Quadratic eigenvalue problems.
Walsh, Timothy Francis; Day, David Minot
2007-04-01
In this report we will describe some nonlinear eigenvalue problems that arise in the areas of solid mechanics, acoustics, and coupled structural acoustics. We will focus mostly on quadratic eigenvalue problems, which are a special case of nonlinear eigenvalue problems. Algorithms for solving the quadratic eigenvalue problem will be presented, along with some example calculations.
Structured eigenvalue problems for rational gauss quadrature
NASA Astrophysics Data System (ADS)
Fasino, Dario; Gemignani, Luca
2007-08-01
The connection between Gauss quadrature rules and the algebraic eigenvalue problem for a Jacobi matrix was first exploited in the now classical paper by Golub and Welsch (Math. Comput. 23(106), 221?230, 1969). From then on many computational problems arising in the construction of (polynomial) Gauss quadrature formulas have been reduced to solving direct and inverse eigenvalue problems for symmetric tridiagonals. Over the last few years (rational) generalizations of the classical Gauss quadrature formulas have been studied, i.e., formulas integrating exactly in spaces of rational functions. This paper wants to illustrate that stable and efficient procedures based on structured numerical linear algebra techniques can also be devised for the solution of the eigenvalue problems arising in the field of rational Gauss quadrature.
Solving Large-scale Eigenvalue Problems in SciDACApplications
Yang, Chao
2005-06-29
Large-scale eigenvalue problems arise in a number of DOE applications. This paper provides an overview of the recent development of eigenvalue computation in the context of two SciDAC applications. We emphasize the importance of Krylov subspace methods, and point out its limitations. We discuss the value of alternative approaches that are more amenable to the use of preconditioners, and report the progression using the multi-level algebraic sub-structuring techniques to speed up eigenvalue calculation. In addition to methods for linear eigenvalue problems, we also examine new approaches to solving two types of non-linear eigenvalue problems arising from SciDAC applications.
Highly indefinite multigrid for eigenvalue problems
Borges, L.; Oliveira, S.
1996-12-31
Eigenvalue problems are extremely important in understanding dynamic processes such as vibrations and control systems. Large scale eigenvalue problems can be very difficult to solve, especially if a large number of eigenvalues and the corresponding eigenvectors need to be computed. For solving this problem a multigrid preconditioned algorithm is presented in {open_quotes}The Davidson Algorithm, preconditioning and misconvergence{close_quotes}. Another approach for solving eigenvalue problems is by developing efficient solutions for highly indefinite problems. In this paper we concentrate on the use of new highly indefinite multigrid algorithms for the eigenvalue problem.
Covariance expressions for eigenvalue and eigenvector problems
NASA Astrophysics Data System (ADS)
Liounis, Andrew J.
There are a number of important scientific and engineering problems whose solutions take the form of an eigenvalue--eigenvector problem. Some notable examples include solutions to linear systems of ordinary differential equations, controllability of linear systems, finite element analysis, chemical kinetics, fitting ellipses to noisy data, and optimal estimation of attitude from unit vectors. In many of these problems, having knowledge of the eigenvalue and eigenvector Jacobians is either necessary or is nearly as important as having the solution itself. For instance, Jacobians are necessary to find the uncertainty in a computed eigenvalue or eigenvector estimate. This uncertainty, which is usually represented as a covariance matrix, has been well studied for problems similar to the eigenvalue and eigenvector problem, such as singular value decomposition. There has been substantially less research on the covariance of an optimal estimate originating from an eigenvalue-eigenvector problem. In this thesis we develop two general expressions for the Jacobians of eigenvalues and eigenvectors with respect to the elements of their parent matrix. The expressions developed make use of only the parent matrix and the eigenvalue and eigenvector pair under consideration. In addition, they are applicable to any general matrix (including complex valued matrices, eigenvalues, and eigenvectors) as long as the eigenvalues are simple. Alongside this, we develop expressions that determine the uncertainty in a vector estimate obtained from an eigenvalue-eigenvector problem given the uncertainty of the terms of the matrix. The Jacobian expressions developed are numerically validated with forward finite, differencing and the covariance expressions are validated using Monte Carlo analysis. Finally, the results from this work are used to determine covariance expressions for a variety of estimation problem examples and are also applied to the design of a dynamical system.
ARPACK: Solving large scale eigenvalue problems
NASA Astrophysics Data System (ADS)
Lehoucq, Rich; Maschhoff, Kristi; Sorensen, Danny; Yang, Chao
2013-11-01
ARPACK is a collection of Fortran77 subroutines designed to solve large scale eigenvalue problems. The package is designed to compute a few eigenvalues and corresponding eigenvectors of a general n by n matrix A. It is most appropriate for large sparse or structured matrices A where structured means that a matrix-vector product w
The Numeric Solution of Eigenvalue Problems.
ERIC Educational Resources Information Center
Bauer, H.; Roth, K.
1980-01-01
Presents the mathematical background for solving eigenvalue problems, with illustrations of the applications in computer programing. The numerical matrix treatment is presented, with a demonstration of the simple HMO theory. (CS)
Some shape optimization problems for eigenvalues
NASA Astrophysics Data System (ADS)
Gasimov, Yusif S.
2008-02-01
In this work we consider some inverse problems with respect to domain for the Laplace operator. The considered problems are reduced to the variational formulation. The equivalency of these problems is obtained under some conditions. The formula is obtained for the eigenvalue in the optimal domain.
Sensitivity analysis and approximation methods for general eigenvalue problems
NASA Technical Reports Server (NTRS)
Murthy, D. V.; Haftka, R. T.
1986-01-01
Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought.
Numerical solution of large nonsymmetric eigenvalue problems
NASA Technical Reports Server (NTRS)
Saad, Youcef
1988-01-01
Several methods are discribed for combinations of Krylov subspace techniques, deflation procedures and preconditionings, for computing a small number of eigenvalues and eigenvectors or Schur vectors of large sparse matrices. The most effective techniques for solving realistic problems from applications are those methods based on some form of preconditioning and one of several Krylov subspace techniques, such as Arnoldi's method or Lanczos procedure. Two forms of preconditioning are considered: shift-and-invert and polynomial acceleration. The latter presents some advantages for parallel/vector processing but may be ineffective if eigenvalues inside the spectrum are sought. Some algorithmic details are provided that improve the reliability and effectiveness of these techniques.
Eigenvalue and eigenvector sensitivity and approximate analysis for repeated eigenvalue problems
NASA Technical Reports Server (NTRS)
Hou, Gene J. W.; Kenny, Sean P.
1991-01-01
A set of computationally efficient equations for eigenvalue and eigenvector sensitivity analysis are derived, and a method for eigenvalue and eigenvector approximate analysis in the presence of repeated eigenvalues is presented. The method developed for approximate analysis involves a reparamaterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem. Examples are given to demonstrate the application of such equations for sensitivity and approximate analysis.
Sparse Regression as a Sparse Eigenvalue Problem
NASA Technical Reports Server (NTRS)
Moghaddam, Baback; Gruber, Amit; Weiss, Yair; Avidan, Shai
2008-01-01
We extend the l0-norm "subspectral" algorithms for sparse-LDA [5] and sparse-PCA [6] to general quadratic costs such as MSE in linear (kernel) regression. The resulting "Sparse Least Squares" (SLS) problem is also NP-hard, by way of its equivalence to a rank-1 sparse eigenvalue problem (e.g., binary sparse-LDA [7]). Specifically, for a general quadratic cost we use a highly-efficient technique for direct eigenvalue computation using partitioned matrix inverses which leads to dramatic x103 speed-ups over standard eigenvalue decomposition. This increased efficiency mitigates the O(n4) scaling behaviour that up to now has limited the previous algorithms' utility for high-dimensional learning problems. Moreover, the new computation prioritizes the role of the less-myopic backward elimination stage which becomes more efficient than forward selection. Similarly, branch-and-bound search for Exact Sparse Least Squares (ESLS) also benefits from partitioned matrix inverse techniques. Our Greedy Sparse Least Squares (GSLS) generalizes Natarajan's algorithm [9] also known as Order-Recursive Matching Pursuit (ORMP). Specifically, the forward half of GSLS is exactly equivalent to ORMP but more efficient. By including the backward pass, which only doubles the computation, we can achieve lower MSE than ORMP. Experimental comparisons to the state-of-the-art LARS algorithm [3] show forward-GSLS is faster, more accurate and more flexible in terms of choice of regularization
Non-conforming finite element methods for transmission eigenvalue problem
NASA Astrophysics Data System (ADS)
Yang, Yidu; Han, Jiayu; Bi, Hai
2016-08-01
The transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory. In this paper, for the Helmholtz transmission eigenvalue problem, we give a weak formulation which is a nonselfadjoint linear eigenvalue problem. Based on the weak formulation, we first discuss the non-conforming finite element approximation, and prove the error estimates of the discrete eigenvalues obtained by the Adini element, Morley-Zienkiewicz element, modified-Zienkiewicz element et. al. And we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem.
Solving large sparse eigenvalue problems on supercomputers
NASA Technical Reports Server (NTRS)
Philippe, Bernard; Saad, Youcef
1988-01-01
An important problem in scientific computing consists in finding a few eigenvalues and corresponding eigenvectors of a very large and sparse matrix. The most popular methods to solve these problems are based on projection techniques on appropriate subspaces. The main attraction of these methods is that they only require the use of the matrix in the form of matrix by vector multiplications. The implementations on supercomputers of two such methods for symmetric matrices, namely Lanczos' method and Davidson's method are compared. Since one of the most important operations in these two methods is the multiplication of vectors by the sparse matrix, methods of performing this operation efficiently are discussed. The advantages and the disadvantages of each method are compared and implementation aspects are discussed. Numerical experiments on a one processor CRAY 2 and CRAY X-MP are reported. Possible parallel implementations are also discussed.
The cumulative reaction probability as eigenvalue problem
NASA Astrophysics Data System (ADS)
Manthe, Uwe; Miller, William H.
1993-09-01
It is shown that the cumulative reaction probability for a chemical reaction can be expressed (absolutely rigorously) as N(E)=∑kpk(E), where {pk} are the eigenvalues of a certain Hermitian matrix (or operator). The eigenvalues {pk} all lie between 0 and 1 and thus have the interpretation as probabilities, eigenreaction probabilities which may be thought of as the rigorous generalization of the transmission coefficients for the various states of the activated complex in transition state theory. The eigenreaction probabilities {pk} can be determined by diagonalizing a matrix that is directly available from the Hamiltonian matrix itself. It is also shown how a very efficient iterative method can be used to determine the eigenreaction probabilities for problems that are too large for a direct diagonalization to be possible. The number of iterations required is much smaller than that of previous methods, approximately the number of eigenreaction probabilities that are significantly different from zero. All of these new ideas are illustrated by application to three model problems—transmission through a one-dimensional (Eckart potential) barrier, the collinear H+H2→H2+H reaction, and the three-dimensional version of this reaction for total angular momentum J=0.
Preconditioned Krylov subspace methods for eigenvalue problems
Wu, Kesheng; Saad, Y.; Stathopoulos, A.
1996-12-31
Lanczos algorithm is a commonly used method for finding a few extreme eigenvalues of symmetric matrices. It is effective if the wanted eigenvalues have large relative separations. If separations are small, several alternatives are often used, including the shift-invert Lanczos method, the preconditioned Lanczos method, and Davidson method. The shift-invert Lanczos method requires direct factorization of the matrix, which is often impractical if the matrix is large. In these cases preconditioned schemes are preferred. Many applications require solution of hundreds or thousands of eigenvalues of large sparse matrices, which pose serious challenges for both iterative eigenvalue solver and preconditioner. In this paper we will explore several preconditioned eigenvalue solvers and identify the ones suited for finding large number of eigenvalues. Methods discussed in this paper make up the core of a preconditioned eigenvalue toolkit under construction.
Differential eigenvalue problems in which the parameter appears nonlinearly
NASA Technical Reports Server (NTRS)
Bridges, T. J.; Morris, P. J.
1984-01-01
Several methods are examined for determining the eigenvalues of a system of equations in which the parameter appears nonlinearly. The equations are the result of the discretization of differential eigenvalue problems using a finite Chebyshev series. Two global methods are considered which determine the spectrum of eigenvalues without an initial estimate. A local iteration scheme with cubic convergence is presented. Calculations are performed for a model second order differential problem and the Orr-Sommerfeld problem for plane Poiseuille flow.
An Implementation and Evaluation of the AMLS Method for SparseEigenvalue Problems
Gao, Weiguo; Li, Xiaoye S.; Yang, Chao; Bai, Zhaojun
2006-02-14
We describe an efficient implementation and present aperformance study of an algebraic multilevel sub-structuring (AMLS)method for sparse eigenvalue problems. We assess the time and memoryrequirements associated with the key steps of the algorithm, and compareitwith the shift-and-invert Lanczos algorithm in computational cost. Oureigenvalue problems come from two very different application areas: theaccelerator cavity design and the normal mode vibrational analysis of thepolyethylene particles. We show that the AMLS method, when implementedcarefully, is very competitive with the traditional method in broadapplication areas, especially when large numbers of eigenvalues aresought.
Extension of the tridiagonal reduction (FEER) method for complex eigenvalue problems in NASTRAN
NASA Technical Reports Server (NTRS)
Newman, M.; Mann, F. I.
1978-01-01
As in the case of real eigenvalue analysis, the eigensolutions closest to a selected point in the eigenspectrum were extracted from a reduced, symmetric, tridiagonal eigenmatrix whose order was much lower than that of the full size problem. The reduction process was effected automatically, and thus avoided the arbitrary lumping of masses and other physical quantities at selected grid points. The statement of the algebraic eigenvalue problem admitted mass, damping, and stiffness matrices which were unrestricted in character, i.e., they might be real, symmetric or nonsymmetric, singular or nonsingular.
Chebyshev polynomials in the spectral Tau method and applications to Eigenvalue problems
NASA Technical Reports Server (NTRS)
Johnson, Duane
1996-01-01
Chebyshev Spectral methods have received much attention recently as a technique for the rapid solution of ordinary differential equations. This technique also works well for solving linear eigenvalue problems. Specific detail is given to the properties and algebra of chebyshev polynomials; the use of chebyshev polynomials in spectral methods; and the recurrence relationships that are developed. These formula and equations are then applied to several examples which are worked out in detail. The appendix contains an example FORTRAN program used in solving an eigenvalue problem.
EvArnoldi: A New Algorithm for Large-Scale Eigenvalue Problems.
Tal-Ezer, Hillel
2016-05-19
Eigenvalues and eigenvectors are an essential theme in numerical linear algebra. Their study is mainly motivated by their high importance in a wide range of applications. Knowledge of eigenvalues is essential in quantum molecular science. Solutions of the Schrödinger equation for the electrons composing the molecule are the basis of electronic structure theory. Electronic eigenvalues compose the potential energy surfaces for nuclear motion. The eigenvectors allow calculation of diople transition matrix elements, the core of spectroscopy. The vibrational dynamics molecule also requires knowledge of the eigenvalues of the vibrational Hamiltonian. Typically in these problems, the dimension of Hilbert space is huge. Practically, only a small subset of eigenvalues is required. In this paper, we present a highly efficient algorithm, named EvArnoldi, for solving the large-scale eigenvalues problem. The algorithm, in its basic formulation, is mathematically equivalent to ARPACK ( Sorensen , D. C. Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations ; Springer , 1997 ; Lehoucq , R. B. ; Sorensen , D. C. SIAM Journal on Matrix Analysis and Applications 1996 , 17 , 789 ; Calvetti , D. ; Reichel , L. ; Sorensen , D. C. Electronic Transactions on Numerical Analysis 1994 , 2 , 21 ) (or Eigs of Matlab) but significantly simpler. PMID:27015379
TWO-GRID METHODS FOR MAXWELL EIGENVALUE PROBLEMS
ZHOU, J.; HU, X.; ZHONG, L.; SHU, S.; CHEN, L.
2015-01-01
Two new two-grid algorithms are proposed for solving the Maxwell eigenvalue problem. The new methods are based on the two-grid methodology recently proposed by Xu and Zhou [Math. Comp., 70 (2001), pp. 17–25] and further developed by Hu and Cheng [Math. Comp., 80 (2011), pp. 1287–1301] for elliptic eigenvalue problems. The new two-grid schemes reduce the solution of the Maxwell eigenvalue problem on a fine grid to one linear indefinite Maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid. The new schemes, therefore, save total computational cost. The error estimates reveals that the two-grid methods maintain asymptotically optimal accuracy, and the numerical experiments presented confirm the theoretical results. PMID:26190866
On 2D bisection method for double eigenvalue problems
Ji, X.
1996-06-01
The two-dimensional bisection method presented in (SIAM J. Matrix Anal. Appl. 13(4), 1085 (1992)) is efficient for solving a class of double eigenvalue problems. This paper further extends the 2D bisection method of full matrix cases and analyses its stability. As in a single parameter case, the 2D bisection method is very stable for the tridiagonal matrix triples satisfying the symmetric-definite condition. Since the double eigenvalue problems arise from two-parameter boundary value problems, an estimate of the discretization error in eigenpairs is also given. Some numerical examples are included. 42 refs., 1 tab.
An analytically solvable eigenvalue problem for the linear elasticity equations.
Day, David Minot; Romero, Louis Anthony
2004-07-01
Analytic solutions are useful for code verification. Structural vibration codes approximate solutions to the eigenvalue problem for the linear elasticity equations (Navier's equations). Unfortunately the verification method of 'manufactured solutions' does not apply to vibration problems. Verification books (for example [2]) tabulate a few of the lowest modes, but are not useful for computations of large numbers of modes. A closed form solution is presented here for all the eigenvalues and eigenfunctions for a cuboid solid with isotropic material properties. The boundary conditions correspond physically to a greased wall.
Dynamic Restarting Schemes for Eigenvalue Problems
Wu, Kesheng; Simon, Horst D.
1999-03-10
In studies of restarted Davidson method, a dynamic thick-restart scheme was found to be excellent in improving the overall effectiveness of the eigen value method. This paper extends the study of the dynamic thick-restart scheme to the Lanczos method for symmetric eigen value problems and systematically explore a range of heuristics and strategies. We conduct a series of numerical tests to determine their relative strength and weakness on a class of electronic structure calculation problems.
NASA Astrophysics Data System (ADS)
Plestenjak, Bor; Gheorghiu, Călin I.; Hochstenbach, Michiel E.
2015-10-01
In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu's system, Lamé's system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently. The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems.
ERIC Educational Resources Information Center
Nyman, Melvin A.; Lapp, Douglas A.; St. John, Dennis; Berry, John S.
2010-01-01
This paper discusses student difficulties in grasping concepts from Linear Algebra--in particular, the connection of eigenvalues and eigenvectors to other important topics in linear algebra. Based on our prior observations from student interviews, we propose technology-enhanced instructional approaches that might positively impact student…
Finite element method for eigenvalue problems in electromagnetics
NASA Technical Reports Server (NTRS)
Reddy, C. J.; Deshpande, Manohar D.; Cockrell, C. R.; Beck, Fred B.
1994-01-01
Finite element method (FEM) has been a very powerful tool to solve many complex problems in electromagnetics. The goal of the current research at the Langley Research Center is to develop a combined FEM/method of moments approach to three-dimensional scattering/radiation problem for objects with arbitrary shape and filled with complex materials. As a first step toward that goal, an exercise is taken to establish the power of FEM, through closed boundary problems. This paper demonstrates the developed of FEM tools for two- and three-dimensional eigenvalue problems in electromagnetics. In section 2, both the scalar and vector finite elements have been used for various waveguide problems to demonstrate the flexibility of FEM. In section 3, vector finite element method has been extended to three-dimensional eigenvalue problems.
The trace minimization method for the symmetric generalized eigenvalue problem
NASA Astrophysics Data System (ADS)
Sameh, Ahmed; Tong, Zhanye
2000-11-01
In this paper, the trace minimization method for the generalized symmetric eigenvalue problems proposed by Sameh and Wisniewski [35] is reviewed. Convergence of an inexact trace minimization algorithm is established and a variant of the algorithm that uses expanding subspaces is introduced and compared with the block Jacobi-Davidson algorithm.
A Projection free method for Generalized Eigenvalue Problem with a nonsmooth Regularizer
Hwang, Seong Jae; Collins, Maxwell D.; Ravi, Sathya N.; Ithapu, Vamsi K.; Adluru, Nagesh; Johnson, Sterling C.; Singh, Vikas
2016-01-01
Eigenvalue problems are ubiquitous in computer vision, covering a very broad spectrum of applications ranging from estimation problems in multi-view geometry to image segmentation. Few other linear algebra problems have a more mature set of numerical routines available and many computer vision libraries leverage such tools extensively. However, the ability to call the underlying solver only as a “black box” can often become restrictive. Many ‘human in the loop’ settings in vision frequently exploit supervision from an expert, to the extent that the user can be considered a subroutine in the overall system. In other cases, there is additional domain knowledge, side or even partial information that one may want to incorporate within the formulation. In general, regularizing a (generalized) eigenvalue problem with such side information remains difficult. Motivated by these needs, this paper presents an optimization scheme to solve generalized eigenvalue problems (GEP) involving a (nonsmooth) regularizer. We start from an alternative formulation of GEP where the feasibility set of the model involves the Stiefel manifold. The core of this paper presents an end to end stochastic optimization scheme for the resultant problem. We show how this general algorithm enables improved statistical analysis of brain imaging data where the regularizer is derived from other ‘views’ of the disease pathology, involving clinical measurements and other image-derived representations. PMID:27081374
Nonlinear eigenvalue problems in Density Functional Theory calculations
Fattebert, J
2009-08-28
Developed in the 1960's by W. Kohn and coauthors, Density Functional Theory (DFT) is a very popular quantum model for First-Principles simulations in chemistry and material sciences. It allows calculations of systems made of hundreds of atoms. Indeed DFT reduces the 3N-dimensional Schroedinger electronic structure problem to the search for a ground state electronic density in 3D. In practice it leads to the search for N electronic wave functions solutions of an energy minimization problem in 3D, or equivalently the solution of an eigenvalue problem with a non-linear operator.
A Many Body Eigenvalue Problem for Quantum Computation
NASA Astrophysics Data System (ADS)
Hershfield, Selman
2008-03-01
A one dimensional many body Hamiltonian is presented whose eigenvalues are related to the order of GN. This is the same order of GN used to decode the RSA algorithm. For some values of N the Hamiltonian is a noninteracting fermion problem. For other values of N the Hamiltonian is a quantum impurity problem with fermions interacting with a spin-like object. However, the generic case has fermions or spins interacting with higher order interactions beyond two body interactions. Because this is a mapping between two different classes of problems, one of interest in quantum computing and the other a more traditional condensed matter physics Hamiltonian, we will show (i) how knowledge of the order of GN can be used to solve some novel one dimensional strongly correlated problems and (ii) how numerical techniques, particularly for quantum impurity limit, can be used to find the order of GN.
NASA Astrophysics Data System (ADS)
Lykke Jacobsen, Jesper
2015-11-01
In previous work with Scullard, we have defined a graph polynomial P B (q, T) that gives access to the critical temperature T c of the q-state Potts model defined on a general two-dimensional lattice {L}. It depends on a basis B, containing n × m unit cells of {L}, and the relevant root T c(n, m) of P B (q, T) was observed to converge quickly to T c in the limit n,m\\to ∞ . Moreover, in exactly solvable cases there is no finite-size dependence at all. In this paper we show how to reformulate this method as an eigenvalue problem within the periodic Temperley-Lieb (TL) algebra. This corresponds to taking m\\to ∞ first, so that the bases B are semi-infinite cylinders of circumference n. The limit implies faster convergence in n, while maintaining the n-independence in exactly solvable cases. In this setup, T c(n) is determined by equating the largest eigenvalues of two topologically distinct sectors of the transfer matrix. Crucially, these two sectors determine the same critical exponent in the continuum limit, and the observed fast convergence is thus corroborated by results of conformal field theory. We obtain similar results for the dense and dilute phases of the O(N) loop model, using now a transfer matrix within the dilute periodic TL algebra. Compared with our previous study, the eigenvalue formulation allows us to double the size n for which T c(n) can be obtained, using the same computational effort. We study in details three significant cases: (i) bond percolation on the kagome lattice, up to n max = 14; (ii) site percolation on the square lattice, to n max = 21; and (iii) self-avoiding polygons on the square lattice, to n max = 19. Convergence properties of T c(n) and extrapolation schemes are studied in details for the first two cases. This leads to rather accurate values for the percolation thresholds: p c = 0.524 404 999 167 439(4) for bond percolation on the kagome lattice, and p c = 0.592 746 050 792 10(2) for site percolation on the square lattice.
Eigenvalue problem of the Liouvillian of open quantum systems
Hatano, Naomichi; Petrosky, Tomio
2015-03-10
It is argued that the Liouvillian that appears in the Liouville-von Neumann equation for open quantum systems can have complex eigenvalues. Attention is paid to the question whether the Liouvillian has an eigenvalue that are not given by the difference of the two Hamiltonian eigenvalues.
NASA Technical Reports Server (NTRS)
Kenny, Sean P.; Hou, Gene J. W.
1994-01-01
A method for eigenvalue and eigenvector approximate analysis for the case of repeated eigenvalues with distinct first derivatives is presented. The approximate analysis method developed involves a reparameterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations to changes in the eigenvalues and the eigenvectors associated with the repeated eigenvalue problem. This work also presents a numerical technique that facilitates the definition of an eigenvector derivative for the case of repeated eigenvalues with repeated eigenvalue derivatives (of all orders). Examples are given which demonstrate the application of such equations for sensitivity and approximate analysis. Emphasis is placed on the application of sensitivity analysis to large-scale structural and controls-structures optimization problems.
NASA Astrophysics Data System (ADS)
Fokas, A. S.; Anderson, R. L.
1982-06-01
We present an algorithmic method for obtaining an hereditary symmetry (the generalized squared-eigenfunction operator) from a given isospectral eigenvalue problem. This method is applied to the n×n eigenvalue problem considered by Ablowitz and Haberman and to the eigenvalue problem considered by Alonso. The relevant Hamiltonian formulations are also determined. Finally, an alternative method is presented in the case two evolution equations are related by a Miura type transformation and their Hamiltonian formulations are known.
NASA Technical Reports Server (NTRS)
Ward, R. C.
1974-01-01
Backward error analyses of the application of Householder transformations to both the standard and the generalized eigenvalue problems are presented. The analysis for the standard eigenvalue problem determines the error from the application of an exact similarity transformation, and the analysis for the generalized eigenvalue problem determines the error from the application of an exact equivalence transformation. Bounds for the norms of the resulting perturbation matrices are presented and compared with existing bounds when known.
Numerical Solution of the k-Eigenvalue Problem
NASA Astrophysics Data System (ADS)
Hamilton, Steven Paul
2011-12-01
Obtaining solutions to the k-eigenvalue form of the radiation transport equation is an important topic in the design and analysis of nuclear reactors. Although this has been an area of active interest in the nuclear engineering community for several decades, to date no truly satisfactory solution strategies exist. In general, existing techniques are either slow to converge for difficult problems or suffer from stability and robustness issues that can cause solvers to diverge for some problems. This work provides a comparison between a variety of methods and introduces a new strategy based on the Davidson method that has been used in other fields for many years but never for this problem. The Davidson method offers an alternative to the nested iteration structure inherent to standard approaches and allows expensive linear solvers to be replaced by a potentially cheap preconditioner. To fill the role of this preconditioner, a strategy based on a multigrid treatment of the energy variable is developed. Numerical experiments using the 2-D NEWT transport package are presented, demonstrating the effectiveness of the proposed strategy.
Discrete Ordinate Quadrature Selection for Reactor-based Eigenvalue Problems
Jarrell, Joshua J; Evans, Thomas M; Davidson, Gregory G
2013-01-01
In this paper we analyze the effect of various quadrature sets on the eigenvalues of several reactor-based problems, including a two-dimensional (2D) fuel pin, a 2D lattice of fuel pins, and a three-dimensional (3D) reactor core problem. While many quadrature sets have been applied to neutral particle discrete ordinate transport calculations, the Level Symmetric (LS) and the Gauss-Chebyshev product (GC) sets are the most widely used in production-level reactor simulations. Other quadrature sets, such as Quadruple Range (QR) sets, have been shown to be more accurate in shielding applications. In this paper, we compare the LS, GC, QR, and the recently developed linear-discontinuous finite element (LDFE) sets, as well as give a brief overview of other proposed quadrature sets. We show that, for a given number of angles, the QR sets are more accurate than the LS and GC in all types of reactor problems analyzed (2D and 3D). We also show that the LDFE sets are more accurate than the LS and GC sets for these problems. We conclude that, for problems where tens to hundreds of quadrature points (directions) per octant are appropriate, QR sets should regularly be used because they have similar integration properties as the LS and GC sets, have no noticeable impact on the speed of convergence of the solution when compared with other quadrature sets, and yield more accurate results. We note that, for very high-order scattering problems, the QR sets exactly integrate fewer angular flux moments over the unit sphere than the GC sets. The effects of those inexact integrations have yet to be analyzed. We also note that the LDFE sets only exactly integrate the zeroth and first angular flux moments. Pin power comparisons and analyses are not included in this paper and are left for future work.
Numerical study of three-parameter matrix eigenvalue problem by Rayleigh quotient method
NASA Astrophysics Data System (ADS)
Bora, Niranjan; Baruah, Arun Kumar
2016-06-01
In this paper, an attempt is done to find approximate eigenvalues and the corresponding eigenvectors of three-parameter matrix eigenvalue problem by extending Rayleigh Quotient Iteration Method (RQIM), which is generally used to solve generalized eigenvalue problems of the form Ax = λBx. Convergence criteria of RQIM will be derived in terms of matrix 2-norm. We will test the computational efficiency of the Method analytically with the help of numerical examples. All calculations are done in MATLAB software.
Eigenvalue inequalities for the buckling problem of the drifting Laplacian on Ricci solitons
NASA Astrophysics Data System (ADS)
Du, Feng; Mao, Jing; Wang, Qiaoling; Wu, Chuanxi
2016-04-01
In this paper, we investigate the buckling problem of the drifting Laplacian and get a general inequality for its eigenvalues on a bounded connected domain in complete Ricci solitons supporting a special function. By applying this general inequality, we obtain some universal inequalities for eigenvalues of the same problem on bounded connected domains in the Gaussian shrinking solitons and some general product solitons.
A parallel algorithm for the non-symmetric eigenvalue problem
Dongarra, J.; Sidani, M. . Dept. of Computer Science Oak Ridge National Lab., TN )
1991-12-01
This paper describes a parallel algorithm for computing the eigenvalues and eigenvectors of a non-symmetric matrix. The algorithm is based on a divide-and-conquer procedure and uses an iterative refinement technique.
Thick-Restart Laczos Method for Symmetric Eigenvalue Problems
1999-01-01
This software package implements the thick-restart Lanczos method. It can be used on either a single address space machine or distributed parallel machine. The user can choose to implement or use a matrix-vector multiplication routine in any form convenient. Most of the arithmetic computations in the software are done through calls to BLAS and LAPACK. The software is written in Fortran 90. Because Fortran 90 offers many utility functions such functions such as dynamic memorymore » management, timing functions, random number generator and so on, the program is easily portable to different machines without modifying the source code. It can also be easily accessed from other language such as C or C-+. Since the software is highly modularized, it is relatively easy to adopt it for different type of situations. For example if the eigenvalue problem may have some symmetry and only a portion of the physical domain is discretized, then the dot-product routine needs to be modified. In this software, this modification is limited to one subroutine. It also can be instructed to write checkpoint files so that it can be restarted at a later time.« less
NASA Astrophysics Data System (ADS)
Qiu, Zhiping; Wang, Xiaojun
2005-04-01
Generalized eigenvalue problems from the modal analysis are often converted to the standard eigenvalue problems. In this paper, it evaluates the upper and lower bounds on the eigenvalues of the standard eigenvalue problem of structures subject to severely deficient information about the structural parameters. Here, we focus on non-probabilistic interval analysis models of uncertainty, which are adapted to the case of severe lack of information on uncertainty. Non-probabilistic, interval analysis method in which uncertainties are defined by interval numbers appears as an alternative to the classical probabilistic models. For the standard eigenvalue problem of structures with uncertain-but-bounded parameters, the vertex solution theorem, the positive semi-definite solution theorem and the parameter decomposition solution theorem for the standard eigenvalue problem are presented, and compared with Deif's solution theorem in numerical examples. It is shown that, for the upper and lower bounds on the eigenvalues of the standard eigenvalue problem with uncertain-but-bounded parameters, the presented vertex solution theorem is unconditional, and the positive semi-definite solution theorem and the parameter decomposition solution theorem have less limitary conditions compared with Deif's solution theorem. The effectiveness of the vertex solution theorem, the positive semi-definite solution theorem and the parameter decomposition solution theorem are illustrated by numerical examples
NASA Technical Reports Server (NTRS)
Costiner, Sorin; Taasan, Shlomo
1994-01-01
This paper presents multigrid (MG) techniques for nonlinear eigenvalue problems (EP) and emphasizes an MG algorithm for a nonlinear Schrodinger EP. The algorithm overcomes the mentioned difficulties combining the following techniques: an MG projection coupled with backrotations for separation of solutions and treatment of difficulties related to clusters of close and equal eigenvalues; MG subspace continuation techniques for treatment of the nonlinearity; an MG simultaneous treatment of the eigenvectors at the same time with the nonlinearity and with the global constraints. The simultaneous MG techniques reduce the large number of self consistent iterations to only a few or one MG simultaneous iteration and keep the solutions in a right neighborhood where the algorithm converges fast.
Willert, Jeffrey; Park, H.; Taitano, William
2015-10-12
High-order/low-order (or moment-based acceleration) algorithms have been used to significantly accelerate the solution to the neutron transport k-eigenvalue problem over the past several years. Recently, the nonlinear diffusion acceleration algorithm has been extended to solve fixed-source problems with anisotropic scattering sources. In this paper, we demonstrate that we can extend this algorithm to k-eigenvalue problems in which the scattering source is anisotropic and a significant acceleration can be achieved. Lastly, we demonstrate that the low-order, diffusion-like eigenvalue problem can be solved efficiently using a technique known as nonlinear elimination.
Efficient solutions to the NDA-NCA low-order eigenvalue problem
Willert, J. A.; Kelley, C. T.
2013-07-01
Recent algorithmic advances combine moment-based acceleration and Jacobian-Free Newton-Krylov (JFNK) methods to accelerate the computation of the dominant eigenvalue in a k-eigenvalue calculation. In particular, NDA-NCA [1], builds a sequence of low-order (LO) diffusion-based eigenvalue problems in which the solution converges to the true eigenvalue solution. Within NDA-NCA, the solution to the LO k-eigenvalue problem is computed by solving a system of nonlinear equation using some variant of Newton's method. We show that we can speed up the solution to the LO problem dramatically by abandoning the JFNK method and exploiting the structure of the Jacobian matrix. (authors)
Numerical linear algebra for reconstruction inverse problems
NASA Astrophysics Data System (ADS)
Nachaoui, Abdeljalil
2004-01-01
Our goal in this paper is to discuss various issues we have encountered in trying to find and implement efficient solvers for a boundary integral equation (BIE) formulation of an iterative method for solving a reconstruction problem. We survey some methods from numerical linear algebra, which are relevant for the solution of this class of inverse problems. We motivate the use of our constructing algorithm, discuss its implementation and mention the use of preconditioned Krylov methods.
NASA Technical Reports Server (NTRS)
Antar, B. N.
1976-01-01
A numerical technique is presented for locating the eigenvalues of two point linear differential eigenvalue problems. The technique is designed to search for complex eigenvalues belonging to complex operators. With this method, any domain of the complex eigenvalue plane could be scanned and the eigenvalues within it, if any, located. For an application of the method, the eigenvalues of the Orr-Sommerfeld equation of the plane Poiseuille flow are determined within a specified portion of the c-plane. The eigenvalues for alpha = 1 and R = 10,000 are tabulated and compared for accuracy with existing solutions.
Nease, Brian R. Ueki, Taro
2009-12-10
A time series approach has been applied to the nuclear fission source distribution generated by Monte Carlo (MC) particle transport in order to calculate the non-fundamental mode eigenvalues of the system. The novel aspect is the combination of the general technical principle of projection pursuit for multivariate data with the neutron multiplication eigenvalue problem in the nuclear engineering discipline. Proof is thoroughly provided that the stationary MC process is linear to first order approximation and that it transforms into one-dimensional autoregressive processes of order one (AR(1)) via the automated choice of projection vectors. The autocorrelation coefficient of the resulting AR(1) process corresponds to the ratio of the desired mode eigenvalue to the fundamental mode eigenvalue. All modern MC codes for nuclear criticality calculate the fundamental mode eigenvalue, so the desired mode eigenvalue can be easily determined. This time series approach was tested for a variety of problems including multi-dimensional ones. Numerical results show that the time series approach has strong potential for three dimensional whole reactor core. The eigenvalue ratio can be updated in an on-the-fly manner without storing the nuclear fission source distributions at all previous iteration cycles for the mean subtraction. Lastly, the effects of degenerate eigenvalues are investigated and solutions are provided.
From Self-consistency to SOAR: Solving Large Scale NonlinearEigenvalue Problems
Bai, Zhaojun; Yang, Chao
2006-02-01
What is common among electronic structure calculation, design of MEMS devices, vibrational analysis of high speed railways, and simulation of the electromagnetic field of a particle accelerator? The answer: they all require solving large scale nonlinear eigenvalue problems. In fact, these are just a handful of examples in which solving nonlinear eigenvalue problems accurately and efficiently is becoming increasingly important. Recognizing the importance of this class of problems, an invited minisymposium dedicated to nonlinear eigenvalue problems was held at the 2005 SIAM Annual Meeting. The purpose of the minisymposium was to bring together numerical analysts and application scientists to showcase some of the cutting edge results from both communities and to discuss the challenges they are still facing. The minisymposium consisted of eight talks divided into two sessions. The first three talks focused on a type of nonlinear eigenvalue problem arising from electronic structure calculations. In this type of problem, the matrix Hamiltonian H depends, in a non-trivial way, on the set of eigenvectors X to be computed. The invariant subspace spanned by these eigenvectors also minimizes a total energy function that is highly nonlinear with respect to X on a manifold defined by a set of orthonormality constraints. In other applications, the nonlinearity of the matrix eigenvalue problem is restricted to the dependency of the matrix on the eigenvalues to be computed. These problems are often called polynomial or rational eigenvalue problems In the second session, Christian Mehl from Technical University of Berlin described numerical techniques for solving a special type of polynomial eigenvalue problem arising from vibration analysis of rail tracks excited by high-speed trains.
The eigenvalue spectrum of the Orr-Sommerfeld problem
NASA Technical Reports Server (NTRS)
Antar, B. N.
1976-01-01
A numerical investigation of the temporal eigenvalue spectrum of the ORR-Sommerfeld equation is presented. Two flow profiles are studied, the plane Poiseuille flow profile and the Blasius boundary layer (parallel): flow profile. In both cases a portion of the complex c-plane bounded by 0 less than or equal to CR sub r 1 and -1 less than or equal to ci sub i 0 is searched and the eigenvalues within it are identified. The spectra for the plane Poiseuille flow at alpha = 1.0 and R = 100, 1000, 6000, and 10000 are determined and compared with existing results where possible. The spectrum for the Blasius boundary layer flow at alpha = 0.308 and R = 998 was found to be infinite and discrete. Other spectra for the Blasius boundary layer at various Reynolds numbers seem to confirm this result. The eigenmodes belonging to these spectra were located and discussed.
Cai, Yunfeng; Bai, Zhaojun; Pask, John E.; Sukumar, N.
2013-12-15
The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for ab initio electronic structure calculations. We propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative diagonalization of such eigenvalue problems. In partition-of-unity finite-element (PUFE) pseudopotential density-functional calculations, employing a nonorthogonal basis, we show that the hybrid preconditioned block steepest descent method is a cost-effective eigensolver, outperforming current state-of-the-art global preconditioning schemes, and comparably efficient for the ill-conditioned generalized eigenvalue problems produced by PUFE as the locally optimal block preconditioned conjugate-gradient method for the well-conditioned standard eigenvalue problems produced by planewave methods.
Lower bounds for eigenvalues of self-adjoint problems
Gundersen, Gary G.
1979-01-01
The equation y″ + [λ - q(x)]y = 0 on (0, ∞) or (-∞, ∞), in which q(x) → ∞ as x → ∞ or x → ± ∞, has a complete set of eigenfunctions with discrete eigenvalues {λn}n=0∞. We derive an inequality that contains λn, by using a quick and elementary method that does not employ a comparison theorem or assume anything special. Explicit lower bounds for λn can often be easily obtained, and three examples are given. The method also gives respectable lower bounds for λn in the classical Sturm—Liouville case. PMID:16592718
NASA Astrophysics Data System (ADS)
Abramov, A. A.; Yukhno, L. F.
2016-07-01
A nonlinear eigenvalue problem for a linear system of ordinary differential equations is examined on a semi-infinite interval. The problem is supplemented by nonlocal conditions specified by a Stieltjes integral. At infinity, the solution must be bounded. In addition to these basic conditions, the solution must satisfy certain redundant conditions, which are also nonlocal. A numerically stable method for solving such a singular overdetermined eigenvalue problem is proposed and analyzed. The essence of the method is that this overdetermined problem is replaced by an auxiliary problem consistent with all the above conditions.
A numerical method for eigenvalue problems in modeling liquid crystals
Baglama, J.; Farrell, P.A.; Reichel, L.; Ruttan, A.; Calvetti, D.
1996-12-31
Equilibrium configurations of liquid crystals in finite containments are minimizers of the thermodynamic free energy of the system. It is important to be able to track the equilibrium configurations as the temperature of the liquid crystals decreases. The path of the minimal energy configuration at bifurcation points can be computed from the null space of a large sparse symmetric matrix. We describe a new variant of the implicitly restarted Lanczos method that is well suited for the computation of extreme eigenvalues of a large sparse symmetric matrix, and we use this method to determine the desired null space. Our implicitly restarted Lanczos method determines adoptively a polynomial filter by using Leja shifts, and does not require factorization of the matrix. The storage requirement of the method is small, and this makes it attractive to use for the present application.
Gene Golub; Kwok Ko
2009-03-30
The solutions of sparse eigenvalue problems and linear systems constitute one of the key computational kernels in the discretization of partial differential equations for the modeling of linear accelerators. The computational challenges faced by existing techniques for solving those sparse eigenvalue problems and linear systems call for continuing research to improve on the algorithms so that ever increasing problem size as required by the physics application can be tackled. Under the support of this award, the filter algorithm for solving large sparse eigenvalue problems was developed at Stanford to address the computational difficulties in the previous methods with the goal to enable accelerator simulations on then the world largest unclassified supercomputer at NERSC for this class of problems. Specifically, a new method, the Hemitian skew-Hemitian splitting method, was proposed and researched as an improved method for solving linear systems with non-Hermitian positive definite and semidefinite matrices.
A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem
Willert, Jeffrey; Park, H.; Knoll, D.A.
2014-10-01
Over the past several years a number of papers have been written describing modern techniques for numerically computing the dominant eigenvalue of the neutron transport criticality problem. These methods fall into two distinct categories. The first category of methods rewrite the multi-group k-eigenvalue problem as a nonlinear system of equations and solve the resulting system using either a Jacobian-Free Newton–Krylov (JFNK) method or Nonlinear Krylov Acceleration (NKA), a variant of Anderson Acceleration. These methods are generally successful in significantly reducing the number of transport sweeps required to compute the dominant eigenvalue. The second category of methods utilize Moment-Based Acceleration (or High-Order/Low-Order (HOLO) Acceleration). These methods solve a sequence of modified diffusion eigenvalue problems whose solutions converge to the solution of the original transport eigenvalue problem. This second class of methods is, in our experience, always superior to the first, as most of the computational work is eliminated by the acceleration from the LO diffusion system. In this paper, we review each of these methods. Our computational results support our claim that the choice of which nonlinear solver to use, JFNK or NKA, should be secondary. The primary computational savings result from the implementation of a HOLO algorithm. We display computational results for a series of challenging multi-dimensional test problems.
A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem
NASA Astrophysics Data System (ADS)
Willert, Jeffrey; Park, H.; Knoll, D. A.
2014-10-01
Over the past several years a number of papers have been written describing modern techniques for numerically computing the dominant eigenvalue of the neutron transport criticality problem. These methods fall into two distinct categories. The first category of methods rewrite the multi-group k-eigenvalue problem as a nonlinear system of equations and solve the resulting system using either a Jacobian-Free Newton-Krylov (JFNK) method or Nonlinear Krylov Acceleration (NKA), a variant of Anderson Acceleration. These methods are generally successful in significantly reducing the number of transport sweeps required to compute the dominant eigenvalue. The second category of methods utilize Moment-Based Acceleration (or High-Order/Low-Order (HOLO) Acceleration). These methods solve a sequence of modified diffusion eigenvalue problems whose solutions converge to the solution of the original transport eigenvalue problem. This second class of methods is, in our experience, always superior to the first, as most of the computational work is eliminated by the acceleration from the LO diffusion system. In this paper, we review each of these methods. Our computational results support our claim that the choice of which nonlinear solver to use, JFNK or NKA, should be secondary. The primary computational savings result from the implementation of a HOLO algorithm. We display computational results for a series of challenging multi-dimensional test problems.
Rees algebras, Monomial Subrings and Linear Optimization Problems
NASA Astrophysics Data System (ADS)
Dupont, Luis A.
2010-06-01
In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut property or the packing property. We also study algebraic properties of symbolic Rees algebras of edge ideals of graphs, edge ideals of clique clutters of comparability graphs, and Stanley-Reisner rings.
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.
Maximum/Minimum Problems Solved Using an Algebraic Way
ERIC Educational Resources Information Center
Modica, Erasmo
2010-01-01
This article describes some problems of the maximum/minimum type, which are generally solved using calculus at secondary school, but which here are solved algebraically. We prove six algebraic properties and then apply them to this kind of problem. This didactic approach allows pupils to solve these problems even at the beginning of secondary…
NASA Technical Reports Server (NTRS)
Wunsche, A.
1993-01-01
The eigenvalue problem of the operator a + zeta(boson creation operator) is solved for arbitrarily complex zeta by applying a nonunitary operator to the vacuum state. This nonunitary approach is compared with the unitary approach leading for the absolute value of zeta less than 1 to squeezed coherent states.
NASA Astrophysics Data System (ADS)
Calef, Matthew T.; Fichtl, Erin D.; Warsa, James S.; Berndt, Markus; Carlson, Neil N.
2013-04-01
We compare a variant of Anderson Mixing with the Jacobian-Free Newton-Krylov and Broyden methods applied to an instance of the k-eigenvalue formulation of the linear Boltzmann transport equation. We present evidence that one variant of Anderson Mixing finds solutions in the fewest number of iterations. We examine and strengthen theoretical results of Anderson Mixing applied to linear problems.
Parallel algorithms for 2-D cylindrical transport equations of Eigenvalue problem
Wei, J.; Yang, S.
2013-07-01
In this paper, aimed at the neutron transport equations of eigenvalue problem under 2-D cylindrical geometry on unstructured grid, the discrete scheme of Sn discrete ordinate and discontinuous finite is built, and the parallel computation for the scheme is realized on MPI systems. Numerical experiments indicate that the designed parallel algorithm can reach perfect speedup, it has good practicality and scalability. (authors)
Two-dimensional frustrated Ising network as an eigenvalue problem
NASA Astrophysics Data System (ADS)
Blackman, J. A.
1982-11-01
The Pfaffian method is used to study the square frustrated Ising network. The formalism is adapted in order to develop a relation with the problem of excitations in random alloys. It is shown that the counterpart of frustrated plaquettes are local modes within a band gap. Properties of the local modes are examined, including questions of gauge invariance and duality. Numerical calculations are done to investigate the way in which the local modes broaden into an impurity band.
NASA Astrophysics Data System (ADS)
Cakoni, Fioralba; Haddar, Houssem
2013-10-01
In inverse scattering theory, transmission eigenvalues can be seen as the extension of the notion of resonant frequencies for impenetrable objects to the case of penetrable dielectrics. The transmission eigenvalue problem is a relatively late arrival to the spectral theory of partial differential equations. Its first appearance was in 1986 in a paper by Kirsch who was investigating the denseness of far-field patterns for scattering solutions of the Helmholtz equation or, in more modern terminology, the injectivity of the far-field operator [1]. The paper of Kirsch was soon followed by a more systematic study by Colton and Monk in the context of developing the dual space method for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium [2]. In this paper they showed that for a spherically stratified media transmission eigenvalues existed and formed a discrete set. Numerical examples were also given showing that in principle transmission eigenvalues could be determined from the far-field data. This first period of interest in transmission eigenvalues was concluded with papers by Colton et al in 1989 [3] and Rynne and Sleeman in 1991 [4] showing that for an inhomogeneous medium (not necessarily spherically stratified) transmission eigenvalues, if they existed, formed a discrete set. For the next seventeen years transmission eigenvalues were ignored. This was mainly due to the fact that, with the introduction of various sampling methods to determine the shape of an inhomogeneous medium from far-field data, transmission eigenvalues were something to be avoided and hence the fact that transmission eigenvalues formed at most a discrete set was deemed to be sufficient. In addition, questions related to the existence of transmission eigenvalues or the structure of associated eigenvectors were recognized as being particularly difficult due to the nonlinearity of the eigenvalue problem and the special structure of the associated transmission
Using parallel banded linear system solvers in generalized eigenvalue problems
NASA Technical Reports Server (NTRS)
Zhang, Hong; Moss, William F.
1993-01-01
Subspace iteration is a reliable and cost effective method for solving positive definite banded symmetric generalized eigenproblems, especially in the case of large scale problems. This paper discusses an algorithm that makes use of two parallel banded solvers in subspace iteration. A shift is introduced to decompose the banded linear systems into relatively independent subsystems and to accelerate the iterations. With this shift, an eigenproblem is mapped efficiently into the memories of a multiprocessor and a high speed-up is obtained for parallel implementations. An optimal shift is a shift that balances total computation and communication costs. Under certain conditions, we show how to estimate an optimal shift analytically using the decay rate for the inverse of a banded matrix, and how to improve this estimate. Computational results on iPSC/2 and iPSC/860 multiprocessors are presented.
NASA Astrophysics Data System (ADS)
Cakoni, Fioralba; Haddar, Houssem
2013-10-01
In inverse scattering theory, transmission eigenvalues can be seen as the extension of the notion of resonant frequencies for impenetrable objects to the case of penetrable dielectrics. The transmission eigenvalue problem is a relatively late arrival to the spectral theory of partial differential equations. Its first appearance was in 1986 in a paper by Kirsch who was investigating the denseness of far-field patterns for scattering solutions of the Helmholtz equation or, in more modern terminology, the injectivity of the far-field operator [1]. The paper of Kirsch was soon followed by a more systematic study by Colton and Monk in the context of developing the dual space method for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium [2]. In this paper they showed that for a spherically stratified media transmission eigenvalues existed and formed a discrete set. Numerical examples were also given showing that in principle transmission eigenvalues could be determined from the far-field data. This first period of interest in transmission eigenvalues was concluded with papers by Colton et al in 1989 [3] and Rynne and Sleeman in 1991 [4] showing that for an inhomogeneous medium (not necessarily spherically stratified) transmission eigenvalues, if they existed, formed a discrete set. For the next seventeen years transmission eigenvalues were ignored. This was mainly due to the fact that, with the introduction of various sampling methods to determine the shape of an inhomogeneous medium from far-field data, transmission eigenvalues were something to be avoided and hence the fact that transmission eigenvalues formed at most a discrete set was deemed to be sufficient. In addition, questions related to the existence of transmission eigenvalues or the structure of associated eigenvectors were recognized as being particularly difficult due to the nonlinearity of the eigenvalue problem and the special structure of the associated transmission
Periodic-parabolic eigenvalue problems with a large parameter and degeneration
NASA Astrophysics Data System (ADS)
Daners, Daniel; Thornett, Christopher
2016-07-01
We consider a periodic-parabolic eigenvalue problem with a non-negative potential λm vanishing on a non-cylindrical domain Dm satisfying conditions similar to those for the parabolic maximum principle. We show that the limit as λ → ∞ leads to a periodic-parabolic problem on Dm having a periodic-parabolic principal eigenvalue and eigenfunction which are unique in some sense. We substantially improve a result from [Du and Peng, Trans. Amer. Math. Soc. 364 (2012), p. 6039-6070]. At the same time we offer a different approach based on a periodic-parabolic initial boundary value problem. The results are motivated by an analysis of the asymptotic behaviour of positive solutions to semilinear logistic periodic-parabolic problems with temporal and spacial degeneracies.
Algebraic Thinking: A Problem Solving Approach
ERIC Educational Resources Information Center
Windsor, Will
2010-01-01
Algebraic thinking is a crucial and fundamental element of mathematical thinking and reasoning. It initially involves recognising patterns and general mathematical relationships among numbers, objects and geometric shapes. This paper will highlight how the ability to think algebraically might support a deeper and more useful knowledge, not only of…
Slower Algebra Students Meet Faster Tools: Solving Algebra Word Problems with Graphing Software
ERIC Educational Resources Information Center
Yerushalmy, Michal
2006-01-01
The article discusses the ways that less successful mathematics students used graphing software with capabilities similar to a basic graphing calculator to solve algebra problems in context. The study is based on interviewing students who learned algebra for 3 years in an environment where software tools were always present. We found differences…
Solving Large Scale Nonlinear Eigenvalue Problem in Next-Generation Accelerator Design
Liao, Ben-Shan; Bai, Zhaojun; Lee, Lie-Quan; Ko, Kwok; /SLAC
2006-09-28
A number of numerical methods, including inverse iteration, method of successive linear problem and nonlinear Arnoldi algorithm, are studied in this paper to solve a large scale nonlinear eigenvalue problem arising from finite element analysis of resonant frequencies and external Q{sub e} values of a waveguide loaded cavity in the next-generation accelerator design. They present a nonlinear Rayleigh-Ritz iterative projection algorithm, NRRIT in short and demonstrate that it is the most promising approach for a model scale cavity design. The NRRIT algorithm is an extension of the nonlinear Arnoldi algorithm due to Voss. Computational challenges of solving such a nonlinear eigenvalue problem for a full scale cavity design are outlined.
Eigenfunctions and Eigenvalues for a Scalar Riemann-Hilbert Problem Associated to Inverse Scattering
NASA Astrophysics Data System (ADS)
Pelinovsky, Dmitry E.; Sulem, Catherine
A complete set of eigenfunctions is introduced within the Riemann-Hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations. In particular, we consider the time-independent and time-dependent Schrödinger problems which are related to the KdV and KPI equations possessing solitons and lumps, respectively. Non-standard scalar products, orthogonality and completeness relations are derived for these problems. The complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations. We classify two different types of bifurcations of new eigenvalues and analyze their characteristic features. One type corresponds to thresholdless generation of solitons in the KdV equation, while the other predicts a threshold for generation of lumps in the KPI equation.
Clifford algebra approach to the coincidence problem for planar lattices.
Rodríguez, M A; Aragón, J L; Verde-Star, L
2005-03-01
The problem of coincidences of planar lattices is analyzed using Clifford algebra. It is shown that an arbitrary coincidence isometry can be decomposed as a product of coincidence reflections and this allows planar coincidence lattices to be characterized algebraically. The cases of square, rectangular and rhombic lattices are worked out in detail. One of the aims of this work is to show the potential usefulness of Clifford algebra in crystallography. The power of Clifford algebra for expressing geometric ideas is exploited here and the procedure presented can be generalized to higher dimensions. PMID:15724067
Reliable use of determinants to solve nonlinear structural eigenvalue problems efficiently
NASA Technical Reports Server (NTRS)
Williams, F. W.; Kennedy, D.
1988-01-01
The analytical derivation, numerical implementation, and performance of a multiple-determinant parabolic interpolation method (MDPIM) for use in solving transcendental eigenvalue (critical buckling or undamped free vibration) problems in structural mechanics are presented. The overall bounding, eigenvalue-separation, qualified parabolic interpolation, accuracy-confirmation, and convergence-recovery stages of the MDPIM are described in detail, and the numbers of iterations required to solve sample plane-frame problems using the MDPIM are compared with those for a conventional bisection method and for the Newtonian method of Simpson (1984) in extensive tables. The MDPIM is shown to use 31 percent less computation time than bisection when accuracy of 0.0001 is required, but 62 percent less when accuracy of 10 to the -8th is required; the time savings over the Newtonian method are about 10 percent.
On the Shape Sensitivity of the First Dirichlet Eigenvalue for Two-Phase Problems
Dambrine, M.; Kateb, D.
2011-02-15
We consider a two-phase problem in thermal conductivity: inclusions filled with a material of conductivity {sigma}{sub 1} are layered in a body of conductivity {sigma}{sub 2}. We address the shape sensitivity of the first eigenvalue associated with Dirichlet boundary conditions when both the boundaries of the inclusions and the body can be modified. We prove a differentiability result and provide the expressions of the first and second order derivatives. We apply the results to the optimal design of an insulated body. We prove the stability of the optimal design thanks to a second order analysis. We also continue the study of an extremal eigenvalue problem for a two-phase conductor in a ball initiated by Conca et al. (Appl. Math. Optim. 60(2):173-184, 2009) and pursued in Conca et al. (CANUM 2008, ESAIM Proc., vol. 27, pp. 311-321, EDP Sci., Les Ulis, 2009).
NASA Astrophysics Data System (ADS)
Hamelinck, Wouter
2008-09-01
Cavity resonators are modelled using a Maxwell eigenvalue problem. In order to obtain a reliable finite element approximation one has to carefully use an appropriate discrete finite element space. In the present paper we extend the known conditions to assure a correct approximation of the spectrum to the case where numerical integration occurs and where curvilinear boundaries are present. We present a set of sufficient conditions which are similar to the case where those so called variational crimes are absent.
NASA Astrophysics Data System (ADS)
Yang, Chuan-Fu; Buterin, Sergey
2016-03-01
The inverse spectral problem of determining a spherically symmetric wave speed v is considered in a bounded spherical region of radius b. A uniqueness theorem for the potential q of the derived Sturm-Liouville problem B (q) is presented from the data involving fractions of the eigenvalues of the problem B (q) on a finite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl-function techniques and properties of zeros of a class of entire functions.
A case against a divide and conquer approach to the nonsymmetric eigenvalue problem
Jessup, E.R.
1991-12-01
Divide and conquer techniques based on rank-one updating have proven fast, accurate, and efficient in parallel for the real symmetric tridiagonal and unitary eigenvalue problems and for the bidiagonal singular value problem. Although the divide and conquer mechanism can also be adapted to the real nonsymmetric eigenproblem in a straightforward way, most of the desirable characteristics of the other algorithms are lost. In this paper, we examine the problems of accuracy and efficiency that can stand in the way of a nonsymmetric divide and conquer eigensolver based on low-rank updating. 31 refs., 2 figs.
The eigenvalue problem associated with the nonlinear buckling of a shear bending column
NASA Astrophysics Data System (ADS)
Nishimura, Isao
2011-04-01
This paper discusses the eigenvalue problem of a nonlinear differential equation that governs the stability of a shear bending column under extremely large deformation. What is taken into consideration is the geometrical nonlinearity while the material is supposed to be linear. The reason of a superbly stable buckling behavior of a slender rubber bearing is physically explained by pointing out the analogy that is similar to the nonlinear wave propagation expressed in KdV equation. The nonlinear boundary condition and the nonlinear term of the differential equation cancel each other and make the associated eigenvalue rather constant. In other words, as far as the material is supposed to be linear, the column does not buckle no matter how large the deformation is. This theoretical prediction is experimentally verified and successfully applied to a base isolation system of a lightweight structure.
White, D; Koning, J
1999-10-21
The authors are interested in determining the electromagnetic fields within closed perfectly conducting cavities that may contain dielectric or magnetic materials. The vector Helmholtz equation is the appropriate partial differential equation for this problem. It is well known that the electromagnetic fields in a cavity can be decomposed into distinct modes that oscillate in time at specific resonant frequencies. These modes are referred to as eigenmodes, and the frequencies of these modes are referred to as eigenfrequencies. The authors' present application is the analysis of linear accelerator components. These components may have a complex geometry; hence numerical methods are require to compute the eigenmodes and the eigenfrequencies of these components. The Implicitly Restarted Arnoldi Method (IRAM) is a robust and efficient method for the numerical solution of the generalized eigenproblem Ax = {lambda}Bx, where A and B are sparse matrices, x is an eigenvector, and {lambda} is an eigenvalue. The IRAM is an iterative method for computing extremal eigenvalues; it is an extension of the classic Lanczos method. The mathematical details of the IRAM are too sophisticated to describe here; instead they refer the reader to [1]. A FORTRAN subroutine library that implements various versions of the IRAM is freely available, both in a serial version named ARPACK and parallel version named PARPACK. In this paper they discretize the vector Helmholtz equation using 1st order H(curl) conforming edge elements (also known as Nedelec elements). This discretization results in a generalized eigenvalue problem which can be solved using the IRAM. The question of so-called spurious modes is discussed, and it is shown that applying a spectral transformation completely eliminates these modes, without any need for an additional constraint equation. Typically they use the IRAM to compute a small set (n < 30) of eigenvalues and eigenmodes for a very large systems (N > 100,000).
ERIC Educational Resources Information Center
Cunningham, Robert F.
2005-01-01
For students to develop an understanding of functions, they must have opportunities to solve problems that require them to transfer between algebraic, numeric, and graphic representations (transfer problems). Research has confirmed student difficulties with certain types of transfer problems and has suggested instructional factors as a possible…
NASA Astrophysics Data System (ADS)
Castro, María Eugenia; Díaz, Javier; Muñoz-Caro, Camelia; Niño, Alfonso
2011-09-01
We present a system of classes, SHMatrix, to deal in a unified way with the computation of eigenvalues and eigenvectors in real symmetric and Hermitian matrices. Thus, two descendant classes, one for the real symmetric and other for the Hermitian cases, override the abstract methods defined in a base class. The use of the inheritance relationship and polymorphism allows handling objects of any descendant class using a single reference of the base class. The system of classes is intended to be the core element of more sophisticated methods to deal with large eigenvalue problems, as those arising in the variational treatment of realistic quantum mechanical problems. The present system of classes allows computing a subset of all the possible eigenvalues and, optionally, the corresponding eigenvectors. Comparison with well established solutions for analogous eigenvalue problems, as those included in LAPACK, shows that the present solution is competitive against them. Program summaryProgram title: SHMatrix Catalogue identifier: AEHZ_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHZ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2616 No. of bytes in distributed program, including test data, etc.: 127 312 Distribution format: tar.gz Programming language: Standard ANSI C++. Computer: PCs and workstations. Operating system: Linux, Windows. Classification: 4.8. Nature of problem: The treatment of problems involving eigensystems is a central topic in the quantum mechanical field. Here, the use of the variational approach leads to the computation of eigenvalues and eigenvectors of real symmetric and Hermitian Hamiltonian matrices. Realistic models with several degrees of freedom leads to large (sometimes very large) matrices. Different techniques, such as divide
The use of Lanczos's method to solve the large generalized symmetric definite eigenvalue problem
NASA Technical Reports Server (NTRS)
Jones, Mark T.; Patrick, Merrell L.
1989-01-01
The generalized eigenvalue problem, Kx = Lambda Mx, is of significant practical importance, especially in structural enginering where it arises as the vibration and buckling problem. A new algorithm, LANZ, based on Lanczos's method is developed. LANZ uses a technique called dynamic shifting to improve the efficiency and reliability of the Lanczos algorithm. A new algorithm for solving the tridiagonal matrices that arise when using Lanczos's method is described. A modification of Parlett and Scott's selective orthogonalization algorithm is proposed. Results from an implementation of LANZ on a Convex C-220 show it to be superior to a subspace iteration code.
Existence of eigenvalues of problem with shift for an equation of parabolic-hyperbolic type
NASA Astrophysics Data System (ADS)
Tengayeva, Aizhan; Dildabek, Gulnar
2016-08-01
In the present paper, a spectral problem for an operator of parabolic-hyperbolic type of I kind with non-classical boundary conditions is considered. The problem is considered in a standard domain. The parabolic part of the space is a rectangle. And the hyperbolic part of the space coincides with a characteristic triangle. We consider a problem with the local boundary condition in the domain of parabolicity and with the boundary condition with displacement in the domain of hyperbolicity. We prove the strong solvability of the considered problem. The main aim of the paper is the research of spectral properties of the problem. The existence of eigenvalues of the problem is proved.
Inhibiting Interference from Prior Knowledge: Arithmetic Intrusions in Algebra Word Problem Solving
ERIC Educational Resources Information Center
Khng, Kiat Hui; Lee, Kerry
2009-01-01
In Singapore, 6-12 year-old students are taught to solve algebra word problems with a mix of arithmetic and pre-algebraic strategies; 13-17 year-olds are typically encouraged to replace these strategies with letter-symbolic algebra. We examined whether algebra problem-solving proficiency amongst beginning learners of letter-symbolic algebra is…
Inverse Modelling Problems in Linear Algebra Undergraduate Courses
ERIC Educational Resources Information Center
Martinez-Luaces, Victor E.
2013-01-01
This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…
Stathopoulos, A.; Fischer, C.F.; Saad, Y.
1994-12-31
The solution of the large, sparse, symmetric eigenvalue problem, Ax = {lambda}x, is central to many scientific applications. Among many iterative methods that attempt to solve this problem, the Lanczos and the Generalized Davidson (GD) are the most widely used methods. The Lanczos method builds an orthogonal basis for the Krylov subspace, from which the required eigenvectors are approximated through a Rayleigh-Ritz procedure. Each Lanczos iteration is economical to compute but the number of iterations may grow significantly for difficult problems. The GD method can be considered a preconditioned version of Lanczos. In each step the Rayleigh-Ritz procedure is solved and explicit orthogonalization of the preconditioned residual ((M {minus} {lambda}I){sup {minus}1}(A {minus} {lambda}I)x) is performed. Therefore, the GD method attempts to improve convergence and robustness at the expense of a more complicated step.
Boundary Concentration for Eigenvalue Problems Related to the Onset of Superconductivity
NASA Astrophysics Data System (ADS)
del Pino, Manuel; Felmer, Patricio L.; Sternberg, Peter
We examine the asymptotic behavior of the eigenvalue μ(h) and corresponding eigenfunction associated with the variational problem
Ji, Xingzhi )
1994-03-01
This paper is concerned with the eigenvalues of Sturm-Liouville problems with periodic and semi-periodic boundary conditions to be approximated by a shooting algorithm. The proposed technique is based on the application of the Floquet theory. Convergence analysis and a general guideline to provide starting values for computed eigenvalues are presented. Some numerical results are also reported. 18 refs., 1 fig., 3 tabs.
NASA Technical Reports Server (NTRS)
Pak, Chan-gi; Lung, Shu
2009-01-01
Modern airplane design is a multidisciplinary task which combines several disciplines such as structures, aerodynamics, flight controls, and sometimes heat transfer. Historically, analytical and experimental investigations concerning the interaction of the elastic airframe with aerodynamic and in retia loads have been conducted during the design phase to determine the existence of aeroelastic instabilities, so called flutter .With the advent and increased usage of flight control systems, there is also a likelihood of instabilities caused by the interaction of the flight control system and the aeroelastic response of the airplane, known as aeroservoelastic instabilities. An in -house code MPASES (Ref. 1), modified from PASES (Ref. 2), is a general purpose digital computer program for the analysis of the closed-loop stability problem. This program used subroutines given in the International Mathematical and Statistical Library (IMSL) (Ref. 3) to compute all of the real and/or complex conjugate pairs of eigenvalues of the Hessenberg matrix. For high fidelity configuration, these aeroelastic system matrices are large and compute all eigenvalues will be time consuming. A subspace iteration method (Ref. 4) for complex eigenvalues problems with nonsymmetric matrices has been formulated and incorporated into the modified program for aeroservoelastic stability (MPASES code). Subspace iteration method only solve for the lowest p eigenvalues and corresponding eigenvectors for aeroelastic and aeroservoelastic analysis. In general, the selection of p is ranging from 10 for wing flutter analysis to 50 for an entire aircraft flutter analysis. The application of this newly incorporated code is an experiment known as the Aerostructures Test Wing (ATW) which was designed by the National Aeronautic and Space Administration (NASA) Dryden Flight Research Center, Edwards, California to research aeroelastic instabilities. Specifically, this experiment was used to study an instability
NASA Astrophysics Data System (ADS)
Ramos, Miguel; Tavares, Hugo; Terracini, Susanna
2016-04-01
Let {Ω subset {R}^N} be an open bounded domain and {m in {N}}. Given {k_1,ldots,k_m in {N}}, we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, of the following form inf{F({λ_{k1}}(ω_1),ldots,λ_{k_m}(ω_m)): (ω_1,ldots, ω_m) in {P}_m(Ω)}, where {λ_{k_i}(ω_i)} denotes the k i -th eigenvalue of {(-Δ,H10(ω_i))} counting multiplicities, and {{P}_m(Ω)} is the set of all open partitions of {Ω}, namely {P}_m(Ω)={(ω_1, ldots, ω_m):ω_i subset Ω open, ωi \\capω_j=emptyset forall i ≠ j }. While the existence of a quasi-open optimal partition {(ω_1,ldots, ω_m)} follows from a general result by Bucur, Buttazzo and Henrot [Adv Math Sci Appl 8(2):571-579, 1998], the aim of this paper is to associate with such minimal partitions and their eigenfunctions some suitable extremality conditions and to exploit them, proving as well the Lipschitz continuity of some eigenfunctions, and the regularity of the partition in the sense that the free boundary {\\cup_{i=1}^m partial ωi \\cap Ω} is, up to a residual set, locally a {C^{1,α}} hypersurface. This last result extends the ones in the paper by Caffarelli and Lin [J Sci Comput 31(1-2):5-18, 2007] to the case of higher eigenvalues.
SEMI-DEFINITE PROGRAMMING TECHNIQUES FOR STRUCTURED QUADRATIC INVERSE EIGENVALUE PROBLEMS
LIN, MATTHEW M.; DONG, BO; CHU, MOODY T.
2014-01-01
In the past decade or so, semi-definite programming (SDP) has emerged as a powerful tool capable of handling a remarkably wide range of problems. This article describes an innovative application of SDP techniques to quadratic inverse eigenvalue problems (QIEPs). The notion of QIEPs is of fundamental importance because its ultimate goal of constructing or updating a vibration system from some observed or desirable dynamical behaviors while respecting some inherent feasibility constraints well suits many engineering applications. Thus far, however, QIEPs have remained challenging both theoretically and computationally due to the great variations of structural constraints that must be addressed. Of notable interest and significance are the uniformity and the simplicity in the SDP formulation that solves effectively many otherwise very difficult QIEPs. PMID:25392603
Multigrid method applied to the solution of an elliptic, generalized eigenvalue problem
Alchalabi, R.M.; Turinsky, P.J.
1996-12-31
The work presented in this paper is concerned with the development of an efficient MG algorithm for the solution of an elliptic, generalized eigenvalue problem. The application is specifically applied to the multigroup neutron diffusion equation which is discretized by utilizing the Nodal Expansion Method (NEM). The underlying relaxation method is the Power Method, also known as the (Outer-Inner Method). The inner iterations are completed using Multi-color Line SOR, and the outer iterations are accelerated using Chebyshev Semi-iterative Method. Furthermore, the MG algorithm utilizes the consistent homogenization concept to construct the restriction operator, and a form function as a prolongation operator. The MG algorithm was integrated into the reactor neutronic analysis code NESTLE, and numerical results were obtained from solving production type benchmark problems.
Optical reflection from planetary surfaces as an operator-eigenvalue problem
Wildey, R.L.
1986-01-01
The understanding of quantum mechanical phenomena has come to rely heavily on theory framed in terms of operators and their eigenvalue equations. This paper investigates the utility of that technique as related to the reciprocity principle in diffuse reflection. The reciprocity operator is shown to be unitary and Hermitian; hence, its eigenvectors form a complete orthonormal basis. The relevant eigenvalue is found to be infinitely degenerate. A superposition of the eigenfunctions found from solution by separation of variables is inadequate to form a general solution that can be fitted to a one-dimensional boundary condition, because the difficulty of resolving the reciprocity operator into a superposition of independent one-dimensional operators has yet to be overcome. A particular lunar application in the form of a failed prediction of limb-darkening of the full Moon from brightness versus phase illustrates this problem. A general solution is derived which fully exploits the determinative powers of the reciprocity operator as an unresolved two-dimensional operator. However, a solution based on a sum of one-dimensional operators, if possible, would be much more powerful. A close association is found between the reciprocity operator and the particle-exchange operator of quantum mechanics, which may indicate the direction for further successful exploitation of the approach based on the operational calculus. ?? 1986 D. Reidel Publishing Company.
Student Difficulties in Mathematizing Word Problems in Algebra
ERIC Educational Resources Information Center
Jupri, Al; Drijvers, Paul
2016-01-01
To investigate student difficulties in solving word problems in algebra, we carried out a teaching experiment involving 51 Indonesian students (12/13 year-old) who used a digital mathematics environment. The findings were backed up by an interview study, in which eighteen students (13/14 year-old) were involved. The perspective of mathematization,…
A hybrid approach to the neutron transport K-eigenvalue problem using NDA-based algorithms
Willert, J. A.; Kelley, C. T.; Knoll, D. A.; Park, H.
2013-07-01
In order to provide more physically accurate solutions to the neutron transport equation it has become increasingly popular to use Monte Carlo simulation to model nuclear reactor dynamics. These Monte Carlo methods can be extremely expensive, so we turn to a class of methods known as hybrid methods, which combine known deterministic and stochastic techniques to solve the transport equation. In our work, we show that we can simulate the action of a transport sweep using a Monte Carlo simulation in order to solve the k-eigenvalue problem. We'll accelerate the solution using nonlinear diffusion acceleration (NDA) as in [1,2]. Our work extends the results in [1] to use Monte Carlo simulation as the high-order solver. (authors)
Graph theory approach to the eigenvalue problem of large space structures
NASA Technical Reports Server (NTRS)
Reddy, A. S. S. R.; Bainum, P. M.
1981-01-01
Graph theory is used to obtain numerical solutions to eigenvalue problems of large space structures (LSS) characterized by a state vector of large dimensions. The LSS are considered as large, flexible systems requiring both orientation and surface shape control. Graphic interpretation of the determinant of a matrix is employed to reduce a higher dimensional matrix into combinations of smaller dimensional sub-matrices. The reduction is implemented by means of a Boolean equivalent of the original matrices formulated to obtain smaller dimensional equivalents of the original numerical matrix. Computation time becomes less and more accurate solutions are possible. An example is provided in the form of a free-free square plate. Linearized system equations and numerical values of a stiffness matrix are presented, featuring a state vector with 16 components.
Parallel computing study for the large-scale generalized eigenvalue problems in modal analysis
NASA Astrophysics Data System (ADS)
Fan, XuanHua; Chen, Pu; Wu, RuiAn; Xiao, ShiFu
2014-03-01
In this paper we study the algorithms and their parallel implementation for solving large-scale generalized eigenvalue problems in modal analysis. Three predominant subspace algorithms, i.e., Krylov-Schur method, implicitly restarted Arnoldi method and Jacobi-Davidson method, are modified with some complementary techniques to make them suitable for modal analysis. Detailed descriptions of the three algorithms are given. Based on these algorithms, a parallel solution procedure is established via the PANDA framework and its associated eigensolvers. Using the solution procedure on a machine equipped with up to 4800 processors, the parallel performance of the three predominant methods is evaluated via numerical experiments with typical engineering structures, where the maximum testing scale attains twenty million degrees of freedom. The speedup curves for different cases are obtained and compared. The results show that the three methods are good for modal analysis in the scale of ten million degrees of freedom with a favorable parallel scalability.
An Extremal Eigenvalue Problem for a Two-Phase Conductor in a Ball
Conca, Carlos Mahadevan, Rajesh Sanz, Leon
2009-10-15
The pioneering works of Murat and Tartar (Topics in the mathematical modeling of composite materials. PNLDE 31. Birkhaeuser, Basel, 1997) go a long way in showing, in general, that problems of optimal design may not admit solutions if microstructural designs are excluded from consideration. Therefore, assuming, tactilely, that the problem of minimizing the first eigenvalue of a two-phase conducting material with the conducting phases to be distributed in a fixed proportion in a given domain has no true solution in general domains, Cox and Lipton only study conditions for an optimal microstructural design (Cox and Lipton in Arch. Ration. Mech. Anal. 136:101-117, 1996). Although, the problem in one dimension has a solution (cf. Krein in AMS Transl. Ser. 2(1):163-187, 1955) and, in higher dimensions, the problem set in a ball can be deduced to have a radially symmetric solution (cf. Alvino et al. in Nonlinear Anal. TMA 13(2):185-220, 1989), these existence results have been regarded so far as being exceptional owing to complete symmetry. It is still not clear why the same problem in domains with partial symmetry should fail to have a solution which does not develop microstructure and respecting the symmetry of the domain. We hope to revive interest in this question by giving a new proof of the result in a ball using a simpler symmetrization result from Alvino and Trombetti (J. Math. Anal. Appl. 94:328-337, 1983)
A nonlinear eigenvalue problem for self-similar spherical force-free magnetic fields
NASA Astrophysics Data System (ADS)
Lerche, I.; Low, B. C.
2014-10-01
An axisymmetric force-free magnetic field B(r, θ) in spherical coordinates is defined by a function r sin θ B φ = Q ( A ) relating its azimuthal component to its poloidal flux-function A. The power law r sin θ B φ = a A | A | 1/ n, n a positive constant, admits separable fields with A = An/(θ)rn, posing a nonlinear boundary-value problem for the constant parameter a as an eigenvalue and An(θ) as its eigenfunction [B. C. Low and Y. Q Lou, Astrophys. J. 352, 343 (1990)]. A complete analysis is presented of the eigenvalue spectrum for a given n, providing a unified understanding of the eigenfunctions and the physical relationship between the field's degree of multi-polarity and rate of radial decay via the parameter n. These force-free fields, self-similar on spheres of constant r, have basic astrophysical applications. As explicit solutions they have, over the years, served as standard benchmarks for testing 3D numerical codes developed to compute general force-free fields in the solar corona. The study presented includes a set of illustrative multipolar field solutions to address the magnetohydrodynamics (MHD) issues underlying the observation that the solar corona has a statistical preference for negative and positive magnetic helicities in its northern and southern hemispheres, respectively; a hemispherical effect, unchanging as the Sun's global field reverses polarity in successive eleven-year cycles. Generalizing these force-free fields to the separable form B = H/(θ ,φ)rn+2 promises field solutions of even richer topological varieties but allowing for φ-dependence greatly complicates the governing equations that have remained intractable. The axisymmetric results obtained are discussed in relation to this generalization and the Parker Magnetostatic Theorem. The axisymmetric solutions are mathematically related to a family of 3D time-dependent ideal MHD solutions for a polytropic fluid of index γ = 4/3 as discussed in the Appendix.
A nonlinear eigenvalue problem for self-similar spherical force-free magnetic fields
Lerche, I.; Low, B. C.
2014-10-15
An axisymmetric force-free magnetic field B(r, θ) in spherical coordinates is defined by a function r sin θB{sub φ}=Q(A) relating its azimuthal component to its poloidal flux-function A. The power law r sin θB{sub φ}=aA|A|{sup 1/n}, n a positive constant, admits separable fields with A=(A{sub n}(θ))/(r{sup n}) , posing a nonlinear boundary-value problem for the constant parameter a as an eigenvalue and A{sub n}(θ) as its eigenfunction [B. C. Low and Y. Q Lou, Astrophys. J. 352, 343 (1990)]. A complete analysis is presented of the eigenvalue spectrum for a given n, providing a unified understanding of the eigenfunctions and the physical relationship between the field's degree of multi-polarity and rate of radial decay via the parameter n. These force-free fields, self-similar on spheres of constant r, have basic astrophysical applications. As explicit solutions they have, over the years, served as standard benchmarks for testing 3D numerical codes developed to compute general force-free fields in the solar corona. The study presented includes a set of illustrative multipolar field solutions to address the magnetohydrodynamics (MHD) issues underlying the observation that the solar corona has a statistical preference for negative and positive magnetic helicities in its northern and southern hemispheres, respectively; a hemispherical effect, unchanging as the Sun's global field reverses polarity in successive eleven-year cycles. Generalizing these force-free fields to the separable form B=(H(θ,φ))/(r{sup n+2}) promises field solutions of even richer topological varieties but allowing for φ-dependence greatly complicates the governing equations that have remained intractable. The axisymmetric results obtained are discussed in relation to this generalization and the Parker Magnetostatic Theorem. The axisymmetric solutions are mathematically related to a family of 3D time-dependent ideal MHD solutions for a polytropic fluid of index γ = 4/3 as
NASA Astrophysics Data System (ADS)
Szalay, Viktor; Smith, Sean C.
1999-01-01
It has been shown that an approximately band-limited function can be reconstructed by using the function's values taken at appropriate equidistant grid points and a generalized Hermite-contracted-continuous-distributed-approximating-function (Hermite-CCDAF) as the reconstruction function. A sampling theorem prescribing the possible choices of grid spacing and DAF parameters has been derived and discussed, and discretized-Hermite-contracted DAFs have been introduced. At certain values of its parameters the generalized Hermite-CCDAF is identical to the Shannon-Gabor-wavelet-DAF (SGWDAF). Simple expressions for constructing the matrix of a vibrational Hamiltonian in the discretized-Hermite-contracted DAF approximation have been given. As a special case the matrix elements corresponding to sinc-DVR (discrete variational representation) are recovered. The usefulness and properties of sinc-DVR and discretized-Hermite-contracted-DAF (or SGWDAF) in bound state calculations have been compared by solving the eigenvalue problem of a number of one- and two-dimensional Hamiltonians. It has been found that if one requires that the same number of energy levels be computed with an error less than or equal to a given value, the SGWDAF method with thresholding is faster than the standard sinc-DVR method. The results obtained with the Barbanis Hamiltonian are described and discussed in detail.
Evaluation of vectorized Monte Carlo algorithms on GPUs for a neutron Eigenvalue problem
Du, X.; Liu, T.; Ji, W.; Xu, X. G.; Brown, F. B.
2013-07-01
Conventional Monte Carlo (MC) methods for radiation transport computations are 'history-based', which means that one particle history at a time is tracked. Simulations based on such methods suffer from thread divergence on the graphics processing unit (GPU), which severely affects the performance of GPUs. To circumvent this limitation, event-based vectorized MC algorithms can be utilized. A versatile software test-bed, called ARCHER - Accelerated Radiation-transport Computations in Heterogeneous Environments - was used for this study. ARCHER facilitates the development and testing of a MC code based on the vectorized MC algorithm implemented on GPUs by using NVIDIA's Compute Unified Device Architecture (CUDA). The ARCHER{sub GPU} code was designed to solve a neutron eigenvalue problem and was tested on a NVIDIA Tesla M2090 Fermi card. We found that although the vectorized MC method significantly reduces the occurrence of divergent branching and enhances the warp execution efficiency, the overall simulation speed is ten times slower than the conventional history-based MC method on GPUs. By analyzing detailed GPU profiling information from ARCHER, we discovered that the main reason was the large amount of global memory transactions, causing severe memory access latency. Several possible solutions to alleviate the memory latency issue are discussed. (authors)
Constructing a Coherent Problem Model to Facilitate Algebra Problem Solving in a Chemistry Context
ERIC Educational Resources Information Center
Ngu, Bing Hiong; Yeung, Alexander Seeshing; Phan, Huy P.
2015-01-01
An experiment using a sample of 11th graders compared text editing and worked examples approaches in learning to solve dilution and molarity algebra word problems in a chemistry context. Text editing requires students to assess the structure of a word problem by specifying whether the problem text contains sufficient, missing, or irrelevant…
Algebraic solution for phase unwrapping problems in multiwavelength interferometry.
Falaggis, Konstantinos; Towers, David P; Towers, Catherine E
2014-06-10
Recent advances in multiwavelength interferometry techniques [Appl. Opt.52, 5758 (2013)] give new insights to phase unwrapping problems and allow the fringe order information contained in the measured phase to be extracted with low computational effort. This work introduces an algebraic solution to the phase unwrapping problem that allows the direct calculation of the unknown integer fringe order. The procedure resembles beat-wavelength approaches, but provides greater flexibility in choosing the measurement wavelengths, a larger measurement range, and a higher robustness against noise, due to the ability to correct for errors during the calculation. PMID:24921139
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.
NASA Technical Reports Server (NTRS)
Warming, Robert F.; Beam, Richard M.
1986-01-01
A hyperbolic initial-boundary-value problem can be approximated by a system of ordinary differential equations (ODEs) by replacing the spatial derivatives by finite-difference approximations. The resulting system of ODEs is called a semidiscrete approximation. A complication is the fact that more boundary conditions are required for the spatially discrete approximation than are specified for the partial differential equation. Consequently, additional numerical boundary conditions are required and improper treatment of these additional conditions can lead to instability. For a linear initial-boundary-value problem (IBVP) with homogeneous analytical boundary conditions, the semidiscrete approximation results in a system of ODEs of the form du/dt = Au whose solution can be written as u(t) = exp(At)u(O). Lax-Richtmyer stability requires that the matrix norm of exp(At) be uniformly bounded for O less than or = t less than or = T independent of the spatial mesh size. Although the classical Lax-Richtmyer stability definition involves a conventional vector norm, there is no known algebraic test for the uniform boundedness of the matrix norm of exp(At) for hyperbolic IBVPs. An alternative but more complicated stability definition is used in the theory developed by Gustafsson, Kreiss, and Sundstrom (GKS). The two methods are compared.
A new Monte Carlo power method for the eigenvalue problem of transfer matrices
Koma, Tohru )
1993-04-01
The author proposes a new Monte Carlo method for calculating eigenvalues of transfer matrices leading to free energies and to correlation lengths of classical and quantum many-body systems. Generally, this method can be applied to the calculation of the maximum eigenvalue of a nonnegative matrix A such that all the matrix elements of A[sup k] are strictly positive for an integer k. This method is based on a new representation of the maximum eigenvalue of the matrix A as the thermal average of a certain observable of a many-body system. Therefore one can easily calculate the maximum eigenvalue of a transfer matrix leading to the free energy in the standard Monte Carlo simulations, such as the Metropolis algorithm. As test cases, the author calculates the free energies of the square-lattice Ising model and of the spin-1/2 XY Heisenberg chain. He also proves two useful theorems on the ergodicity in quantum Monte Carlo algorithms, or more generally, on the ergodicity of Monte Carlo algorithms using the new representation of the maximum eigenvalue of the matrix A. 39 refs., 5 figs., 2 tabs.
Top Element Problem and Macneille Completions of Generalized Effect Algebras
NASA Astrophysics Data System (ADS)
RieČanová, Z.; Kalina, M.
2014-10-01
Effect algebras (EAs), introduced by D. J. Foulis and M. K. Bennett, as common generalizations of Boolean algebras, orthomodular lattices and MV-algebras, are nondistributive algebraic structures including unsharp elements. Their unbounded versions, called generalized effect algebras, are posets which may have or may have not an EA-MacNeille completion, or cannot be embedded into any complete effect algebra. We give a necessary and sufficient condition for a generalized effect algebra to have an EA-MacNeille completion. Some examples are provided.
Coulomb problem in an angular-momentum basis: An algebraic formulation
NASA Astrophysics Data System (ADS)
de Lange, O. L.; Raab, R. E.
1988-03-01
We show that a representation-independent, spectrum-generating algebra for the Coulomb problem in an angular momentum basis can be obtained by quantizing two complex, time-dependent, classical vectors, Dc=Fc+iGc and D*c. The approach is based on an analogy with a treatment of the isotropic harmonic oscillator [A. J. Bracken and H. I. Leemon, J. Math. Phys. 21, 2170 (1980)], and on work in which classical constants of the motion were quantized to yield shift operators for angular momentum in the Coulomb problem [O. L. de Lange and R. E. Raab, Phys. Rev. A 34, 1650 (1986)]. By construction Fc and Gc are orthogonal to the orbital angular momentum L, their moduli have equal, constant magnitude, and they rotate about L. In this construction we use Ac (the Laplace-Runge-Lenz vector) and Ac×L^ as basis vectors. Fc and Gc contain an undetermined phase factor exp(iδ). Dc and D*c are quantized by requiring that the resulting operators should be shift operators for energy and angular momentum in the bound-state kets ||nlm>. This determines the operators Δ+/- corresponding to the classical phase factors exp(+/-iδ). In the coordinate and momentum representations of wave mechanics respectively, Δ+/- are the dilatation operators for coordinate-space and momentum-space wave functions. The shift operators can be factorized to yield 20 abstract operators. Apart from their dependence on Δ+/- and constants of the motion, ten of these are linear in p, eight are linear in r, and two are quadratic in r. Apart from Δ+/-, these operators can be linearized by replacing constants of the motion with their eigenvalues: In the coordinate and momentum representations of wave mechanics they are first-order differential operators. The shift operators are part of a Hermitian basis for a spectrum-generating algebra which is shown to be SO(2,1)⊕SO(3,2).
The spatial isomorphism problem for close separable nuclear C*-algebras
Christensen, Erik; Sinclair, Allan M.; Smith, Roger R.; White, Stuart A.; Winter, Wilhelm
2010-01-01
The Kadison–Kastler problem asks whether close C*-algebras on a Hilbert space must be spatially isomorphic. We establish this when one of the algebras is separable and nuclear. We also apply our methods to the study of near inclusions of C*-algebras. PMID:20080723
Primary School Students' Strategies in Early Algebra Problem Solving Supported by an Online Game
ERIC Educational Resources Information Center
van den Heuvel-Panhuizen, Marja; Kolovou, Angeliki; Robitzsch, Alexander
2013-01-01
In this study we investigated the role of a dynamic online game on students' early algebra problem solving. In total 253 students from grades 4, 5, and 6 (10-12 years old) used the game at home to solve a sequence of early algebra problems consisting of contextual problems addressing covarying quantities. Special software monitored the…
Paving a Way to Algebraic Word Problems Using a Nonalgebraic Route
ERIC Educational Resources Information Center
Amit, Miriam; Klass-Tsirulnikov, Bella
2005-01-01
A three-stage model for algebraic word problem solving is developed in which students' understanding of the intrinsic logical structure of word problems is strengthened by connecting real-life problems and formal mathematics. (Contains 3 figure.)
Bottcher, C.; Strayer, M.R.; Werby, M.F.
1993-10-01
The Helmholtz-Poincare Wave Equation (H-PWE) arises in many areas of classical wave scattering theory. In particular it can be found for the cases of acoustical scattering from submerged bounded objects and electromagnetic scattering from objects. The extended boundary integral equations (EBIE) method is derived from considering both the exterior and interior solutions of the H-PWE`s. This coupled set of expressions has the advantage of not only offering a prescription for obtaining a solution for the exterior scattering problem, but it also obviates the problem of irregular values corresponding to fictitious interior eigenvalues. Once the coupled equations are derived, they can by obtained in matrix form be expanding all relevant terms in partial wave expansions, including a biorthogonal expansion of the Green function. However some freedom of choice in the choice of the surface expansion is available since the unknown surface quantities may be expanded in a variety of ways to long as closure is obtained. Out of many possible choices, we develop an optimal method to obtain such expansions which is based on the optimum eigenfunctions related to the surface of the object. In effect, we convert part of the problem (that associated with the Fredholms integral equation of the first kind) an eigenvalue problem of a related Hermition operator. The methodology will be explained in detail and examples will be presented.
Massively Parallel, Three-Dimensional Transport Solutions for the k-Eigenvalue Problem
Davidson, Gregory G; Evans, Thomas M; Jarrell, Joshua J; Pandya, Tara M; Slaybaugh, R
2014-01-01
We have implemented a new multilevel parallel decomposition in the Denovo dis- crete ordinates radiation transport code. In concert with Krylov subspace iterative solvers, the multilevel decomposition allows concurrency over energy in addition to space-angle, enabling scalability beyond the limits imposed by the traditional KBA space-angle partitioning. Furthermore, a new Arnoldi-based k-eigenvalue solver has been implemented. The added phase-space concurrency combined with the high- performance Krylov and Arnoldi solvers has enabled weak scaling to O(100K) cores on the Jaguar XK6 supercomputer. The multilevel decomposition provides sucient parallelism to scale to exascale computing and beyond.
Working Memory and Literacy as Predictors of Performance on Algebraic Word Problems
ERIC Educational Resources Information Center
Lee, Kerry; Ng, Swee-Fong; Ng, Ee-Lynn; Lim, Zee-Ying
2004-01-01
Previous studies on individual differences in mathematical abilities have shown that working memory contributes to early arithmetic performance. In this study, we extended the investigation to algebraic word problem solving. A total of 151 10-year-olds were administered algebraic word problems and measures of working memory, intelligence quotient…
A Schematic-Theoretic View of Problem Solving and Development of Algebraic Thinking
ERIC Educational Resources Information Center
Steele, Diana F.; Johanning, Debra I.
2004-01-01
This study explored the problem-solving schemas developed by 7th-grade pre-algebra students as they participated in a teaching experiment that was designed to help students develop effective schemas for solving algebraic problem situations involving contexts of (1) growth and change and (2) size and shape. This article describes the qualities and…
ERIC Educational Resources Information Center
Merriweather, Michelle; Tharp, Marcia L.
1999-01-01
Focuses on changes in attitude toward mathematics and calculator use and changes in how general mathematics students naturalistically solve algebraic problems. Uses a survey to determine whether a student is rule-based. Concludes that the rule-based students used an equation to solve the algebraic word problem whereas the non-rule-based students…
Algebraic Sub-Structuring for Electromagnetic Applications
Yang, C.; Gao, W.G.; Bai, Z.J.; Li, X.Y.S.; Lee, L.Q.; Husbands, P.; Ng, E.G.; /LBL, Berkeley /UC, Davis /SLAC
2006-06-30
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, they show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Algebraic sub-structuring for electromagnetic applications
Yang, Chao; Gao, Weiguo; Bai, Zhaojun; Li, Xiaoye; Lee, Lie-Quan; Husbands, Parry; Ng, Esmond G.
2004-09-14
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, we show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Complex eigenvalue extraction in NASTRAN by the tridiagonal reduction (FEER) method
NASA Technical Reports Server (NTRS)
Newman, M.; Mann, F. I.
1977-01-01
An extension of the Tridiagonal Reduction (FEER) method to complex eigenvalue analysis in NASTRAN is described. As in the case of real eigenvalue analysis, the eigensolutions closest to a selected point in the eigenspectrum are extracted from a reduced, symmetric, tridiagonal eigenmatrix whose order is much lower than that of the full size problem. The reduction process is effected automatically, and thus avoids the arbitrary lumping of masses and other physical quantities at selected grid points. The statement of the algebraic eigenvalue problem admits mass, damping and stiffness matrices which are unrestricted in character, i.e., they may be real, complex, symmetric or unsymmetric, singular or non-singular.
Persistent and Pernicious Errors in Algebraic Problem Solving
ERIC Educational Resources Information Center
Booth, Julie L.; Barbieri, Christina; Eyer, Francie; Paré-Blagoev, E. Juliana
2014-01-01
Students hold many misconceptions as they transition from arithmetic to algebraic thinking, and these misconceptions can hinder their performance and learning in the subject. To identify the errors in Algebra I which are most persistent and pernicious in terms of predicting student difficulty on standardized test items, the present study assessed…
Solving the eigenvalue problem of the nuclear Yukawa-folded mean-field Hamiltonian
NASA Astrophysics Data System (ADS)
Dobrowolski, A.; Pomorski, K.; Bartel, J.
2016-02-01
The nuclear Hamiltonian with a Yukawa-folded mean-field potential is diagonalized within the basis of a deformed harmonic-oscillator in Cartesian coordinates. The nuclear shape is characterized by the equivalent sharp surface described either by the well known Funny-Hills or the Trentalange-Koonin-Sierk parametrizations. They are both able to describe a very vast variety of nuclear deformations, including necked-in shapes, left-right asymmetry and non-axiality. The only imposed limitation on the nuclear shape is the z-signature symmetry, which corresponds to a symmetry of the shape with respect to a rotation by an angle π around the z-axis. On output, the computer code produces for a given nucleus with mass number A and charge number Z the energy eigenvalues and eigenfunctions of the mean-field Hamiltonian at chosen deformation.
Solving Our Algebra Problem: Getting All Students through Algebra I to Improve Graduation Rates
ERIC Educational Resources Information Center
Schachter, Ron
2013-01-01
graduation as well as admission to most colleges. But taking algebra also can turn into a pathway for failure, from which some students never recover. In 2010, a national U.S. Department of Education study…
The Hochschild cohomology problem for von Neumann algebras
Sinclair, Allan M.; Smith, Roger R.
1998-01-01
In 1967, when Kadison and Ringrose began the development of continuous cohomology theory for operator algebras, they conjectured that the cohomology groups Hn(M, M), n ≥ 1, for a von Neumann algebra M, should all be zero. This conjecture, which has important structural implications for von Neumann algebras, has been solved affirmatively in the type I, II∞, and III cases, leaving open only the type II1 case. In this paper, we describe a positive solution when M is type II1 and has a Cartan subalgebra and a separable predual. PMID:9520373
The hochschild cohomology problem for von neumann algebras.
Sinclair, A M; Smith, R R
1998-03-31
In 1967, when Kadison and Ringrose began the development of continuous cohomology theory for operator algebras, they conjectured that the cohomology groups Hn(M, M), n >/= 1, for a von Neumann algebra M, should all be zero. This conjecture, which has important structural implications for von Neumann algebras, has been solved affirmatively in the type I, IIinfinity, and III cases, leaving open only the type II1 case. In this paper, we describe a positive solution when M is type II1 and has a Cartan subalgebra and a separable predual. PMID:9520373
NASA Astrophysics Data System (ADS)
Brinkmeier, Maik; Nackenhorst, Udo
2008-03-01
The transient dynamic response of rolling tires is of essential importance for comfort questions, e.g. noise radiation. Whereas finite element models are well established for stationary rolling simulations, it lacks computational methods for the treatment of the high frequency response. One challenge is the large mode density of tire structures that is up to the acoustic frequency domain and another lies on the physically correct description of rolling (gyroscopic) structures. Despite that the eigenvalue analysis of gyroscopic systems, described by complex-valued quadratic eigenvalue systems, seems to be well understood in general, specific problems arise for the computability of large scale three-dimensional tire models. In this presentation an overall computational strategy for the high frequency response of FE-tire models is outlined, where special emphasis is placed upon the efficient numerical treatment of the complex-valued eigenproblems for large scale gyroscopic systems. The practicability of the proposed approach will be demonstrated by the analysis of detailed finite element tire models. The physical interpretation of the computational results is also discussed in detail.
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…
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
Some properties of eigenvalues and generalized eigenvectors of one boundary-value problem
NASA Astrophysics Data System (ADS)
Olgar, Hayati; Mukhtarov, Oktay; Aydemir, Kadriye
2016-08-01
We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piece-wise continuous potential together with eigenparameter-dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular it is shown that the generalized eigen-functions form a Riesz basis of the adequate Hilbert space.
An Evaluation of Interventions to Facilitate Algebra Problem Solving
ERIC Educational Resources Information Center
Mayfield, Kristin H.; Glenn, Irene M.
2008-01-01
Three participants were trained on 6 target algebra skills and subsequently received a series of 5 instructional interventions (cumulative practice, tiered feedback, feedback plus solution sequence instruction, review practice, and transfer training) in a multiple baseline across skills design. The effects of the interventions on the performance…
Excel Spreadsheets for Algebra: Improving Mental Modeling for Problem Solving
ERIC Educational Resources Information Center
Engerman, Jason; Rusek, Matthew; Clariana, Roy
2014-01-01
This experiment investigates the effectiveness of Excel spreadsheets in a high school algebra class. Students in the experiment group convincingly outperformed the control group on a post lesson assessment. The student responses, teacher observations involving Excel spreadsheet revealed that it operated as a mindtool, which formed the users'…
Student Strategy Choices on a Constructed Response Algebra Problem
ERIC Educational Resources Information Center
Ross, Dan; Reys, Robert; Chavez, Oscar; McNaught, Melissa D.; Grouws, Douglas A.
2011-01-01
A central goal of secondary mathematics is for students to learn to use powerful algebraic strategies appropriately. Research has demonstrated student difficulties in the transition to using such strategies. We examined strategies used by several thousand 8th-, 9th-, and 10th-grade students in five different school systems over three consecutive…
Cognitive Load and Modelling of an Algebra Problem
ERIC Educational Resources Information Center
Chinnappan, Mohan
2010-01-01
In the present study, I examine a modelling strategy as employed by a teacher in the context of an algebra lesson. The actions of this teacher suggest that a modelling approach will have a greater impact on enriching student learning if we do not lose sight of the need to manage associated cognitive loads that could either aid or hinder the…
NASA Astrophysics Data System (ADS)
Movassagh, Ramis
2016-02-01
We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. As an application we consider random perturbations of a fixed matrix M. If M is Normal, the total expected force on any eigenvalue is shown to be only the attraction of its c.c. (Eq. 24) and when M is circulant the strength of interaction can be related to the power spectrum of white noise. We extend this by calculating the expected force (Eq. 41) for real stochastic processes with zero-mean and independent intervals. To quantify the dominance of the c.c. attraction, we calculate the variance of other forces. We apply the results to the Hatano-Nelson model and provide other numerical illustrations. It is our hope that the simple dynamical perspective herein might help better understanding of the aggregation and low density of the eigenvalues of real random matrices on and near the real line respectively. In the appendix we provide a Matlab code for plotting the trajectories of the eigenvalues.
Acceleration of k-Eigenvalue / Criticality Calculations using the Jacobian-Free Newton-Krylov Method
Dana Knoll; HyeongKae Park; Chris Newman
2011-02-01
We present a new approach for the $k$--eigenvalue problem using a combination of classical power iteration and the Jacobian--free Newton--Krylov method (JFNK). The method poses the $k$--eigenvalue problem as a fully coupled nonlinear system, which is solved by JFNK with an effective block preconditioning consisting of the power iteration and algebraic multigrid. We demonstrate effectiveness and algorithmic scalability of the method on a 1-D, one group problem and two 2-D two group problems and provide comparison to other efforts using silmilar algorithmic approaches.
Application of symbolic and algebraic manipulation software in solving applied mechanics problems
NASA Technical Reports Server (NTRS)
Tsai, Wen-Lang; Kikuchi, Noboru
1993-01-01
As its name implies, symbolic and algebraic manipulation is an operational tool which not only can retain symbols throughout computations but also can express results in terms of symbols. This report starts with a history of symbolic and algebraic manipulators and a review of the literatures. With the help of selected examples, the capabilities of symbolic and algebraic manipulators are demonstrated. These applications to problems of applied mechanics are then presented. They are the application of automatic formulation to applied mechanics problems, application to a materially nonlinear problem (rigid-plastic ring compression) by finite element method (FEM) and application to plate problems by FEM. The advantages and difficulties, contributions, education, and perspectives of symbolic and algebraic manipulation are discussed. It is well known that there exist some fundamental difficulties in symbolic and algebraic manipulation, such as internal swelling and mathematical limitation. A remedy for these difficulties is proposed, and the three applications mentioned are solved successfully. For example, the closed from solution of stiffness matrix of four-node isoparametrical quadrilateral element for 2-D elasticity problem was not available before. Due to the work presented, the automatic construction of it becomes feasible. In addition, a new advantage of the application of symbolic and algebraic manipulation found is believed to be crucial in improving the efficiency of program execution in the future. This will substantially shorten the response time of a system. It is very significant for certain systems, such as missile and high speed aircraft systems, in which time plays an important role.
Perturbation of eigenvalues of preconditioned Navier-Stokes operators
Elman, H.C.
1996-12-31
We study the sensitivity of algebraic eigenvalue problems associated with matrices arising from linearization and discretization of the steady-state Navier-Stokes equations. In particular, for several choices of preconditioners applied to the system of discrete equations, we derive upper bounds on perturbations of eigenvalues as functions of the viscosity and discretization mesh size. The bounds suggest that the sensitivity of the eigenvalues is at worst linear in the inverse of the viscosity and quadratic in the inverse of the mesh size, and that scaling can be used to decrease the sensitivity in some cases. Experimental results supplement these results and confirm the relatively mild dependence on viscosity. They also indicate a dependence on the mesh size of magnitude smaller than the analysis suggests.
The Poincaré problem, algebraic integrability and dicritical divisors
NASA Astrophysics Data System (ADS)
Galindo, C.; Monserrat, F.
We solve the Poincaré problem for plane foliations with only one dicritical divisor. Moreover, in this case, we give a simple algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case. We also provide an algorithm to compute a rational first integral of prefixed genus g≠1 of any type of plane foliation F. When the number of dicritical divisors dic(F) is larger than 2, this algorithm depends on suitable families of invariant curves. When dic(F)=2, it proves that the degree of the rational first integral can be bounded only in terms of g, the degree of F and the local analytic type of the dicritical singularities of F. The degree d of a general integral invariant curve is less than or equal to 4. Therefore, the Poincaré problem is solved in this case. There exists a valueλ∈Z>0such thatPF:=|λΔF|is a pencil and the rational mapP2⋯→P1that it defines is a rational first integral ofF. Moreover λ is the minimum of the set{α∈Z>0|dim|αΔF|⩾1}. The above clause (b) supports a very simple algorithm, our forthcoming Algorithm 2, which decides about the existence of a rational first integral of F (and computes it in the positive case) whenever dic(F)=1. Other alternative algorithms are treated in Section 4. Our remaining main results are: Assume thatFhas a rational first integral of genus g. Then, there exists a bound on the degree of the first integral depending only on the degree ofF, the genus g and the local analytic type of the dicritical singularities ofF. There exists an algorithm to decide whetherFhas a rational first integral of genus g (and to compute it, in the affirmative case) whose inputs are: g, a homogeneous 1-form definingFand the minimal resolution of the dicritical singularities ofF. Assume thatFhas a rational first integral of genus g. Then there exists a bound on the degree of the first integral which depends on the degree ofF, the genus g, the local analytic type of the
ERIC Educational Resources Information Center
Hernandez, Andrea C.
2013-01-01
This dissertation analyzes differences found in Spanish-speaking middle school and high school students in algebra-based problem solving. It identifies the accuracy differences between word problems presented in English, Spanish and numerically based problems. The study also explores accuracy differences between each subgroup of Spanish-speaking…
Inverse eigenvalue problems in vibration absorption: Passive modification and active control
NASA Astrophysics Data System (ADS)
Mottershead, John E.; Ram, Yitshak M.
2006-01-01
The abiding problem of vibration absorption has occupied engineering scientists for over a century and there remain abundant examples of the need for vibration suppression in many industries. For example, in the automotive industry the resolution of noise, vibration and harshness (NVH) problems is of extreme importance to customer satisfaction. In rotorcraft it is vital to avoid resonance close to the blade passing speed and its harmonics. An objective of the greatest importance, and extremely difficult to achieve, is the isolation of the pilot's seat in a helicopter. It is presently impossible to achieve the objectives of vibration absorption in these industries at the design stage because of limitations inherent in finite element models. Therefore, it is necessary to develop techniques whereby the dynamic of the system (possibly a car or a helicopter) can be adjusted after it has been built. There are two main approaches: structural modification by passive elements and active control. The state of art of the mathematical theory of vibration absorption is presented and illustrated for the benefit of the reader with numerous simple examples.
Yamazaki, Ichitaro; Wu, Kesheng; Simon, Horst
2008-10-27
The original software package TRLan, [TRLan User Guide], page 24, implements the thick restart Lanczos method, [Wu and Simon 2001], page 24, for computing eigenvalues {lambda} and their corresponding eigenvectors v of a symmetric matrix A: Av = {lambda}v. Its effectiveness in computing the exterior eigenvalues of a large matrix has been demonstrated, [LBNL-42982], page 24. However, its performance strongly depends on the user-specified dimension of a projection subspace. If the dimension is too small, TRLan suffers from slow convergence. If it is too large, the computational and memory costs become expensive. Therefore, to balance the solution convergence and costs, users must select an appropriate subspace dimension for each eigenvalue problem at hand. To free users from this difficult task, nu-TRLan, [LNBL-1059E], page 23, adjusts the subspace dimension at every restart such that optimal performance in solving the eigenvalue problem is automatically obtained. This document provides a user guide to the nu-TRLan software package. The original TRLan software package was implemented in Fortran 90 to solve symmetric eigenvalue problems using static projection subspace dimensions. nu-TRLan was developed in C and extended to solve Hermitian eigenvalue problems. It can be invoked using either a static or an adaptive subspace dimension. In order to simplify its use for TRLan users, nu-TRLan has interfaces and features similar to those of TRLan: (1) Solver parameters are stored in a single data structure called trl-info, Chapter 4 [trl-info structure], page 7. (2) Most of the numerical computations are performed by BLAS, [BLAS], page 23, and LAPACK, [LAPACK], page 23, subroutines, which allow nu-TRLan to achieve optimized performance across a wide range of platforms. (3) To solve eigenvalue problems on distributed memory systems, the message passing interface (MPI), [MPI forum], page 23, is used. The rest of this document is organized as follows. In Chapter 2 [Installation
An Eigenvalue Analysis of finite-difference approximations for hyperbolic IBVPs
NASA Technical Reports Server (NTRS)
Warming, Robert F.; Beam, Richard M.
1989-01-01
The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L(sub 2) stability on a finite domain.
An eigenvalue analysis of finite-difference approximations for hyperbolic IBVPs
NASA Technical Reports Server (NTRS)
Warming, Robert F.; Beam, Richard M.
1990-01-01
The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L (sub 2) stability on a finite domain.
NASA Astrophysics Data System (ADS)
Luukko, P. J. J.; Räsänen, E.
2013-03-01
We present a code for solving the single-particle, time-independent Schrödinger equation in two dimensions. Our program utilizes the imaginary time propagation (ITP) algorithm, and it includes the most recent developments in the ITP method: the arbitrary order operator factorization and the exact inclusion of a (possibly very strong) magnetic field. Our program is able to solve thousands of eigenstates of a two-dimensional quantum system in reasonable time with commonly available hardware. The main motivation behind our work is to allow the study of highly excited states and energy spectra of two-dimensional quantum dots and billiard systems with a single versatile code, e.g., in quantum chaos research. In our implementation we emphasize a modern and easily extensible design, simple and user-friendly interfaces, and an open-source development philosophy. Catalogue identifier: AENR_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AENR_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License version 3 No. of lines in distributed program, including test data, etc.: 11310 No. of bytes in distributed program, including test data, etc.: 97720 Distribution format: tar.gz Programming language: C++ and Python. Computer: Tested on x86 and x86-64 architectures. Operating system: Tested under Linux with the g++ compiler. Any POSIX-compliant OS with a C++ compiler and the required external routines should suffice. Has the code been vectorised or parallelized?: Yes, with OpenMP. RAM: 1 MB or more, depending on system size. Classification: 7.3. External routines: FFTW3 (http://www.fftw.org), CBLAS (http://netlib.org/blas), LAPACK (http://www.netlib.org/lapack), HDF5 (http://www.hdfgroup.org/HDF5), OpenMP (http://openmp.org), TCLAP (http://tclap.sourceforge.net), Python (http://python.org), Google Test (http://code.google.com/p/googletest/) Nature of problem: Numerical calculation
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…
ERIC Educational Resources Information Center
Ngu, Bing Hiong; Yeung, Alexander Seeshing
2012-01-01
Holyoak and Koh (1987) and Holyoak (1984) propose four critical tasks for analogical transfer to occur in problem solving. A study was conducted to test this hypothesis by comparing a multiple components (MC) approach against worked examples (WE) in helping students to solve algebra word problems in chemistry classes. The MC approach incorporated…
Does Calculation or Word-Problem Instruction Provide A Stronger Route to Pre-Algebraic Knowledge?
Fuchs, Lynn S.; Powell, Sarah R.; Cirino, Paul T.; Schumacher, Robin F.; Marrin, Sarah; Hamlett, Carol L.; Fuchs, Douglas; Compton, Donald L.; Changas, Paul C.
2014-01-01
The focus of this study was connections among 3 aspects of mathematical cognition at 2nd grade: calculations, word problems, and pre-algebraic knowledge. We extended the literature, which is dominated by correlational work, by examining whether intervention conducted on calculations or word problems contributes to improved performance in the other domain and whether intervention in either or both domains contributes to pre-algebraic knowledge. Participants were 1102 children in 127 2nd-grade classrooms in 25 schools. Teachers were randomly assigned to 3 conditions: calculation intervention, word-problem intervention, and business-as-usual control. Intervention, which lasted 17 weeks, was designed to provide research-based linkages between arithmetic calculations or arithmetic word problems (depending on condition) to pre-algebraic knowledge. Multilevel modeling suggested calculation intervention improved calculation but not word-problem outcomes; word-problem intervention enhanced word-problem but not calculation outcomes; and word-problem intervention provided a stronger route than calculation intervention to pre-algebraic knowledge. PMID:25541565
The Model Method: Singapore Children's Tool for Representing and Solving Algebraic Word Problems
ERIC Educational Resources Information Center
Ng, Swee Fong; Lee, Kerry
2009-01-01
Solving arithmetic and algebraic word problems is a key component of the Singapore elementary mathematics curriculum. One heuristic taught, the model method, involves drawing a diagram to represent key information in the problem. We describe the model method and a three-phase theoretical framework supporting its use. We conducted 2 studies to…
A spectral projection method for transmission eigenvalues
NASA Astrophysics Data System (ADS)
Zeng, Fang; Sun, JiGuang; Xu, LiWei
2016-08-01
In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples.
Geist, G.A.; Howell, G.W.; Watkins, D.S.
1997-11-01
The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. It is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR algorithm, it is well adapted to computing the eigenvalues of the narrowband, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the QR algorithm by a factor of 30--60 in computing time and a factor of over 100 in matrix storage space.
The Schrodinger Eigenvalue March
ERIC Educational Resources Information Center
Tannous, C.; Langlois, J.
2011-01-01
A simple numerical method for the determination of Schrodinger equation eigenvalues is introduced. It is based on a marching process that starts from an arbitrary point, proceeds in two opposite directions simultaneously and stops after a tolerance criterion is met. The method is applied to solving several 1D potential problems including symmetric…
Promoting Problem Solving across Geometry and Algebra by Using Technology
ERIC Educational Resources Information Center
Erbas, A. Kursat; Ledford, Sara D.; Orrill, Chandra Hawley; Polly, Drew
2005-01-01
Technology is a powerful tool in assisting students in problem solving by allowing for multiple representations. The vignette offered in this article provides insight into ways to solve open-ended problems using multiple technologies.
An efficient algorithm for the contig ordering problem under algebraic rearrangement distance.
Lu, Chin Lung
2015-11-01
Assembling a genome from short reads currently obtained by next-generation sequencing techniques often results in a collection of contigs, whose relative position and orientation along the genome being sequenced are unknown. Given two sets of contigs, the contig ordering problem is to order and orient the contigs in each set such that the genome rearrangement distance between the resulting sets of ordered and oriented contigs is minimized. In this article, we utilize the permutation groups in algebra to propose a near-linear time algorithm for solving the contig ordering problem under algebraic rearrangement distance, where the algebraic rearrangement distance between two sets of ordered and oriented contigs is the minimum weight of applicable rearrangement operations required to transform one set into the other. PMID:26247343
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.
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…
Effects of Graphic Organiser on Students' Achievement in Algebraic Word Problems
ERIC Educational Resources Information Center
Owolabi, Josiah; Adaramati, Tobiloba Faith
2015-01-01
This study investigated the effects of graphic organiser and gender on students' academic achievement in algebraic word problem. Three research questions and three null hypotheses were used in guiding this study. Quasi experimental research was employed and Non-equivalent pre and post test design was used. The study involved the Senior Secondary…
ERIC Educational Resources Information Center
Lee, Kerry; Khng, Kiat Hui; Ng, Swee Fong; Ng Lan Kong, Jeremy
2013-01-01
In Singapore, primary school students are taught to use bar diagrams to represent known and unknown values in algebraic word problems. However, little is known about students' understanding of these graphical representations. We investigated whether students use and think of the bar diagrams in a concrete or a more abstract fashion. We also…
CREUTZ, M.
2006-01-26
It is popular to discuss low energy physics in lattice gauge theory ill terms of the small eigenvalues of the lattice Dirac operator. I play with some ensuing pitfalls in the interpretation of these eigenvalue spectra. In short, thinking about the eigenvalues of the Dirac operator in the presence of gauge fields can give some insight, for example the elegant Banks-Casher picture for chiral symmetry breaking. Nevertheless, care is necessary because the problem is highly non-linear. This manifests itself in the non-intuitive example of how adding flavors enhances rather than suppresses low eigenvalues. Issues involving zero mode suppression represent one facet of a set of connected unresolved issues. Are there non-perturbative ambiguities in quantities such as the topological susceptibility? How essential are rough gauge fields, i.e. gauge fields on which the winding number is ambiguous? How do these issues interplay with the quark masses? I hope the puzzles presented here will stimulate more thought along these lines.
Henson, V E
2003-02-06
The purpose of this research project was to investigate, design, and implement new algebraic multigrid (AMG) algorithms to enable the effective use of AMG in large-scale multiphysics simulation codes. These problems are extremely large; storage requirements and excessive run-time make direct solvers infeasible. The problems are highly ill-conditioned, so that existing iterative solvers either fail or converge very slowly. While existing AMG algorithms have been shown to be robust and stable for a large class of problems, there are certain problems of great interest to the Laboratory for which no effective algorithm existed prior to this research.
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…
Algebraic Approach to the Minimum-Cost Multi-Impulse Orbit-Transfer Problem
NASA Astrophysics Data System (ADS)
Avendaño, M.; Martín-Molina, V.; Martín-Morales, J.; Ortigas-Galindo, J.
2016-08-01
We present a purely algebraic formulation (i.e. polynomial equations only) of the minimum-cost multi-impulse orbit transfer problem without time constraints, while keeping all the variables with a precise physical meaning. We apply general algebraic techniques to solve these equations (resultants, Gr\\"obner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, we provide a proof of the optimality of the Hohmann transfer for the minimum fuel 2-impulse circular to circular orbit transfer problem, and we provide a general formula for the optimal 2-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which we conjecture that can be removed).
Fuchs, Lynn S; Compton, Donald L; Fuchs, Douglas; Hollenbeck, Kurstin N; Hamlett, Carol L; Seethaler, Pamela M
2011-01-01
The purpose of this study was to explore the utility of a dynamic assessment (DA) of algebraic learning in predicting third graders' development of mathematics word-problem difficulty. In the fall, 122 third-grade students were assessed on a test of math word-problem skill and DA of algebraic learning. In the spring, they were assessed on word-problem performance. Logistic regression was conducted to contrast two models. One relied exclusively on the fall test of math word-problem skill to predict word-problem difficulty on the spring outcome (less than the 25th percentile). The second model relied on a combination of the fall test of math word-problem skill and the fall DA to predict the same outcome. Holding sensitivity at 87.5%, the universal screener alone resulted in a high proportion of false positives, which was practically reduced when DA was included in the prediction model. Findings are discussed in terms of a two-stage process for screening students within a responsiveness-to-intervention prevention model. PMID:21685352
NASA Astrophysics Data System (ADS)
Liu, Tianyu; Du, Xining; Ji, Wei; Xu, X. George; Brown, Forrest B.
2014-06-01
For nuclear reactor analysis such as the neutron eigenvalue calculations, the time consuming Monte Carlo (MC) simulations can be accelerated by using graphics processing units (GPUs). However, traditional MC methods are often history-based, and their performance on GPUs is affected significantly by the thread divergence problem. In this paper we describe the development of a newly designed event-based vectorized MC algorithm for solving the neutron eigenvalue problem. The code was implemented using NVIDIA's Compute Unified Device Architecture (CUDA), and tested on a NVIDIA Tesla M2090 GPU card. We found that although the vectorized MC algorithm greatly reduces the occurrence of thread divergence thus enhancing the warp execution efficiency, the overall simulation speed is roughly ten times slower than the history-based MC code on GPUs. Profiling results suggest that the slow speed is probably due to the memory access latency caused by the large amount of global memory transactions. Possible solutions to improve the code efficiency are discussed.
NASA Astrophysics Data System (ADS)
Zhao, Ye; Gu, Zhuquan; Liu, Yafeng
2012-07-01
In this paper, the Neumann system for the 4th-order eigenvalue problem Ly = (∂4+ q∂2+∂2 q+ ip∂+ i∂ p+ y = Λy) has been given. By means of the Neumann constraint condition, the perfect constraint set Γ and the relations between the potentials { q, p, r} and the eigenvector y are obtained. Then, based on the Euler-Lagrange function and Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system has been found, which can be equal to the real Hamiltonian canonical coordinate system in R 8 N . Using Cao's method and Moser's constraint manifold, the Lax pairs of the evolution equation hierarchy with the 4th-order eigenvalue problems are nonlinearized. So a new finite-dimensional integrable Hamilton system on the constraint submanifold R 8 N-4 is generated. Moreover, the solutions of the evolution equations for the infinite-dimensional soliton systems are obtained by the involutive flow of the finite-dimensional completely integrable systems.
Kulkarni, A.K.; Paranjape, S.D.; Kumar, V.; Sahni, D.C.
1994-12-31
Nonmonotonic variation of the {open_quotes}C{close_quotes} eigenvalue (average number of secondaries per collision) with increasing {alpha}, the strength of forward scattering, has been observed earlier for one-dimensional infinite homogeneous slabs and infinitely long homogeneous cylinders. The authors have developed the Integral Transform (IT) method, an accurate semi-analytical method to obtain the C eigenvalue for a homogeneous cylinder (two-dimensional system). They are thus able to detect any nonmonotonic variation of C (with {alpha}) using the Sahni and Sjoestrand criterion. Along with the IT method, the authors also present the results obtained by the well-known numerical techniques like the discrete ordinates method using a high quadrature order and the Monte Carlo method for the same problem. The S{sub N} results show disagreement with the other two methods when one of the dimensions is very small (<0.05{lambda}{sub t}). They believe that even the 16th order quadrature set cannot integrate the angular flux accurately in these extreme situations. 12 refs., 9 tabs.
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…
Individualized Math Problems in Algebra. Oregon Vo-Tech Mathematics Problem Sets.
ERIC Educational Resources Information Center
Cosler, Norma, Ed.
This is one of eighteen sets of individualized mathematics problems developed by the Oregon Vo-Tech Math Project. Each of these problem packages is organized around a mathematical topic, and contains problems related to diverse vocations. Solutions are provided for all problems. Problems presented in this package concern ratios used in food…
NASA Technical Reports Server (NTRS)
Sidi, Avram
1992-01-01
Let F(z) be a vectored-valued function F: C approaches C sup N, which is analytic at z=0 and meromorphic in a neighborhood of z=0, and let its Maclaurin series be given. We use vector-valued rational approximation procedures for F(z) that are based on its Maclaurin series in conjunction with power iterations to develop bona fide generalizations of the power method for an arbitrary N X N matrix that may be diagonalizable or not. These generalizations can be used to obtain simultaneously several of the largest distinct eigenvalues and the corresponding invariant subspaces, and present a detailed convergence theory for them. In addition, it is shown that the generalized power methods of this work are equivalent to some Krylov subspace methods, among them the methods of Arnoldi and Lanczos. Thus, the theory provides a set of completely new results and constructions for these Krylov subspace methods. This theory suggests at the same time a new mode of usage for these Krylov subspace methods that were observed to possess computational advantages over their common mode of usage.
Algebraic analysis of the phase-calibration problem in the self-calibration procedures
NASA Astrophysics Data System (ADS)
Lannes, A.; Prieur, J.-L.
2011-10-01
This paper presents an analysis of the phase-calibration problem encountered in astronomy when mapping incoherent sources with aperture-synthesis devices. More precisely, this analysis concerns the phase-calibration operation involved in the self-calibration procedures of phase-closure imaging. The paper revisits and completes a previous analysis presented by Lannes in the Journal of the Optical Society of America A in 2005. It also benefits from some recent developments made for solving similar problems encountered in global navigation satellite systems. In radio-astronomy, the related optimization problems have been stated and solved hitherto at the phasor level. We present here an analysis conducted at the phase level, from which we derive a method for diagnosing and solving the difficulties of the phasor approach. In the most general case, the techniques to be implemented appeal to the algebraic graph theory and the algebraic number theory. The minima of the objective functionals to be minimized are identified by raising phase-closure integer ambiguities. We also show that in some configurations, to benefit from all the available information, closure phases of order greater than three are to be introduced. In summary, this study leads to a better understanding of the difficulties related to the very principle of phase-closure imaging. To circumvent these difficulties, we propose a strategy both simple and robust.
Voila: A visual object-oriented iterative linear algebra problem solving environment
Edwards, H.C.; Hayes, L.J.
1994-12-31
Application of iterative methods to solve a large linear system of equations currently involves writing a program which calls iterative method subprograms from a large software package. These subprograms have complex interfaces which are difficult to use and even more difficult to program. A problem solving environment specifically tailored to the development and application of iterative methods is needed. This need will be fulfilled by Voila, a problem solving environment which provides a visual programming interface to object-oriented iterative linear algebra kernels. Voila will provide several quantum improvements over current iterative method problem solving environments. First, programming and applying iterative methods is considerably simplified through Voila`s visual programming interface. Second, iterative method algorithm implementations are independent of any particular sparse matrix data structure through Voila`s object-oriented kernels. Third, the compile-link-debug process is eliminated as Voila operates as an interpreter.
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.
NASA Astrophysics Data System (ADS)
Konovalov, Y. V.
2015-09-01
Ice-shelf forced vibration modelling is performed using a full 3-D finite-difference elastic model, which also takes into account sub-ice seawater flow. The ocean flow in the cavity is described by the wave equation; therefore, ice-shelf flexures result from hydrostatic pressure perturbations in sub-ice seawater layer. Numerical experiments have been carried out for idealized rectangular and trapezoidal ice-shelf geometries. The ice-plate vibrations are modelled for harmonic ingoing pressure perturbations and for high-frequency spectra of the ocean swells. The spectra show distinct resonance peaks, which demonstrate the ability to model a resonant-like motion in the suitable conditions of forcing. The spectra and ice-shelf deformations obtained by the developed full 3-D model are compared with the spectra and the deformations modelled by the thin-plate Holdsworth and Glynn model (1978). The main resonance peaks and ice-shelf deformations in the corresponding modes, derived by the full 3-D model, are in agreement with the peaks and deformations obtained by the Holdsworth and Glynn model. The relative deviation between the eigenvalues (periodicities) in the two compared models is about 10 %. In addition, the full model allows observation of 3-D effects, for instance, the vertical distribution of the stress components in the plate. In particular, the full model reveals an increase in shear stress, which is neglected in the thin-plate approximation, from the terminus towards the grounding zone with a maximum at the grounding line in the case of the considered high-frequency forcing. Thus, the high-frequency forcing can reinforce the tidal impact on the ice-shelf grounding zone causing an ice fracture therein.
Fuchs, Lynn S; Compton, Donald L; Fuchs, Douglas; Hollenbeck, Kurstin N; Craddock, Caitlin F; Hamlett, Carol L
2008-11-01
Dynamic assessment (DA) involves helping students learn a task and indexing responsiveness to that instruction as a measure of learning potential. The purpose of this study was to explore the utility of a DA of algebraic learning in predicting 3(rd) graders' development of mathematics problem solving. In the fall, 122 3(rd)-grade students were assessed on language, nonverbal reasoning, attentive behavior, calculations, word-problem skill, and DA. On the basis of random assignment, students received 16 weeks of validated instruction on word problems or received 16 weeks of conventional instruction on word problems. Then, students were assessed on word-problem measures proximal and distal to instruction. Structural equation measurement models showed that DA measured a distinct dimension of pretreatment ability and that proximal and distal word-problem measures were needed to account for outcome. Structural equation modeling showed that instruction (conventional vs. validated) was sufficient to account for math word-problem outcome proximal to instruction; by contrast, language, pretreatment math skill, and DA were needed to forecast learning on word-problem outcomes more distal to instruction. Findings are discussed in terms of responsiveness-to-intervention models for preventing and identifying learning disabilities. PMID:19884957
Marek, A; Blum, V; Johanni, R; Havu, V; Lang, B; Auckenthaler, T; Heinecke, A; Bungartz, H-J; Lederer, H
2014-05-28
Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N(3)) with the size of the investigated problem, N (e.g. the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on a few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the Eigenvalue soLvers for Petascale Applications (ELPA) library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the Scalable Linear Algebra PACKage (ScaLAPACK) library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g. Intel's MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPI-OpenMPI implementation available as well. Scalability beyond 10,000 CPU cores for problem
Implicity restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations
NASA Technical Reports Server (NTRS)
Sorensen, Danny C.
1996-01-01
Eigenvalues and eigenfunctions of linear operators are important to many areas of applied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fueled interest in the development of new methods and software for the numerical solution of large-scale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now available, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for large-scale nonsymmetric problems was virtually non-existent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of large-scale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The well-known Lanczos method is the premier member of this class. The Arnoldi method generalizes the Lanczos method to the nonsymmetric case. A recently developed variant of the Arnoldi/Lanczos scheme called the Implicitly Restarted Arnoldi Method is presented here in some depth. This method is highlighted because of its suitability as a basis for software development.
Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems
NASA Technical Reports Server (NTRS)
Vanek, Petr; Mandel, Jan; Brezina, Marian
1996-01-01
Multigrid methods are very efficient iterative solvers for system of algebraic equations arising from finite element and finite difference discretization of elliptic boundary value problems. The main principle of multigrid methods is to complement the local exchange of information in point-wise iterative methods by a global one utilizing several related systems, called coarse levels, with a smaller number of variables. The coarse levels are often obtained as a hierarchy of discretizations with different characteristic meshsizes, but this requires that the discretization is controlled by the iterative method. To solve linear systems produced by existing finite element software, one needs to create an artificial hierarchy of coarse problems. The principal issue is then to obtain computational complexity and approximation properties similar to those for nested meshes, using only information in the matrix of the system and as little extra information as possible. Such algebraic multigrid method that uses the system matrix only was developed by Ruge. The prolongations were based on the matrix of the system by partial solution from given values at selected coarse points. The coarse grid points were selected so that each point would be interpolated to via so-called strong connections. Our approach is based on smoothed aggregation introduced recently by Vanek. First the set of nodes is decomposed into small mutually disjoint subsets. A tentative piecewise constant interpolation (in the discrete sense) is then defined on those subsets as piecewise constant for second order problems, and piecewise linear for fourth order problems. The prolongation operator is then obtained by smoothing the output of the tentative prolongation and coarse level operators are defined variationally.
AEST: Adaptive Eigenvalue Stability Code
NASA Astrophysics Data System (ADS)
Zheng, L.-J.; Kotschenreuther, M.; Waelbroeck, F.; van Dam, J. W.; Berk, H.
2002-11-01
An adaptive eigenvalue linear stability code is developed. The aim is on one hand to include the non-ideal MHD effects into the global MHD stability calculation for both low and high n modes and on the other hand to resolve the numerical difficulty involving MHD singularity on the rational surfaces at the marginal stability. Our code follows some parts of philosophy of DCON by abandoning relaxation methods based on radial finite element expansion in favor of an efficient shooting procedure with adaptive gridding. The δ W criterion is replaced by the shooting procedure and subsequent matrix eigenvalue problem. Since the technique of expanding a general solution into a summation of the independent solutions employed, the rank of the matrices involved is just a few hundreds. This makes easier to solve the eigenvalue problem with non-ideal MHD effects, such as FLR or even full kinetic effects, as well as plasma rotation effect, taken into account. To include kinetic effects, the approach of solving for the distribution function as a local eigenvalue ω problem as in the GS2 code will be employed in the future. Comparison of the ideal MHD version of the code with DCON, PEST, and GATO will be discussed. The non-ideal MHD version of the code will be employed to study as an application the transport barrier physics in tokamak discharges.
NASA Astrophysics Data System (ADS)
Leukhin, Anatolii N.
2005-08-01
The algebraic solution of a 'complex' problem of synthesis of phase-coded (PC) sequences with the zero level of side lobes of the cyclic autocorrelation function (ACF) is proposed. It is shown that the solution of the synthesis problem is connected with the existence of difference sets for a given code dimension. The problem of estimating the number of possible code combinations for a given code dimension is solved. It is pointed out that the problem of synthesis of PC sequences is related to the fundamental problems of discrete mathematics and, first of all, to a number of combinatorial problems, which can be solved, as the number factorisation problem, by algebraic methods by using the theory of Galois fields and groups.
Algebraic methods for the identification problem with short arcs of observations.
NASA Astrophysics Data System (ADS)
Gronchi, G. F.
The identification problem of short arcs of asteroid observations is related with the determination of the orbits of the observed asteroids. Recently this problem has been faced with algebraic methods using the first integrals of Kepler's problem. These methods allow us to solve the problem in an efficient way, keeping under control also alternative solutions, that may occur. However, the huge and continuously increasing amount of data produced by the new asteroid surveys suggests us to search for new algorithms, with shorter computation times. In this communication I'll review the known methods \\cite{p1}, \\cite{p2}, that lead to polynomial equations of degree 48 and 20 respectively. Then I'll present a new algorithm \\cite{p3}, that we are currently studying, allowing to deal with this problem with a polynomial of degree 9, thus decreasing the computation times in a significant way. Finally, I'll show some examples of computation of asteroid orbits using these methods.
The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl-1(2)
NASA Astrophysics Data System (ADS)
Genest, Vincent X.; Vinet, Luc; Zhedanov, Alexei
2013-02-01
The algebra H of the dual -1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl-1(2). The dual -1 Hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q → -1 limit of the dual q-Hahn polynomials. The Hopf algebra sl-1(2) has four generators including an involution, it is also a q → -1 limit of the quantum algebra slq(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra H, a two-parameter generalization of {u}(2) with an involution as additional generator, is first derived from the recurrence relation of the -1 Hahn polynomials. It is then shown that H can be realized in terms of the generators of two added sl-1(2) algebras, so that the Clebsch-Gordan coefficients of sl-1(2) are dual -1 Hahn polynomials. An irreducible representation of H involving five-diagonal matrices and connected to the difference equation of the dual -1 Hahn polynomials is constructed.
The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl{sub -1}(2)
Genest, Vincent X.; Vinet, Luc; Zhedanov, Alexei
2013-02-15
The algebra H of the dual -1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl{sub -1}(2). The dual -1 Hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q{yields}-1 limit of the dual q-Hahn polynomials. The Hopf algebra sl{sub -1}(2) has four generators including an involution, it is also a q{yields}-1 limit of the quantum algebra sl{sub q}(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra H, a two-parameter generalization of u(2) with an involution as additional generator, is first derived from the recurrence relation of the -1 Hahn polynomials. It is then shown that H can be realized in terms of the generators of two added sl{sub -1}(2) algebras, so that the Clebsch-Gordan coefficients of sl{sub -1}(2) are dual -1 Hahn polynomials. An irreducible representation of H involving five-diagonal matrices and connected to the difference equation of the dual -1 Hahn polynomials is constructed.
A new mathematical evaluation of smoking problem based of algebraic statistical method.
Mohammed, Maysaa J; Rakhimov, Isamiddin S; Shitan, Mahendran; Ibrahim, Rabha W; Mohammed, Nadia F
2016-01-01
Smoking problem is considered as one of the hot topics for many years. In spite of overpowering facts about the dangers, smoking is still a bad habit widely spread and socially accepted. Many people start smoking during their gymnasium period. The discovery of the dangers of smoking gave a warning sign of danger for individuals. There are different statistical methods used to analyze the dangers of smoking. In this study, we apply an algebraic statistical method to analyze and classify real data using Markov basis for the independent model on the contingency table. Results show that the Markov basis based classification is able to distinguish different date elements. Moreover, we check our proposed method via information theory by utilizing the Shannon formula to illustrate which one of these alternative tables is the best in term of independent. PMID:26858555
ERIC Educational Resources Information Center
Bull, Elizabeth Kay
The goal of this study was to find a way to quantify three criteria of representational quality, described by Greeno, so that it would be possible to examine statistically the relationship between representational quality and other variables related to problem solution. The sample consisted of 18 college students, 84 percent of whom had…
SO(4) algebraic approach to the three-body bound state problem in two dimensions
NASA Astrophysics Data System (ADS)
Dmitrašinović, V.; Salom, Igor
2014-08-01
We use the permutation symmetric hyperspherical three-body variables to cast the non-relativistic three-body Schrödinger equation in two dimensions into a set of (possibly decoupled) differential equations that define an eigenvalue problem for the hyper-radial wave function depending on an SO(4) hyper-angular matrix element. We express this hyper-angular matrix element in terms of SO(3) group Clebsch-Gordan coefficients and use the latter's properties to derive selection rules for potentials with different dynamical/permutation symmetries. Three-body potentials acting on three identical particles may have different dynamical symmetries, in order of increasing symmetry, as follows: (1) S3 ⊗ OL(2), the permutation times rotational symmetry, that holds in sums of pairwise potentials, (2) O(2) ⊗ OL(2), the so-called "kinematic rotations" or "democracy symmetry" times rotational symmetry, that holds in area-dependent potentials, and (3) O(4) dynamical hyper-angular symmetry, that holds in hyper-radial three-body potentials. We show how the different residual dynamical symmetries of the non-relativistic three-body Hamiltonian lead to different degeneracies of certain states within O(4) multiplets.
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.
ERIC Educational Resources Information Center
Shoecraft, Paul Joseph
Three instructional approaches on translating selected types of algebra word problems were investigated: direct translations, high imagery with materials, and high imagery with drawings. Participating were 366 seventh grade and 336 ninth grade students. Treatment effects by grade used multivariate analysis of covariance for student scores and…
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, Inc., Reston, VA.
This is a reprint of the historical capsules dealing with algebra from the 31st Yearbook of NCTM,"Historical Topics for the Mathematics Classroom." Included are such themes as the change from a geometric to an algebraic solution of problems, the development of algebraic symbolism, the algebraic contributions of different countries, the origin and…
NASA Astrophysics Data System (ADS)
Hendrikse, Anne; Veldhuis, Raymond; Spreeuwers, Luuk
2013-12-01
Second-order statistics play an important role in data modeling. Nowadays, there is a tendency toward measuring more signals with higher resolution (e.g., high-resolution video), causing a rapid increase of dimensionality of the measured samples, while the number of samples remains more or less the same. As a result the eigenvalue estimates are significantly biased as described by the Marčenko Pastur equation for the limit of both the number of samples and their dimensionality going to infinity. By introducing a smoothness factor, we show that the Marčenko Pastur equation can be used in practical situations where both the number of samples and their dimensionality remain finite. Based on this result we derive methods, one already known and one new to our knowledge, to estimate the sample eigenvalues when the population eigenvalues are known. However, usually the sample eigenvalues are known and the population eigenvalues are required. We therefore applied one of the these methods in a feedback loop, resulting in an eigenvalue bias correction method. We compare this eigenvalue correction method with the state-of-the-art methods and show that our method outperforms other methods particularly in real-life situations often encountered in biometrics: underdetermined configurations, high-dimensional configurations, and configurations where the eigenvalues are exponentially distributed.
ERIC Educational Resources Information Center
Lukas, George; And Others
In order to provide high school students with general problem-solving skills, two LOGO computer-assisted instruction units were developed--one on the methods and strategies for solution and a second on the relation between formal and informal representations of problems. In both cases specific problem contexts were used to give definition and…
NASA Astrophysics Data System (ADS)
Chuluunbaatar, O.; Gusev, A. A.; Vinitsky, S. I.; Abrashkevich, A. G.
2009-08-01
A FORTRAN 77 program is presented for calculating with the given accuracy eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program calculates also potential matrix elements - integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. Eigenvalues and matrix elements computed by the ODPEVP program can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649-675; O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Commun. 179 (2008) 685-693]. As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials, a 3D-model of a hydrogen atom in a homogeneous magnetic field and a hydrogen atom on a three-dimensional sphere. Program summaryProgram title: ODPEVP Catalogue identifier: AEDV_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDV_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 3001 No. of bytes in distributed program, including test data, etc.: 24 195 Distribution format: tar.gz Programming language: FORTRAN 77 Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IV Operating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XP RAM: depends on the number and order of finite
ERIC Educational Resources Information Center
Booth, Julie L.; Lange, Karin E.; Koedinger, Kenneth R.; Newton, Kristie J.
2013-01-01
In a series of two in vivo experiments, we examine whether correct and incorrect examples with prompts for self-explanation can be effective for improving students' conceptual understanding and procedural skill in Algebra when combined with guided practice. In Experiment 1, students working with the Algebra I Cognitive Tutor were randomly assigned…
ERIC Educational Resources Information Center
Booth, Julie L.; Lange, Karin E.; Koedinger, Kenneth R.; Newton, Kristie J.
2013-01-01
In a series of two "in vivo" experiments, we examine whether correct and incorrect examples with prompts for self-explanation can be effective for improving students' conceptual understanding and procedural skill in Algebra when combined with guided practice. In Experiment 1, students working with the Algebra I Cognitive Tutor were randomly…
Acceleration of multiple solution of a boundary value problem involving a linear algebraic system
NASA Astrophysics Data System (ADS)
Gazizov, Talgat R.; Kuksenko, Sergey P.; Surovtsev, Roman S.
2016-06-01
Multiple solution of a boundary value problem that involves a linear algebraic system is considered. New approach to acceleration of the solution is proposed. The approach uses the structure of the linear system matrix. Particularly, location of entries in the right columns and low rows of the matrix, which undergo variation due to the computing in the range of parameters, is used to apply block LU decomposition. Application of the approach is considered on the example of multiple computing of the capacitance matrix by method of moments used in numerical electromagnetics. Expressions for analytic estimation of the acceleration are presented. Results of the numerical experiments for solution of 100 linear systems with matrix orders of 1000, 2000, 3000 and different relations of variated and constant entries of the matrix show that block LU decomposition can be effective for multiple solution of linear systems. The speed up compared to pointwise LU factorization increases (up to 15) for larger number and order of considered systems with lower number of variated entries.
Geometric and algebraic multigrid techniques for fluid dynamics problems on unstructured grids
NASA Astrophysics Data System (ADS)
Volkov, K. N.; Emel'yanov, V. N.; Teterina, I. V.
2016-02-01
Issues concerning the implementation and practical application of geometric and algebraic multigrid techniques for solving systems of difference equations generated by the finite volume discretization of the Euler and Navier-Stokes equations on unstructured grids are studied. The construction of prolongation and interpolation operators, as well as grid levels of various resolutions, is discussed. The results of the application of geometric and algebraic multigrid techniques for the simulation of inviscid and viscous compressible fluid flows over an airfoil are compared. Numerical results show that geometric methods ensure faster convergence and weakly depend on the method parameters, while the efficiency of algebraic methods considerably depends on the input parameters.
ERIC Educational Resources Information Center
Green, Jan
2009-01-01
In recent years, the learning of algebra by all students has become a significant national priority (Moses & Cobb, 2001; National Council of Teachers of Mathematics, 2000). Algebra is considered to be a foundational topic in mathematics (Usiskin, 1988) and some have argued that an understanding of algebra is fundamental to success in today's…
Eigenvalues and musical instruments
NASA Astrophysics Data System (ADS)
Howle, V. E.; Trefethen, Lloyd N.
2001-10-01
Most musical instruments are built from physical systems that oscillate at certain natural frequencies. The frequencies are the imaginary parts of the eigenvalues of a linear operator, and the decay rates are the negatives of the real parts, so it ought to be possible to give an approximate idea of the sound of a musical instrument by a single plot of points in the complex plane. Nevertheless, the authors are unaware of any such picture that has ever appeared in print. This paper attempts to fill that gap by plotting eigenvalues for simple models of a guitar string, a flute, a clarinet, a kettledrum, and a musical bell. For the drum and the bell, simple idealized models have eigenvalues that are irrationally related, but as the actual instruments have evolved over the generations, the leading five or six eigenvalues have moved around the complex plane so that their relative positions are musically pleasing.
ERIC Educational Resources Information Center
Rubio, Guillermo; del Valle, Rafael
2004-01-01
The study proves that a didactical model based in a method to solve word problems of increasing complexity which uses a numerical approach was essential to develop the analytical ability and the competent use of the algebraic language with students from three different performance levels in elementary algebra. It is shown that before using the…
Ellouz, Hanen; Feki, Ines; Jeribi, Aref
2013-11-15
In the present paper, we prove that the family of exponentials associated to the eigenvalues of the perturbed operator T(ε) ≔ T{sub 0} + εT{sub 1} + ε{sup 2}T{sub 2} + … + ε{sup k}T{sub k} + … forms a Riesz basis in L{sup 2}(0, T), T > 0, where ε∈C, T{sub 0} is a closed densely defined linear operator on a separable Hilbert space H with domain D(T{sub 0}) having isolated eigenvalues with multiplicity one, while T{sub 1}, T{sub 2}, … are linear operators on H having the same domain D⊃D(T{sub 0}) and satisfying a specific growing inequality. After that, we generalize this result using a H-Lipschitz function. As application, we consider a non-selfadjoint problem deduced from a perturbation method for sound radiation.
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.
Strategies Used by Second-Year Algebra Students to Solve Problems
ERIC Educational Resources Information Center
Senk, Sharon L.; Thompson, Denisse R.
2006-01-01
This Brief Report describes a secondary analysis of the solutions written by 306 second-year algebra students to four constructed-response items representative of content at this level. The type of solution (symbolic, graphical, or numerical) used most frequently varied by item. Curriculum effects were observed. Students studying from the second…
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…
Higher level twisted Zhu algebras
Ekeren, Jethro van
2011-05-15
The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper, we consider the general setup of a vertex algebra V, graded by {Gamma}/Z for some subgroup {Gamma} of R containing Z, and with a Hamiltonian operator H having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level p Zhu algebras Zhu{sub p,{Gamma}}(V), and we prove the following theorems: For each p, there is a bijection between the irreducible Zhu{sub p,{Gamma}}(V)-modules and the irreducible {Gamma}-twisted positive energy V-modules, and V is ({Gamma}, H)-rational if and only if all its Zhu algebras Zhu{sub p,{Gamma}}(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for H. We provide an explicit description of the level p Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra Vir{sup c} and the universal affine Kac-Moody vertex algebra V{sup k}(g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.
The Coulomb problem on a 3-sphere and Heun polynomials
NASA Astrophysics Data System (ADS)
Bellucci, Stefano; Yeghikyan, Vahagn
2013-08-01
The paper studies the quantum mechanical Coulomb problem on a 3-sphere. We present a special parametrization of the ellipto-spheroidal coordinate system suitable for the separation of variables. After quantization we get the explicit form of the spectrum and present an algebraic equation for the eigenvalues of the Runge-Lentz vector. We also present the wave functions expressed via Heun polynomials.
The Coulomb problem on a 3-sphere and Heun polynomials
Bellucci, Stefano; Yeghikyan, Vahagn
2013-08-15
The paper studies the quantum mechanical Coulomb problem on a 3-sphere. We present a special parametrization of the ellipto-spheroidal coordinate system suitable for the separation of variables. After quantization we get the explicit form of the spectrum and present an algebraic equation for the eigenvalues of the Runge-Lentz vector. We also present the wave functions expressed via Heun polynomials.
A Problem-Centered Approach to Canonical Matrix Forms
ERIC Educational Resources Information Center
Sylvestre, Jeremy
2014-01-01
This article outlines a problem-centered approach to the topic of canonical matrix forms in a second linear algebra course. In this approach, abstract theory, including such topics as eigenvalues, generalized eigenspaces, invariant subspaces, independent subspaces, nilpotency, and cyclic spaces, is developed in response to the patterns discovered…
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.
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.
Symmetry preserving eigenvalue embedding in finite-element model updating of vibrating structures
NASA Astrophysics Data System (ADS)
Carvalho, Joao B.; Datta, Biswa N.; Lin, Wen-Wei; Wang, Chern-Shuh
2006-03-01
The eigenvalue embedding problem addressed in this paper is the one of reassigning a few troublesome eigenvalues of a symmetric finite-element model to some suitable chosen ones, in such a way that the updated model remains symmetric and the remaining large number of eigenvalues and eigenvectors of the original model is to remain unchanged. The problem naturally arises in stabilizing a large-scale system or combating dangerous vibrations, which can be responsible for undesired phenomena such as resonance, in large vibrating structures. A new computationally efficient and symmetry preserving method and associated theories are presented in this paper. The model is updated using low-rank symmetric updates and other computational requirements of the method include only simple operations such as matrix multiplications and solutions of low-order algebraic linear systems. These features make the method practical for large-scale applications. The results of numerical experiments on the simulated data obtained from the Boeing company and on some benchmark examples are presented to show the accuracy of the method. Computable error bounds for the updated matrices are also given by means of rigorous mathematical analysis.
The calculation of the eigenvalues and eigenfunctions of Mathieu's equation
NASA Technical Reports Server (NTRS)
Hodge, D. B.
1972-01-01
The eigenfunctions of Mathieu's equation are expanded in trigonometric series, and the resulting eigenvalue problem is cast in matrix form. This matrix is found to be a symmetric, triagonal matrix, and the eigenvalues are computed using the bisection method. The eigenfunction expansion coefficients are obtained by the standard recursion method. This computational technique for the eigenvalues and eigenfunctions of Mathieu's equation is both rapid and accurate.
Eigenvalue asymptotics for Dirac-Bessel operators
NASA Astrophysics Data System (ADS)
Hryniv, Rostyslav O.; Mykytyuk, Yaroslav V.
2016-06-01
In this paper, we establish the eigenvalue asymptotics for non-self-adjoint Dirac-Bessel operators on (0, 1) with arbitrary real angular momenta and square integrable potentials, which gives the first step for solution of the related inverse problem. The approach is based on a careful examination of the corresponding characteristic functions and their zero distribution.
NASA Technical Reports Server (NTRS)
Newman, M.; Flanagan, P. F.
1976-01-01
The development of the tridiagonal reduction method and its implementation in NASTRAN are described for real eigenvalue analysis as typified by structural vibration and buckling problems. This method is an automatic matrix reduction scheme whereby the eigensolutions in the neighborhood of a specified point in the eigenspectrum can be accurately extracted from a tridiagonal eigenvalue problem whose order is much lower than that of the full problem. The process is effected without orbitrary lumping of masses or other physical quantities at selected node points and thus avoids one of the basic weaknesses of other techniques.
Quantum cluster algebras and quantum nilpotent algebras
Goodearl, Kenneth R.; Yakimov, Milen T.
2014-01-01
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197
A Clifford Algebra Approach to the Classical Problem of a Charge in a Magnetic Monopole Field
NASA Astrophysics Data System (ADS)
Vaz, Jayme
2013-05-01
The motion of an electric charge in the field of a magnetic monopole is described by means of a Lagrangian model written in terms of the Clifford algebra of the physical space. The equations of motion are written in terms of a radial equation (involving r=| r|, where r( t) is the charge trajectory) and a rotor equation (written in terms of an unitary operator spinor R). The solution corresponding to the charge trajectory in the field of a magnetic monopole is given in parametric form. The model can be generalized in order to describe the motion of a charge in the field of a magnetic monopole and other additional central forces, and as an example, we discuss the classical ones involving linear and inverse square interactions.
Computational method for transmission eigenvalues for a spherically stratified medium.
Cheng, Xiaoliang; Yang, Jing
2015-07-01
We consider a computational method for the interior transmission eigenvalue problem that arises in acoustic and electromagnetic scattering. The transmission eigenvalues contain useful information about some physical properties, such as the index of refraction. Instead of the existence and estimation of the spectral property of the transmission eigenvalues, we focus on the numerical calculation, especially for spherically stratified media in R^{3}. Due to the nonlinearity and the special structure of the interior transmission eigenvalue problem, there are not many numerical methods to date. First, we reduce the problem into a second-order ordinary differential equation. Then, we apply the Hermite finite element to the weak formulation of the equation. With proper rewriting of the matrix-vector form, we change the original nonlinear eigenvalue problem into a quadratic eigenvalue problem, which can be written as a linear system and solved by the eigs function in MATLAB. This numerical method is fast, effective, and can calculate as many transmission eigenvalues as needed at a time. PMID:26367151
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…
ERIC Educational Resources Information Center
Nobre, Sandra; Amado, Nelia; Carreira, Susana
2012-01-01
In this article we report and discuss a contextual problem solving task that was proposed to a class of 8th grade (13-14-year-old) students. These students had been developing a reasonable experience in the use of the spreadsheet to model relations within contextual problems and chose to use this tool to solve the mentioned problem, engaging in…
Facilitating Case Reuse during Problem Solving in Algebra-Based Physics
ERIC Educational Resources Information Center
Mateycik, Frances Ann
2010-01-01
This research project investigates students' development of problem solving schemata while using strategies that facilitate the process of using solved examples to assist with a new problem (case reuse). Focus group learning interviews were used to explore students' perceptions and understanding of several problem solving strategies. Individual…
A Comparison of Two Mathematics Problem-Solving Strategies: Facilitate Algebra-Readiness
ERIC Educational Resources Information Center
Xin, Yan Ping; Zhang, Dake; Park, Joo Young; Tom, Kinsey; Whipple, Amanda; Si, Luo
2011-01-01
The authors compared a conceptual model-based problem-solving (COMPS) approach with a general heuristic instructional approach for teaching multiplication-division word-problem solving to elementary students with learning problems (LP). The results indicate that only the COMPS group significantly improved, from pretests to posttests, their…
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…
Asymptotic formula for eigenvalues of one dimensional Dirac system
NASA Astrophysics Data System (ADS)
Ulusoy, Ismail; Penahlı, Etibar
2016-06-01
In this paper, we study the spectral problem for one dimensional Dirac system with Dirichlet boundary conditions. By using Counting lemma, we give an asymptotic formulas of eigenvalues of Dirac system.
On the design derivatives of eigenvalues and eigenvectors for distributed parameter systems
NASA Technical Reports Server (NTRS)
Reiss, R.
1985-01-01
In this paper, analytic expressions are obtained for the design derivatives of eigenvalues and eigenfunctions of self-adjoint linear distributed parameter systems. Explicit treatment of boundary conditions is avoided by casting the eigenvalue equation into integral form. Results are expressed in terms of the linear operators defining the eigenvalue problem, and are therefore quite general. Sufficiency conditions appropriate to structural optimization of eigenvalues are obtained.
Thinking and Writing Mathematically: "Achilles and the Tortoise" as an Algebraic Word Problem.
ERIC Educational Resources Information Center
Martinez, Joseph G. R.
2001-01-01
Introduces Hogben's adaptation of Zeno's paradox, "Achilles and the Tortoise", as a thinking and writing exercise. Emphasizes engaging students' imagination with creative, thought-provoking problems and involving students in evaluating their word problem-solving strategies. Describes the paradox, logical solutions, and students' mathematical…
ERIC Educational Resources Information Center
González-Calero, José Antonio; Arnau, David; Puig, Luis; Arevalillo-Herráez, Miguel
2015-01-01
The term intensive scaffolding refers to any set of conceptual scaffolding strategies that always allow the user to find the solution to a problem. Despite the many benefits of scaffolding, some negative effects have also been reported. These are mainly related to the possibility that a student solves the problems without actually engaging in…
A robust multilevel simultaneous eigenvalue solver
NASA Technical Reports Server (NTRS)
Costiner, Sorin; Taasan, Shlomo
1993-01-01
Multilevel (ML) algorithms for eigenvalue problems are often faced with several types of difficulties such as: the mixing of approximated eigenvectors by the solution process, the approximation of incomplete clusters of eigenvectors, the poor representation of solution on coarse levels, and the existence of close or equal eigenvalues. Algorithms that do not treat appropriately these difficulties usually fail, or their performance degrades when facing them. These issues motivated the development of a robust adaptive ML algorithm which treats these difficulties, for the calculation of a few eigenvectors and their corresponding eigenvalues. The main techniques used in the new algorithm include: the adaptive completion and separation of the relevant clusters on different levels, the simultaneous treatment of solutions within each cluster, and the robustness tests which monitor the algorithm's efficiency and convergence. The eigenvectors' separation efficiency is based on a new ML projection technique generalizing the Rayleigh Ritz projection, combined with a technique, the backrotations. These separation techniques, when combined with an FMG formulation, in many cases lead to algorithms of O(qN) complexity, for q eigenvectors of size N on the finest level. Previously developed ML algorithms are less focused on the mentioned difficulties. Moreover, algorithms which employ fine level separation techniques are of O(q(sub 2)N) complexity and usually do not overcome all these difficulties. Computational examples are presented where Schrodinger type eigenvalue problems in 2-D and 3-D, having equal and closely clustered eigenvalues, are solved with the efficiency of the Poisson multigrid solver. A second order approximation is obtained in O(qN) work, where the total computational work is equivalent to only a few fine level relaxations per eigenvector.
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…
Facilitating case reuse during problem solving in algebra-based physics
NASA Astrophysics Data System (ADS)
Mateycik, Frances Ann
This research project investigates students' development of problem solving schemata while using strategies that facilitate the process of using solved examples to assist with a new problem (case reuse). Focus group learning interviews were used to explore students' perceptions and understanding of several problem solving strategies. Individual clinical interviews were conducted and quantitative examination data were collected to assess students' conceptual understanding, knowledge organization, and problem solving performance on a variety of problem tasks. The study began with a short one-time treatment of two independent, research-based strategies chosen to facilitate case reuse. Exploration of students' perceptions and use of the strategies lead investigators to select one of the two strategies to be implemented over a full semester of focus group interviews. The strategy chosen was structure mapping. Structure maps are defined as visual representations of quantities and their associations. They were created by experts to model the appropriate mental organization of knowledge elements for a given physical concept. Students were asked to use these maps as they were comfortable while problem solving. Data obtained from this phase of our study (Phase I) offered no evidence of improved problem solving schema. The 11 contact hour study was barely sufficient time for students to become comfortable using the maps. A set of simpler strategies were selected for their more explicit facilitation of analogical reasoning, and were used together during two more semester long focus group treatments (Phase II and Phase III of this study). These strategies included the use of a step-by-step process aimed at reducing cognitive load associated with mathematical procedure, direct reflection of principles involved in a given set of problems, and the direct comparison of problem pairs designed to be void of surface similarities (similar objects or object orientations) and sharing
Graphic and algebraic solutions of the discordant lead-uranium age problem
Stieff, L.R.; Stern, T.W.
1961-01-01
for the contaminating common Pb206 and Pb207. The linear relationships noted in this graphical procedure have been extended to plots of the mole ratios of total Pb206 U238 ( tN206 N238) vs. total Pb207 U235 ( tN207 N235). This modification permits the calculation of concordant ages for unaltered samples using only the Pb207 Pb206 ratio of the contaminating common lead. If isotopic data are available for two samples of the same age, x and y, from the same or related deposits or outcrops, graphs of the normalized difference ratios [ ( N206 N204)x - ( N206 N204)y ( N238 N204)x -( N238 N204)y] vs. [ ( N207 N204)x - ( N207 N204)y ( N235 N204)x -( N235 N204)y] can give concordant ages corrected for unknown amounts of a common lead with an unknown Pb207/ Pb206 ratio. (If thorium is absent the difference ratios may be normalized with the more abundant index isotope, Pb208.) Similar plots of tho normalized, difference ratios for three genetically related samples (x - y) and(x - z), will give concordant ages corrected, in addition, for either one unknown period of past alteration or initial contamination by an older generation of radiogenic lead of unknown Pb207/Pb206 ratio. Practical numerical solutions for many of tho concordant age calculations are not currently available. However, the algebraic equivalents of these new graphical methods give equations which may be programmed for computing machines. For geologically probable parameters the equations of higher order have two positive real roots that rapidly converge on the exact concordant ages corrected for original radiogenic lead and for loss or gain of lead or uranium. Modifications of these general age equations expanded only to the second degree have been derived for use with desk calculators. These graphical and algebraic methods clearly suggest both the type and minimum number of samples necessary for adequate mathematical analysis of discordant lead isotope age data. This mathematical treatment also makes it clear t
Maximizing algebraic connectivity in interconnected networks
NASA Astrophysics Data System (ADS)
Shakeri, Heman; Albin, Nathan; Darabi Sahneh, Faryad; Poggi-Corradini, Pietro; Scoglio, Caterina
2016-03-01
Algebraic connectivity, the second eigenvalue of the Laplacian matrix, is a measure of node and link connectivity on networks. When studying interconnected networks it is useful to consider a multiplex model, where the component networks operate together with interlayer links among them. In order to have a well-connected multilayer structure, it is necessary to optimally design these interlayer links considering realistic constraints. In this work, we solve the problem of finding an optimal weight distribution for one-to-one interlayer links under budget constraint. We show that for the special multiplex configurations with identical layers, the uniform weight distribution is always optimal. On the other hand, when the two layers are arbitrary, increasing the budget reveals the existence of two different regimes. Up to a certain threshold budget, the second eigenvalue of the supra-Laplacian is simple, the optimal weight distribution is uniform, and the Fiedler vector is constant on each layer. Increasing the budget past the threshold, the optimal weight distribution can be nonuniform. The interesting consequence of this result is that there is no need to solve the optimization problem when the available budget is less than the threshold, which can be easily found analytically.
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
A popular humorist and avowed mathphobe once declared that in real life, there's no such thing as algebra. Kathie Wilson knows better. Most of the students in her 8th grade class will be thrust into algebra, the definitive course that heralds the beginning of high school mathematics, next school year. The problem: Many of them are about three…
Lowest eigenvalues of random Hamiltonians
Shen, J. J.; Zhao, Y. M.; Arima, A.; Yoshinaga, N.
2008-05-15
In this article we study the lowest eigenvalues of random Hamiltonians for both fermion and boson systems. We show that an empirical formula of evaluating the lowest eigenvalues of random Hamiltonians in terms of energy centroids and widths of eigenvalues is applicable to many different systems. We improve the accuracy of the formula by considering the third central moment. We show that these formulas are applicable not only to the evaluation of the lowest energy but also to the evaluation of excited energies of systems under random two-body interactions.
NASA Astrophysics Data System (ADS)
Koldan, Jelena; Puzyrev, Vladimir; de la Puente, Josep; Houzeaux, Guillaume; Cela, José María
2014-06-01
We present an elaborate preconditioning scheme for Krylov subspace methods which has been developed to improve the performance and reduce the execution time of parallel node-based finite-element (FE) solvers for 3-D electromagnetic (EM) numerical modelling in exploration geophysics. This new preconditioner is based on algebraic multigrid (AMG) that uses different basic relaxation methods, such as Jacobi, symmetric successive over-relaxation (SSOR) and Gauss-Seidel, as smoothers and the wave front algorithm to create groups, which are used for a coarse-level generation. We have implemented and tested this new preconditioner within our parallel nodal FE solver for 3-D forward problems in EM induction geophysics. We have performed series of experiments for several models with different conductivity structures and characteristics to test the performance of our AMG preconditioning technique when combined with biconjugate gradient stabilized method. The results have shown that, the more challenging the problem is in terms of conductivity contrasts, ratio between the sizes of grid elements and/or frequency, the more benefit is obtained by using this preconditioner. Compared to other preconditioning schemes, such as diagonal, SSOR and truncated approximate inverse, the AMG preconditioner greatly improves the convergence of the iterative solver for all tested models. Also, when it comes to cases in which other preconditioners succeed to converge to a desired precision, AMG is able to considerably reduce the total execution time of the forward-problem code-up to an order of magnitude. Furthermore, the tests have confirmed that our AMG scheme ensures grid-independent rate of convergence, as well as improvement in convergence regardless of how big local mesh refinements are. In addition, AMG is designed to be a black-box preconditioner, which makes it easy to use and combine with different iterative methods. Finally, it has proved to be very practical and efficient in the
Algebraic rings of integers and some 2D lattice problems in physics
NASA Astrophysics Data System (ADS)
Nanxian, Chen; Zhaodou, Chen; Shaojun, Liu; Yanan, Shen; Xijin, Ge
1996-09-01
This paper develops the Möbius inversion formula for the Gaussian integers and Eisenstein's integers, and gives two applications. The first application is to the two-dimensional arithmetic Fourier transform (AFT), which is suitable for parallel processing. The second application is to two-dimensional inverse lattice problems, and is illustrated with the recovery of interatomic potentials from the cohesive energy for monolayer graphite. The paper demonstrates the potential application in the physical science of integral domains other than the standard integers.
Elimination of spurious eigenvalues in the Chebyshev tau spectral method
NASA Technical Reports Server (NTRS)
Mcfadden, G. B.; Murray, B. T.; Boisvert, R. F.
1990-01-01
A very simple modification is presented for the Chebyshev tau method which can eliminate spurious eigenvalues, proceeding from a consideration of the vorticity-streamfunction reformulation of the Chebyshev tau method and the Chebyshev-Galerkin method, which have no spurious modes. Consideration of a model problem indicates that these two approaches are equivalent, and that they reduce to the present modification of the tau method. This modified tau method also eliminates spurious eigenvalues from the Orr-Sommerfeld equation.
ERIC Educational Resources Information Center
Schmidt, Sylvine; Bednarz, Nadine
1997-01-01
Discusses the difficulties observed in the transition from teaching arithmetic to teaching algebra. Future teachers (n=164) were questioned regarding to what extent they were able to shift back and forth between teaching methods within the context of problem solving. Interviews were conducted individually and in a dyad format. (AIM)
NASA Astrophysics Data System (ADS)
Dul, Franciszek A.; Arczewski, Krzysztof
1994-03-01
Although it has been stated that "an attempt to solve (very large problems) by subspace iterations seems futile" (H. G. Matthies, Comput. Struct.21 (1985), p. 324), we will show that the statement is not true, especially for extremely large eigenproblems. In this paper a new two-phase subspace iteration/Rayleigh quotient/conjugate gradient method for generalized, large, symmetric eigenproblems Ax = λBx is presented. It has the ability of solving extremely large eigenproblems, N = 216,000, for example, and finding a large number of leftmost or rightmost eigenpairs, up to 1000 or more. Multiple eigenpairs, even those with multiplicity 100, can be easily found. The use of the proposed method for solving the big full eigenproblems ( N ˜ 10 3), as well as for large weakly non-symmetric eigenproblems, have been considered also. The proposed method is fully iterative; thus the factorization of matrices is avoided. The key idea consists in joining two methods: subspace and Rayleigh quotient iterations. The systems of indefinite and almost singular linear equations ( A - σ B) x = By are solved by various iterative conjugate gradient/Lanczos methods. It will be shown that the standard conjugate gradient method can be used without danger of breaking down due to its property that may be called "self-correction towards the eigenvector," discovered recently by us. The use of various preconditioners (SSOR and IC) has also been considered. The main features of the proposed method have been analyzed in detail. Comparisons with other methods, such as, accelerated subspace iteration, Lanczos, Davidson, TLIME, TRACMN, and SRQMCG, are presented. The results of numerical tests for various physical problems (acoustic, vibrations of structures, quantum chemistry) are presented as well. The final conclusion is that our new method is usually much faster than other iterative methods, especially for very large eigenproblems arising from 3D elliptic or biharmonic problems defined on
Computing estimates of material properties from transmission eigenvalues
NASA Astrophysics Data System (ADS)
Giorgi, Giovanni; Haddar, Houssem
2012-05-01
This work is motivated by inverse scattering problems, those problems where one is interested in reconstructing the shape and the material properties of an inclusion from electromagnetic farfield measurements. More precisely, we are interested in complementing the so-called sampling methods by providing an estimate of the material properties of the sought inclusion. We use for this purpose a measure of the first transmission eigenvalue. Our method is then based on computing the desired estimate by reformulating the so-called interior transmission eigenvalue problem as an eigenvalue problem for the material coefficients. We will restrict ourselves to the two-dimensional setting of the problem and treat the cases of both transverse electric and transverse magnetic polarizations. We present a number of numerical experiments that validate our methodology for homogeneous and inhomogeneous inclusions and backgrounds. We also treat the case of a background with absorption and the case of scatterers with multiple connected components of different refractive indices.
NASA Astrophysics Data System (ADS)
Arczewski, Krzysztof; Dul, Franciszek A.
1994-03-01
In this paper a new two-phase subspace iteration/Rayleigh quotient/conjugate gradient method for generalized, large, symmetric eigenproblems Ax = lambda Bx is presented. It has the ability of solving extremely large eigenproblems, N = 216,000, for example, and finding a large number of leftmost or rightmost eigenpairs, up to 1000 or more. Multiple eigenpairs, even those with multiplicity 100, can be easily found. The use of the proposed method for solving the big full eigenproblems N approximately 10(exp 3), as well as for large weakly non-symmetric eigenproblems, have been considered also. The proposed method is fully iterative; thus the factorization of matrices is avoided. The key idea consists in joining two methods: subspace and Rayleigh quotient iterations. The systems of indefinite and almost singular linear equations (A - sigma B)x = By are solved by various iterative conjugate gradient/Lanczos methods. It will be shown that the standard conjugate gradient method can be used without danger of breaking down due to its property that may be called 'self-correction towards the eigenvector,' discovered recently by us. The use of various preconditions (SSOR and IC) has also been considered. The main features of the proposed method have been analyzed in detail. Comparisons with other methods, such as, accelerated subspace iteration, Lanczos, Davidson, TLIME, TRACMN, and SRQMCG, are presented. The results of numerical tests for various physical problems (acoustic, vibrations of structures, quantum chemistry) are presented as well. The final conclusion is that our new method is usually much faster than other iterative methods, especially for very large eigenproblems arising from 3D elliptic or biharmonic problems defined on irregular, multiply-connected domains, discretized by the finite element (FEM) or finite difference (FDM) methods.
NASA Astrophysics Data System (ADS)
Mathai, Pramod P.
the uncertainty in the parameters of the differential equations. There is a clear need to design better experiments for IEF without the current overhead of expensive chemicals and labor. We show how with a simpler modeling of the underlying chemistry, we can still achieve the accuracy that has been achieved in existing literature for modeling small ranges of pH (hydrogen ion concentration) in IEF, but with far less computational time. We investigate a further reduction of time by modeling the IEF problem using the Proper Orthogonal Decomposition (POD) technique and show why POD may not be sufficient due to the underlying constraints. The final problem that we address in this thesis addresses a certain class of dynamics with high stiffness - in particular, differential algebraic equations. With the help of simple examples, we show how the traditional POD procedure will fail to model certain high stiffness problems due to a particular behavior of the vector field which we will denote as twist. We further show how a novel augmentation to the traditional POD algorithm can model-reduce problems with twist in a computationally cheap manner without any additional data requirements.
NASA Astrophysics Data System (ADS)
Connes, Alain; Kreimer, Dirk
This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra which is commutative as an algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of . We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop
Determination of eigenvalues of dynamical systems by symbolic computation
NASA Technical Reports Server (NTRS)
Howard, J. C.
1982-01-01
A symbolic computation technique for determining the eigenvalues of dynamical systems is described wherein algebraic operations, symbolic differentiation, matrix formulation and inversion, etc., can be performed on a digital computer equipped with a formula-manipulation compiler. An example is included that demonstrates the facility with which the system dynamics matrix and the control distribution matrix from the state space formulation of the equations of motion can be processed to obtain eigenvalue loci as a function of a system parameter. The example chosen to demonstrate the technique is a fourth-order system representing the longitudinal response of a DC 8 aircraft to elevator inputs. This simplified system has two dominant modes, one of which is lightly damped and the other well damped. The loci may be used to determine the value of the controlling parameter that satisfied design requirements. The results were obtained using the MACSYMA symbolic manipulation system.
Software for computing eigenvalue bounds for iterative subspace matrix methods
NASA Astrophysics Data System (ADS)
Shepard, Ron; Minkoff, Michael; Zhou, Yunkai
2005-07-01
This paper describes software for computing eigenvalue bounds to the standard and generalized hermitian eigenvalue problem as described in [Y. Zhou, R. Shepard, M. Minkoff, Computing eigenvalue bounds for iterative subspace matrix methods, Comput. Phys. Comm. 167 (2005) 90-102]. The software discussed in this manuscript applies to any subspace method, including Lanczos, Davidson, SPAM, Generalized Davidson Inverse Iteration, Jacobi-Davidson, and the Generalized Jacobi-Davidson methods, and it is applicable to either outer or inner eigenvalues. This software can be applied during the subspace iterations in order to truncate the iterative process and to avoid unnecessary effort when converging specific eigenvalues to a required target accuracy, and it can be applied to the final set of Ritz values to assess the accuracy of the converged results. Program summaryTitle of program: SUBROUTINE BOUNDS_OPT Catalogue identifier: ADVE Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVE Computers: any computer that supports a Fortran 90 compiler Operating systems: any computer that supports a Fortran 90 compiler Programming language: Standard Fortran 90 High speed storage required:5m+5 working-precision and 2m+7 integer for m Ritz values No. of bits in a word: The floating point working precision is parameterized with the symbolic constant WP No. of lines in distributed program, including test data, etc.: 2452 No. of bytes in distributed program, including test data, etc.: 281 543 Distribution format: tar.gz Nature of physical problem: The computational solution of eigenvalue problems using iterative subspace methods has widespread applications in the physical sciences and engineering as well as other areas of mathematical modeling (economics, social sciences, etc.). The accuracy of the solution of such problems and the utility of those errors is a fundamental problem that is of
Lehoucq, Richard B.; Salinger, Andrew G.
1999-08-01
We present an approach for determining the linear stability of steady states of PDEs on massively parallel computers. Linearizing the transient behavior around a steady state leads to a generalized eigenvalue problem. The eigenvalues with largest real part are calculated using Arnoldi's iteration driven by a novel implementation of the Cayley transformation to recast the problem as an ordinary eigenvalue problem. The Cayley transformation requires the solution of a linear system at each Arnoldi iteration, which must be done iteratively for the algorithm to scale with problem size. A representative model problem of 3D incompressible flow and heat transfer in a rotating disk reactor is used to analyze the effect of algorithmic parameters on the performance of the eigenvalue algorithm. Successful calculations of leading eigenvalues for matrix systems of order up to 4 million were performed, identifying the critical Grashof number for a Hopf bifurcation.
Eigenvalue methods for unimolecular rate calculations with several products.
Pritchard, Huw O
2007-10-25
When the calculation of a unimolecular reaction rate constant is cast in the form of a master equation eigenvalue problem, the magnitude of that rate is often smaller than the rounding error of the trace of the corresponding reaction matrix. Here, a previously published procedure (Pritchard, H. O. J. Phys. Chem. A 2004, 108, 5249-5252) for solving this problem is extended to the case of more than one reaction product. An Appendix notes the occurrence of avoided crossings between eigenvalues of the master equation in reversible, in mixed reversible-irreversible, and in multiwell unimolecular reaction calculations. PMID:17914776
Learning Algebra from Worked Examples
ERIC Educational Resources Information Center
Lange, Karin E.; Booth, Julie L.; Newton, Kristie J.
2014-01-01
For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is…
NASA Astrophysics Data System (ADS)
Nara, T.; Koiwa, K.; Takagi, S.; Oyama, D.; Uehara, G.
2014-05-01
This paper presents an algebraic reconstruction method for dipole-quadrupole sources using magnetoencephalography data. Compared to the conventional methods with the equivalent current dipoles source model, our method can more accurately reconstruct two close, oppositely directed sources. Numerical simulations show that two sources on both sides of the longitudinal fissure of cerebrum are stably estimated. The method is verified using a quadrupolar source phantom, which is composed of two isosceles-triangle-coils with parallel bases.
Efficient, massively parallel eigenvalue computation
NASA Technical Reports Server (NTRS)
Huo, Yan; Schreiber, Robert
1993-01-01
In numerical simulations of disordered electronic systems, one of the most common approaches is to diagonalize random Hamiltonian matrices and to study the eigenvalues and eigenfunctions of a single electron in the presence of a random potential. An effort to implement a matrix diagonalization routine for real symmetric dense matrices on massively parallel SIMD computers, the Maspar MP-1 and MP-2 systems, is described. Results of numerical tests and timings are also presented.
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.
ERIC Educational Resources Information Center
Schaufele, Christopher; Zumoff, Nancy
Earth Algebra is an entry level college algebra course that incorporates the spirit of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics at the college level. The context of the course places mathematics at the center of one of the major current concerns of the world. Through…
ERIC Educational Resources Information Center
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
NASA Technical Reports Server (NTRS)
Sloss, J. M.; Kranzler, S. K.
1972-01-01
The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.
A study of eigenvalue sensitivity for hydrodynamic stability operators
NASA Technical Reports Server (NTRS)
Schmid, Peter J.; Henningson, Dan S.; Khorrami, Mehdi R.; Malik, Mujeeb R.
1993-01-01
The eigenvalue sensitivity for hydrodynamic stability operators is investigated. Classical matrix perturbation techniques as well as the concept of epsilon-pseudospectra are applied to show that parts of the spectrum are highly sensitive to small perturbations. Applications are drawn from incompressible plane Couette flow, trailing line vortex flow, and compressible Blasius boundary-layer flow. Parameter studies indicate a monotonically increasing effect of the Reynolds number on the sensitivity. The phenomenon of eigenvalue sensitivity is due to the nonnormality of the operators and their discrete matrix analogs and may be associated with large transient growth of the corresponding initial value problem.
A subspace preconditioning algorithm for eigenvector/eigenvalue computation
Bramble, J.H.; Knyazev, A.V.; Pasciak, J.E.
1996-12-31
We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigen-spaces of a symmetric positive definite matrix. In our applications, the dimension of a matrix is large and the cost of its inverting is prohibitive. In this paper, we shall develop an effective parallelizable technique for computing these eigenvalues and eigenvectors utilizing subspace iteration and preconditioning. Estimates will be provided which show that the preconditioned method converges linearly and uniformly in the matrix dimension when used with a uniform preconditioner under the assumption that the approximating subspace is close enough to the span of desired eigenvectors.
New algorithms for the symmetric tridiagonal eigenvalue computation
Pan, V. |
1994-12-31
The author presents new algorithms that accelerate the bisection method for the symmetric eigenvalue problem. The algorithms rely on some new techniques, which include acceleration of Newton`s iteration and can also be further applied to acceleration of some other iterative processes, in particular, of iterative algorithms for approximating polynomial zeros.
The Algebra of Complex Numbers.
ERIC Educational Resources Information Center
LePage, Wilbur R.
This programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex number…
Perfetti, C.; Martin, W.; Rearden, B.; Williams, M.
2012-07-01
This study introduced two new approaches for calculating the F*(r) importance weighting function for Contributon and CLUTCH eigenvalue sensitivity coefficient calculations, and compared them in terms of accuracy and applicability. The necessary levels of F*(r) mesh refinement and mesh convergence for obtaining accurate eigenvalue sensitivity coefficients were determined for two preliminary problems through two parametric studies, and the results of these studies suggest that a sufficiently accurate F*(r) mesh for calculating eigenvalue sensitivity coefficients can be obtained for these problems with only a small increase in problem runtime. (authors)
Calculating alpha Eigenvalues in a Continuous-Energy Infinite Medium with Monte Carlo
Betzler, Benjamin R.; Kiedrowski, Brian C.; Brown, Forrest B.; Martin, William R.
2012-09-04
The {alpha} eigenvalue has implications for time-dependent problems where the system is sub- or supercritical. We present methods and results from calculating the {alpha}-eigenvalue spectrum for a continuous-energy infinite medium with a simplified Monte Carlo transport code. We formulate the {alpha}-eigenvalue problem, detail the Monte Carlo code physics, and provide verification and results. We have a method for calculating the {alpha}-eigenvalue spectrum in a continuous-energy infinite-medium. The continuous-time Markov process described by the transition rate matrix provides a way of obtaining the {alpha}-eigenvalue spectrum and kinetic modes. These are useful for the approximation of the time dependence of the system.
Multi-input partial eigenvalue assignment for high order control systems with time delay
NASA Astrophysics Data System (ADS)
Zhang, Lei
2016-05-01
In this paper, we consider the partial eigenvalue assignment problem for high order control systems with time delay. Ram et al. (2011) [1] have shown that a hybrid method can be used to solve partial quadratic eigenvalue assignment problem of single-input vibratory system. Based on this theory, a rather simple algorithm for solving multi-input partial eigenvalue assignment for high order control systems with time delay is proposed. Our method can assign the expected eigenvalues and keep the no spillover property. The solution can be implemented with only partial information of the eigenvalues and the corresponding eigenvectors of the matrix polynomial. Numerical examples are given to illustrate the efficiency of our approach.
Preconditioned iterations to calculate extreme eigenvalues
Brand, C.W.; Petrova, S.
1994-12-31
Common iterative algorithms to calculate a few extreme eigenvalues of a large, sparse matrix are Lanczos methods or power iterations. They converge at a rate proportional to the separation of the extreme eigenvalues from the rest of the spectrum. Appropriate preconditioning improves the separation of the eigenvalues. Davidson`s method and its generalizations exploit this fact. The authors examine a preconditioned iteration that resembles a truncated version of Davidson`s method with a different preconditioning strategy.
An algebraic sub-structuring method for large-scale eigenvaluecalculation
Yang, C.; Gao, W.; Bai, Z.; Li, X.; Lee, L.; Husbands, P.; Ng, E.
2004-05-26
We examine sub-structuring methods for solving large-scalegeneralized eigenvalue problems from a purely algebraic point of view. Weuse the term "algebraic sub-structuring" to refer to the process ofapplying matrix reordering and partitioning algorithms to divide a largesparse matrix into smaller submatrices from which a subset of spectralcomponents are extracted and combined to provide approximate solutions tothe original problem. We are interested in the question of which spectralcomponentsone should extract from each sub-structure in order to producean approximate solution to the original problem with a desired level ofaccuracy. Error estimate for the approximation to the small esteigen pairis developed. The estimate leads to a simple heuristic for choosingspectral components (modes) from each sub-structure. The effectiveness ofsuch a heuristic is demonstrated with numerical examples. We show thatalgebraic sub-structuring can be effectively used to solve a generalizedeigenvalue problem arising from the simulation of an acceleratorstructure. One interesting characteristic of this application is that thestiffness matrix produced by a hierarchical vector finite elements schemecontains a null space of large dimension. We present an efficient schemeto deflate this null space in the algebraic sub-structuringprocess.
ERIC Educational Resources Information Center
Glick, David
1995-01-01
Presents a technique that helps students concentrate more on the science and less on the mechanics of algebra while dealing with introductory physics formulas. Allows the teacher to do complex problems at a lower level and not be too concerned about the mathematical abilities of the students. (JRH)
ERIC Educational Resources Information Center
Ketterlin-Geller, Leanne R.; Jungjohann, Kathleen; Chard, David J.; Baker, Scott
2007-01-01
Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…
Computer Algebra versus Manipulation
ERIC Educational Resources Information Center
Zand, Hossein; Crowe, David
2004-01-01
In the UK there is increasing concern about the lack of skill in algebraic manipulation that is evident in students entering mathematics courses at university level. In this note we discuss how the computer can be used to ameliorate some of the problems. We take as an example the calculations needed in three dimensional vector analysis in polar…
Cierniak, Robert; Lorent, Anna
2016-09-01
The main aim of this paper is to investigate properties of our originally formulated statistical model-based iterative approach applied to the image reconstruction from projections problem which are related to its conditioning, and, in this manner, to prove a superiority of this approach over ones recently used by other authors. The reconstruction algorithm based on this conception uses a maximum likelihood estimation with an objective adjusted to the probability distribution of measured signals obtained from an X-ray computed tomography system with parallel beam geometry. The analysis and experimental results presented here show that our analytical approach outperforms the referential algebraic methodology which is explored widely in the literature and exploited in various commercial implementations. PMID:27289536
The quantum Casimir operators of {U}_q{(\\mathfrak {gl}_{n})} and their eigenvalues
NASA Astrophysics Data System (ADS)
Li, Junbo
2010-08-01
We show that the quantum Casimir operators of the quantum linear group constructed in early work of Bracken, Gould and Zhang together with one obvious central element generate the entire center of {U}_q{(\\mathfrak {gl}_{n})}. As a byproduct of the proof, we obtain intriguing new formulae for eigenvalues of these quantum Casimir operators, which are expressed in terms of the characters of a class of finite-dimensional irreducible representations of the classical general linear algebra.
A comparison of matrix methods for calculating eigenvalues in acoustically lined ducts
NASA Technical Reports Server (NTRS)
Watson, W.; Lansing, D. L.
1976-01-01
Three approximate methods - finite differences, weighted residuals, and finite elements - were used to solve the eigenvalue problem which arises in finding the acoustic modes and propagation constants in an absorptively lined two-dimensional duct without airflow. The matrix equations derived for each of these methods were solved for the eigenvalues corresponding to various values of wall impedance. Two matrix orders, 20 x 20 and 40 x 40, were used. The cases considered included values of wall admittance for which exact eigenvalues were known and for which several nearly equal roots were present. Ten of the lower order eigenvalues obtained from the three approximate methods were compared with solutions calculated from the exact characteristic equation in order to make an assessment of the relative accuracy and reliability of the three methods. The best results were given by the finite element method using a cubic polynomial. Excellent accuracy was consistently obtained, even for nearly equal eigenvalues, by using a 20 x 20 order matrix.
Kinetic applications of the ArbiTER eigenvalue code
NASA Astrophysics Data System (ADS)
Baver, D. A.; Myra, J. R.; Umansky, M. V.
2014-10-01
ArbiTER is a flexible eigenvalue code designed for linear fluid or kinetic plasma models is various dimensionalities and topologies. This flexibility derives from the use of specialized equation and topology parsers, which permit run-time specification of a particular linearized physics model, geometry, and grid connectivity, which in turn determine how a particular equation set will be discretized. The resulting matrix form of the problem is then solved using the SLEPc eigensolver package, and can be solved either as a generalized eigenvalue problem, or as a matrix solve in the case of source-driven problems. While the ArbiTER code and its predecessor 2DX have demonstrated significant utility in tokamak edge fluid problems due to their inherent flexibility, the primary aim of its development is to solve kinetic eigenvalue problems. To address this goal, we present first results from implementation of a gyrokinetic model in slab geometry. These results are compared to known solutions for limiting cases. Work supported by the U.S. DOE grant DE-SC0006562.
Higher dimensional nonclassical eigenvalue asymptotics
NASA Astrophysics Data System (ADS)
Camus, Brice; Rautenberg, Nils
2015-02-01
In this article, we extend Simon's construction and results [B. Simon, J. Funct. Anal. 53(1), 84-98 (1983)] for leading order eigenvalue asymptotics to n-dimensional Schrödinger operators with non-confining potentials given by Hn α = - Δ + ∏ i = 1 n |x i| α i on ℝn (n > 2), α ≔ ( α 1 , … , α n ) ∈ ( R+ ∗ ) n . We apply the results to also derive the leading order spectral asymptotics in the case of the Dirichlet Laplacian -ΔD on domains Ωn α = { x ∈ R n : ∏ j = 1 n }x j| /α j α n < 1 } .
Elimination of spurious eigenvalues in the Chebyshev tau spectral method
NASA Technical Reports Server (NTRS)
Mcfadden, G. B.; Murray, B. T.; Boisvert, R. F.
1989-01-01
Spectral methods have been used to great advantage in hydrodynamic stability calculations; the concepts are described in Orszag's seminal application of the Chebyshev tau method to the Orr-Sommerfeld equation for plane Poiseuille flow in 1971. Orszag discusses both the Chebyshev Galerkin and the Chebyshev tau methods, but presents results for the tau method, which is easier to implement than the Galerkin method. The tau method has the disadvantage that two unstable eigenvalues are produced that are artifacts of the discretization. An extremely simple modification to the Chebyshev tau method is presented which eliminates the spurious eigenvalues. First a simplified model of the Orr-Sommerfeld equation discussed by Gottlieb and Orszag was studied. Then the Chebyshev tau method is considered, which has two spurious eigenvalues, and then a modification which eliminates them is described. Finally, results for the Orr-Sommerfeld equation are considered where the modified tau method also eliminates the spurious eigenvalues. The simplicity of the modification makes it a convenient alternative to other approaches to the problem.
A method to stabilize linear systems using eigenvalue gradient information
NASA Technical Reports Server (NTRS)
Wieseman, C. D.
1985-01-01
Formal optimization methods and eigenvalue gradient information are used to develop a stabilizing control law for a closed loop linear system that is initially unstable. The method was originally formulated by using direct, constrained optimization methods with the constraints being the real parts of the eigenvalues. However, because of problems in trying to achieve stabilizing control laws, the problem was reformulated to be solved differently. The method described uses the Davidon-Fletcher-Powell minimization technique to solve an indirect, constrained minimization problem in which the performance index is the Kreisselmeier-Steinhauser function of the real parts of all the eigenvalues. The method is applied successfully to solve two different problems: the determination of a fourth-order control law stabilizes a single-input single-output active flutter suppression system and the determination of a second-order control law for a multi-input multi-output lateral-directional flight control system. Various sets of design variables and initial starting points were chosen to show the robustness of the method.
On the behavior of the leading eigenvalue of Eigen's evolutionary matrices.
Semenov, Yuri S; Bratus, Alexander S; Novozhilov, Artem S
2014-12-01
We study general properties of the leading eigenvalue w¯(q) of Eigen's evolutionary matrices depending on the replication fidelity q. This is a linear algebra problem that has various applications in theoretical biology, including such diverse fields as the origin of life, evolution of cancer progression, and virus evolution. We present the exact expressions for w¯(q),w¯(')(q),w¯('')(q) for q = 0, 0.5, 1 and prove that the absolute minimum of w¯(q), which always exists, belongs to the interval (0, 0.5]. For the specific case of a single peaked landscape we also find lower and upper bounds on w¯(q), which are used to estimate the critical mutation rate, after which the distribution of the types of individuals in the population becomes almost uniform. This estimate is used as a starting point to conjecture another estimate, valid for any fitness landscape, and which is checked by numerical calculations. The last estimate stresses the fact that the inverse dependence of the critical mutation rate on the sequence length is not a generally valid fact. PMID:25445764
Feature Surfaces in Symmetric Tensor Fields Based on Eigenvalue Manifold.
Palacios, Jonathan; Yeh, Harry; Wang, Wenping; Zhang, Yue; Laramee, Robert S; Sharma, Ritesh; Schultz, Thomas; Zhang, Eugene
2016-03-01
Three-dimensional symmetric tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D symmetric tensor fields focus on degenerate tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces, into tensor field analysis, based on the notion of eigenvalue manifold. Neutral surfaces are the boundary between linear tensors and planar tensors, and the traceless surfaces are the boundary between tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of tensor modes, tensor isotropy, and tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can cause the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches, to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis. PMID:26441450
NASA Astrophysics Data System (ADS)
Aghaei, S.; Chenaghlou, A.
2014-02-01
The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E8 (case (3b)), S4 and S2, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.
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
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.
Perfetti, Christopher M; Martin, William R; Rearden, Bradley T; Williams, Mark L
2012-01-01
This study introduced three approaches for calculating the importance weighting function for Contributon and CLUTCH eigenvalue sensitivity coefficient calculations, and compared them in terms of accuracy and applicability. The necessary levels of mesh refinement and mesh convergence for obtaining accurate eigenvalue sensitivity coefficients were determined through two parametric studies, and the results of these studies suggest that a sufficiently-accurate mesh for calculating eigenvalue sensitivity coefficients can be obtained for the Contributon and CLUTCH methods with only a small increase in problem runtime.
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…
Investigating a hybrid perturbation-Galerkin technique using computer algebra
NASA Technical Reports Server (NTRS)
Andersen, Carl M.; Geer, James F.
1988-01-01
A two-step hybrid perturbation-Galerkin method is presented for the solution of a variety of differential equations type problems which involve a scalar parameter. The resulting (approximate) solution has the form of a sum where each term consists of the product of two functions. The first function is a function of the independent field variable(s) x, and the second is a function of the parameter lambda. In step one the functions of x are determined by forming a perturbation expansion in lambda. In step two the functions of lambda are determined through the use of the classical Bubnov-Gelerkin method. The resulting hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Bubnov-Galerkin methods applied separately, while combining some of the good features of each. In particular, the results can be useful well beyond the radius of convergence associated with the perturbation expansion. The hybrid method is applied with the aid of computer algebra to a simple two-point boundary value problem where the radius of convergence is finite and to a quantum eigenvalue problem where the radius of convergence is zero. For both problems the hybrid method apparently converges for an infinite range of the parameter lambda. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed.
Perfetti, Christopher M; Martin, William R; Rearden, Bradley T; Williams, Mark L
2012-01-01
Three methods for calculating continuous-energy eigenvalue sensitivity coefficients were developed and implemented into the SHIFT Monte Carlo code within the Scale code package. The methods were used for several simple test problems and were evaluated in terms of speed, accuracy, efficiency, and memory requirements. A promising new method for calculating eigenvalue sensitivity coefficients, known as the CLUTCH method, was developed and produced accurate sensitivity coefficients with figures of merit that were several orders of magnitude larger than those from existing methods.
Arbitrary eigenvalue assignments for linear time-varying multivariable control systems
NASA Technical Reports Server (NTRS)
Nguyen, Charles C.
1987-01-01
The problem of eigenvalue assignments for a class of linear time-varying multivariable systems is considered. Using matrix operators and canonical transformations, it is shown that a time-varying system that is 'lexicography-fixedly controllable' can be made via state feedback to be equivalent to a time-invariant system whose eigenvalues are arbitrarily assignable. A simple algorithm for the design of the state feedback is provided.
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.
Twisted Quantum Toroidal Algebras
NASA Astrophysics Data System (ADS)
Jing, Naihuan; Liu, Rongjia
2014-09-01
We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.
Philip, Bobby; Chartier, Dr Timothy
2012-01-01
methods based on Local Sensitivity Analysis (LSA). The method can be used in the context of geometric and algebraic multigrid methods for constructing smoothers, and in the context of Krylov methods for constructing block preconditioners. It is suitable for both constant and variable coecient problems. Furthermore, the method can be applied to systems arising from both scalar and coupled system partial differential equations (PDEs), as well as linear systems that do not arise from PDEs. The simplicity of the method will allow it to be easily incorporated into existing multigrid and Krylov solvers while providing a powerful tool for adaptively constructing methods tuned to a problem.
Fu, Zhongtao; Yang, Wenyu; Yang, Zhen
2013-08-01
In this paper, we present an efficient method based on geometric algebra for computing the solutions to the inverse kinematics problem (IKP) of the 6R robot manipulators with offset wrist. Due to the fact that there exist some difficulties to solve the inverse kinematics problem when the kinematics equations are complex, highly nonlinear, coupled and multiple solutions in terms of these robot manipulators stated mathematically, we apply the theory of Geometric Algebra to the kinematic modeling of 6R robot manipulators simply and generate closed-form kinematics equations, reformulate the problem as a generalized eigenvalue problem with symbolic elimination technique, and then yield 16 solutions. Finally, a spray painting robot, which conforms to the type of robot manipulators, is used as an example of implementation for the effectiveness and real-time of this method. The experimental results show that this method has a large advantage over the classical methods on geometric intuition, computation and real-time, and can be directly extended to all serial robot manipulators and completely automatized, which provides a new tool on the analysis and application of general robot manipulators. PMID:23918347
Algebraic vs physical N = 6 3-algebras
Cantarini, Nicoletta; Kac, Victor G.
2014-01-15
In our previous paper, we classified linearly compact algebraic simple N = 6 3-algebras. In the present paper, we classify their “physical” counterparts, which actually appear in the N = 6 supersymmetric 3-dimensional Chern-Simons theories.
Advanced Variance Reduction for Global k-Eigenvalue Simulations in MCNP
Edward W. Larsen
2008-06-01
to the correlations between fission source estimates. In the new FMC method, the eigenvalue problem (expressed in terms of the Boltzmann equation) is integrated over the energy and direction variables. Then these equations are multiplied by J special "tent" functions in space and integrated over the spatial variable. This yields J equations that are exactly satisfied by the eigenvalue k and J space-angle-energy moments of the eigenfunction. Multiplying and dividing by suitable integrals of the eigenfunction, one obtains J algebraic equations for k and the space-angle-energy moments of the eigenfunction, which contain nonlinear functionals that depend weakly on the eigenfunction. In the FMC method, information from the standard Monte Carlo solution for each active cycle is used to estimate the functionals, and at the end of each cycle the J equations for k and the space-angle-energy moments of the eigenfunction are solved. Finally, these results are averaged over N active cycles to obtain estimated means and standard deviations for k and the space-angle-energy moments of the eigenfunction. Our limited testing shows that for large single fissile systems such as a commercial reactor core, (i) the FMC estimate of the eigenvalue is at least one order of magnitude more accurate than estimates obtained from the standard Monte Carlo approach, (ii) the FMC estimate of the eigenfunction converges and is several orders of magnitude more accurate than the standard estimate, and (iii) the FMC estimate of the standard deviation in k is at least one order of magnitude closer to the correct standard deviation than the standard estimate. These advances occur because: (i) the Monte Carlo estimates of the nonlinear functionals are much more accurate than the direct Monte Carlo estimates of the eigenfunction, (ii) the system of discrete equations that determines the FMC estimates of k is robust, and (iii) the functionals are only very weakly correlated between different fission
Application of polynomial su(1, 1) algebra to Pöschl-Teller potentials
Zhang, Hong-Biao Lu, Lu
2013-12-15
Two novel polynomial su(1, 1) algebras for the physical systems with the first and second Pöschl-Teller (PT) potentials are constructed, and their specific representations are presented. Meanwhile, these polynomial su(1, 1) algebras are used as an algebraic technique to solve eigenvalues and eigenfunctions of the Hamiltonians associated with the first and second PT potentials. The algebraic approach explores an appropriate new pair of raising and lowing operators K-circumflex{sub ±} of polynomial su(1, 1) algebra as a pair of shift operators of our Hamiltonians. In addition, two usual su(1, 1) algebras associated with the first and second PT potentials are derived naturally from the polynomial su(1, 1) algebras built by us.
Euclidean supergravity in terms of Dirac eigenvalues
NASA Astrophysics Data System (ADS)
Vancea, Ion V.
1998-08-01
It has been recently shown that the eigenvalues of the Dirac operator can be considered as dynamical variables of Euclidean gravity. The purpose of this paper is to explore the possibility that the eigenvalues of the Dirac operator might play the same role in the case of supergravity. It is shown that for this purpose some primary constraints on covariant phase space as well as secondary constraints on the eigenspinors must be imposed. The validity of primary constraints under covariant transport is further analyzed. It is shown that in this case restrictions on the tangent bundle and on the spinor bundle of spacetime arise. The form of these restrictions is determined under some simplifying assumptions. It is also shown that manifolds with flat curvature of tangent bundle and spinor bundle satisfy these restrictions and thus they support the Dirac eigenvalues as global observables.
Eigenvalue analysis of uncertain ODE systems.
Sonday, Benjamin; Berry, Robert Dan; Debusschere, Bert J.; Najm, Habib N.
2010-07-01
The Polynomial chaos expansion provides a means of representing any L2 random variable as a sum of polynomials that are orthogonal with respect to a chosen measure. Examples include the Hermite polynomials with Gaussian measure on the real line and the Legendre polynomials with uniform measure on an interval. Polynomial chaos can be used to reformulate an uncertain ODE system, using Galerkin projection, as a new, higher-dimensional, deterministic ODE system which describes the evolution of each mode of the polynomial chaos expansion. It is of interest to explore the eigenstructure of the original and reformulated ODE systems by studying the eigenvalues and eigenvectors of their Jacobians. In this talk, we study the distribution of the eigenvalues of the two Jacobians. We outline in general the location of the eigenvalues of the new system with respect to those of the original system, and examine the effect of expansion order on this distribution.
Upgrades to the ArbiTER edge plasma eigenvalue code
NASA Astrophysics Data System (ADS)
Baver, D. A.; Myra, J. R.; Umansky, M. V.
2012-10-01
The Arbitrary Topology Equation Reader, or ArbiTER, is a flexible eigenvalue code that is under continued development for plasma kinetic problems. The preliminary stage of ArbiTER development has demonstrated its capability in handling complicated geometries (such as multiple X-points) as well as simple kinetic problems. Planned upgrades (such as parallelization and unstructured grids) are expected to expand its range of potential applications. In order to handle large eigenvalue problems produced by realistic kinetic problems, parallelization is necessary. ArbiTER uses the SLEPc [1] eigensolver package, which already has parallel capability, however, early versions of the code lack the structures needed to exploit this capability. Integrating parallel SLEPc into the ArbiTER code is therefore a high priority. In addition, we will also present first physics studies using ArbiTER. This will be analysis of surface-localized phenomena such as coaxial modes, which are relevant to RF heating and current drive in devices such as NSTX. Work supported by the U.S. DOE. [4pt] [1] http://www.grycap.upv.es/slepc/
Eigenvalue Detonation of Combined Effects Aluminized Explosives
NASA Astrophysics Data System (ADS)
Capellos, C.; Baker, E. L.; Nicolich, S.; Balas, W.; Pincay, J.; Stiel, L. I.
2007-12-01
Theory and performance for recently developed combined—effects aluminized explosives are presented. Our recently developed combined-effects aluminized explosives (PAX-29C, PAX-30, PAX-42) are capable of achieving excellent metal pushing, as well as high blast energies. Metal pushing capability refers to the early volume expansion work produced during the first few volume expansions associated with cylinder and wall velocities and Gurney energies. Eigenvalue detonation explains the observed detonation states achieved by these combined effects explosives. Cylinder expansion data and thermochemical calculations (JAGUAR and CHEETAH) verify the eigenvalue detonation behavior.
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…
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.
Cornell interaction in the two-body semi-relativistic framework: The Lie algebraic approach
NASA Astrophysics Data System (ADS)
Panahi, H.; Zarrinkamar, S.; Baradaran, M.
2016-02-01
We consider an approximation to the two-body spinless Salpeter equation which is valid for the case of heavy quarks with the Cornell potential. We then replace the square of kinetic term with its nonrelativistic equivalent and obtain an equation which can be alternatively viewed as the generalization of the Schrödinger equation into the relativistic regime. In the calculations, we use the Lie algebraic approach within the framework of quasi-exact solvability. With the help of the representation theory of sl(2) , the ( n+1 -dimensional matrix equation of the problem is constructed in a quite detailed manner and thereby the quasi-exact expressions for the energy eigenvalues and the corresponding wave functions as well as the allowed values of the potential parameters are obtained.
Numerical Linear Algebra On The CEDAR Multiprocessor
NASA Astrophysics Data System (ADS)
Meier, Ulrike; Sameh, Ahmed
1988-01-01
In this paper we describe in some detail the architectural features of the CEDAR. multiprocessor. We also discuss strategies for implementation of dense matrix computations, and present performance results on one cluster for a variety of linear system solvers, eigenvalue problem solvers, as well as algorithms for solving linear least squares problems.
Parallel eigensolver for H(curl) problems using H1-auxiliary space AMG preconditioning
Kolev, T V; Vassilevski, P S
2006-11-15
This report describes an application of the recently developed H{sup 1}-auxiliary space preconditioner for H(curl) problems to the Maxwell eigenvalue problem. The auxiliary space method based on the new (HX) finite element space decomposition introduced in [7], was implemented in the hypre library, [10, 11] under the name AMS. The eigensolver considered in the present paper, referred to as the AME, is an extension of the AMS. It is based on the locally optimal block eigensolver LOBPCG [9] and the parallel AMG (algebraic multigrid) solver BoomerAMG [2] from the hypre library. AME is designed to compute a block of few minimal nonzero eigenvalues and eigenvectors, for general unstructured finite element discretizations utilizing the lowest order Nedelec elements. The main goal of the current report is to document the usage of AME and to illustrate its parallel scalability.
Koc, Ramazan . E-mail: koc@gantep.edu.tr; Tuetuencueler, Hayriye; Koca, Mehmet; Olgar, Eser
2005-10-01
We consider solutions of the 2 x 2 matrix Hamiltonians of the physical systems within the context of the su (2) and su (1, 1) Lie algebras. Our technique is relatively simple when compared with those of others and treats those Hamiltonians which can be treated in a unified framework of the Sp (4, R) algebra. The systematic study presented here reproduces a number of earlier results in a natural way as well as leads to a novel finding. Possible generalizations of the method are also suggested.
The nth root of sequential effect algebras
NASA Astrophysics Data System (ADS)
Shen, Jun; Wu, Junde
2010-06-01
In 2005, Gudder [Int. J. Theor. Phys. 44, 2219 (2005)] presented 25 problems of sequential effect algebras, the 20th problem asked: In a sequential effect algebra, if the square root of some element exists, is it unique? In this paper, we show that for each given positive integer n >1, there is a sequential effect algebra such that the nth root of its some element c is not unique, and the nth root of c is not the kth root of c (k
First Bloch eigenvalue in high contrast media
NASA Astrophysics Data System (ADS)
Briane, Marc; Vanninathan, Muthusamy
2014-01-01
This paper deals with the asymptotic behavior of the first Bloch eigenvalue in a heterogeneous medium with a high contrast ɛY-periodic conductivity. When the conductivity is bounded in L1 and the constant of the Poincaré-Wirtinger weighted by the conductivity is very small with respect to ɛ-2, the first Bloch eigenvalue converges as ɛ → 0 to a limit which preserves the second-order expansion with respect to the Bloch parameter. In dimension two the expansion of the limit can be improved until the fourth-order under the same hypotheses. On the contrary, in dimension three a fibers reinforced medium combined with a L1-unbounded conductivity leads us to a discontinuity of the limit first Bloch eigenvalue as the Bloch parameter tends to zero but remains not orthogonal to the direction of the fibers. Therefore, the high contrast conductivity of the microstructure induces an anomalous effect, since for a given low-contrast conductivity the first Bloch eigenvalue is known to be analytic with respect to the Bloch parameter around zero.
Classical versus Computer Algebra Methods in Elementary Geometry
ERIC Educational Resources Information Center
Pech, Pavel
2005-01-01
Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic…
Figueroa-O'Farrill, Jose Miguel
2009-11-15
We phrase deformations of n-Leibniz algebras in terms of the cohomology theory of the associated Leibniz algebra. We do the same for n-Lie algebras and for the metric versions of n-Leibniz and n-Lie algebras. We place particular emphasis on the case of n=3 and explore the deformations of 3-algebras of relevance to three-dimensional superconformal Chern-Simons theories with matter.
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.
A new direction in hydrodynamic stability: Beyond eigenvalues
NASA Technical Reports Server (NTRS)
Trefethen, Lloyd N.; Trefethen, Anne E.; Reddy, Satish C.; Driscoll, Tobin A.
1992-01-01
Fluid flows that are smooth at low speeds become unstable and then turbulent at higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow and looking for unstable eigenvalues of the linearized problem, but the results agree poorly in many cases with experiments. Nevertheless, it has become clear in recent years that linear effects play a central role in hydrodynamic instability. A reconciliation of these findings with the traditional analysis can be obtained by considering the 'pseudospectra' of the linearized problem, which reveals that small perturbations to the smooth flow in the form of streamwise vortices may be amplified by factors on the order of 10(exp 5) by a linear mechanism, even though all the eigenmodes are stable. The same principles apply also to other problems in the mathematical sciences that involve non-orthogonal eigenfunctions.
Eigenvalue routines in NASTRAN: A comparison with the Block Lanczos method
NASA Technical Reports Server (NTRS)
Tischler, V. A.; Venkayya, Vipperla B.
1993-01-01
The NASA STRuctural ANalysis (NASTRAN) program is one of the most extensively used engineering applications software in the world. It contains a wealth of matrix operations and numerical solution techniques, and they were used to construct efficient eigenvalue routines. The purpose of this paper is to examine the current eigenvalue routines in NASTRAN and to make efficiency comparisons with a more recent implementation of the Block Lanczos algorithm by Boeing Computer Services (BCS). This eigenvalue routine is now available in the BCS mathematics library as well as in several commercial versions of NASTRAN. In addition, CRAY maintains a modified version of this routine on their network. Several example problems, with a varying number of degrees of freedom, were selected primarily for efficiency bench-marking. Accuracy is not an issue, because they all gave comparable results. The Block Lanczos algorithm was found to be extremely efficient, in particular, for very large size problems.
Eigenvalue routines in NASTRAN: A comparison with the Block Lanczos method
NASA Astrophysics Data System (ADS)
Tischler, V. A.; Venkayya, Vipperla B.
1993-09-01
The NASA STRuctural ANalysis (NASTRAN) program is one of the most extensively used engineering applications software in the world. It contains a wealth of matrix operations and numerical solution techniques, and they were used to construct efficient eigenvalue routines. The purpose of this paper is to examine the current eigenvalue routines in NASTRAN and to make efficiency comparisons with a more recent implementation of the Block Lanczos algorithm by Boeing Computer Services (BCS). This eigenvalue routine is now available in the BCS mathematics library as well as in several commercial versions of NASTRAN. In addition, CRAY maintains a modified version of this routine on their network. Several example problems, with a varying number of degrees of freedom, were selected primarily for efficiency bench-marking. Accuracy is not an issue, because they all gave comparable results. The Block Lanczos algorithm was found to be extremely efficient, in particular, for very large size problems.
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.
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.
Spectral properties of sums of Hermitian matrices and algebraic geometry
NASA Astrophysics Data System (ADS)
Chau Huu-Tai, P.; Van Isacker, P.
2016-04-01
It is shown that all the eigenvectors of a sum of Hermitian matrices belong to the same algebraic variety. A polynomial system characterizing this variety is given and a set of nonlinear equations is derived which allows the construction of the variety. Moreover, in some specific cases, explicit expressions for the eigenvectors and eigenvalues can be obtained. Explicit solutions of selected models are also derived.
Generalization of Richardson-Gaudin models to rank-2 algebras
Errea, B; Lerma, S; Dukelsky, J; Dimitrova, S S; Pittel, S; Van Isacker, P; Gueorguiev, V G
2006-07-20
A generalization of Richardson-Gaudin models to the rank-2 SO(5) and SO(3,2) algebras is used to describe systems of two kinds of fermions or bosons interacting through a pairing force. They are applied to the proton-neutron neutron isovector pairing model and to the Interacting Boson Model 2, in the transition from vibration to gamma-soft nuclei, respectively. In both cases, the integrals of motion and their eigenvalues are obtained.
On computational complexity of Clifford algebra
NASA Astrophysics Data System (ADS)
Budinich, Marco
2009-05-01
After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced by Budinich and Budinich ["A spinorial formulation of the maximum clique problem of a graph," J. Math. Phys. 47, 043502 (2006)].
Edge covers and independence: Algebraic approach
NASA Astrophysics Data System (ADS)
Kalinina, E. A.; Khitrov, G. M.; Pogozhev, S. V.
2016-06-01
In this paper, linear algebra methods are applied to solve some problems of graph theory. For ordinary connected graphs, edge coverings and independent sets are considered. Some results concerning minimum edge covers and maximum matchings are proved with the help of linear algebraic approach. The problem of finding a maximum matching of a graph is fundamental both practically and theoretically, and has numerous applications, e.g., in computational chemistry and mathematical chemistry.
The arithmetic theory of algebraic groups
NASA Astrophysics Data System (ADS)
Platonov, V. P.
1982-06-01
CONTENTS Introduction § 1. Arithmetic groups § 2. Adèle groups § 3. Tamagawa numbers § 4. Approximations in algebraic groups § 5. Class numbers and class groups of algebraic groups § 6. The genus problem in arithmetic groups § 7. Classification of maximal arithmetic subgroups § 8. The congruence problem § 9. Groups of rational points over global fields § 10. Galois cohomology and the Hasse principle § 11. Cohomology of arithmetic groups References
Algorithmic Questions for Linear Algebraic Groups. Ii
NASA Astrophysics Data System (ADS)
Sarkisjan, R. A.
1982-04-01
It is proved that, given a linear algebraic group defined over an algebraic number field and satisfying certain conditions, there exists an algorithm which determines whether or not two double cosets of a special type coincide in its adele group, and which enumerates all such double cosets. This result is applied to the isomorphism problem for finitely generated nilpotent groups, and also to other problems.Bibliography: 18 titles.
Feasible eigenvalue sensitivity for large power systems
Smed, T. . Dept. of Electric Power Systems)
1993-05-01
Traditional eigenvalue sensitivity for power systems requires the formulation of the system matrix, which lacks sparsity. In this paper, a new sensitivity analysis, derived for a sparse formulation, is presented. Variables that are computed as intermediate results in established eigen value programs for power systems, but not used further, are given a new interpretation. The effect of virtually any control action can be assessed based on a single eigenvalue-eigenvector calculation. In particular, the effect of active and reactive power modulation can be found as a multiplication of two or three complex numbers. The method is illustrated in an example for a large power system when applied to the control design for an HVDC-link.
Boolean Algebra. Geometry Module for Use in a Mathematics Laboratory Setting.
ERIC Educational Resources Information Center
Brotherton, Sheila; And Others
This module is recommended as an honors unit to follow a unit on logic. There are four basic parts: (1) What is a Boolean Algebra; (2) Using Boolean Algebra to Prove Theorems; (3) Using Boolean Algebra to Simplify Logical Statements; and (4) Circuit Problems with Logic and Boolean Algebra. Of these, sections 1, 2, and 3 are primarily written…
NASA Astrophysics Data System (ADS)
Caldwell, Curtis Irvin
This is a theoretical thesis. The goal is to determine how many signal sources exist in the medium when constrained to using only a few samples. The need to make decisions based on only a few samples is motivated by the slow sound propagation speed and the time urgency to make decisions. This research treats the problem from the point of view of classical hypothesis testing assuming complex multivariate Gaussian random variables. This is the small sample complex principal components analysis problem. The critical issue is the derivation of probability density functions of appropriate test statistics. The goal has been partially achieved. The probability density functions for several important distributions have been derived. In particular, these include the distribution for the set of eigenvalues satisfying the generalized eigenvalue problem of two complex Wishart matrices, the matrix complex Gaussian distribution, a joint distribution needed to derive the density for the sphericity test statistic, the density function for the ratio of averages of disjoint sums of sequential eigenvalues of a complex Wishart matrix, and several tests based on the ratio of an arbitrary eigenvalue to the maximum, minimum, average, or sum of all the eigenvalues for a special case of the complex Wishart matrix. This thesis includes a derivation completely in the context of complex variables of the density function of the complex Wishart distribution and the distribution of its eigenvalues. It also includes a few minor results regarding zonal polynomials of complex matrix argument. A comprehensive development of the tools of statistics of complex variables for engineers and physicists is provided. This includes a study of complex matrix derivatives, changes of complex variables, and properties of the characteristic function of a complex multivariate random variable. A derivation of the complex Hotelling's T^2 test statistic and distribution useful for tests on means is given. A tutorial
Highly accurate eigenvalues for the distorted Coulomb potential
NASA Astrophysics Data System (ADS)
Ixaru, L. Gr.; de Meyer, H.; vanden Berghe, G.
2000-03-01
We consider the eigenvalue problem for the radial Schrödinger equation with potentials of the form V(r)=S(r)/r+R(r) where S(r) and R(r) are well behaved functions which tend to some (not necessarily equal) constants when r-->0 and r-->∞. Formulas (14.4.5)-(14.4.8) of Abramowitz and Stegun [Handbook of Mathematical Functions, 8th ed. (Dover, New York, 1972)], corresponding to the pure Coulomb case, are here generalized for this distorted case. We also present a complete procedure for the numerical solution of the problem. Our procedure is robust, very economic and particularly suited for very large n. Numerical illustrations for n up to 2000 are given.
Variational principles for eigenvalues of the Klein-Gordon equation
Langer, Matthias; Tretter, Christiane
2006-10-15
In this paper variational principles for eigenvalues of an abstract model of the Klein-Gordon equation with electromagnetic potential are established. They are used to characterize and estimate eigenvalues in cases where the essential spectrum has a gap around 0, even in the presence of complex eigenvalues. As a consequence, a comparison between eigenvalues of the Klein-Gordon equation in R{sup d} and eigenvalues of certain Schroedinger operators is obtained. The results are illustrated on examples including the Klein-Gordon equation with Coulomb and square-well potential.
Perfetti, C.; Martin, W.; Rearden, B.; Williams, M.
2012-07-01
Three methods for calculating continuous-energy eigenvalue sensitivity coefficients were developed and implemented into the Shift Monte Carlo code within the SCALE code package. The methods were used for two small-scale test problems and were evaluated in terms of speed, accuracy, efficiency, and memory requirements. A promising new method for calculating eigenvalue sensitivity coefficients, known as the CLUTCH method, was developed and produced accurate sensitivity coefficients with figures of merit that were several orders of magnitude larger than those from existing methods. (authors)
Optimal lower bound for the first eigenvalue of the fourth order equation
NASA Astrophysics Data System (ADS)
Meng, Gang; Yan, Ping
2016-09-01
In this paper we will find optimal lower bound for the first eigenvalue of the fourth order equation with integrable potentials when the L1 norm of potentials is known. We establish the minimization characterization for the first eigenvalue of the measure differential equation, which plays an important role in the extremal problem of ordinary differential equation. The conclusion of this paper will illustrate a new and very interesting phenomenon that the minimizing measures will no longer be located at the center of the interval when the norm is large enough.
Lower bounds for sums of eigenvalues of elliptic operators and systems
Ilyin, Aleksei A
2013-04-30
Two-term lower bounds of Berzin-Li-Yau type are obtained for the sums of eigenvalues of elliptic operators and systems with constant coefficients and Dirichlet boundary conditions. The polyharmonic operator, the Stokes system and its generalizations, the two-dimensional buckling problem, and also the Klein-Gordon operator are considered. Bibliography: 32 titles.
Two novel classes of solvable many-body problems of goldfish type with constraints
NASA Astrophysics Data System (ADS)
Calogero, F.; Gómez-Ullate, D.
2007-05-01
Two novel classes of many-body models with nonlinear interactions 'of goldfish type' are introduced. They are solvable provided the initial data satisfy a single constraint (in one case; in the other, two constraints), i.e., for such initial data the solution of their initial-value problem can be achieved via algebraic operations, such as finding the eigenvalues of given matrices or equivalently the zeros of known polynomials. Entirely isochronous versions of some of these models are also exhibited, i.e., versions of these models whose nonsingular solutions are all completely periodic with the same period.
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…
Realizations of Galilei algebras
NASA Astrophysics Data System (ADS)
Nesterenko, Maryna; Pošta, Severin; Vaneeva, Olena
2016-03-01
All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations.
A Decentralized Eigenvalue Computation Method for Spectrum Sensing Based on Average Consensus
NASA Astrophysics Data System (ADS)
Mohammadi, Jafar; Limmer, Steffen; Stańczak, Sławomir
2016-07-01
This paper considers eigenvalue estimation for the decentralized inference problem for spectrum sensing. We propose a decentralized eigenvalue computation algorithm based on the power method, which is referred to as generalized power method GPM; it is capable of estimating the eigenvalues of a given covariance matrix under certain conditions. Furthermore, we have developed a decentralized implementation of GPM by splitting the iterative operations into local and global computation tasks. The global tasks require data exchange to be performed among the nodes. For this task, we apply an average consensus algorithm to efficiently perform the global computations. As a special case, we consider a structured graph that is a tree with clusters of nodes at its leaves. For an accelerated distributed implementation, we propose to use computation over multiple access channel (CoMAC) as a building block of the algorithm. Numerical simulations are provided to illustrate the performance of the two algorithms.
Algebraic Bethe Ansatz for Open XXX Model with Triangular Boundary Matrices
NASA Astrophysics Data System (ADS)
Belliard, Samuel; Crampé, Nicolas; Ragoucy, Eric
2013-05-01
We consider an open XXX spin chain with two general boundary matrices whose entries obey a relation, which is equivalent to the possibility to put simultaneously the two matrices in a upper-triangular form. We construct Bethe vectors by means of a generalized algebraic Bethe ansatz. As usual, the method uses Bethe equations and provides transfer matrix eigenvalues.
Orientation in operator algebras
Alfsen, Erik M.; Shultz, Frederic W.
1998-01-01
A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics. PMID:9618457
Developing Thinking in Algebra
ERIC Educational Resources Information Center
Mason, John; Graham, Alan; Johnson-Wilder, Sue
2005-01-01
This book is for people with an interest in algebra whether as a learner, or as a teacher, or perhaps as both. It is concerned with the "big ideas" of algebra and what it is to understand the process of thinking algebraically. The book has been structured according to a number of pedagogic principles that are exposed and discussed along the way,…
Connecting Arithmetic to Algebra
ERIC Educational Resources Information Center
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
Applied Algebra Curriculum Modules.
ERIC Educational Resources Information Center
Texas State Technical Coll., Marshall.
This collection of 11 applied algebra curriculum modules can be used independently as supplemental modules for an existing algebra curriculum. They represent diverse curriculum styles that should stimulate the teacher's creativity to adapt them to other algebra concepts. The selected topics have been determined to be those most needed by students…
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…
Ternary Virasoro - Witt algebra.
Zachos, C.; Curtright, T.; Fairlie, D.; High Energy Physics; Univ. of Miami; Univ. of Durham
2008-01-01
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.
Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type
NASA Astrophysics Data System (ADS)
Dey, Sanjib; Fring, Andreas; Mathanaranjan, Thilagarajah
2015-11-01
We propose a noncommutative version of the Euclidean Lie algebra E 2. Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra are investigated. Using the breakdown of the explicitly constructed Dyson maps as a criterium, we identify the domains in the parameter space in which the Hamiltonians have real energy spectra and determine the exceptional points signifying the crossover into the different types of spontaneously broken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find exceptional points which remain invariant under the deformation as well as exceptional points becoming dependent on the deformation parameter of the algebra.
3D transient eddy current fields using the u-v integral-eigenvalue formulation
NASA Astrophysics Data System (ADS)
Davey, Kent R.; Han, Hsiu Chi; Turner, Larry
1988-02-01
The three-dimensional eddy current transient field problem is formulated using the u-v method. This method breaks the vector Helmholtz equation into two scalar Helmholtz equations. Null field integral equations and the appropriate boundary conditions are used to set up an identification matrix which is independent of null field point locations. Embedded in the identification matrix are the unknown eigenvalues of the problem representing its impulse response in time. These eigenvalues are found by equating the determinant of the identification matrix to zero. When the initial transient forcing function is Fourier decomposed into its spatial harmonics, each Fourier component can be associated with a unique eigenvalue by this technique. The true transient solution comes through a convolution of the impulse response, so obtained with the particular external field decay governing the problem at hand. The technique is applied to the FELIX (fusion electromagnetic induction experiments) medium cylinder experiment; computed results are compared with data. A pseudoanalytic confirmation of the eigenvalues so obtained is formulated to validate the procedure.
3D transient eddy current fields using the u-v integral-eigenvalue formulation
Davey, K.R.; Han, H.C.; Turner, L.
1988-02-15
The three-dimensional eddy current transient field problem is formulated using the u-v method. This method breaks the vector Helmholtz equation into two scalar Helmholtz equations. Null field integral equations and the appropriate boundary conditions are used to set up an identification matrix which is independent of null field point locations. Embedded in the identification matrix are the unknown eigenvalues of the problem representing its impulse response in time. These eigenvalues are found by equating the determinant of the identification matrix to zero. When the initial transient forcing function is Fourier decomposed into its spatial harmonics, each Fourier component can be associated with a unique eigenvalue by this technique. The true transient solution comes through a convolution of the impulse response, so obtained with the particular external field decay governing the problem at hand. The technique is applied to the FELIX (fusion electromagnetic induction experiments) medium cylinder experiment; computed results are compared with data. A pseudoanalytic confirmation of the eigenvalues so obtained is formulated to validate the procedure.
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.
Approximation methods in relativistic eigenvalue perturbation theory
NASA Astrophysics Data System (ADS)
Noble, Jonathan Howard
In this dissertation, three questions, concerning approximation methods for the eigenvalues of quantum mechanical systems, are investigated: (i) What is a pseudo--Hermitian Hamiltonian, and how can its eigenvalues be approximated via numerical calculations? This is a fairly broad topic, and the scope of the investigation is narrowed by focusing on a subgroup of pseudo--Hermitian operators, namely, PT--symmetric operators. Within a numerical approach, one projects a PT--symmetric Hamiltonian onto an appropriate basis, and uses a straightforward two--step algorithm to diagonalize the resulting matrix, leading to numerically approximated eigenvalues. (ii) Within an analytic ansatz, how can a relativistic Dirac Hamiltonian be decoupled into particle and antiparticle degrees of freedom, in appropriate kinematic limits? One possible answer is the Foldy--Wouthuysen transform; however, there are alter- native methods which seem to have some advantages over the time--tested approach. One such method is investigated by applying both the traditional Foldy--Wouthuysen transform and the "chiral" Foldy--Wouthuysen transform to a number of Dirac Hamiltonians, including the central-field Hamiltonian for a gravitationally bound system; namely, the Dirac-(Einstein-)Schwarzschild Hamiltonian, which requires the formal- ism of general relativity. (iii) Are there are pseudo--Hermitian variants of Dirac Hamiltonians that can be approximated using a decoupling transformation? The tachyonic Dirac Hamiltonian, which describes faster-than-light spin-1/2 particles, is gamma5--Hermitian, i.e., pseudo-Hermitian. Superluminal particles remain faster than light upon a Lorentz transformation, and hence, the Foldy--Wouthuysen program is unsuited for this case. Thus, inspired by the Foldy--Wouthuysen program, a decoupling transform in the ultrarelativistic limit is proposed, which is applicable to both sub- and superluminal particles.
Toward robust scalable algebraic multigrid solvers.
Waisman, Haim; Schroder, Jacob; Olson, Luke; Hiriyur, Badri; Gaidamour, Jeremie; Siefert, Christopher; Hu, Jonathan Joseph; Tuminaro, Raymond Stephen
2010-10-01
This talk highlights some multigrid challenges that arise from several application areas including structural dynamics, fluid flow, and electromagnetics. A general framework is presented to help introduce and understand algebraic multigrid methods based on energy minimization concepts. Connections between algebraic multigrid prolongators and finite element basis functions are made to explored. It is shown how the general algebraic multigrid framework allows one to adapt multigrid ideas to a number of different situations. Examples are given corresponding to linear elasticity and specifically in the solution of linear systems associated with extended finite elements for fracture problems.
The first eigenvalue of the Laplace operator
NASA Astrophysics Data System (ADS)
Kanguzhin, Baltabek E.; Dauitbek, Dostilek
2016-08-01
We consider a self-adjoint differential operator in the Hilbert space. The domain of the operator is changed by the perturbation of the boundary conditions so that a given neighborhood "there are no eigenvalues on neighborhood of zero" from the points of the spectrum of the perturbed operator. For the Sturm-Liouville operator on the segment and the Laplace operator on the square such a possibility is achieved through integral perturbations of boundary conditions. These statements are given with full proofs, and with a possible extension.
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.
Fuzzy-algebra uncertainty assessment
Cooper, J.A.; Cooper, D.K.
1994-12-01
A significant number of analytical problems (for example, abnormal-environment safety analysis) depend on data that are partly or mostly subjective. Since fuzzy algebra depends on subjective operands, we have been investigating its applicability to these forms of assessment, particularly for portraying uncertainty in the results of PRA (probabilistic risk analysis) and in risk-analysis-aided decision-making. Since analysis results can be a major contributor to a safety-measure decision process, risk management depends on relating uncertainty to only known (not assumed) information. The uncertainties due to abnormal environments are even more challenging than those in normal-environment safety assessments; and therefore require an even more judicious approach. Fuzzy algebra matches these requirements well. One of the most useful aspects of this work is that we have shown the potential for significant differences (especially in perceived margin relative to a decision threshold) between fuzzy assessment and probabilistic assessment based on subtle factors inherent in the choice of probability distribution models. We have also shown the relation of fuzzy algebra assessment to ``bounds`` analysis, as well as a description of how analyses can migrate from bounds analysis to fuzzy-algebra analysis, and to probabilistic analysis as information about the process to be analyzed is obtained. Instructive examples are used to illustrate the points.
Eigenvalue error analysis of viscously damped structures using a Ritz reduction method
NASA Technical Reports Server (NTRS)
Chu, Cheng-Chih; Milman, Mark H.
1992-01-01
The efficient solution of the eigenvalue problem that results from inserting passive dampers with variable stiffness and damping coefficients into a structure is addressed. Eigenanalysis of reduced models obtained by retaining a number of normal modes augmented with Ritz vectors corresponding to the static solutions resulting from the load patterns introduced by the dampers has been empirically shown to yield excellent approximations to the full eigenvalue problem. An analysis of this technique in the case of a single damper is presented. A priori and a posteriori error estimates are generated and tested on numerical examples. Comparison theorems with modally truncated models and a Markov parameter matching reduced-order model are derived. These theorems corroborate the heuristic that residual flexibility methods improve low-frequency approximation of the system. The analysis leads to other techniques for eigenvalue approximation. Approximate closed-form solutions are derived that include a refinement to eigenvalue derivative methods for approximation. An efficient Newton scheme is also developed. A numerical example is presented demonstrating the effectiveness of each of these methods.
Eigenvalues distribution for products of independent spherical ensembles
NASA Astrophysics Data System (ADS)
Zeng, Xingyuan
2016-06-01
We consider the product of independent spherical ensembles. By the special structure of eigenvalues as a rotation-invariant determinant point process, we show that the empirical spectral distribution of the product converges, with probability one, to a non-random distribution. And the limiting eigenvalue distribution is a power of spherical law. We also present an interesting correspondence between the eigenvalues of three classes of random matrix ensembles and zeros of random polynomials.
Mohan, R.; Ahmed, F.; Kothari, L.S.
1985-05-01
The results of a detailed study of a fast neutron diffusion length and pulsed problem in depleted and enriched subcritical uranium assemblies (0.2 to 4% /sup 235/U) are reported. The multigroup space- and time-dependent equations are solved using the eigenfunction expansion method. The effect of /sup 235/U concentration on space (diffusion length problem) and time (pulsed problem) eigenvalues and eigenfunctions, particularly on the ''discrete'' eigenvalue and eigenfunction, is discussed. The approach to equilibrium (both in space and in time) of fast neutrons changes with changing /sup 235/U concentration.
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…
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)
Saldin, E.L.; Schneidmiller, E.A.; Ulyanov, Yu.N.
1995-12-31
The paper presents analysis of the eigenvalue problem of the FEL amplifier with axisymmetric electron beam and diaphragm focusing line. An FEL model is discussed wherein diffraction effects, space charge fields and energy spread of electrons in the beam are taken into account. To take into account diffraction effects at the diaphragms we apply the rigorous impedance boundary conditions proposed by Veinstein. The rigorous solutions of the eigenvalue problem leave been found for the stepped and bounded parabolic electron beam profiles. Analytical expressions for eigenfunctions of active open waveguide and formulae of their expansion in eigenfunctions of passive open waveguide, are derived, too. Asymptotic behaviour of the obtained solutions is studied in details. The multilayer approximation method has been used to solve the eigenvalue problem for the beams with an arbitrary gradient profile of current density. This novel type of an FEL amplifier has perspective to be used for applications where high average and peak radiation power is required.
Eigenvalue spacings for quantized cat maps
NASA Astrophysics Data System (ADS)
Gamburd, Alex; Lafferty, John; Rockmore, Dan
2003-03-01
According to one of the basic conjectures in quantum chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms - 'cat maps'. Our numerical experiments indicate that for 'generic' choices of cat maps, the unfolded consecutive spacing distribution in the irreducible components of the Nth quantization (given by the N-dimensional Weil representation) approaches the GOE/GSE law of random matrix theory. For certain special 'arithmetic' transformations, related to the Ramanujan graphs of Lubotzky, Phillips and Sarnak, the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.
Shifted power method for computing tensor eigenvalues.
Mayo, Jackson R.; Kolda, Tamara Gibson
2010-07-01
Recent work on eigenvalues and eigenvectors for tensors of order m >= 3 has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor eigenpairs of the form Ax{sup m-1} = lambda x subject to ||x||=1, which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a shifted symmetric higher-order power method (SS-HOPM), which we show is guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to finding complex eigenpairs.
Eigenvalue Detonation of Combined Effects Aluminized Explosives
NASA Astrophysics Data System (ADS)
Capellos, Christos; Baker, Ernest; Balas, Wendy; Nicolich, Steven; Stiel, Leonard
2007-06-01
This paper reports on the development of theory and performance for recently developed combined effects aluminized explosives. Traditional high energy explosives used for metal pushing incorporate high loading percentages of HMX or RDX, whereas blast explosives incorporate some percentage of aluminum. However, the high blast explosives produce increased blast energies, with reduced metal pushing capability due to late time aluminum reaction. Metal pushing capability refers to the early volume expansion work produced during the first few volume expansions associated with cylinder wall velocities and Gurney energies. Our Recently developed combined effects aluminized explosives (PAX-29C, PAX-30, PAX-42) are capable of achieving excellent metal pushing and high blast energies. Traditional Chapman-Jouguet detonation theory does not explain the observed detonation states achieved by these combined effects explosives. This work demonstrates, with the use of cylinder expansion data and thermochemical code calculations (JAGUAR and CHEETAH), that eigenvalue detonation theory explains the observed behavior.
Semigroups And Computer Algebra In Discrete Structures
NASA Astrophysics Data System (ADS)
Bijev, G.
2010-10-01
Some concepts in semigroup theory are interpreted in discrete structures such as finite lattices, binary relations, and finite semilattices. An algebraic approach to the pseudoinverse generalization problem in Boolean vector spaces is used. By analogy with the linear spaces in the linear algebra semilattice homomorphisms, isomorphisms, projections on Boolean vector spaces are defined and some properties of them are investigated in detail. Maps, corresponding to them in the linear algebra, are connected with matrices and their pseudouinverse. Important properties of these maps, which are essential for solving linear systems, remain the same in the Boolean vector spaces. Stochastic experiments using the maps defined and computer algebra methods have been made for solving linear equations Ax = b. The Hamming distance between b and the projection p(b) = Ax of b is equal or close to the least possible one, if the system has no solutions.
Digital Maps, Matrices and Computer Algebra
ERIC Educational Resources Information Center
Knight, D. G.
2005-01-01
The way in which computer algebra systems, such as Maple, have made the study of complex problems accessible to undergraduate mathematicians with modest computational skills is illustrated by some large matrix calculations, which arise from representing the Earth's surface by digital elevation models. Such problems are often considered to lie in…
A Linear Algebraic Approach to Teaching Interpolation
ERIC Educational Resources Information Center
Tassa, Tamir
2007-01-01
A novel approach for teaching interpolation in the introductory course in numerical analysis is presented. The interpolation problem is viewed as a problem in linear algebra, whence the various forms of interpolating polynomial are seen as different choices of a basis to the subspace of polynomials of the corresponding degree. This approach…
NASA Astrophysics Data System (ADS)
Castro, Carlos
2006-11-01
A novel Weyl-Heisenberg algebra in Clifford spaces is constructed that is based on a matrix-valued {\\cal H}^{AB} extension of Planck's constant. As a result of this modified Weyl-Heisenberg algebra one will no longer be able to measure, simultaneously, the pairs of variables (x, px), (x, py), (x, pz), (y, px), ... with absolute precision. New Klein-Gordon and Dirac wave equations and dispersion relations in Clifford spaces are presented. The latter Dirac equation is a generalization of the Dirac-Lanczos-Barut-Hestenes equation. We display the explicit isomorphism between Yang's noncommutative spacetime algebra and the area-coordinates algebra associated with Clifford spaces. The former Yang's algebra involves noncommuting coordinates and momenta with a minimum Planck scale λ (ultraviolet cutoff) and a minimum momentum p = planck/R (maximal length R, infrared cutoff). The double-scaling limit of Yang's algebra λ → 0, R → ∞, in conjunction with the large n → ∞ limit, leads naturally to the area quantization condition λR = L2 = nλ2 (in Planck area units) given in terms of the discrete angular-momentum eigenvalues n. It is shown how modified Newtonian dynamics is also a consequence of Yang's algebra resulting from the modified Poisson brackets. Finally, another noncommutative algebra which differs from Yang's algebra and related to the minimal length uncertainty relations is presented. We conclude with a discussion of the implications of noncommutative QM and QFT's in Clifford spaces.
Lefrancois, Daniel; Wormit, Michael; Dreuw, Andreas
2015-09-28
For the investigation of molecular systems with electronic ground states exhibiting multi-reference character, a spin-flip (SF) version of the algebraic diagrammatic construction (ADC) scheme for the polarization propagator up to third order perturbation theory (SF-ADC(3)) is derived via the intermediate state representation and implemented into our existing ADC computer program adcman. The accuracy of these new SF-ADC(n) approaches is tested on typical situations, in which the ground state acquires multi-reference character, like bond breaking of H{sub 2} and HF, the torsional motion of ethylene, and the excited states of rectangular and square-planar cyclobutadiene. Overall, the results of SF-ADC(n) reveal an accurate description of these systems in comparison with standard multi-reference methods. Thus, the spin-flip versions of ADC are easy-to-use methods for the calculation of “few-reference” systems, which possess a stable single-reference triplet ground state.
An eigenvalue/eigenvector assignment algorithm using output feedback
NASA Technical Reports Server (NTRS)
Mielke, R. R.; Liberty, S. R.
1983-01-01
An eigenvalue/eigenvector assignment algorithm using stationary output feedback is presented. The algorithm permits assignment of min (n, m + r - 1) eigenvalues and max (m-1, r-1) eigenvectors, where n, m, r refer to the system state, input and output dimensions, respectively. An example is given to illustrate the design procedures.
ERIC Educational Resources Information Center
Merlin, Ethan M.
2013-01-01
This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this…
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
Algebraic models of flexible manufacturing systems
NASA Astrophysics Data System (ADS)
Leskin, Aleksei Alekseevich
Various aspects of the use of mathematical methods in the development of flexible manufacturing systems are examined. Attention is given to dynamical and structural models of flexible manufacturing systems developed by using methods of algebraic and differential geometry, topology, polynomial algebra, and extreme value problem theory. The principles of model integration are discussed, and approaches are proposed for solving problems related to the selection of flexible manufacturing equipment, real-time modeling of the manufacturing process, and optimization of local automation systems. The discussion is illustrated by examples.
A Lanczos eigenvalue method on a parallel computer
NASA Technical Reports Server (NTRS)
Bostic, Susan W.; Fulton, Robert E.
1987-01-01
Eigenvalue analyses of complex structures is a computationally intensive task which can benefit significantly from new and impending parallel computers. This study reports on a parallel computer implementation of the Lanczos method for free vibration analysis. The approach used here subdivides the major Lanczos calculation tasks into subtasks and introduces parallelism down to the subtask levels such as matrix decomposition and forward/backward substitution. The method was implemented on a commercial parallel computer and results were obtained for a long flexible space structure. While parallel computing efficiency for the Lanczos method was good for a moderate number of processors for the test problem, the greatest reduction in time was realized for the decomposition of the stiffness matrix, a calculation which took 70 percent of the time in the sequential program and which took 25 percent of the time on eight processors. For a sample calculation of the twenty lowest frequencies of a 486 degree of freedom problem, the total sequential computing time was reduced by almost a factor of ten using 16 processors.
Applications of the ArbiTER edge plasma eigenvalue code
NASA Astrophysics Data System (ADS)
Baver, D. A.; Myra, J. R.; Umansky, M. V.
2013-10-01
ArbiTER is a flexible eigenvalue code designed for plasma physics applications. This code uses an equation and topology parser to determine how a particular set of linearized model equations is spatially discretized. The resulting matrix form is then solved using the SLEPc eigensolver package. The equation and topology parsers permit a wide variety of capabilities, including variable numbers of dimensions, both finite difference and finite element methods, and irregular boundary conditions. Recent upgrades also permit parallel operation and the solution of source-driven problems. Two applications of this code will be presented, both as demonstrations of capability and as benchmark cases. One of these is the calculation of resistive ballooning modes with fully kinetic electrons. This will demonstrate the capacity for solving kinetic problems. The other is the use of extended spatial domains for ballooning stability analysis. This will demonstrate the utility of extra dimensions in calculations with fluid models. Work supported by the U.S. DOE grant DE-SC0006562.
Infinitesimal deformations of filiform Lie algebras of order 3
NASA Astrophysics Data System (ADS)
Navarro, R. M.
2015-12-01
The Lie algebras of order F have important applications for the fractional supersymmetry, and on the other hand the filiform Lie (super)algebras have very important properties into the Lie Theory. Thus, the aim of this work is to study filiform Lie algebras of order F which were introduced in Navarro (2014). In this work we obtain new families of filiform Lie algebras of order 3, in which the complexity of the problem rises considerably respecting to the cases considered in Navarro (2014).
NASA Astrophysics Data System (ADS)
Senovilla, José M. M.
2010-11-01
The algebraic classification of the Weyl tensor in the arbitrary dimension n is recovered by means of the principal directions of its 'superenergy' tensor. This point of view can be helpful in order to compute the Weyl aligned null directions explicitly, and permits one to obtain the algebraic type of the Weyl tensor by computing the principal eigenvalue of rank-2 symmetric future tensors. The algebraic types compatible with states of intrinsic gravitational radiation can then be explored. The underlying ideas are general, so that a classification of arbitrary tensors in the general dimension can be achieved.
Semigroups and computer algebra in algebraic structures
NASA Astrophysics Data System (ADS)
Bijev, G.
2012-11-01
Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.
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.
Numerical computations of interior transmission eigenvalues for scattering objects with cavities
NASA Astrophysics Data System (ADS)
Peters, Stefan; Kleefeld, Andreas
2016-04-01
In this article we extend the inside-outside duality for acoustic transmission eigenvalue problems by allowing scattering objects that may contain cavities. In this context we provide the functional analytical framework necessary to transfer the techniques that have been used in Kirsch and Lechleiter (2013 Inverse Problems, 29 104011) to derive the inside-outside duality. Additionally, extensive numerical results are presented to show that we are able to successfully detect interior transmission eigenvalues with the inside-outside duality approach for a variety of obstacles with and without cavities in three dimensions. In this context, we also discuss the advantages and disadvantages of the inside-outside duality approach from a numerical point of view. Furthermore we derive the integral equations necessary to extend the algorithm in Kleefeld (2013 Inverse Problems, 29 104012) to compute highly accurate interior transmission eigenvalues for scattering objects with cavities, which we will then use as reference values to examine the accuracy of the inside-outside duality algorithm.
Robbins algebra : conditions that make a near-Boolean algebra Boolean.
Winker, S.; Mathematics and Computer Science
1990-01-01
Some problems posed years ago remain challenging today. In particular, the Robbins problem, which is still open and which is the focus of attention in this paper, offers interesting challenges for attack with the assistance of an automated reasoning program; for the study presented here, we used the program OTTER. For example, when one submits this problem, which asks for a proof that every Robbins algebra is a Boolean algebra, a large number of deduced clauses results. One must, therefore, consider the possibility that there exists a Robbins algebra that is not Boolean; such an algebra would have to be infinite. One can instead search for properties that, if adjoined to those of a Robbins algebra, guarantee that the algebra is Boolean. Here we present a number of such properties, and we show how an automated reasoning program was used to obtain the corresponding proofs. Additional properties have been identified, and we include here examples of using such a program to check that the corresponding hand-proofs are correct. We present the appropriate input for many of the examples and also include the resulting proofs in clause notation.
Characterizing repulsive gravity with curvature eigenvalues
NASA Astrophysics Data System (ADS)
Luongo, Orlando; Quevedo, Hernando
2014-10-01
Repulsive gravity has been investigated in several scenarios near compact objects by using different intuitive approaches. Here, we propose an invariant method to characterize regions of repulsive gravity, associated to black holes and naked singularities. Our method is based upon the behavior of the curvature tensor eigenvalues, and leads to an invariant definition of a repulsion radius. The repulsion radius determines a physical region, which can be interpreted as a repulsion sphere, where the effects due to repulsive gravity naturally arise. Further, we show that the use of effective masses to characterize repulsion regions can lead to coordinate-dependent results whereas, in our approach, repulsion emerges as a consequence of the spacetime geometry in a completely invariant way. Our definition is tested in the spacetime of an electrically charged Kerr naked singularity and in all its limiting cases. We show that a positive mass can generate repulsive gravity if it is equipped with an electric charge or an angular momentum. We obtain reasonable results for the spacetime regions contained inside the repulsion sphere whose size and shape depend on the value of the mass, charge and angular momentum. Consequently, we define repulsive gravity as a classical relativistic effect by using the geometry of spacetime only.
Eigenvalue assignment strategies in rotor systems
NASA Technical Reports Server (NTRS)
Youngblood, J. N.; Welzyn, K. J.
1986-01-01
The work done to establish the control and direction of effective eigenvalue excursions of lightly damped, speed dependent rotor systems using passive control is discussed. Both second order and sixth order bi-axis, quasi-linear, speed dependent generic models were investigated. In every case a single, bi-directional control bearing was used in a passive feedback stabilization loop to resist modal destabilization above the rotor critical speed. Assuming incomplete state measurement, sub-optimal control strategies were used to define the preferred location of the control bearing, the most effective measurement locations, and the best set of control gains to extend the speed range of stable operation. Speed dependent control gains were found by Powell's method to maximize the minimum modal damping ratio for the speed dependent linear model. An increase of 300 percent in stable speed operation was obtained for the sixth order linear system using passive control. Simulations were run to examine the effectiveness of the linear control law on nonlinear rotor models with bearing deadband. The maximum level of control effort (force) required by the control bearing to stabilize the rotor at speeds above the critical was determined for the models with bearing deadband.
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.
Four Lie algebras associated with R6 and their applications
NASA Astrophysics Data System (ADS)
Zhang, Yufeng; Tam, Honwah
2010-09-01
The first part in the paper reads that a three-dimensional Lie algebra is first introduced, whose corresponding loop algebra is constructed, for which isospectral problems are established. By employing zero curvature equations, a modified Kaup-Newell (mKN) soliton hierarchy of evolution equations is obtained. The corresponding hereditary operator and Hamiltonian structure are worked out, respectively. Then two types of enlarging semisimple Lie algebras isomorphic to the linear space R6 are followed to construct, one of them is a complex Lie algebra. Their corresponding loop algebras are also given so that two types of new isospectral problems are introduced to generate two kinds of integrable couplings of the above mKN hierarchy. The hereditary operators, Hamiltonian structures of the hierarchies are produced again, respectively. The exact computing formulas of the constant γ appearing in the trace identity and the variational identity are derived under the semisimple algebras. The second part of this paper is devoted to constructing two kinds of Lie algebras by using product of complex vectors, which are also isomorphic to the linear space R6. Then we make use of the corresponding loop algebras to produce two integrable hierarchies along with bi-Hamiltonian structures. From various aspects, we give some ways for constructing Lie algebras which have extensive applications in generating integrable Hamiltonian systems.
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.
NASA Astrophysics Data System (ADS)
Kuzmin, Dmitri; Möller, Matthias; Gurris, Marcel
Flux limiting for hyperbolic systems requires a careful generalization of the design principles and algorithms introduced in the context of scalar conservation laws. In this chapter, we develop FCT-like algebraic flux correction schemes for the Euler equations of gas dynamics. In particular, we discuss the construction of artificial viscosity operators, the choice of variables to be limited, and the transformation of antidiffusive fluxes. An a posteriori control mechanism is implemented to make the limiter failsafe. The numerical treatment of initial and boundary conditions is discussed in some detail. The initialization is performed using an FCT-constrained L 2 projection. The characteristic boundary conditions are imposed in a weak sense, and an approximate Riemann solver is used to evaluate the fluxes on the boundary. We also present an unconditionally stable semi-implicit time-stepping scheme and an iterative solver for the fully discrete problem. The results of a numerical study indicate that the nonlinearity and non-differentiability of the flux limiter do not inhibit steady state convergence even in the case of strongly varying Mach numbers. Moreover, the convergence rates improve as the pseudo-time step is increased.
The first eigenvalue of the p- Laplacian on quantum graphs
NASA Astrophysics Data System (ADS)
Del Pezzo, Leandro M.; Rossi, Julio D.
2016-01-01
We study the first eigenvalue of the p- Laplacian (with 1
eigenvalue when we prescribe the total sum of the lengths of the edges and the number of Dirichlet nodes of the graph. Also we find a formula for the shape derivative of the first eigenvalue (assuming that it is simple) when we perturb the graph by changing the length of an edge. Finally, we study in detail the limit cases p→ ∞ and p→ 1.
Coreflections in Algebraic Quantum Logic
NASA Astrophysics Data System (ADS)
Jacobs, Bart; Mandemaker, Jorik
2012-07-01
Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.
Not each sequential effect algebra is sharply dominating
NASA Astrophysics Data System (ADS)
Shen, Jun; Wu, Junde
2009-04-01
Let E be an effect algebra and E be the set of all sharp elements of E. E is said to be sharply dominating if for each a∈E there exists a smallest element aˆ∈E such that a⩽aˆ. In 2002, Professors Gudder and Greechie proved that each σ-sequential effect algebra is sharply dominating. In 2005, Professor Gudder presented 25 open problems in [S. Gudder, Int. J. Theory Phys. 44 (2005) 2219], the 3rd problem asked: Is each sequential effect algebra sharply dominating? Now, we construct an example to answer the problem negatively.
NASA Astrophysics Data System (ADS)
Lannes, A.; Teunissen, P. J. G.
2011-05-01
The first objective of this paper is to show that some basic concepts used in global navigation satellite systems (GNSS) are similar to those introduced in Fourier synthesis for handling some phase calibration problems. In experimental astronomy, the latter are at the heart of what is called `phase closure imaging.' In both cases, the analysis of the related structures appeals to the algebraic graph theory and the algebraic number theory. For example, the estimable functions of carrier-phase ambiguities, which were introduced in GNSS to correct some rank defects of the undifferenced equations, prove to be `closure-phase ambiguities:' the so-called `closure-delay' (CD) ambiguities. The notion of closure delay thus generalizes that of double difference (DD). The other estimable functional variables involved in the phase and code undifferenced equations are the receiver and satellite pseudo-clock biases. A related application, which corresponds to the second objective of this paper, concerns the definition of the clock information to be broadcasted to the network users for their precise point positioning (PPP). It is shown that this positioning can be achieved by simply having access to the satellite pseudo-clock biases. For simplicity, the study is restricted to relatively small networks. Concerning the phase for example, these biases then include five components: a frequency-dependent satellite-clock error, a tropospheric satellite delay, an ionospheric satellite delay, an initial satellite phase, and an integer satellite ambiguity. The form of the PPP equations to be solved by the network user is then similar to that of the traditional PPP equations. As soon as the CD ambiguities are fixed and validated, an operation which can be performed in real time via appropriate decorrelation techniques, estimates of these float biases can be immediately obtained. No other ambiguity is to be fixed. The satellite pseudo-clock biases can thus be obtained in real time. This is
ERIC Educational Resources Information Center
Arnau, David; Arevalillo-Herraez, Miguel; Puig, Luis; Gonzalez-Calero, Jose Antonio
2013-01-01
Designers of interactive learning environments with a focus on word problem solving usually have to compromise between the amount of resolution paths that a user is allowed to follow and the quality of the feedback provided. We have built an intelligent tutoring system (ITS) that is able to both track the user's actions and provide adequate…
The Taylor spectrum and transversality for a Heisenberg algebra of operators
Dosi, Anar A
2010-05-11
A problem on noncommutative holomorphic functional calculus is considered for a Banach module over a finite-dimensional nilpotent Lie algebra. As the main result, the transversality property of algebras of noncommutative holomorphic functions with respect to the Taylor spectrum is established for a family of bounded linear operators generating a Heisenberg algebra. Bibliography: 25 titles.
The Algebra of Lexical Semantics
NASA Astrophysics Data System (ADS)
Kornai, András
The current generative theory of the lexicon relies primarily on tools from formal language theory and mathematical logic. Here we describe how a different formal apparatus, taken from algebra and automata theory, resolves many of the known problems with the generative lexicon. We develop a finite state theory of word meaning based on machines in the sense of Eilenberg [11], a formalism capable of describing discrepancies between syntactic type (lexical category) and semantic type (number of arguments). This mechanism is compared both to the standard linguistic approaches and to the formalisms developed in AI/KR.
Algebraic structure of general electromagnetic fields and energy flow
Hacyan, Shahen
2011-08-15
Highlights: > Algebraic structure of general electromagnetic fields in stationary spacetime. > Eigenvalues and eigenvectors of the electomagnetic field tensor. > Energy-momentum in terms of eigenvectors and Killing vector. > Explicit form of reference frame with vanishing Poynting vector. > Application of formalism to Bessel beams. - Abstract: The algebraic structures of a general electromagnetic field and its energy-momentum tensor in a stationary space-time are analyzed. The explicit form of the reference frame in which the energy of the field appears at rest is obtained in terms of the eigenvectors of the electromagnetic tensor and the existing Killing vector. The case of a stationary electromagnetic field is also studied and a comparison is made with the standard short-wave approximation. The results can be applied to the general case of a structured light beams, in flat or curved spaces. Bessel beams are worked out as example.
Developing Algebraic Thinking.
ERIC Educational Resources Information Center
Alejandre, Suzanne
2002-01-01
Presents a teaching experience that resulted in students getting to a point of full understanding of the kinesthetic activity and the algebra behind it. Includes a lesson plan for a traffic jam activity. (KHR)
Algebraic 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
Algebraic Semantics for Narrative
ERIC Educational Resources Information Center
Kahn, E.
1974-01-01
This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)
Aprepro - Algebraic Preprocessor
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.
Network extreme eigenvalue: From mutimodal to scale-free networks
NASA Astrophysics Data System (ADS)
Chung, N. N.; Chew, L. Y.; Lai, C. H.
2012-03-01
The extreme eigenvalues of adjacency matrices are important indicators on the influence of topological structures to the collective dynamical behavior of complex networks. Recent findings on the ensemble averageability of the extreme eigenvalue have further authenticated its applicability to the study of network dynamics. However, the ensemble average of extreme eigenvalue has only been solved analytically up to the second order correction. Here, we determine the ensemble average of the extreme eigenvalue and characterize its deviation across the ensemble through the discrete form of random scale-free network. Remarkably, the analytical approximation derived from the discrete form shows significant improvement over previous results, which implies a more accurate prediction of the epidemic threshold. In addition, we show that bimodal networks, which are more robust against both random and targeted removal of nodes, are more vulnerable to the spreading of diseases.
NASA Astrophysics Data System (ADS)
Ossandón, Sebastián; Reyes, Camilo
2016-02-01
A new numerical method is presented with the purpose to calculate the Lamé coefficients, associated with an elastic material, through eigenvalues of the elasticity operator. The finite element method is used to solve repeatedly, using different Lamé coefficients values, the direct problem by training a direct radial basis neural network. A map of eigenvalues, as a function of the Lamé constants, is then obtained. This relationship is later inverted and refined by training an inverse radial basis neural network, allowing calculation of mentioned coefficients. A numerical example is presented to prove the effectiveness of this novel method.
Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice
NASA Astrophysics Data System (ADS)
Sasaki, Ryu
2014-06-01
We address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice. As Kay and Moses showed in 1956, reflectionless potentials in one dimensional quantum mechanics are exactly solvable. With an additional time dependence these potentials are identified as the soliton solutions of the Korteweg de Vries (KdV) hierarchy. An N-soliton potential has the time t and 2N positive parameters, k1 < ⋯ < kN and {cj}, j = 1, …, N, corresponding to N discrete eigenvalues lbrace -k_j^2rbrace. The eigenfunctions are elementary functions expressed by the ratio of determinants. The Darboux-Crum-Krein-Adler transformations or the Abraham-Moses transformations based on eigenfunction deletions produce lower soliton number potentials with modified parameters lbrace c^' }_jrbrace. We explore various identities satisfied by the eigenfunctions of the soliton potentials, which reflect the uniqueness theorem of Gel'fand-Levitan-Marchenko equations for separable (degenerate) kernels.
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.
Quantum eigenvalue estimation for irreducible non-negative matrices
NASA Astrophysics Data System (ADS)
Daskin, Anmer
2016-04-01
Quantum phase estimation algorithm (PEA) has been successfully adapted as a sub frame of many other algorithms applied to a wide variety of applications in different fields. However, the requirement of a good approximate eigenvector given as an input to the algorithm hinders the application of the algorithm to the problems where we do not have any prior knowledge about the eigenvector. In this paper, we show that the principal eigenvalue of an irreducible non-negative operator can be determined by using an equal superposition initial state in the PEA. This removes the necessity of the existence of an initial good approximate eigenvector. Moreover, we show that the success probability of the algorithm is related to the closeness of the operator to a stochastic matrix. Therefore, we draw an estimate for the success probability by using the variance of the column sums of the operator. This provides a priori information which can be used to know the success probability of the algorithm beforehand for the non-negative matrices and apply the algorithm only in cases when the estimated probability is reasonably high. Finally, we discuss the possible applications and show the results for random symmetric matrices and 3-local Hamiltonians with non-negative off-diagonal elements.
Overview of the ArbiTER edge plasma eigenvalue code
NASA Astrophysics Data System (ADS)
Baver, Derek; Myra, James; Umansky, Maxim
2011-10-01
The Arbitrary Topology Equation Reader, or ArbiTER, is a flexible eigenvalue solver that is currently under development for plasma physics applications. The ArbiTER code builds on the equation parser framework of the existing 2DX code, extending it to include a topology parser. This will give the code the capability to model problems with complicated geometries (such as multiple X-points and scrape-off layers) or model equations with arbitrary numbers of dimensions (e.g. for kinetic analysis). In the equation parser framework, model equations are not included in the program's source code. Instead, an input file contains instructions for building a matrix from profile functions and elementary differential operators. The program then executes these instructions in a sequential manner. These instructions may also be translated into analytic form, thus giving the code transparency as well as flexibility. We will present an overview of how the ArbiTER code is to work, as well as preliminary results from early versions of this code. Work supported by the U.S. DOE.
PERTURB: A program for calculating vibrational energies by generalized algebraic quantization
NASA Astrophysics Data System (ADS)
Fried, Laurence E.; Ezra, Gregory S.
1988-09-01
We describe PERTURB, a special purpose algebraic manipulation program which calculates vibrational eigenvalues in coupled oscillator systems. PERTURB implements the method of generalized algebraic quantization (AQ), in which Van Vleck perturbation theory is formulated in a mock phase space. The phase space formulation enables quantum and classical perturbation theory to be treated on the same footing, and allows the systematic calculation of corrections to classical perturbation results in powers of h̷. Generalized AQ is a powerful and efficient technique for calculating semiclassical vibrational energy levels. In many cases, including just the first correction to classical perturbation theory yields highly accurate energies.
DG Poisson algebra and its universal enveloping algebra
NASA Astrophysics Data System (ADS)
Lü, JiaFeng; Wang, XingTing; Zhuang, GuangBin
2016-05-01
In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.
Approximating the largest eigenvalue of network adjacency matrices
NASA Astrophysics Data System (ADS)
Restrepo, Juan G.; Ott, Edward; Hunt, Brian R.
2007-11-01
The largest eigenvalue of the adjacency matrix of a network plays an important role in several network processes (e.g., synchronization of oscillators, percolation on directed networks, and linear stability of equilibria of network coupled systems). In this paper we develop approximations to the largest eigenvalue of adjacency matrices and discuss the relationships between these approximations. Numerical experiments on simulated networks are used to test our results.
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…
Parallel Algebraic Multigrids for Structural mechanics
Brezina, M; Tong, C; Becker, R
2004-05-11
This paper presents the results of a comparison of three parallel algebraic multigrid (AMG) preconditioners for structural mechanics applications. In particular, they are interested in investigating both the scalability and robustness of the preconditioners. Numerical results are given for a range of structural mechanics problems with various degrees of difficulty.
Algebra 1Q, Mathematics: 5215.12.
ERIC Educational Resources Information Center
Hirigoyen, Hector
This is the second of the six guidebooks on minimum course content for first-year algebra; it includes the ordered field properties of the real number system, solution of linear equations and inequalities, verbal problems, exponents and operations with polynomials. Overall goals for the course are stated; performance objectives for each unit, a…
Pauli spinors and Hestenes' geometric algebra
NASA Astrophysics Data System (ADS)
Hamilton, J. Dwayne
1984-01-01
Hestenes' geometric algebra and Pauli's two-component spinors are reviewed and are united into a simple mathematical system. The resulting formalism is used to develop a new method for spin 1/2 projection calculations and is also applied to a spin 1/2 electron magnetic resonance problem.
NASA Astrophysics Data System (ADS)
Ott, Edward; Pomerance, Andrew
2009-05-01
Motivated by its relevance to various types of dynamical behavior of network systems, the maximum eigenvalue λA of the adjacency matrix A of a network has been considered and mean-field-type approximations to λA have been developed for different kinds of networks. Here A is defined by Aij=1 (Aij=0) if there is (is not) a directed network link to i from j . However, in at least two recent problems involving networks with heterogeneous node properties (percolation on a directed network and the stability of Boolean models of gene networks), an analogous but different eigenvalue problem arises, namely, that of finding the largest eigenvalue λQ of the matrix Q , where Qij=qiAij and the “bias” qi may be different at each node i . (In the previously mentioned percolation and gene network contexts, qi is a probability and so lies in the range 0≤qi≤1 .) The purposes of this paper are to extend the previous considerations of the maximum eigenvalue λA of A to λQ , to develop suitable analytic approximations to λQ , and to test these approximations with numerical experiments. In particular, three issues considered are (i) the effect of the correlation (or anticorrelation) between the value of qi and the number of links to and from node i , (ii) the effect of correlation between the properties of two nodes at either end of a network link (“assortativity”), and (iii) the effect of community structure allowing for a situation in which different q values are associated with different communities.
Ott, Edward; Pomerance, Andrew
2009-05-01
Motivated by its relevance to various types of dynamical behavior of network systems, the maximum eigenvalue lambdaA of the adjacency matrix A of a network has been considered and mean-field-type approximations to lambdaA have been developed for different kinds of networks. Here A is defined by Aij=1 (Aij=0) if there is (is not) a directed network link to i from j. However, in at least two recent problems involving networks with heterogeneous node properties (percolation on a directed network and the stability of Boolean models of gene networks), an analogous but different eigenvalue problem arises, namely, that of finding the largest eigenvalue lambdaQ of the matrix Q, where Qij=qiAij and the "bias" qi may be different at each node i. (In the previously mentioned percolation and gene network contexts, qi is a probability and so lies in the range 0
SCALE Continuous-Energy Eigenvalue Sensitivity Coefficient Calculations
Perfetti, Christopher M.; Rearden, Bradley T.; Martin, William R.
2016-02-25
Sensitivity coefficients describe the fractional change in a system response that is induced by changes to system parameters and nuclear data. The Tools for Sensitivity and UNcertainty Analysis Methodology Implementation (TSUNAMI) code within the SCALE code system makes use of eigenvalue sensitivity coefficients for an extensive number of criticality safety applications, including quantifying the data-induced uncertainty in the eigenvalue of critical systems, assessing the neutronic similarity between different critical systems, and guiding nuclear data adjustment studies. The need to model geometrically complex systems with improved fidelity and the desire to extend TSUNAMI analysis to advanced applications has motivated the developmentmore » of a methodology for calculating sensitivity coefficients in continuous-energy (CE) Monte Carlo applications. The Contributon-Linked eigenvalue sensitivity/Uncertainty estimation via Tracklength importance CHaracterization (CLUTCH) and Iterated Fission Probability (IFP) eigenvalue sensitivity methods were recently implemented in the CE-KENO framework of the SCALE code system to enable TSUNAMI-3D to perform eigenvalue sensitivity calculations using continuous-energy Monte Carlo methods. This work provides a detailed description of the theory behind the CLUTCH method and describes in detail its implementation. This work explores the improvements in eigenvalue sensitivity coefficient accuracy that can be gained through the use of continuous-energy sensitivity methods and also compares several sensitivity methods in terms of computational efficiency and memory requirements.« less
Piecewise lexsegment ideals in exterior algebras
NASA Astrophysics Data System (ADS)
Shakin, D. A.
2005-02-01
The problem of describing the Hilbert functions of homogeneous ideals of an exterior algebra over a field containing a fixed monomial ideal I is considered. For this purpose the notion of a piecewise lexsegment ideal in an exterior algebra is introduced generalizing the notion of a lexsegment ideal. It is proved that if I is a piecewise lexsegment ideal, then it is possible to describe the Hilbert functions of the homogeneous ideals containing I in a way similar to that suggested by Kruskal and Katona for the situation I=0. Moreover, a generalization of the extremal properties of lexsegment ideals is obtained (the inequality for the Betti numbers).
SLAPP: A systolic linear algebra parallel processor
Drake, B.L.; Luk, F.T.; Speiser, J.M.; Symanski, J.J.
1987-07-01
Systolic array computer architectures provide a means for fast computation of the linear algebra algorithms that form the building blocks of many signal-processing algorithms, facilitating their real-time computation. For applications to signal processing, the systolic array operates on matrices, an inherently parallel view of the data, using numerical linear algebra algorithms that have been suitably parallelized to efficiently utilize the available hardware. This article describes work currently underway at the Naval Ocean Systems Center, San Diego, California, to build a two-dimensional systolic array, SLAPP, demonstrating efficient and modular parallelization of key matric computations for real-time signal- and image-processing problems.
The Dirac equation and Hestenes' geometric algebra
NASA Astrophysics Data System (ADS)
Hamilton, J. Dwayne
1984-06-01
Hestenes' geometric algebra and Dirac spinors are reviewed and united into a common mathematical formalism, a unification that establishes the Dirac equation as being manifestly covariant under the Lorentz group, and one that needs no matrix representation of the Dirac algebra. New and simple methods of amplitude or ``trace'' calculations are then described. A number of problems are then considered within the context of the new approach, such as relativistic spin projections, new and covariant C and T-transformations and spinors for massless and Majorana fields.
Algebraic Bethe ansatz for Q-operators: the Heisenberg spin chain
NASA Astrophysics Data System (ADS)
Frassek, Rouven
2015-07-01
We diagonalize Q-operators for rational homogeneous {sl}(2)-invariant Heisenberg spin chains using the algebraic Bethe ansatz. After deriving the fundamental commutation relations relevant for this case from the Yang-Baxter equation we demonstrate that the Q-operators act diagonally on the Bethe vectors if the Bethe equations are satisfied. In this way we provide a direct proof that the eigenvalues of the Q-operators studied here are given by Baxter's Q-functions.
Quantum integrable systems, non-skew-symmetric r-matrices and algebraic Bethe ansatz
Skrypnyk, T.
2007-02-15
We prove the integrability of the general quantum Hamiltonian systems governed by an arbitrary non-skew-symmetric, so(3)-valued, nondynamical classical r-matrix with spectral parameters. We consider the most interesting example of these quantum integrable systems, namely, the so(3) 'generalized Gaudin systems' in detail. In the case of an arbitrary r-matrix which is 'diagonal' in the sl(2) basis we calculate the spectrum and the eigenvalues of the corresponding Hamiltonians using the algebraic Bethe ansatz technique.
Clifford Algebra Cℓ 3(ℂ) for Applications to Field Theories
NASA Astrophysics Data System (ADS)
Panicaud, B.
2011-10-01
The multivectorial algebras present yet both an academic and a technological interest. Difficulties can occur for their use. Indeed, in all applications care is taken to distinguish between polar and axial vectors and between scalars and pseudo scalars. Then a total of eight elements are often considered even if they are not given the correct name of multivectors. Eventually because of their simplicity, only the vectorial algebra or the quaternions algebra are explicitly used for physical applications. Nevertheless, it should be more convenient to use directly more complex algebras in order to have a wider range of application. The aim of this paper is to inquire into one particular Clifford algebra which could solve this problem. The present study is both didactic concerning its construction and pragmatic because of the introduced applications. The construction method is not an original one. But this latter allows to build up the associated real algebra as well as a peculiar formalism that enables a formal analogy with the classical vectorial algebra. Finally several fields of the theoretical physics will be described thanks to this algebra, as well as a more applied case in general relativity emphasizing simultaneously its relative validity in this particular domain and the easiness of modeling some physical problems.
An eigenvalue correction due to scattering by a rough wall of an acoustic waveguide.
Krynkin, Anton; Horoshenkov, Kirill V; Tait, Simon J
2013-08-01
In this paper a derivation of the attenuation factor in a waveguide with stochastic walls is presented. The perturbation method and Fourier analysis are employed to derive asymptotically consistent boundary-value problems at each asymptotic order. The derived approximation predicts the attenuation of the propagating mode in a rough waveguide through a correction to the eigenvalue corresponding to smooth walls. The proposed approach can be used to derive results that are consistent with those obtained by Bass et al. [IEEE Trans. Antennas Propag. 22, 278-288 (1974)]. The novelty of the method is that it does not involve the integral Dyson-type equation and, as a result, the large number of statistical moments included in the equation in the form of the mass operator of the volume scattering theory. The derived eigenvalue correction is described by the correlation function of the randomly rough surface. The averaged solution in the plane wave regime is approximated by the exponential function dependent on the derived eigenvalue correction. The approximations are compared with numerical results obtained using the finite element method (FEM). An approach to retrieve the correct deviation in roughness height and correlation length from multiple numerical realizations of the stochastic surface is proposed to account for the oversampling of the rough surface occurring in the FEM meshing procedure. PMID:23927093
Parameter estimation of structural dynamic models using eigenvalue and eigenvector information
Allen, J.J.; Martinez, D.R.
1990-11-01
Structural system identification methods are analytical techniques for reconciling test data with analytical models. The response data frequently used to compare a finite element model and test data are the eigenvalues of the system. However, eigenvalues alone cannot assure an adequate model. Eigenvectors also provide valuable information for the process of updating finite element models. For large order, complex finite element models, ad-hoc procedures have often proven inadequate for model parameter updating. Therefore, parameter estimation techniques such as Bayes estimation or mathematical programming have been applied. Mathematical programming techniques can be use for parameter estimation allowing a very general definition of the objective function and constraints. This paper will present the application of mathematical programming techniques of parameter estimation to the updating of a finite element model of an electronic package. The following topics will be discussed in the paper. The mathematical programming formulation of the parameter estimation problem, which uses both eigenvalue and eigenvector response data. The software implementation of this technique. The application of this methodology to the estimation of parameters of an electronics package model.
NASA Astrophysics Data System (ADS)
Roitman, Michael
2008-08-01
In this paper we prove that for any commutative (but in general non-associative) algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V0 Å V2 Å V3 Å ¼, such that dim V0 = 1 and V2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
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.
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…
Unstable transient response of gyroscopic systems with stable eigenvalues
NASA Astrophysics Data System (ADS)
Giannini, O.
2016-06-01
Gyroscopic conservative dynamical systems may exhibit flutter instability that leads to a pair of complex conjugate eigenvalues, one of which has a positive real part and thus leads to a divergent free response of the system. When dealing with non-conservative systems, the pitch fork bifurcation shifts toward the negative real part of the root locus, presenting a pair of eigenvalues with equal imaginary parts, while the real parts may or may not be negative. Several works study the stability of these systems for relevant engineering applications such as the flutter in airplane wings or suspended bridges, brake squeal, etc. and a common approach to detect the stability is the complex eigenvalue analysis that considers systems with all negative real part eigenvalues as stable systems. This paper studies analytically and numerically the cases where the free response of these systems exhibits a transient divergent time history even if all the eigenvalues have negative real part thus usually considered as stable, and relates such a behaviour to the non orthogonality of the eigenvectors. Finally, a numerical method to evaluate the presence of such instability is proposed.
NASA Astrophysics Data System (ADS)
Jönsthövel, T. B.; van Gijzen, M. B.; MacLachlan, S.; Vuik, C.; Scarpas, A.
2012-09-01
Many applications in computational science and engineering concern composite materials, which are characterized by large discontinuities in the material properties. Such applications require fine-scale finite-element meshes, which lead to large linear systems that are challenging to solve with current direct and iterative solutions algorithms. In this paper, we consider the simulation of asphalt concrete, which is a mixture of components with large differences in material stiffness. The discontinuities in material stiffness give rise to many small eigenvalues that negatively affect the convergence of iterative solution algorithms such as the preconditioned conjugate gradient (PCG) method. This paper considers the deflated preconditioned conjugate gradient (DPCG) method in which the rigid body modes of sets of elements with homogeneous material properties are used as deflation vectors. As preconditioner we consider several variants of the algebraic multigrid smoothed aggregation method. We evaluate the performance of the DPCG method on a parallel computer using up to 64 processors. Our test problems are derived from real asphalt core samples, obtained using CT scans. We show that the DPCG method is an efficient and robust technique for solving these challenging linear systems.
Computer Program For Linear Algebra
NASA Technical Reports Server (NTRS)
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
NASA Technical Reports Server (NTRS)
Shahshahani, M.
1991-01-01
The performance characteristics are discussed of certain algebraic geometric codes. Algebraic geometric codes have good minimum distance properties. On many channels they outperform other comparable block codes; therefore, one would expect them eventually to replace some of the block codes used in communications systems. It is suggested that it is unlikely that they will become useful substitutes for the Reed-Solomon codes used by the Deep Space Network in the near future. However, they may be applicable to systems where the signal to noise ratio is sufficiently high so that block codes would be more suitable than convolutional or concatenated codes.
NASA Astrophysics Data System (ADS)
Bouwknegt, Peter
1988-06-01
We investigate extensions of the Virasoro algebra by a single primary field of integer or halfinteger conformal dimension Δ. We argue that for vanishing structure constant CΔΔΔ, the extended conformal algebra can only be associative for a generic c-value if Δ=1/2, 1, 3/2, 2 or 3. For the other Δ<=5 we compute the finite set of allowed c-values and identify the rational solutions. The case CΔΔΔ≠0 is also briefly discussed. I would like to thank Kareljan Schoutens for discussions and Sander Bais for a careful reading of the manuscript.
Fibonacci's Triangle: A Vehicle for Problem Solving.
ERIC Educational Resources Information Center
Ouellette, Hugh
1979-01-01
A method for solving certain types of problems is illustrated by problems related to Fibonacci's triangle. The method involves pattern recognition, generalizing, algebraic manipulation, and mathematical induction. (MP)
Teaching Arithmetic and Algebraic Expressions
ERIC Educational Resources Information Center
Subramaniam, K.; Banerjee, Rakhi
2004-01-01
A teaching intervention study was conducted with sixth grade students to explore the interconnections between students' growing understanding of arithmetic expressions and beginning algebra. Three groups of students were chosen, with two groups receiving instruction in arithmetic and algebra, and one group in algebra without arithmetic. Students…
Assessing Elementary Algebra with STACK
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2007-01-01
This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…
Spinors in the hyperbolic algebra
NASA Astrophysics Data System (ADS)
Ulrych, S.
2006-01-01
The three-dimensional universal complex Clifford algebra Cbar3,0 is used to represent relativistic vectors in terms of paravectors. In analogy to the Hestenes spacetime approach spinors are introduced in an algebraic form. This removes the dependance on an explicit matrix representation of the algebra.
Situating the Debate on "Geometrical Algebra" within the Framework of Premodern Algebra.
Sialaros, Michalis; Christianidis, Jean
2016-06-01
Argument The aim of this paper is to employ the newly contextualized historiographical category of "premodern algebra" in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on "geometrical algebra." Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called "semi-algebraic" alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing "premodern algebra," and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition. PMID:27171890
ERIC Educational Resources Information Center
Boiteau, Denise; Stansfield, David
This document describes mathematical programs on the basic concepts of algebra produced by Louisiana Public Broadcasting. Programs included are: (1) "Inverse Operations"; (2) "The Order of Operations"; (3) "Basic Properties" (addition and multiplication of numbers and variables); (4) "The Positive and Negative Numbers"; and (5) "Using Positive…
Thinking Visually about Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2015-01-01
Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…
ERIC Educational Resources Information Center
Kennedy, John
This text provides information and exercises on arithmetic topics which should be mastered before a student enrolls in an Elementary Algebra course. Section I describes the fundamental properties and relationships of whole numbers, focusing on basic operations, divisibility tests, exponents, order of operations, prime numbers, greatest common…
ERIC Educational Resources Information Center
Nwabueze, Kenneth K.
2004-01-01
The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the…
Campoamor-Stursberg, R.
2008-05-15
By means of contractions of Lie algebras, we obtain new classes of indecomposable quasiclassical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise naturally from noncompact real simple algebras with nonsimple complexification, where we impose that a nondegenerate quadratic Casimir operator is preserved by the limiting process. We further consider the converse problem and obtain sufficient conditions on integrable cocycles of quasiclassical Lie algebras in order to preserve nondegenerate quadratic Casimir operators by the associated linear deformations.
Clusters of eigenvalues for the magnetic Laplacian with Robin condition
NASA Astrophysics Data System (ADS)
Goffeng, Magnus; Kachmar, Ayman; Persson Sundqvist, Mikael
2016-06-01
We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in Euclidean space. Functions in the domain of the operator are subject to a boundary condition of the third type (a magnetic Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which is independent of the boundary condition.
Laouar, A; Guerziz, A; Boussaha, A
2016-01-01
This paper focuses on the mathematical study of the existence of solitary gravity waves (solitons) and their characteristics (amplitude, velocity, [Formula: see text]) generated by a piston wave maker lying upstream of a horizontal channel. The mathematical model requires both incompressibility condition, irrotational flow of no viscous fluid and Lagrange coordinates. By using both the inverse scattering method and a given initial potential [Formula: see text] we can transform the KdV equation into Sturm-Liouville spectral problem. The latter problem amounts to find negative discrete eigenvalues [Formula: see text] and associated eigenfunctions [Formula: see text], where each calculated eigenvalue [Formula: see text] gives a soliton and the profile of the free surface. For solving this problem, we can use the Runge-Kutta method. For illustration, two examples of the wave maker movement are proposed. The numerical simulations show that the perturbation of wave maker with hyperbolic tangent displacement under physical conditions affect the number of solitons emitted. PMID:27606157
NASA Astrophysics Data System (ADS)
Zhang, Ming; Yao, JingTao
2004-04-01
The XML is a new standard for data representation and exchange on the Internet. There are studies on XML query languages as well as XML algebras in literature. However, attention has not been paid to research on XML algebras for data mining due to partially the fact that there is no widely accepted definition of XML mining tasks. This paper tries to examine the XML mining tasks and provide guidelines to design XML algebras for data mining. Some summarization and comparison have been done to existing XML algebras. We argue that by adding additional operators for mining tasks, XML algebras may work well for data mining with XML documents.
Algebraic Approach to the Computation of the Defining Polynomial of the Algebraic Riccati Equation
NASA Astrophysics Data System (ADS)
Kitamoto, Takuya
The algebraic Riccati equation, which we denote by ’ARE’ in the rest of the paper, is one of the most important equations of the post modern control theory. It plays important role for solving H 2 and H ∞ optimal control problems.
Analyzing modal behavior of guided waves using high order eigenvalue derivatives.
Krome, Fabian; Gravenkamp, Hauke
2016-09-01
This paper presents a mode-tracing approach for elastic guided waves based on analytically computed derivatives and includes a study of interesting phenomena in the dispersion curve representation. Numerical simulation is done by means of the Scaled Boundary Finite Element Method (SBFEM). Two approaches are used to identify the characteristics of the resulting wave modes: Taylor approximation and Padé approximation. Higher order differentials of the underlying eigenvalue problem are the basis for these approaches. Remarkable phenomena in potentially critical frequency regions are identified and the tracing approach is adapted to these regions. Additionally, a stabilization of the solution process is suggested. PMID:27286265
NASA Astrophysics Data System (ADS)
Colombo, V.; Ravetto, P.; Sumini, M.
1988-08-01
An approximate determination of the critical eigenvalue of the neutron transport equation in integral form, within both the one speed and energy multigroup models, for a homogeneous medium, is achieved by means of a variational technique. The space asymptotic solutions for both the direct and adjoint problems are used as trial functions. A variational procedure is also developed and numerically exploited within the Fourier transformed domain, where noticeable theoretical features can be demonstrated. It is evidenced that excellent results can be obtained with little computational effort, and a set of critical calculations in plane geometry is presented and discussed.
Colombo, V.; Ravetto, P.; Sumini, M.
1988-08-01
An approximate determination of the critical eigenvalue of the neutron transport equation in integral form, within both the one speed and energy multigroup models, for a homogeneous medium, is achieved by means of a variational technique. The space asymptotic solutions for both the direct and adjoint problems are used as trial functions. A variational procedure is also developed and numerically exploited within the Fourier transformed domain, where noticeable theoretical features can be demonstrated. It is evidenced that excellent results can be obtained with little computational effort, and a set of critical calculations in plane geometry is presented and discussed. copyright 1988 Academic Press, Inc.
Continuous-energy eigenvalue sensitivity coefficient calculations in TSUNAMI-3D
Perfetti, C. M.; Rearden, B. T.
2013-07-01
Two methods for calculating eigenvalue sensitivity coefficients in continuous-energy Monte Carlo applications were implemented in the KENO code within the SCALE code package. The methods were used to calculate sensitivity coefficients for several test problems and produced sensitivity coefficients that agreed well with both reference sensitivities and multigroup TSUNAMI-3D sensitivity coefficients. The newly developed CLUTCH method was observed to produce sensitivity coefficients with high figures of merit and a low memory footprint, and both continuous-energy sensitivity methods met or exceeded the accuracy of the multigroup TSUNAMI-3D calculations. (authors)
Development of a SCALE Tool for Continuous-Energy Eigenvalue Sensitivity Coefficient Calculations
Perfetti, Christopher M; Rearden, Bradley T
2013-01-01
Two methods for calculating eigenvalue sensitivity coefficients in continuous-energy Monte Carlo applications were implemented in the KENO code within the SCALE code package. The methods were used to calculate sensitivity coefficients for several criticality safety problems and produced sensitivity coefficients that agreed well with both reference sensitivities and multigroup TSUNAMI-3D sensitivity coefficients. The newly developed CLUTCH method was observed to produce sensitivity coefficients with high figures of merit and low memory requirements, and both continuous-energy sensitivity methods met or exceeded the accuracy of the multigroup TSUNAMI-3D calculations.
Development of a SCALE Tool for Continuous-Energy Eigenvalue Sensitivity Coefficient Calculations
NASA Astrophysics Data System (ADS)
Perfetti, Christopher M.; Rearden, Bradley T.
2014-06-01
Two methods for calculating eigenvalue sensitivity coefficients in continuous-energy Monte Carlo applications were implemented in the KENO code within the SCALE code package. The methods were used to calculate sensitivity coefficients for several criticality safety problems and produced sensitivity coefficients that agreed well with both reference sensitivities and multigroup TSUNAMI-3D sensitivity coefficients. The newly developed CLUTCH method was observed to produce sensitivity coefficients with high figures of merit and low memory requirements, and both continuous-energy sensitivity methods met or exceeded the accuracy of the multigroup TSUNAMI-3D calculations.
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.
On Dunkl angular momenta algebra
NASA Astrophysics Data System (ADS)
Feigin, Misha; Hakobyan, Tigran
2015-11-01
We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl( N ) version of the subalge-bra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.
Maximizing algebraic connectivity in air transportation networks
NASA Astrophysics Data System (ADS)
Wei, Peng
In air transportation networks the robustness of a network regarding node and link failures is a key factor for its design. An experiment based on the real air transportation network is performed to show that the algebraic connectivity is a good measure for network robustness. Three optimization problems of algebraic connectivity maximization are then formulated in order to find the most robust network design under different constraints. The algebraic connectivity maximization problem with flight routes addition or deletion is first formulated. Three methods to optimize and analyze the network algebraic connectivity are proposed. The Modified Greedy Perturbation Algorithm (MGP) provides a sub-optimal solution in a fast iterative manner. The Weighted Tabu Search (WTS) is designed to offer a near optimal solution with longer running time. The relaxed semi-definite programming (SDP) is used to set a performance upper bound and three rounding techniques are discussed to find the feasible solution. The simulation results present the trade-off among the three methods. The case study on two air transportation networks of Virgin America and Southwest Airlines show that the developed methods can be applied in real world large scale networks. The algebraic connectivity maximization problem is extended by adding the leg number constraint, which considers the traveler's tolerance for the total connecting stops. The Binary Semi-Definite Programming (BSDP) with cutting plane method provides the optimal solution. The tabu search and 2-opt search heuristics can find the optimal solution in small scale networks and the near optimal solution in large scale networks. The third algebraic connectivity maximization problem with operating cost constraint is formulated. When the total operating cost budget is given, the number of the edges to be added is not fixed. Each edge weight needs to be calculated instead of being pre-determined. It is illustrated that the edge addition and the
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
Reachability analysis of rational eigenvalue linear systems
NASA Astrophysics Data System (ADS)
Xu, Ming; Chen, Liangyu; Zeng, Zhenbing; Li, Zhi-bin
2010-12-01
One of the key problems in the safety analysis of control systems is the exact computation of reachable state spaces for continuous-time systems. Issues related to the controllability and observability of these systems are well-studied in systems theory. However, there are not many results on reachability, even for general linear systems. In this study, we present a large class of linear systems with decidable reachable state spaces. This is approached by reducing the reachability analysis to real root isolation of exponential polynomials. Furthermore, we have implemented this method in a Maple package based on symbolic computation and applied to several examples successfully.
Non-Real Eigenvalues for {{{PT}}} -Symmetric Double Wells
NASA Astrophysics Data System (ADS)
Benbernou, Amina; Boussekkine, Naima; Mecherout, Nawal; Ramond, Thierry; Sjöstrand, Johannes
2016-05-01
We study small, {{{PT}}} -symmetric perturbations of self-adjoint double-well Schrödinger operators in dimension {n≥ 1} . We prove that the eigenvalues stay real for a very small perturbation, then bifurcate to the complex plane as the perturbation gets stronger.
The Fourier analysis technique and epsilon-pseudo-eigenvalues
Donato, J.M.
1993-07-01
The spectral radii of iteration matrices and the spectra and condition numbers of preconditioned systems are important in forecasting the convergence rates for iterative methods. Unfortunately, the spectra of iteration matrices or preconditioned systems is rarely easily available. The Fourier analysis technique has been shown to be a useful tool in studying the effectiveness of iterative methods by determining approximate expressions for the eigenvalues or condition numbers of matrix systems. For non-symmetric matrices the eigenvalues may be highly sensitive to perturbations. The spectral radii of nonsymmetric iteration matrices may not give a numerically realistic indication of the convergence of the iterative method. Trefethen and others have presented a theory on the use of {epsilon}-pseudo-eigenvalues in the study of matrix equations. For Toeplitz matrices, we show that the theory of c-pseudo-eigenvalues includes the Fourier analysis technique as a limiting case. For non-Toeplitz matrices, the relationship is not clear. We shall examine this relationship for non-Toeplitz matrices that arise when studying preconditioned systems for methods applied to a two-dimensional discretized elliptic differential equation.
Error estimations and their biases in Monte Carlo eigenvalue calculations
Ueki, Taro; Mori, Takamasa; Nakagawa, Masayuki
1997-01-01
In the Monte Carlo eigenvalue calculation of neutron transport, the eigenvalue is calculated as the average of multiplication factors from cycles, which are called the cycle k{sub eff}`s. Biases in the estimators of the variance and intercycle covariances in Monte Carlo eigenvalue calculations are analyzed. The relations among the real and apparent values of variances and intercycle covariances are derived, where real refers to a true value that is calculated from independently repeated Monte Carlo runs and apparent refers to the expected value of estimates from a single Monte Carlo run. Next, iterative methods based on the foregoing relations are proposed to estimate the standard deviation of the eigenvalue. The methods work well for the cases in which the ratios of the real to apparent values of variances are between 1.4 and 3.1. Even in the case where the foregoing ratio is >5, >70% of the standard deviation estimates fall within 40% from the true value.
DIFFERENTIAL PHYTOPLANKTON SINKING- AND GROWTH-RATES: AN EIGENVALUE ANALYSIS
An eigenvalue analysis of the vertical phytoplankton biomass equation is applied to calculate the differential sinking- and loss-rates of phytoplankton for different taxonomic groups in Lake Lyndon B. Johnson (Texas) and in Lake Erie. The analysis includes factors determining the...
Eigenvalues of Rectangular Waveguide Using FEM With Hybrid Elements
NASA Technical Reports Server (NTRS)
Deshpande, Manohar D.; Hall, John M.
2002-01-01
A finite element analysis using hybrid triangular-rectangular elements is developed to estimate eigenvalues of a rectangular waveguide. Use of rectangular vector-edge finite elements in the vicinity of the PEC boundary and triangular elements in the interior region more accurately models the physical nature of the electromagnetic field, and consequently quicken the convergence.
Statistics of complex eigenvalues in friction-induced vibration
NASA Astrophysics Data System (ADS)
Nobari, Amir; Ouyang, Huajiang; Bannister, Paul
2015-03-01
Self-excited vibrations appear in many mechanical systems with sliding contacts. There are several mechanisms whereby friction can cause the self-excited vibration to become unstable. Of these mechanisms, mode coupling is thought to be responsible for generating annoying high-frequency noise and vibration in brakes. Conventionally, in order to identify whether a system is stable or not, complex eigenvalue analysis is performed. However, what has recently received much attention of researchers is the variability and uncertainty of input variables in the stability analysis of self-excited vibrations. For this purpose, a second-order perturbation method is extended and employed in the current study. The moments of the output distribution along with its joint moment generating function are used for quantifying the statistics of the complex eigenvalues. Moreover, the eigen-derivatives required for the perturbation method are presented in a way that they can deal with the asymmetry of the stiffness matrix and non-proportional damping. Since the eigen-derivatives of such systems are complex-valued numbers, it is mathematically more informative and convenient to derive the statistics of the eigenvalues in a complex form, without decomposing them into two real-valued real and imaginary parts. Then, the variance and pseudo-variance of the complex eigenvalues are used for determining the statistics of the real and imaginary parts. The reliability and robustness of the system in terms of stability can also be quantified by the approximated output distribution.
Selecting reusable components using algebraic specifications
NASA Technical Reports Server (NTRS)
Eichmann, David A.
1992-01-01
A significant hurdle confronts the software reuser attempting to select candidate components from a software repository - discriminating between those components without resorting to inspection of the implementation(s). We outline a mixed classification/axiomatic approach to this problem based upon our lattice-based faceted classification technique and Guttag and Horning's algebraic specification techniques. This approach selects candidates by natural language-derived classification, by their interfaces, using signatures, and by their behavior, using axioms. We briefly outline our problem domain and related work. Lattice-based faceted classifications are described; the reader is referred to surveys of the extensive literature for algebraic specification techniques. Behavioral support for reuse queries is presented, followed by the conclusions.
Linear Algebraic Method for Non-Linear Map Analysis
Yu,L.; Nash, B.
2009-05-04
We present a newly developed method to analyze some non-linear dynamics problems such as the Henon map using a matrix analysis method from linear algebra. Choosing the Henon map as an example, we analyze the spectral structure, the tune-amplitude dependence, the variation of tune and amplitude during the particle motion, etc., using the method of Jordan decomposition which is widely used in conventional linear algebra.
Marquette, Ian
2013-07-15
We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.
NASA Astrophysics Data System (ADS)
Dankova, T. S.; Rosensteel, G.
1998-10-01
Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of coadjoint orbits for the groups U(n) and O(2n). The general theory of mean fields is formulated for an arbitrary Lie algebra L of fermion operators. The moment map provides the correspondence between the Hilbert space of microscopic wave functions and the dual space L^* of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The mean field theories for the Elliott SU(3) and symplectic Sp(3,R) algebras are constructed explicitly in the coadjoint orbit framework.
ERIC Educational Resources Information Center
Beigie, Darin
2014-01-01
Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…
Vertex Algebras, Kac-Moody Algebras, and the Monster
NASA Astrophysics Data System (ADS)
Borcherds, Richard E.
1986-05-01
It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The ``Moonshine'' representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.
NASA Astrophysics Data System (ADS)
Palmkvist, Jakob
2014-01-01
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D - 2 - p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
NASA Technical Reports Server (NTRS)
Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.
1999-01-01
This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.
Palmkvist, Jakob
2014-01-15
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
NASA Astrophysics Data System (ADS)
Ahmed, Hassan Yousif; Nisar, K. S.
2013-08-01
Code with ideal in-phase cross correlation (CC) and practical code length to support high number of users are required in spectral amplitude coding-optical code division multiple access (SAC-OCDMA) systems. SAC systems are getting more attractive in the field of OCDMA because of its ability to eliminate the influence of multiple access interference (MAI) and also suppress the effect of phase induced intensity noise (PIIN). In this paper, we have proposed new Diagonal Eigenvalue Unity (DEU) code families with ideal in-phase CC based on Jordan block matrix with simple algebraic ways. Four sets of DEU code families based on the code weight W and number of users N for the combination (even, even), (even, odd), (odd, odd) and (odd, even) are constructed. This combination gives DEU code more flexibility in selection of code weight and number of users. These features made this code a compelling candidate for future optical communication systems. Numerical results show that the proposed DEU system outperforms reported codes. In addition, simulation results taken from a commercial optical systems simulator, Virtual Photonic Instrument (VPI™) shown that, using point to multipoint transmission in passive optical network (PON), DEU has better performance and could support long span with high data rate.
NASA Astrophysics Data System (ADS)
Dresvyannikov, M. A.; Chernyaev, A. P.; Karuzskii, A. L.; Mityagin, Yu. A.; Perestoronin, A. V.; Volchkov, N. A.
2016-02-01
An operator of the permittivity can completely describe alone a microwave response of conductors with the spatial dispersion. A wave problem is formulated to search the eigenvalues of the permittivity operator, similar to the problem of the wave propagation in hollow waveguides and resonators, but non-self conjugated. Dispersion relations and general solutions are obtained. A significant role of the spatial-type force resonances is considered. Due to the self-consistency of a kinetics problem, the spatial-type force resonances are added to and usually dominate over the influence of boundary conditions. The obtained resonances include particular solutions corresponding to the surface impedances for the anomalous skin effect, for superconductors, as well as four novel solutions. The general frequency dependence of the surface impedance is derived for all solutions except that for a superconductor.
Graphs and matroids weighted in a bounded incline algebra.
Lu, Ling-Xia; Zhang, Bei
2014-01-01
Firstly, for a graph weighted in a bounded incline algebra (or called a dioid), a longest path problem (LPP, for short) is presented, which can be considered the uniform approach to the famous shortest path problem, the widest path problem, and the most reliable path problem. The solutions for LPP and related algorithms are given. Secondly, for a matroid weighted in a linear matroid, the maximum independent set problem is studied. PMID:25126607
Compactly Generated de Morgan Lattices, Basic Algebras and Effect Algebras
NASA Astrophysics Data System (ADS)
Paseka, Jan; Riečanová, Zdenka
2010-12-01
We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.
Locally finite dimensional Lie algebras
NASA Astrophysics Data System (ADS)
Hennig, Johanna
We prove that in a locally finite dimensional Lie algebra L, any maximal, locally solvable subalgebra is the stabilizer of a maximal, generalized flag in an integrable, faithful module over L. Then we prove two structure theorems for simple, locally finite dimensional Lie algebras over an algebraically closed field of characteristic p which give sufficient conditions for the algebras to be of the form [K(R, *), K( R, *)] / (Z(R) ∩ [ K(R, *), K(R, *)]) for a simple, locally finite dimensional associative algebra R with involution *. Lastly, we explore the noncommutative geometry of locally simple representations of the diagonal locally finite Lie algebras sl(ninfinity), o( ninfinity), and sp(n infinity).
Quantum computation using geometric algebra
NASA Astrophysics Data System (ADS)
Matzke, Douglas James
This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.
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
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.
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…
A novel iris segmentation algorithm based on small eigenvalue analysis
NASA Astrophysics Data System (ADS)
Harish, B. S.; Aruna Kumar, S. V.; Guru, D. S.; Ngo, Minh Ngoc
2015-12-01
In this paper, a simple and robust algorithm is proposed for iris segmentation. The proposed method consists of two steps. In first step, iris and pupil is segmented using Robust Spatial Kernel FCM (RSKFCM) algorithm. RSKFCM is based on traditional Fuzzy-c-Means (FCM) algorithm, which incorporates spatial information and uses kernel metric as distance measure. In second step, small eigenvalue transformation is applied to localize iris boundary. The transformation is based on statistical and geometrical properties of the small eigenvalue of the covariance matrix of a set of edge pixels. Extensive experimentations are carried out on standard benchmark iris dataset (viz. CASIA-IrisV4 and UBIRIS.v2). We compared our proposed method with existing iris segmentation methods. Our proposed method has the least time complexity of O(n(i+p)) . The result of the experiments emphasizes that the proposed algorithm outperforms the existing iris segmentation methods.
Eigenvalues of the Neumann Laplacian in symmetric regions
NASA Astrophysics Data System (ADS)
Marrocos, Marcus A. M.; Pereira, Antônio L.
2015-11-01
In this work, we are concerned with the multiplicity of the eigenvalues of the Neumann Laplacian in regions of ℝn which are invariant under the natural action of a compact subgroup G of O(n). We give a partial positive answer (in the Neumann case) to a conjecture of Arnol'd [Funct. Anal. Appl. 6, 94-101 (1972)] on the transversality of the transformation given by the Dirichlet integral to the stratification in the space of quadratic forms according to the multiplicities of the eigenvalues. We show, for some classes of subgroups of O(n) that, generically in the set of G - invariant, C 2 -regions, the action is irreducible in each eigenspace Ker(Δ + λ). These classes include finite subgroups with irreducible representations of dimension not greater than 2 and, in the case n = 2, any compact subgroup of O(2). We also obtain some partial results for general compact subgroups of O(n).
Network and eigenvalue analysis of financial transaction networks
NASA Astrophysics Data System (ADS)
Kyriakopoulos, F.; Thurner, S.; Puhr, C.; Schmitz, S. W.
2009-10-01
We study a dataset containing all financial transactions between the accounts of practically all major financial players within Austria over one year. We empirically analyze transaction networks of money (in and out) flows and report the characteristic network parameters. We observe a significant dependence of network topology on the time scales of observation, and remarkably low correlation between node degrees and transaction volume. We further use transaction timeseries of the financial agents to compute covariance matrices and their eigenvalue spectra. Eigenvectors corresponding to eigenvalues deviating from the Marcenko-Pastur law are analyzed in detail. The potential for practical use as an automated detection mechanism for abnormal behavior of financial players is discussed. The opinion expressed in this paper is that of the authors and does not necessarily reflect the opinion of the OeNB or the ESCB. in here
An asymptotic expansion for energy eigenvalues of anharmonic oscillators
Gaudreau, Philippe; Slevinsky, Richard M.; Safouhi, Hassan
2013-10-15
In the present contribution, we derive an asymptotic expansion for the energy eigenvalues of anharmonic oscillators for potentials of the form V(x)=κx{sup 2q}+ωx{sup 2},q=2,3,… as the energy level n approaches infinity. The asymptotic expansion is obtained using the WKB theory and series reversion. Furthermore, we construct an algorithm for computing the coefficients of the asymptotic expansion for quartic anharmonic oscillators, leading to an efficient and accurate computation of the energy values for n≥6. -- Highlights: •We derived the asymptotic expansion for energy eigenvalues of anharmonic oscillators. •A highly efficient recursive algorithm for computing S{sub k}{sup ′}(z) for WKB. •We contributed to series reversion theory by reverting a new form of asymptotic series. •Our numerical algorithm achieves high accuracy for higher energy levels.
Hidden algebra method (quasi-exact-solvability in quantum mechanics)
Turbiner, Alexander
1996-02-20
A general introduction to quasi-exactly-solvable problems of quantum mechanics is presented. Main attention is given to multidimensional quasi-exactly-solvable and exactly-solvable Schroedinger operators. Exact-solvability of the Calogero and Sutherland N-body problems ass ociated with an existence of the hidden algebra slN is discussed extensively.
Questions, Reflections, Messages,...Do We Really Need Algebra?
ERIC Educational Resources Information Center
Broekman, Harrie; Hoffmann, Agata
2002-01-01
One of the workshops at the 10th SNM Stowarzyszenie Nauczcieli Matematyki (SNM) conference was entitled: Do we really need algebra? The workshop members were given several problems to consider. By doing these problems and reflecting on them the participants explored some important aspects of the question posed. Moreover they became aware of a…
Clearing the Fog from the Undergraduate Course in Linear Algebra
ERIC Educational Resources Information Center
Scott, Damon
2007-01-01
For over a decade it has been a common observation that a "fog" passes over the course in linear algebra once abstract vector spaces are presented. See [2, 3]. We show how this fog may be cleared by having the students translate "abstract" vector-space problems to isomorphic "concrete" settings, solve the "concrete" problem either by hand or with…
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…
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…
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…
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
Integrability of Hamiltonian systems with algebraic potentials
NASA Astrophysics Data System (ADS)
Maciejewski, Andrzej J.; Przybylska, Maria
2016-01-01
Problem of integrability for Hamiltonian systems with potentials that are algebraic thus multivalued functions of coordinates is discussed. Introducing potential as a new variable the original Hamiltonian system on 2n dimensional phase space is extended to 2 n + 1 dimensional system with rational right-hand sides. For extended system its non-canonical degenerated Poisson structure of constant rank 2n and rational Hamiltonian is identified. For algebraic homogeneous potentials of non-zero rational homogeneity degree necessary integrability conditions are formulated. These conditions are deduced from an analysis of the differential Galois group of variational equations around particular solutions of a straight line type. Obtained integrability obstructions are applied to the class of monomial homogeneous potentials. Some integrable potentials satisfying these conditions are found.
On Fluctuations of Eigenvalues of Random Band Matrices
NASA Astrophysics Data System (ADS)
Shcherbina, M.
2015-10-01
We consider the fluctuations of linear eigenvalue statistics of random band matrices whose entries have the form with i.i.d. possessing the th moment, where the function u has a finite support , so that M has only nonzero diagonals. The parameter b (called the bandwidth) is assumed to grow with n in a way such that . Without any additional assumptions on the growth of b we prove CLT for linear eigenvalue statistics for a rather wide class of test functions. Thus we improve and generalize the results of the previous papers (Jana et al., arXiv:1412.2445; Li et al. Random Matrices 2:04, 2013), where CLT was proven under the assumption . Moreover, we develop a method which allows to prove automatically the CLT for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse matrices, diluted random matrices, matrices with heavy tales etc.
Diagnosing Undersampling in Monte Carlo Eigenvalue and Flux Tally Estimates
Perfetti, Christopher M; Rearden, Bradley T
2015-01-01
This study explored the impact of undersampling on the accuracy of tally estimates in Monte Carlo (MC) calculations. Steady-state MC simulations were performed for models of several critical systems with varying degrees of spatial and isotopic complexity, and the impact of undersampling on eigenvalue and fuel pin flux/fission estimates was examined. This study observed biases in MC eigenvalue estimates as large as several percent and biases in fuel pin flux/fission tally estimates that exceeded tens, and in some cases hundreds, of percent. This study also investigated five statistical metrics for predicting the occurrence of undersampling biases in MC simulations. Three of the metrics (the Heidelberger-Welch RHW, the Geweke Z-Score, and the Gelman-Rubin diagnostics) are commonly used for diagnosing the convergence of Markov chains, and two of the methods (the Contributing Particles per Generation and Tally Entropy) are new convergence metrics developed in the course of this study. These metrics were implemented in the KENO MC code within the SCALE code system and were evaluated for their reliability at predicting the onset and magnitude of undersampling biases in MC eigenvalue and flux tally estimates in two of the critical models. Of the five methods investigated, the Heidelberger-Welch RHW, the Gelman-Rubin diagnostics, and Tally Entropy produced test metrics that correlated strongly to the size of the observed undersampling biases, indicating their potential to effectively predict the size and prevalence of undersampling biases in MC simulations.
Conformal algebra on Fock space and conjugate pairs of operators
Sibold, Klaus; Burkhard, Eden
2010-11-15
Using the moment construction, we represent the generators of the conformal algebra as bilinear products of creation and annihiliation operators on the Fock space of the massless real scalar field in four dimensions. A complete set of one-particle eigenstates of the dilatation generator is given. Next, a complete set of one-particle eigenstates of the conformal generator is constructed in two distinct ways, once directly and once through an expansion in terms of dilatation eigenstates. The second approach uses an analytic continuation of the dilatation eigenvalue away from the real axis; the validity of the method is illustrated by the consistency with the first approach. Drawing upon this technique, we finally ponder the idea of building conjugates to the four components of the momentum operator by suitably modifying the action of the conformal generators on dilatation eigenstates. The construction of eigenstates of these new operators proceeds as for the conformal generator itself.
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.
COMMENT: Comment on `Dirac theory in spacetime algebra'
NASA Astrophysics Data System (ADS)
Baylis, William E.
2002-06-01
In contrast to formulations of the Dirac theory by Hestenes and by the present author, the formulation recently presented by Joyce (Joyce W P 2001 J. Phys. A: Math. Gen. 34 1991-2005) is equivalent to the usual Dirac equation only in the case of vanishing mass. For nonzero mass, solutions to Joyce's equation can be solutions either of the Dirac equation in the Hestenes form or of the same equation with the sign of the mass reversed, and in general they are mixtures of the two possibilities. Because of this relationship, Joyce obtains twice as many linearly independent plane-wave solutions for a given momentum eigenvalue as exist in the conventional theory. A misconception about the symmetry of the Hestenes equation and the geometric significance of the algebraic spinors is also briefly discussed.
Algebraic Squares: Complete and Incomplete.
ERIC Educational Resources Information Center
Gardella, Francis J.
2000-01-01
Illustrates ways of using algebra tiles to give students a visual model of competing squares that appear in algebra as well as in higher mathematics. Such visual representations give substance to the symbolic manipulation and give students who do not learn symbolically a way of understanding the underlying concepts of completing the square. (KHR)
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…
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.)
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…
Algebraic Thinking in Adult Education
ERIC Educational Resources Information Center
Manly, Myrna; Ginsburg, Lynda
2010-01-01
In adult education, algebraic thinking can be a sense-making tool that introduces coherence among mathematical concepts for those who previously have had trouble learning math. Further, a modeling approach to algebra connects mathematics and the real world, demonstrating the usefulness of math to those who have seen it as just an academic…
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.)
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…
Exploring Algebraic Patterns through Literature.
ERIC Educational Resources Information Center
Austin, Richard A.; Thompson, Denisse R.
1997-01-01
Presents methods for using literature to develop algebraic thinking in an environment that connects algebra to various situations. Activities are based on the book "Anno's Magic Seeds" with additional resources listed. Students express a constant function, exponential function, and a recursive function in their own words as well as writing about…
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.
2004-04-21
Version 04 NESTLE solves the few-group neutron diffusion equation utilizing the NEM. The NESTLE code can solve the eigenvalue (criticality), eigenvalue adjoint, external fixed-source steady-state, and external fixed-source or eigenvalue initiated transient problems. The eigenvalue problem allows criticality searches to be completed, and the external fixed-source steady-state problem can search to achieve a specified power level. Transient problems model delayed neutrons via precursor groups. Several core properties can be input as time dependent. Two- ormore » four-energy groups can be utilized, with all energy groups being thermal groups (i.e., upscatter exits) if desired. Core geometries modeled include Cartesian and hexagonal. Three-, two-, and one-dimensional models can be utilized with various symmetries. The thermal conditions predicted by the thermal-hydraulic model of the core are used to correct cross sections for temperature and density effects. Cross sections are parameterized by color, control rod state (i.e., in or out), and burnup, allowing fuel depletion to be modeled. Either a macroscopic or microscopic model may be employed.« less
NASA Astrophysics Data System (ADS)
Huang, Zhenghua; Zhang, Tianxu; Deng, Lihua; Fang, Hao; Li, Qian
2015-12-01
Total variation(TV) based on regularization has been proven as a popular and effective model for image restoration, because of its ability of edge preserved. However, as the TV favors a piece-wise constant solution, the processing results in the flat regions of the image are easily produced "staircase effects", and the amplitude of the edges will be underestimated; the underlying cause of the problem is that the regularization parameter can not be changeable with spatial local information of image. In this paper, we propose a novel Scatter-matrix eigenvalues-based TV(SMETV) regularization with image blind restoration algorithm for deblurring medical images. The spatial information in different image regions is incorporated into regularization by using the edge indicator called difference eigenvalue to distinguish edges from flat areas. The proposed algorithm can effectively reduce the noise in flat regions as well as preserve the edge and detailed information. Moreover, it becomes more robust with the change of the regularization parameter. Extensive experiments demonstrate that the proposed approach produces results superior to most methods in both visual image quality and quantitative measures.
Invariants of triangular Lie algebras
NASA Astrophysics Data System (ADS)
Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman
2007-07-01
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of Boyko et al (2006 J. Phys. A: Math. Gen.39 5749 (Preprint math-ph/0602046)), developed further in Boyko et al (2007 J. Phys. A: Math. Theor.40 113 (Preprint math-ph/0606045)), is used to determine the invariants. A conjecture of Tremblay and Winternitz (2001 J. Phys. A: Math. Gen.34 9085), concerning the number of independent invariants and their form, is corroborated.
Element Agglomeration Algebraic Multilevel Monte-Carlo Library
2015-02-19
ElagMC is a parallel C++ library for Multilevel Monte Carlo simulations with algebraically constructed coarse spaces. ElagMC enables Multilevel variance reduction techniques in the context of general unstructured meshes by using the specialized element-based agglomeration techniques implemented in ELAG (the Element-Agglomeration Algebraic Multigrid and Upscaling Library developed by U. Villa and P. Vassilevski and currently under review for public release). The ElabMC library can support different type of deterministic problems, including mixed finite element discretizations of subsurface flow problems.