Using Technology to Facilitate Reasoning: Lifting the Fog from Linear Algebra

ERIC Educational Resources Information Center

Berry, John S.; Lapp, Douglas A.; Nyman, Melvin A.

2008-01-01

This article discusses student difficulties in grasping concepts from linear algebra. Using an example from an interview with a student, we propose changes that might positively impact student understanding of concepts within a problem-solving context. In particular, we illustrate barriers to student understanding and suggest technological…

Students' Use of Computational Thinking in Linear Algebra

ERIC Educational Resources Information Center

Bagley, Spencer; Rabin, Jeffrey M.

2016-01-01

In this work, we examine students' ways of thinking when presented with a novel linear algebra problem. Our intent was to explore how students employ and coordinate three modes of thinking, which we call computational, abstract, and geometric, following similar frameworks proposed by Hillel (2000) and Sierpinska (2000). However, the undergraduate…

Undergraduate Mathematics Students' Emotional Experiences in Linear Algebra Courses

ERIC Educational Resources Information Center

Martínez-Sierra, Gustavo; García-González, María del Socorro

2016-01-01

Little is known about students' emotions in the field of Mathematics Education that go beyond students' emotions in problem solving. To start filling this gap this qualitative research has the aim to identify emotional experiences of undergraduate mathematics students in Linear Algebra courses. In order to obtain data, retrospective focus group…

Contextualizing symbol, symbolizing context

NASA Astrophysics Data System (ADS)

Maudy, Septiani Yugni; Suryadi, Didi; Mulyana, Endang

2017-08-01

When students learn algebra for the first time, inevitably they are experiencing transition from arithmetic to algebraic thinking. Once students could apprehend this essential mathematical knowledge, they are cultivating their ability in solving daily life problems by applying algebra. However, as we dig into this transitional stage, we identified possible students' learning obstacles to be dealt with seriously in order to forestall subsequent hindrance in studying more advance algebra. We come to realize this recurring problem as we undertook the processes of re-personalization and re-contextualization in which we scrutinize the very basic questions: 1) what is variable, linear equation with one variable and their relationship with the arithmetic-algebraic thinking? 2) Why student should learn such concepts? 3) How to teach those concepts to students? By positioning ourselves as a seventh grade student, we address the possibility of children to think arithmetically when confronted with the problems of linear equation with one variable. To help them thinking algebraically, Bruner's modes of representation developed contextually from concrete to abstract were delivered to enhance their interpretation toward the idea of variables. Hence, from the outset we designed the context for student to think symbolically initiated by exploring various symbols that could be contextualized in order to bridge student traversing the arithmetic-algebraic fruitfully.

A new application of algebraic geometry to systems theory

NASA Technical Reports Server (NTRS)

Martin, C. F.; Hermann, R.

1976-01-01

Following an introduction to algebraic geometry, the dominant morphism theorem is stated, and the application of this theorem to systems-theoretic problems, such as the feedback problem, is discussed. The Gaussian elimination method used for solving linear equations is shown to be an example of a dominant morphism.

W-algebra for solving problems with fuzzy parameters

NASA Astrophysics Data System (ADS)

Shevlyakov, A. O.; Matveev, M. G.

2018-03-01

A method of solving the problems with fuzzy parameters by means of a special algebraic structure is proposed. The structure defines its operations through operations on real numbers, which simplifies its use. It avoids deficiencies limiting applicability of the other known structures. Examples for solution of a quadratic equation, a system of linear equations and a network planning problem are given.

Mathematical modelling in engineering: an alternative way to teach Linear Algebra

NASA Astrophysics Data System (ADS)

Domínguez-García, S.; García-Planas, M. I.; Taberna, J.

2016-10-01

Technological advances require that basic science courses for engineering, including Linear Algebra, emphasize the development of mathematical strengths associated with modelling and interpretation of results, which are not limited only to calculus abilities. Based on this consideration, we have proposed a project-based learning, giving a dynamic classroom approach in which students modelled real-world problems and turn gain a deeper knowledge of the Linear Algebra subject. Considering that most students are digital natives, we use the e-portfolio as a tool of communication between students and teachers, besides being a good place making the work visible. In this article, we present an overview of the design and implementation of a project-based learning for a Linear Algebra course taught during the 2014-2015 at the 'ETSEIB'of Universitat Politècnica de Catalunya (UPC).

Graphs and matroids weighted in a bounded incline algebra.

PubMed

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.

Bicycles, Birds, Bats and Balloons: New Applications for Algebra Classes.

ERIC Educational Resources Information Center

Yoshiwara, Bruce; Yoshiwara, Kathy

This collection of activities is intended to enhance the teaching of college algebra through the use of modeling. The problems use real data and involve the representation and interpretation of the data. The concepts addressed include rates of change, linear and quadratic regression, and functions. The collection consists of eight problems, four…

Packing a Box with Bricks.

ERIC Educational Resources Information Center

Jepsen, Charles H.

1991-01-01

Presented are solutions to variations of a combinatorics problem from a recent International Mathematics Olympiad. In particular, the matrix algebra solution illustrates an interaction among the undergraduate areas of geometry, combinatorics, linear algebra, and group theory. (JJK)

Some Applications Of Semigroups And Computer Algebra In Discrete Structures

NASA Astrophysics Data System (ADS)

Bijev, G.

2009-11-01

An algebraic approach to the pseudoinverse generalization problem in Boolean vector spaces is used. A map (p) is defined, which is similar to an orthogonal projection in linear vector spaces. Some other important maps with properties similar to those of the generalized inverses (pseudoinverses) of linear transformations and matrices corresponding to them are also defined and investigated. Let Ax = b be an equation with matrix A and vectors x and b Boolean. Stochastic experiments for solving the equation, which involves the maps defined and use computer algebra methods, have been made. As a result, the Hamming distance between vectors Ax = p(b) and b is equal or close to the least possible. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra and research. Some examples for computations with binary relations using Maple are given.

Class and Homework Problems: The Break-Even Radius of Insulation Computed Using Excel Solver and WolframAlpha

ERIC Educational Resources Information Center

Foley, Greg

2014-01-01

A problem that illustrates two ways of computing the break-even radius of insulation is outlined. The problem is suitable for students who are taking an introductory module in heat transfer or transport phenomena and who have some previous knowledge of the numerical solution of non- linear algebraic equations. The potential for computer algebra,…

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

DOE Office of Scientific and Technical Information (OSTI.GOV)

Campoamor-Stursberg, R., E-mail: rutwig@mat.ucm.e

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.

Communication Avoiding and Overlapping for Numerical Linear Algebra

DTIC Science & Technology

2012-05-08

future exascale systems, communication cost must be avoided or overlapped. Communication-avoiding 2.5D algorithms improve scalability by reducing...linear algebra problems to future exascale systems, communication cost must be avoided or overlapped. Communication-avoiding 2.5D algorithms improve...will continue to grow relative to the cost of computation. With exascale computing as the long-term goal, the community needs to develop techniques

Multiple shooting algorithms for jump-discontinuous problems in optimal control and estimation

NASA Technical Reports Server (NTRS)

Mook, D. J.; Lew, Jiann-Shiun

1991-01-01

Multiple shooting algorithms are developed for jump-discontinuous two-point boundary value problems arising in optimal control and optimal estimation. Examples illustrating the origin of such problems are given to motivate the development of the solution algorithms. The algorithms convert the necessary conditions, consisting of differential equations and transversality conditions, into algebraic equations. The solution of the algebraic equations provides exact solutions for linear problems. The existence and uniqueness of the solution are proved.

Design of Linear Quadratic Regulators and Kalman Filters

NASA Technical Reports Server (NTRS)

Lehtinen, B.; Geyser, L.

1986-01-01

AESOP solves problems associated with design of controls and state estimators for linear time-invariant systems. Systems considered are modeled in state-variable form by set of linear differential and algebraic equations with constant coefficients. Two key problems solved by AESOP are linear quadratic regulator (LQR) design problem and steady-state Kalman filter design problem. AESOP is interactive. User solves design problems and analyzes solutions in single interactive session. Both numerical and graphical information available to user during the session.

Linear and Quadratic Change: A Problem from Japan

ERIC Educational Resources Information Center

Peterson, Blake E.

2006-01-01

In the fall of 2003, the author conducted research on the student teaching process in Japan. The basis for most of the lessons observed was rich mathematics problems. Upon returning to the US, the author used one such problem while teaching an algebra 2 class. This article introduces that problem, which gives rise to both linear and quadratic…

A Nonlinear, Multiinput, Multioutput Process Control Laboratory Experiment

ERIC Educational Resources Information Center

Young, Brent R.; van der Lee, James H.; Svrcek, William Y.

2006-01-01

Experience in using a user-friendly software, Mathcad, in the undergraduate chemical reaction engineering course is discussed. Example problems considered for illustration deal with simultaneous solution of linear algebraic equations (kinetic parameter estimation), nonlinear algebraic equations (equilibrium calculations for multiple reactions and…

ORACLS: A system for linear-quadratic-Gaussian control law design

NASA Technical Reports Server (NTRS)

Armstrong, E. S.

1978-01-01

A modern control theory design package (ORACLS) for constructing controllers and optimal filters for systems modeled by linear time-invariant differential or difference equations is described. Numerical linear-algebra procedures are used to implement the linear-quadratic-Gaussian (LQG) methodology of modern control theory. Algorithms are included for computing eigensystems of real matrices, the relative stability of a matrix, factored forms for nonnegative definite matrices, the solutions and least squares approximations to the solutions of certain linear matrix algebraic equations, the controllability properties of a linear time-invariant system, and the steady state covariance matrix of an open-loop stable system forced by white noise. Subroutines are provided for solving both the continuous and discrete optimal linear regulator problems with noise free measurements and the sampled-data optimal linear regulator problem. For measurement noise, duality theory and the optimal regulator algorithms are used to solve the continuous and discrete Kalman-Bucy filter problems. Subroutines are also included which give control laws causing the output of a system to track the output of a prescribed model.

Exact solution of some linear matrix equations using algebraic methods

NASA Technical Reports Server (NTRS)

Djaferis, T. E.; Mitter, S. K.

1979-01-01

Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.

A high-accuracy optical linear algebra processor for finite element applications

NASA Technical Reports Server (NTRS)

Casasent, D.; Taylor, B. K.

1984-01-01

Optical linear processors are computationally efficient computers for solving matrix-matrix and matrix-vector oriented problems. Optical system errors limit their dynamic range to 30-40 dB, which limits their accuray to 9-12 bits. Large problems, such as the finite element problem in structural mechanics (with tens or hundreds of thousands of variables) which can exploit the speed of optical processors, require the 32 bit accuracy obtainable from digital machines. To obtain this required 32 bit accuracy with an optical processor, the data can be digitally encoded, thereby reducing the dynamic range requirements of the optical system (i.e., decreasing the effect of optical errors on the data) while providing increased accuracy. This report describes a new digitally encoded optical linear algebra processor architecture for solving finite element and banded matrix-vector problems. A linear static plate bending case study is described which quantities the processor requirements. Multiplication by digital convolution is explained, and the digitally encoded optical processor architecture is advanced.

Asymptotic identity in min-plus algebra: a report on CPNS.

PubMed

Li, Ming; Zhao, Wei

2012-01-01

Network calculus is a theory initiated primarily in computer communication networks, especially in the aspect of real-time communications, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast and models in data flows as well as systems in CPNS are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNS as can be seen from its applications to the Internet computing, there are tough problems remaining unsolved in this regard. The identity in min-plus algebra is one problem we shall address. We shall point out the confusions about the conventional identity in the min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusions.

Asymptotic Identity in Min-Plus Algebra: A Report on CPNS

PubMed Central

Li, Ming; Zhao, Wei

2012-01-01

Network calculus is a theory initiated primarily in computer communication networks, especially in the aspect of real-time communications, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast and models in data flows as well as systems in CPNS are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNS as can be seen from its applications to the Internet computing, there are tough problems remaining unsolved in this regard. The identity in min-plus algebra is one problem we shall address. We shall point out the confusions about the conventional identity in the min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusions. PMID:21822446

The algebraic-hyperbolic approach to the linearized gravitational constraints on a Minkowski background

NASA Astrophysics Data System (ADS)

Winicour, Jeffrey

2017-08-01

An algebraic-hyperbolic method for solving the Hamiltonian and momentum constraints has recently been shown to be well posed for general nonlinear perturbations of the initial data for a Schwarzschild black hole. This is a new approach to solving the constraints of Einstein’s equations which does not involve elliptic equations and has potential importance for the construction of binary black hole data. In order to shed light on the underpinnings of this approach, we consider its application to obtain solutions of the constraints for linearized perturbations of Minkowski space. In that case, we find the surprising result that there are no suitable Cauchy hypersurfaces in Minkowski space for which the linearized algebraic-hyperbolic constraint problem is well posed.

Ermakov's Superintegrable Toy and Nonlocal Symmetries

NASA Astrophysics Data System (ADS)

Leach, P. G. L.; Karasu Kalkanli, A.; Nucci, M. C.; Andriopoulos, K.

2005-11-01

We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2, R). The number of point symmetries is insufficient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties which can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.

Numerical Solution of Systems of Loaded Ordinary Differential Equations with Multipoint Conditions

NASA Astrophysics Data System (ADS)

Assanova, A. T.; Imanchiyev, A. E.; Kadirbayeva, Zh. M.

2018-04-01

A system of loaded ordinary differential equations with multipoint conditions is considered. The problem under study is reduced to an equivalent boundary value problem for a system of ordinary differential equations with parameters. A system of linear algebraic equations for the parameters is constructed using the matrices of the loaded terms and the multipoint condition. The conditions for the unique solvability and well-posedness of the original problem are established in terms of the matrix made up of the coefficients of the system of linear algebraic equations. The coefficients and the righthand side of the constructed system are determined by solving Cauchy problems for linear ordinary differential equations. The solutions of the system are found in terms of the values of the desired function at the initial points of subintervals. The parametrization method is numerically implemented using the fourth-order accurate Runge-Kutta method as applied to the Cauchy problems for ordinary differential equations. The performance of the constructed numerical algorithms is illustrated by examples.

Exact solution of some linear matrix equations using algebraic methods

NASA Technical Reports Server (NTRS)

Djaferis, T. E.; Mitter, S. K.

1977-01-01

A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.

USSR and Eastern Europe Scientific Abstracts, Electronics and Electrical Engineering, Number 33.

DTIC Science & Technology

1977-09-27

reduces to an infinite system of linear homogeneous algebraic equations and leads to Mathieu functions of the k-th order. The solution is convergent in...cylinder walls to be infinitesimally thin ideal conductors. The problem is reduced to a system of Fredholm linear algebraic equations of the second...EXPECTED DEVELOPMENTS OF TRANSISTORIZED LOW-NOISE MICROWAVE AMPLIFIERS Prague SDELOVACI TECHNIKA in Czech Vol 25, No 2, Feb 77 pp 47-49 TALLO, ANTON

Finite Element Based Structural Damage Detection Using Artificial Boundary Conditions

DTIC Science & Technology

2007-09-01

C. (2005). Elementary Linear Algebra . New York: John Wiley and Sons. Avitable, Peter (2001, January) Experimental Modal Analysis, A Simple Non...variables under consideration. 3 Frequency sensitivities are the basis for a linear approximation to compute the change in the natural frequencies of a...THEORY The general problem statement for a non- linear constrained optimization problem is: To minimize ( )f x Objective Function Subject to

A note on probabilistic models over strings: the linear algebra approach.

PubMed

Bouchard-Côté, Alexandre

2013-12-01

Probabilistic models over strings have played a key role in developing methods that take into consideration indels as phylogenetically informative events. There is an extensive literature on using automata and transducers on phylogenies to do inference on these probabilistic models, in which an important theoretical question is the complexity of computing the normalization of a class of string-valued graphical models. This question has been investigated using tools from combinatorics, dynamic programming, and graph theory, and has practical applications in Bayesian phylogenetics. In this work, we revisit this theoretical question from a different point of view, based on linear algebra. The main contribution is a set of results based on this linear algebra view that facilitate the analysis and design of inference algorithms on string-valued graphical models. As an illustration, we use this method to give a new elementary proof of a known result on the complexity of inference on the "TKF91" model, a well-known probabilistic model over strings. Compared to previous work, our proving method is easier to extend to other models, since it relies on a novel weak condition, triangular transducers, which is easy to establish in practice. The linear algebra view provides a concise way of describing transducer algorithms and their compositions, opens the possibility of transferring fast linear algebra libraries (for example, based on GPUs), as well as low rank matrix approximation methods, to string-valued inference problems.

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

ERIC Educational Resources Information Center

Gonzalez-Vega, Laureano

1999-01-01

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

Investigating High-School Students' Reasoning Strategies when They Solve Linear Equations

ERIC Educational Resources Information Center

Huntley, Mary Ann; Marcus, Robin; Kahan, Jeremy; Miller, Jane Lincoln

2007-01-01

A cross-curricular structured-probe task-based clinical interview study with 44 pairs of third-year high-school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from one problem that involved solving a set of three linear equations of…

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

ERIC Educational Resources Information Center

Foley, Greg

2011-01-01

Continuous feed and bleed ultrafiltration, modeled with the gel polarization model for the limiting flux, is shown to provide a rich source of non-linear algebraic equations that can be readily solved using numerical and graphical techniques familiar to undergraduate students. We present a variety of numerical problems in the design, analysis, and…

Asymptotic aspect of derivations in Banach algebras.

PubMed

Roh, Jaiok; Chang, Ick-Soon

2017-01-01

We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.

Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation

NASA Astrophysics Data System (ADS)

Bokhari, Ashfaque H.; Mahomed, F. M.; Zaman, F. D.

2010-05-01

The complete symmetry group classification of the fourth-order Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry.

Numerical methods in Markov chain modeling

NASA Technical Reports Server (NTRS)

Philippe, Bernard; Saad, Youcef; Stewart, William J.

1989-01-01

Several methods for computing stationary probability distributions of Markov chains are described and compared. The main linear algebra problem consists of computing an eigenvector of a sparse, usually nonsymmetric, matrix associated with a known eigenvalue. It can also be cast as a problem of solving a homogeneous singular linear system. Several methods based on combinations of Krylov subspace techniques are presented. The performance of these methods on some realistic problems are compared.

Block iterative restoration of astronomical images with the massively parallel processor

NASA Technical Reports Server (NTRS)

Heap, Sara R.; Lindler, Don J.

1987-01-01

A method is described for algebraic image restoration capable of treating astronomical images. For a typical 500 x 500 image, direct algebraic restoration would require the solution of a 250,000 x 250,000 linear system. The block iterative approach is used to reduce the problem to solving 4900 121 x 121 linear systems. The algorithm was implemented on the Goddard Massively Parallel Processor, which can solve a 121 x 121 system in approximately 0.06 seconds. Examples are shown of the results for various astronomical images.

LAPACKrc: Fast linear algebra kernels/solvers for FPGA accelerators

NASA Astrophysics Data System (ADS)

Gonzalez, Juan; Núñez, Rafael C.

2009-07-01

We present LAPACKrc, a family of FPGA-based linear algebra solvers able to achieve more than 100x speedup per commodity processor on certain problems. LAPACKrc subsumes some of the LAPACK and ScaLAPACK functionalities, and it also incorporates sparse direct and iterative matrix solvers. Current LAPACKrc prototypes demonstrate between 40x-150x speedup compared against top-of-the-line hardware/software systems. A technology roadmap is in place to validate current performance of LAPACKrc in HPC applications, and to increase the computational throughput by factors of hundreds within the next few years.

Discrete Methods and their Applications

DTIC Science & Technology

1993-02-03

problem of finding all near-optimal solutions to a linear program. In paper [18], we give a brief and elementary proof of a result of Hoffman [1952) about...relies only on linear programming duality; second, we obtain geometric and algebraic representations of the bounds that are determined explicitly in...same. We have studied the problem of finding the minimum n such that a given unit interval graph is an n--graph. A linear time algorithm to compute

An improved error bound for linear complementarity problems for B-matrices.

PubMed

Gao, Lei; Li, Chaoqian

2017-01-01

A new error bound for the linear complementarity problem when the matrix involved is a B -matrix is presented, which improves the corresponding result in (Li et al. in Electron. J. Linear Algebra 31(1):476-484, 2016). In addition some sufficient conditions such that the new bound is sharper than that in (García-Esnaola and Peña in Appl. Math. Lett. 22(7):1071-1075, 2009) are provided.

Variational data assimilation system "INM RAS - Black Sea"

NASA Astrophysics Data System (ADS)

Parmuzin, Eugene; Agoshkov, Valery; Assovskiy, Maksim; Giniatulin, Sergey; Zakharova, Natalia; Kuimov, Grigory; Fomin, Vladimir

2013-04-01

Development of Informational-Computational Systems (ICS) for Data Assimilation Procedures is one of multidisciplinary problems. To study and solve these problems one needs to apply modern results from different disciplines and recent developments in: mathematical modeling; theory of adjoint equations and optimal control; inverse problems; numerical methods theory; numerical algebra and scientific computing. The problems discussed above are studied in the Institute of Numerical Mathematics of the Russian Academy of Science (INM RAS) in ICS for Personal Computers (PC). Special problems and questions arise while effective ICS versions for PC are being developed. These problems and questions can be solved with applying modern methods of numerical mathematics and by solving "parallelism problem" using OpenMP technology and special linear algebra packages. In this work the results on the ICS development for PC-ICS "INM RAS - Black Sea" are presented. In the work the following problems and questions are discussed: practical problems that can be studied by ICS; parallelism problems and their solutions with applying of OpenMP technology and the linear algebra packages used in ICS "INM - Black Sea"; Interface of ICS. The results of ICS "INM RAS - Black Sea" testing are presented. Efficiency of technologies and methods applied are discussed. The work was supported by RFBR, grants No. 13-01-00753, 13-05-00715 and by The Ministry of education and science of Russian Federation, project 8291, project 11.519.11.1005 References: [1] V.I. Agoshkov, M.V. Assovskii, S.A. Lebedev, Numerical simulation of Black Sea hydrothermodynamics taking into account tide-forming forces. Russ. J. Numer. Anal. Math. Modelling (2012) 27, No.1, 5-31 [2] E.I. Parmuzin, V.I. Agoshkov, Numerical solution of the variational assimilation problem for sea surface temperature in the model of the Black Sea dynamics. Russ. J. Numer. Anal. Math. Modelling (2012) 27, No.1, 69-94 [3] V.B. Zalesny, N.A. Diansky, V.V. Fomin, S.N. Moshonkin, S.G. Demyshev, Numerical model of the circulation of Black Sea and Sea of Azov. Russ. J. Numer. Anal. Math. Modelling (2012) 27, No.1, 95-111 [4] V.I. Agoshkov, S.V. Giniatulin, G.V. Kuimov. OpenMP technology and linear algebra packages in the variation data assimilation systems. - Abstracts of the 1-st China-Russia Conference on Numerical Algebra with Applications in Radiactive Hydrodynamics, Beijing, China, October 16-18, 2012. [5] Zakharova N.B., Agoshkov V.I., Parmuzin E.I., The new method of ARGO buoys system observation data interpolation. Russian Journal of Numerical Analysis and Mathematical Modelling. Vol. 28, Issue 1, 2013.

A Characterization of a Unified Notion of Mathematical Function: The Case of High School Function and Linear Transformation

ERIC Educational Resources Information Center

Zandieh, Michelle; Ellis, Jessica; Rasmussen, Chris

2017-01-01

As part of a larger study of student understanding of concepts in linear algebra, we interviewed 10 university linear algebra students as to their conceptions of functions from high school algebra and linear transformation from their study of linear algebra. An overarching goal of this study was to examine how linear algebra students see linear…

The Effects of the Concrete-Representational-Abstract Integration Strategy on the Ability of Students with Learning Disabilities to Multiply Linear Expressions within Area Problems

ERIC Educational Resources Information Center

Strickland, Tricia K.; Maccini, Paula

2013-01-01

We examined the effects of the Concrete-Representational-Abstract Integration strategy on the ability of secondary students with learning disabilities to multiply linear algebraic expressions embedded within contextualized area problems. A multiple-probe design across three participants was used. Results indicated that the integration of the…

Many-core graph analytics using accelerated sparse linear algebra routines

NASA Astrophysics Data System (ADS)

Kozacik, Stephen; Paolini, Aaron L.; Fox, Paul; Kelmelis, Eric

2016-05-01

Graph analytics is a key component in identifying emerging trends and threats in many real-world applications. Largescale graph analytics frameworks provide a convenient and highly-scalable platform for developing algorithms to analyze large datasets. Although conceptually scalable, these techniques exhibit poor performance on modern computational hardware. Another model of graph computation has emerged that promises improved performance and scalability by using abstract linear algebra operations as the basis for graph analysis as laid out by the GraphBLAS standard. By using sparse linear algebra as the basis, existing highly efficient algorithms can be adapted to perform computations on the graph. This approach, however, is often less intuitive to graph analytics experts, who are accustomed to vertex-centric APIs such as Giraph, GraphX, and Tinkerpop. We are developing an implementation of the high-level operations supported by these APIs in terms of linear algebra operations. This implementation is be backed by many-core implementations of the fundamental GraphBLAS operations required, and offers the advantages of both the intuitive programming model of a vertex-centric API and the performance of a sparse linear algebra implementation. This technology can reduce the number of nodes required, as well as the run-time for a graph analysis problem, enabling customers to perform more complex analysis with less hardware at lower cost. All of this can be accomplished without the requirement for the customer to make any changes to their analytics code, thanks to the compatibility with existing graph APIs.

The Growing Importance of Linear Algebra in Undergraduate Mathematics.

ERIC Educational Resources Information Center

Tucker, Alan

1993-01-01

Discusses the theoretical and practical importance of linear algebra. Presents a brief history of linear algebra and matrix theory and describes the place of linear algebra in the undergraduate curriculum. (MDH)

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.

Lie-algebraic Approach to Dynamics of Closed Quantum Systems and Quantum-to-Classical Correspondence

NASA Astrophysics Data System (ADS)

Galitski, Victor

2012-02-01

I will briefly review our recent work on a Lie-algebraic approach to various non-equilibrium quantum-mechanical problems, which has been motivated by continuous experimental advances in the field of cold atoms. First, I will discuss non-equilibrium driven dynamics of a generic closed quantum system. It will be emphasized that mathematically a non-equilibrium Hamiltonian represents a trajectory in a Lie algebra, while the evolution operator is a trajectory in a Lie group generated by the underlying algebra via exponentiation. This turns out to be a constructive statement that establishes, in particular, the fact that classical and quantum unitary evolutions are two sides of the same coin determined uniquely by the same dynamic generators in the group. An equation for these generators - dubbed dual Schr"odinger-Bloch equation - will be derived and analyzed for a few of specific examples. This non-linear equation allows one to construct new exact non-linear solutions to quantum-dynamical systems. An experimentally-relevant example of a family of exact solutions to the many-body Landau-Zener problem will be presented. One practical application of the latter result includes dynamical means to optimize molecular production rate following a quench across the Feshbach resonance.

Linear systems with structure group and their feedback invariants

NASA Technical Reports Server (NTRS)

Martin, C.; Hermann, R.

1977-01-01

A general method described by Hermann and Martin (1976) for the study of the feedback invariants of linear systems is considered. It is shown that this method, which makes use of ideas of topology and algebraic geometry, is very useful in the investigation of feedback problems for which the classical methods are not suitable. The transfer function as a curve in the Grassmanian is examined. The general concepts studied in the context of specific systems and applications are organized in terms of the theory of Lie groups and algebraic geometry. Attention is given to linear systems which have a structure group, linear mechanical systems, and feedback invariants. The investigation shows that Lie group techniques are powerful and useful tools for analysis of the feedback structure of linear systems.

An Inquiry-Oriented Approach to Span and Linear Independence: The Case of the Magic Carpet Ride Sequence

ERIC Educational Resources Information Center

Wawro, Megan; Rasmussen, Chris; Zandieh, Michelle; Sweeney, George Franklin; Larson, Christine

2012-01-01

In this paper we present an innovative instructional sequence for an introductory linear algebra course that supports students' reinvention of the concepts of span, linear dependence, and linear independence. Referred to as the Magic Carpet Ride sequence, the problems begin with an imaginary scenario that allows students to build rich imagery and…

An Extended Microcomputer-Based Network Optimization Package.

DTIC Science & Technology

1982-10-01

Analysis, Laxenberq, Austria, 1981, pp. 781-808. 9. Anton , H., Elementary Linear Algebra , John Wiley & Sons, New York, 1977. 10. Koopmans, T. C...fCaRUlue do leVee. aide It 001100"M OW eedea9f’ OF Nooke~e Network, generalized network, microcomputer, optimization, network with gains, linear ...Oboe &111111041 network problem, in turn, can be viewed as a specialization of a linear programuing problem having at most two non-zero entries in each

Extended Decentralized Linear-Quadratic-Gaussian Control

NASA Technical Reports Server (NTRS)

Carpenter, J. Russell

2000-01-01

A straightforward extension of a solution to the decentralized linear-Quadratic-Gaussian problem is proposed that allows its use for commonly encountered classes of problems that are currently solved with the extended Kalman filter. This extension allows the system to be partitioned in such a way as to exclude the nonlinearities from the essential algebraic relationships that allow the estimation and control to be optimally decentralized.

Algebraic multigrid preconditioners for two-phase flow in porous media with phase transitions [Algebraic multigrid preconditioners for multiphase flow in porous media with phase transitions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bui, Quan M.; Wang, Lu; Osei-Kuffuor, Daniel

Multiphase flow is a critical process in a wide range of applications, including oil and gas recovery, carbon sequestration, and contaminant remediation. Numerical simulation of multiphase flow requires solving of a large, sparse linear system resulting from the discretization of the partial differential equations modeling the flow. In the case of multiphase multicomponent flow with miscible effect, this is a very challenging task. The problem becomes even more difficult if phase transitions are taken into account. A new approach to handle phase transitions is to formulate the system as a nonlinear complementarity problem (NCP). Unlike in the primary variable switchingmore » technique, the set of primary variables in this approach is fixed even when there is phase transition. Not only does this improve the robustness of the nonlinear solver, it opens up the possibility to use multigrid methods to solve the resulting linear system. The disadvantage of the complementarity approach, however, is that when a phase disappears, the linear system has the structure of a saddle point problem and becomes indefinite, and current algebraic multigrid (AMG) algorithms cannot be applied directly. In this study, we explore the effectiveness of a new multilevel strategy, based on the multigrid reduction technique, to deal with problems of this type. We demonstrate the effectiveness of the method through numerical results for the case of two-phase, two-component flow with phase appearance/disappearance. In conclusion, we also show that the strategy is efficient and scales optimally with problem size.« less

Algebraic multigrid preconditioners for two-phase flow in porous media with phase transitions [Algebraic multigrid preconditioners for multiphase flow in porous media with phase transitions

DOE PAGES

Bui, Quan M.; Wang, Lu; Osei-Kuffuor, Daniel

2018-02-06

Multiphase flow is a critical process in a wide range of applications, including oil and gas recovery, carbon sequestration, and contaminant remediation. Numerical simulation of multiphase flow requires solving of a large, sparse linear system resulting from the discretization of the partial differential equations modeling the flow. In the case of multiphase multicomponent flow with miscible effect, this is a very challenging task. The problem becomes even more difficult if phase transitions are taken into account. A new approach to handle phase transitions is to formulate the system as a nonlinear complementarity problem (NCP). Unlike in the primary variable switchingmore » technique, the set of primary variables in this approach is fixed even when there is phase transition. Not only does this improve the robustness of the nonlinear solver, it opens up the possibility to use multigrid methods to solve the resulting linear system. The disadvantage of the complementarity approach, however, is that when a phase disappears, the linear system has the structure of a saddle point problem and becomes indefinite, and current algebraic multigrid (AMG) algorithms cannot be applied directly. In this study, we explore the effectiveness of a new multilevel strategy, based on the multigrid reduction technique, to deal with problems of this type. We demonstrate the effectiveness of the method through numerical results for the case of two-phase, two-component flow with phase appearance/disappearance. In conclusion, we also show that the strategy is efficient and scales optimally with problem size.« less

Particle-like structure of coaxial Lie algebras

NASA Astrophysics Data System (ADS)

Vinogradov, A. M.

2018-01-01

This paper is a natural continuation of Vinogradov [J. Math. Phys. 58, 071703 (2017)] where we proved that any Lie algebra over an algebraically closed field or over R can be assembled in a number of steps from two elementary constituents, called dyons and triadons. Here we consider the problems of the construction and classification of those Lie algebras which can be assembled in one step from base dyons and triadons, called coaxial Lie algebras. The base dyons and triadons are Lie algebra structures that have only one non-trivial structure constant in a given basis, while coaxial Lie algebras are linear combinations of pairwise compatible base dyons and triadons. We describe the maximal families of pairwise compatible base dyons and triadons called clusters, and, as a consequence, we give a complete description of the coaxial Lie algebras. The remarkable fact is that dyons and triadons in clusters are self-organised in structural groups which are surrounded by casings and linked by connectives. We discuss generalisations and applications to the theory of deformations of Lie algebras.

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.

A look at scalable dense linear algebra libraries

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dongarra, J.J.; Van de Geijn, R.A.; Walker, D.W.

1992-01-01

We discuss the essential design features of a library of scalable software for performing dense linear algebra computations on distributed memory concurrent computers. The square block scattered decomposition is proposed as a flexible and general-purpose way of decomposing most, if not all, dense matrix problems. An object- oriented interface to the library permits more portable applications to be written, and is easy to learn and use, since details of the parallel implementation are hidden from the user. Experiments on the Intel Touchstone Delta system with a prototype code that uses the square block scattered decomposition to perform LU factorization aremore » presented and analyzed. It was found that the code was both scalable and efficient, performing at about 14 GFLOPS (double precision) for the largest problem considered.« less

A look at scalable dense linear algebra libraries

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dongarra, J.J.; Van de Geijn, R.A.; Walker, D.W.

1992-08-01

We discuss the essential design features of a library of scalable software for performing dense linear algebra computations on distributed memory concurrent computers. The square block scattered decomposition is proposed as a flexible and general-purpose way of decomposing most, if not all, dense matrix problems. An object- oriented interface to the library permits more portable applications to be written, and is easy to learn and use, since details of the parallel implementation are hidden from the user. Experiments on the Intel Touchstone Delta system with a prototype code that uses the square block scattered decomposition to perform LU factorization aremore » presented and analyzed. It was found that the code was both scalable and efficient, performing at about 14 GFLOPS (double precision) for the largest problem considered.« less

Investigating Students' Modes of Thinking in Linear Algebra: The Case of Linear Independence

ERIC Educational Resources Information Center

Çelik, Derya

2015-01-01

Linear algebra is one of the most challenging topics to learn and teach in many countries. To facilitate the teaching and learning of linear algebra, priority should be given to epistemologically analyze the concepts that the undergraduate students have difficulty in conceptualizing and to define their ways of reasoning in linear algebra. After…

Linear Equations. [Student Worksheets for Vocational Agricultural Courses].

ERIC Educational Resources Information Center

Jewell, Larry R.

This learning module provides students with practice in applying algebraic operations to vocational agriculture. The module consists of unit objectives, definitions, information, problems to solve, worksheets suitable for various levels of vocational agriculture instruction, and answer keys for the problems and worksheets. This module, which…

Implicit Plasma Kinetic Simulation Using The Jacobian-Free Newton-Krylov Method

NASA Astrophysics Data System (ADS)

Taitano, William; Knoll, Dana; Chacon, Luis

2009-11-01

The use of fully implicit time integration methods in kinetic simulation is still area of algorithmic research. A brute-force approach to simultaneously including the field equations and the particle distribution function would result in an intractable linear algebra problem. A number of algorithms have been put forward which rely on an extrapolation in time. They can be thought of as linearly implicit methods or one-step Newton methods. However, issues related to time accuracy of these methods still remain. We are pursuing a route to implicit plasma kinetic simulation which eliminates extrapolation, eliminates phase-space from the linear algebra problem, and converges the entire nonlinear system within a time step. We accomplish all this using the Jacobian-Free Newton-Krylov algorithm. The original research along these lines considered particle methods to advance the distribution function [1]. In the current research we are advancing the Vlasov equations on a grid. Results will be presented which highlight algorithmic details for single species electrostatic problems and coupled ion-electron electrostatic problems. [4pt] [1] H. J. Kim, L. Chac'on, G. Lapenta, ``Fully implicit particle in cell algorithm,'' 47th Annual Meeting of the Division of Plasma Physics, Oct. 24-28, 2005, Denver, CO

On ``Overestimation-free Computational Version of Interval Analysis''

NASA Astrophysics Data System (ADS)

Popova, Evgenija D.

2013-10-01

The transformation of interval parameters into trigonometric functions, proposed in Int. J. Comput. Meth. Eng. Sci. Mech., vol. 13, pp. 319-328 (2012), is not motivated in comparison to the infinitely many equivalent algebraic transformations. The conclusions about the efficacy of the methodology used are based on incorrect comparisons between solutions of different problems. We show theoretically, and in the examples considered in the commented article, that changing the number of the parameters in a system of linear algebraic equations may change the initial problem, respectively, its solution set. We also correct various misunderstandings and bugs that appear in the article noted above.

The Shock and Vibration Digest, Volume 18, Number 3

DTIC Science & Technology

1986-03-01

Linear Distributed Parameter Des., Proc. Intl. Symp., 11th ONR Naval Struc. Systems by Shifted Legendre Polynomial Func- Mech. Symp., Tucson, AZ, pp...University, Atlanta, Georgia nonlinear problems with elementary algebra . It J. Sound Vib., 102 (2), pp 247-257 (Sept 22, uses i = -1, the Pascal’s...eigenvalues specified. The optimal avoid failure due to resonance under the action control problem of a linear distributed parameter 0School of Mechanical

Computer-Based Feedback in Linear Algebra: Effects on Transfer Performance and Motivation

ERIC Educational Resources Information Center

Corbalan, Gemma; Paas, Fred; Cuypers, Hans

2010-01-01

Two studies investigated the effects on students' perceptions (Study 1) and learning and motivation (Study 2) of different levels of feedback in mathematical problems. In these problems, an error made in one step of the problem-solving procedure will carry over to the following steps and consequently to the final solution. Providing immediate…

Matrix with Prescribed Eigenvectors

ERIC Educational Resources Information Center

Ahmad, Faiz

2011-01-01

It is a routine matter for undergraduates to find eigenvalues and eigenvectors of a given matrix. But the converse problem of finding a matrix with prescribed eigenvalues and eigenvectors is rarely discussed in elementary texts on linear algebra. This problem is related to the "spectral" decomposition of a matrix and has important technical…

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…

Algebraic Thinking in Solving Linier Program at High School Level: Female Student’s Field Independent Cognitive Style

NASA Astrophysics Data System (ADS)

Hardiani, N.; Budayasa, I. K.; Juniati, D.

2018-01-01

The aim of this study was to describe algebraic thinking of high school female student’s field independent cognitive style in solving linier program problem by revealing deeply the female students’ responses. Subjects in this study were 7 female students having field independent cognitive style in class 11. The type of this research was descriptive qualitative. The method of data collection used was observation, documentation, and interview. Data analysis technique was by reduction, presentation, and conclusion. The results of this study showed that the female students with field independent cognitive style in solving the linier program problem had the ability to represent algebraic ideas from the narrative question that had been read by manipulating symbols and variables presented in tabular form, creating and building mathematical models in two variables linear inequality system which represented algebraic ideas, and interpreting the solutions as variables obtained from the point of intersection in the solution area to obtain maximum benefit.

Discrete Applied Mathematics

DTIC Science & Technology

1993-05-31

program. In paper [28], we give a brief and elementary proof of a result of Hoffman [1952) about approximate solutions to systems, of linear inequalities...UCLA, Vestvood, CA, February 1993. " Linear Problems: Formulation and Solution," International Linear Algebra Society, Pensacola, FL, May 1993. Denise S...thresAold If there is a number h and a linear k-separator w assigning a real number to each vertex so that for any subset S of vertices, the sum of w

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.

Analysis of junior high school students' attempt to solve a linear inequality problem

NASA Astrophysics Data System (ADS)

Taqiyuddin, Muhammad; Sumiaty, Encum; Jupri, Al

2017-08-01

Linear inequality is one of fundamental subjects within junior high school mathematics curricula. Several studies have been conducted to asses students' perform on linear inequality. However, it can hardly be found that linear inequality problems are in the form of "ax + b < dx + e" with "a, d ≠ 0", and "a ≠ d" as it can be seen on the textbook used by Indonesian students and several studies. This condition leads to the research questions concerning students' attempt on solving a simple linear inequality problem in this form. In order to do so, the written test was administered to 58 students from two schools in Bandung followed by interviews. The other sources of the data are from teachers' interview and mathematics books used by students. After that, the constant comparative method was used to analyse the data. The result shows that the majority approached the question by doing algebraic operations. Interestingly, most of them did it incorrectly. In contrast, algebraic operations were correctly used by some of them. Moreover, the others performed expected-numbers solution, rewriting the question, translating the inequality into words, and blank answer. Furthermore, we found that there is no one who was conscious of the existence of all-numbers solution. It was found that this condition is reasonably due to how little the learning components concern about why a procedure of solving a linear inequality works and possibilities of linear inequality solution.

Those Do What? Connecting Eigenvectors and Eigenvalues to the Rest of Linear Algebra: Using Visual Enhancements to Help Students Connect Eigenvectors to the Rest of Linear Algebra

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…

Interpolation problem for the solutions of linear elasticity equations based on monogenic functions

NASA Astrophysics Data System (ADS)

Grigor'ev, Yuri; Gürlebeck, Klaus; Legatiuk, Dmitrii

2017-11-01

Interpolation is an important tool for many practical applications, and very often it is beneficial to interpolate not only with a simple basis system, but rather with solutions of a certain differential equation, e.g. elasticity equation. A typical example for such type of interpolation are collocation methods widely used in practice. It is known, that interpolation theory is fully developed in the framework of the classical complex analysis. However, in quaternionic analysis, which shows a lot of analogies to complex analysis, the situation is more complicated due to the non-commutative multiplication. Thus, a fundamental theorem of algebra is not available, and standard tools from linear algebra cannot be applied in the usual way. To overcome these problems, a special system of monogenic polynomials the so-called Pseudo Complex Polynomials, sharing some properties of complex powers, is used. In this paper, we present an approach to deal with the interpolation problem, where solutions of elasticity equations in three dimensions are used as an interpolation basis.

On differential operators generating iterative systems of linear ODEs of maximal symmetry algebra

NASA Astrophysics Data System (ADS)

Ndogmo, J. C.

2017-06-01

Although every iterative scalar linear ordinary differential equation is of maximal symmetry algebra, the situation is different and far more complex for systems of linear ordinary differential equations, and an iterative system of linear equations need not be of maximal symmetry algebra. We illustrate these facts by examples and derive families of vector differential operators whose iterations are all linear systems of equations of maximal symmetry algebra. Some consequences of these results are also discussed.

Knowledge Building and Mathematics: Shifting the Responsibility for Knowledge Advancement and Engagement

ERIC Educational Resources Information Center

Moss, Joan; Beatty, Ruth

2010-01-01

Three classrooms of Grade 4 students from different schools and diverse backgrounds collaborated in early algebra research to solve a series of linear and quadratic generalizing problems. Results revealed that high- and low-achieving students were able to solve problems of recognized difficulty. We discuss Knowledge Building principles and…

Linear homotopy solution of nonlinear systems of equations in geodesy

NASA Astrophysics Data System (ADS)

Paláncz, Béla; Awange, Joseph L.; Zaletnyik, Piroska; Lewis, Robert H.

2010-01-01

A fundamental task in geodesy is solving systems of equations. Many geodetic problems are represented as systems of multivariate polynomials. A common problem in solving such systems is improper initial starting values for iterative methods, leading to convergence to solutions with no physical meaning, or to convergence that requires global methods. Though symbolic methods such as Groebner bases or resultants have been shown to be very efficient, i.e., providing solutions for determined systems such as 3-point problem of 3D affine transformation, the symbolic algebra can be very time consuming, even with special Computer Algebra Systems (CAS). This study proposes the Linear Homotopy method that can be implemented easily in high-level computer languages like C++ and Fortran that are faster than CAS by at least two orders of magnitude. Using Mathematica, the power of Homotopy is demonstrated in solving three nonlinear geodetic problems: resection, GPS positioning, and affine transformation. The method enlarging the domain of convergence is found to be efficient, less sensitive to rounding of numbers, and has lower complexity compared to other local methods like Newton-Raphson.

Improved Linear Algebra Methods for Redshift Computation from Limited Spectrum Data - II

NASA Technical Reports Server (NTRS)

Foster, Leslie; Waagen, Alex; Aijaz, Nabella; Hurley, Michael; Luis, Apolo; Rinsky, Joel; Satyavolu, Chandrika; Gazis, Paul; Srivastava, Ashok; Way, Michael

2008-01-01

Given photometric broadband measurements of a galaxy, Gaussian processes may be used with a training set to solve the regression problem of approximating the redshift of this galaxy. However, in practice solving the traditional Gaussian processes equation is too slow and requires too much memory. We employed several methods to avoid this difficulty using algebraic manipulation and low-rank approximation, and were able to quickly approximate the redshifts in our testing data within 17 percent of the known true values using limited computational resources. The accuracy of one method, the V Formulation, is comparable to the accuracy of the best methods currently used for this problem.

Study of coherent synchrotron radiation effects by means of a new simulation code based on the non-linear extension of the operator splitting method

NASA Astrophysics Data System (ADS)

Dattoli, G.; Migliorati, M.; Schiavi, A.

2007-05-01

The coherent synchrotron radiation (CSR) is one of the main problems limiting the performance of high-intensity electron accelerators. The complexity of the physical mechanisms underlying the onset of instabilities due to CSR demands for accurate descriptions, capable of including the large number of features of an actual accelerating device. A code devoted to the analysis of these types of problems should be fast and reliable, conditions that are usually hardly achieved at the same time. In the past, codes based on Lie algebraic techniques have been very efficient to treat transport problems in accelerators. The extension of these methods to the non-linear case is ideally suited to treat CSR instability problems. We report on the development of a numerical code, based on the solution of the Vlasov equation, with the inclusion of non-linear contribution due to wake field effects. The proposed solution method exploits an algebraic technique that uses the exponential operators. We show that the integration procedure is capable of reproducing the onset of instability and the effects associated with bunching mechanisms leading to the growth of the instability itself. In addition, considerations on the threshold of the instability are also developed.

An Algebraic Construction of the First Integrals of the Stationary KdV Hierarchy

NASA Astrophysics Data System (ADS)

Matsushima, Masatomo; Ohmiya, Mayumi

2009-09-01

The stationary KdV hierarchy is constructed using a kind of recursion operator called Λ-operator. The notion of the maximal solution of the n-th stationary KdV equation is introduced. Using this maximal solution, a specific differential polynomial with the auxiliary spectral parameter called the spectral M-function is constructed as the quadratic form of the fundamental system of the eigenvalue problem for the 2-nd order linear ordinary differential equation which is related to the linearizing operator of the hierarchy. By calculating a perfect square condition of the quadratic form by an elementary algebraic method, the complete set of first integrals of this hierarchy is constructed.

AN EFFICIENT HIGHER-ORDER FAST MULTIPOLE BOUNDARY ELEMENT SOLUTION FOR POISSON-BOLTZMANN BASED MOLECULAR ELECTROSTATICS

PubMed Central

Bajaj, Chandrajit; Chen, Shun-Chuan; Rand, Alexander

2011-01-01

In order to compute polarization energy of biomolecules, we describe a boundary element approach to solving the linearized Poisson-Boltzmann equation. Our approach combines several important features including the derivative boundary formulation of the problem and a smooth approximation of the molecular surface based on the algebraic spline molecular surface. State of the art software for numerical linear algebra and the kernel independent fast multipole method is used for both simplicity and efficiency of our implementation. We perform a variety of computational experiments, testing our method on a number of actual proteins involved in molecular docking and demonstrating the effectiveness of our solver for computing molecular polarization energy. PMID:21660123

Linear decentralized systems with special structure. [for twin lift helicopters

NASA Technical Reports Server (NTRS)

Martin, C. F.

1982-01-01

Certain fundamental structures associated with linear systems having internal symmetries are outlined. It is shown that the theory of finite-dimensional algebras and their representations are closely related to such systems. It is also demonstrated that certain problems in the decentralized control of symmetric systems are equivalent to long-standing problems of linear systems theory. Even though the structure imposed arose in considering the problems of twin-lift helicopters, any large system composed of several identical intercoupled control systems can be modeled by a linear system that satisfies the constraints imposed. Internal symmetry can be exploited to yield new system-theoretic invariants and a better understanding of the way in which the underlying structure affects overall system performance.

A spatial operator algebra for manipulator modeling and control

NASA Technical Reports Server (NTRS)

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

1989-01-01

A recently developed spatial operator algebra, useful for modeling, control, and trajectory design of manipulators is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high level framework for describing the dynamic and kinematic behavior of a manipulator and control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanizaton of specific algorithms is greatly simplified. The analytical formulation of the operator algebra, as well as its implementation in the Ada programming language are discussed.

Algebraic approach to electronic spectroscopy and dynamics.

PubMed

Toutounji, Mohamad

2008-04-28

Lie algebra, Zassenhaus, and parameter differentiation techniques are utilized to break up the exponential of a bilinear Hamiltonian operator into a product of noncommuting exponential operators by the virtue of the theory of Wei and Norman [J. Math. Phys. 4, 575 (1963); Proc. Am. Math. Soc., 15, 327 (1964)]. There are about three different ways to find the Zassenhaus exponents, namely, binomial expansion, Suzuki formula, and q-exponential transformation. A fourth, and most reliable method, is provided. Since linearly displaced and distorted (curvature change upon excitation/emission) Hamiltonian and spin-boson Hamiltonian may be classified as bilinear Hamiltonians, the presented algebraic algorithm (exponential operator disentanglement exploiting six-dimensional Lie algebra case) should be useful in spin-boson problems. The linearly displaced and distorted Hamiltonian exponential is only treated here. While the spin-boson model is used here only as a demonstration of the idea, the herein approach is more general and powerful than the specific example treated. The optical linear dipole moment correlation function is algebraically derived using the above mentioned methods and coherent states. Coherent states are eigenvectors of the bosonic lowering operator a and not of the raising operator a(+). While exp(a(+)) translates coherent states, exp(a(+)a(+)) operation on coherent states has always been a challenge, as a(+) has no eigenvectors. Three approaches, and the results, of that operation are provided. Linear absorption spectra are derived, calculated, and discussed. The linear dipole moment correlation function for the pure quadratic coupling case is expressed in terms of Legendre polynomials to better show the even vibronic transitions in the absorption spectrum. Comparison of the present line shapes to those calculated by other methods is provided. Franck-Condon factors for both linear and quadratic couplings are exactly accounted for by the herein calculated linear absorption spectra. This new methodology should easily pave the way to calculating the four-point correlation function, F(tau(1),tau(2),tau(3),tau(4)), of which the optical nonlinear response function may be procured, as evaluating F(tau(1),tau(2),tau(3),tau(4)) is only evaluating the optical linear dipole moment correlation function iteratively over different time intervals, which should allow calculating various optical nonlinear temporal/spectral signals.

Derive Workshop Matrix Algebra and Linear Algebra.

ERIC Educational Resources Information Center

Townsley Kulich, Lisa; Victor, Barbara

This document presents the course content for a workshop that integrates the use of the computer algebra system Derive with topics in matrix and linear algebra. The first section is a guide to using Derive that provides information on how to write algebraic expressions, make graphs, save files, edit, define functions, differentiate expressions,…

Computing Gröbner Bases within Linear Algebra

NASA Astrophysics Data System (ADS)

Suzuki, Akira

In this paper, we present an alternative algorithm to compute Gröbner bases, which is based on computations on sparse linear algebra. Both of S-polynomial computations and monomial reductions are computed in linear algebra simultaneously in this algorithm. So it can be implemented to any computational system which can handle linear algebra. For a given ideal in a polynomial ring, it calculates a Gröbner basis along with the corresponding term order appropriately.

Using Example Generation to Explore Students' Understanding of the Concepts of Linear Dependence/Independence in Linear Algebra

ERIC Educational Resources Information Center

Aydin, Sinan

2014-01-01

Linear algebra is a basic mathematical subject taught in mathematics and science depar-tments of universities. The teaching and learning of this course has always been difficult. This study aims to contribute to the research in linear algebra education, focusing on linear dependence and independence concepts. This was done by introducing…

Students’ difficulties in solving linear equation problems

NASA Astrophysics Data System (ADS)

Wati, S.; Fitriana, L.; Mardiyana

2018-03-01

A linear equation is an algebra material that exists in junior high school to university. It is a very important material for students in order to learn more advanced mathematics topics. Therefore, linear equation material is essential to be mastered. However, the result of 2016 national examination in Indonesia showed that students’ achievement in solving linear equation problem was low. This fact became a background to investigate students’ difficulties in solving linear equation problems. This study used qualitative descriptive method. An individual written test on linear equation tasks was administered, followed by interviews. Twenty-one sample students of grade VIII of SMPIT Insan Kamil Karanganyar did the written test, and 6 of them were interviewed afterward. The result showed that students with high mathematics achievement donot have difficulties, students with medium mathematics achievement have factual difficulties, and students with low mathematics achievement have factual, conceptual, operational, and principle difficulties. Based on the result there is a need of meaningfulness teaching strategy to help students to overcome difficulties in solving linear equation problems.

Monte Carlo, Probability, Algebra, and Pi.

ERIC Educational Resources Information Center

Hinders, Duane C.

1981-01-01

The uses of random number generators are illustrated in three ways: (1) the solution of a probability problem using a coin; (2) the solution of a system of simultaneous linear equations using a die; and (3) the approximation of pi using darts. (MP)

Cayley transform on Stiefel manifolds

NASA Astrophysics Data System (ADS)

Macías-Virgós, Enrique; Pereira-Sáez, María José; Tanré, Daniel

2018-01-01

The Cayley transform for orthogonal groups is a well known construction with applications in real and complex analysis, linear algebra and computer science. In this work, we construct Cayley transforms on Stiefel manifolds. Applications to the Lusternik-Schnirelmann category and optimization problems are presented.

Linear-Algebra Programs

NASA Technical Reports Server (NTRS)

Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.

1982-01-01

The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.

Non-algebraic integrability of the Chew-Low reversible dynamical system of the Cremona type and the relation with the 7th Hilbert problem (non-resonant case)

NASA Astrophysics Data System (ADS)

Rerikh, K. V.

A smooth reversible dynamical system (SRDS) and a system of nonlinear functional equations, defined by a certain rational quadratic Cremona mapping and arising from the static model of the dispersion approach in the theory of strong interactions (the Chew-Low equations for p- wave πN- scattering) are considered. This SRDS is splitted into 1- and 2-dimensional ones. An explicit Cremona transformation that completely determines the exact solution of the two-dimensional system is found. This solution depends on an odd function satisfying a nonlinear autonomous 3-point functional equation. Non-algebraic integrability of SRDS under consideration is proved using the method of Poincaré normal forms and the Siegel theorem on biholomorphic linearization of a mapping at a non-resonant fixed point. The proof is based on the classical Feldman-Baker theorem on linear forms of logarithms of algebraic numbers, which, in turn, relies upon solving the 7th Hilbert problem by A.I. Gel'fond and T. Schneider and new powerful methods of A. Baker in the theory of transcendental numbers. The general theorem, following from the Feldman-Baker theorem, on applicability of the Siegel theorem to the set of the eigenvalues λ ɛ Cn of a mapping at a non-resonant fixed point which belong to the algebraic number field A is formulated and proved. The main results are presented in Theorems 1-3, 5, 7, 8 and Remarks 3, 7.

Short Round Sub-Linear Zero-Knowledge Argument for Linear Algebraic Relations

NASA Astrophysics Data System (ADS)

Seo, Jae Hong

Zero-knowledge arguments allows one party to prove that a statement is true, without leaking any other information than the truth of the statement. In many applications such as verifiable shuffle (as a practical application) and circuit satisfiability (as a theoretical application), zero-knowledge arguments for mathematical statements related to linear algebra are essentially used. Groth proposed (at CRYPTO 2009) an elegant methodology for zero-knowledge arguments for linear algebraic relations over finite fields. He obtained zero-knowledge arguments of the sub-linear size for linear algebra using reductions from linear algebraic relations to equations of the form z = x *' y, where x, y ∈ Fnp are committed vectors, z ∈ Fp is a committed element, and *' : Fnp × Fnp → Fp is a bilinear map. These reductions impose additional rounds on zero-knowledge arguments of the sub-linear size. The round complexity of interactive zero-knowledge arguments is an important measure along with communication and computational complexities. We focus on minimizing the round complexity of sub-linear zero-knowledge arguments for linear algebra. To reduce round complexity, we propose a general transformation from a t-round zero-knowledge argument, satisfying mild conditions, to a (t - 2)-round zero-knowledge argument; this transformation is of independent interest.

An Inquiry-Based Linear Algebra Class

ERIC Educational Resources Information Center

Wang, Haohao; Posey, Lisa

2011-01-01

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

Numerical Linear Algebra.

DTIC Science & Technology

1980-09-08

February 1979 through 31 March 1980 Title of Research: NUMERICAL LINEAR ALGEBRA Principal Investigators: Gene H. Golub James H. Wilkinson Research...BEFORE COMPLETING FORM 2 OTAgSSION NO. 3. RECIPIENT’S CATALOG NUMBER ITE~ btitle) ~qEE NUMERICAL LINEAR ALGEBRA #I ~ f#7&/8 PER.ORMING ORG. REPORT NUM 27R 7

Linear {GLP}-algebras and their elementary theories

NASA Astrophysics Data System (ADS)

Pakhomov, F. N.

2016-12-01

The polymodal provability logic {GLP} was introduced by Japaridze in 1986. It is the provability logic of certain chains of provability predicates of increasing strength. Every polymodal logic corresponds to a variety of polymodal algebras. Beklemishev and Visser asked whether the elementary theory of the free {GLP}-algebra generated by the constants \\mathbf{0}, \\mathbf{1} is decidable [1]. For every positive integer n we solve the corresponding question for the logics {GLP}_n that are the fragments of {GLP} with n modalities. We prove that the elementary theory of the free {GLP}_n-algebra generated by the constants \\mathbf{0}, \\mathbf{1} is decidable for all n. We introduce the notion of a linear {GLP}_n-algebra and prove that all free {GLP}_n-algebras generated by the constants \\mathbf{0}, \\mathbf{1} are linear. We also consider the more general case of the logics {GLP}_α whose modalities are indexed by the elements of a linearly ordered set α: we define the notion of a linear algebra and prove the latter result in this case.

The method of Ritz applied to the equation of Hamilton. [for pendulum systems

NASA Technical Reports Server (NTRS)

Bailey, C. D.

1976-01-01

Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.

A Comparison of Solver Performance for Complex Gastric Electrophysiology Models

PubMed Central

Sathar, Shameer; Cheng, Leo K.; Trew, Mark L.

2016-01-01

Computational techniques for solving systems of equations arising in gastric electrophysiology have not been studied for efficient solution process. We present a computationally challenging problem of simulating gastric electrophysiology in anatomically realistic stomach geometries with multiple intracellular and extracellular domains. The multiscale nature of the problem and mesh resolution required to capture geometric and functional features necessitates efficient solution methods if the problem is to be tractable. In this study, we investigated and compared several parallel preconditioners for the linear systems arising from tetrahedral discretisation of electrically isotropic and anisotropic problems, with and without stimuli. The results showed that the isotropic problem was computationally less challenging than the anisotropic problem and that the application of extracellular stimuli increased workload considerably. Preconditioning based on block Jacobi and algebraic multigrid solvers were found to have the best overall solution times and least iteration counts, respectively. The algebraic multigrid preconditioner would be expected to perform better on large problems. PMID:26736543

Some estimation formulae for continuous time-invariant linear systems

NASA Technical Reports Server (NTRS)

Bierman, G. J.; Sidhu, G. S.

1975-01-01

In this brief paper we examine a Riccati equation decomposition due to Reid and Lainiotis and apply the result to the continuous time-invariant linear filtering problem. Exploitation of the time-invariant structure leads to integration-free covariance recursions which are of use in covariance analyses and in filter implementations. A super-linearly convergent iterative solution to the algebraic Riccati equation (ARE) is developed. The resulting algorithm, arranged in a square-root form, is thought to be numerically stable and competitive with other ARE solution methods. Certain covariance relations that are relevant to the fixed-point and fixed-lag smoothing problems are also discussed.

Visualizing the inner product space ℝm×n in a MATLAB-assisted linear algebra classroom

NASA Astrophysics Data System (ADS)

Caglayan, Günhan

2018-05-01

This linear algebra note offers teaching and learning ideas in the treatment of the inner product space ? in a technology-supported learning environment. Classroom activities proposed in this note demonstrate creative ways of integrating MATLAB technology into various properties of Frobenius inner product as visualization tools that complement the algebraic approach. As implemented in linear algebra lessons in a university in the Unites States, the article also incorporates algebraic and visual work of students who experienced these activities with MATLAB software. The connection between the Frobenius norm and the Euclidean norm is also emphasized.

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.)

Resources for Teaching Linear Algebra. MAA Notes Volume 42.

ERIC Educational Resources Information Center

Carlson, David, Ed.; And Others

This book takes the position that the teaching of elementary linear algebra can be made more effective by emphasizing applications, exposition, and pedagogy. It includes the recommendations of the Linear Algebra Curriculum Study Group with their core syllabus for the first course, and the thoughts of mathematics faculty who have taught linear…

Emphasizing Language and Visualization in Teaching Linear Algebra

ERIC Educational Resources Information Center

Hannah, John; Stewart, Sepideh; Thomas, Mike

2013-01-01

Linear algebra with its rich theoretical nature is a first step towards advanced mathematical thinking for many undergraduate students. In this paper, we consider the teaching approach of an experienced mathematician as he attempts to engage his students with the key ideas embedded in a second-year course in linear algebra. We describe his…

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

NASA Astrophysics Data System (ADS)

Moraes Rêgo, Patrícia Helena; Viana da Fonseca Neto, João; Ferreira, Ernesto M.

2015-08-01

The main focus of this article is to present a proposal to solve, via UDUT factorisation, the convergence and numerical stability problems that are related to the covariance matrix ill-conditioning of the recursive least squares (RLS) approach for online approximations of the algebraic Riccati equation (ARE) solution associated with the discrete linear quadratic regulator (DLQR) problem formulated in the actor-critic reinforcement learning and approximate dynamic programming context. The parameterisations of the Bellman equation, utility function and dynamic system as well as the algebra of Kronecker product assemble a framework for the solution of the DLQR problem. The condition number and the positivity parameter of the covariance matrix are associated with statistical metrics for evaluating the approximation performance of the ARE solution via RLS-based estimators. The performance of RLS approximators is also evaluated in terms of consistence and polarisation when associated with reinforcement learning methods. The used methodology contemplates realisations of online designs for DLQR controllers that is evaluated in a multivariable dynamic system model.

Dynamical basis sets for algebraic variational calculations in quantum-mechanical scattering theory

NASA Technical Reports Server (NTRS)

Sun, Yan; Kouri, Donald J.; Truhlar, Donald G.; Schwenke, David W.

1990-01-01

New basis sets are proposed for linear algebraic variational calculations of transition amplitudes in quantum-mechanical scattering problems. These basis sets are hybrids of those that yield the Kohn variational principle (KVP) and those that yield the generalized Newton variational principle (GNVP) when substituted in Schlessinger's stationary expression for the T operator. Trial calculations show that efficiencies almost as great as that of the GNVP and much greater than the KVP can be obtained, even for basis sets with the majority of the members independent of energy.

Dual-scale topology optoelectronic processor.

PubMed

Marsden, G C; Krishnamoorthy, A V; Esener, S C; Lee, S H

1991-12-15

The dual-scale topology optoelectronic processor (D-STOP) is a parallel optoelectronic architecture for matrix algebraic processing. The architecture can be used for matrix-vector multiplication and two types of vector outer product. The computations are performed electronically, which allows multiplication and summation concepts in linear algebra to be generalized to various nonlinear or symbolic operations. This generalization permits the application of D-STOP to many computational problems. The architecture uses a minimum number of optical transmitters, which thereby reduces fabrication requirements while maintaining area-efficient electronics. The necessary optical interconnections are space invariant, minimizing space-bandwidth requirements.

Algebraic multigrid preconditioners for two-phase flow in porous media with phase transitions

NASA Astrophysics Data System (ADS)

Bui, Quan M.; Wang, Lu; Osei-Kuffuor, Daniel

2018-04-01

Multiphase flow is a critical process in a wide range of applications, including oil and gas recovery, carbon sequestration, and contaminant remediation. Numerical simulation of multiphase flow requires solving of a large, sparse linear system resulting from the discretization of the partial differential equations modeling the flow. In the case of multiphase multicomponent flow with miscible effect, this is a very challenging task. The problem becomes even more difficult if phase transitions are taken into account. A new approach to handle phase transitions is to formulate the system as a nonlinear complementarity problem (NCP). Unlike in the primary variable switching technique, the set of primary variables in this approach is fixed even when there is phase transition. Not only does this improve the robustness of the nonlinear solver, it opens up the possibility to use multigrid methods to solve the resulting linear system. The disadvantage of the complementarity approach, however, is that when a phase disappears, the linear system has the structure of a saddle point problem and becomes indefinite, and current algebraic multigrid (AMG) algorithms cannot be applied directly. In this study, we explore the effectiveness of a new multilevel strategy, based on the multigrid reduction technique, to deal with problems of this type. We demonstrate the effectiveness of the method through numerical results for the case of two-phase, two-component flow with phase appearance/disappearance. We also show that the strategy is efficient and scales optimally with problem size.

Analysis of algebraic reasoning ability of cognitive style perspectives on field dependent field independent and gender

NASA Astrophysics Data System (ADS)

Rosita, N. T.

2018-03-01

The purpose of this study is to analyse algebraic reasoning ability using the SOLO model as a theoretical framework to assess students’ algebraic reasoning abilities of Field Dependent cognitive (FD), Field Independent (FI) and Gender perspectives. The method of this study is a qualitative research. The instrument of this study is the researcher himself assisted with algebraic reasoning tests, the problems have been designed based on NCTM indicators and algebraic reasoning according to SOLO model. While the cognitive style of students is determined using Group Embedded Figure Test (GEFT), as well as interviews on the subject as triangulation. The subjects are 15 female and 15 males of the sixth semester students of mathematics education, STKIP Sebelas April. The results of the qualitative data analysis is that most subjects are at the level of unistructural and multi-structural, subjects at the relational level have difficulty in forming a new linear pattern. While the subjects at the extended abstract level are able to meet all the indicators of algebraic reasoning ability even though some of the answers are not perfect yet. Subjects of FI tend to have higher algebraic reasoning abilities than of the subject of FD.

Multicriterion problem of allocation of resources in the heterogeneous distributed information processing systems

NASA Astrophysics Data System (ADS)

Antamoshkin, O. A.; Kilochitskaya, T. R.; Ontuzheva, G. A.; Stupina, A. A.; Tynchenko, V. S.

2018-05-01

This study reviews the problem of allocation of resources in the heterogeneous distributed information processing systems, which may be formalized in the form of a multicriterion multi-index problem with the linear constraints of the transport type. The algorithms for solution of this problem suggest a search for the entire set of Pareto-optimal solutions. For some classes of hierarchical systems, it is possible to significantly speed up the procedure of verification of a system of linear algebraic inequalities for consistency due to the reducibility of them to the stream models or the application of other solution schemes (for strongly connected structures) that take into account the specifics of the hierarchies under consideration.

Predicting the Underwater Sound of Moderate and Heavy Rainfall from Laboratory Measurements of Radiation from Single Large Raindrops

DTIC Science & Technology

1992-03-01

Elementary Linear Algebra with Applications, pp. 301- 323, John Wiley and Sons Inc., 1987. Atlas, D., and Ulbrich, C. E. W., "The Physical Basis for...vector drd In this case, the linear system is said to be inconsistent ( Anton and Rorres, 1987). In contrast, for an underdetermined system (where the...ocean acoustical tomography and seismology. In simplest terms, the general linear inverse problem consists of fimding the desired solution to a set of m

Algebraic approach to electronic spectroscopy and dynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Toutounji, Mohamad

Lie algebra, Zassenhaus, and parameter differentiation techniques are utilized to break up the exponential of a bilinear Hamiltonian operator into a product of noncommuting exponential operators by the virtue of the theory of Wei and Norman [J. Math. Phys. 4, 575 (1963); Proc. Am. Math. Soc., 15, 327 (1964)]. There are about three different ways to find the Zassenhaus exponents, namely, binomial expansion, Suzuki formula, and q-exponential transformation. A fourth, and most reliable method, is provided. Since linearly displaced and distorted (curvature change upon excitation/emission) Hamiltonian and spin-boson Hamiltonian may be classified as bilinear Hamiltonians, the presented algebraic algorithm (exponentialmore » operator disentanglement exploiting six-dimensional Lie algebra case) should be useful in spin-boson problems. The linearly displaced and distorted Hamiltonian exponential is only treated here. While the spin-boson model is used here only as a demonstration of the idea, the herein approach is more general and powerful than the specific example treated. The optical linear dipole moment correlation function is algebraically derived using the above mentioned methods and coherent states. Coherent states are eigenvectors of the bosonic lowering operator a and not of the raising operator a{sup +}. While exp(a{sup +}) translates coherent states, exp(a{sup +}a{sup +}) operation on coherent states has always been a challenge, as a{sup +} has no eigenvectors. Three approaches, and the results, of that operation are provided. Linear absorption spectra are derived, calculated, and discussed. The linear dipole moment correlation function for the pure quadratic coupling case is expressed in terms of Legendre polynomials to better show the even vibronic transitions in the absorption spectrum. Comparison of the present line shapes to those calculated by other methods is provided. Franck-Condon factors for both linear and quadratic couplings are exactly accounted for by the herein calculated linear absorption spectra. This new methodology should easily pave the way to calculating the four-point correlation function, F({tau}{sub 1},{tau}{sub 2},{tau}{sub 3},{tau}{sub 4}), of which the optical nonlinear response function may be procured, as evaluating F({tau}{sub 1},{tau}{sub 2},{tau}{sub 3},{tau}{sub 4}) is only evaluating the optical linear dipole moment correlation function iteratively over different time intervals, which should allow calculating various optical nonlinear temporal/spectral signals.« less

Advanced Mathematics Online: Assessing Particularities in the Online Delivery of a Second Linear Algebra Course

ERIC Educational Resources Information Center

Montiel, Mariana; Bhatti, Uzma

2010-01-01

This article presents an overview of some issues that were confronted when delivering an online second Linear Algebra course (assuming a previous Introductory Linear Algebra course) to graduate students enrolled in a Secondary Mathematics Education program. The focus is on performance in one particular aspect of the course: "change of basis" and…

Supporting Students' Understanding of Linear Equations with One Variable Using Algebra Tiles

ERIC Educational Resources Information Center

Saraswati, Sari; Putri, Ratu Ilma Indra; Somakim

2016-01-01

This research aimed to describe how algebra tiles can support students' understanding of linear equations with one variable. This article is a part of a larger research on learning design of linear equations with one variable using algebra tiles combined with balancing method. Therefore, it will merely discuss one activity focused on how students…

Linear Algebra Revisited: An Attempt to Understand Students' Conceptual Difficulties

ERIC Educational Resources Information Center

Britton, Sandra; Henderson, Jenny

2009-01-01

This article looks at some of the conceptual difficulties that students have in a linear algebra course. An overview of previous research in this area is given, and the various theories that have been espoused regarding the reasons that students find linear algebra so difficult are discussed. Student responses to two questions testing the ability…

Newton's method: A link between continuous and discrete solutions of nonlinear problems

NASA Technical Reports Server (NTRS)

Thurston, G. A.

1980-01-01

Newton's method for nonlinear mechanics problems replaces the governing nonlinear equations by an iterative sequence of linear equations. When the linear equations are linear differential equations, the equations are usually solved by numerical methods. The iterative sequence in Newton's method can exhibit poor convergence properties when the nonlinear problem has multiple solutions for a fixed set of parameters, unless the iterative sequences are aimed at solving for each solution separately. The theory of the linear differential operators is often a better guide for solution strategies in applying Newton's method than the theory of linear algebra associated with the numerical analogs of the differential operators. In fact, the theory for the differential operators can suggest the choice of numerical linear operators. In this paper the method of variation of parameters from the theory of linear ordinary differential equations is examined in detail in the context of Newton's method to demonstrate how it might be used as a guide for numerical solutions.

Convergence analysis of modulus-based matrix splitting iterative methods for implicit complementarity problems.

PubMed

Wang, An; Cao, Yang; Shi, Quan

2018-01-01

In this paper, we demonstrate a complete version of the convergence theory of the modulus-based matrix splitting iteration methods for solving a class of implicit complementarity problems proposed by Hong and Li (Numer. Linear Algebra Appl. 23:629-641, 2016). New convergence conditions are presented when the system matrix is a positive-definite matrix and an [Formula: see text]-matrix, respectively.

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

NASA Technical Reports Server (NTRS)

Ito, Kazufumi

1987-01-01

The linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces is considered. The optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence pi sup N of finite dimensional approximations of the solution to ARE. A sufficient condition that shows pi sup N converges strongly to pi is obtained. Under this condition, a formula is derived which can be used to obtain a rate of convergence of pi sup N to pi. The results of the Galerkin approximation is demonstrated and applied for parabolic systems and the averaging approximation for hereditary differential systems.

Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-Categories

NASA Astrophysics Data System (ADS)

Cheng, Tao; Huang, Hua-Lin; Yang, Yuping

2016-01-01

By viewing Clifford algebras as algebras in some suitable symmetric Gr-categories, Albuquerque and Majid were able to give a new derivation of some well known results about Clifford algebras and to generalize them. Along the same line, Bulacu observed that Clifford algebras are weak Hopf algebras in the aforementioned categories and obtained other interesting properties. The aim of this paper is to study generalized Clifford algebras in a similar manner and extend the results of Albuquerque, Majid and Bulacu to the generalized setting. In particular, by taking full advantage of the gauge transformations in symmetric linear Gr-categories, we derive the decomposition theorem and provide categorical weak Hopf structures for generalized Clifford algebras in a conceptual and simpler manner.

The Linear Algebra Curriculum Study Group Recommendations for the First Course in Linear Algebra.

ERIC Educational Resources Information Center

Carlson, David; And Others

1993-01-01

Presents five recommendations of the Linear Algebra Curriculum Study Group: (1) The syllabus must respond to the client disciplines; (2) The first course should be matrix oriented; (3) Faculty should consider the needs and interests of students; (4) Faculty should use technology; and (5) At least one follow-up course should be required. Provides a…

Chinese Algebra: Using Historical Problems to Think about Current Curricula

ERIC Educational Resources Information Center

Tillema, Erik

2005-01-01

The Chinese used the idea of generating equivalent expressions for solving problems where the problems from a historical Chinese text are studied to understand the ways in which the ideas can lead into algebraic calculations and help students to learn algebra. The texts unify algebraic problem solving through complex algebraic thought and afford…

Linear frictional forces cause orbits to neither circularize nor precess

NASA Astrophysics Data System (ADS)

Hamilton, B.; Crescimanno, M.

2008-06-01

For the undamped Kepler potential the lack of precession has historically been understood in terms of the Runge-Lenz symmetry. For the damped Kepler problem this result may be understood in terms of the generalization of Poisson structure to damped systems suggested recently by Tarasov (2005 J. Phys. A: Math. Gen. 38 2145). In this generalized algebraic structure the orbit-averaged Runge-Lenz vector remains a constant in the linearly damped Kepler problem to leading order in the damping coefficient. Beyond Kepler, we prove that, for any potential proportional to a power of the radius, the orbit shape and precession angle remain constant to leading order in the linear friction coefficient.

Fahrenheit to Celsius: An Exploration in College Algebra.

ERIC Educational Resources Information Center

Fay, Temple H.; Hardie, Keith A.

2003-01-01

Suggests that the classical exact formula for the conversion of degrees Celsius and degrees Fahrenheit is not user-friendly. Offers an approximate linear transformation that is easier to remember and use. Investigates both the exact conversion and the approximate conversion and provides interesting and relevant problems for small group…

Individual and Collective Analyses of the Genesis of Student Reasoning Regarding the Invertible Matrix Theorem in Linear Algebra

ERIC Educational Resources Information Center

Wawro, Megan Jean

2011-01-01

In this study, I considered the development of mathematical meaning related to the Invertible Matrix Theorem (IMT) for both a classroom community and an individual student over time. In this particular linear algebra course, the IMT was a core theorem in that it connected many concepts fundamental to linear algebra through the notion of…

A spatial operator algebra for manipulator modeling and control

NASA Technical Reports Server (NTRS)

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

1989-01-01

A spatial operator algebra for modeling the control and trajectory design of manipulation is discussed, with emphasis on its analytical formulation and implementation in the Ada programming language. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of the manipulator. Inversion is obtained using techniques of recursive filtering and smoothing. The operator alegbra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and control and trajectory design algorithms. Implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection, thus greatly simplifying the transition from an abstract problem formulation and solution to the detailed mechanization of a specific algorithm.

Homotopy approach to optimal, linear quadratic, fixed architecture compensation

NASA Technical Reports Server (NTRS)

Mercadal, Mathieu

1991-01-01

Optimal linear quadratic Gaussian compensators with constrained architecture are a sensible way to generate good multivariable feedback systems meeting strict implementation requirements. The optimality conditions obtained from the constrained linear quadratic Gaussian are a set of highly coupled matrix equations that cannot be solved algebraically except when the compensator is centralized and full order. An alternative to the use of general parameter optimization methods for solving the problem is to use homotopy. The benefit of the method is that it uses the solution to a simplified problem as a starting point and the final solution is then obtained by solving a simple differential equation. This paper investigates the convergence properties and the limitation of such an approach and sheds some light on the nature and the number of solutions of the constrained linear quadratic Gaussian problem. It also demonstrates the usefulness of homotopy on an example of an optimal decentralized compensator.

Bisimulation equivalence of differential-algebraic systems

NASA Astrophysics Data System (ADS)

Megawati, Noorma Yulia; Schaft, Arjan van der

2018-01-01

In this paper, the notion of bisimulation relation for linear input-state-output systems is extended to general linear differential-algebraic (DAE) systems. Geometric control theory is used to derive a linear-algebraic characterisation of bisimulation relations, and an algorithm for computing the maximal bisimulation relation between two linear DAE systems. The general definition is specialised to the case where the matrix pencil sE - A is regular. Furthermore, by developing a one-sided version of bisimulation, characterisations of simulation and abstraction are obtained.

Parallel Algorithms for Least Squares and Related Computations.

DTIC Science & Technology

1991-03-22

for dense computations in linear algebra . The work has recently been published in a general reference book on parallel algorithms by SIAM. AFO SR...written his Ph.D. dissertation with the principal investigator. (See publication 6.) • Parallel Algorithms for Dense Linear Algebra Computations. Our...and describe and to put into perspective a selection of the more important parallel algorithms for numerical linear algebra . We give a major new

University of Chicago School Mathematics Project (UCSMP) Algebra. WWC Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2009

2009-01-01

University of Chicago School Mathematics Project (UCSMP) Algebra is a one-year course covering three primary topics: (1) linear and quadratic expressions, sentences, and functions; (2) exponential expressions and functions; and (3) linear systems. Topics from geometry, probability, and statistics are integrated with the appropriate algebra.…

Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence.

PubMed

Ozaktas, Haldun M; Yüksel, Serdar; Kutay, M Alper

2002-08-01

A linear algebraic theory of partial coherence is presented that allows precise mathematical definitions of concepts such as coherence and incoherence. This not only provides new perspectives and insights but also allows us to employ the conceptual and algebraic tools of linear algebra in applications. We define several scalar measures of the degree of partial coherence of an optical field that are zero for full incoherence and unity for full coherence. The mathematical definitions are related to our physical understanding of the corresponding concepts by considering them in the context of Young's experiment.

Gender differences in algebraic thinking ability to solve mathematics problems

NASA Astrophysics Data System (ADS)

Kusumaningsih, W.; Darhim; Herman, T.; Turmudi

2018-05-01

This study aimed to conduct a gender study on students' algebraic thinking ability in solving a mathematics problem, polyhedron concept, for grade VIII. This research used a qualitative method. The data was collected using: test and interview methods. The subjects in this study were eight male and female students with different level of abilities. It was found that the algebraic thinking skills of male students reached high group of five categories. They were superior in terms of reasoning and quick understanding in solving problems. Algebraic thinking ability of high-achieving group of female students also met five categories of algebraic thinking indicators. They were more diligent, tenacious and thorough in solving problems. Algebraic thinking ability of male students in medium category only satisfied three categories of algebraic thinking indicators. They were sufficient in terms of reasoning and understanding in solving problems. Algebraic thinking ability group of female students in medium group also satisfied three categories of algebraic thinking indicators. They were fairly diligent, tenacious and meticulous on working on the problems.

An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher-KPP equation

NASA Astrophysics Data System (ADS)

Shapovalov, A. V.; Trifonov, A. Yu.

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher-KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

smoothG

DOE Office of Scientific and Technical Information (OSTI.GOV)

Barker, Andrew T.; Gelever, Stephan A.; Lee, Chak S.

2017-12-12

smoothG is a collection of parallel C++ classes/functions that algebraically constructs reduced models of different resolutions from a given high-fidelity graph model. In addition, smoothG also provides efficient linear solvers for the reduced models. Other than pure graph problem, the software finds its application in subsurface flow and power grid simulations in which graph Laplacians are found

Successfully Transitioning to Linear Equations

ERIC Educational Resources Information Center

Colton, Connie; Smith, Wendy M.

2014-01-01

The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…

Student Performance and Attitudes in a Collaborative and Flipped Linear Algebra Course

ERIC Educational Resources Information Center

Murphy, Julia; Chang, Jen-Mei; Suaray, Kagba

2016-01-01

Flipped learning is gaining traction in K-12 for enhancing students' problem-solving skills at an early age; however, there is relatively little large-scale research showing its effectiveness in promoting better learning outcomes in higher education, especially in mathematics classes. In this study, we examined the data compiled from both…

A Comprehensive Probability Project for the Upper Division One-Semester Probability Course Using Yahtzee

ERIC Educational Resources Information Center

Wilson, Jason; Lawman, Joshua; Murphy, Rachael; Nelson, Marissa

2011-01-01

This article describes a probability project used in an upper division, one-semester probability course with third-semester calculus and linear algebra prerequisites. The student learning outcome focused on developing the skills necessary for approaching project-sized math/stat application problems. These skills include appropriately defining…

Numerical linear algebra in data mining

NASA Astrophysics Data System (ADS)

Eldén, Lars

Ideas and algorithms from numerical linear algebra are important in several areas of data mining. We give an overview of linear algebra methods in text mining (information retrieval), pattern recognition (classification of handwritten digits), and PageRank computations for web search engines. The emphasis is on rank reduction as a method of extracting information from a data matrix, low-rank approximation of matrices using the singular value decomposition and clustering, and on eigenvalue methods for network analysis.

Sampled-Data Kalman Filtering and Multiple Model Adaptive Estimation for Infinite-Dimensional Continuous-Time Systems

DTIC Science & Technology

2007-03-01

mathematical frame- 1-6 work of linear algebra and functional analysis [122, 33], while Kalman-Bucy filtering [96, 32] is an especially important...Engineering, Air Force Institute of Technology (AU), Wright- Patterson AFB, Ohio, March 2002. 85. Hoffman, Kenneth and Ray Kunze. Linear Algebra (Second Edition...Engineering, Air Force Institute of Technology (AU), Wright- Patterson AFB, Ohio, December 1989. 189. Strang, Gilbert. Linear Algebra and Its Applications

Calculating Required Substructure Damping to Meet Prescribed System Damping Levels

DTIC Science & Technology

2007-06-01

Rorres, Elementary Linear Algebra . New Jersey: John Wiley & Sons, 2005. 2. Klaus-Jurgen Bathe, Finite Element Procedures. New Jersey: Prentice Hall...will be covered in the explanation of orthogonal complement. The definitions are extracted from the book “ Linear Algebra and its Applications” by...TA = left nullspace of A; dimension m-r Applying the first part of the fundamental theorem of Linear Algebra we can now talk about the orthogonal

Emphasizing language and visualization in teaching linear algebra

NASA Astrophysics Data System (ADS)

Hannah, John; Stewart, Sepideh; Thomas, Mike

2013-06-01

Linear algebra with its rich theoretical nature is a first step towards advanced mathematical thinking for many undergraduate students. In this paper, we consider the teaching approach of an experienced mathematician as he attempts to engage his students with the key ideas embedded in a second-year course in linear algebra. We describe his approach in both lectures and tutorials, and how he employed visualization and an emphasis on language to encourage conceptual thinking. We use Tall's framework of three worlds of mathematical thinking to reflect on the effect of these activities in students' learning. An analysis of students' attitudes to the course and their test and examination results help to answer questions about the value of such an approach, suggesting ways forward in teaching linear algebra.

Robustness of linear quadratic state feedback designs in the presence of system uncertainty. [application to Augmentor Wing Jet STOL Research Aircraft flare control autopilot design

NASA Technical Reports Server (NTRS)

Patel, R. V.; Toda, M.; Sridhar, B.

1977-01-01

The paper deals with the problem of expressing the robustness (stability) property of a linear quadratic state feedback (LQSF) design quantitatively in terms of bounds on the perturbations (modeling errors or parameter variations) in the system matrices so that the closed-loop system remains stable. Nonlinear time-varying and linear time-invariant perturbations are considered. The only computation required in obtaining a measure of the robustness of an LQSF design is to determine the eigenvalues of two symmetric matrices determined when solving the algebraic Riccati equation corresponding to the LQSF design problem. Results are applied to a complex dynamic system consisting of the flare control of a STOL aircraft. The design of the flare control is formulated as an LQSF tracking problem.

Commentary on A General Curriculum in Mathematics for Colleges.

ERIC Educational Resources Information Center

Committee on the Undergraduate Program in Mathematics, Berkeley, CA.

This document constitutes a complete revision of the report of the same name first published in 1965. A new list of basic courses is described, consisting of Calculus I, Calculus II, Elementary Linear Algebra, Multivariable Calculus I, Linear Algebra, and Introductory Modern Algebra. Commentaries outline the content and spirit of these courses in…

Teaching the "Diagonalization Concept" in Linear Algebra with Technology: A Case Study at Galatasaray University

ERIC Educational Resources Information Center

Yildiz Ulus, Aysegul

2013-01-01

This paper examines experimental and algorithmic contributions of advanced calculators (graphing and computer algebra system, CAS) in teaching the concept of "diagonalization," one of the key topics in Linear Algebra courses taught at the undergraduate level. Specifically, the proposed hypothesis of this study is to assess the effective…

The Role of Proof in Comprehending and Teaching Elementary Linear Algebra.

ERIC Educational Resources Information Center

Uhlig, Frank

2002-01-01

Describes how elementary linear algebra can be taught successfully while introducing students to the concept and practice of mathematical proof. Suggests exploring the concept of solvability of linear systems first via the row echelon form (REF). (Author/KHR)

Constitutive relations in optics in terms of geometric algebra

NASA Astrophysics Data System (ADS)

Dargys, A.

2015-11-01

To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p-form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium.

Basic Research in the Mathematical Foundations of Stability Theory, Control Theory and Numerical Linear Algebra.

DTIC Science & Technology

1979-09-01

without determinantal divisors, Linear and Multilinear Algebra 7(1979), 107-109. 4. The use of integral operators in number theory (with C. Ryavec and...Gersgorin revisited, to appear in Letters in Linear Algebra. 15. A surprising determinantal inequality for real matrices (with C.R. Johnson), to appear in...Analysis: An Essay Concerning the Limitations of Some Mathematical Methods in the Social , Political and Biological Sciences, David Berlinski, MIT Press

A Comparison Study between a Traditional and Experimental Program.

ERIC Educational Resources Information Center

Dogan, Hamide

This paper is part of a dissertation defended in January 2001 as part of the author's Ph.D. requirement. The study investigated the effects of use of Mathematica, a computer algebra system, in learning basic linear algebra concepts, It was done by means of comparing two first year linear algebra classes, one traditional and one Mathematica…

Efficient hybrid-symbolic methods for quantum mechanical calculations

NASA Astrophysics Data System (ADS)

Scott, T. C.; Zhang, Wenxing

2015-06-01

We present hybrid symbolic-numerical tools to generate optimized numerical code for rapid prototyping and fast numerical computation starting from a computer algebra system (CAS) and tailored to any given quantum mechanical problem. Although a major focus concerns the quantum chemistry methods of H. Nakatsuji which has yielded successful and very accurate eigensolutions for small atoms and molecules, the tools are general and may be applied to any basis set calculation with a variational principle applied to its linear and non-linear parameters.

Time domain convergence properties of Lyapunov stable penalty methods

NASA Technical Reports Server (NTRS)

Kurdila, A. J.; Sunkel, John

1991-01-01

Linear hyperbolic partial differential equations are analyzed using standard techniques to show that a sequence of solutions generated by the Liapunov stable penalty equations approaches the solution of the differential-algebraic equations governing the dynamics of multibody problems arising in linear vibrations. The analysis does not require that the system be conservative and does not impose any specific integration scheme. Variational statements are derived which bound the error in approximation by the norm of the constraint violation obtained in the approximate solutions.

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets

DTIC Science & Technology

2014-11-01

linear hybrid systems by linear algebraic methods. In SAS, volume 6337 of LNCS, pages 373–389. Springer, 2010. [19] E. W. Mayr. Membership in polynomial...383–394, 2009. [31] A. Tarski. A decision method for elementary algebra and geometry. Bull. Amer. Math. Soc., 59, 1951. [32] A. Tiwari. Abstractions...A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets Khalil Ghorbal1 Andrew Sogokon2 André Platzer1 November 2014 CMU

Linear-algebraic bath transformation for simulating complex open quantum systems

DOE PAGES

Huh, Joonsuk; Mostame, Sarah; Fujita, Takatoshi; ...

2014-12-02

In studying open quantum systems, the environment is often approximated as a collection of non-interacting harmonic oscillators, a configuration also known as the star-bath model. It is also well known that the star-bath can be transformed into a nearest-neighbor interacting chain of oscillators. The chain-bath model has been widely used in renormalization group approaches. The transformation can be obtained by recursion relations or orthogonal polynomials. Based on a simple linear algebraic approach, we propose a bath partition strategy to reduce the system-bath coupling strength. As a result, the non-interacting star-bath is transformed into a set of weakly coupled multiple parallelmore » chains. Furthermore, the transformed bath model allows complex problems to be practically implemented on quantum simulators, and it can also be employed in various numerical simulations of open quantum dynamics.« less

Linear algebraic methods applied to intensity modulated radiation therapy.

PubMed

Crooks, S M; Xing, L

2001-10-01

Methods of linear algebra are applied to the choice of beam weights for intensity modulated radiation therapy (IMRT). It is shown that the physical interpretation of the beam weights, target homogeneity and ratios of deposited energy can be given in terms of matrix equations and quadratic forms. The methodology of fitting using linear algebra as applied to IMRT is examined. Results are compared with IMRT plans that had been prepared using a commercially available IMRT treatment planning system and previously delivered to cancer patients.

Numerical Methods for Forward and Inverse Problems in Discontinuous Media

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chartier, Timothy P.

The research emphasis under this grant's funding is in the area of algebraic multigrid methods. The research has two main branches: 1) exploring interdisciplinary applications in which algebraic multigrid can make an impact and 2) extending the scope of algebraic multigrid methods with algorithmic improvements that are based in strong analysis.The work in interdisciplinary applications falls primarily in the field of biomedical imaging. Work under this grant demonstrated the effectiveness and robustness of multigrid for solving linear systems that result from highly heterogeneous finite element method models of the human head. The results in this work also give promise tomore » medical advances possible with software that may be developed. Research to extend the scope of algebraic multigrid has been focused in several areas. In collaboration with researchers at the University of Colorado, Lawrence Livermore National Laboratory, and Los Alamos National Laboratory, the PI developed an adaptive multigrid with subcycling via complementary grids. This method has very cheap computing costs per iterate and is showing promise as a preconditioner for conjugate gradient. Recent work with Los Alamos National Laboratory concentrates on developing algorithms that take advantage of the recent advances in adaptive multigrid research. The results of the various efforts in this research could ultimately have direct use and impact to researchers for a wide variety of applications, including, astrophysics, neuroscience, contaminant transport in porous media, bi-domain heart modeling, modeling of tumor growth, and flow in heterogeneous porous media. This work has already led to basic advances in computational mathematics and numerical linear algebra and will continue to do so into the future.« less

The Baker-Akhiezer Function and Factorization of the Chebotarev-Khrapkov Matrix

NASA Astrophysics Data System (ADS)

Antipov, Yuri A.

2014-10-01

A new technique is proposed for the solution of the Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient {G(t) = α1(t)I + α2(t)Q(t)} , {α1(t), α2(t) in H(L)} , I = diag{1, 1}, Q(t) is a {2×2} zero-trace polynomial matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of essential singularities of the solution to the associated homogeneous scalar Riemann-Hilbert problem on the hyperelliptic surface of an algebraic function by means of the Baker-Akhiezer function. The consequent application of this function for the derivation of the general solution to the vector Riemann-Hilbert problem requires the finding of the {ρ} zeros of the Baker-Akhiezer function ({ρ} is the genus of the surface). These zeros are recovered through the solution to the associated Jacobi problem of inversion of abelian integrals or, equivalently, the determination of the zeros of the associated degree-{ρ} polynomial and solution of a certain linear algebraic system of {ρ} equations.

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

PubMed

Benhammouda, Brahim

2016-01-01

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.

Where Am I? A Change of Basis Project

ERIC Educational Resources Information Center

Selby, Christina

2016-01-01

Linear algebra students are typically introduced to the problem of how to convert from one coordinate system to another in a very abstract way. Often, two bases for a given vector space are provided, and students are taught how to determine a transition matrix to be used for changing coordinates. If students are successful in memorizing this…

Methods, Software and Tools for Three Numerical Applications. Final report

DOE Office of Scientific and Technical Information (OSTI.GOV)

E. R. Jessup

2000-03-01

This is a report of the results of the authors work supported by DOE contract DE-FG03-97ER25325. They proposed to study three numerical problems. They are: (1) the extension of the PMESC parallel programming library; (2) the development of algorithms and software for certain generalized eigenvalue and singular value (SVD) problems, and (3) the application of techniques of linear algebra to an information retrieval technique known as latent semantic indexing (LSI).

Efficient computer algebra algorithms for polynomial matrices in control design

NASA Technical Reports Server (NTRS)

Baras, J. S.; Macenany, D. C.; Munach, R.

1989-01-01

The theory of polynomial matrices plays a key role in the design and analysis of multi-input multi-output control and communications systems using frequency domain methods. Examples include coprime factorizations of transfer functions, cannonical realizations from matrix fraction descriptions, and the transfer function design of feedback compensators. Typically, such problems abstract in a natural way to the need to solve systems of Diophantine equations or systems of linear equations over polynomials. These and other problems involving polynomial matrices can in turn be reduced to polynomial matrix triangularization procedures, a result which is not surprising given the importance of matrix triangularization techniques in numerical linear algebra. Matrices with entries from a field and Gaussian elimination play a fundamental role in understanding the triangularization process. In the case of polynomial matrices, matrices with entries from a ring for which Gaussian elimination is not defined and triangularization is accomplished by what is quite properly called Euclidean elimination. Unfortunately, the numerical stability and sensitivity issues which accompany floating point approaches to Euclidean elimination are not very well understood. New algorithms are presented which circumvent entirely such numerical issues through the use of exact, symbolic methods in computer algebra. The use of such error-free algorithms guarantees that the results are accurate to within the precision of the model data--the best that can be hoped for. Care must be taken in the design of such algorithms due to the phenomenon of intermediate expressions swell.

Libraries for Software Use on Peregrine | High-Performance Computing | NREL

Science.gov Websites

-specific libraries. Libraries List Name Description BLAS Basic Linear Algebra Subroutines, libraries only managing hierarchically structured data. LAPACK Standard Netlib offering for computational linear algebra

The Effects of Formalism on Teacher Trainees' Algebraic and Geometric Interpretation of the Notions of Linear Dependency/Independency

ERIC Educational Resources Information Center

Ertekin, E.; Solak, S.; Yazici, E.

2010-01-01

The aim of this study is to identify the effects of formalism in teaching on primary and secondary school mathematics teacher trainees' algebraic and geometric interpretations of the notions of linear dependency/independency. Quantitative research methods are drawn in order to determine differences in success levels between algebraic and geometric…

Implementing Linear Algebra Related Algorithms on the TI-92+ Calculator.

ERIC Educational Resources Information Center

Alexopoulos, John; Abraham, Paul

2001-01-01

Demonstrates a less utilized feature of the TI-92+: its natural and powerful programming language. Shows how to implement several linear algebra related algorithms including the Gram-Schmidt process, Least Squares Approximations, Wronskians, Cholesky Decompositions, and Generalized Linear Least Square Approximations with QR Decompositions.…

Lie algebras and linear differential equations.

NASA Technical Reports Server (NTRS)

Brockett, R. W.; Rahimi, A.

1972-01-01

Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

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.

Ada Linear-Algebra Program

NASA Technical Reports Server (NTRS)

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

1988-01-01

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

Chiropractic biophysics technique: a linear algebra approach to posture in chiropractic.

PubMed

Harrison, D D; Janik, T J; Harrison, G R; Troyanovich, S; Harrison, D E; Harrison, S O

1996-10-01

This paper discusses linear algebra as applied to human posture in chiropractic, specifically chiropractic biophysics technique (CBP). Rotations, reflections and translations are geometric functions studied in vector spaces in linear algebra. These mathematical functions are termed rigid body transformations and are applied to segmental spinal movement in the literature. Review of the literature indicates that these linear algebra concepts have been used to describe vertebral motion. However, these rigid body movers are presented here as applying to the global postural movements of the head, thoracic cage and pelvis. The unique inverse functions of rotations, reflections and translations provide a theoretical basis for making postural corrections in neutral static resting posture. Chiropractic biophysics technique (CBP) uses these concepts in examination procedures, manual spinal manipulation, instrument assisted spinal manipulation, postural exercises, extension traction and clinical outcome measures.

Discrete Inverse and State Estimation Problems

NASA Astrophysics Data System (ADS)

Wunsch, Carl

2006-06-01

The problems of making inferences about the natural world from noisy observations and imperfect theories occur in almost all scientific disciplines. This book addresses these problems using examples taken from geophysical fluid dynamics. It focuses on discrete formulations, both static and time-varying, known variously as inverse, state estimation or data assimilation problems. Starting with fundamental algebraic and statistical ideas, the book guides the reader through a range of inference tools including the singular value decomposition, Gauss-Markov and minimum variance estimates, Kalman filters and related smoothers, and adjoint (Lagrange multiplier) methods. The final chapters discuss a variety of practical applications to geophysical flow problems. Discrete Inverse and State Estimation Problems is an ideal introduction to the topic for graduate students and researchers in oceanography, meteorology, climate dynamics, and geophysical fluid dynamics. It is also accessible to a wider scientific audience; the only prerequisite is an understanding of linear algebra. Provides a comprehensive introduction to discrete methods of inference from incomplete information Based upon 25 years of practical experience using real data and models Develops sequential and whole-domain analysis methods from simple least-squares Contains many examples and problems, and web-based support through MIT opencourseware

Students’ Algebraic Reasonsing In Solving Mathematical Problems With Adversity Quotient

NASA Astrophysics Data System (ADS)

Aryani, F.; Amin, S. M.; Sulaiman, R.

2018-01-01

Algebraic reasoning is a process in which students generalize mathematical ideas from a set of particular instances and express them in increasingly formal and age-appropriate ways. Using problem solving approach to develop algebraic reasoning of mathematics may enhace the long-term learning trajectory of the majority students. The purpose of this research was to describe the algebraic reasoning of quitter, camper, and climber junior high school students in solving mathematical problems. This research used qualitative descriptive method. Subjects were determined by purposive sampling. The technique of collecting data was done by task-based interviews.The results showed that the algebraic reasoning of three students in the process of pattern seeking by identifying the things that are known and asked in a similar way. But three students found the elements of pattern recognition in different ways or method. So, they are generalize the problem of pattern formation with different ways. The study of algebraic reasoning and problem solving can be a learning paradigm in the improve students’ knowledge and skills in algebra work. The goal is to help students’ improve academic competence, develop algebraic reasoning in problem solving.

Embodied, Symbolic and Formal Thinking in Linear Algebra

ERIC Educational Resources Information Center

Stewart, Sepideh; Thomas, Michael O. J.

2007-01-01

Students often find their first university linear algebra experience very challenging. While coping with procedural aspects of the subject, solving linear systems and manipulating matrices, they may struggle with crucial conceptual ideas underpinning them, making it very difficult to progress in more advanced courses. This research has sought to…

Generating a New Higher-Dimensional Coupled Integrable Dispersionless System: Algebraic Structures, Bäcklund Transformation and Hidden Structural Symmetries

NASA Astrophysics Data System (ADS)

Souleymanou, Abbagari; Thomas, B. Bouetou; Timoleon, C. Kofane

2013-08-01

The prolongation structure methodologies of Wahlquist—Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 (1975) 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Bäcklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new (2 + 1)-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.

αAMG based on Weighted Matching for Systems of Elliptic PDEs Arising From Displacement and Mixed Methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

D'Ambra, P.; Vassilevski, P. S.

2014-05-30

Adaptive Algebraic Multigrid (or Multilevel) Methods (αAMG) are introduced to improve robustness and efficiency of classical algebraic multigrid methods in dealing with problems where no a-priori knowledge or assumptions on the near-null kernel of the underlined matrix are available. Recently we proposed an adaptive (bootstrap) AMG method, αAMG, aimed to obtain a composite solver with a desired convergence rate. Each new multigrid component relies on a current (general) smooth vector and exploits pairwise aggregation based on weighted matching in a matrix graph to define a new automatic, general-purpose coarsening process, which we refer to as “the compatible weighted matching”. Inmore » this work, we present results that broaden the applicability of our method to different finite element discretizations of elliptic PDEs. In particular, we consider systems arising from displacement methods in linear elasticity problems and saddle-point systems that appear in the application of the mixed method to Darcy problems.« less

New modified multi-level residue harmonic balance method for solving nonlinearly vibrating double-beam problem

NASA Astrophysics Data System (ADS)

Rahman, Md. Saifur; Lee, Yiu-Yin

2017-10-01

In this study, a new modified multi-level residue harmonic balance method is presented and adopted to investigate the forced nonlinear vibrations of axially loaded double beams. Although numerous nonlinear beam or linear double-beam problems have been tackled and solved, there have been few studies of this nonlinear double-beam problem. The geometric nonlinear formulations for a double-beam model are developed. The main advantage of the proposed method is that a set of decoupled nonlinear algebraic equations is generated at each solution level. This heavily reduces the computational effort compared with solving the coupled nonlinear algebraic equations generated in the classical harmonic balance method. The proposed method can generate the higher-level nonlinear solutions that are neglected by the previous modified harmonic balance method. The results from the proposed method agree reasonably well with those from the classical harmonic balance method. The effects of damping, axial force, and excitation magnitude on the nonlinear vibrational behaviour are examined.

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

PubMed Central

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 2nd- 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

Oasis: A high-level/high-performance open source Navier-Stokes solver

NASA Astrophysics Data System (ADS)

Mortensen, Mikael; Valen-Sendstad, Kristian

2015-03-01

Oasis is a high-level/high-performance finite element Navier-Stokes solver written from scratch in Python using building blocks from the FEniCS project (fenicsproject.org). The solver is unstructured and targets large-scale applications in complex geometries on massively parallel clusters. Oasis utilizes MPI and interfaces, through FEniCS, to the linear algebra backend PETSc. Oasis advocates a high-level, programmable user interface through the creation of highly flexible Python modules for new problems. Through the high-level Python interface the user is placed in complete control of every aspect of the solver. A version of the solver, that is using piecewise linear elements for both velocity and pressure, is shown to reproduce very well the classical, spectral, turbulent channel simulations of Moser et al. (1999). The computational speed is strongly dominated by the iterative solvers provided by the linear algebra backend, which is arguably the best performance any similar implicit solver using PETSc may hope for. Higher order accuracy is also demonstrated and new solvers may be easily added within the same framework.

Model predictive control of P-time event graphs

NASA Astrophysics Data System (ADS)

Hamri, H.; Kara, R.; Amari, S.

2016-12-01

This paper deals with model predictive control of discrete event systems modelled by P-time event graphs. First, the model is obtained by using the dater evolution model written in the standard algebra. Then, for the control law, we used the finite-horizon model predictive control. For the closed-loop control, we used the infinite-horizon model predictive control (IH-MPC). The latter is an approach that calculates static feedback gains which allows the stability of the closed-loop system while respecting the constraints on the control vector. The problem of IH-MPC is formulated as a linear convex programming subject to a linear matrix inequality problem. Finally, the proposed methodology is applied to a transportation system.

Blogging and the Learning of Linear Algebra Concepts through Contextual Mathematics

ERIC Educational Resources Information Center

Nehme, Zeina

2011-01-01

Contextual mathematics is an area of mathematics teaching and learning through which researchers and educators believe that mathematics is better taught, and learned, if connected to real-life situations and problems. It is also very helpful if it makes sense in the students' world. Thus, the author decided to start a project by creating a blog,…

A scalable approach to solving dense linear algebra problems on hybrid CPU-GPU systems

DOE PAGES

Song, Fengguang; Dongarra, Jack

2014-10-01

Aiming to fully exploit the computing power of all CPUs and all graphics processing units (GPUs) on hybrid CPU-GPU systems to solve dense linear algebra problems, in this paper we design a class of heterogeneous tile algorithms to maximize the degree of parallelism, to minimize the communication volume, and to accommodate the heterogeneity between CPUs and GPUs. The new heterogeneous tile algorithms are executed upon our decentralized dynamic scheduling runtime system, which schedules a task graph dynamically and transfers data between compute nodes automatically. The runtime system uses a new distributed task assignment protocol to solve data dependencies between tasksmore » without any coordination between processing units. By overlapping computation and communication through dynamic scheduling, we are able to attain scalable performance for the double-precision Cholesky factorization and QR factorization. Finally, our approach demonstrates a performance comparable to Intel MKL on shared-memory multicore systems and better performance than both vendor (e.g., Intel MKL) and open source libraries (e.g., StarPU) in the following three environments: heterogeneous clusters with GPUs, conventional clusters without GPUs, and shared-memory systems with multiple GPUs.« less

A scalable approach to solving dense linear algebra problems on hybrid CPU-GPU systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Song, Fengguang; Dongarra, Jack

Aiming to fully exploit the computing power of all CPUs and all graphics processing units (GPUs) on hybrid CPU-GPU systems to solve dense linear algebra problems, in this paper we design a class of heterogeneous tile algorithms to maximize the degree of parallelism, to minimize the communication volume, and to accommodate the heterogeneity between CPUs and GPUs. The new heterogeneous tile algorithms are executed upon our decentralized dynamic scheduling runtime system, which schedules a task graph dynamically and transfers data between compute nodes automatically. The runtime system uses a new distributed task assignment protocol to solve data dependencies between tasksmore » without any coordination between processing units. By overlapping computation and communication through dynamic scheduling, we are able to attain scalable performance for the double-precision Cholesky factorization and QR factorization. Finally, our approach demonstrates a performance comparable to Intel MKL on shared-memory multicore systems and better performance than both vendor (e.g., Intel MKL) and open source libraries (e.g., StarPU) in the following three environments: heterogeneous clusters with GPUs, conventional clusters without GPUs, and shared-memory systems with multiple GPUs.« less

ParaExp Using Leapfrog as Integrator for High-Frequency Electromagnetic Simulations

NASA Astrophysics Data System (ADS)

Merkel, M.; Niyonzima, I.; Schöps, S.

2017-12-01

Recently, ParaExp was proposed for the time integration of linear hyperbolic problems. It splits the time interval of interest into subintervals and computes the solution on each subinterval in parallel. The overall solution is decomposed into a particular solution defined on each subinterval with zero initial conditions and a homogeneous solution propagated by the matrix exponential applied to the initial conditions. The efficiency of the method depends on fast approximations of this matrix exponential based on recent results from numerical linear algebra. This paper deals with the application of ParaExp in combination with Leapfrog to electromagnetic wave problems in time domain. Numerical tests are carried out for a simple toy problem and a realistic spiral inductor model discretized by the Finite Integration Technique.

The algebraic criteria for the stability of control systems

NASA Technical Reports Server (NTRS)

Cremer, H.; Effertz, F. H.

1986-01-01

This paper critically examines the standard algebraic criteria for the stability of linear control systems and their proofs, reveals important previously unnoticed connections, and presents new representations. Algebraic stability criteria have also acquired significance for stability studies of non-linear differential equation systems by the Krylov-Bogoljubov-Magnus Method, and allow realization conditions to be determined for classes of broken rational functions as frequency characteristics of electrical network.

Population Projection. Applications of Linear Algebra to Population Studies. Modules and Monographs in Undergraduate Mathematics and Its Applications. UMAP Module 345.

ERIC Educational Resources Information Center

Keller, Edward L.

This unit, which looks at applications of linear algebra to population studies, is designed to help pupils: (1) understand an application of matrix algebra to the study of populations; (2) see how knowledge of eigen values and eigen vectors is useful in studying powers of matrices; and (3) be briefly exposed to some difficult but interesting…

Gauss Elimination: Workhorse of Linear Algebra.

DTIC Science & Technology

1995-08-05

linear algebra computation for solving systems, computing determinants and determining the rank of matrix. All of these are discussed in varying contexts. These include different arithmetic or algebraic setting such as integer arithmetic or polynomial rings as well as conventional real (floating-point) arithmetic. These have effects on both accuracy and complexity analyses of the algorithm. These, too, are covered here. The impact of modern parallel computer architecture on GE is also

Linear maps preserving maximal deviation and the Jordan structure of quantum systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hamhalter, Jan

2012-12-15

In the algebraic approach to quantum theory, a quantum observable is given by an element of a Jordan algebra and a state of the system is modelled by a normalized positive functional on the underlying algebra. Maximal deviation of a quantum observable is the largest statistical deviation one can obtain in a particular state of the system. The main result of the paper shows that each linear bijective transformation between JBW algebras preserving maximal deviations is formed by a Jordan isomorphism or a minus Jordan isomorphism perturbed by a linear functional multiple of an identity. It shows that only onemore » numerical statistical characteristic has the power to determine the Jordan algebraic structure completely. As a consequence, we obtain that only very special maps can preserve the diameter of the spectra of elements. Nonlinear maps preserving the pseudometric given by maximal deviation are also described. The results generalize hitherto known theorems on preservers of maximal deviation in the case of self-adjoint parts of von Neumann algebras proved by Molnar.« less

Efficient linear algebra routines for symmetric matrices stored in packed form.

PubMed

Ahlrichs, Reinhart; Tsereteli, Kakha

2002-01-30

Quantum chemistry methods require various linear algebra routines for symmetric matrices, for example, diagonalization or Cholesky decomposition for positive matrices. We present a small set of these basic routines that are efficient and minimize memory requirements.

High-Order Automatic Differentiation of Unmodified Linear Algebra Routines via Nilpotent Matrices

NASA Astrophysics Data System (ADS)

Dunham, Benjamin Z.

This work presents a new automatic differentiation method, Nilpotent Matrix Differentiation (NMD), capable of propagating any order of mixed or univariate derivative through common linear algebra functions--most notably third-party sparse solvers and decomposition routines, in addition to basic matrix arithmetic operations and power series--without changing data-type or modifying code line by line; this allows differentiation across sequences of arbitrarily many such functions with minimal implementation effort. NMD works by enlarging the matrices and vectors passed to the routines, replacing each original scalar with a matrix block augmented by derivative data; these blocks are constructed with special sparsity structures, termed "stencils," each designed to be isomorphic to a particular multidimensional hypercomplex algebra. The algebras are in turn designed such that Taylor expansions of hypercomplex function evaluations are finite in length and thus exactly track derivatives without approximation error. Although this use of the method in the "forward mode" is unique in its own right, it is also possible to apply it to existing implementations of the (first-order) discrete adjoint method to find high-order derivatives with lowered cost complexity; for example, for a problem with N inputs and an adjoint solver whose cost is independent of N--i.e., O(1)--the N x N Hessian can be found in O(N) time, which is comparable to existing second-order adjoint methods that require far more problem-specific implementation effort. Higher derivatives are likewise less expensive--e.g., a N x N x N rank-three tensor can be found in O(N2). Alternatively, a Hessian-vector product can be found in O(1) time, which may open up many matrix-based simulations to a range of existing optimization or surrogate modeling approaches. As a final corollary in parallel to the NMD-adjoint hybrid method, the existing complex-step differentiation (CD) technique is also shown to be capable of finding the Hessian-vector product. All variants are implemented on a stochastic diffusion problem and compared in-depth with various cost and accuracy metrics.

Algebra for Gifted Third Graders.

ERIC Educational Resources Information Center

Borenson, Henry

1987-01-01

Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)

SD-CAS: Spin Dynamics by Computer Algebra System.

PubMed

Filip, Xenia; Filip, Claudiu

2010-11-01

A computer algebra tool for describing the Liouville-space quantum evolution of nuclear 1/2-spins is introduced and implemented within a computational framework named Spin Dynamics by Computer Algebra System (SD-CAS). A distinctive feature compared with numerical and previous computer algebra approaches to solving spin dynamics problems results from the fact that no matrix representation for spin operators is used in SD-CAS, which determines a full symbolic character to the performed computations. Spin correlations are stored in SD-CAS as four-entry nested lists of which size increases linearly with the number of spins into the system and are easily mapped into analytical expressions in terms of spin operator products. For the so defined SD-CAS spin correlations a set of specialized functions and procedures is introduced that are essential for implementing basic spin algebra operations, such as the spin operator products, commutators, and scalar products. They provide results in an abstract algebraic form: specific procedures to quantitatively evaluate such symbolic expressions with respect to the involved spin interaction parameters and experimental conditions are also discussed. Although the main focus in the present work is on laying the foundation for spin dynamics symbolic computation in NMR based on a non-matrix formalism, practical aspects are also considered throughout the theoretical development process. In particular, specific SD-CAS routines have been implemented using the YACAS computer algebra package (http://yacas.sourceforge.net), and their functionality was demonstrated on a few illustrative examples. Copyright © 2010 Elsevier Inc. All rights reserved.

A Lie algebraic condition for exponential stability of discrete hybrid systems and application to hybrid synchronization.

PubMed

Zhao, Shouwei

2011-06-01

A Lie algebraic condition for global exponential stability of linear discrete switched impulsive systems is presented in this paper. By considering a Lie algebra generated by all subsystem matrices and impulsive matrices, when not all of these matrices are Schur stable, we derive new criteria for global exponential stability of linear discrete switched impulsive systems. Moreover, simple sufficient conditions in terms of Lie algebra are established for the synchronization of nonlinear discrete systems using a hybrid switching and impulsive control. As an application, discrete chaotic system's synchronization is investigated by the proposed method.

Symmetric linear systems - An application of algebraic systems theory

NASA Technical Reports Server (NTRS)

Hazewinkel, M.; Martin, C.

1983-01-01

Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.

Capelli bitableaux and Z-forms of general linear Lie superalgebras.

PubMed Central

Brini, A; Teolis, A G

1990-01-01

The combinatorics of the enveloping algebra UQ(pl(L)) of the general linear Lie superalgebra of a finite dimensional Z2-graded Q-vector space is studied. Three non-equivalent Z-forms of UQ(pl(L)) are introduced: one of these Z-forms is a version of the Kostant Z-form and the others are Lie algebra analogs of Rota and Stein's straightening formulae for the supersymmetric algebra Super[L P] and for its dual Super[L* P*]. The method is based on an extension of Capelli's technique of variabili ausiliarie to algebras containing positively and negatively signed elements. PMID:11607048

On conforming mixed finite element methods for incompressible viscous flow problems

NASA Technical Reports Server (NTRS)

Gunzburger, M. D; Nicolaides, R. A.; Peterson, J. S.

1982-01-01

The application of conforming mixed finite element methods to obtain approximate solutions of linearized Navier-Stokes equations is examined. Attention is given to the convergence rates of various finite element approximations of the pressure and the velocity field. The optimality of the convergence rates are addressed in terms of comparisons of the approximation convergence to a smooth solution in relation to the best approximation available for the finite element space used. Consideration is also devoted to techniques for efficient use of a Gaussian elimination algorithm to obtain a solution to a system of linear algebraic equations derived by finite element discretizations of linear partial differential equations.

Student Learning of Basis, Span and Linear Independence in Linear Algebra

ERIC Educational Resources Information Center

Stewart, Sepideh; Thomas, Michael O. J.

2010-01-01

One of the earlier, more challenging concepts in linear algebra at university is that of basis. Students are often taught procedurally how to find a basis for a subspace using matrix manipulation, but may struggle with understanding the construct of basis, making further progress harder. We believe one reason for this is because students have…

Application of laser speckle to randomized numerical linear algebra

NASA Astrophysics Data System (ADS)

Valley, George C.; Shaw, Thomas J.; Stapleton, Andrew D.; Scofield, Adam C.; Sefler, George A.; Johannson, Leif

2018-02-01

We propose and simulate integrated optical devices for accelerating numerical linear algebra (NLA) calculations. Data is modulated on chirped optical pulses and these propagate through a multimode waveguide where speckle provides the random projections needed for NLA dimensionality reduction.

Constructive Learning in Undergraduate Linear Algebra

ERIC Educational Resources Information Center

Chandler, Farrah Jackson; Taylor, Dewey T.

2008-01-01

In this article we describe a project that we used in our undergraduate linear algebra courses to help our students successfully master fundamental concepts and definitions and generate interest in the course. We describe our philosophy and discuss the projects overall success.

UCSMP Algebra. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2007

2007-01-01

"University of Chicago School Mathematics Project (UCSMP) Algebra," designed to increase students' skills in algebra, is appropriate for students in grades 7-10, depending on the students' incoming knowledge. This one-year course highlights applications, uses statistics and geometry to develop the algebra of linear equations and inequalities, and…

Continuity in Representation between Children and Adults: Arithmetic Knowledge Hinders Undergraduates' Algebraic Problem Solving

ERIC Educational Resources Information Center

McNeil, Nicole M.; Rittle-Johnson, Bethany; Hattikudur, Shanta; Petersen, Lori A.

2010-01-01

This study examined if solving arithmetic problems hinders undergraduates' accuracy on algebra problems. The hypothesis was that solving arithmetic problems would hinder accuracy because it activates an operational view of equations, even in educated adults who have years of experience with algebra. In three experiments, undergraduates (N = 184)…

Algebraic reasoning and bat-and-ball problem variants: Solving isomorphic algebra first facilitates problem solving later.

PubMed

Hoover, Jerome D; Healy, Alice F

2017-12-01

The classic bat-and-ball problem is used widely to measure biased and correct reasoning in decision-making. University students overwhelmingly tend to provide the biased answer to this problem. To what extent might reasoners be led to modify their judgement, and, more specifically, is it possible to facilitate problem solution by prompting participants to consider the problem from an algebraic perspective? One hundred ninety-seven participants were recruited to investigate the effect of algebraic cueing as a debiasing strategy on variants of the bat-and-ball problem. Participants who were cued to consider the problem algebraically were significantly more likely to answer correctly relative to control participants. Most of this cueing effect was confined to a condition that required participants to solve isomorphic algebra equations corresponding to the structure of bat-and-ball question types. On a subsequent critical question with differing item and dollar amounts presented without a cue, participants were able to generalize the learned information to significantly reduce overall bias. Math anxiety was also found to be significantly related to bat-and-ball problem accuracy. These results suggest that, under specific conditions, algebraic reasoning is an effective debiasing strategy on bat-and-ball problem variants, and provide the first documented evidence for the influence of math anxiety on Cognitive Reflection Test performance.

Generalizations of Tikhonov's regularized method of least squares to non-Euclidean vector norms

NASA Astrophysics Data System (ADS)

Volkov, V. V.; Erokhin, V. I.; Kakaev, V. V.; Onufrei, A. Yu.

2017-09-01

Tikhonov's regularized method of least squares and its generalizations to non-Euclidean norms, including polyhedral, are considered. The regularized method of least squares is reduced to mathematical programming problems obtained by "instrumental" generalizations of the Tikhonov lemma on the minimal (in a certain norm) solution of a system of linear algebraic equations with respect to an unknown matrix. Further studies are needed for problems concerning the development of methods and algorithms for solving reduced mathematical programming problems in which the objective functions and admissible domains are constructed using polyhedral vector norms.

A Proposed Algebra Assessment for Use in a Problem-Analysis Framework

ERIC Educational Resources Information Center

Walick, Christopher M.; Burns, Matthew K.

2017-01-01

Algebra is critical to high school graduation and college success, but student achievement in algebra frequently falls significantly below expected proficiency levels. While existing research emphasizes the importance of quality algebra instruction, there is little research about how to conduct problem analysis for struggling secondary students.…

Amesos2 and Belos: Direct and Iterative Solvers for Large Sparse Linear Systems

DOE PAGES

Bavier, Eric; Hoemmen, Mark; Rajamanickam, Sivasankaran; ...

2012-01-01

Solvers for large sparse linear systems come in two categories: direct and iterative. Amesos2, a package in the Trilinos software project, provides direct methods, and Belos, another Trilinos package, provides iterative methods. Amesos2 offers a common interface to many different sparse matrix factorization codes, and can handle any implementation of sparse matrices and vectors, via an easy-to-extend C++ traits interface. It can also factor matrices whose entries have arbitrary “Scalar” type, enabling extended-precision and mixed-precision algorithms. Belos includes many different iterative methods for solving large sparse linear systems and least-squares problems. Unlike competing iterative solver libraries, Belos completely decouples themore » algorithms from the implementations of the underlying linear algebra objects. This lets Belos exploit the latest hardware without changes to the code. Belos favors algorithms that solve higher-level problems, such as multiple simultaneous linear systems and sequences of related linear systems, faster than standard algorithms. The package also supports extended-precision and mixed-precision algorithms. Together, Amesos2 and Belos form a complete suite of sparse linear solvers.« less

Simulations of Coherent Synchrotron Radiation Effects in Electron Machines

NASA Astrophysics Data System (ADS)

Migliorati, M.; Schiavi, A.; Dattoli, G.

2007-09-01

Coherent synchrotron radiation (CSR) generated by high intensity electron beams can be a source of undesirable effects limiting the performance of storage rings. The complexity of the physical mechanisms underlying the interplay between the electron beam and the CSR demands for reliable simulation codes. In the past, codes based on Lie algebraic techniques have been very efficient to treat transport problems in accelerators. The extension of these methods to the non linear case is ideally suited to treat wakefields - beam interaction. In this paper we report on the development of a numerical code, based on the solution of the Vlasov equation, which includes the non linear contribution due to wakefields. The proposed solution method exploits an algebraic technique that uses the exponential operators. We show that, in the case of CSR wakefields, the integration procedure is capable of reproducing the onset of an instability which leads to microbunching of the beam thus increasing the CSR at short wavelengths. In addition, considerations on the threshold of the instability for Gaussian bunches is also reported.

Simulations of Coherent Synchrotron Radiation Effects in Electron Machines

NASA Astrophysics Data System (ADS)

Migliorati, M.; Schiavi, A.; Dattoli, G.

Coherent synchrotron radiation (CSR) generated by high intensity electron beams can be a source of undesirable effects limiting the performance of storage rings. The complexity of the physical mechanisms underlying the interplay between the electron beam and the CSR demands for reliable simulation codes. In the past, codes based on Lie algebraic techniques have been very efficient to treat transport problems in accelerators. The extension of these methods to the non linear case is ideally suited to treat wakefields - beam interaction. In this paper we report on the development of a numerical code, based on the solution of the Vlasov equation, which includes the non linear contribution due to wakefields. The proposed solution method exploits an algebraic technique that uses the exponential operators. We show that, in the case of CSR wakefields, the integration procedure is capable of reproducing the onset of an instability which leads to microbunching of the beam thus increasing the CSR at short wavelengths. In addition, considerations on the threshold of the instability for Gaussian bunches is also reported.

Computer Algebra Systems in Undergraduate Instruction.

ERIC Educational Resources Information Center

Small, Don; And Others

1986-01-01

Computer algebra systems (such as MACSYMA and muMath) can carry out many of the operations of calculus, linear algebra, and differential equations. Use of them with sketching graphs of rational functions and with other topics is discussed. (MNS)

Conical Lens for 5-Inch/54 Gun Launched Missile

DTIC Science & Technology

1981-06-01

Propagation, Interferenceand Diffraction of Light, 2nd ed. (revised), p. 121-124, Pergamon Press, 1964. 10. Anton , Howard, Elementary Linear Algebra , p. 1-21...equations is nonlinear in x, but is linear in the coefficients. Therefore, the techniques of linear algebra can be used on equation (F-13). The method...This thesis assumes the air to be homogenous, isotropic, linear , time indepen- dent (HILT) and free of shock waves in order to investigate the

An Integrity Framework for Image-Based Navigation Systems

DTIC Science & Technology

2010-06-01

Anton H. and Rorres C. Elementary Linear Algebra . New York, NY: John Wiley & Sons, Inc., 2000. 4. Arthur T. “The Disparity of Parity, Determining...107. Spilker , James J.J. Digital Communications by Satellite. Englewood Cliffs NJ: Prentice Hall, 1977. 108. Strang G. Linear Algebra and its...2.3 The Linearized and Extended Kalman Filters . . . . . . 22 2.3.1 State and Measurement Model Equations . . . 23 2.3.2 The Linearized Kalman Filter

Journal Writing: Enlivening Elementary Linear Algebra.

ERIC Educational Resources Information Center

Meel, David E.

1999-01-01

Examines the various issues surrounding the implementation of journal writing in an undergraduate linear algebra course. Identifies the benefits of incorporating journal writing into an undergraduate mathematics course, which are supported with students' comments from their journals and their reflections on the process. Contains 14 references.…

Deconvolutions based on singular value decomposition and the pseudoinverse: a guide for beginners.

PubMed

Hendler, R W; Shrager, R I

1994-01-01

Singular value decomposition (SVD) is deeply rooted in the theory of linear algebra, and because of this is not readily understood by a large group of researchers who could profit from its application. In this paper, we discuss the subject on a level that should be understandable to scientists who are not well versed in linear algebra. However, because it is necessary that certain key concepts in linear algebra be appreciated in order to comprehend what is accomplished by SVD, we present the section, 'Bare basics of linear algebra'. This is followed by a discussion of the theory of SVD. Next we present step-by-step examples to illustrate how SVD is applied to deconvolute a titration involving a mixture of three pH indicators. One noiseless case is presented as well as two cases where either a fixed or varying noise level is present. Finally, we discuss additional deconvolutions of mixed spectra based on the use of the pseudoinverse.

Does Calculation or Word-Problem Instruction Provide A Stronger Route to Pre-Algebraic Knowledge?

PubMed Central

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

Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

NASA Astrophysics Data System (ADS)

Saveliev, M. V.; Vershik, A. M.

1989-12-01

We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations of Z-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras “continuum Lie algebras.” The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.

Invariants of polarization transformations.

PubMed

Sadjadi, Firooz A

2007-05-20

The use of polarization-sensitive sensors is being explored in a variety of applications. Polarization diversity has been shown to improve the performance of the automatic target detection and recognition in a significant way. However, it also brings out the problems associated with processing and storing more data and the problem of polarization distortion during transmission. We present a technique for extracting attributes that are invariant under polarization transformations. The polarimetric signatures are represented in terms of the components of the Stokes vectors. Invariant algebra is then used to extract a set of signature-related attributes that are invariant under linear transformation of the Stokes vectors. Experimental results using polarimetric infrared signatures of a number of manmade and natural objects undergoing systematic linear transformations support the invariancy of these attributes.

Solving the Problem of Bending of Multiply Connected Plates with Elastic Inclusions

NASA Astrophysics Data System (ADS)

Kaloerov, S. A.; Koshkin, A. A.

2017-11-01

This paper describes a method for determining the strain state of a thin anisotropic plate with elastic arbitrarily arranged elliptical inclusions. Complex potentials are used to reduce the problem to determining functions of generalized complex variables, which, in turn, comes down to an overdetermined system of linear algebraic equations, solved by singular expansions. This paper presents the results of numerical calculations that helped establish the influence of rigidity of elastic inclusions, distances between inclusions, and their geometric characteristics on the bending moments occurring in the plate. It is found that the specific properties of distribution of moments near the apexes of linear elastic inclusions, characterized by moment intensity coefficients, occur only in the case of sufficiently rigid and elastic inclusions.

Meanings Given to Algebraic Symbolism in Problem-Posing

ERIC Educational Resources Information Center

Cañadas, María C.; Molina, Marta; del Río, Aurora

2018-01-01

Some errors in the learning of algebra suggest that students might have difficulties giving meaning to algebraic symbolism. In this paper, we use problem posing to analyze the students' capacity to assign meaning to algebraic symbolism and the difficulties that students encounter in this process, depending on the characteristics of the algebraic…

Optical Linear Algebra for Computational Light Transport

NASA Astrophysics Data System (ADS)

O'Toole, Matthew

Active illumination refers to optical techniques that use controllable lights and cameras to analyze the way light propagates through the world. These techniques confer many unique imaging capabilities (e.g. high-precision 3D scanning, image-based relighting, imaging through scattering media), but at a significant cost; they often require long acquisition and processing times, rely on predictive models for light transport, and cease to function when exposed to bright ambient sunlight. We develop a mathematical framework for describing and analyzing such imaging techniques. This framework is deeply rooted in numerical linear algebra, and models the transfer of radiant energy through an unknown environment with the so-called light transport matrix. Performing active illumination on a scene equates to applying a numerical operator on this unknown matrix. The brute-force approach to active illumination follows a two-step procedure: (1) optically measure the light transport matrix and (2) evaluate the matrix operator numerically. This approach is infeasible in general, because the light transport matrix is often much too large to measure, store, and analyze directly. Using principles from optical linear algebra, we evaluate these matrix operators in the optical domain, without ever measuring the light transport matrix in the first place. Specifically, we explore numerical algorithms that can be implemented partially or fully with programmable optics. These optical algorithms provide solutions to many longstanding problems in computer vision and graphics, including the ability to (1) photo-realistically change the illumination conditions of a given photo with only a handful of measurements, (2) accurately capture the 3D shape of objects in the presence of complex transport properties and strong ambient illumination, and (3) overcome the multipath interference problem associated with time-of-flight cameras. Most importantly, we introduce an all-new imaging regime---optical probing---that provides unprecedented control over which light paths contribute to a photo.

Analytical Solution of Steady State Equations for Chemical Reaction Networks with Bilinear Rate Laws

PubMed Central

Halász, Ádám M.; Lai, Hong-Jian; McCabe, Meghan M.; Radhakrishnan, Krishnan; Edwards, Jeremy S.

2014-01-01

True steady states are a rare occurrence in living organisms, yet their knowledge is essential for quasi-steady state approximations, multistability analysis, and other important tools in the investigation of chemical reaction networks (CRN) used to describe molecular processes on the cellular level. Here we present an approach that can provide closed form steady-state solutions to complex systems, resulting from CRN with binary reactions and mass-action rate laws. We map the nonlinear algebraic problem of finding steady states onto a linear problem in a higher dimensional space. We show that the linearized version of the steady state equations obeys the linear conservation laws of the original CRN. We identify two classes of problems for which complete, minimally parameterized solutions may be obtained using only the machinery of linear systems and a judicious choice of the variables used as free parameters. We exemplify our method, providing explicit formulae, on CRN describing signal initiation of two important types of RTK receptor-ligand systems, VEGF and EGF-ErbB1. PMID:24334389

Symbolic-numeric interface: A review

NASA Technical Reports Server (NTRS)

Ng, E. W.

1980-01-01

A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach.

A Note on Multigrid Theory for Non-nested Grids and/or Quadrature

NASA Technical Reports Server (NTRS)

Douglas, C. C.; Douglas, J., Jr.; Fyfe, D. E.

1996-01-01

We provide a unified theory for multilevel and multigrid methods when the usual assumptions are not present. For example, we do not assume that the solution spaces or the grids are nested. Further, we do not assume that there is an algebraic relationship between the linear algebra problems on different levels. What we provide is a computationally useful theory for adaptively changing levels. Theory is provided for multilevel correction schemes, nested iteration schemes, and one way (i.e., coarse to fine grid with no correction iterations) schemes. We include examples showing the applicability of this theory: finite element examples using quadrature in the matrix assembly and finite volume examples with non-nested grids. Our theory applies directly to other discretizations as well.

Dirac delta representation by exact parametric equations.. Application to impulsive vibration systems

NASA Astrophysics Data System (ADS)

Chicurel-Uziel, Enrique

2007-08-01

A pair of closed parametric equations are proposed to represent the Heaviside unit step function. Differentiating the step equations results in two additional parametric equations, that are also hereby proposed, to represent the Dirac delta function. These equations are expressed in algebraic terms and are handled by means of elementary algebra and elementary calculus. The proposed delta representation complies exactly with the values of the definition. It complies also with the sifting property and the requisite unit area and its Laplace transform coincides with the most general form given in the tables. Furthermore, it leads to a very simple method of solution of impulsive vibrating systems either linear or belonging to a large class of nonlinear problems. Two example solutions are presented.

Error-Detecting Identification Codes for Algebra Students.

ERIC Educational Resources Information Center

Sutherland, David C.

1990-01-01

Discusses common error-detecting identification codes using linear algebra terminology to provide an interesting application of algebra. Presents examples from the International Standard Book Number, the Universal Product Code, bank identification numbers, and the ZIP code bar code. (YP)

Regularization strategies for hyperplane classifiers: application to cancer classification with gene expression data.

PubMed

Andries, Erik; Hagstrom, Thomas; Atlas, Susan R; Willman, Cheryl

2007-02-01

Linear discrimination, from the point of view of numerical linear algebra, can be treated as solving an ill-posed system of linear equations. In order to generate a solution that is robust in the presence of noise, these problems require regularization. Here, we examine the ill-posedness involved in the linear discrimination of cancer gene expression data with respect to outcome and tumor subclasses. We show that a filter factor representation, based upon Singular Value Decomposition, yields insight into the numerical ill-posedness of the hyperplane-based separation when applied to gene expression data. We also show that this representation yields useful diagnostic tools for guiding the selection of classifier parameters, thus leading to improved performance.

Applications of Maple To Algebraic Cryptography.

ERIC Educational Resources Information Center

Sigmon, Neil P.

1997-01-01

Demonstrates the use of technology to enhance the appreciation of applications involving abstract algebra. The symbolic manipulator Maple can perform computations required for a linear cryptosystem. One major benefit of this process is that students can encipher and decipher messages using a linear cryptosystem without becoming confused and…

Mathematical Modelling in Engineering: A Proposal to Introduce Linear Algebra Concepts

ERIC Educational Resources Information Center

Cárcamo Bahamonde, Andrea; Gómez Urgelles, Joan; Fortuny Aymemí, Josep

2016-01-01

The modern dynamic world requires that basic science courses for engineering, including linear algebra, emphasise the development of mathematical abilities primarily associated with modelling and interpreting, which are not exclusively calculus abilities. Considering this, an instructional design was created based on mathematical modelling and…

Noise limitations in optical linear algebra processors.

PubMed

Batsell, S G; Jong, T L; Walkup, J F; Krile, T F

1990-05-10

A general statistical noise model is presented for optical linear algebra processors. A statistical analysis which includes device noise, the multiplication process, and the addition operation is undertaken. We focus on those processes which are architecturally independent. Finally, experimental results which verify the analytical predictions are also presented.

Modules as Learning Tools in Linear Algebra

ERIC Educational Resources Information Center

Cooley, Laurel; Vidakovic, Draga; Martin, William O.; Dexter, Scott; Suzuki, Jeff; Loch, Sergio

2014-01-01

This paper reports on the experience of STEM and mathematics faculty at four different institutions working collaboratively to integrate learning theory with curriculum development in a core undergraduate linear algebra context. The faculty formed a Professional Learning Community (PLC) with a focus on learning theories in mathematics and…

Efficient Solution of Three-Dimensional Problems of Acoustic and Electromagnetic Scattering by Open Surfaces

NASA Technical Reports Server (NTRS)

Turc, Catalin; Anand, Akash; Bruno, Oscar; Chaubell, Julian

2011-01-01

We present a computational methodology (a novel Nystrom approach based on use of a non-overlapping patch technique and Chebyshev discretizations) for efficient solution of problems of acoustic and electromagnetic scattering by open surfaces. Our integral equation formulations (1) Incorporate, as ansatz, the singular nature of open-surface integral-equation solutions, and (2) For the Electric Field Integral Equation (EFIE), use analytical regularizes that effectively reduce the number of iterations required by iterative linear-algebra solution based on Krylov-subspace iterative solvers.

Bounding solutions of geometrically nonlinear viscoelastic problems

NASA Technical Reports Server (NTRS)

Stubstad, J. M.; Simitses, G. J.

1985-01-01

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

Bounding solutions of geometrically nonlinear viscoelastic problems

NASA Technical Reports Server (NTRS)

Stubstad, J. M.; Simitses, G. J.

1986-01-01

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

The algebra of supertraces for 2+1 super de Sitter gravity

NASA Technical Reports Server (NTRS)

Urrutia, L. F.; Waelbroeck, H.; Zertuche, F.

1993-01-01

The algebra of the observables for 2+1 super de Sitter gravity, for one genus of the spatial surface is calculated. The algebra turns out to be an infinite Lie algebra subject to non-linear constraints. The constraints are solved explicitly in terms of five independent complex supertraces. These variables are the true degrees of freedom of the system and their quantized algebra generates a new structure which is referred to as a 'central extension' of the quantum algebra SU(2)q.

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

NASA Astrophysics Data System (ADS)

Man, Yiu-Kwong

2010-10-01

In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle-Singer method for computing first integrals of differential equations. Some examples on using this approach are provided.

Identification of Large Space Structures on Orbit

DTIC Science & Technology

1986-09-01

requires only the eigenvector corresponding to the eigenvector 93 .:. ,S --- k’.’ L derivative being calculated. However, a set of linear algebraic ...Journal of Guidance, Control and Dynamics. 204. Noble, B. and J. W. Daniel, Applied Linear Algebra , Prentice-Hall, Inc., 1977. 205. Nurre, G. S., R. S...4.2.1. Linear Relationships . . . . . . . . . . 114 4.2.2. Nonlinear Relationships . . . . . . . . . 120 4.3. Series Expansion Methods

Contributions of Domain-General Cognitive Resources and Different Forms of Arithmetic Development to Pre-Algebraic Knowledge

PubMed Central

Fuchs, Lynn S.; Compton, Donald L.; Fuchs, Douglas; Powell, Sarah R.; Schumacher, Robin F.; Hamlett, Carol L.; Vernier, Emily; Namkung, Jessica M.; Vukovic, Rose K.

2012-01-01

The purpose of this study was to investigate the contributions of domain-general cognitive resources and different forms of arithmetic development to individual differences in pre-algebraic knowledge. Children (n=279; mean age=7.59 yrs) were assessed on 7 domain-general cognitive resources as well as arithmetic calculations and word problems at start of 2nd grade and on calculations, word problems, and pre-algebraic knowledge at end of 3rd grade. Multilevel path analysis, controlling for instructional effects associated with the sequence of classrooms in which students were nested across grades 2–3, indicated arithmetic calculations and word problems are foundational to pre-algebraic knowledge. Also, results revealed direct contributions of nonverbal reasoning and oral language to pre-algebraic knowledge, beyond indirect effects that are mediated via arithmetic calculations and word problems. By contrast, attentive behavior, phonological processing, and processing speed contributed to pre-algebraic knowledge only indirectly via arithmetic calculations and word problems. PMID:22409764

Working memory, worry, and algebraic ability.

PubMed

Trezise, Kelly; Reeve, Robert A

2014-05-01

Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship. Copyright © 2013 Elsevier Inc. All rights reserved.

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…

Just-in-Time Algebra: A Problem Solving Approach Including Multimedia and Animation.

ERIC Educational Resources Information Center

Hofmann, Roseanne S.; Hunter, Walter R.

2003-01-01

Describes a beginning algebra course that places stronger emphasis on learning to solve problems and introduces topics using real world applications. Students learn estimating, graphing, and algebraic algorithms for the purpose of solving problems. Indicates that applications motivate students by appearing to be a more relevant topic as well as…

Motivating the Concept of Eigenvectors via Cryptography

ERIC Educational Resources Information Center

Siap, Irfan

2008-01-01

New methods of teaching linear algebra in the undergraduate curriculum have attracted much interest lately. Most of this work is focused on evaluating and discussing the integration of special computer software into the Linear Algebra curriculum. In this article, I discuss my approach on introducing the concept of eigenvectors and eigenvalues,…

A Linear Algebra Measure of Cluster Quality.

ERIC Educational Resources Information Center

Mather, Laura A.

2000-01-01

Discussion of models for information retrieval focuses on an application of linear algebra to text clustering, namely, a metric for measuring cluster quality based on the theory that cluster quality is proportional to the number of terms that are disjoint across the clusters. Explains term-document matrices and clustering algorithms. (Author/LRW)

The Transformation App Redux: The Notion of Linearity

ERIC Educational Resources Information Center

Domenick, Anthony

2015-01-01

The notion of linearity is perhaps the most fundamental idea in algebraic thinking. It sets the transition to functions and culminates with the instantaneous rate of change in calculus. Despite its simplicity, this concept poses complexities to a considerable number of first semester college algebra students. The purpose of this observational…

Optical linear algebra processors: noise and error-source modeling.

PubMed

Casasent, D; Ghosh, A

1985-06-01

The modeling of system and component noise and error sources in optical linear algebra processors (OLAP's) are considered, with attention to the frequency-multiplexed OLAP. General expressions are obtained for the output produced as a function of various component errors and noise. A digital simulator for this model is discussed.

Mathematical Modelling in Engineering: An Alternative Way to Teach Linear Algebra

ERIC Educational Resources Information Center

Domínguez-García, S.; García-Planas, M. I.; Taberna, J.

2016-01-01

Technological advances require that basic science courses for engineering, including Linear Algebra, emphasize the development of mathematical strengths associated with modelling and interpretation of results, which are not limited only to calculus abilities. Based on this consideration, we have proposed a project-based learning, giving a dynamic…

Space and frequency-multiplexed optical linear algebra processor - Fabrication and initial tests

NASA Technical Reports Server (NTRS)

Casasent, D.; Jackson, J.

1986-01-01

A new optical linear algebra processor architecture is described. Space and frequency-multiplexing are used to accommodate bipolar and complex-valued data. A fabricated laboratory version of this processor is described, the electronic support system used is discussed, and initial test data obtained on it are presented.

Optical linear algebra processors - Noise and error-source modeling

NASA Technical Reports Server (NTRS)

Casasent, D.; Ghosh, A.

1985-01-01

The modeling of system and component noise and error sources in optical linear algebra processors (OLAPs) are considered, with attention to the frequency-multiplexed OLAP. General expressions are obtained for the output produced as a function of various component errors and noise. A digital simulator for this model is discussed.

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…

A splitting algorithm for the wavelet transform of cubic splines on a nonuniform grid

NASA Astrophysics Data System (ADS)

Sulaimanov, Z. M.; Shumilov, B. M.

2017-10-01

For cubic splines with nonuniform nodes, splitting with respect to the even and odd nodes is used to obtain a wavelet expansion algorithm in the form of the solution to a three-diagonal system of linear algebraic equations for the coefficients. Computations by hand are used to investigate the application of this algorithm for numerical differentiation. The results are illustrated by solving a prediction problem.

SATA Stochastic Algebraic Topology and Applications

DTIC Science & Technology

2017-01-23

Harris et al. Selective sampling after solving a convex problem". arXiv:1609.05609 [ math , stat] (Sept. 2016). arXiv: 1609.05609. 13. Baryshnikov...Functions, Adv. Math . 245, 573-586, 2014. 15. Y. Baryshnikov, Liberzon, Daniel,Robust stability conditions for switched linear systems: Commutator bounds...Consistency via Kernel Estimation, arXiv:1407.5272 [ math , stat] (July 2014) arXiv: 1407.5272. to appear in Bernoulli 18. O.Bobrowski and S.Weinberger

Symmetries of the Space of Linear Symplectic Connections

NASA Astrophysics Data System (ADS)

Fox, Daniel J. F.

2017-01-01

There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen-Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle polynomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their! linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term.

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.

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

PubMed Central

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 taught students to represent the structural, defining features of word problems with overarching equations. Intervention lasted 16 weeks. We pretested and posttested 270 students on measures of word-problem skill; analyses that accounted for the nested structure of the data indicated superior word-problem learning for SBI students. Descriptive analyses of students’ word-problem work indicated that SBI helped students represent the structure of word problems with algebraic equations, suggesting that SBI promoted this aspect of students’ emerging algebraic reasoning. PMID:20539822

Mathematics in the Real World.

ERIC Educational Resources Information Center

Borenstein, Matt

1997-01-01

The abstract nature of algebra causes difficulties for many students. Describes "Real-World Data," an algebra course designed for students with low grades in algebra and provides multidisciplinary experiments (linear functions and variations; quadratic, square-root, and inverse relations; and exponential and periodic variation)…

Linear, multivariable robust control with a mu perspective

NASA Technical Reports Server (NTRS)

Packard, Andy; Doyle, John; Balas, Gary

1993-01-01

The structured singular value is a linear algebra tool developed to study a particular class of matrix perturbation problems arising in robust feedback control of multivariable systems. These perturbations are called linear fractional, and are a natural way to model many types of uncertainty in linear systems, including state-space parameter uncertainty, multiplicative and additive unmodeled dynamics uncertainty, and coprime factor and gap metric uncertainty. The structured singular value theory provides a natural extension of classical SISO robustness measures and concepts to MIMO systems. The structured singular value analysis, coupled with approximate synthesis methods, make it possible to study the tradeoff between performance and uncertainty that occurs in all feedback systems. In MIMO systems, the complexity of the spatial interactions in the loop gains make it difficult to heuristically quantify the tradeoffs that must occur. This paper examines the role played by the structured singular value (and its computable bounds) in answering these questions, as well as its role in the general robust, multivariable control analysis and design problem.

Linking Computer Algebra Systems and Paper-and-Pencil Techniques To Support the Teaching of Mathematics.

ERIC Educational Resources Information Center

van Herwaarden, Onno A.; Gielen, Joseph L. W.

2002-01-01

Focuses on students showing a lack of conceptual insight while using computer algebra systems (CAS) in the setting of an elementary calculus and linear algebra course for first year university students in social sciences. The use of a computer algebra environment has been incorporated into a more traditional course but with special attention on…

The pre-image problem in kernel methods.

PubMed

Kwok, James Tin-yau; Tsang, Ivor Wai-hung

2004-11-01

In this paper, we address the problem of finding the pre-image of a feature vector in the feature space induced by a kernel. This is of central importance in some kernel applications, such as on using kernel principal component analysis (PCA) for image denoising. Unlike the traditional method which relies on nonlinear optimization, our proposed method directly finds the location of the pre-image based on distance constraints in the feature space. It is noniterative, involves only linear algebra and does not suffer from numerical instability or local minimum problems. Evaluations on performing kernel PCA and kernel clustering on the USPS data set show much improved performance.

Propagating Qualitative Values Through Quantitative Equations

NASA Technical Reports Server (NTRS)

Kulkarni, Deepak

1992-01-01

In most practical problems where traditional numeric simulation is not adequate, one need to reason about a system with both qualitative and quantitative equations. In this paper, we address the problem of propagating qualitative values represented as interval values through quantitative equations. Previous research has produced exponential-time algorithms for approximate solution of the problem. These may not meet the stringent requirements of many real time applications. This paper advances the state of art by producing a linear-time algorithm that can propagate a qualitative value through a class of complex quantitative equations exactly and through arbitrary algebraic expressions approximately. The algorithm was found applicable to Space Shuttle Reaction Control System model.

Effects of van der Waals Force and Thermal Stresses on Pull-in Instability of Clamped Rectangular Microplates

PubMed Central

Batra, Romesh C.; Porfiri, Maurizio; Spinello, Davide

2008-01-01

We study the influence of von Kármán nonlinearity, van der Waals force, and thermal stresses on pull-in instability and small vibrations of electrostatically actuated microplates. We use the Galerkin method to develop a tractable reduced-order model for electrostatically actuated clamped rectangular microplates in the presence of van der Waals forces and thermal stresses. More specifically, we reduce the governing two-dimensional nonlinear transient boundary-value problem to a single nonlinear ordinary differential equation. For the static problem, the pull-in voltage and the pull-in displacement are determined by solving a pair of nonlinear algebraic equations. The fundamental vibration frequency corresponding to a deflected configuration of the microplate is determined by solving a linear algebraic equation. The proposed reduced-order model allows for accurately estimating the combined effects of van der Waals force and thermal stresses on the pull-in voltage and the pull-in deflection profile with an extremely limited computational effort. PMID:27879752

Effects of van der Waals Force and Thermal Stresses on Pull-in Instability of Clamped Rectangular Microplates.

PubMed

Batra, Romesh C; Porfiri, Maurizio; Spinello, Davide

2008-02-15

We study the influence of von Karman nonlinearity, van der Waals force, and a athermal stresses on pull-in instability and small vibrations of electrostatically actuated mi-croplates. We use the Galerkin method to develop a tractable reduced-order model for elec-trostatically actuated clamped rectangular microplates in the presence of van der Waals forcesand thermal stresses. More specifically, we reduce the governing two-dimensional nonlineartransient boundary-value problem to a single nonlinear ordinary differential equation. For thestatic problem, the pull-in voltage and the pull-in displacement are determined by solving apair of nonlinear algebraic equations. The fundamental vibration frequency corresponding toa deflected configuration of the microplate is determined by solving a linear algebraic equa-tion. The proposed reduced-order model allows for accurately estimating the combined effectsof van der Waals force and thermal stresses on the pull-in voltage and the pull-in deflectionprofile with an extremely limited computational effort.

A new exact method for line radiative transfer

NASA Astrophysics Data System (ADS)

Elitzur, Moshe; Asensio Ramos, Andrés

2006-01-01

We present a new method, the coupled escape probability (CEP), for exact calculation of line emission from multi-level systems, solving only algebraic equations for the level populations. The CEP formulation of the classical two-level problem is a set of linear equations, and we uncover an exact analytic expression for the emission from two-level optically thick sources that holds as long as they are in the `effectively thin' regime. In a comparative study of a number of standard problems, the CEP method outperformed the leading line transfer methods by substantial margins. The algebraic equations employed by our new method are already incorporated in numerous codes based on the escape probability approximation. All that is required for an exact solution with these existing codes is to augment the expression for the escape probability with simple zone-coupling terms. As an application, we find that standard escape probability calculations generally produce the correct cooling emission by the CII 158-μm line but not by the 3P lines of OI.

An LMI approach for the Integral Sliding Mode and H∞ State Feedback Control Problem

NASA Astrophysics Data System (ADS)

Bezzaoucha, Souad; Henry, David

2015-11-01

This paper deals with the state feedback control problem for linear uncertain systems subject to both matched and unmatched perturbations. The proposed control law is based on an the Integral Sliding Mode Control (ISMC) approach to tackle matched perturbations as well as the H∞ paradigm for robustness against unmatched perturbations. The proposed method also parallels the work presented in [1] which addressed the same problem and proposed a solution involving an Algebraic Riccati Equation (ARE)-based formulation. The contribution of this paper is concerned by the establishment of a Linear Matrix Inequality (LMI)-based solution which offers the possibility to consider other types of constraints such as 𝓓-stability constraints (pole assignment-like constraints). The proposed methodology is applied to a pilot three-tank system and experiment results illustrate the feasibility. Note that only a few real experiments have been rarely considered using SMC in the past. This is due to the high energetic behaviour of the control signal. It is important to outline that the paper does not aim at proposing a LMI formulation of an ARE. This is done since 1971 [2] and further discussed in [3] where the link between AREs and ARIs (algebraic Riccati inequality) is established for the H∞ control problem. The main contribution of this paper is to establish the adequate LMI-based methodology (changes of matrix variables) so that the ARE that corresponds to the particular structure of the mixed ISMC/H∞ structure proposed by [1] can be re-formulated within the LMI paradigm.

A Brief Historical Introduction to Matrices and Their Applications

ERIC Educational Resources Information Center

Debnath, L.

2014-01-01

This paper deals with the ancient origin of matrices, and the system of linear equations. Included are algebraic properties of matrices, determinants, linear transformations, and Cramer's Rule for solving the system of algebraic equations. Special attention is given to some special matrices, including matrices in graph theory and electrical…

Avoiding Communication in Dense Linear Algebra

DTIC Science & Technology

2013-08-16

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Asymptotic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6...and parallelizing Strassen’s matrix multiplication algorithm (Chapter 11). 6 Chapter 2 Preliminaries 2.1 Notation and Definitions In this section we...between computations and algo- rithms). The following definition is based on [56]: Definition 2.1. A classical algorithm in linear algebra is one that

Principal Component Analysis: Resources for an Essential Application of Linear Algebra

ERIC Educational Resources Information Center

Pankavich, Stephen; Swanson, Rebecca

2015-01-01

Principal Component Analysis (PCA) is a highly useful topic within an introductory Linear Algebra course, especially since it can be used to incorporate a number of applied projects. This method represents an essential application and extension of the Spectral Theorem and is commonly used within a variety of fields, including statistics,…

Secondary Pre-Service Teachers' Algebraic Reasoning about Linear Equation Solving

ERIC Educational Resources Information Center

Alvey, Christina; Hudson, Rick A.; Newton, Jill; Males, Lorraine M.

2016-01-01

This study analyzes the responses of 12 secondary pre-service teachers on two tasks focused on reasoning when solving linear equations. By documenting the choices PSTs made while engaging in these tasks, we gain insight into how new teachers work mathematically, reason algebraically, communicate their thinking, and make pedagogical decisions. We…

Student Connections of Linear Algebra Concepts: An Analysis of Concept Maps

ERIC Educational Resources Information Center

Lapp, Douglas A.; Nyman, Melvin A.; Berry, John S.

2010-01-01

This article examines the connections of linear algebra concepts in a first course at the undergraduate level. The theoretical underpinnings of this study are grounded in the constructivist perspective (including social constructivism), Vernaud's theory of conceptual fields and Pirie and Kieren's model for the growth of mathematical understanding.…

All Talk and More Action

ERIC Educational Resources Information Center

Williams-Candek, Maryellen

2016-01-01

How better to begin the study of linear equations in an algebra class than to determine what students already know about the subject? A seventh-grade algebra class in a suburban school undertook a project early in the school year that was completed before they began studying linear relations and functions. The project, which might have been…

Transforming an Introductory Linear Algebra Course with a TI-92 Hand-Held Computer.

ERIC Educational Resources Information Center

Quesada, Antonio R.

2003-01-01

Describes how the introduction of the TI-92 transformed a traditional first semester linear algebra course into a matrix-oriented course that emphasized conceptual understanding, relevant applications, and numerical issues. Indicates an increase in students' overall performance as they found the calculator very useful, believed it helped them…

Subspace in Linear Algebra: Investigating Students' Concept Images and Interactions with the Formal Definition

ERIC Educational Resources Information Center

Wawro, Megan; Sweeney, George F.; Rabin, Jeffrey M.

2011-01-01

This paper reports on a study investigating students' ways of conceptualizing key ideas in linear algebra, with the particular results presented here focusing on student interactions with the notion of subspace. In interviews conducted with eight undergraduates, we found students' initial descriptions of subspace often varied substantially from…

Advanced Linear Algebra: A Call for the Early Introduction of Complex Numbers

ERIC Educational Resources Information Center

Garcia, Stephan Ramon

2017-01-01

A second course in linear algebra that goes beyond the traditional lower-level curriculum is increasingly important for students of the mathematical sciences. Although many applications involve only real numbers, a solid understanding of complex arithmetic often sheds significant light. Many instructors are unaware of the opportunities afforded by…

An Authentic Task That Models Quadratics

ERIC Educational Resources Information Center

Baron, Lorraine M.

2015-01-01

As students develop algebraic reasoning in grades 5 to 9, they learn to recognize patterns and understand expressions, equations, and variables. Linear functions are a focus in eighth-grade mathematics, and by algebra 1, students must make sense of functions that are not linear. This article describes how students worked through a classroom task…

Lack of Set Theory Relevant Prerequisite Knowledge

ERIC Educational Resources Information Center

Dogan-Dunlap, Hamide

2006-01-01

Many students struggle with college mathematics topics due to a lack of mastery of prerequisite knowledge. Set theory language is one such prerequisite for linear algebra courses. Many students' mistakes on linear algebra questions reveal a lack of mastery of set theory knowledge. This paper reports the findings of a qualitative analysis of a…

Mat-Rix-Toe: Improving Writing through a Game-Based Project in Linear Algebra

ERIC Educational Resources Information Center

Graham-Squire, Adam; Farnell, Elin; Stockton, Julianna Connelly

2014-01-01

The Mat-Rix-Toe project utilizes a matrix-based game to deepen students' understanding of linear algebra concepts and strengthen students' ability to express themselves mathematically. The project was administered in three classes using slightly different approaches, each of which included some editing component to encourage the…

Using Cognitive Tutor Software in Learning Linear Algebra Word Concept

ERIC Educational Resources Information Center

Yang, Kai-Ju

2015-01-01

This paper reports on a study of twelve 10th grade students using Cognitive Tutor, a math software program, to learn linear algebra word concept. The study's purpose was to examine whether students' mathematics performance as it is related to using Cognitive Tutor provided evidence to support Koedlinger's (2002) four instructional principles used…

Student Reactions to Learning Theory Based Curriculum Materials in Linear Algebra--A Survey Analysis

ERIC Educational Resources Information Center

Cooley, Laurel; Vidakovic, Draga; Martin, William O.; Dexter, Scott; Suzuki, Jeff

2016-01-01

In this report we examine students' perceptions of the implementation of carefully designed curriculum materials (called modules) in linear algebra courses at three different universities. The curricular materials were produced collaboratively by STEM and mathematics education faculty as members of a professional learning community (PLC) over…

A Framework for Mathematical Thinking: The Case of Linear Algebra

ERIC Educational Resources Information Center

Stewart, Sepideh; Thomas, Michael O. J.

2009-01-01

Linear algebra is one of the unavoidable advanced courses that many mathematics students encounter at university level. The research reported here was part of the first author's recent PhD study, where she created and applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate the…

Partially Flipped Linear Algebra: A Team-Based Approach

ERIC Educational Resources Information Center

Carney, Debra; Ormes, Nicholas; Swanson, Rebecca

2015-01-01

In this article we describe a partially flipped Introductory Linear Algebra course developed by three faculty members at two different universities. We give motivation for our partially flipped design and describe our implementation in detail. Two main features of our course design are team-developed preview videos and related in-class activities.…

Definitions Are Important: The Case of Linear Algebra

ERIC Educational Resources Information Center

Berman, Abraham; Shvartsman, Ludmila

2016-01-01

In this paper we describe an experiment in a linear algebra course. The aim of the experiment was to promote the students' understanding of the studied concepts focusing on their definitions. It seems to be a given that students should understand concepts' definitions before working substantially with them. Unfortunately, in many cases they do…

Developing CORE model-based worksheet with recitation task to facilitate students’ mathematical communication skills in linear algebra course

NASA Astrophysics Data System (ADS)

Risnawati; Khairinnisa, S.; Darwis, A. H.

2018-01-01

The purpose of this study was to develop a CORE model-based worksheet with recitation task that were valid and practical and could facilitate students’ communication skills in Linear Algebra course. This study was conducted in mathematics education department of one public university in Riau, Indonesia. Participants of the study were media and subject matter experts as validators as well as students from mathematics education department. The objects of this study are students’ worksheet and students’ mathematical communication skills. The results of study showed that: (1) based on validation of the experts, the developed students’ worksheet was valid and could be applied for students in Linear Algebra courses; (2) based on the group trial, the practicality percentage was 92.14% in small group and 90.19% in large group, so the worksheet was very practical and could attract students to learn; and (3) based on the post test, the average percentage of ideals was 87.83%. In addition, the results showed that the students’ worksheet was able to facilitate students’ mathematical communication skills in linear algebra course.

The Effect of Using Concept Maps in Elementary Linear Algebra Course on Students’ Learning

NASA Astrophysics Data System (ADS)

Syarifuddin, H.

2018-04-01

This paper presents the results of a classroom action research that was done in Elementary Linear Algebra course at Universitas Negeri Padang. The focus of the research want to see the effect of using concept maps in the course on students’ learning. Data in this study were collected through classroom observation, students’ reflective journal and concept maps that were created by students. The result of the study was the using of concept maps in Elementary Linera Algebra course gave positive effect on students’ learning.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chiang, Nai-Yuan; Zavala, Victor M.

We present a filter line-search algorithm that does not require inertia information of the linear system. This feature enables the use of a wide range of linear algebra strategies and libraries, which is essential to tackle large-scale problems on modern computing architectures. The proposed approach performs curvature tests along the search step to detect negative curvature and to trigger convexification. We prove that the approach is globally convergent and we implement the approach within a parallel interior-point framework to solve large-scale and highly nonlinear problems. Our numerical tests demonstrate that the inertia-free approach is as efficient as inertia detection viamore » symmetric indefinite factorizations. We also demonstrate that the inertia-free approach can lead to reductions in solution time because it reduces the amount of convexification needed.« less

Algebraic special functions and SO(3,2)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Celeghini, E., E-mail: celeghini@fi.infn.it; Olmo, M.A. del, E-mail: olmo@fta.uva.es

2013-06-15

A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra so(3,2) with quadratic Casimir equals to −5/4. As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be homomorphic to the space of linear operators acting on the L{sup 2} functions defined on (−1,1)×Z and on the sphere S{sup 2}, respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining inmore » this way the “algebraic special functions” that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the L{sup 2} functions is homomorphic to the universal enveloping algebra. The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group SO(3,2). -- Highlights: •The algebraic ladder structure is constructed for the associated Legendre polynomials (ALP). •ALP and spherical harmonics support a unitary irreducible SO(3,2)-representation. •A ladder structure is the condition to get a Lie group representation defining “algebraic special functions”. •The “algebraic special functions” connect Lie algebras and L{sup 2} functions.« less

What Students Choose to Do and Have to Say about Use of Multiple Representations in College Algebra

ERIC Educational Resources Information Center

Herman, Marlena

2007-01-01

This report summarizes findings on strategies chosen by students (n=38) when solving algebra problems related to various functions with the freedom to use a TI-83 graphing calculator, influences on student problem-solving strategy choices, student ability to approach algebra problems with use of multiple representations, and student beliefs on how…

Realization of preconditioned Lanczos and conjugate gradient algorithms on optical linear algebra processors.

PubMed

Ghosh, A

1988-08-01

Lanczos and conjugate gradient algorithms are important in computational linear algebra. In this paper, a parallel pipelined realization of these algorithms on a ring of optical linear algebra processors is described. The flow of data is designed to minimize the idle times of the optical multiprocessor and the redundancy of computations. The effects of optical round-off errors on the solutions obtained by the optical Lanczos and conjugate gradient algorithms are analyzed, and it is shown that optical preconditioning can improve the accuracy of these algorithms substantially. Algorithms for optical preconditioning and results of numerical experiments on solving linear systems of equations arising from partial differential equations are discussed. Since the Lanczos algorithm is used mostly with sparse matrices, a folded storage scheme to represent sparse matrices on spatial light modulators is also described.

Student Logical Implications and Connections between Symbolic Representations of a Linear System within the Context of an Introductory Linear Algebra Course Employing Inquiry-Oriented Teaching and Traditional Lecture

ERIC Educational Resources Information Center

Payton, Spencer D.

2017-01-01

This study aimed to explore how inquiry-oriented teaching could be implemented in an introductory linear algebra course that, due to various constraints, may not lend itself to inquiry-oriented teaching. In particular, the course in question has a traditionally large class size, limited amount of class time, and is often coordinated with other…

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…

The method of perturbation-harmonic balance for analysing nonlinear free vibration of MDOF systems and structures

NASA Astrophysics Data System (ADS)

Tang, Qiangang; Sun, Shixian

1992-03-01

In this paper, the perturbation technique is introduced into the method of harmonic balance. A new method used for analyzing nonlinear free vibration of multidegree-of-freedom systems and structures is obtained. The form of solution is expanded into a series of small parameters and harmonics, so no term will be lost in the solution and the algebraic equations are linear. With the linear transformations, the matrices of the equations become diagonal. As soon as the modes related to linear vibration are found, the solution can be obtained. This method is superior to the method of linearized iteration. The examples show that the method has high accuracy for small-amplitude problems and the results for rather large amplitudes are satisfactory.

General algebraic method applied to control analysis of complex engine types

NASA Technical Reports Server (NTRS)

Boksenbom, Aaron S; Hood, Richard

1950-01-01

A general algebraic method of attack on the problem of controlling gas-turbine engines having any number of independent variables was utilized employing operational functions to describe the assumed linear characteristics for the engine, the control, and the other units in the system. Matrices were used to describe the various units of the system, to form a combined system showing all effects, and to form a single condensed matrix showing the principal effects. This method directly led to the conditions on the control system for noninteraction so that any setting disturbance would affect only its corresponding controlled variable. The response-action characteristics were expressed in terms of the control system and the engine characteristics. The ideal control-system characteristics were explicitly determined in terms of any desired response action.

Reduction by invariants and projection of linear representations of Lie algebras applied to the construction of nonlinear realizations

NASA Astrophysics Data System (ADS)

Campoamor-Stursberg, R.

2018-03-01

A procedure for the construction of nonlinear realizations of Lie algebras in the context of Vessiot-Guldberg-Lie algebras of first-order systems of ordinary differential equations (ODEs) is proposed. The method is based on the reduction of invariants and projection of lowest-dimensional (irreducible) representations of Lie algebras. Applications to the description of parameterized first-order systems of ODEs related by contraction of Lie algebras are given. In particular, the kinematical Lie algebras in (2 + 1)- and (3 + 1)-dimensions are realized simultaneously as Vessiot-Guldberg-Lie algebras of parameterized nonlinear systems in R3 and R4, respectively.

Some Applications of Algebraic System Solving

ERIC Educational Resources Information Center

Roanes-Lozano, Eugenio

2011-01-01

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

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

ERIC Educational Resources Information Center

Lin, Cheng-Yao; Kuo, Yu-Chun; Ko, Yi-Yin

2015-01-01

The purpose of this study was to investigate elementary pre-service teachers' content knowledge in algebra (Linear Equation, Quadratic Equation, Functions, System Equations and Polynomials) as well as their technological pedagogical content knowledge (TPACK) in teaching algebra. Participants were 79 undergraduate pre-service teachers who were…

Ten-Year-Old Students Solving Linear Equations

ERIC Educational Resources Information Center

Brizuela, Barbara; Schliemann, Analucia

2004-01-01

In this article, the authors seek to re-conceptualize the perspective regarding students' difficulties with algebra. While acknowledging that students "do" have difficulties when learning algebra, they also argue that the generally espoused criteria for algebra as the ability to work with the syntactical rules for solving equations is…

Measuring the Readability of Elementary Algebra Using the Cloze Technique.

ERIC Educational Resources Information Center

Kulm, Gerald

The relationship to readability of ten variables characterizing structural properties of mathematical prose was investigated in elementary algebra textbooks. Readability was measured by algebra student's responses to two forms of cloze tests. Linear and currilinear correlations were calculated between each structural variable and the cloze test.…

Case Studies Listening to Students Using Kinesthetic Movement While Learning to Graph Linear Functions

ERIC Educational Resources Information Center

Novak, Melissa A.

2017-01-01

The purpose of this qualitative practitioner research study was to describe middle school algebra students' experiences of learning linear functions through kinesthetic movement. Participants were comprised of 8th grade algebra students. Practitioner research was used because I wanted to improve my teaching so students will have more success in…

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

PubMed

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

2013-01-01

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

A Practical Approach to Inquiry-Based Learning in Linear Algebra

ERIC Educational Resources Information Center

Chang, J.-M.

2011-01-01

Linear algebra has become one of the most useful fields of mathematics since last decade, yet students still have trouble seeing the connection between some of the abstract concepts and real-world applications. In this article, we propose the use of thought-provoking questions in lesson designs to allow two-way communications between instructors…

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

PubMed Central

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

2013-01-01

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

Instructional Supports for Representational Fluency in Solving Linear Equations with Computer Algebra Systems and Paper-and-Pencil

ERIC Educational Resources Information Center

Fonger, Nicole L.; Davis, Jon D.; Rohwer, Mary Lou

2018-01-01

This research addresses the issue of how to support students' representational fluency--the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is…

Flipping an Algebra Classroom: Analyzing, Modeling, and Solving Systems of Linear Equations

ERIC Educational Resources Information Center

Kirvan, Rebecca; Rakes, Christopher R.; Zamora, Regie

2015-01-01

The present study investigated whether flipping an algebra classroom led to a stronger focus on conceptual understanding and improved learning of systems of linear equations for 54 seventh- and eighth-grade students using teacher journal data and district-mandated unit exam items. Multivariate analysis of covariance was used to compare scores on…

Some Comments on 'The Role of Proof in Comprehending and Teaching Elementary Linear Algebra' by F. Uhlig.

ERIC Educational Resources Information Center

Dorier, Jean-Luc; Robert, Aline; Rogalski, Marc

2002-01-01

Underlines the common points in F. Uhlig's approach published in an earlier issue of this journal about the question of proof in linear algebra. Describes some of his ideas in a new light and gives perspective for a further didactical development of Uhlig's first experiments. (Author/KHR)

Mathematical Modelling and the Learning Trajectory: Tools to Support the Teaching of Linear Algebra

ERIC Educational Resources Information Center

Cárcamo Bahamonde, Andrea Dorila; Fortuny Aymemí, Josep Maria; Gómez i Urgellés, Joan Vicenç

2017-01-01

In this article we present a didactic proposal for teaching linear algebra based on two compatible theoretical models: emergent models and mathematical modelling. This proposal begins with a problematic situation related to the creation and use of secure passwords, which leads students toward the construction of the concepts of spanning set and…

A Modified Approach to Team-Based Learning in Linear Algebra Courses

ERIC Educational Resources Information Center

Nanes, Kalman M.

2014-01-01

This paper documents the author's adaptation of team-based learning (TBL), an active learning pedagogy developed by Larry Michaelsen and others, in the linear algebra classroom. The paper discusses the standard components of TBL and the necessary changes to those components for the needs of the course in question. There is also an empirically…

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

ERIC Educational Resources Information Center

Gasyna, Zbigniew L.

2008-01-01

Computational experiment is proposed in which a linear algebra method is applied to the solution of the Schrodinger equation for a diatomic oscillator. Calculations of the vibration-rotation spectrum for the HCl molecule are presented and the results show excellent agreement with experimental data. (Contains 1 table and 1 figure.)

Developing Conceptual Understanding and Definitional Clarity in Linear Algebra through the Three Worlds of Mathematical Thinking

ERIC Educational Resources Information Center

Hannah, John; Stewart, Sepideh; Thomas, Michael

2016-01-01

Linear algebra is one of the first abstract mathematics courses that students encounter at university. Research shows that many students find the dense presentation of definitions, theorems and proofs difficult to comprehend. Using a case study approach, we report on a teaching intervention based on Tall's three worlds (embodied, symbolic and…

Creating Discussions with Classroom Voting in Linear Algebra

ERIC Educational Resources Information Center

Cline, Kelly; Zullo, Holly; Duncan, Jonathan; Stewart, Ann; Snipes, Marie

2013-01-01

We present a study of classroom voting in linear algebra, in which the instructors posed multiple-choice questions to the class and then allowed a few minutes for consideration and small-group discussion. After each student in the class voted on the correct answer using a classroom response system, a set of clickers, the instructor then guided a…

An Example of Inquiry in Linear Algebra: The Roles of Symbolizing and Brokering

ERIC Educational Resources Information Center

Zandieh, Michelle; Wawro, Megan; Rasmussen, Chris

2017-01-01

In this paper we address practical questions such as: How do symbols appear and evolve in an inquiry-oriented classroom? How can an instructor connect students with traditional notation and vocabulary without undermining their sense of ownership of the material? We tender an example from linear algebra that highlights the roles of the instructor…

Linear Algebra and the Experiences of a "Flipper"

ERIC Educational Resources Information Center

Wright, Sarah E.

2015-01-01

This paper describes the linear algebra class I taught during Spring 2014 semester at Adelphi University. I discuss the details of how I flipped the class and incorporated elements of inquiry-based learning as well as the reasoning behind specific decisions I made. I give feedback from the students on the success of the course and provide my own…

DOE Office of Scientific and Technical Information (OSTI.GOV)

Spotz, William F.

PyTrilinos is a set of Python interfaces to compiled Trilinos packages. This collection supports serial and parallel dense linear algebra, serial and parallel sparse linear algebra, direct and iterative linear solution techniques, algebraic and multilevel preconditioners, nonlinear solvers and continuation algorithms, eigensolvers and partitioning algorithms. Also included are a variety of related utility functions and classes, including distributed I/O, coloring algorithms and matrix generation. PyTrilinos vector objects are compatible with the popular NumPy Python package. As a Python front end to compiled libraries, PyTrilinos takes advantage of the flexibility and ease of use of Python, and the efficiency of themore » underlying C++, C and Fortran numerical kernels. This paper covers recent, previously unpublished advances in the PyTrilinos package.« less

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.

Discretization and Numerical Solution of a Plane Problem in the Mechanics of Interfacial Cracks

NASA Astrophysics Data System (ADS)

Khoroshun, L. P.

2017-01-01

The Fourier transform is used to reduce the linear plane problem of the tension of a body with an interfacial crack to a system of dual equations for the transformed stresses and, then, to a system of integro-differential equations for the difference of displacements of the crack faces. After discretization, this latter system transforms into a system of algebraic equations for displacements of the crack faces. The effect of the bielastic constant and the number of discretization points on the half-length of the crack faces and the distribution of stresses at the interface is studied

The exact fundamental solution for the Benes tracking problem

NASA Astrophysics Data System (ADS)

Balaji, Bhashyam

2009-05-01

The universal continuous-discrete tracking problem requires the solution of a Fokker-Planck-Kolmogorov forward equation (FPKfe) for an arbitrary initial condition. Using results from quantum mechanics, the exact fundamental solution for the FPKfe is derived for the state model of arbitrary dimension with Benes drift that requires only the computation of elementary transcendental functions and standard linear algebra techniques- no ordinary or partial differential equations need to be solved. The measurement process may be an arbitrary, discrete-time nonlinear stochastic process, and the time step size can be arbitrary. Numerical examples are included, demonstrating its utility in practical implementation.

On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations

NASA Technical Reports Server (NTRS)

Gunzburger, M. D.; Nicolaides, R. A.

1986-01-01

Substructuring methods are in common use in mechanics problems where typically the associated linear systems of algebraic equations are positive definite. Here these methods are extended to problems which lead to nonpositive definite, nonsymmetric matrices. The extension is based on an algorithm which carries out the block Gauss elimination procedure without the need for interchanges even when a pivot matrix is singular. Examples are provided wherein the method is used in connection with finite element solutions of the stationary Stokes equations and the Helmholtz equation, and dual methods for second-order elliptic equations.

Resolving Phase Ambiguities in the Calibration of Redundant Interferometric Arrays: Implications for Array Design

DTIC Science & Technology

2016-03-04

summary of the linear algebra involved. As we have seen, the RSC process begins with the interferometric phase measurement β, which due to wrapping will...mentary Divisors) in Section 2 and the following defi- nition of the matrix determinant. This definition is given in many linear algebra texts (see...principle solve for a particular solution of this system by arbitrarily setting two object phases (whose spatial frequencies are not co- linear ) and one

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 the…

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kitanidis, Peter

As large-scale, commercial storage projects become operational, the problem of utilizing information from diverse sources becomes more critically important. In this project, we developed, tested, and applied an advanced joint data inversion system for CO 2 storage modeling with large data sets for use in site characterization and real-time monitoring. Emphasis was on the development of advanced and efficient computational algorithms for joint inversion of hydro-geophysical data, coupled with state-of-the-art forward process simulations. The developed system consists of (1) inversion tools using characterization data, such as 3D seismic survey (amplitude images), borehole log and core data, as well as hydraulic,more » tracer and thermal tests before CO 2 injection, (2) joint inversion tools for updating the geologic model with the distribution of rock properties, thus reducing uncertainty, using hydro-geophysical monitoring data, and (3) highly efficient algorithms for directly solving the dense or sparse linear algebra systems derived from the joint inversion. The system combines methods from stochastic analysis, fast linear algebra, and high performance computing. The developed joint inversion tools have been tested through synthetic CO 2 storage examples.« less

CSpace: an integrated workplace for the graphical and algebraic analysis of phase assemblages on 32-bit wintel platforms

NASA Astrophysics Data System (ADS)

Torres-Roldan, Rafael L.; Garcia-Casco, Antonio; Garcia-Sanchez, Pedro A.

2000-08-01

CSpace is a program for the graphical and algebraic analysis of composition relations within chemical systems. The program is particularly suited to the needs of petrologists, but could also prove useful for mineralogists, geochemists and other environmental scientists. A few examples of what can be accomplished with CSpace are the mapping of compositions into some desired set of system/phase components, the estimation of reaction/mixing coefficients and assessment of phase-rule compatibility relations within or between complex mineral assemblages. The program also allows dynamic inspection of compositional relations by means of barycentric plots. CSpace provides an integrated workplace for data management, manipulation and plotting. Data management is done through a built-in spreadsheet-like editor, which also acts as a data repository for the graphical and algebraic procedures. Algebraic capabilities are provided by a mapping engine and a matrix analysis tool, both of which are based on singular-value decomposition. The mapping engine uses a general approach to linear mapping, capable of handling determined, underdetermined and overdetermined problems. The matrix analysis tool is implemented as a task "wizard" that guides the user through a number of steps to perform matrix approximation (finding nearest rank-deficient models of an input composition matrix), and inspection of null-reaction space relationships (i.e. of implicit linear relations among the elements of the composition matrix). Graphical capabilities are provided by a graph engine that directly links with the contents of the data editor. The graph engine can generate sophisticated 2-D ternary (triangular) and 3D quaternary (tetrahedral) barycentric plots and includes features such as interactive re-sizing and rotation, on-the-fly coordinate scaling and support for automated drawing of tie lines.

A generalized Lyapunov theory for robust root clustering of linear state space models with real parameter uncertainty

NASA Technical Reports Server (NTRS)

Yedavalli, R. K.

1992-01-01

The problem of analyzing and designing controllers for linear systems subject to real parameter uncertainty is considered. An elegant, unified theory for robust eigenvalue placement is presented for a class of D-regions defined by algebraic inequalities by extending the nominal matrix root clustering theory of Gutman and Jury (1981) to linear uncertain time systems. The author presents explicit conditions for matrix root clustering for different D-regions and establishes the relationship between the eigenvalue migration range and the parameter range. The bounds are all obtained by one-shot computation in the matrix domain and do not need any frequency sweeping or parameter gridding. The method uses the generalized Lyapunov theory for getting the bounds.

Elastic collisions of classical point particles on a finite frictionless linear track with perfectly reflecting endpoints

NASA Astrophysics Data System (ADS)

DeLuca, R.

2006-03-01

Repeated elastic collisions of point particles on a finite frictionless linear track with perfectly reflecting endpoints are considered. The problem is analysed by means of an elementary linear algebra approach. It is found that, starting with a state consisting of a projectile particle in motion at constant velocity and a target particle at rest in a fixed known position, the points at which collisions occur on track, when plotted versus progressive numerals, corresponding to the collisions themselves, show periodic patterns for a rather large choice of values of the initial position x(0) and on the mass ratio r. For certain values of these parameters, however, only regular behaviour over a large number of collisions is detected.

Statistics of the stochastically forced Lorenz attractor by the Fokker-Planck equation and cumulant expansions.

PubMed

Allawala, Altan; Marston, J B

2016-11-01

We investigate the Fokker-Planck description of the equal-time statistics of the three-dimensional Lorenz attractor with additive white noise. The invariant measure is found by computing the zero (or null) mode of the linear Fokker-Planck operator as a problem of sparse linear algebra. Two variants are studied: a self-adjoint construction of the linear operator and the replacement of diffusion with hyperdiffusion. We also access the low-order statistics of the system by a perturbative expansion in equal-time cumulants. A comparison is made to statistics obtained by the standard approach of accumulation via direct numerical simulation. Theoretical and computational aspects of the Fokker-Planck and cumulant expansion methods are discussed.

Algebraic methods for the solution of some linear matrix equations

NASA Technical Reports Server (NTRS)

Djaferis, T. E.; Mitter, S. K.

1979-01-01

The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.

Control and stabilization of decentralized systems

NASA Technical Reports Server (NTRS)

Byrnes, Christopher I.; Gilliam, David; Martin, Clyde F.

1989-01-01

Proceeding from the problem posed by the need to stabilize the motion of two helicopters maneuvering a single load, a methodology is developed for the stabilization of classes of decentralized systems based on a more algebraic approach, which involves the external symmetries of decentralized systems. Stabilizing local-feedback laws are derived for any class of decentralized systems having a semisimple algebra of symmetries; the helicopter twin-lift problem, as well as certain problems involving the stabilization of discretizations of distributed parameter problems, have just such algebras of symmetries.

Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

NASA Astrophysics Data System (ADS)

Parand, K.; Latifi, S.; Moayeri, M. M.; Delkhosh, M.

2018-05-01

In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective, reliable and does not require any restrictive assumptions for nonlinear terms.

Form in Algebra: Reflecting, with Peacock, on Upper Secondary School Teaching.

ERIC Educational Resources Information Center

Menghini, Marta

1994-01-01

Discusses algebra teaching by looking back into the history of algebra and the work of George Peacock, who considered algebra from two points of view: symbolic and instrumental. Claims that, to be meaningful, algebra must be linked to real-world problems. (18 references) (MKR)

Deriving the Regression Line with Algebra

ERIC Educational Resources Information Center

Quintanilla, John A.

2017-01-01

Exploration with spreadsheets and reliance on previous skills can lead students to determine the line of best fit. To perform linear regression on a set of data, students in Algebra 2 (or, in principle, Algebra 1) do not have to settle for using the mysterious "black box" of their graphing calculators (or other classroom technologies).…

Exact Baker-Campbell-Hausdorff formula for the contact Heisenberg algebra

NASA Astrophysics Data System (ADS)

Bravetti, Alessandro; Garcia-Chung, Angel; Tapias, Diego

2017-03-01

In this work we introduce the contact Heisenberg algebra which is the restriction of the Jacobi algebra on contact manifolds to the linear and constant functions. We give the exact expression of its corresponding Baker-Campbell-Hausdorff formula. We argue that this result is relevant to the quantization of contact systems.

Algebraic Generalization Strategies Used by Kuwaiti Pre-Service Teachers

ERIC Educational Resources Information Center

Alajmi, Amal Hussain

2016-01-01

This study reports on the algebraic generalization strategies used by elementary and middle/high school pre-service mathematics teachers in Kuwait. They were presented with 9 tasks that involved linear, exponential, and quadratic situations. The results showed that these pre-service teachers had difficulty in generalizing algebraic rules in all 3…

Introduction to Matrix Algebra, Student's Text, Unit 23.

ERIC Educational Resources Information Center

Allen, Frank B.; And Others

Unit 23 in the SMSG secondary school mathematics series is a student text covering the following topics in matrix algebra: matrix operations, the algebra of 2 X 2 matrices, matrices and linear systems, representation of column matrices as geometric vectors, and transformations of the plane. Listed in the appendix are four research exercises in…

Implementing dense linear algebra algorithms using multitasking on the CRAY X-MP-4 (or approaching the gigaflop)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dongarra, J.J.; Hewitt, T.

1985-08-01

This note describes some experiments on simple, dense linear algebra algorithms. These experiments show that the CRAY X-MP is capable of small-grain multitasking arising from standard implementations of LU and Cholesky decomposition. The implementation described here provides the ''fastest'' execution rate for LU decomposition, 718 MFLOPS for a matrix of order 1000.

Visualizing the Inner Product Space R[superscript m x n] in a MATLAB-Assisted Linear Algebra Classroom

ERIC Educational Resources Information Center

Caglayan, Günhan

2018-01-01

This linear algebra note offers teaching and learning ideas in the treatment of the inner product space R[superscript m x n] in a technology-supported learning environment. Classroom activities proposed in this note demonstrate creative ways of integrating MATLAB technology into various properties of Frobenius inner product as visualization tools…

A Method for Using Adjacency Matrices to Analyze the Connections Students Make within and between Concepts: The Case of Linear Algebra

ERIC Educational Resources Information Center

Selinski, Natalie E.; Rasmussen, Chris; Wawro, Megan; Zandieh, Michelle

2014-01-01

The central goals of most introductory linear algebra courses are to develop students' proficiency with matrix techniques, to promote their understanding of key concepts, and to increase their ability to make connections between concepts. In this article, we present an innovative method using adjacency matrices to analyze students' interpretation…

BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS

NASA Technical Reports Server (NTRS)

Krogh, F. T.

1994-01-01

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

Comparative analysis of different variants of the Uzawa algorithm in problems of the theory of elasticity for incompressible materials.

PubMed

Styopin, Nikita E; Vershinin, Anatoly V; Zingerman, Konstantin M; Levin, Vladimir A

2016-09-01

Different variants of the Uzawa algorithm are compared with one another. The comparison is performed for the case in which this algorithm is applied to large-scale systems of linear algebraic equations. These systems arise in the finite-element solution of the problems of elasticity theory for incompressible materials. A modification of the Uzawa algorithm is proposed. Computational experiments show that this modification improves the convergence of the Uzawa algorithm for the problems of solid mechanics. The results of computational experiments show that each variant of the Uzawa algorithm considered has its advantages and disadvantages and may be convenient in one case or another.

GPU-accelerated computation of electron transfer.

PubMed

Höfinger, Siegfried; Acocella, Angela; Pop, Sergiu C; Narumi, Tetsu; Yasuoka, Kenji; Beu, Titus; Zerbetto, Francesco

2012-11-05

Electron transfer is a fundamental process that can be studied with the help of computer simulation. The underlying quantum mechanical description renders the problem a computationally intensive application. In this study, we probe the graphics processing unit (GPU) for suitability to this type of problem. Time-critical components are identified via profiling of an existing implementation and several different variants are tested involving the GPU at increasing levels of abstraction. A publicly available library supporting basic linear algebra operations on the GPU turns out to accelerate the computation approximately 50-fold with minor dependence on actual problem size. The performance gain does not compromise numerical accuracy and is of significant value for practical purposes. Copyright © 2012 Wiley Periodicals, Inc.

Analysis of the stress field in a wedge using the fast expansions with pointwise determined coefficients

NASA Astrophysics Data System (ADS)

Chernyshov, A. D.; Goryainov, V. V.; Danshin, A. A.

2018-03-01

The stress problem for the elastic wedge-shaped cutter of finite dimensions with mixed boundary conditions is considered. The differential problem is reduced to the system of linear algebraic equations by applying twice the fast expansions with respect to the angular and radial coordinate. In order to determine the unknown coefficients of fast expansions, the pointwise method is utilized. The problem solution derived has explicit analytical form and it’s valid for the entire domain including its boundary. The computed profiles of the displacements and stresses in a cross-section of the cutter are provided. The stress field is investigated for various values of opening angle and cusp’s radius.

A Comprehensive Analytical Model of Rotorcraft Aerodynamics and Dynamics. Part 1. Analysis Development

DTIC Science & Technology

1980-06-01

sufficient. Dropping the time lag terms, the equations for Xu, Xx’, and X reduce to linear algebraic equations.Y Hence in the quasistatic case the...quasistatic variables now are not described by differential equations but rather by linear algebraic equations. The solution for x0 then is simply -365...matrices for two-bladed rotor 414 7. LINEAR SYSTEM ANALYSIS 425 7,1 State Variable Form 425 7.2 Constant Coefficient System 426 7.2. 1 Eigen-analysis 426

Resolving phase ambiguities in the calibration of redundant interferometric arrays: implications for array design

DTIC Science & Technology

2015-11-30

matrix determinant. This definition is given in many linear algebra texts (see e.g. Bretscher (2001)). Definition 3.1 : Suppose we have an n-by-n...Processing, 2, 767 Blanchard P., Greenaway A., Anderton R., Appleby R., 1996, J. Opt. Soc. Am. A, 13, 1593 Bretscher O., 2001, Linear Algebra with...frequencies are not co- linear ) and one piston phase. This particular solution will then differ from the true solution by a phase ramp in the Fourier

Analyzing Algebraic Thinking Using "Guess My Number" Problems

ERIC Educational Resources Information Center

Patton, Barba; De Los Santos, Estella

2012-01-01

The purpose of this study was to assess student knowledge of numeric, visual and algebraic representations. A definite gap between arithmetic and algebra has been documented in the research. The researchers' goal was to identify a link between the two. Using four "Guess My Number" problems, seventh and tenth grade students were asked to write…

Spectral geometry of {kappa}-Minkowski space

DOE Office of Scientific and Technical Information (OSTI.GOV)

D'Andrea, Francesco

After recalling Snyder's idea [Phys. Rev. 71, 38 (1947)] of using vector fields over a smooth manifold as 'coordinates on a noncommutative space', we discuss a two-dimensional toy-model whose 'dual' noncommutative coordinates form a Lie algebra: this is the well-known {kappa}-Minkowski space [Phys. Lett. B 334, 348 (1994)]. We show how to improve Snyder's idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of {kappa}-Minkowski as linear operators on an Hilbert space (a major problem in the construction of a physical theory), study its 'spectral properties', and discuss how tomore » obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of Dimitrijevic et al. [Eur. Phys. J. C 31, 129 (2003)] can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbounded commutators with the algebra.« less

Entanglement classification with algebraic geometry

NASA Astrophysics Data System (ADS)

Sanz, M.; Braak, D.; Solano, E.; Egusquiza, I. L.

2017-05-01

We approach multipartite entanglement classification in the symmetric subspace in terms of algebraic geometry, its natural language. We show that the class of symmetric separable states has the structure of a Veronese variety and that its k-secant varieties are SLOCC invariants. Thus SLOCC classes gather naturally into families. This classification presents useful properties such as a linear growth of the number of families with the number of particles, and nesting, i.e. upward consistency of the classification. We attach physical meaning to this classification through the required interaction length of parent Hamiltonians. We show that the states W N and GHZ N are in the same secant family and that, effectively, the former can be obtained in a limit from the latter. This limit is understood in terms of tangents, leading to a refinement of the previous families. We compute explicitly the classification of symmetric states with N≤slant4 qubits in terms of both secant families and its refinement using tangents. This paves the way to further use of projective varieties in algebraic geometry to solve open problems in entanglement theory.

Computational Science in Armenia (Invited Talk)

NASA Astrophysics Data System (ADS)

Marandjian, H.; Shoukourian, Yu.

This survey is devoted to the development of informatics and computer science in Armenia. The results in theoretical computer science (algebraic models, solutions to systems of general form recursive equations, the methods of coding theory, pattern recognition and image processing), constitute the theoretical basis for developing problem-solving-oriented environments. As examples can be mentioned: a synthesizer of optimized distributed recursive programs, software tools for cluster-oriented implementations of two-dimensional cellular automata, a grid-aware web interface with advanced service trading for linear algebra calculations. In the direction of solving scientific problems that require high-performance computing resources, examples of completed projects include the field of physics (parallel computing of complex quantum systems), astrophysics (Armenian virtual laboratory), biology (molecular dynamics study of human red blood cell membrane), meteorology (implementing and evaluating the Weather Research and Forecast Model for the territory of Armenia). The overview also notes that the Institute for Informatics and Automation Problems of the National Academy of Sciences of Armenia has established a scientific and educational infrastructure, uniting computing clusters of scientific and educational institutions of the country and provides the scientific community with access to local and international computational resources, that is a strong support for computational science in Armenia.

A new S-type eigenvalue inclusion set for tensors and its applications.

PubMed

Huang, Zheng-Ge; Wang, Li-Gong; Xu, Zhong; Cui, Jing-Jing

2016-01-01

In this paper, a new S -type eigenvalue localization set for a tensor is derived by dividing [Formula: see text] into disjoint subsets S and its complement. It is proved that this new set is sharper than those presented by Qi (J. Symb. Comput. 40:1302-1324, 2005), Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and Li et al. (Linear Algebra Appl. 481:36-53, 2015). As applications of the results, new bounds for the spectral radius of nonnegative tensors and the minimum H -eigenvalue of strong M -tensors are established, and we prove that these bounds are tighter than those obtained by Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and He and Huang (J. Inequal. Appl. 2014:114, 2014).

Tracking problem solving by multivariate pattern analysis and Hidden Markov Model algorithms.

PubMed

Anderson, John R

2012-03-01

Multivariate pattern analysis can be combined with Hidden Markov Model algorithms to track the second-by-second thinking as people solve complex problems. Two applications of this methodology are illustrated with a data set taken from children as they interacted with an intelligent tutoring system for algebra. The first "mind reading" application involves using fMRI activity to track what students are doing as they solve a sequence of algebra problems. The methodology achieves considerable accuracy at determining both what problem-solving step the students are taking and whether they are performing that step correctly. The second "model discovery" application involves using statistical model evaluation to determine how many substates are involved in performing a step of algebraic problem solving. This research indicates that different steps involve different numbers of substates and these substates are associated with different fluency in algebra problem solving. Copyright © 2011 Elsevier Ltd. All rights reserved.

Finite-analytic numerical solution of heat transfer in two-dimensional cavity flow

NASA Technical Reports Server (NTRS)

Chen, C.-J.; Naseri-Neshat, H.; Ho, K.-S.

1981-01-01

Heat transfer in cavity flow is numerically analyzed by a new numerical method called the finite-analytic method. The basic idea of the finite-analytic method is the incorporation of local analytic solutions in the numerical solutions of linear or nonlinear partial differential equations. In the present investigation, the local analytic solutions for temperature, stream function, and vorticity distributions are derived. When the local analytic solution is evaluated at a given nodal point, it gives an algebraic relationship between a nodal value in a subregion and its neighboring nodal points. A system of algebraic equations is solved to provide the numerical solution of the problem. The finite-analytic method is used to solve heat transfer in the cavity flow at high Reynolds number (1000) for Prandtl numbers of 0.1, 1, and 10.

Stabilization of Inviscid Vortex Sheets

NASA Astrophysics Data System (ADS)

Protas, Bartosz; Sakajo, Takashi

2017-11-01

In this study we investigate the problem of stabilizing inviscid vortex sheets via feedback control. Such models, expressed in terms of the Birkhoff-Rott equation, are often used to describe the Kevin-Helmholtz instability of shear layers and are known to be strongly unstable to small-scale perturbations. First, we consider the linear stability of a straight vortex sheet in the periodic setting with actuation in the form of an array of point vortices or sources located a certain distance away from the sheet. We establish conditions under which this system is controllable and observable. Next, using methods of the linear control theory, we synthesize a feedback control strategy which stabilizes a straight vortex sheet in the linear regime. Given the poor conditioning of the discretized problem, reliable solution of the resulting algebraic Riccati equation requires the use of high-precision arithmetic. Finally, we demonstrate that this control approach also succeeds in the nonlinear regime, provided the magnitude of the initial perturbation is sufficiently small.

Geometry of quantum state manifolds generated by the Lie algebra operators

NASA Astrophysics Data System (ADS)

Kuzmak, A. R.

2018-03-01

The Fubini-Study metric of quantum state manifold generated by the operators which satisfy the Heisenberg Lie algebra is calculated. The similar problem is studied for the manifold generated by the so(3) Lie algebra operators. Using these results, we calculate the Fubini-Study metrics of state manifolds generated by the position and momentum operators. Also the metrics of quantum state manifolds generated by some spin systems are obtained. Finally, we generalize this problem for operators of an arbitrary Lie algebra.

An Investigation into Challenges Faced by Secondary School Teachers and Pupils in Algebraic Linear Equations: A Case of Mufulira District, Zambia

ERIC Educational Resources Information Center

Samuel, Koji; Mulenga, H. M.; Angel, Mukuka

2016-01-01

This paper investigates the challenges faced by secondary school teachers and pupils in the teaching and learning of algebraic linear equations. The study involved 80 grade 11 pupils and 15 teachers of mathematics, drawn from 4 selected secondary schools in Mufulira district, Zambia in Central Africa. A descriptive survey method was employed to…

A set for relational reasoning: Facilitation of algebraic modeling by a fraction task.

PubMed

DeWolf, Melissa; Bassok, Miriam; Holyoak, Keith J

2016-12-01

Recent work has identified correlations between early mastery of fractions and later math achievement, especially in algebra. However, causal connections between aspects of reasoning with fractions and improved algebra performance have yet to be established. The current study investigated whether relational reasoning with fractions facilitates subsequent algebraic reasoning using both pre-algebra students and adult college students. Participants were first given either a relational reasoning fractions task or a fraction algebra procedures control task. Then, all participants solved word problems and constructed algebraic equations in either multiplication or division format. The word problems and the equation construction tasks involved simple multiplicative comparison statements such as "There are 4 times as many students as teachers in a classroom." Performance on the algebraic equation construction task was enhanced for participants who had previously completed the relational fractions task compared with those who completed the fraction algebra procedures task. This finding suggests that relational reasoning with fractions can establish a relational set that promotes students' tendency to model relations using algebraic expressions. Copyright © 2016 Elsevier Inc. All rights reserved.

Mesh-free data transfer algorithms for partitioned multiphysics problems: Conservation, accuracy, and parallelism

DOE PAGES

Slattery, Stuart R.

2015-12-02

In this study we analyze and extend mesh-free algorithms for three-dimensional data transfer problems in partitioned multiphysics simulations. We first provide a direct comparison between a mesh-based weighted residual method using the common-refinement scheme and two mesh-free algorithms leveraging compactly supported radial basis functions: one using a spline interpolation and one using a moving least square reconstruction. Through the comparison we assess both the conservation and accuracy of the data transfer obtained from each of the methods. We do so for a varying set of geometries with and without curvature and sharp features and for functions with and without smoothnessmore » and with varying gradients. Our results show that the mesh-based and mesh-free algorithms are complementary with cases where each was demonstrated to perform better than the other. We then focus on the mesh-free methods by developing a set of algorithms to parallelize them based on sparse linear algebra techniques. This includes a discussion of fast parallel radius searching in point clouds and restructuring the interpolation algorithms to leverage data structures and linear algebra services designed for large distributed computing environments. The scalability of our new algorithms is demonstrated on a leadership class computing facility using a set of basic scaling studies. Finally, these scaling studies show that for problems with reasonable load balance, our new algorithms for both spline interpolation and moving least square reconstruction demonstrate both strong and weak scalability using more than 100,000 MPI processes with billions of degrees of freedom in the data transfer operation.« less

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.

Mathematical and computational aspects of nonuniform frictional slip modeling

NASA Astrophysics Data System (ADS)

Gorbatikh, Larissa

2004-07-01

A mechanics-based model of non-uniform frictional sliding is studied from the mathematical/computational analysis point of view. This problem is of a key importance for a number of applications (particularly geomechanical ones), where materials interfaces undergo partial frictional sliding under compression and shear. We show that the problem is reduced to Dirichlet's problem for monotonic loading and to Riemman's problem for cyclic loading. The problem may look like a traditional crack interaction problem, however, it is confounded by the fact that locations of n sliding intervals are not known. They are to be determined from the condition for the stress intensity factors: KII=0 at the ends of the sliding zones. Computationally, it reduces to solving a system of 2n coupled non-linear algebraic equations involving singular integrals with unknown limits of integration.

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras

PubMed Central

Yu, Zhang; Zhang, Yufeng

2009-01-01

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings. PMID:20084092

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras.

PubMed

Yu, Zhang; Zhang, Yufeng

2009-01-15

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra smu(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra smu(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras smu(6) and E is used to directly construct integrable couplings.

Reflective thinking in solving an algebra problem: a case study of field independent-prospective teacher

NASA Astrophysics Data System (ADS)

Agustan, S.; Juniati, Dwi; Yuli Eko Siswono, Tatag

2017-10-01

Nowadays, reflective thinking is one of the important things which become a concern in learning mathematics, especially in solving a mathematical problem. The purpose of this paper is to describe how the student used reflective thinking when solved an algebra problem. The subject of this research is one female student who has field independent cognitive style. This research is a descriptive exploratory study with data analysis using qualitative approach to describe in depth reflective thinking of prospective teacher in solving an algebra problem. Four main categories are used to analyse the reflective thinking in solving an algebra problem: (1) formulation and synthesis of experience, (2) orderliness of experience, (3) evaluating the experience and (4) testing the selected solution based on the experience. The results showed that the subject described the problem by using another word and the subject also found the difficulties in making mathematical modelling. The subject analysed two concepts used in solving problem. For instance, geometry related to point and line while algebra is related to algebra arithmetic operation. The subject stated that solution must have four aspect to get effective solution, specifically the ability to (a) understand the meaning of every words; (b) make mathematical modelling; (c) calculate mathematically; (d) interpret solution obtained logically. To test the internal consistency or error in solution, the subject checked and looked back related procedures and operations used. Moreover, the subject tried to resolve the problem in a different way to compare the answers which had been obtained before. The findings supported the assertion that reflective thinking provides an opportunity for the students in improving their weakness in mathematical problem solving. It can make a grow accuracy and concentration in solving a mathematical problem. Consequently, the students will get the right and logic answer by reflective thinking.

Algebraic theory of molecules

NASA Technical Reports Server (NTRS)

Iachello, Franco

1995-01-01

An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.

Low-Rank Correction Methods for Algebraic Domain Decomposition Preconditioners

DOE PAGES

Li, Ruipeng; Saad, Yousef

2017-08-01

This study presents a parallel preconditioning method for distributed sparse linear systems, based on an approximate inverse of the original matrix, that adopts a general framework of distributed sparse matrices and exploits domain decomposition (DD) and low-rank corrections. The DD approach decouples the matrix and, once inverted, a low-rank approximation is applied by exploiting the Sherman--Morrison--Woodbury formula, which yields two variants of the preconditioning methods. The low-rank expansion is computed by the Lanczos procedure with reorthogonalizations. Numerical experiments indicate that, when combined with Krylov subspace accelerators, this preconditioner can be efficient and robust for solving symmetric sparse linear systems. Comparisonsmore » with pARMS, a DD-based parallel incomplete LU (ILU) preconditioning method, are presented for solving Poisson's equation and linear elasticity problems.« less

A hypothetical learning trajectory for conceptualizing matrices as linear transformations

NASA Astrophysics Data System (ADS)

Andrews-Larson, Christine; Wawro, Megan; Zandieh, Michelle

2017-08-01

In this paper, we present a hypothetical learning trajectory (HLT) aimed at supporting students in developing flexible ways of reasoning about matrices as linear transformations in the context of introductory linear algebra. In our HLT, we highlight the integral role of the instructor in this development. Our HLT is based on the 'Italicizing N' task sequence, in which students work to generate, compose, and invert matrices that correspond to geometric transformations specified within the problem context. In particular, we describe the ways in which the students develop local transformation views of matrix multiplication (focused on individual mappings of input vectors to output vectors) and extend these local views to more global views in which matrices are conceptualized in terms of how they transform a space in a coordinated way.

Low-Rank Correction Methods for Algebraic Domain Decomposition Preconditioners

DOE Office of Scientific and Technical Information (OSTI.GOV)

Li, Ruipeng; Saad, Yousef

This study presents a parallel preconditioning method for distributed sparse linear systems, based on an approximate inverse of the original matrix, that adopts a general framework of distributed sparse matrices and exploits domain decomposition (DD) and low-rank corrections. The DD approach decouples the matrix and, once inverted, a low-rank approximation is applied by exploiting the Sherman--Morrison--Woodbury formula, which yields two variants of the preconditioning methods. The low-rank expansion is computed by the Lanczos procedure with reorthogonalizations. Numerical experiments indicate that, when combined with Krylov subspace accelerators, this preconditioner can be efficient and robust for solving symmetric sparse linear systems. Comparisonsmore » with pARMS, a DD-based parallel incomplete LU (ILU) preconditioning method, are presented for solving Poisson's equation and linear elasticity problems.« less

Graph C ∗-algebras and Z2-quotients of quantum spheres

NASA Astrophysics Data System (ADS)

Hajac, Piotr M.; Matthes, Rainer; Szymański, Wojciech

2003-06-01

We consider two Z2-actions on the Podleś generic quantum spheres. They yield, as noncommutative quotient spaces, the Klimek-Lesmewski q-disc and the quantum real projective space, respectively. The C ∗-algebas of all these quantum spaces are described as graph C ∗-algebras. The K-groups of the thus presented C ∗-algebras are then easily determined from the general theory of graph C ∗-algebas. For the quantum real projective space, we also recall the classification of the classes of irreducible ∗-representations of its algebra and give a linear basis for this algebra.

Breaking Megrelishvili protocol using matrix diagonalization

NASA Astrophysics Data System (ADS)

Arzaki, Muhammad; Triantoro Murdiansyah, Danang; Adi Prabowo, Satrio

2018-03-01

In this article we conduct a theoretical security analysis of Megrelishvili protocol—a linear algebra-based key agreement between two participants. We study the computational complexity of Megrelishvili vector-matrix problem (MVMP) as a mathematical problem that strongly relates to the security of Megrelishvili protocol. In particular, we investigate the asymptotic upper bounds for the running time and memory requirement of the MVMP that involves diagonalizable public matrix. Specifically, we devise a diagonalization method for solving the MVMP that is asymptotically faster than all of the previously existing algorithms. We also found an important counterintuitive result: the utilization of primitive matrix in Megrelishvili protocol makes the protocol more vulnerable to attacks.

On controllability of homogeneous and inhomogeneous discrete-time multi-input bilinear systems in dimension two

NASA Astrophysics Data System (ADS)

Tie, Lin

2017-08-01

In this paper, the controllability problem of two-dimensional discrete-time multi-input bilinear systems is completely solved. The homogeneous and the inhomogeneous cases are studied separately and necessary and sufficient conditions for controllability are established by using a linear algebraic method, which are easy to apply. Moreover, for the uncontrollable systems, near-controllability is considered and similar necessary and sufficient conditions are also obtained. Finally, examples are provided to demonstrate the results of this paper.

GHM method for obtaining rationalsolutions of nonlinear differential equations.

PubMed

Vazquez-Leal, Hector; Sarmiento-Reyes, Arturo

2015-01-01

In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30.

Factors Related to Problem Solving by College Students in Developmental Algebra.

ERIC Educational Resources Information Center

Schonberger, Ann K.

A study was conducted to contrast the characteristics of three groups of college students who completed a developmental algebra course at the University of Maine at Orono during 1980-81. On the basis of a two-part final examination, involving a multiple-choice test of algebraic concepts and skills and a free-response test of problem-solving…

Towards classical spectrum generating algebras for f-deformations

NASA Astrophysics Data System (ADS)

Kullock, Ricardo; Latini, Danilo

2016-01-01

In this paper we revise the classical analog of f-oscillators, a generalization of q-oscillators given in Man'ko et al. (1997) [8], in the framework of classical spectrum generating algebras (SGA) introduced in Kuru and Negro (2008) [9]. We write down the deformed Poisson algebra characterizing the entire family of non-linear oscillators and construct its general solution algebraically. The latter, covering the full range of f-deformations, shows an energy dependence both in the amplitude and the frequency of the motion.

A Quantum Groups Primer

NASA Astrophysics Data System (ADS)

Majid, Shahn

2002-05-01

Here is a self-contained introduction to quantum groups as algebraic objects. Based on the author's lecture notes for the Part III pure mathematics course at Cambridge University, the book is suitable as a primary text for graduate courses in quantum groups or supplementary reading for modern courses in advanced algebra. The material assumes knowledge of basic and linear algebra. Some familiarity with semisimple Lie algebras would also be helpful. The volume is a primer for mathematicians but it will also be useful for mathematical physicists.

Identities of Finitely Generated Algebras Over AN Infinite Field

NASA Astrophysics Data System (ADS)

Kemer, A. R.

1991-02-01

It is proved that for each finitely generated associative PI-algebra U over an infinite field F, there is a finite-dimensional F-algebra C such that the ideals of identities of the algebras U and C coincide. This yields a positive solution to the local problem of Specht for algebras over an infinite field: A finitely generated free associative algebra satisfies the maximum condition for T-ideals.

An automatic multigrid method for the solution of sparse linear systems

NASA Technical Reports Server (NTRS)

Shapira, Yair; Israeli, Moshe; Sidi, Avram

1993-01-01

An automatic version of the multigrid method for the solution of linear systems arising from the discretization of elliptic PDE's is presented. This version is based on the structure of the algebraic system solely, and does not use the original partial differential operator. Numerical experiments show that for the Poisson equation the rate of convergence of our method is equal to that of classical multigrid methods. Moreover, the method is robust in the sense that its high rate of convergence is conserved for other classes of problems: non-symmetric, hyperbolic (even with closed characteristics) and problems on non-uniform grids. No double discretization or special treatment of sub-domains (e.g. boundaries) is needed. When supplemented with a vector extrapolation method, high rates of convergence are achieved also for anisotropic and discontinuous problems and also for indefinite Helmholtz equations. A new double discretization strategy is proposed for finite and spectral element schemes and is found better than known strategies.

On squares of representations of compact Lie algebras

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zeier, Robert, E-mail: robert.zeier@ch.tum.de; Zimborás, Zoltán, E-mail: zimboras@gmail.com

We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the summore » of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.« less

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…

Computation of indirect nuclear spin-spin couplings with reduced complexity in pure and hybrid density functional approximations.

PubMed

Luenser, Arne; Kussmann, Jörg; Ochsenfeld, Christian

2016-09-28

We present a (sub)linear-scaling algorithm to determine indirect nuclear spin-spin coupling constants at the Hartree-Fock and Kohn-Sham density functional levels of theory. Employing efficient integral algorithms and sparse algebra routines, an overall (sub)linear scaling behavior can be obtained for systems with a non-vanishing HOMO-LUMO gap. Calculations on systems with over 1000 atoms and 20 000 basis functions illustrate the performance and accuracy of our reference implementation. Specifically, we demonstrate that linear algebra dominates the runtime of conventional algorithms for 10 000 basis functions and above. Attainable speedups of our method exceed 6 × in total runtime and 10 × in the linear algebra steps for the tested systems. Furthermore, a convergence study of spin-spin couplings of an aminopyrazole peptide upon inclusion of the water environment is presented: using the new method it is shown that large solvent spheres are necessary to converge spin-spin coupling values.

Preconditioned conjugate gradient methods for the compressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Venkatakrishnan, V.

1990-01-01

The compressible Navier-Stokes equations are solved for a variety of two-dimensional inviscid and viscous problems by preconditioned conjugate gradient-like algorithms. Roe's flux difference splitting technique is used to discretize the inviscid fluxes. The viscous terms are discretized by using central differences. An algebraic turbulence model is also incorporated. The system of linear equations which arises out of the linearization of a fully implicit scheme is solved iteratively by the well known methods of GMRES (Generalized Minimum Residual technique) and Chebyschev iteration. Incomplete LU factorization and block diagonal factorization are used as preconditioners. The resulting algorithm is competitive with the best current schemes, but has wide applications in parallel computing and unstructured mesh computations.

Instability of the cored barotropic disc: the linear eigenvalue formulation

NASA Astrophysics Data System (ADS)

Polyachenko, E. V.

2018-05-01

Gaseous rotating razor-thin discs are a testing ground for theories of spiral structure that try to explain appearance and diversity of disc galaxy patterns. These patterns are believed to arise spontaneously under the action of gravitational instability, but calculations of its characteristics in the gas are mostly obscured. The paper suggests a new method for finding the spiral patterns based on an expansion of small amplitude perturbations over Lagrange polynomials in small radial elements. The final matrix equation is extracted from the original hydrodynamical equations without the use of an approximate theory and has a form of the linear algebraic eigenvalue problem. The method is applied to a galactic model with the cored exponential density profile.

Algebraic criteria for positive realness relative to the unit circle.

NASA Technical Reports Server (NTRS)

Siljak, D. D.

1973-01-01

A definition is presented of the circle positive realness of real rational functions relative to the unit circle in the complex variable plane. The problem of testing this kind of positive reality is reduced to the algebraic problem of determining the distribution of zeros of a real polynomial with respect to and on the unit circle. Such reformulation of the problem avoids the search for explicit information about imaginary poles of rational functions. The stated algebraic problem is solved by applying the polynomial criteria of Marden (1966) and Jury (1964), and a completely recursive algorithm for circle positive realness is obtained.

Rupture or Continuity: The Arithmetico-Algebraic Thinking as an Alternative in a Modelling Process in a Paper and Pencil and Technology Environment

ERIC Educational Resources Information Center

Hitt, Fernando; Saboya, Mireille; Zavala, Carlos Cortés

2017-01-01

Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which…

Decomposition Theory in the Teaching of Elementary Linear Algebra.

ERIC Educational Resources Information Center

London, R. R.; Rogosinski, H. P.

1990-01-01

Described is a decomposition theory from which the Cayley-Hamilton theorem, the diagonalizability of complex square matrices, and functional calculus can be developed. The theory and its applications are based on elementary polynomial algebra. (KR)

Manifolds, Tensors, and Forms

NASA Astrophysics Data System (ADS)

Renteln, Paul

2013-11-01

Preface; 1. Linear algebra; 2. Multilinear algebra; 3. Differentiation on manifolds; 4. Homotopy and de Rham cohomology; 5. Elementary homology theory; 6. Integration on manifolds; 7. Vector bundles; 8. Geometric manifolds; 9. The degree of a smooth map; Appendixes; References; Index.

Pole-placement Predictive Functional Control for under-damped systems with real numbers algebra.

PubMed

Zabet, K; Rossiter, J A; Haber, R; Abdullah, M

2017-11-01

This paper presents the new algorithm of PP-PFC (Pole-placement Predictive Functional Control) for stable, linear under-damped higher-order processes. It is shown that while conventional PFC aims to get first-order exponential behavior, this is not always straightforward with significant under-damped modes and hence a pole-placement PFC algorithm is proposed which can be tuned more precisely to achieve the desired dynamics, but exploits complex number algebra and linear combinations in order to deliver guarantees of stability and performance. Nevertheless, practical implementation is easier by avoiding complex number algebra and hence a modified formulation of the PP-PFC algorithm is also presented which utilises just real numbers while retaining the key attributes of simple algebra, coding and tuning. The potential advantages are demonstrated with numerical examples and real-time control of a laboratory plant. Copyright © 2017 ISA. All rights reserved.

FOURTH SEMINAR TO THE MEMORY OF D.N. KLYSHKO: Algebraic solution of the synthesis problem for coded sequences

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.

Eigenmodes of Multilayer Slit Structures

NASA Astrophysics Data System (ADS)

Kovalenko, A. N.

2017-12-01

We generalize the high-efficiency numerical-analytical method of calculating the eigenmodes of a microstrip line, which was proposed in [1], to multilayer slit structures. The obtained relationships make it possible to allow for the multilayer nature of the medium on the basis of solving the electrodynamic problem for a two-layer structure. The algebraic models of a single line and coupled slit lines in a multilayer dielectric medium are constructed. The matrix elements of the system of linear algebraic equations, which is used to determine the expansion coefficients of the electric field inside the slits in a Chebyshev basis, are converted to rapidly convergent series. The constructed models allow one to use computer simulation to obtain numerical results with high speed and accuracy, regardless of the number of dielectric layers. The presented results of a numerical study of the method convergence confirm high efficiency of the method.

A least-squares finite element method for 3D incompressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Lin, T. L.; Hou, Lin-Jun; Povinelli, Louis A.

1993-01-01

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.

Using Student Work to Develop Teachers' Knowledge of Algebra

ERIC Educational Resources Information Center

Herbel-Eisenmann, Beth A.; Phillips, Elizabeth Difanis

2005-01-01

This article describes a set of learning activities that use algebraic problems and written student work to help preservice and in-service teachers understand students' algebraic thinking. (Contains 4 figures.)

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…

Using CAS to Solve Classical Mathematics Problems

ERIC Educational Resources Information Center

Burke, Maurice J.; Burroughs, Elizabeth A.

2009-01-01

Historically, calculus has displaced many algebraic methods for solving classical problems. This article illustrates an algebraic method for finding the zeros of polynomial functions that is closely related to Newton's method (devised in 1669, published in 1711), which is encountered in calculus. By exploring this problem, precalculus students…

An Algebraic Approach to Inference in Complex Networked Structures

DTIC Science & Technology

2015-07-09

44], [45],[46] where the shift is the elementary non-trivial filter that generates, under an appropriate notion of shift invariance, all linear ... elementary filter, and its output is a graph signal with the value at vertex n of the graph given approximately by a weighted linear combination of...AFRL-AFOSR-VA-TR-2015-0265 An Algebraic Approach to Inference in Complex Networked Structures Jose Moura CARNEGIE MELLON UNIVERSITY Final Report 07

Labeled trees and the efficient computation of derivations

NASA Technical Reports Server (NTRS)

Grossman, Robert; Larson, Richard G.

1989-01-01

The effective parallel symbolic computation of operators under composition is discussed. Examples include differential operators under composition and vector fields under the Lie bracket. Data structures consisting of formal linear combinations of rooted labeled trees are discussed. A multiplication on rooted labeled trees is defined, thereby making the set of these data structures into an associative algebra. An algebra homomorphism is defined from the original algebra of operators into this algebra of trees. An algebra homomorphism from the algebra of trees into the algebra of differential operators is then described. The cancellation which occurs when noncommuting operators are expressed in terms of commuting ones occurs naturally when the operators are represented using this data structure. This leads to an algorithm which, for operators which are derivations, speeds up the computation exponentially in the degree of the operator. It is shown that the algebra of trees leads naturally to a parallel version of the algorithm.

Inverse solutions for electrical impedance tomography based on conjugate gradients methods

NASA Astrophysics Data System (ADS)

Wang, M.

2002-01-01

A multistep inverse solution for two-dimensional electric field distribution is developed to deal with the nonlinear inverse problem of electric field distribution in relation to its boundary condition and the problem of divergence due to errors introduced by the ill-conditioned sensitivity matrix and the noise produced by electrode modelling and instruments. This solution is based on a normalized linear approximation method where the change in mutual impedance is derived from the sensitivity theorem and a method of error vector decomposition. This paper presents an algebraic solution of the linear equations at each inverse step, using a generalized conjugate gradients method. Limiting the number of iterations in the generalized conjugate gradients method controls the artificial errors introduced by the assumption of linearity and the ill-conditioned sensitivity matrix. The solution of the nonlinear problem is approached using a multistep inversion. This paper also reviews the mathematical and physical definitions of the sensitivity back-projection algorithm based on the sensitivity theorem. Simulations and discussion based on the multistep algorithm, the sensitivity coefficient back-projection method and the Newton-Raphson method are given. Examples of imaging gas-liquid mixing and a human hand in brine are presented.

A Mathematics Software Database Update.

ERIC Educational Resources Information Center

Cunningham, R. S.; Smith, David A.

1987-01-01

Contains an update of an earlier listing of software for mathematics instruction at the college level. Topics are: advanced mathematics, algebra, calculus, differential equations, discrete mathematics, equation solving, general mathematics, geometry, linear and matrix algebra, logic, statistics and probability, and trigonometry. (PK)

College Algebra I.

ERIC Educational Resources Information Center

Benjamin, Carl; And Others

Presented are student performance objectives, a student progress chart, and assignment sheets with objective and diagnostic measures for the stated performance objectives in College Algebra I. Topics covered include: sets; vocabulary; linear equations; inequalities; real numbers; operations; factoring; fractions; formulas; ratio, proportion, and…

Gender Differences in Solution of Algebraic Word Problems Containing Irrelevant Information.

ERIC Educational Resources Information Center

Low, Renae; Over, Ray

1993-01-01

Female tenth graders (n=217) were less likely than male tenth graders (n=219) to identify missing or irrelevant information in algebra problems. Female eleventh graders (n=234) were less likely than male eleventh graders (n=287) to solve problems with irrelevant information. Results indicate sex differences in knowledge of problem structure. (SLD)

Deformation Theory and Physics Model Building

NASA Astrophysics Data System (ADS)

Sternheimer, Daniel

2006-08-01

The mathematical theory of deformations has proved to be a powerful tool in modeling physical reality. We start with a short historical and philosophical review of the context and concentrate this rapid presentation on a few interrelated directions where deformation theory is essential in bringing a new framework - which has then to be developed using adapted tools, some of which come from the deformation aspect. Minkowskian space-time can be deformed into Anti de Sitter, where massless particles become composite (also dynamically): this opens new perspectives in particle physics, at least at the electroweak level, including prediction of new mesons. Nonlinear group representations and covariant field equations, coming from interactions, can be viewed as some deformation of their linear (free) part: recognizing this fact can provide a good framework for treating problems in this area, in particular global solutions. Last but not least, (algebras associated with) classical mechanics (and field theory) on a Poisson phase space can be deformed to (algebras associated with) quantum mechanics (and quantum field theory). That is now a frontier domain in mathematics and theoretical physics called deformation quantization, with multiple ramifications, avatars and connections in both mathematics and physics. These include representation theory, quantum groups (when considering Hopf algebras instead of associative or Lie algebras), noncommutative geometry and manifolds, algebraic geometry, number theory, and of course what is regrouped under the name of M-theory. We shall here look at these from the unifying point of view of deformation theory and refer to a limited number of papers as a starting point for further study.

Student performance and attitudes in a collaborative and flipped linear algebra course

NASA Astrophysics Data System (ADS)

Murphy, Julia; Chang, Jen-Mei; Suaray, Kagba

2016-07-01

Flipped learning is gaining traction in K-12 for enhancing students' problem-solving skills at an early age; however, there is relatively little large-scale research showing its effectiveness in promoting better learning outcomes in higher education, especially in mathematics classes. In this study, we examined the data compiled from both quantitative and qualitative measures such as item scores on a common final and attitude survey results between a flipped and a traditional Introductory Linear Algebra class taught by two individual instructors at a state university in California in Fall 2013. Students in the flipped class were asked to watch short video lectures made by the instructor and complete a short online quiz prior to each class attendance. The class time was completely devoted to problem solving in group settings where students were prompted to communicate their reasoning with proper mathematical terms and structured sentences verbally and in writing. Examination of the quality and depth of student responses from the common final exam showed that students in the flipped class produced more comprehensive and well-explained responses to the questions that required reasoning, creating examples, and more complex use of mathematical objects. Furthermore, students in the flipped class performed superiorly in the overall comprehension of the content with a 21% increase in the median final exam score. Overall, students felt more confident about their ability to learn mathematics independently, showed better retention of materials over time, and enjoyed the flipped experience.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Slattery, Stuart R.

In this study we analyze and extend mesh-free algorithms for three-dimensional data transfer problems in partitioned multiphysics simulations. We first provide a direct comparison between a mesh-based weighted residual method using the common-refinement scheme and two mesh-free algorithms leveraging compactly supported radial basis functions: one using a spline interpolation and one using a moving least square reconstruction. Through the comparison we assess both the conservation and accuracy of the data transfer obtained from each of the methods. We do so for a varying set of geometries with and without curvature and sharp features and for functions with and without smoothnessmore » and with varying gradients. Our results show that the mesh-based and mesh-free algorithms are complementary with cases where each was demonstrated to perform better than the other. We then focus on the mesh-free methods by developing a set of algorithms to parallelize them based on sparse linear algebra techniques. This includes a discussion of fast parallel radius searching in point clouds and restructuring the interpolation algorithms to leverage data structures and linear algebra services designed for large distributed computing environments. The scalability of our new algorithms is demonstrated on a leadership class computing facility using a set of basic scaling studies. Finally, these scaling studies show that for problems with reasonable load balance, our new algorithms for both spline interpolation and moving least square reconstruction demonstrate both strong and weak scalability using more than 100,000 MPI processes with billions of degrees of freedom in the data transfer operation.« less

Redesigning the Quantum Mechanics Curriculum to Incorporate Problem Solving Using a Computer Algebra System

NASA Astrophysics Data System (ADS)

Roussel, Marc R.

1999-10-01

One of the traditional obstacles to learning quantum mechanics is the relatively high level of mathematical proficiency required to solve even routine problems. Modern computer algebra systems are now sufficiently reliable that they can be used as mathematical assistants to alleviate this difficulty. In the quantum mechanics course at the University of Lethbridge, the traditional three lecture hours per week have been replaced by two lecture hours and a one-hour computer-aided problem solving session using a computer algebra system (Maple). While this somewhat reduces the number of topics that can be tackled during the term, students have a better opportunity to familiarize themselves with the underlying theory with this course design. Maple is also available to students during examinations. The use of a computer algebra system expands the class of feasible problems during a time-limited exercise such as a midterm or final examination. A modern computer algebra system is a complex piece of software, so some time needs to be devoted to teaching the students its proper use. However, the advantages to the teaching of quantum mechanics appear to outweigh the disadvantages.

Solving Optimization Problems with Spreadsheets

ERIC Educational Resources Information Center

Beigie, Darin

2017-01-01

Spreadsheets provide a rich setting for first-year algebra students to solve problems. Individual spreadsheet cells play the role of variables, and creating algebraic expressions for a spreadsheet to perform a task allows students to achieve a glimpse of how mathematics is used to program a computer and solve problems. Classic optimization…

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 contemporary…

Mathematical Techniques for Nonlinear System Theory.

DTIC Science & Technology

1981-09-01

This report deals with research results obtained in the following areas: (1) Finite-dimensional linear system theory by algebraic methods--linear...Infinite-dimensional linear systems--realization theory of infinite-dimensional linear systems; (3) Nonlinear system theory --basic properties of

A tensor Banach algebra approach to abstract kinetic equations

NASA Astrophysics Data System (ADS)

Greenberg, W.; van der Mee, C. V. M.

The study deals with a concrete algebraic construction providing the existence theory for abstract kinetic equation boundary-value problems, when the collision operator A is an accretive finite-rank perturbation of the identity operator in a Hilbert space H. An algebraic generalization of the Bochner-Phillips theorem is utilized to study solvability of the abstract boundary-value problem without any regulatory condition. A Banach algebra in which the convolution kernel acts is obtained explicitly, and this result is used to prove a perturbation theorem for bisemigroups, which then plays a vital role in solving the initial equations.

The detection and stabilisation of limit cycle for deterministic finite automata

NASA Astrophysics Data System (ADS)

Han, Xiaoguang; Chen, Zengqiang; Liu, Zhongxin; Zhang, Qing

2018-04-01

In this paper, the topological structure properties of deterministic finite automata (DFA), under the framework of the semi-tensor product of matrices, are investigated. First, the dynamics of DFA are converted into a new algebraic form as a discrete-time linear system by means of Boolean algebra. Using this algebraic description, the approach of calculating the limit cycles of different lengths is given. Second, we present two fundamental concepts, namely, domain of attraction of limit cycle and prereachability set. Based on the prereachability set, an explicit solution of calculating domain of attraction of a limit cycle is completely characterised. Third, we define the globally attractive limit cycle, and then the necessary and sufficient condition for verifying whether all state trajectories of a DFA enter a given limit cycle in a finite number of transitions is given. Fourth, the problem of whether a DFA can be stabilised to a limit cycle by the state feedback controller is discussed. Criteria for limit cycle-stabilisation are established. All state feedback controllers which implement the minimal length trajectories from each state to the limit cycle are obtained by using the proposed algorithm. Finally, an illustrative example is presented to show the theoretical results.

Mathematical biology modules based on modern molecular biology and modern discrete mathematics.

PubMed

Robeva, Raina; Davies, Robin; Hodge, Terrell; Enyedi, Alexander

2010-01-01

We describe an ongoing collaborative curriculum materials development project between Sweet Briar College and Western Michigan University, with support from the National Science Foundation. We present a collection of modules under development that can be used in existing mathematics and biology courses, and we address a critical national need to introduce students to mathematical methods beyond the interface of biology with calculus. Based on ongoing research, and designed to use the project-based-learning approach, the modules highlight applications of modern discrete mathematics and algebraic statistics to pressing problems in molecular biology. For the majority of projects, calculus is not a required prerequisite and, due to the modest amount of mathematical background needed for some of the modules, the materials can be used for an early introduction to mathematical modeling. At the same time, most modules are connected with topics in linear and abstract algebra, algebraic geometry, and probability, and they can be used as meaningful applied introductions into the relevant advanced-level mathematics courses. Open-source software is used to facilitate the relevant computations. As a detailed example, we outline a module that focuses on Boolean models of the lac operon network.

Mathematical Biology Modules Based on Modern Molecular Biology and Modern Discrete Mathematics

PubMed Central

Davies, Robin; Hodge, Terrell; Enyedi, Alexander

2010-01-01

We describe an ongoing collaborative curriculum materials development project between Sweet Briar College and Western Michigan University, with support from the National Science Foundation. We present a collection of modules under development that can be used in existing mathematics and biology courses, and we address a critical national need to introduce students to mathematical methods beyond the interface of biology with calculus. Based on ongoing research, and designed to use the project-based-learning approach, the modules highlight applications of modern discrete mathematics and algebraic statistics to pressing problems in molecular biology. For the majority of projects, calculus is not a required prerequisite and, due to the modest amount of mathematical background needed for some of the modules, the materials can be used for an early introduction to mathematical modeling. At the same time, most modules are connected with topics in linear and abstract algebra, algebraic geometry, and probability, and they can be used as meaningful applied introductions into the relevant advanced-level mathematics courses. Open-source software is used to facilitate the relevant computations. As a detailed example, we outline a module that focuses on Boolean models of the lac operon network. PMID:20810955

Serre duality, Abel's theorem, and Jacobi inversion for supercurves over a thick superpoint

NASA Astrophysics Data System (ADS)

Rothstein, Mitchell J.; Rabin, Jeffrey M.

2015-04-01

The principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1 | q over a finite-dimensional local supercommutative C-algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincaré duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic groups and class fields, combined with an invariance result for the topology on the dual of the space of répartitions. Our Abel map, taking Cartier divisors of degree zero to the dual of the space of sections of the Berezinian sheaf, modulo periods, is defined via Penkov's characterization of the Berezinian sheaf as the cohomology of the de Rham complex of the sheaf D of differential operators. We discuss the Jacobi inversion problem for the Abel map and give an example demonstrating that if n is an integer sufficiently large that the generic divisor of degree n is linearly equivalent to an effective divisor, this need not be the case for all divisors of degree n.

A rigorous approach to investigating common assumptions about disease transmission: Process algebra as an emerging modelling methodology for epidemiology.

PubMed

McCaig, Chris; Begon, Mike; Norman, Rachel; Shankland, Carron

2011-03-01

Changing scale, for example, the ability to move seamlessly from an individual-based model to a population-based model, is an important problem in many fields. In this paper, we introduce process algebra as a novel solution to this problem in the context of models of infectious disease spread. Process algebra allows us to describe a system in terms of the stochastic behaviour of individuals, and is a technique from computer science. We review the use of process algebra in biological systems, and the variety of quantitative and qualitative analysis techniques available. The analysis illustrated here solves the changing scale problem: from the individual behaviour we can rigorously derive equations to describe the mean behaviour of the system at the level of the population. The biological problem investigated is the transmission of infection, and how this relates to individual interactions.

Situating the Debate on "Geometrical Algebra" within the Framework of Premodern Algebra.

PubMed

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.

The Role of Cognitive Processes, Foundational Math Skill, and Calculation Accuracy and Fluency in Word-Problem Solving versus Pre-Algebraic Knowledge

PubMed Central

Fuchs, Lynn S.; Gilbert, Jennifer K.; Powell, Sarah R.; Cirino, Paul T.; Fuchs, Douglas; Hamlett, Carol L.; Seethaler, Pamela M.; Tolar, Tammy D.

2016-01-01

The purpose of this study was to examine child-level pathways in development of pre-algebraic knowledge versus word-problem solving, while evaluating the contribution of calculation accuracy and fluency as mediators of foundational skills/processes. Children (n = 962; mean 7.60 years) were assessed on general cognitive processes and early calculation, word-problem, and number knowledge at start of grade 2; calculation accuracy and calculation fluency at end of grade 2; and pre-algebraic knowledge and word-problem solving at end of grade 4. Important similarities in pathways were identified, but path analysis also indicated that language comprehension is more critical for later word-problem solving than pre-algebraic knowledge. We conclude that pathways in development of these forms of 4th-grade mathematics performance are more alike than different, but demonstrate the need to fine-tune instruction for strands of the mathematics curriculum in ways that address individual students’ foundational mathematics skills or cognitive processes. PMID:27786534

On representations of the filiform Lie superalgebra Lm,n

NASA Astrophysics Data System (ADS)

Wang, Qi; Chen, Hongjia; Liu, Wende

2015-11-01

In this paper, we study the representations for the filiform Lie superalgebras Lm,n, a particular class of nilpotent Lie superalgebras. We determine the minimal dimension of a faithful module over Lm,n using the theory of linear algebra. In addition, using the method of Feingold and Frenkel (1985), we construct some finite and infinite dimensional modules over Lm,n on the Grassmann algebra and the mixed Clifford-Weyl algebra.

Special Year on Numerical Linear Algebra

DTIC Science & Technology

1988-09-01

ORNL) Worley, Pat (ORNL) A special acknowledgement should go to Mary Drake (UT) and Mitzy Denson (ORNL) who carried the burden of making the innumerable...a time step appropriate for the regular cells with no stability restriction. Entrance to Y-12 requires a pass. Contact Mitzy Denson (615) 574-3125 to...requires a pass. Contact Mitzy Denson (615) 574-3125 to obtain one. ’This seminar is part of the Special Year on Numerical Linear Algebra sponsored by the

Generation of Custom DSP Transform IP Cores: Case Study Walsh-Hadamard Transform

DTIC Science & Technology

2002-09-01

mathematics and hardware design What I know: Finite state machine Pipelining Systolic array … What I know: Linear algebra Digital signal processing...state machine Pipelining Systolic array … What I know: Linear algebra Digital signal processing Adaptive filter theory … A math guy A hardware engineer...Synthesis Technology Libary Bit-width (8) HF factor (1,2,3,6) VF factor (1,2,4, ... 32) Xilinx FPGA Place&Route Xilinx FPGA Place&Route Performance

GPU Linear Algebra Libraries and GPGPU Programming for Accelerating MOPAC Semiempirical Quantum Chemistry Calculations.

PubMed

Maia, Julio Daniel Carvalho; Urquiza Carvalho, Gabriel Aires; Mangueira, Carlos Peixoto; Santana, Sidney Ramos; Cabral, Lucidio Anjos Formiga; Rocha, Gerd B

2012-09-11

In this study, we present some modifications in the semiempirical quantum chemistry MOPAC2009 code that accelerate single-point energy calculations (1SCF) of medium-size (up to 2500 atoms) molecular systems using GPU coprocessors and multithreaded shared-memory CPUs. Our modifications consisted of using a combination of highly optimized linear algebra libraries for both CPU (LAPACK and BLAS from Intel MKL) and GPU (MAGMA and CUBLAS) to hasten time-consuming parts of MOPAC such as the pseudodiagonalization, full diagonalization, and density matrix assembling. We have shown that it is possible to obtain large speedups just by using CPU serial linear algebra libraries in the MOPAC code. As a special case, we show a speedup of up to 14 times for a methanol simulation box containing 2400 atoms and 4800 basis functions, with even greater gains in performance when using multithreaded CPUs (2.1 times in relation to the single-threaded CPU code using linear algebra libraries) and GPUs (3.8 times). This degree of acceleration opens new perspectives for modeling larger structures which appear in inorganic chemistry (such as zeolites and MOFs), biochemistry (such as polysaccharides, small proteins, and DNA fragments), and materials science (such as nanotubes and fullerenes). In addition, we believe that this parallel (GPU-GPU) MOPAC code will make it feasible to use semiempirical methods in lengthy molecular simulations using both hybrid QM/MM and QM/QM potentials.

Math Ties: Problem Solving, Logic Teasers, and Math Puzzles All "Tied" To the Math Curriculum. Book B1.

ERIC Educational Resources Information Center

Santi, Terri

This book contains a classroom-tested approach to the teaching of problem solving to all students in Grades 6-8, regardless of ability. Information on problem solving in general is provided, then mathematical problems on logic, exponents, fractions, pre-algebra, algebra, geometry, number theory, set theory, ratio, proportion, percent, probability,…

The Contributions of Working Memory and Executive Functioning to Problem Representation and Solution Generation in Algebraic Word Problems

ERIC Educational Resources Information Center

Lee, Kerry; Ng, Ee Lynn; Ng, Swee Fong

2009-01-01

Solving algebraic word problems involves multiple cognitive phases. The authors used a multitask approach to examine the extent to which working memory and executive functioning are associated with generating problem models and producing solutions. They tested 255 11-year-olds on working memory (Counting Recall, Letter Memory, and Keep Track),…

High-School Students' Approaches to Solving Algebra Problems that Are Posed Symbolically: Results from an Interview Study

ERIC Educational Resources Information Center

Huntley, Mary Ann; Davis, Jon D.

2008-01-01

A cross-curricular structured-probe task-based clinical interview study with 44 pairs of third year high-school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from three problems that were posed in symbolic form. Two problems are…

Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis

NASA Astrophysics Data System (ADS)

Martin, J.; Shore, B. W.; Bergmann, K.

1995-07-01

We extend previous theoretical work on coherent population transfer by stimulated Raman adiabatic passage for states involving nonzero angular momentum. The pump and Stokes fields are either copropagating or counterpropagating with the corresponding linearly polarized electric-field vectors lying in a common plane with the magnetic-field direction. Zeeman splitting lifts the magnetic sublevel degeneracy. We present an algebraic analysis of dressed-state properties to explain the behavior noted in numerical studies. In particular, we discuss conditions which are likely to lead to a failure of complete population transfer. The applied strategy, based on simple methods of linear algebra, will also be successful for other types of discrete multilevel systems, provided the rotating-wave and adiabatic approximation are valid.

Associations of Students' Beliefs with Self-Regulated Problem Solving in College Algebra

ERIC Educational Resources Information Center

Cifarelli, Victor; Goodson-Espy, Tracy; Chae, Jeong-Lim

2010-01-01

This paper reports results from a study of self-regulated problem solving actions of students enrolled in College Algebra (N = 139). The study examined the associations between the expressed mathematical beliefs of students and the students' self-regulated actions in solving mathematics problems. The research questions are: (a) What are some…

Progress on a Taylor weak statement finite element algorithm for high-speed aerodynamic flows

NASA Technical Reports Server (NTRS)

Baker, A. J.; Freels, J. D.

1989-01-01

A new finite element numerical Computational Fluid Dynamics (CFD) algorithm has matured to the point of efficiently solving two-dimensional high speed real-gas compressible flow problems in generalized coordinates on modern vector computer systems. The algorithm employs a Taylor Weak Statement classical Galerkin formulation, a variably implicit Newton iteration, and a tensor matrix product factorization of the linear algebra Jacobian under a generalized coordinate transformation. Allowing for a general two-dimensional conservation law system, the algorithm has been exercised on the Euler and laminar forms of the Navier-Stokes equations. Real-gas fluid properties are admitted, and numerical results verify solution accuracy, efficiency, and stability over a range of test problem parameters.

Fem Formulation for Heat and Mass Transfer in Porous Medium

NASA Astrophysics Data System (ADS)

Azeem; Soudagar, Manzoor Elahi M.; Salman Ahmed, N. J.; Anjum Badruddin, Irfan

2017-08-01

Heat and mass transfer in porous medium can be modelled using three partial differential equations namely, momentum equation, energy equation and mass diffusion. These three equations are coupled to each other by some common terms that turn the whole phenomenon into a complex problem with inter-dependable variables. The current article describes the finite element formulation of heat and mass transfer in porous medium with respect to Cartesian coordinates. The problem under study is formulated into algebraic form of equations by using Galerkin's method with the help of two-node linear triangular element having three nodes. The domain is meshed with smaller sized elements near the wall region and bigger size away from walls.

Initial values for the integration scheme to compute the eigenvalues for propagation in ducts

NASA Technical Reports Server (NTRS)

Eversman, W.

1977-01-01

A scheme for the calculation of eigenvalues in the problem of acoustic propagation in a two-dimensional duct is described. The computation method involves changing the coupled transcendental nonlinear algebraic equations into an initial value problem involving a nonlinear ordinary differential equation. The simplest approach is to use as initial values the hardwall eigenvalues and to integrate away from these values as the admittance varies from zero to its actual value with a linear variation. The approach leads to a powerful root finding routine capable of computing the transverse and axial wave numbers for two-dimensional ducts for any frequency, lining, admittance and Mach number without requiring initial guesses or starting points.

On improvement of the series convergence in the problem of the vibrations of orhotropic rectangular prism

NASA Astrophysics Data System (ADS)

Lyashko, A. D.

2017-11-01

A new analytical presentation of the solution for steady-state oscillations of orthotopic rectangular prism is found. The corresponding infinite system of linear algebraic equations has been deduced by the superposition method. A countable set of precise eigenfrequencies and elementary eigenforms is found. The identities are found which make it possible to improve the convergence of all the infinite series in the solution of the problem. All the infinite series in presentation of solution are analytically summed up. Numerical calculations of stresses in the rectangular orthotropic prism with a uniform along the border and harmonic in time load on two opposite faces have been performed.

The complexity of divisibility.

PubMed

Bausch, Johannes; Cubitt, Toby

2016-09-01

We address two sets of long-standing open questions in linear algebra and probability theory, from a computational complexity perspective: stochastic matrix divisibility, and divisibility and decomposability of probability distributions. We prove that finite divisibility of stochastic matrices is an NP-complete problem, and extend this result to nonnegative matrices, and completely-positive trace-preserving maps, i.e. the quantum analogue of stochastic matrices. We further prove a complexity hierarchy for the divisibility and decomposability of probability distributions, showing that finite distribution divisibility is in P, but decomposability is NP-hard. For the former, we give an explicit polynomial-time algorithm. All results on distributions extend to weak-membership formulations, proving that the complexity of these problems is robust to perturbations.

Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type

NASA Astrophysics Data System (ADS)

Doha, E. H.; Abd-Elhameed, W. M.; Youssri, Y. H.

2013-10-01

In this paper, we present a new second kind Chebyshev (S2KC) operational matrix of derivatives. With the aid of S2KC, an algorithm is described to obtain numerical solutions of a class of linear and nonlinear Lane-Emden type singular initial value problems (IVPs). The idea of obtaining such solutions is essentially based on reducing the differential equation with its initial conditions to a system of algebraic equations. Two illustrative examples concern relevant physical problems (the Lane-Emden equations of the first and second kind) are discussed to demonstrate the validity and applicability of the suggested algorithm. Numerical results obtained are comparing favorably with the analytical known solutions.

Elliptic biquaternion algebra

NASA Astrophysics Data System (ADS)

Özen, Kahraman Esen; Tosun, Murat

2018-01-01

In this study, we define the elliptic biquaternions and construct the algebra of elliptic biquaternions over the elliptic number field. Also we give basic properties of elliptic biquaternions. An elliptic biquaternion is in the form A0 + A1i + A2j + A3k which is a linear combination of {1, i, j, k} where the four components A0, A1, A2 and A3 are elliptic numbers. Here, 1, i, j, k are the quaternion basis of the elliptic biquaternion algebra and satisfy the same multiplication rules which are satisfied in both real quaternion algebra and complex quaternion algebra. In addition, we discuss the terms; conjugate, inner product, semi-norm, modulus and inverse for elliptic biquaternions.

A spatial operator algebra for manipulator modeling and control

NASA Technical Reports Server (NTRS)

Rodriguez, G.; Jain, A.; Kreutz-Delgado, K.

1991-01-01

A recently developed spatial operator algebra for manipulator modeling, control, and trajectory design is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and for control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics.

Grade 11 Students' Interconnected Use of Conceptual Knowledge, Procedural Skills, and Strategic Competence in Algebra: A Mixed Method Study of Error Analysis

ERIC Educational Resources Information Center

Egodawatte, Gunawardena; Stoilescu, Dorian

2015-01-01

The purpose of this mixed-method study was to investigate grade 11 university/college stream mathematics students' difficulties in applying conceptual knowledge, procedural skills, strategic competence, and algebraic thinking in solving routine (instructional) algebraic problems. A standardized algebra test was administered to thirty randomly…

MILP model for integrated balancing and sequencing mixed-model two-sided assembly line with variable launching interval and assignment restrictions

NASA Astrophysics Data System (ADS)

Azmi, N. I. L. Mohd; Ahmad, R.; Zainuddin, Z. M.

2017-09-01

This research explores the Mixed-Model Two-Sided Assembly Line (MMTSAL). There are two interrelated problems in MMTSAL which are line balancing and model sequencing. In previous studies, many researchers considered these problems separately and only few studied them simultaneously for one-sided line. However in this study, these two problems are solved simultaneously to obtain more efficient solution. The Mixed Integer Linear Programming (MILP) model with objectives of minimizing total utility work and idle time is generated by considering variable launching interval and assignment restriction constraint. The problem is analysed using small-size test cases to validate the integrated model. Throughout this paper, numerical experiment was conducted by using General Algebraic Modelling System (GAMS) with the solver CPLEX. Experimental results indicate that integrating the problems of model sequencing and line balancing help to minimise the proposed objectives function.

The preconditioned Gauss-Seidel method faster than the SOR method

NASA Astrophysics Data System (ADS)

Niki, Hiroshi; Kohno, Toshiyuki; Morimoto, Munenori

2008-09-01

In recent years, a number of preconditioners have been applied to linear systems [A.D. Gunawardena, S.K. Jain, L. Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl. 154-156 (1991) 123-143; T. Kohno, H. Kotakemori, H. Niki, M. Usui, Improving modified Gauss-Seidel method for Z-matrices, Linear Algebra Appl. 267 (1997) 113-123; H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner (I+Smax), J. Comput. Appl. Math. 145 (2002) 373-378; H. Kotakemori, H. Niki, N. Okamoto, Accelerated iteration method for Z-matrices, J. Comput. Appl. Math. 75 (1996) 87-97; M. Usui, H. Niki, T.Kohno, Adaptive Gauss-Seidel method for linear systems, Internat. J. Comput. Math. 51(1994)119-125 [10

Closed-form solutions for linear regulator-design of mechanical systems including optimal weighting matrix selection

NASA Technical Reports Server (NTRS)

Hanks, Brantley R.; Skelton, Robert E.

1991-01-01

This paper addresses the restriction of Linear Quadratic Regulator (LQR) solutions to the algebraic Riccati Equation to design spaces which can be implemented as passive structural members and/or dampers. A general closed-form solution to the optimal free-decay control problem is presented which is tailored for structural-mechanical systems. The solution includes, as subsets, special cases such as the Rayleigh Dissipation Function and total energy. Weighting matrix selection is a constrained choice among several parameters to obtain desired physical relationships. The closed-form solution is also applicable to active control design for systems where perfect, collocated actuator-sensor pairs exist. Some examples of simple spring mass systems are shown to illustrate key points.

Application of the Finite Element Method to Rotary Wing Aeroelasticity

NASA Technical Reports Server (NTRS)

Straub, F. K.; Friedmann, P. P.

1982-01-01

A finite element method for the spatial discretization of the dynamic equations of equilibrium governing rotary-wing aeroelastic problems is presented. Formulation of the finite element equations is based on weighted Galerkin residuals. This Galerkin finite element method reduces algebraic manipulative labor significantly, when compared to the application of the global Galerkin method in similar problems. The coupled flap-lag aeroelastic stability boundaries of hingeless helicopter rotor blades in hover are calculated. The linearized dynamic equations are reduced to the standard eigenvalue problem from which the aeroelastic stability boundaries are obtained. The convergence properties of the Galerkin finite element method are studied numerically by refining the discretization process. Results indicate that four or five elements suffice to capture the dynamics of the blade with the same accuracy as the global Galerkin method.

Systems of conservation laws with third-order Hamiltonian structures

NASA Astrophysics Data System (ADS)

Ferapontov, Evgeny V.; Pavlov, Maxim V.; Vitolo, Raffaele F.

2018-06-01

We investigate n-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in P^{n+2} satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space W of dimension n+2, classify n-tuples of skew-symmetric 2-forms A^{α } \\in Λ^2(W) such that φ _{β γ }A^{β }\\wedge A^{γ }=0, for some non-degenerate symmetric φ.

Intrinsic problems of the gravitational baryogenesis

NASA Astrophysics Data System (ADS)

Arbuzova, E. V.; Dolgov, A. D.

2017-06-01

Modification of gravity due to the curvature dependent term in the gravitational baryogenesis scenario is considered. It is shown that this term leads to the fourth order differential equation of motion for the curvature scalar instead of the algebraic one of General Relativity (GR). The fourth order gravitational equations are generically unstable with respect to small perturbations. Non-linear in curvature terms may stabilize the solution but the magnitude of the stabilized curvature scalar would be much larger than that dictated by GR, so the standard cosmology would be strongly distorted.

An efficient computer based wavelets approximation method to solve Fuzzy boundary value differential equations

NASA Astrophysics Data System (ADS)

Alam Khan, Najeeb; Razzaq, Oyoon Abdul

2016-03-01

In the present work a wavelets approximation method is employed to solve fuzzy boundary value differential equations (FBVDEs). Essentially, a truncated Legendre wavelets series together with the Legendre wavelets operational matrix of derivative are utilized to convert FB- VDE into a simple computational problem by reducing it into a system of fuzzy algebraic linear equations. The capability of scheme is investigated on second order FB- VDE considered under generalized H-differentiability. Solutions are represented graphically showing competency and accuracy of this method.

On the existence of mosaic-skeleton approximations for discrete analogues of integral operators

NASA Astrophysics Data System (ADS)

Kashirin, A. A.; Taltykina, M. Yu.

2017-09-01

Exterior three-dimensional Dirichlet problems for the Laplace and Helmholtz equations are considered. By applying methods of potential theory, they are reduced to equivalent Fredholm boundary integral equations of the first kind, for which discrete analogues, i.e., systems of linear algebraic equations (SLAEs) are constructed. The existence of mosaic-skeleton approximations for the matrices of the indicated systems is proved. These approximations make it possible to reduce the computational complexity of an iterative solution of the SLAEs. Numerical experiments estimating the capabilities of the proposed approach are described.

Numerical algebraic geometry: a new perspective on gauge and string theories

NASA Astrophysics Data System (ADS)

Mehta, Dhagash; He, Yang-Hui; Hauensteine, Jonathan D.

2012-07-01

There is a rich interplay between algebraic geometry and string and gauge theories which has been recently aided immensely by advances in computational algebra. However, symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these shortcomings. The so-called `embarrassing parallelizability' allows us to solve many problems and extract physical information which elude symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.

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

NASA Technical Reports Server (NTRS)

Byrnes, C. I.

1980-01-01

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

BLAS (Basic Linear Algebra Subroutines), Linear Algebra Modules and Supercomputers.

DTIC Science & Technology

1984-12-31

the BLAS, Dodson and Lewis C.Remarks on "A. Proposal for a New Set of BLAS", Hanson D. Standard MSC/ NASTRAN Kernels, Komzsik E. Summary of Functions...Fortran names and that character string arguments for the BLAS could provide incr-ased naturalrness in the n3aL,’cs. D ’:andard MSC/ NASTRAN Kernels. Louis...Komnzsik, 8 pages. NASTRAN is a very large structural engineering system marketed by MacNeal- Schwvrdler Corp. (MSC). They are interested in

Computer programs for the solution of systems of linear algebraic equations

NASA Technical Reports Server (NTRS)

Sequi, W. T.

1973-01-01

FORTRAN subprograms for the solution of systems of linear algebraic equations are described, listed, and evaluated in this report. Procedures considered are direct solution, iteration, and matrix inversion. Both incore methods and those which utilize auxiliary data storage devices are considered. Some of the subroutines evaluated require the entire coefficient matrix to be in core, whereas others account for banding or sparceness of the system. General recommendations relative to equation solving are made, and on the basis of tests, specific subprograms are recommended.

[Relations between biomedical variables: mathematical analysis or linear algebra?].

PubMed

Hucher, M; Berlie, J; Brunet, M

1977-01-01

The authors, after a short reminder of one pattern's structure, stress on the possible double approach of relations uniting the variables of this pattern: use of fonctions, what is within the mathematical analysis sphere, use of linear algebra profiting by matricial calculation's development and automatiosation. They precise the respective interests on these methods, their bounds and the imperatives for utilization, according to the kind of variables, of data, and the objective for work, understanding phenomenons or helping towards decision.

Optical linear algebra processors - Architectures and algorithms

NASA Technical Reports Server (NTRS)

Casasent, David

1986-01-01

Attention is given to the component design and optical configuration features of a generic optical linear algebra processor (OLAP) architecture, as well as the large number of OLAP architectures, number representations, algorithms and applications encountered in current literature. Number-representation issues associated with bipolar and complex-valued data representations, high-accuracy (including floating point) performance, and the base or radix to be employed, are discussed, together with case studies on a space-integrating frequency-multiplexed architecture and a hybrid space-integrating and time-integrating multichannel architecture.

On alphabetic presentations of Clifford algebras and their possible applications

NASA Astrophysics Data System (ADS)

Toppan, Francesco; Verbeek, Piet W.

2009-12-01

In this paper, we address the problem of constructing a class of representations of Clifford algebras that can be named "alphabetic (re)presentations." The Clifford algebra generators are expressed as m-letter words written with a three-character or a four-character alphabet. We formulate the problem of the alphabetic presentations, deriving the main properties and some general results. At the end, we briefly discuss the motivations of this work and outline some possible applications.

Invariant algebraic surfaces for a virus dynamics

NASA Astrophysics Data System (ADS)

Valls, Claudia

2015-08-01

In this paper, we provide a complete classification of the invariant algebraic surfaces and of the rational first integrals for a well-known virus system. In the proofs, we use the weight-homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.

Secondary School Mathematics Curriculum Improvement Study Information Bulletin 7.

ERIC Educational Resources Information Center

Secondary School Mathematics Curriculum Improvement Study, New York, NY.

The background, objectives, and design of Secondary School Mathematics Curriculum Improvement Study (SSMCIS) are summarized. Details are given of the content of the text series, "Unified Modern Mathematics," in the areas of algebra, geometry, linear algebra, probability and statistics, analysis (calculus), logic, and computer…

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…

DISTING: A web application for fast algorithmic computation of alternative indistinguishable linear compartmental models.

PubMed

Davidson, Natalie R; Godfrey, Keith R; Alquaddoomi, Faisal; Nola, David; DiStefano, Joseph J

2017-05-01

We describe and illustrate use of DISTING, a novel web application for computing alternative structurally identifiable linear compartmental models that are input-output indistinguishable from a postulated linear compartmental model. Several computer packages are available for analysing the structural identifiability of such models, but DISTING is the first to be made available for assessing indistinguishability. The computational algorithms embedded in DISTING are based on advanced versions of established geometric and algebraic properties of linear compartmental models, embedded in a user-friendly graphic model user interface. Novel computational tools greatly speed up the overall procedure. These include algorithms for Jacobian matrix reduction, submatrix rank reduction, and parallelization of candidate rank computations in symbolic matrix analysis. The application of DISTING to three postulated models with respectively two, three and four compartments is given. The 2-compartment example is used to illustrate the indistinguishability problem; the original (unidentifiable) model is found to have two structurally identifiable models that are indistinguishable from it. The 3-compartment example has three structurally identifiable indistinguishable models. It is found from DISTING that the four-compartment example has five structurally identifiable models indistinguishable from the original postulated model. This example shows that care is needed when dealing with models that have two or more compartments which are neither perturbed nor observed, because the numbering of these compartments may be arbitrary. DISTING is universally and freely available via the Internet. It is easy to use and circumvents tedious and complicated algebraic analysis previously done by hand. Copyright © 2017 Elsevier B.V. All rights reserved.

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…

An Algebraic Method for Exploring Quantum Monodromy and Quantum Phase Transitions in Non-Rigid Molecules

NASA Astrophysics Data System (ADS)

Larese, D.; Iachello, F.

2011-06-01

A simple algebraic Hamiltonian has been used to explore the vibrational and rotational spectra of the skeletal bending modes of HCNO, BrCNO, NCNCS, and other ``floppy`` (quasi-linear or quasi-bent) molecules. These molecules have large-amplitude, low-energy bending modes and champagne-bottle potential surfaces, making them good candidates for observing quantum phase transitions (QPT). We describe the geometric phase transitions from bent to linear in these and other non-rigid molecules, quantitatively analysing the spectroscopy signatures of ground state QPT, excited state QPT, and quantum monodromy.The algebraic framework is ideal for this work because of its small calculational effort yet robust results. Although these methods have historically found success with tri- and four-atomic molecules, we now address five-atomic and simple branched molecules such as CH_3NCO and GeH_3NCO. Extraction of potential functions is completed for several molecules, resulting in predictions of barriers to linearity and equilibrium bond angles.

Asymptotic theory of neutral stability of the Couette flow of a vibrationally excited gas

NASA Astrophysics Data System (ADS)

Grigor'ev, Yu. N.; Ershov, I. V.

2017-01-01

An asymptotic theory of the neutral stability curve for a supersonic plane Couette flow of a vibrationally excited gas is developed. The initial mathematical model consists of equations of two-temperature viscous gas dynamics, which are used to derive a spectral problem for a linear system of eighth-order ordinary differential equations within the framework of the classical linear stability theory. Unified transformations of the system for all shear flows are performed in accordance with the classical Lin scheme. The problem is reduced to an algebraic secular equation with separation into the "inviscid" and "viscous" parts, which is solved numerically. It is shown that the thus-calculated neutral stability curves agree well with the previously obtained results of the direct numerical solution of the original spectral problem. In particular, the critical Reynolds number increases with excitation enhancement, and the neutral stability curve is shifted toward the domain of higher wave numbers. This is also confirmed by means of solving an asymptotic equation for the critical Reynolds number at the Mach number M ≤ 4.

Now & Then: Roger Whitmore, Police Officer.

ERIC Educational Resources Information Center

Barnes, Sue; Michalowicz, Karen Dee

1995-01-01

Discusses police officers' use of mathematics when reconstructing an accident scene; and the history of algebra, including al-Khwarizmi's works on the theory of equations, the Rhind Papyrus, a Chinese and an Indian manuscript on systems of linear and quadratic equations, and Diophantus'"syncopated algebra." (10 references) (EK)

Smooth function approximation using neural networks.

PubMed

Ferrari, Silvia; Stengel, Robert F

2005-01-01

An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the approximation of smooth batch data containing the function's input, output, and possibly, gradient information. The training set is associated to the network adjustable parameters by nonlinear weight equations. The cascade structure of these equations reveals that they can be treated as sets of linear systems. Hence, the training process and the network approximation properties can be investigated via linear algebra. Four algorithms are developed to achieve exact or approximate matching of input-output and/or gradient-based training sets. Their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques.

Structure of Lie point and variational symmetry algebras for a class of odes

NASA Astrophysics Data System (ADS)

Ndogmo, J. C.

2018-04-01

It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation y(n)=0. For arbitrary systems of ordinary differential equations of order n ≥ 3 reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras. As a corollary, we obtain the maximal dimension of the Lie point symmetry algebra for any system of linear or nonlinear ordinary differential equations.

Brain activity associated with translation from a visual to a symbolic representation in algebra and geometry.

PubMed

Leikin, Mark; Waisman, Ilana; Shaul, Shelley; Leikin, Roza

2014-03-01

This paper presents a small part of a larger interdisciplinary study that investigates brain activity (using event related potential methodology) of male adolescents when solving mathematical problems of different types. The study design links mathematics education research with neurocognitive studies. In this paper we performed a comparative analysis of brain activity associated with the translation from visual to symbolic representations of mathematical objects in algebra and geometry. Algebraic tasks require translation from graphical to symbolic representation of a function, whereas tasks in geometry require translation from a drawing of a geometric figure to a symbolic representation of its property. The findings demonstrate that electrical activity associated with the performance of geometrical tasks is stronger than that associated with solving algebraic tasks. Additionally, we found different scalp topography of the brain activity associated with algebraic and geometric tasks. Based on these results, we argue that problem solving in algebra and geometry is associated with different patterns of brain activity.

Genetic algorithms in teaching artificial intelligence (automated generation of specific algebras)

NASA Astrophysics Data System (ADS)

Habiballa, Hashim; Jendryscik, Radek

2017-11-01

The problem of teaching essential Artificial Intelligence (AI) methods is an important task for an educator in the branch of soft-computing. The key focus is often given to proper understanding of the principle of AI methods in two essential points - why we use soft-computing methods at all and how we apply these methods to generate reasonable results in sensible time. We present one interesting problem solved in the non-educational research concerning automated generation of specific algebras in the huge search space. We emphasize above mentioned points as an educational case study of an interesting problem in automated generation of specific algebras.

Effects of Argumentation on Group Micro-Creativity: Statistical Discourse Analyses of Algebra Students' Collaborative Problem Solving

ERIC Educational Resources Information Center

Chiu, Ming Ming

2008-01-01

The micro-time context of group processes (such as argumentation) can affect a group's micro-creativity (new ideas). Eighty high school students worked in groups of four on an algebra problem. Groups with higher mathematics grades showed greater micro-creativity, and both were linked to better problem solving outcomes. Dynamic multilevel analyses…

Fostering Analogical Transfer: The Multiple Components Approach to Algebra Word Problem Solving in a Chemistry Context

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…

Characterizing the Nature of Students' Feature Noticing-and-Using with Respect to Mathematical Symbols across Different Levels of Algebra Exposure

ERIC Educational Resources Information Center

Sullivan, Patrick

2013-01-01

The purpose of this study is to examine the nature of what students notice about symbols and use as they solve unfamiliar algebra problems based on familiar algebra concepts and involving symbolic inscriptions. The researcher conducted a study of students at three levels of algebra exposure: (a) students enrolled in a high school pre-calculus…

Change in Peer Ability as a Mediator and Moderator of the Effect of the Algebra-For-All Policy on Ninth Graders' Math Outcomes

ERIC Educational Resources Information Center

Hong, Guanglei; Nomi, Takako

2011-01-01

A recent report by the Mathematics Advisory Panel referred to algebra as a "gateway" to later achievement (National Mathematics Advisory Panel, 2008). To address the problem of low academic performance in algebra, an increasing number of states and districts have started to implement a policy of requiring algebra for all students in…

On the scalability of the Albany/FELIX first-order Stokes approximation ice sheet solver for large-scale simulations of the Greenland and Antarctic ice sheets

DOE PAGES

Tezaur, Irina K.; Tuminaro, Raymond S.; Perego, Mauro; ...

2015-01-01

We examine the scalability of the recently developed Albany/FELIX finite-element based code for the first-order Stokes momentum balance equations for ice flow. We focus our analysis on the performance of two possible preconditioners for the iterative solution of the sparse linear systems that arise from the discretization of the governing equations: (1) a preconditioner based on the incomplete LU (ILU) factorization, and (2) a recently-developed algebraic multigrid (AMG) preconditioner, constructed using the idea of semi-coarsening. A strong scalability study on a realistic, high resolution Greenland ice sheet problem reveals that, for a given number of processor cores, the AMG preconditionermore » results in faster linear solve times but the ILU preconditioner exhibits better scalability. In addition, a weak scalability study is performed on a realistic, moderate resolution Antarctic ice sheet problem, a substantial fraction of which contains floating ice shelves, making it fundamentally different from the Greenland ice sheet problem. We show that as the problem size increases, the performance of the ILU preconditioner deteriorates whereas the AMG preconditioner maintains scalability. This is because the linear systems are extremely ill-conditioned in the presence of floating ice shelves, and the ill-conditioning has a greater negative effect on the ILU preconditioner than on the AMG preconditioner.« less

Processes and Reasoning in Representations of Linear Functions

ERIC Educational Resources Information Center

Adu-Gyamfi, Kwaku; Bossé, Michael J.

2014-01-01

This study examined student actions, interpretations, and language in respect to questions raised regarding tabular, graphical, and algebraic representations in the context of functions. The purpose was to investigate students' interpretations and specific ways of working within table, graph, and the algebraic on notions fundamental to a…

Teaching Algebraic Equations to Middle School Students with Intellectual Disabilities

ERIC Educational Resources Information Center

Baker, Joshua N.; Rivera, Christopher J.; Morgan, Joseph John; Reese, Noelle

2015-01-01

The purpose of this study was to replicate similar instructional techniques of Jimenez, Browder, and Courtade (2008) using a single-subject multiple-probe across participants design to investigate the effects of task analytic instruction coupled with semi-concrete representations to teach linear algebraic equations to middle school students with…

Generalized Heisenberg algebra and (non linear) pseudo-bosons

NASA Astrophysics Data System (ADS)

Bagarello, F.; Curado, E. M. F.; Gazeau, J. P.

2018-04-01

We propose a deformed version of the generalized Heisenberg algebra by using techniques borrowed from the theory of pseudo-bosons. In particular, this analysis is relevant when non self-adjoint Hamiltonians are needed to describe a given physical system. We also discuss relations with nonlinear pseudo-bosons. Several examples are discussed.

The Jukes-Cantor Model of Molecular Evolution

ERIC Educational Resources Information Center

Erickson, Keith

2010-01-01

The material in this module introduces students to some of the mathematical tools used to examine molecular evolution. This topic is standard fare in many mathematical biology or bioinformatics classes, but could also be suitable for classes in linear algebra or probability. While coursework in matrix algebra, Markov processes, Monte Carlo…

Racing against Time: Using Technology To Explore Distance, Rate, and Time.

ERIC Educational Resources Information Center

Essex, N. Kathryn; Lambdin, Diana V.; McGraw, Rebecca H.

2002-01-01

Investigates ways to analyze change in various contexts. Focuses on computer technology providing contexts for children's investigations of patterns of change and helping to develop foundational ideas of algebra and calculus. Discusses relationships between patterns of change, fundamental algebraic notions as linear and nonlinear functions, and…

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,…

Prospective Mathematics Teachers' Sense Making of Polynomial Multiplication and Factorization Modeled with Algebra Tiles

ERIC Educational Resources Information Center

Caglayan, Günhan

2013-01-01

This study is about prospective secondary mathematics teachers' understanding and sense making of representational quantities generated by algebra tiles, the quantitative units (linear vs. areal) inherent in the nature of these quantities, and the quantitative addition and multiplication operations--referent preserving versus referent…

Implementing the Curriculum and Evaluation Standards: First-Year Algebra.

ERIC Educational Resources Information Center

Kysh, Judith

1991-01-01

Described is an alternative first year algebra program developed to bridge the gap between the NCTM's Curriculum and Evaluation Standards and institutional demands of schools. Increased attention is given to graphing as a context for algebra, calculator use, solving "memorable problems," and incorporating geometry concepts, while…

AN ADA LINEAR ALGEBRA PACKAGE MODELED AFTER HAL/S

NASA Technical Reports Server (NTRS)

Klumpp, A. R.

1994-01-01

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

Efficient solvers for coupled models in respiratory mechanics.

PubMed

Verdugo, Francesc; Roth, Christian J; Yoshihara, Lena; Wall, Wolfgang A

2017-02-01

We present efficient preconditioners for one of the most physiologically relevant pulmonary models currently available. Our underlying motivation is to enable the efficient simulation of such a lung model on high-performance computing platforms in order to assess mechanical ventilation strategies and contributing to design more protective patient-specific ventilation treatments. The system of linear equations to be solved using the proposed preconditioners is essentially the monolithic system arising in fluid-structure interaction (FSI) extended by additional algebraic constraints. The introduction of these constraints leads to a saddle point problem that cannot be solved with usual FSI preconditioners available in the literature. The key ingredient in this work is to use the idea of the semi-implicit method for pressure-linked equations (SIMPLE) for getting rid of the saddle point structure, resulting in a standard FSI problem that can be treated with available techniques. The numerical examples show that the resulting preconditioners approach the optimal performance of multigrid methods, even though the lung model is a complex multiphysics problem. Moreover, the preconditioners are robust enough to deal with physiologically relevant simulations involving complex real-world patient-specific lung geometries. The same approach is applicable to other challenging biomedical applications where coupling between flow and tissue deformations is modeled with additional algebraic constraints. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.

Super-Laplacians and their symmetries

NASA Astrophysics Data System (ADS)

Howe, P. S.; Lindström, U.

2017-05-01

A super-Laplacian is a set of differential operators in superspace whose highestdimensional component is given by the spacetime Laplacian. Symmetries of super-Laplacians are given by linear differential operators of arbitrary finite degree and are determined by superconformal Killing tensors. We investigate these in flat superspaces. The differential operators determining the symmetries give rise to algebras which can be identified in many cases with the tensor algebras of the relevant superconformal Lie algebras modulo certain ideals. They have applications to Higher Spin theories.

Anti-commutative Gröbner-Shirshov basis of a free Lie algebra

NASA Astrophysics Data System (ADS)

Bokut, L. A.; Chen, Yuqun; Li, Yu

2009-03-01

One of the natural ways to prove that the Hall words (Philip Hall, 1933) consist of a basis of a free Lie algebra is a direct construction: to start with a linear space spanned by Hall words, to define the Lie product of Hall words, and then to check that the product yields the Lie identities (Marshall Hall, 1950). Here we suggest another way using the Composition-Diamond lemma for free anti-commutative (non-associative) algebras (A.I. Shirshov, 1962).

Cryptographic Properties of Monotone Boolean Functions

DTIC Science & Technology

2016-01-01

Algebraic attacks on stream ciphers with linear feedback, in: Advances in Cryptology (Eurocrypt 2003), Lecture Notes in Comput. Sci. 2656, Springer, Berlin...spectrum, algebraic immu- nity MSC 2010: 06E30, 94C10, 94A60, 11T71, 05E99 || Communicated by: Carlo Blundo 1 Introduction Let F 2 be the prime eld of...7]. For the reader’s convenience, we recall some basic notions below. Any f ∈ Bn can be expressed in algebraic normal form (ANF) as f(x 1 , x 2

Effect of Worked Examples and Cognitive Tutor Training on Constructing Equations

ERIC Educational Resources Information Center

Reed, Stephen K.; Corbett, Albert; Hoffman, Bob; Wagner, Angela; MacLaren, Ben

2013-01-01

Algebra students studied either static-table, static-graphics, or interactive-graphics instructional worked examples that alternated with Algebra Cognitive Tutor practice problems. A control group did not study worked examples but solved both the instructional and practice problems on the Cognitive Tutor (CT). Students in the control group…

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…

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…

DOE Office of Scientific and Technical Information (OSTI.GOV)

Heroux, Michael Allen; Marker, Bryan

This report summarizes the progress made as part of a one year lab-directed research and development (LDRD) project to fund the research efforts of Bryan Marker at the University of Texas at Austin. The goal of the project was to develop new techniques for automatically tuning the performance of dense linear algebra kernels. These kernels often represent the majority of computational time in an application. The primary outcome from this work is a demonstration of the value of model driven engineering as an approach to accurately predict and study performance trade-offs for dense linear algebra computations.

Basic linear algebra subprograms for FORTRAN usage

NASA Technical Reports Server (NTRS)

Lawson, C. L.; Hanson, R. J.; Kincaid, D. R.; Krogh, F. T.

1977-01-01

A package of 38 low level subprograms for many of the basic operations of numerical linear algebra is presented. The package is intended to be used with FORTRAN. The operations in the package are dot products, elementary vector operations, Givens transformations, vector copy and swap, vector norms, vector scaling, and the indices of components of largest magnitude. The subprograms and a test driver are available in portable FORTRAN. Versions of the subprograms are also provided in assembly language for the IBM 360/67, the CDC 6600 and CDC 7600, and the Univac 1108.

Convergence Estimates for Multidisciplinary Analysis and Optimization

NASA Technical Reports Server (NTRS)

Arian, Eyal

1997-01-01

A quantitative analysis of coupling between systems of equations is introduced. This analysis is then applied to problems in multidisciplinary analysis, sensitivity, and optimization. For the sensitivity and optimization problems both multidisciplinary and single discipline feasibility schemes are considered. In all these cases a "convergence factor" is estimated in terms of the Jacobians and Hessians of the system, thus it can also be approximated by existing disciplinary analysis and optimization codes. The convergence factor is identified with the measure for the "coupling" between the disciplines in the system. Applications to algorithm development are discussed. Demonstration of the convergence estimates and numerical results are given for a system composed of two non-linear algebraic equations, and for a system composed of two PDEs modeling aeroelasticity.

Asymptotically Exact Solution of the Problem of Harmonic Vibrations of an Elastic Parallelepiped

NASA Astrophysics Data System (ADS)

Papkov, S. O.

2017-11-01

An asymptotically exact solution of the classical problem of elasticity about the steadystate forced vibrations of an elastic rectangular parallelepiped is constructed. The general solution of the vibration equations is constructed in the form of double Fourier series with undetermined coefficients, and an infinite system of linear algebraic equations is obtained for determining these coefficients. An analysis of the infinite system permits determining the asymptotics of the unknowns which are used to convolve the double series in both equations of the infinite systems and the displacement and stress components. The efficiency of this approach is illustrated by numerical examples and comparison with known solutions. The spectrum of the parallelepiped symmetric vibrations is studied for various ratios of its sides.

Algebraic Reasoning in Solving Mathematical Problem Based on Learning Style

NASA Astrophysics Data System (ADS)

Indraswari, N. F.; Budayasa, I. K.; Ekawati, R.

2018-01-01

This study aimed to describe algebraic reasoning of secondary school’s pupils with different learning styles in solving mathematical problem. This study begins by giving the questionnaire to find out the learning styles and followed by mathematical ability test to get three subjects of 8th-grade whereas the learning styles of each pupil is visual, auditory, kinesthetic and had similar mathematical abilities. Then it continued with given algebraic problems and interviews. The data is validated using triangulation of time. The result showed that in the pattern of seeking indicator, subjects identified the things that were known and asked based on them observations. The visual and kinesthetic learners represented the known information in a chart, whereas the auditory learner in a table. In addition, they found the elements which makes the pattern and made a relationship between two quantities. In the pattern recognition indicator, they created conjectures on the relationship between two quantities and proved it. In the generalization indicator, they were determining the general rule of pattern found on each element of pattern using algebraic symbols and created a mathematical model. Visual and kinesthetic learners determined the general rule of equations which was used to solve problems using algebraic symbols, but auditory learner in a sentence.

Multiple solution of linear algebraic systems by an iterative method with recomputed preconditioner in the analysis of microstrip structures

NASA Astrophysics Data System (ADS)

Ahunov, Roman R.; Kuksenko, Sergey P.; Gazizov, Talgat R.

2016-06-01

A multiple solution of linear algebraic systems with dense matrix by iterative methods is considered. To accelerate the process, the recomputing of the preconditioning matrix is used. A priory condition of the recomputing based on change of the arithmetic mean of the current solution time during the multiple solution is proposed. To confirm the effectiveness of the proposed approach, the numerical experiments using iterative methods BiCGStab and CGS for four different sets of matrices on two examples of microstrip structures are carried out. For solution of 100 linear systems the acceleration up to 1.6 times, compared to the approach without recomputing, is obtained.

Tracking control of concentration profiles in a fed-batch bioreactor using a linear algebra methodology.

PubMed

Rómoli, Santiago; Serrano, Mario Emanuel; Ortiz, Oscar Alberto; Vega, Jorge Rubén; Eduardo Scaglia, Gustavo Juan

2015-07-01

Based on a linear algebra approach, this paper aims at developing a novel control law able to track reference profiles that were previously-determined in the literature. A main advantage of the proposed strategy is that the control actions are obtained by solving a system of linear equations. The optimal controller parameters are selected through Monte Carlo Randomized Algorithm in order to minimize a proposed cost index. The controller performance is evaluated through several tests, and compared with other controller reported in the literature. Finally, a Monte Carlo Randomized Algorithm is conducted to assess the performance of the proposed controller. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Numerical evaluation of mobile robot navigation in static indoor environment via EGAOR Iteration

NASA Astrophysics Data System (ADS)

Dahalan, A. A.; Saudi, A.; Sulaiman, J.; Din, W. R. W.

2017-09-01

One of the key issues in mobile robot navigation is the ability for the robot to move from an arbitrary start location to a specified goal location without colliding with any obstacles while traveling, also known as mobile robot path planning problem. In this paper, however, we examined the performance of a robust searching algorithm that relies on the use of harmonic potentials of the environment to generate smooth and safe path for mobile robot navigation in a static known indoor environment. The harmonic potentials will be discretized by using Laplacian’s operator to form a system of algebraic approximation equations. This algebraic linear system will be computed via 4-Point Explicit Group Accelerated Over-Relaxation (4-EGAOR) iterative method for rapid computation. The performance of the proposed algorithm will then be compared and analyzed against the existing algorithms in terms of number of iterations and execution time. The result shows that the proposed algorithm performed better than the existing methods.

Modeling Stochastic Complexity in Complex Adaptive Systems: Non-Kolmogorov Probability and the Process Algebra Approach.

PubMed

Sulis, William H

2017-10-01

Walter Freeman III pioneered the application of nonlinear dynamical systems theories and methodologies in his work on mesoscopic brain dynamics.Sadly, mainstream psychology and psychiatry still cling to linear correlation based data analysis techniques, which threaten to subvert the process of experimentation and theory building. In order to progress, it is necessary to develop tools capable of managing the stochastic complexity of complex biopsychosocial systems, which includes multilevel feedback relationships, nonlinear interactions, chaotic dynamics and adaptability. In addition, however, these systems exhibit intrinsic randomness, non-Gaussian probability distributions, non-stationarity, contextuality, and non-Kolmogorov probabilities, as well as the absence of mean and/or variance and conditional probabilities. These properties and their implications for statistical analysis are discussed. An alternative approach, the Process Algebra approach, is described. It is a generative model, capable of generating non-Kolmogorov probabilities. It has proven useful in addressing fundamental problems in quantum mechanics and in the modeling of developing psychosocial systems.

Statistical quasi-particle theory for open quantum systems

NASA Astrophysics Data System (ADS)

Zhang, Hou-Dao; Xu, Rui-Xue; Zheng, Xiao; Yan, YiJing

2018-04-01

This paper presents a comprehensive account on the recently developed dissipaton-equation-of-motion (DEOM) theory. This is a statistical quasi-particle theory for quantum dissipative dynamics. It accurately describes the influence of bulk environments, with a few number of quasi-particles, the dissipatons. The novel dissipaton algebra is then followed, which readily bridges the Schrödinger equation to the DEOM theory. As a fundamental theory of quantum mechanics in open systems, DEOM characterizes both the stationary and dynamic properties of system-and-bath interferences. It treats not only the quantum dissipative systems of primary interest, but also the hybrid environment dynamics that could be experimentally measurable. Examples are the linear or nonlinear Fano interferences and the Herzberg-Teller vibronic couplings in optical spectroscopies. This review covers the DEOM construction, the underlying dissipaton algebra and theorems, the physical meanings of dynamical variables, the possible identifications of dissipatons, and some recent advancements in efficient DEOM evaluations on various problems. The relations of the present theory to other nonperturbative methods are also critically presented.

A differential operator realisation approach for constructing Casimir operators of non-semisimple Lie algebras

NASA Astrophysics Data System (ADS)

Alshammari, Fahad; Isaac, Phillip S.; Marquette, Ian

2018-02-01

We introduce a search algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. To demonstrate the algorithm, we look at two classes of examples: (1) the model filiform Lie algebras and (2) the Schrödinger Lie algebras. We find that an abstract form of dimensional analysis assists us in our algorithm, and greatly reduces the complexity of the problem.

Effects of Modified Schema-Based Instruction on Real-World Algebra Problem Solving of Students with Autism Spectrum Disorder and Moderate Intellectual Disability

ERIC Educational Resources Information Center

Root, Jenny Rose

2016-01-01

The current study evaluated the effects of modified schema-based instruction (SBI) on the algebra problem solving skills of three middle school students with autism spectrum disorder and moderate intellectual disability (ASD/ID). Participants learned to solve two types of group word problems: missing-whole and missing-part. The themes of the word…

The Effect of Using the TI-92 on Basic College Algebra Students' Ability To Solve Word Problems.

ERIC Educational Resources Information Center

Runde, Dennis C.

As part of an effort to improve community college algebra students' ability to solve word problems, a study was undertaken at Florida's Manatee Community College to determine the effects of using heuristic instruction (i.e., providing general rules for solving different types of math problems) in combination with the TI-92 calculator. The TI-92…

ORACLS- OPTIMAL REGULATOR ALGORITHMS FOR THE CONTROL OF LINEAR SYSTEMS (CDC VERSION)

NASA Technical Reports Server (NTRS)

Armstrong, E. S.

1994-01-01

This control theory design package, called Optimal Regulator Algorithms for the Control of Linear Systems (ORACLS), was developed to aid in the design of controllers and optimal filters for systems which can be modeled by linear, time-invariant differential and difference equations. Optimal linear quadratic regulator theory, currently referred to as the Linear-Quadratic-Gaussian (LQG) problem, has become the most widely accepted method of determining optimal control policy. Within this theory, the infinite duration time-invariant problems, which lead to constant gain feedback control laws and constant Kalman-Bucy filter gains for reconstruction of the system state, exhibit high tractability and potential ease of implementation. A variety of new and efficient methods in the field of numerical linear algebra have been combined into the ORACLS program, which provides for the solution to time-invariant continuous or discrete LQG problems. The ORACLS package is particularly attractive to the control system designer because it provides a rigorous tool for dealing with multi-input and multi-output dynamic systems in both continuous and discrete form. The ORACLS programming system is a collection of subroutines which can be used to formulate, manipulate, and solve various LQG design problems. The ORACLS program is constructed in a manner which permits the user to maintain considerable flexibility at each operational state. This flexibility is accomplished by providing primary operations, analysis of linear time-invariant systems, and control synthesis based on LQG methodology. The input-output routines handle the reading and writing of numerical matrices, printing heading information, and accumulating output information. The basic vector-matrix operations include addition, subtraction, multiplication, equation, norm construction, tracing, transposition, scaling, juxtaposition, and construction of null and identity matrices. The analysis routines provide for the following computations: the eigenvalues and eigenvectors of real matrices; the relative stability of a given matrix; matrix factorization; the solution of linear constant coefficient vector-matrix algebraic equations; the controllability properties of a linear time-invariant system; the steady-state covariance matrix of an open-loop stable system forced by white noise; and the transient response of continuous linear time-invariant systems. The control law design routines of ORACLS implement some of the more common techniques of time-invariant LQG methodology. For the finite-duration optimal linear regulator problem with noise-free measurements, continuous dynamics, and integral performance index, a routine is provided which implements the negative exponential method for finding both the transient and steady-state solutions to the matrix Riccati equation. For the discrete version of this problem, the method of backwards differencing is applied to find the solutions to the discrete Riccati equation. A routine is also included to solve the steady-state Riccati equation by the Newton algorithms described by Klein, for continuous problems, and by Hewer, for discrete problems. Another routine calculates the prefilter gain to eliminate control state cross-product terms in the quadratic performance index and the weighting matrices for the sampled data optimal linear regulator problem. For cases with measurement noise, duality theory and optimal regulator algorithms are used to calculate solutions to the continuous and discrete Kalman-Bucy filter problems. Finally, routines are included to implement the continuous and discrete forms of the explicit (model-in-the-system) and implicit (model-in-the-performance-index) model following theory. These routines generate linear control laws which cause the output of a dynamic time-invariant system to track the output of a prescribed model. In order to apply ORACLS, the user must write an executive (driver) program which inputs the problem coefficients, formulates and selects the routines to be used to solve the problem, and specifies the desired output. There are three versions of ORACLS source code available for implementation: CDC, IBM, and DEC. The CDC version has been implemented on a CDC 6000 series computer with a central memory of approximately 13K (octal) of 60 bit words. The CDC version is written in FORTRAN IV, was developed in 1978, and last updated in 1989. The IBM version has been implemented on an IBM 370 series computer with a central memory requirement of approximately 300K of 8 bit bytes. The IBM version is written in FORTRAN IV and was generated in 1981. The DEC version has been implemented on a VAX series computer operating under VMS. The VAX version is written in FORTRAN 77 and was generated in 1986.

ORACLS- OPTIMAL REGULATOR ALGORITHMS FOR THE CONTROL OF LINEAR SYSTEMS (DEC VAX VERSION)

NASA Technical Reports Server (NTRS)

Frisch, H.

1994-01-01

This control theory design package, called Optimal Regulator Algorithms for the Control of Linear Systems (ORACLS), was developed to aid in the design of controllers and optimal filters for systems which can be modeled by linear, time-invariant differential and difference equations. Optimal linear quadratic regulator theory, currently referred to as the Linear-Quadratic-Gaussian (LQG) problem, has become the most widely accepted method of determining optimal control policy. Within this theory, the infinite duration time-invariant problems, which lead to constant gain feedback control laws and constant Kalman-Bucy filter gains for reconstruction of the system state, exhibit high tractability and potential ease of implementation. A variety of new and efficient methods in the field of numerical linear algebra have been combined into the ORACLS program, which provides for the solution to time-invariant continuous or discrete LQG problems. The ORACLS package is particularly attractive to the control system designer because it provides a rigorous tool for dealing with multi-input and multi-output dynamic systems in both continuous and discrete form. The ORACLS programming system is a collection of subroutines which can be used to formulate, manipulate, and solve various LQG design problems. The ORACLS program is constructed in a manner which permits the user to maintain considerable flexibility at each operational state. This flexibility is accomplished by providing primary operations, analysis of linear time-invariant systems, and control synthesis based on LQG methodology. The input-output routines handle the reading and writing of numerical matrices, printing heading information, and accumulating output information. The basic vector-matrix operations include addition, subtraction, multiplication, equation, norm construction, tracing, transposition, scaling, juxtaposition, and construction of null and identity matrices. The analysis routines provide for the following computations: the eigenvalues and eigenvectors of real matrices; the relative stability of a given matrix; matrix factorization; the solution of linear constant coefficient vector-matrix algebraic equations; the controllability properties of a linear time-invariant system; the steady-state covariance matrix of an open-loop stable system forced by white noise; and the transient response of continuous linear time-invariant systems. The control law design routines of ORACLS implement some of the more common techniques of time-invariant LQG methodology. For the finite-duration optimal linear regulator problem with noise-free measurements, continuous dynamics, and integral performance index, a routine is provided which implements the negative exponential method for finding both the transient and steady-state solutions to the matrix Riccati equation. For the discrete version of this problem, the method of backwards differencing is applied to find the solutions to the discrete Riccati equation. A routine is also included to solve the steady-state Riccati equation by the Newton algorithms described by Klein, for continuous problems, and by Hewer, for discrete problems. Another routine calculates the prefilter gain to eliminate control state cross-product terms in the quadratic performance index and the weighting matrices for the sampled data optimal linear regulator problem. For cases with measurement noise, duality theory and optimal regulator algorithms are used to calculate solutions to the continuous and discrete Kalman-Bucy filter problems. Finally, routines are included to implement the continuous and discrete forms of the explicit (model-in-the-system) and implicit (model-in-the-performance-index) model following theory. These routines generate linear control laws which cause the output of a dynamic time-invariant system to track the output of a prescribed model. In order to apply ORACLS, the user must write an executive (driver) program which inputs the problem coefficients, formulates and selects the routines to be used to solve the problem, and specifies the desired output. There are three versions of ORACLS source code available for implementation: CDC, IBM, and DEC. The CDC version has been implemented on a CDC 6000 series computer with a central memory of approximately 13K (octal) of 60 bit words. The CDC version is written in FORTRAN IV, was developed in 1978, and last updated in 1986. The IBM version has been implemented on an IBM 370 series computer with a central memory requirement of approximately 300K of 8 bit bytes. The IBM version is written in FORTRAN IV and was generated in 1981. The DEC version has been implemented on a VAX series computer operating under VMS. The VAX version is written in FORTRAN 77 and was generated in 1986.

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…

Promoting Quantitative Literacy in an Online College Algebra Course

ERIC Educational Resources Information Center

Tunstall, Luke; Bossé, Michael J.

2016-01-01

College algebra (a university freshman level algebra course) fulfills the quantitative literacy requirement of many college's general education programs and is a terminal course for most who take it. An online problem-based learning environment provides a unique means of engaging students in quantitative discussions and research. This article…

Algebraic Concepts: What's Really New in New Curricula?

ERIC Educational Resources Information Center

Star, Jon R.; Herbel-Eisenmann, Beth A.; Smith, John P., III

2000-01-01

Examines 8th grade units from the Connected Mathematics Project (CMP). Identifies differences in older and newer conceptions, fundamental objects of study, typical problems, and typical solution methods in algebra. Also discusses where the issue of what is new in algebra is relevant to many other innovative middle school curricula. (KHR)

Vector Potential Generation for Numerical Relativity Simulations

NASA Astrophysics Data System (ADS)

Silberman, Zachary; Faber, Joshua; Adams, Thomas; Etienne, Zachariah; Ruchlin, Ian

2017-01-01

Many different numerical codes are employed in studies of highly relativistic magnetized accretion flows around black holes. Based on the formalisms each uses, some codes evolve the magnetic field vector B, while others evolve the magnetic vector potential A, the two being related by the curl: B=curl(A). Here, we discuss how to generate vector potentials corresponding to specified magnetic fields on staggered grids, a surprisingly difficult task on finite cubic domains. The code we have developed solves this problem in two ways: a brute-force method, whose scaling is nearly linear in the number of grid cells, and a direct linear algebra approach. We discuss the success both algorithms have in generating smooth vector potential configurations and how both may be extended to more complicated cases involving multiple mesh-refinement levels. NSF ACI-1550436

A globally well-posed finite element algorithm for aerodynamics applications

NASA Technical Reports Server (NTRS)

Iannelli, G. S.; Baker, A. J.

1991-01-01

A finite element CFD algorithm is developed for Euler and Navier-Stokes aerodynamic applications. For the linear basis, the resultant approximation is at least second-order-accurate in time and space for synergistic use of three procedures: (1) a Taylor weak statement, which provides for derivation of companion conservation law systems with embedded dispersion-error control mechanisms; (2) a stiffly stable second-order-accurate implicit Rosenbrock-Runge-Kutta temporal algorithm; and (3) a matrix tensor product factorization that permits efficient numerical linear algebra handling of the terminal large-matrix statement. Thorough analyses are presented regarding well-posed boundary conditions for inviscid and viscous flow specifications. Numerical solutions are generated and compared for critical evaluation of quasi-one- and two-dimensional Euler and Navier-Stokes benchmark test problems.

Choosing order of operations to accelerate strip structure analysis in parameter range

NASA Astrophysics Data System (ADS)

Kuksenko, S. P.; Akhunov, R. R.; Gazizov, T. R.

2018-05-01

The paper considers the issue of using iteration methods in solving the sequence of linear algebraic systems obtained in quasistatic analysis of strip structures with the method of moments. Using the analysis of 4 strip structures, the authors have proved that additional acceleration (up to 2.21 times) of the iterative process can be obtained during the process of solving linear systems repeatedly by means of choosing a proper order of operations and a preconditioner. The obtained results can be used to accelerate the process of computer-aided design of various strip structures. The choice of the order of operations to accelerate the process is quite simple, universal and could be used not only for strip structure analysis but also for a wide range of computational problems.

Investigating Integer Restrictions in Linear Programming

ERIC Educational Resources Information Center

Edwards, Thomas G.; Chelst, Kenneth R.; Principato, Angela M.; Wilhelm, Thad L.

2015-01-01

Linear programming (LP) is an application of graphing linear systems that appears in many Algebra 2 textbooks. Although not explicitly mentioned in the Common Core State Standards for Mathematics, linear programming blends seamlessly into modeling with mathematics, the fourth Standard for Mathematical Practice (CCSSI 2010, p. 7). In solving a…