer Algebra Systems, Pedagogy, and Epistemology
ERIC Educational Resources Information Center
Bosse, Michael J.; Nandakumar, N. R.
2004-01-01
The advent of powerful Computer Algebra Systems (CAS) continues to dramatically affect curricula, pedagogy, and epistemology in secondary and college algebra classrooms. However, epistemological and pedagogical research regarding the role and effectiveness of CAS in the learning of algebra lags behind. This paper investigates concerns regarding…
Teaching Strategies to Improve Algebra Learning
ERIC Educational Resources Information Center
Zbiek, Rose Mary; Larson, Matthew R.
2015-01-01
Improving student learning is the primary goal of every teacher of algebra. Teachers seek strategies to help all students learn important algebra content and develop mathematical practices. The new Institute of Education Sciences[IES] practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students"…
Leapfrog variants of iterative methods for linear algebra equations
NASA Technical Reports Server (NTRS)
Saylor, Paul E.
1988-01-01
Two iterative methods are considered, Richardson's method and a general second order method. For both methods, a variant of the method is derived for which only even numbered iterates are computed. The variant is called a leapfrog method. Comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large. In the case of Richardson's method, it is possible to express the final iterate in terms of only the initial approximation, a variant of the iteration called the grand-leap method. In the case of the grand-leap variant, a set of parameters is required. An algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for Gaussian quadrature. General algorithms to implement the leapfrog and grand-leap methods are presented. Algorithms for the important special case of the Chebyshev method are also given.
Powell, Sarah R; Fuchs, Lynn S
2014-08-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2(nd)- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty.
Powell, Sarah R; Fuchs, Lynn S
2014-08-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2(nd)- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty. PMID:25309044
Lie algebras for time evolution with applications from chaos studies to spintronics
NASA Astrophysics Data System (ADS)
Wendler, Tim G.; Berrondo, Manuel; Beus, Ty; Sayer, Ryan T.; van Huele, Jean-Francois S.
2012-10-01
We illustrate the power of Lie algebras in computing the time evolution of quantum systems with time-dependent physical parameters. By factorizing the quantum mechanical time evolution operator and using the linear independence of the Lie algebra generators, we reduce the operator equations to systems of coupled ordinary differential equations of scalar functions applicable to a variety of dynamical systems. We use the results to explore the possibility of detecting chaos in quantum nonlinear oscillators based on criteria from classical chaos studies and to follow spin currents in time-dependent spin-orbit coupled media.
Finding the Axis of Revolution of an Algebraic Surface of Revolution.
Alcazar, Juan G; Goldman, Ron
2016-09-01
We present an algorithm for extracting the axis of revolution from the implicit equation of an algebraic surface of revolution based on three distinct computational methods: factoring the highest order form into quadrics, contracting the tensor of the highest order form, and using univariate resultants and gcds. We compare and contrast the advantages and disadvantages of each of these three techniques and we derive conditions under which each technique is most appropriate. In addition, we provide several necessary conditions for an implicit algebraic equation to represent a surface of revolution.
The Clifford algebra of physical space and Dirac theory
NASA Astrophysics Data System (ADS)
Vaz, Jayme, Jr.
2016-09-01
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein–Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.
The Clifford algebra of physical space and Dirac theory
NASA Astrophysics Data System (ADS)
Vaz, Jayme, Jr.
2016-09-01
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.
Entropy algebras and Birkhoff factorization
NASA Astrophysics Data System (ADS)
Marcolli, Matilde; Tedeschi, Nicolas
2015-11-01
We develop notions of Rota-Baxter structures and associated Birkhoff factorizations, in the context of min-plus semirings and their thermodynamic deformations, including deformations arising from quantum information measures such as the von Neumann entropy. We consider examples related to Manin's renormalization and computation program, to Markov random fields and to counting functions and zeta functions of algebraic varieties.
Algebraic Activities Aid Discovery Lessons
ERIC Educational Resources Information Center
Wallace-Gomez, Patricia
2013-01-01
After a unit on the rules for positive and negative numbers and the order of operations for evaluating algebraic expressions, many students believe that they understand these principles well enough, but they really do not. They clearly need more practice, but not more of the same kind of drill. Wallace-Gomez provides three graphing activities that…
ERIC Educational Resources Information Center
Bosse, Michael J.; Ries, Heather; Chandler, Kayla
2012-01-01
Secondary school mathematics teachers often need to answer the "Why do we do that?" question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students' conceptual knowledge of secondary mathematics. This…
Dimension independence in exterior algebra.
Hawrylycz, M
1995-01-01
The identities between homogeneous expressions in rank 1 vectors and rank n - 1 covectors in a Grassmann-Cayley algebra of rank n, in which one set occurs multilinearly, are shown to represent a set of dimension-independent identities. The theorem yields an infinite set of nontrivial geometric identities from a given identity. PMID:11607520
ERIC Educational Resources Information Center
Oishi, Lindsay
2011-01-01
"Solve for x." While many people first encountered this enigmatic instruction in high school, the last 20 years have seen a strong push to get students to take algebra in eighth grade or even before. Today, concerns about the economy highlight a familiar worry: American eighth-graders trailed their peers in five Asian countries on the 2007 TIMSS…
Weaving Geometry and Algebra Together
ERIC Educational Resources Information Center
Cetner, Michelle
2015-01-01
When thinking about student reasoning and sense making, teachers must consider the nature of tasks given to students along with how to plan to use the tasks in the classroom. Students should be presented with tasks in a way that encourages them to draw connections between algebraic and geometric concepts. This article focuses on the idea that it…
Algebraic methods in system theory
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Willems, J. C.; Willsky, A. S.
1975-01-01
Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.
Algebra from Chips and Chopsticks
ERIC Educational Resources Information Center
Yun, Jeong Oak; Flores, Alfinio
2012-01-01
Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…
Celestial mechanics with geometric algebra
NASA Technical Reports Server (NTRS)
Hestenes, D.
1983-01-01
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.
Algebra for All. Research Brief
ERIC Educational Resources Information Center
Bleyaert, Barbara
2009-01-01
The call for "algebra for all" is not a recent phenomenon. Concerns about the inadequacy of math (and science) preparation in America's high schools have been a steady drumbeat since the 1957 launch of Sputnik; a call for raising standards and the number of math (and science) courses required for graduation has been a part of countless national…
Adventures in Flipping College Algebra
ERIC Educational Resources Information Center
Van Sickle, Jenna
2015-01-01
This paper outlines the experience of a university professor who implemented flipped learning in two sections of college algebra courses for two semesters. It details how the courses were flipped, what technology was used, advantages, challenges, and results. It explains what students do outside of class, what they do inside class, and discusses…
ERIC Educational Resources Information Center
Deakin, Michael A. B.
1974-01-01
Euler's famous formula, e to the (i, pi) power equals -1, is developed by a purely algebraic method that avoids the use of both trigonometry and calculus. A heuristic outline is given followed by the rigorous theory. Pedagogical considerations for classroom presentation are suggested. (LS)
Inequalities, Assessment and Computer Algebra
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2015-01-01
The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in…
Algebra, Home Mortgages, and Recessions
ERIC Educational Resources Information Center
Mariner, Jean A. Miller; Miller, Richard A.
2009-01-01
The current financial crisis and recession in the United States present an opportunity to discuss relevant applications of some topics in typical first-and second-year algebra and precalculus courses. Real-world applications of percent change, exponential functions, and sums of finite geometric sequences can help students understand the problems…
Math Sense: Algebra and Geometry.
ERIC Educational Resources Information Center
Howett, Jerry
This book is designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four strands: (1) skill lessons…
NASA Astrophysics Data System (ADS)
Kuipers, J.
2012-06-01
New features of the symbolic algebra package Form 4 are discussed. Most importantly, these features include polynomial factorization and polynomial gcd computation. Examples of their use are shown. One of them is an exact version of Mincer which gives answers in terms of rational polynomials and 5 master integrals.
An algebraic criterion for the onset of chaos in nonlinear dynamic systems
NASA Technical Reports Server (NTRS)
Unal, A.; Tobak, M.
1987-01-01
The correspondence between iterated integrals and a noncommutative algebra is used to recast the given dynamical system from the time domain to the Laplace-Borel transform domain. It is then shown that the following algebraic criterion has to be satisfied for the outset of chaos: the limit (as tau approaches infinity and x sub 0 approaches infinity) of ((sigma(k=0) (tau sup k) / (k* x sub 0 sup k)) G II G = 0, where G is the generating power series of the trajectories, the symbol II is the shuffle product (le melange) of the noncommutative algebra, x sub 0 is a noncommutative variable, and tau is the correlation parameter. In the given equation, symbolic forms for both G and II can be obtained by use of one of the currently available symbolic languages such as PLI, REDUCE, and MACSYMA. Hence, the criterion is a computer-algebraic one.
Array algebra estimation in signal processing
NASA Astrophysics Data System (ADS)
Rauhala, U. A.
A general theory of linear estimators called array algebra estimation is interpreted in some terms of multidimensional digital signal processing, mathematical statistics, and numerical analysis. The theory has emerged during the past decade from the new field of a unified vector, matrix and tensor algebra called array algebra. The broad concepts of array algebra and its estimation theory cover several modern computerized sciences and technologies converting their established notations and terminology into one common language. Some concepts of digital signal processing are adopted into this language after a review of the principles of array algebra estimation and its predecessors in mathematical surveying sciences.
Filiform Lie algebras of order 3
Navarro, R. M.
2014-04-15
The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.
Atomic effect algebras with compression bases
Caragheorgheopol, Dan; Tkadlec, Josef
2011-01-15
Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.
NASA Astrophysics Data System (ADS)
Essén, Hanno; Nordmark, Arne B.
2016-09-01
The canonical Poisson bracket algebra of four-dimensional relativistic mechanics is used to derive the equation of motion for a charged particle, with the Lorentz force, and the homogeneous Maxwell equations.
NASA Astrophysics Data System (ADS)
Essén, Hanno; Nordmark, Arne B.
2016-09-01
The canonical Poisson bracket algebra of four-dimensional relativistic mechanics is used to derive the equation of motion for a charged particle, with the Lorentz force, and the homogeneous Maxwell equations.
Generalized quantum statistics and Lie (super)algebras
NASA Astrophysics Data System (ADS)
Stoilova, N. I.
2016-03-01
Generalized quantum statistics, such as paraboson and parafermion statistics, are characterized by triple relations which are related to Lie (super)algebras of type B. The correspondence of the Fock spaces of parabosons, parafermions as well as the Fock space of a system of parafermions and parabosons to irreducible representations of (super)algebras of type B will be pointed out. Example of generalized quantum statistics connected to the basic classical Lie superalgebra B(1|1) ≡ osp(3|2) with interesting physical properties, such as noncommutative coordinates, will be given. Therefore the article focuses on the question, addressed already in 1950 by Wigner: do the equation of motion determine the quantum mechanical commutation relation?
A Unified Introduction to Ordinary Differential Equations
ERIC Educational Resources Information Center
Lutzer, Carl V.
2006-01-01
This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)
Synthesizing Strategies Creatively: Solving Linear Equations
ERIC Educational Resources Information Center
Ponce, Gregorio A.; Tuba, Imre
2015-01-01
New strategies can ignite teachers' imagination to create new lessons or adapt lessons created by others. In this article, the authors present the experience of an algebra teacher and his students solving linear and literal equations and explain how the use of ideas found in past NCTM journals helped bring this lesson to life. The…
A new model for algebraic Rossby solitary waves in rotation fluid and its solution
NASA Astrophysics Data System (ADS)
Chen, Yao-Deng; Yang, Hong-Wei; Gao, Yu-Fang; Yin, Bao-Shu; Feng, Xing-Ru
2015-09-01
A generalized Boussinesq equation that includes the dissipation effect is derived to describe a kind of algebraic Rossby solitary waves in a rotating fluid by employing perturbation expansions and stretching transformations of time and space. Using this equation, the conservation laws of algebraic Rossby solitary waves are discussed. It is found that the mass, the momentum, the energy, and the velocity of center of gravity of the algebraic solitary waves are conserved in the propagation process. Finally, the analytical solution of the equation is generated. Based on the analytical solution, the properties of the algebraic solitary waves and the dissipation effect are discussed. The results point out that, similar to classic solitary waves, the dissipation can cause the amplitude and the speed of solitary waves to decrease; however, unlike classic solitary waves, the algebraic solitary waves can split during propagation and the decrease of the detuning parameter can accelerate the occurrence of the solitary waves fission phenomenon. Project supported by the Shandong Provincial Key Laboratory of Marine Ecology and Environment and Disaster Prevention and Mitigation Project, China (Grant No. 2012010), the National Natural Science Foundation of China (Grant Nos. 41205082 and 41476019), the Special Funds for Theoretical Physics of the National Natural Science Foundation of China (Grant No. 11447205), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), China.
A Z{sub 3} generalization of Pauli's principle, quark algebra and the Lorentz invariance
Kerner, Richard
2012-09-24
The fundamental difference between bosons and fermions is that they obey two alternative representations of the Z{sub 2} group, resulting in symmetric or anti-symmetric binary commutation relations. Our aim is to explore possibilities offered by ternary Z{sub 3} generalization commutation relations. This leads to cubic and ternary algebras which are a direct generalization of usual commutation relations, with Z{sub 3}-grading replacing the usual Z{sub 2}-grading. Properties and structure of such algebras are discussed, with special interest in a low-dimensional one, with two generators. Invariant cubic forms on such algebras are introduced, and it is shown how the SL(2,C) group arises naturally as the symmetry group preserving these forms. In the case of lowest dimension, with only two generators, it is shown how the cubic combinations of elements of the same Z{sub 3} grade behave like Lorentz spinors, while binary products of elements of this algebra with an element of the conjugate algebra behave like Lorentz vectors. The wave equation generalizing the Dirac operator to the Z{sub 3}-graded case is introduced, whose diagonalization leads to a third-order equation. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry obeyed by quark states.
Simple derivation of the Lindblad equation
NASA Astrophysics Data System (ADS)
Pearle, Philip
2012-07-01
The Lindblad equation is an evolution equation for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is Hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad equation are given. The derivation of the Lindblad equation presented here is ‘simple’ in that all it uses is the expression of a Hermitian matrix in terms of its orthonormal eigenvectors and real eigenvalues. Thus, it is appropriate for students who have learned the algebra of quantum theory. Where helpful, arguments are first given in a two-dimensional Hilbert space.
Analytical solution of tt dilepton equations
Sonnenschein, Lars
2006-03-01
The top quark antiquark production system in the dilepton decay channel is described by a set of equations which is nonlinear in the unknown neutrino momenta. Its most precise and least time consuming solution is of major importance for measurements of top quark properties like the top quark mass and tt spin correlations. The initial system of equations can be transformed into two polynomial equations with two unknowns by means of elementary algebraic operations. These two polynomials of multidegree two can be reduced to one univariate polynomial of degree four by means of resultants. The obtained quartic equation is solved analytically.
Linear algebra algorithms for divisors on an algebraic curve
NASA Astrophysics Data System (ADS)
Khuri-Makdisi, Kamal
We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and matrices of size $O(g^2)\\times O(g)$. When the base field $k$ is finite, or if $C$ has a rational point over $k$, these give algorithms for working on the Jacobian of $C$ that require $O(g^4)$ field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to work with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Hess' 1999 Ph.D. thesis, which works with function fields as extensions of $k[x]$. However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor), superelliptic curves (Galbraith, Paulus, and Smart), and $C_{ab}$ curves (Harasawa and Suzuki); in all those cases, one can attain a complexity of $O(g^2)$.
Using trees to compute approximate solutions to ordinary differential equations exactly
NASA Technical Reports Server (NTRS)
Grossman, Robert
1991-01-01
Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.
Alternative algebraic approaches in quantum chemistry
Mezey, Paul G.
2015-01-22
Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed.
The algebras of large N matrix mechanics
Halpern, M.B.; Schwartz, C.
1999-09-16
Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.
BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1994-01-01
The Basic Linear Algebra Subprogram (BLAS) library is a collection of FORTRAN callable routines for employing standard techniques in performing the basic operations of numerical linear algebra. The BLAS library was developed to provide a portable and efficient source of basic operations for designers of programs involving linear algebraic computations. The subprograms available in the library cover the operations of dot product, multiplication of a scalar and a vector, vector plus a scalar times a vector, Givens transformation, modified Givens transformation, copy, swap, Euclidean norm, sum of magnitudes, and location of the largest magnitude element. Since these subprograms are to be used in an ANSI FORTRAN context, the cases of single precision, double precision, and complex data are provided for. All of the subprograms have been thoroughly tested and produce consistent results even when transported from machine to machine. BLAS contains Assembler versions and FORTRAN test code for any of the following compilers: Lahey F77L, Microsoft FORTRAN, or IBM Professional FORTRAN. It requires the Microsoft Macro Assembler and a math co-processor. The PC implementation allows individual arrays of over 64K. The BLAS library was developed in 1979. The PC version was made available in 1986 and updated in 1988.
Introduction to Image Algebra Ada
NASA Astrophysics Data System (ADS)
Wilson, Joseph N.
1991-07-01
Image Algebra Ada (IAA) is a superset of the Ada programming language designed to support use of the Air Force Armament Laboratory's image algebra in the development of computer vision application programs. The IAA language differs from other computer vision languages is several respects. It is machine independent, and an IAA translator has been implemented in the military standard Ada language. Its image operands and operations can be used to program a range of both low- and high-level vision algorithms. This paper provides an overview of the image algebra constructs supported in IAA and describes the embodiment of these constructs in the IAA extension of Ada. Examples showing the use of IAA for a range of computer vision tasks are given. The design of IAA as a superset of Ada and the implementation of the initial translator in Ada represent critical choices. The authors discuss the reasoning behind these choices as well as the benefits and drawbacks associated with them. Implementation strategies associated with the use of Ada as an implementation language for IAA are also discussed. While one can look on IAA as a program design language (PDL) for specifying Ada programs, it is useful to consider IAA as a separate language superset of Ada. This admits the possibility of directly translating IAA for implementation on special purpose architectures. This paper explores strategies for porting IAA to various architectures and notes the critical language and implementation features for porting to different architectures.
Modular properties of characters of the W3 algebra
NASA Astrophysics Data System (ADS)
Iles, Nicholas J.; Watts, Gérard M. T.
2016-01-01
In a previous work, exact formulae and differential equations were found for traces of powers of the zero mode in the W 3 algebra. In this paper we investigate their modular properties, in particular we find the exact result for the modular transformations of traces of W 0 n for n = 1 , 2 , 3, solving exactly the problem studied approximately by Gaberdiel, Hartman and Jin. We also find modular differential equations satisfied by traces with a single W 0 inserted, and relate them to differential equations studied by Mathur et al. We find that, remarkably, these all seem to be related to weight 0 modular forms with expansions with non-negative integer coefficients.
Polarization ellipse and Stokes parameters in geometric algebra.
Santos, Adler G; Sugon, Quirino M; McNamara, Daniel J
2012-01-01
In this paper, we use geometric algebra to describe the polarization ellipse and Stokes parameters. We show that a solution to Maxwell's equation is a product of a complex basis vector in Jackson and a linear combination of plane wave functions. We convert both the amplitudes and the wave function arguments from complex scalars to complex vectors. This conversion allows us to separate the electric field vector and the imaginary magnetic field vector, because exponentials of imaginary scalars convert vectors to imaginary vectors and vice versa, while exponentials of imaginary vectors only rotate the vector or imaginary vector they are multiplied to. We convert this expression for polarized light into two other representations: the Cartesian representation and the rotated ellipse representation. We compute the conversion relations among the representation parameters and their corresponding Stokes parameters. And finally, we propose a set of geometric relations between the electric and magnetic fields that satisfy an equation similar to the Poincaré sphere equation.
A Unified Approach to Teaching Quadratic and Cubic Equations.
ERIC Educational Resources Information Center
Ward, A. J. B.
2003-01-01
Presents a simple method for teaching the algebraic solution of cubic equations via completion of the cube. Shows that this method is readily accepted by students already familiar with completion of the square as a method for quadratic equations. (Author/KHR)
XXZ-type Bethe ansatz equations and quasi-polynomials
NASA Astrophysics Data System (ADS)
Li, Jian Rong; Tarasov, Vitaly
2013-01-01
We study solutions of the Bethe ansatz equation for the XXZ-type integrable model associated with the Lie algebra fraktur sfraktur lN. We give a correspondence between solutions of the Bethe ansatz equations and collections of quasi-polynomials. This extends the results of E. Mukhin and A. Varchenko for the XXX-type model and the trigonometric Gaudin model.
Algebra: A Challenge at the Crossroads of Policy and Practice
ERIC Educational Resources Information Center
Stein, Mary Kay; Kaufman, Julia Heath; Sherman, Milan; Hillen, Amy F.
2011-01-01
The authors review what is known about early and universal algebra, including who is getting access to algebra and student outcomes associated with algebra course taking in general and specifically with universal algebra policies. The findings indicate that increasing numbers of students, some of whom are underprepared, are taking algebra earlier.…
Loop equations and KDV hierarchy in 2-D quantum gravity
Fucito, F. ); Martellini, M. )
1992-04-20
In this paper a derivation of the loop equation for two-dimensional quantum gravity from the KdV equations and the string equation of the one-matrix model is given. The loop equation was found to be equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the virasoro algebra. Starting form the equations expressing these constraints. The authors are able to rederive the equations of the KdV hierarchy using the vertex operator construction of the A{sup (I)}{sub I} infinite dimensional twisted Kac-Moody algebra. From these considerations it follows that the solutions of the string equation of the one-matrix model are given by a subset of the solutions of the KdV hierarchy.
Software package and API in MATLAB for working with fuzzy algebras
NASA Astrophysics Data System (ADS)
Zahariev, Zlatko
2009-11-01
New software package for fuzzy calculus is presented in this paper. Most important feature of this package is solving fuzzy linear systems of equations and inequalities in fuzzy algebras. Together with this, some other functionality is implemented. An example is also given.
Mathematics for High School, First Course in Algebra, Part 3. Preliminary Edition.
ERIC Educational Resources Information Center
Allen, Frank B.; And Others
This is the third part of a three-part SMSG algebra text for high school students. Chapter titles include: Truth Sets of Open Sentences; Graphs of Open Sentences in Two Variables; Systems of Equations and Inequalities; Quadratic Polynomials; and Functions. (MK)
NASA Technical Reports Server (NTRS)
Rostand, Philippe
1988-01-01
The incorporation of algebraic turbulence models in a solver for the 2-D compressible Navier-Stokes equations using triangular grids is described. A practical way to use the Cebeci Smith model, and to modify it in separated regions is proposed. The ability of the model to predict high speed, perfect gas boundary layers is investigated from a numerical point of view.
Cognitive Tutor[R] Algebra I. What Works Clearinghouse Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2007
2007-01-01
"Cognitive Tutor[R] Algebra I," a full year course, delivers instruction in single variable data, simplifying linear expressions, mathematical modeling, solving systems with linear equations, problem solving using proportional reasoning, and powers and exponents. Students work at their own pace to develop problem-solving skills. The duration of…
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…
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…
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.
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…
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…
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…
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
Kumjian-Pask algebras of desourcification
NASA Astrophysics Data System (ADS)
Rosjanuardi, Rizky; Yusnitha, Isnie
2016-02-01
Kumjian-Pask algebra which was introduced by Pino, Clark, an Huef and Raeburn [1] in 2013, gives a purely algebraic version of a k-graph algebra. Rosjanuardi [2] gave necessary and sufficient condition of finitely dimensional complex Kumjian-Pask algebra of row-finite k-graph without sources. We will improve the previous results which allows us to deal with sources. We will consider Kumjian-Pask algebra for locally convex row-finite k-graph which was introduced by Clark, Flynn and an Huef [3], and use the desourcification of the graph to get conditions which characterise when the complex Kumjian-Pask algebra of locally convex row-finite k-graph is finite dimensional.
Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
NASA Astrophysics Data System (ADS)
Zhang, Tianjie; Gao, Xing; Guo, Li
2016-10-01
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.
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.
Nonlinear algebra and Bogoliubov's recursion
NASA Astrophysics Data System (ADS)
Morozov, A. Yu.; Serbyn, M. N.
2008-02-01
We give many examples of applying Bogoliubov’s forest formula to iterative solutions of various nonlinear equations. The same formula describes an extremely wide class of objects, from an ordinary quadratic equation to renormalization in quantum field theory.
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…
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.
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.
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.
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.
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.
Entanglement and algebraic independence in fermion systems
NASA Astrophysics Data System (ADS)
Benatti, Fabio; Floreanini, Roberto
2014-04-01
In the case of systems composed of identical particles, a typical instance in quantum statistical mechanics, the standard approach to separability and entanglement ought to be reformulated and rephrased in terms of correlations between operators from subalgebras localized in spatially disjoint regions. While this algebraic approach is straightforward for bosons, in the case of fermions it is subtler since one has to distinguish between micro-causality, that is the anti-commutativity of the basic creation and annihilation operators, and algebraic independence that is the commutativity of local observables. We argue that a consistent algebraic formulation of separability and entanglement should be compatible with micro-causality rather than with algebraic independence.
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.
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.
ERIC Educational Resources Information Center
Hitt, Fernando; Saboya, Mireille; Cortés Zavala, Carlos
2016-01-01
This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…
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.
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…
Cluster automorphism groups of cluster algebras with coefficients
NASA Astrophysics Data System (ADS)
Chang, Wen; Zhu, Bin
2016-10-01
We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra (i.e. the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients, cluster algebras with universal geometric coefficients, and cluster algebras from surfaces (except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.
The Structure of Parafermion Vertex Operator Algebras: General Case
NASA Astrophysics Data System (ADS)
Dong, Chongying; Wang, Qing
2010-11-01
The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined.
Primordial fluctuations from deformed quantum algebras
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.
Gene algebra from a genetic code algebraic structure.
Sanchez, R; Morgado, E; Grau, R
2005-10-01
By considering two important factors involved in the codon-anticodon interactions, the hydrogen bond number and the chemical type of bases, a codon array of the genetic code table as an increasing code scale of interaction energies of amino acids in proteins was obtained. Next, in order to consecutively obtain all codons from the codon AAC, a sum operation has been introduced in the set of codons. The group obtained over the set of codons is isomorphic to the group (Z(64), +) of the integer module 64. On the Z(64)-algebra of the set of 64(N) codon sequences of length N, gene mutations are described by means of endomorphisms f:(Z(64))(N)-->(Z(64))(N). Endomorphisms and automorphisms helped us describe the gene mutation pathways. For instance, 77.7% mutations in 749 HIV protease gene sequences correspond to unique diagonal endomorphisms of the wild type strain HXB2. In particular, most of the reported mutations that confer drug resistance to the HIV protease gene correspond to diagonal automorphisms of the wild type. What is more, in the human beta-globin gene a similar situation appears where most of the single codon mutations correspond to automorphisms. Hence, in the analyses of molecular evolution process on the DNA sequence set of length N, the Z(64)-algebra will help us explain the quantitative relationships between genes.
Dirac matrices as elements of a superalgebraic matrix algebra
NASA Astrophysics Data System (ADS)
Monakhov, V. V.
2016-08-01
The paper considers a Clifford extension of the Grassmann algebra, in which operators are built from Grassmann variables and by the derivatives with respect to them. It is shown that a subalgebra which is isomorphic to the usual matrix algebra exists in this algebra, the Clifford exten-sion of the Grassmann algebra is a generalization of the matrix algebra and contains superalgebraic operators expanding matrix algebra and produces supersymmetric transformations.
Automated Angular Momentum Recoupling Algebra
NASA Astrophysics Data System (ADS)
Williams, H. T.; Silbar, Richard R.
1992-04-01
We present a set of heuristic rules for algebraic solution of angular momentum recoupling problems. The general problem reduces to that of finding an optimal path from one binary tree (representing the angular momentum coupling scheme for the reduced matrix element) to another (representing the sub-integrals and spin sums to be done). The method lends itself to implementation on a microcomputer, and we have developed such an implementation using a dialect of LISP. We describe both how our code, called RACAH, works and how it appears to the user. We illustrate the use of RACAH for several transition and scattering amplitude matrix elements occurring in atomic, nuclear, and particle physics.
Constitutive equations for discrete electromagnetic problems over polyhedral grids
Codecasa, Lorenzo . E-mail: codecasa@elet.polimi.it; Trevisan, Francesco . E-mail: trevisan@uniud.it
2007-08-10
In this paper a novel approach is proposed for constructing discrete counterparts of constitutive equations over polyhedral grids which ensure both consistency and stability of the algebraic equations discretizing an electromagnetic field problem. The idea is to construct discrete constitutive equations preserving the thermodynamic relations for constitutive equations. In this way, consistency and stability of the discrete equations are ensured. At the base, a purely geometric condition between the primal and the dual grids has to be satisfied for a given primal polyhedral grid, by properly choosing the dual grid. Numerical experiments demonstrate that the proposed discrete constitutive equations lead to accurate approximations of the electromagnetic field.
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.
Infinite-dimensional Estabrook-Wahlquist prolongations for the sine-Gordon equation
NASA Astrophysics Data System (ADS)
Finley, J. D., III; McIver, John K.
1995-10-01
We are looking for the universal covering algebra for all symmetries of a given partial differential equation (PDE), using the sine-Gordon equation as a typical example for a nonevolution equation. For nonevolution equations, Estabrook-Wahlquist prolongation structures for nonlocal symmetries depend on the choice of a specific subideal of the contact module to define the PDE. For each inequivalent such choice we determine the most general solution of the prolongation equations, as subalgebras of the (infinite-dimensional) algebra of all vector fields over the space of nonlocal variables associated with the PDE, in the style of Vinogradov covering spaces. We show explicitly how previously known prolongation structures, known to lie within the Kac-Moody algebra, A(1)1, are special cases of these general solutions, although we are unable to identify the most general solutions with previously studied algebras. We show the existence of gauge transformations between prolongation structures, viewed as determining connections over the solution space, and use these to relate (otherwise) distinct algebras. Faithful realizations of the universal algebra allow integral representations of the prolongation structure, opening up interesting connections with algebras of Toeplitz operators over Banach spaces, an area that has only begun to be explored.
Contact Geometry of Hyperbolic Equations of Generic Type
NASA Astrophysics Data System (ADS)
The, Dennis
2008-08-01
We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampère (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the generic case. An intriguing feature of this class of equations is that every generic hyperbolic equation admits at most a nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp: normal forms for the contact-equivalence classes of these maximally symmetric generic hyperbolic equations are derived and explicit symmetry algebras are presented. Moreover, these maximally symmetric equations are Darboux integrable. An enumeration of several submaximally symmetric (eight and seven-dimensional) generic hyperbolic structures is also given.
Generalized Morse and Poeschl-Teller potentials: The connection via Schroedinger equation
Yahiaoui, S.-A.; Hattou, S.; Bentaiba, M.
2007-11-15
A systematic and unified treatment to connect the Schroedinger equation for generalized Morse and Poeschl-Teller potentials, generated by supersymmetry quantum mechanics, is used. An algebraic treatment of bound-state problems is presented.
ERIC Educational Resources Information Center
Eastman, Phillip M.; Salhab, Mohammed
1978-01-01
One group of elementary education majors received an algebraic instructional treatment of a unit on linear absolute value equations while another received a geometric treatment. The hypothesis of an aptitude-by-treatment interaction was statistically confirmed. (MN)
Algebraic Thinking through Koch Snowflake Constructions
ERIC Educational Resources Information Center
Ghosh, Jonaki B.
2016-01-01
Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…
Investigating Algebraic Procedures Using Discussion and Writing
ERIC Educational Resources Information Center
Harper, Jonathan; Ford, Jeffrey
2012-01-01
This study reports on the implementation of an intermediate algebra curriculum centered on a framework of student-centered questions designed to investigate algebraic procedures. Instructional activities were designed to build discourse in the small-group discussion meetings of the course. Students were assigned writing prompts to emphasize the…
Practicing Algebraic Skills: A Conceptual Approach
ERIC Educational Resources Information Center
Friedlander, Alex; Arcavi, Abraham
2012-01-01
Traditionally, a considerable part of teaching and learning algebra has focused on routine practice and the application of rules, procedures, and techniques. Although today's computerized environments may have decreased the need to master algebraic skills, procedural competence is still a central component in any mathematical activity. However,…
Using Students' Interests as Algebraic Models
ERIC Educational Resources Information Center
Whaley, Kenneth A.
2012-01-01
Fostering algebraic thinking is an important goal for middle-grades mathematics teachers. Developing mathematical reasoning requires that teachers cultivate students' habits of mind. Teachers develop students' understanding of algebra by engaging them in tasks that involve modeling and representation. This study was designed to investigate how…
THE RADICAL OF A JORDAN ALGEBRA
McCrimmon, Kevin
1969-01-01
In this paper we define a Jacobson radical for Jordan algebras analogous to that for associative algebras and show that it enjoys many of the properties of the associative radical. We then relate the corresponding notion of “semisimplicity” to the previously defined notion of “nondegeneracy” (Jacobson, N., these Proceedings, 55, 243-251 (1966)). PMID:16591736
The operator algebra approach to quantum groups
Kustermans, Johan; Vaes, Stefaan
2000-01-01
A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory. PMID:10639116
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…
Predicting Turkish Ninth Grade Students' Algebra Performance
ERIC Educational Resources Information Center
Erbas, Ayhan Kursat
2005-01-01
The prediction of students' achievement in algebra in eighth and ninth grades has become a research interest for practical issues of placement. A group of simple, easily accessible variables was used to predict student performance in algebra after completion of eighth grade. The three variables of school type, grade level, and previous year…
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…
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…
How To Prepare Students for Algebra.
ERIC Educational Resources Information Center
Wu, H.
2001-01-01
Suggests that no matter how much algebraic thinking is introduced in the early grades, and no matter how worthwhile this might be, the failure rate in algebra will continue unless the teaching of fractions and decimals is radically revamped. The proper study of fractions provides a ramp that leads students gently from whole number arithmetic up to…
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…
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.
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.
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.
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…
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…
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…
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…
Parabolas: Connection between Algebraic and Geometrical Representations
ERIC Educational Resources Information Center
Shriki, Atara
2011-01-01
A parabola is an interesting curve. What makes it interesting at the secondary school level is the fact that this curve is presented in both its contexts: algebraic and geometric. Being one of Apollonius' conic sections, the parabola is basically a geometric entity. It is, however, typically known for its algebraic characteristics, in particular…
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.
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…
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…
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"…
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)
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…
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)
NASA Astrophysics Data System (ADS)
Ho, Choon-Lin; Hosotani, Yutaka
Starting from the quantum field theory of nonrelativistic matter on a torus interacting with Chern-Simons gauge fields, we derive the Schrödinger equation for an anyon system. The nonintegrable phases of the Wilson line integrals on a torus play an essential role. In addition to generating degenerate vacua, they enter in the definition of a many-body Schrödinger wave function in quantum mechanics, which can be defined as a regular function of the coordinates of anyons. It obeys a non-Abelian representation of the braid group algebra, being related to Einarsson’s wave function by a singular gauge transformation.
On multisoliton solutions of the constant astigmatism equation
NASA Astrophysics Data System (ADS)
Hlaváč, Adam
2015-09-01
We introduce an algebraic formula producing infinitely many exact solutions of the constant astigmatism equation {z}{yy}+{(1/z)}{xx}+2=0 from a given seed. A construction of corresponding surfaces of constant astigmatism is then a matter of routine. As a special case, we consider multisoliton solutions of the constant astigmatism equation defined as counterparts of famous multisoliton solutions of the sine-Gordon equation. A few particular examples are surveyed as well.
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.
Generalization of n-ary Nambu algebras and beyond
Ataguema, H.; Makhlouf, A.; Silvestrov, S.
2009-08-15
The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.
Efficient computer algebra algorithms for polynomial matrices in control design
NASA Technical Reports Server (NTRS)
Baras, J. S.; Macenany, D. C.; Munach, R.
1989-01-01
The theory of polynomial matrices plays a key role in the design and analysis of multi-input multi-output control and communications systems using frequency domain methods. Examples include coprime factorizations of transfer functions, cannonical realizations from matrix fraction descriptions, and the transfer function design of feedback compensators. Typically, such problems abstract in a natural way to the need to solve systems of Diophantine equations or systems of linear equations over polynomials. These and other problems involving polynomial matrices can in turn be reduced to polynomial matrix triangularization procedures, a result which is not surprising given the importance of matrix triangularization techniques in numerical linear algebra. Matrices with entries from a field and Gaussian elimination play a fundamental role in understanding the triangularization process. In the case of polynomial matrices, matrices with entries from a ring for which Gaussian elimination is not defined and triangularization is accomplished by what is quite properly called Euclidean elimination. Unfortunately, the numerical stability and sensitivity issues which accompany floating point approaches to Euclidean elimination are not very well understood. New algorithms are presented which circumvent entirely such numerical issues through the use of exact, symbolic methods in computer algebra. The use of such error-free algorithms guarantees that the results are accurate to within the precision of the model data--the best that can be hoped for. Care must be taken in the design of such algorithms due to the phenomenon of intermediate expressions swell.
Finite-Dimensional Lie Algebras for Fast Diffeomorphic Image Registration.
Zhang, Miaomiao; Fletcher, P Thomas
2015-01-01
This paper presents a fast geodesic shooting algorithm for diffeomorphic image registration. We first introduce a novel finite-dimensional Lie algebra structure on the space of bandlimited velocity fields. We then show that this space can effectively represent initial velocities for diffeomorphic image registration at much lower dimensions than typically used, with little to no loss in registration accuracy. We then leverage the fact that the geodesic evolution equations, as well as the adjoint Jacobi field equations needed for gradient descent methods, can be computed entirely in this finite-dimensional Lie algebra. The result is a geodesic shooting method for large deformation metric mapping (LDDMM) that is dramatically faster and less memory intensive than state-of-the-art methods. We demonstrate the effectiveness of our model to register 3D brain images and compare its registration accuracy, run-time, and memory consumption with leading LDDMM methods. We also show how our algorithm breaks through the prohibitive time and memory requirements of diffeomorphic atlas building.
Non-Galerkin Coarse Grids for Algebraic Multigrid
Falgout, Robert D.; Schroder, Jacob B.
2014-06-26
Algebraic multigrid (AMG) is a popular and effective solver for systems of linear equations that arise from discretized partial differential equations. And while AMG has been effectively implemented on large scale parallel machines, challenges remain, especially when moving to exascale. Particularly, stencil sizes (the number of nonzeros in a row) tend to increase further down in the coarse grid hierarchy, and this growth leads to more communication. Therefore, as problem size increases and the number of levels in the hierarchy grows, the overall efficiency of the parallel AMG method decreases, sometimes dramatically. This growth in stencil size is due to the standard Galerkin coarse grid operator, $P^T A P$, where $P$ is the prolongation (i.e., interpolation) operator. For example, the coarse grid stencil size for a simple three-dimensional (3D) seven-point finite differencing approximation to diffusion can increase into the thousands on present day machines, causing an associated increase in communication costs. We therefore consider algebraically truncating coarse grid stencils to obtain a non-Galerkin coarse grid. First, the sparsity pattern of the non-Galerkin coarse grid is determined by employing a heuristic minimal “safe” pattern together with strength-of-connection ideas. Second, the nonzero entries are determined by collapsing the stencils in the Galerkin operator using traditional AMG techniques. The result is a reduction in coarse grid stencil size, overall operator complexity, and parallel AMG solve phase times.
Acoustooptic linear algebra processors - Architectures, algorithms, and applications
NASA Technical Reports Server (NTRS)
Casasent, D.
1984-01-01
Architectures, algorithms, and applications for systolic processors are described with attention to the realization of parallel algorithms on various optical systolic array processors. Systolic processors for matrices with special structure and matrices of general structure, and the realization of matrix-vector, matrix-matrix, and triple-matrix products and such architectures are described. Parallel algorithms for direct and indirect solutions to systems of linear algebraic equations and their implementation on optical systolic processors are detailed with attention to the pipelining and flow of data and operations. Parallel algorithms and their optical realization for LU and QR matrix decomposition are specifically detailed. These represent the fundamental operations necessary in the implementation of least squares, eigenvalue, and SVD solutions. Specific applications (e.g., the solution of partial differential equations, adaptive noise cancellation, and optimal control) are described to typify the use of matrix processors in modern advanced signal processing.
Eye Movements Reveal Students' Strategies in Simple Equation Solving
ERIC Educational Resources Information Center
Susac, Ana; Bubic, Andreja; Kaponja, Jurica; Planinic, Maja; Palmovic, Marijan
2014-01-01
Equation rearrangement is an important skill required for problem solving in mathematics and science. Eye movements of 40 university students were recorded while they were rearranging simple algebraic equations. The participants also reported on their strategies during equation solving in a separate questionnaire. The analysis of the behavioral…
The Effect of using two variables when there are two unknowns in solving algebraic word problems
NASA Astrophysics Data System (ADS)
Mathews, Susann M.
1997-09-01
This article reports an experiment in which Algebra I students learned to translate word problems with two unknowns from the prose representation to symbolic representation using two variables (one to represent each unknown) when they first started solving word problems with two unknowns. Their performance on a test of word problems with two unknowns was compared with the results on the same test taken by students who had learned to solve word problems with two unknowns the traditional way, using only one variable to translate from prose to an algebraic equation. Four algebra teachers and 181 of their students participated in the study. A block-randomised factorial design was used. An analysis of covariance showed a statistically significant difference in the mean scores of the experimental group and the control group on this word problem test with the experimental group scoring substantially higher.
Complex oscillator and Painlevé IV equation
Fernández C, David J. González, J.C.
2015-08-15
Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. In this article the first- and second-order supersymmetric transformations will be used to obtain new exactly solvable potentials departing from the complex oscillator. The corresponding Hamiltonians turn out to be ruled by polynomial Heisenberg algebras. By applying a mechanism to reduce to second the order of these algebras, the connection with the Painlevé IV equation is achieved, thus giving place to new solutions for the Painlevé IV equation.
Dispersive deformations of the Hamiltonian structure of Euler's equations
NASA Astrophysics Data System (ADS)
Casati, M.
2016-09-01
Euler's equations for a two-dimensional fluid can be written in the Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated with the Lie algebra of divergence-free vector fields. For the two-dimensional hydrodynamics of ideal fluids, we propose a derivation of the Poisson brackets using a reduction from the bracket associated with the full algebra of vector fields. Taking the results of some recent studies of the deformations of Lie-Poisson brackets of vector fields into account, we investigate the dispersive deformations of the Poisson brackets of Euler's equation: we show that they are trivial up to the second order.
Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*
Bank, R.; Falgout, R. D.; Jones, T.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J. W.
2015-10-29
In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods inmore » Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.« less
Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*
Bank, R.; Falgout, R. D.; Jones, T.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J. W.
2015-10-29
In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.
Conformal symmetry algebra of the quark potential and degeneracies in the hadron spectra
NASA Astrophysics Data System (ADS)
Kirchbach, M.
2012-10-01
The essence of the potential algebra concept [Y. Alhassid, F. Gürsey, F. Yachello. Phys. Rev. Lett. 50 (1983)] is that quantum mechanical free motions of scalar particles on curved surfaces of given isometry algebras can be mapped on 1D Schrödinger equations with particular potentials. As long as the Laplace-Beltrami operator on a curved surface is proportional to one of the Casimir invariants of the isometry algebra, free motion on the surface is described by means of the eigenvalue problem of that very Casimir operator. In effect, the excitation modes considered are classified according to the irreducible representations of the algebra of interest and are characterized by typical degeneracies. In consequence, also the spectra of the equivalent Schrödinger operators are classified according to the same irreducible representations and carry the same typical degeneracies. A subtle point concerns the representation of the algebra elements which may or may not be unitarily equivalent to the standard one generating classical groups like SO(n), SO(p,q), etc. To be specific, any similarity transformations of an algebra that underlies, say, an orthogonal group, always conserve the commutators among the elements, but a non-unitarily transformed algebra must not generate same group. One can then consider the parameters of the non-unitary similarity transformation as group symmetry breaking scales and seek to identify them with physical observables. We here use the potential algebra concept as a guidance in the search for an interaction describing conformal degeneracies. For this purpose we subject the so(4) ⊂ so(2,4) isometry algebra of the S3 ball to a particular non-unitary similarity transformation and obtain a deformed isometry copy to S3 such that free motion on the copy is equivalent to a cotangent perturbed motion on S3, and to the 1D Schrödinger operator with the trigonometric Rosen-Morse potential as well. The latter presents itself especially well suited for
Working memory, worry, and algebraic ability.
Trezise, Kelly; Reeve, Robert A
2014-05-01
Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship.
Weak homological dimensions and biflat Koethe algebras
Pirkovskii, A Yu
2008-06-30
The homological properties of metrizable Koethe algebras {lambda}(P) are studied. A criterion for an algebra A={lambda}(P) to be biflat in terms of the Koethe set P is obtained, which implies, in particular, that for such algebras the properties of being biprojective, biflat, and flat on the left are equivalent to the surjectivity of the multiplication operator A otimes-hat A{yields}A. The weak homological dimensions (the weak global dimension w.dg and the weak bidimension w.db) of biflat Koethe algebras are calculated. Namely, it is shown that the conditions w.db {lambda}(P)<=1 and w.dg {lambda}(P)<=1 are equivalent to the nuclearity of {lambda}(P); and if {lambda}(P) is non-nuclear, then w.dg {lambda}(P)=w.db {lambda}(P)=2. It is established that the nuclearity of a biflat Koethe algebra {lambda}(P), under certain additional conditions on the Koethe set P, implies the stronger estimate db {lambda}(P), where db is the (projective) bidimension. On the other hand, an example is constructed of a nuclear biflat Koethe algebra {lambda}(P) such that db {lambda}(P)=2 (while w.db {lambda}(P)=1). Finally, it is shown that many biflat Koethe algebras, while not being amenable, have trivial Hochschild homology groups in positive degrees (with arbitrary coefficients). Bibliography: 37 titles.
Algebraic curves of maximal cyclicity
NASA Astrophysics Data System (ADS)
Caubergh, Magdalena; Dumortier, Freddy
2006-01-01
The paper deals with analytic families of planar vector fields, studying methods to detect the cyclicity of a non-isolated closed orbit, i.e. the maximum number of limit cycles that can locally bifurcate from it. It is known that this multi-parameter problem can be reduced to a single-parameter one, in the sense that there exist analytic curves in parameter space along which the maximal cyclicity can be attained. In that case one speaks about a maximal cyclicity curve (mcc) in case only the number is considered and of a maximal multiplicity curve (mmc) in case the multiplicity is also taken into account. In view of obtaining efficient algorithms for detecting the cyclicity, we investigate whether such mcc or mmc can be algebraic or even linear depending on certain general properties of the families or of their associated Bautin ideal. In any case by well chosen examples we show that prudence is appropriate.
Inequalities, assessment and computer algebra
NASA Astrophysics Data System (ADS)
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 contemporary curricula. We consider the formal mathematical processes by which such inequalities are solved, and we consider the notation and syntax through which solutions are expressed. We review the extent to which current CAS can accurately solve these inequalities, and the form given to the solutions by the designers of this software. Finally, we discuss the functionality needed to deal with students' answers, i.e. to establish equivalence (or otherwise) of expressions representing unions of intervals. We find that while contemporary CAS accurately solve inequalities there is a wide variety of notation used.
PC Basic Linear Algebra Subroutines
1992-03-09
PC-BLAS is a highly optimized version of the Basic Linear Algebra Subprograms (BLAS), a standardized set of thirty-eight routines that perform low-level operations on vectors of numbers in single and double-precision real and complex arithmetic. Routines are included to find the index of the largest component of a vector, apply a Givens or modified Givens rotation, multiply a vector by a constant, determine the Euclidean length, perform a dot product, swap and copy vectors, andmore » find the norm of a vector. The BLAS have been carefully written to minimize numerical problems such as loss of precision and underflow and are designed so that the computation is independent of the interface with the calling program. This independence is achieved through judicious use of Assembly language macros. Interfaces are provided for Lahey Fortran 77, Microsoft Fortran 77, and Ryan-McFarland IBM Professional Fortran.« less
2011-08-22
miniXyce uses a modified nodal analysis to approximate the governing differential algebraic equations (DAEs) for electronic devices and their coupling via a network and Kirchoff's conservation laws. The resulting nonlinear problem is solved iteratively using a fully coupled Newton method,which in turn results in a linear system that is solved using an iterative method.
Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz
NASA Astrophysics Data System (ADS)
Belliard, Samuel; Crampé, Nicolas
2013-11-01
We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al. The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain. We prove this result for the chains with small length. We obtain also an off-shell equation (i.e. satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.
Differential equations, associators, and recurrences for amplitudes
NASA Astrophysics Data System (ADS)
Puhlfürst, Georg; Stieberger, Stephan
2016-01-01
We provide new methods to straightforwardly obtain compact and analytic expressions for ɛ-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for recurrence relations connecting different ɛ-orders of a power series solution in ɛ of a differential equation. This strategy generalizes the usual iteration by Picard's method. Our tools are demonstrated for generalized hypergeometric functions. Furthermore, we match the ɛ-expansion of specific generalized hypergeometric functions with the underlying Drinfeld associator with proper Lie algebra and monodromy representations. We also apply our tools for computing ɛ-expansions for solutions to generic first-order Fuchsian equations (Schlesinger system). Finally, we set up our methods to systematically get compact and explicit α‧-expansions of tree-level superstring amplitudes to any order in α‧.
Short Round Sub-Linear Zero-Knowledge Argument for Linear Algebraic Relations
NASA Astrophysics Data System (ADS)
Seo, Jae Hong
Zero-knowledge arguments allows one party to prove that a statement is true, without leaking any other information than the truth of the statement. In many applications such as verifiable shuffle (as a practical application) and circuit satisfiability (as a theoretical application), zero-knowledge arguments for mathematical statements related to linear algebra are essentially used. Groth proposed (at CRYPTO 2009) an elegant methodology for zero-knowledge arguments for linear algebraic relations over finite fields. He obtained zero-knowledge arguments of the sub-linear size for linear algebra using reductions from linear algebraic relations to equations of the form z = x *' y, where x, y ∈ Fnp are committed vectors, z ∈ Fp is a committed element, and *' : Fnp × Fnp → Fp is a bilinear map. These reductions impose additional rounds on zero-knowledge arguments of the sub-linear size. The round complexity of interactive zero-knowledge arguments is an important measure along with communication and computational complexities. We focus on minimizing the round complexity of sub-linear zero-knowledge arguments for linear algebra. To reduce round complexity, we propose a general transformation from a t-round zero-knowledge argument, satisfying mild conditions, to a (t - 2)-round zero-knowledge argument; this transformation is of independent interest.
NASA Astrophysics Data System (ADS)
Orantin, N.
2007-09-01
The 2-matrix model has been introduced to study Ising model on random surfaces. Since then, the link between matrix models and combinatorics of discrete surfaces has strongly tightened. This manuscript aims to investigate these deep links and extend them beyond the matrix models, following my work's evolution. First, I take care to define properly the hermitian 2 matrix model which gives rise to generating functions of discrete surfaces equipped with a spin structure. Then, I show how to compute all the terms in the topological expansion of any observable by using algebraic geometry tools. They are obtained as differential forms on an algebraic curve associated to the model: the spectral curve. In a second part, I show how to define such differentials on any algebraic curve even if it does not come from a matrix model. I then study their numerous symmetry properties under deformations of the algebraic curve. In particular, I show that these objects coincide with the topological expansion of the observable of a matrix model if the algebraic curve is the spectral curve of this model. Finally, I show that fine tuning the parameters ensure that these objects can be promoted to modular invariants and satisfy the holomorphic anomaly equation of the Kodaira-Spencer theory. This gives a new hint that the Dijkgraaf-Vafa conjecture is correct.
Jucys-Murphy elements for Birman-Murakami-Wenzl algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Ogievetsky, O. V.
2011-05-01
The Burman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys-Murphy elements. We show that the set of Jucys-Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.
Kinematical superalgebras and Lie algebras of order 3
Campoamor-Stursberg, R.; Rausch de Traubenberg, M.
2008-06-15
We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order 3. All these algebras are related through generalized Inonue-Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order 3.
Becchi-Rouet-Stora-Tyutin operators for W algebras
Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.
2008-07-15
The study of quantum Lie algebras motivates a use of noncanonical ghosts and antighosts for nonlinear algebras, such as W-algebras. This leads, for the W{sub 3} and W{sub 3}{sup (2)} algebras, to the Becchi-Rouet-Stora-Tyutin operator having the conventional cubic form.