GENERAL PURPOSE ADA PACKAGES

NASA Technical Reports Server (NTRS)

Klumpp, A. R.

1994-01-01

Ten families of subprograms are bundled together for the General-Purpose Ada Packages. The families bring to Ada many features from HAL/S, PL/I, FORTRAN, and other languages. These families are: string subprograms (INDEX, TRIM, LOAD, etc.); scalar subprograms (MAX, MIN, REM, etc.); array subprograms (MAX, MIN, PROD, SUM, GET, and PUT); numerical subprograms (EXP, CUBIC, etc.); service subprograms (DATE_TIME function, etc.); Linear Algebra II; Runge-Kutta integrators; and three text I/O families of packages. In two cases, a family consists of a single non-generic package. In all other cases, a family comprises a generic package and its instances for a selected group of scalar types. All generic packages are designed to be easily instantiated for the types declared in the user facility. The linear algebra package is LINRAG2. This package includes subprograms supplementing those in NPO-17985, An Ada Linear Algebra Package Modeled After HAL/S (LINRAG). Please note that LINRAG2 cannot be compiled without LINRAG. Most packages have widespread applicability, although some are oriented for avionics applications. All are designed to facilitate writing new software in Ada. Several of the packages use conventions introduced by other programming languages. A package of string subprograms is based on HAL/S (a language designed for the avionics software in the Space Shuttle) and PL/I. Packages of scalar and array subprograms are taken from HAL/S or generalized current Ada subprograms. A package of Runge-Kutta integrators is patterned after a built-in MAC (MIT Algebraic Compiler) integrator. Those packages modeled after HAL/S make it easy to translate existing HAL/S software to Ada. The General-Purpose Ada Packages program source code is available on two 360K 5.25" MS-DOS format diskettes. The software was developed using VAX Ada v1.5 under DEC VMS v4.5. It should be portable to any validated Ada compiler and it should execute either interactively or in batch. The largest package requires 205K of main memory on a DEC VAX running VMS. The software was developed in 1989, and is a copyrighted work with all copyright vested in NASA.

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

Advanced complex trait analysis.

PubMed

Gray, A; Stewart, I; Tenesa, A

2012-12-01

The Genome-wide Complex Trait Analysis (GCTA) software package can quantify the contribution of genetic variation to phenotypic variation for complex traits. However, as those datasets of interest continue to increase in size, GCTA becomes increasingly computationally prohibitive. We present an adapted version, Advanced Complex Trait Analysis (ACTA), demonstrating dramatically improved performance. We restructure the genetic relationship matrix (GRM) estimation phase of the code and introduce the highly optimized parallel Basic Linear Algebra Subprograms (BLAS) library combined with manual parallelization and optimization. We introduce the Linear Algebra PACKage (LAPACK) library into the restricted maximum likelihood (REML) analysis stage. For a test case with 8999 individuals and 279,435 single nucleotide polymorphisms (SNPs), we reduce the total runtime, using a compute node with two multi-core Intel Nehalem CPUs, from ∼17 h to ∼11 min. The source code is fully available under the GNU Public License, along with Linux binaries. For more information see http://www.epcc.ed.ac.uk/software-products/acta. a.gray@ed.ac.uk Supplementary data are available at Bioinformatics online.

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)

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.

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).

Image-algebraic design of multispectral target recognition algorithms

NASA Astrophysics Data System (ADS)

Schmalz, Mark S.; Ritter, Gerhard X.

1994-06-01

In this paper, we discuss methods for multispectral ATR (Automated Target Recognition) of small targets that are sensed under suboptimal conditions, such as haze, smoke, and low light levels. In particular, we discuss our ongoing development of algorithms and software that effect intelligent object recognition by selecting ATR filter parameters according to ambient conditions. Our algorithms are expressed in terms of IA (image algebra), a concise, rigorous notation that unifies linear and nonlinear mathematics in the image processing domain. IA has been implemented on a variety of parallel computers, with preprocessors available for the Ada and FORTRAN languages. An image algebra C++ class library has recently been made available. Thus, our algorithms are both feasible implementationally and portable to numerous machines. Analyses emphasize the aspects of image algebra that aid the design of multispectral vision algorithms, such as parameterized templates that facilitate the flexible specification of ATR filters.

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

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

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 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)

QuBiLS-MAS, open source multi-platform software for atom- and bond-based topological (2D) and chiral (2.5D) algebraic molecular descriptors computations.

PubMed

Valdés-Martiní, José R; Marrero-Ponce, Yovani; García-Jacas, César R; Martinez-Mayorga, Karina; Barigye, Stephen J; Vaz d'Almeida, Yasser Silveira; Pham-The, Hai; Pérez-Giménez, Facundo; Morell, Carlos A

2017-06-07

In previous reports, Marrero-Ponce et al. proposed algebraic formalisms for characterizing topological (2D) and chiral (2.5D) molecular features through atom- and bond-based ToMoCoMD-CARDD (acronym for Topological Molecular Computational Design-Computer Aided Rational Drug Design) molecular descriptors. These MDs codify molecular information based on the bilinear, quadratic and linear algebraic forms and the graph-theoretical electronic-density and edge-adjacency matrices in order to consider atom- and bond-based relations, respectively. These MDs have been successfully applied in the screening of chemical compounds of different therapeutic applications ranging from antimalarials, antibacterials, tyrosinase inhibitors and so on. To compute these MDs, a computational program with the same name was initially developed. However, this in house software barely offered the functionalities required in contemporary molecular modeling tasks, in addition to the inherent limitations that made its usability impractical. Therefore, the present manuscript introduces the QuBiLS-MAS (acronym for Quadratic, Bilinear and N-Linear mapS based on graph-theoretic electronic-density Matrices and Atomic weightingS) software designed to compute topological (0-2.5D) molecular descriptors based on bilinear, quadratic and linear algebraic forms for atom- and bond-based relations. The QuBiLS-MAS module was designed as standalone software, in which extensions and generalizations of the former ToMoCoMD-CARDD 2D-algebraic indices are implemented, considering the following aspects: (a) two new matrix normalization approaches based on double-stochastic and mutual probability formalisms; (b) topological constraints (cut-offs) to take into account particular inter-atomic relations; (c) six additional atomic properties to be used as weighting schemes in the calculation of the molecular vectors; (d) four new local-fragments to consider molecular regions of interest; (e) number of lone-pair electrons in chemical structure defined by diagonal coefficients in matrix representations; and (f) several aggregation operators (invariants) applied over atom/bond-level descriptors in order to compute global indices. This software permits the parallel computation of the indices, contains a batch processing module and data curation functionalities. This program was developed in Java v1.7 using the Chemistry Development Kit library (version 1.4.19). The QuBiLS-MAS software consists of two components: a desktop interface (GUI) and an API library allowing for the easy integration of the latter in chemoinformatics applications. The relevance of the novel extensions and generalizations implemented in this software is demonstrated through three studies. Firstly, a comparative Shannon's entropy based variability study for the proposed QuBiLS-MAS and the DRAGON indices demonstrates superior performance for the former. A principal component analysis reveals that the QuBiLS-MAS approach captures chemical information orthogonal to that codified by the DRAGON descriptors. Lastly, a QSAR study for the binding affinity to the corticosteroid-binding globulin using Cramer's steroid dataset is carried out. From these analyses, it is revealed that the QuBiLS-MAS approach for atom-pair relations yields similar-to-superior performance with regard to other QSAR methodologies reported in the literature. Therefore, the QuBiLS-MAS approach constitutes a useful tool for the diversity analysis of chemical compound datasets and high-throughput screening of structure-activity data.

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…

An effective automatic procedure for testing parameter identifiability of HIV/AIDS models.

PubMed

Saccomani, Maria Pia

2011-08-01

Realistic HIV models tend to be rather complex and many recent models proposed in the literature could not yet be analyzed by traditional identifiability testing techniques. In this paper, we check a priori global identifiability of some of these nonlinear HIV models taken from the recent literature, by using a differential algebra algorithm based on previous work of the author. The algorithm is implemented in a software tool, called DAISY (Differential Algebra for Identifiability of SYstems), which has been recently released (DAISY is freely available on the web site http://www.dei.unipd.it/~pia/ ). The software can be used to automatically check global identifiability of (linear and) nonlinear models described by polynomial or rational differential equations, thus providing a general and reliable tool to test global identifiability of several HIV models proposed in the literature. It can be used by researchers with a minimum of mathematical background.

Schwarz maps of algebraic linear ordinary differential equations

NASA Astrophysics Data System (ADS)

Sanabria Malagón, Camilo

2017-12-01

A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard-Vessiot Theory.

Cognitive Foundry v. 3.0 (OSS)

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

Basilico, Justin; Dixon, Kevin; McClain, Jonathan

2009-11-18

The Cognitive Foundry is a unified collection of tools designed for research and applications that use cognitive modeling, machine learning, or pattern recognition. The software library contains design patterns, interface definitions, and default implementations of reusable software components and algorithms designed to support a wide variety of research and development needs. The library contains three main software packages: the Common package that contains basic utilities and linear algebraic methods, the Cognitive Framework package that contains tools to assist in implementing and analyzing theories of cognition, and the Machine Learning package that provides general algorithms and methods for populating Cognitive Frameworkmore » components from domain-relevant data.« less

Genten: Software for Generalized Tensor Decompositions v. 1.0.0

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

Phipps, Eric T.; Kolda, Tamara G.; Dunlavy, Daniel

Tensors, or multidimensional arrays, are a powerful mathematical means of describing multiway data. This software provides computational means for decomposing or approximating a given tensor in terms of smaller tensors of lower dimension, focusing on decomposition of large, sparse tensors. These techniques have applications in many scientific areas, including signal processing, linear algebra, computer vision, numerical analysis, data mining, graph analysis, neuroscience and more. The software is designed to take advantage of parallelism present emerging computer architectures such has multi-core CPUs, many-core accelerators such as the Intel Xeon Phi, and computation-oriented GPUs to enable efficient processing of large tensors.

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.

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.

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…

Construction of Orthonormal Wavelets Using Symbolic Algebraic Methods

NASA Astrophysics Data System (ADS)

Černá, Dana; Finěk, Václav

2009-09-01

Our contribution is concerned with the solution of nonlinear algebraic equations systems arising from the computation of scaling coefficients of orthonormal wavelets with compact support. Specifically Daubechies wavelets, symmlets, coiflets, and generalized coiflets. These wavelets are defined as a solution of equation systems which are partly linear and partly nonlinear. The idea of presented methods consists in replacing those equations for scaling coefficients by equations for scaling moments. It enables us to eliminate some quadratic conditions in the original system and then simplify it. The simplified system is solved with the aid of the Gröbner basis method. The advantage of our approach is that in some cases, it provides all possible solutions and these solutions can be computed to arbitrary precision. For small systems, we are even able to find explicit solutions. The computation was carried out by symbolic algebra software Maple.

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

ERIC Educational Resources Information Center

What Works Clearinghouse, 2009

2009-01-01

The "Cognitive Tutor[R] Algebra I" curriculum, published by Carnegie Learning, is an approach that combines algebra textbooks with interactive software. The software is developed around an artificial intelligence model that identifies strengths and weaknesses in each individual student's mastery of mathematical concepts. It then customizes prompts…

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.

Sixth SIAM conference on applied linear algebra: Final program and abstracts. Final technical report

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

NONE

1997-12-31

Linear algebra plays a central role in mathematics and applications. The analysis and solution of problems from an amazingly wide variety of disciplines depend on the theory and computational techniques of linear algebra. In turn, the diversity of disciplines depending on linear algebra also serves to focus and shape its development. Some problems have special properties (numerical, structural) that can be exploited. Some are simply so large that conventional approaches are impractical. New computer architectures motivate new algorithms, and fresh ways to look at old ones. The pervasive nature of linear algebra in analyzing and solving problems means that peoplemore » from a wide spectrum--universities, industrial and government laboratories, financial institutions, and many others--share an interest in current developments in linear algebra. This conference aims to bring them together for their mutual benefit. Abstracts of papers presented are included.« less

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

Software Reviews.

ERIC Educational Resources Information Center

Mathematics and Computer Education, 1988

1988-01-01

Presents reviews of six software packages. Includes (1) "Plain Vanilla Statistics"; (2) "MathCAD 2.0"; (3) "GrFx"; (4) "Trigonometry"; (5) "Algebra II"; (6) "Algebra Drill and Practice I, II, and III." (PK)

Generic, Type-Safe and Object Oriented Computer Algebra Software

NASA Astrophysics Data System (ADS)

Kredel, Heinz; Jolly, Raphael

Advances in computer science, in particular object oriented programming, and software engineering have had little practical impact on computer algebra systems in the last 30 years. The software design of existing systems is still dominated by ad-hoc memory management, weakly typed algorithm libraries and proprietary domain specific interactive expression interpreters. We discuss a modular approach to computer algebra software: usage of state-of-the-art memory management and run-time systems (e.g. JVM) usage of strongly typed, generic, object oriented programming languages (e.g. Java) and usage of general purpose, dynamic interactive expression interpreters (e.g. Python) To illustrate the workability of this approach, we have implemented and studied computer algebra systems in Java and Scala. In this paper we report on the current state of this work by presenting new examples.

Assessing non-uniqueness: An algebraic approach

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

Vasco, Don W.

Geophysical inverse problems are endowed with a rich mathematical structure. When discretized, most differential and integral equations of interest are algebraic (polynomial) in form. Techniques from algebraic geometry and computational algebra provide a means to address questions of existence and uniqueness for both linear and non-linear inverse problem. In a sense, the methods extend ideas which have proven fruitful in treating linear inverse problems.

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…

DAISY: a new software tool to test global identifiability of biological and physiological systems.

PubMed

Bellu, Giuseppina; Saccomani, Maria Pia; Audoly, Stefania; D'Angiò, Leontina

2007-10-01

A priori global identifiability is a structural property of biological and physiological models. It is considered a prerequisite for well-posed estimation, since it concerns the possibility of recovering uniquely the unknown model parameters from measured input-output data, under ideal conditions (noise-free observations and error-free model structure). Of course, determining if the parameters can be uniquely recovered from observed data is essential before investing resources, time and effort in performing actual biomedical experiments. Many interesting biological models are nonlinear but identifiability analysis for nonlinear system turns out to be a difficult mathematical problem. Different methods have been proposed in the literature to test identifiability of nonlinear models but, to the best of our knowledge, so far no software tools have been proposed for automatically checking identifiability of nonlinear models. In this paper, we describe a software tool implementing a differential algebra algorithm to perform parameter identifiability analysis for (linear and) nonlinear dynamic models described by polynomial or rational equations. Our goal is to provide the biological investigator a completely automatized software, requiring minimum prior knowledge of mathematical modelling and no in-depth understanding of the mathematical tools. The DAISY (Differential Algebra for Identifiability of SYstems) software will potentially be useful in biological modelling studies, especially in physiology and clinical medicine, where research experiments are particularly expensive and/or difficult to perform. Practical examples of use of the software tool DAISY are presented. DAISY is available at the web site http://www.dei.unipd.it/~pia/.

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.

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.

Teaching Linear Algebra: Must the Fog Always Roll In?

ERIC Educational Resources Information Center

Carlson, David

1993-01-01

Proposes methods to teach the more difficult concepts of linear algebra. Examines features of the Linear Algebra Curriculum Study Group Core Syllabus, and presents problems from the core syllabus that utilize the mathematical process skills of making conjectures, proving the results, and communicating the results to colleagues. Presents five…

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

Using the Interactive Learning Environment Aplusix for Teaching and Learning School Algebra: A Research Experiment in a Middle School

ERIC Educational Resources Information Center

Hadjerrouit, Said

2011-01-01

Most software tools that have been developed with the aim of helping students to learn school algebra have not yet achieved successful results in classroom. Almost all of them are menu-based systems that provide transformation rules in menus and buttons. Aplusix is a new interactive software tool for learning school algebra. In contrast to…

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.

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…

A Richer Understanding of Algebra

ERIC Educational Resources Information Center

Foy, Michelle

2008-01-01

Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number,…

Automated ILA design for synchronous sequential circuits

NASA Technical Reports Server (NTRS)

Liu, M. N.; Liu, K. Z.; Maki, G. K.; Whitaker, S. R.

1991-01-01

An iterative logic array (ILA) architecture for synchronous sequential circuits is presented. This technique utilizes linear algebra to produce the design equations. The ILA realization of synchronous sequential logic can be fully automated with a computer program. A programmable design procedure is proposed to fullfill the design task and layout generation. A software algorithm in the C language has been developed and tested to generate 1 micron CMOS layouts using the Hewlett-Packard FUNGEN module generator shell.

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…

DAISY: a new software tool to test global identifiability of biological and physiological systems

PubMed Central

Bellu, Giuseppina; Saccomani, Maria Pia; Audoly, Stefania; D’Angiò, Leontina

2009-01-01

A priori global identifiability is a structural property of biological and physiological models. It is considered a prerequisite for well-posed estimation, since it concerns the possibility of recovering uniquely the unknown model parameters from measured input-output data, under ideal conditions (noise-free observations and error-free model structure). Of course, determining if the parameters can be uniquely recovered from observed data is essential before investing resources, time and effort in performing actual biomedical experiments. Many interesting biological models are nonlinear but identifiability analysis for nonlinear system turns out to be a difficult mathematical problem. Different methods have been proposed in the literature to test identifiability of nonlinear models but, to the best of our knowledge, so far no software tools have been proposed for automatically checking identifiability of nonlinear models. In this paper, we describe a software tool implementing a differential algebra algorithm to perform parameter identifiability analysis for (linear and) nonlinear dynamic models described by polynomial or rational equations. Our goal is to provide the biological investigator a completely automatized software, requiring minimum prior knowledge of mathematical modelling and no in-depth understanding of the mathematical tools. The DAISY (Differential Algebra for Identifiability of SYstems) software will potentially be useful in biological modelling studies, especially in physiology and clinical medicine, where research experiments are particularly expensive and/or difficult to perform. Practical examples of use of the software tool DAISY are presented. DAISY is available at the web site http://www.dei.unipd.it/~pia/. PMID:17707944

Efficient and Robust Optimization for Building Energy Simulation

PubMed Central

Pourarian, Shokouh; Kearsley, Anthony; Wen, Jin; Pertzborn, Amanda

2016-01-01

Efficiently, robustly and accurately solving large sets of structured, non-linear algebraic and differential equations is one of the most computationally expensive steps in the dynamic simulation of building energy systems. Here, the efficiency, robustness and accuracy of two commonly employed solution methods are compared. The comparison is conducted using the HVACSIM+ software package, a component based building system simulation tool. The HVACSIM+ software presently employs Powell’s Hybrid method to solve systems of nonlinear algebraic equations that model the dynamics of energy states and interactions within buildings. It is shown here that the Powell’s method does not always converge to a solution. Since a myriad of other numerical methods are available, the question arises as to which method is most appropriate for building energy simulation. This paper finds considerable computational benefits result from replacing the Powell’s Hybrid method solver in HVACSIM+ with a solver more appropriate for the challenges particular to numerical simulations of buildings. Evidence is provided that a variant of the Levenberg-Marquardt solver has superior accuracy and robustness compared to the Powell’s Hybrid method presently used in HVACSIM+. PMID:27325907

Efficient and Robust Optimization for Building Energy Simulation.

PubMed

Pourarian, Shokouh; Kearsley, Anthony; Wen, Jin; Pertzborn, Amanda

2016-06-15

Efficiently, robustly and accurately solving large sets of structured, non-linear algebraic and differential equations is one of the most computationally expensive steps in the dynamic simulation of building energy systems. Here, the efficiency, robustness and accuracy of two commonly employed solution methods are compared. The comparison is conducted using the HVACSIM+ software package, a component based building system simulation tool. The HVACSIM+ software presently employs Powell's Hybrid method to solve systems of nonlinear algebraic equations that model the dynamics of energy states and interactions within buildings. It is shown here that the Powell's method does not always converge to a solution. Since a myriad of other numerical methods are available, the question arises as to which method is most appropriate for building energy simulation. This paper finds considerable computational benefits result from replacing the Powell's Hybrid method solver in HVACSIM+ with a solver more appropriate for the challenges particular to numerical simulations of buildings. Evidence is provided that a variant of the Levenberg-Marquardt solver has superior accuracy and robustness compared to the Powell's Hybrid method presently used in HVACSIM+.

Matrix Algebra for GPU and Multicore Architectures (MAGMA) for Large Petascale Systems

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

Dongarra, Jack J.; Tomov, Stanimire

2014-03-24

The goal of the MAGMA project is to create a new generation of linear algebra libraries that achieve the fastest possible time to an accurate solution on hybrid Multicore+GPU-based systems, using all the processing power that future high-end systems can make available within given energy constraints. Our efforts at the University of Tennessee achieved the goals set in all of the five areas identified in the proposal: 1. Communication optimal algorithms; 2. Autotuning for GPU and hybrid processors; 3. Scheduling and memory management techniques for heterogeneity and scale; 4. Fault tolerance and robustness for large scale systems; 5. Building energymore » efficiency into software foundations. The University of Tennessee’s main contributions, as proposed, were the research and software development of new algorithms for hybrid multi/many-core CPUs and GPUs, as related to two-sided factorizations and complete eigenproblem solvers, hybrid BLAS, and energy efficiency for dense, as well as sparse, operations. Furthermore, as proposed, we investigated and experimented with various techniques targeting the five main areas outlined.« less

Open Source Software in Teaching Physics: A Case Study on Vector Algebra and Visual Representations

ERIC Educational Resources Information Center

Cataloglu, Erdat

2006-01-01

This study aims to report the effort on teaching vector algebra using free open source software (FOSS). Recent studies showed that students have difficulties in learning basic physics concepts. Constructivist learning theories suggest the use of visual and hands-on activities in learning. We will report on the software used for this purpose. The…

IBM system/360 assembly language interval arithmetic software

NASA Technical Reports Server (NTRS)

Phillips, E. J.

1972-01-01

Computer software designed to perform interval arithmetic is described. An interval is defined as the set of all real numbers between two given numbers including or excluding one or both endpoints. Interval arithmetic consists of the various elementary arithmetic operations defined on the set of all intervals, such as interval addition, subtraction, union, etc. One of the main applications of interval arithmetic is in the area of error analysis of computer calculations. For example, it has been used sucessfully to compute bounds on sounding errors in the solution of linear algebraic systems, error bounds in numerical solutions of ordinary differential equations, as well as integral equations and boundary value problems. The described software enables users to implement algorithms of the type described in references efficiently on the IBM 360 system.

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…

Heterogeneous Software System Interoperability Through Computer-Aided Resolution of Modeling Differences

DTIC Science & Technology

2002-06-01

techniques for addressing the software component retrieval problem. Steigerwald [Ste91] introduced the use of algebraic specifications for defining the...provided in terms of a specification written using Luqi’s Prototype Specification Description Language (PSDL) [LBY88] augmented with an algebraic

Component-based integration of chemistry and optimization software.

PubMed

Kenny, Joseph P; Benson, Steven J; Alexeev, Yuri; Sarich, Jason; Janssen, Curtis L; McInnes, Lois Curfman; Krishnan, Manojkumar; Nieplocha, Jarek; Jurrus, Elizabeth; Fahlstrom, Carl; Windus, Theresa L

2004-11-15

Typical scientific software designs make rigid assumptions regarding programming language and data structures, frustrating software interoperability and scientific collaboration. Component-based software engineering is an emerging approach to managing the increasing complexity of scientific software. Component technology facilitates code interoperability and reuse. Through the adoption of methodology and tools developed by the Common Component Architecture Forum, we have developed a component architecture for molecular structure optimization. Using the NWChem and Massively Parallel Quantum Chemistry packages, we have produced chemistry components that provide capacity for energy and energy derivative evaluation. We have constructed geometry optimization applications by integrating the Toolkit for Advanced Optimization, Portable Extensible Toolkit for Scientific Computation, and Global Arrays packages, which provide optimization and linear algebra capabilities. We present a brief overview of the component development process and a description of abstract interfaces for chemical optimizations. The components conforming to these abstract interfaces allow the construction of applications using different chemistry and mathematics packages interchangeably. Initial numerical results for the component software demonstrate good performance, and highlight potential research enabled by this platform.

A Relational Algebra Query Language for Programming Relational Databases

ERIC Educational Resources Information Center

McMaster, Kirby; Sambasivam, Samuel; Anderson, Nicole

2011-01-01

In this paper, we describe a Relational Algebra Query Language (RAQL) and Relational Algebra Query (RAQ) software product we have developed that allows database instructors to teach relational algebra through programming. Instead of defining query operations using mathematical notation (the approach commonly taken in database textbooks), students…

HAL/S - The programming language for Shuttle

NASA Technical Reports Server (NTRS)

Martin, F. H.

1974-01-01

HAL/S is a higher order language and system, now operational, adopted by NASA for programming Space Shuttle on-board software. Program reliability is enhanced through language clarity and readability, modularity through program structure, and protection of code and data. Salient features of HAL/S include output orientation, automatic checking (with strictly enforced compiler rules), the availability of linear algebra, real-time control, a statement-level simulator, and compiler transferability (for applying HAL/S to additional object and host computers). The compiler is described briefly.

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…

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

I CAN Learn[R] Pre-Algebra and Algebra. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2009

2009-01-01

The I CAN Learn[R] Education System is an interactive, self-paced, mastery-based software system that includes the I CAN Learn[R] Fundamentals of Math (5th-6th grade math) curriculum, the I CAN Learn[R] Pre-Algebra curriculum, and the I CAN Learn[R] Algebra curriculum. College algebra credit is also available to students in participating schools…

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.

Examples of testing global identifiability of biological and biomedical models with the DAISY software.

PubMed

Saccomani, Maria Pia; Audoly, Stefania; Bellu, Giuseppina; D'Angiò, Leontina

2010-04-01

DAISY (Differential Algebra for Identifiability of SYstems) is a recently developed computer algebra software tool which can be used to automatically check global identifiability of (linear and) nonlinear dynamic models described by differential equations involving polynomial or rational functions. Global identifiability is a fundamental prerequisite for model identification which is important not only for biological or medical systems but also for many physical and engineering systems derived from first principles. Lack of identifiability implies that the parameter estimation techniques may not fail but any obtained numerical estimates will be meaningless. The software does not require understanding of the underlying mathematical principles and can be used by researchers in applied fields with a minimum of mathematical background. We illustrate the DAISY software by checking the a priori global identifiability of two benchmark nonlinear models taken from the literature. The analysis of these two examples includes comparison with other methods and demonstrates how identifiability analysis is simplified by this tool. Thus we illustrate the identifiability analysis of other two examples, by including discussion of some specific aspects related to the role of observability and knowledge of initial conditions in testing identifiability and to the computational complexity of the software. The main focus of this paper is not on the description of the mathematical background of the algorithm, which has been presented elsewhere, but on illustrating its use and on some of its more interesting features. DAISY is available on the web site http://www.dei.unipd.it/ approximately pia/. 2010 Elsevier Ltd. All rights reserved.

An empirical investigation of methods for nonsymmetric linear systems

NASA Technical Reports Server (NTRS)

Sherman, A. H.

1981-01-01

The present investigation is concerned with a comparison of methods for solving linear algebraic systems which arise from finite difference discretizations of the elliptic convection-diffusion equation in a planar region Omega with Dirichlet boundary conditions. Such linear systems are typically of the form Ax = b where A is an N x N sparse nonsymmetric matrix. In a discussion of discretizations, it is assumed that a regular rectilinear mesh of width h has been imposed on Omega. The discretizations considered include central differences, upstream differences, and modified upstream differences. Six methods for solving Ax = b are considered. Three variants of Gaussian elimination have been chosen as representatives of state-of-the-art software for direct methods under different assumptions about pivoting. Three iterative methods are also included.

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.

Using Group Explorer in teaching abstract algebra

NASA Astrophysics Data System (ADS)

Schubert, Claus; Gfeller, Mary; Donohue, Christopher

2013-04-01

This study explores the use of Group Explorer in an undergraduate mathematics course in abstract algebra. The visual nature of Group Explorer in representing concepts in group theory is an attractive incentive to use this software in the classroom. However, little is known about students' perceptions on this technology in learning concepts in abstract algebra. A total of 26 participants in an undergraduate course studying group theory were surveyed regarding their experiences using Group Explorer. Findings indicate that all participants believed that the software was beneficial to their learning and described their attitudes regarding the software in terms of using the technology and its helpfulness in learning concepts. A multiple regression analysis reveals that representational fluency of concepts with the software correlated significantly with participants' understanding of group concepts yet, participants' attitudes about Group Explorer and technology in general were not significant factors.

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…

Elucidation of covariant proofs in general relativity: example of the use of algebraic software in the shear-free conjecture in MAPLE

NASA Astrophysics Data System (ADS)

Huf, P. A.; Carminati, J.

2018-01-01

In this paper we explore the use of a new algebraic software package in providing independent covariant proof of a conjecture in general relativity. We examine the proof of two sub-cases of the shear-free conjecture σ =0 => ω Θ =0 by Senovilla et al. (Gen. Relativ. Gravit 30:389-411, 1998): case 1: for dust; case 2: for acceleration parallel to vorticity. We use TensorPack, a software package recently released for the Maple environment. In this paper, we briefly summarise the key features of the software and then demonstrate its use by providing and discussing examples of independent proofs of the paper in question. A full set of our completed proofs is available online at http://www.bach2roq.com/science/maths/GR/ShearFreeProofs.html. We are in agreeance with the equations provided in the original paper, noting that the proofs often require many steps. Furthermore, in our proofs we provide fully worked algebraic steps in such a way that the proofs can be examined systematically, and avoiding hand calculation. It is hoped that the elucidated proofs may be of use to other researchers in verifying the algebraic consistency of the expressions in the paper in question, as well as related literature. Furthermore we suggest that the appropriate use of algebraic software in covariant formalism could be useful for developing research and teaching in GR theory.

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

Image Algebra Matlab language version 2.3 for image processing and compression research

NASA Astrophysics Data System (ADS)

Schmalz, Mark S.; Ritter, Gerhard X.; Hayden, Eric

2010-08-01

Image algebra is a rigorous, concise notation that unifies linear and nonlinear mathematics in the image domain. Image algebra was developed under DARPA and US Air Force sponsorship at University of Florida for over 15 years beginning in 1984. Image algebra has been implemented in a variety of programming languages designed specifically to support the development of image processing and computer vision algorithms and software. The University of Florida has been associated with development of the languages FORTRAN, Ada, Lisp, and C++. The latter implementation involved a class library, iac++, that supported image algebra programming in C++. Since image processing and computer vision are generally performed with operands that are array-based, the Matlab™ programming language is ideal for implementing the common subset of image algebra. Objects include sets and set operations, images and operations on images, as well as templates and image-template convolution operations. This implementation, called Image Algebra Matlab (IAM), has been found to be useful for research in data, image, and video compression, as described herein. Due to the widespread acceptance of the Matlab programming language in the computing community, IAM offers exciting possibilities for supporting a large group of users. The control over an object's computational resources provided to the algorithm designer by Matlab means that IAM programs can employ versatile representations for the operands and operations of the algebra, which are supported by the underlying libraries written in Matlab. In a previous publication, we showed how the functionality of IAC++ could be carried forth into a Matlab implementation, and provided practical details of a prototype implementation called IAM Version 1. In this paper, we further elaborate the purpose and structure of image algebra, then present a maturing implementation of Image Algebra Matlab called IAM Version 2.3, which extends the previous implementation of IAM to include polymorphic operations over different point sets, as well as recursive convolution operations and functional composition. We also show how image algebra and IAM can be employed in image processing and compression research, as well as algorithm development and analysis.

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…

Discovering Theorems in Abstract Algebra Using the Software "GAP"

ERIC Educational Resources Information Center

Blyth, Russell D.; Rainbolt, Julianne G.

2010-01-01

A traditional abstract algebra course typically consists of the professor stating and then proving a sequence of theorems. As an alternative to this classical structure, the students could be expected to discover some of the theorems even before they are motivated by classroom examples. This can be done by using a software system to explore a…

Using Graphing Software to Teach about Algebraic Forms: A Study of Technology-Supported Practice in Secondary-School Mathematics

ERIC Educational Resources Information Center

Ruthven, Kenneth; Deaney, Rosemary; Hennessy, Sara

2009-01-01

From preliminary analysis of teacher-nominated examples of successful technology-supported practice in secondary-school mathematics, the use of graphing software to teach about algebraic forms was identified as being an important archetype. Employing evidence from lesson observation and teacher interview, such practice was investigated in greater…

Stability of Linear Equations--Algebraic Approach

ERIC Educational Resources Information Center

Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.

2012-01-01

This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…

Optical systolic solutions of linear algebraic equations

NASA Technical Reports Server (NTRS)

Neuman, C. P.; Casasent, D.

1984-01-01

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

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 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.

SPARSKIT: A basic tool kit for sparse matrix computations

NASA Technical Reports Server (NTRS)

Saad, Youcef

1990-01-01

Presented here are the main features of a tool package for manipulating and working with sparse matrices. One of the goals of the package is to provide basic tools to facilitate the exchange of software and data between researchers in sparse matrix computations. The starting point is the Harwell/Boeing collection of matrices for which the authors provide a number of tools. Among other things, the package provides programs for converting data structures, printing simple statistics on a matrix, plotting a matrix profile, and performing linear algebra operations with sparse matrices.

Software Development Of XML Parser Based On Algebraic Tools

NASA Astrophysics Data System (ADS)

Georgiev, Bozhidar; Georgieva, Adriana

2011-12-01

In this paper, is presented one software development and implementation of an algebraic method for XML data processing, which accelerates XML parsing process. Therefore, the proposed in this article nontraditional approach for fast XML navigation with algebraic tools contributes to advanced efforts in the making of an easier user-friendly API for XML transformations. Here the proposed software for XML documents processing (parser) is easy to use and can manage files with strictly defined data structure. The purpose of the presented algorithm is to offer a new approach for search and restructuring hierarchical XML data. This approach permits fast XML documents processing, using algebraic model developed in details in previous works of the same authors. So proposed parsing mechanism is easy accessible to the web consumer who is able to control XML file processing, to search different elements (tags) in it, to delete and to add a new XML content as well. The presented various tests show higher rapidity and low consumption of resources in comparison with some existing commercial parsers.

Stability Analysis of Finite Difference Schemes for Hyperbolic Systems, and Problems in Applied and Computational Linear Algebra.

DTIC Science & Technology

FINITE DIFFERENCE THEORY, * LINEAR ALGEBRA , APPLIED MATHEMATICS, APPROXIMATION(MATHEMATICS), BOUNDARY VALUE PROBLEMS, COMPUTATIONS, HYPERBOLAS, MATHEMATICAL MODELS, NUMERICAL ANALYSIS, PARTIAL DIFFERENTIAL EQUATIONS, STABILITY.

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.

Early Algebra with Graphics Software as a Type II Application of Technology

ERIC Educational Resources Information Center

Abramovich, Sergei

2006-01-01

This paper describes the use of Kid Pix-graphics software for creative activities of young children--in the context of early algebra as determined by the mathematics core curriculum of New York state. It shows how grade-two appropriate pedagogy makes it possible to bring about a qualitative change in the learning process of those commonly…

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).

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.

Application of Computer Graphics to Graphing in Algebra and Trigonometry. Final Report.

ERIC Educational Resources Information Center

Morris, J. Richard

This project was designed to improve the graphing competency of students in elementary algebra, intermediate algebra, and trigonometry courses at Virginia Commonwealth University. Computer graphics programs were designed using an Apple II Plus computer and implemented using Pascal. The software package is interactive and gives students control…

Astro Algebra [CD-ROM].

ERIC Educational Resources Information Center

1997

Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…

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…

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.

An Example of Competence-Based Learning: Use of Maxima in Linear Algebra for Engineers

ERIC Educational Resources Information Center

Diaz, Ana; Garcia, Alfonsa; de la Villa, Agustin

2011-01-01

This paper analyses the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is…

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

Carnegie Learning Curricula and Cognitive Tutor™. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2013

2013-01-01

"Carnegie Learning Curricula and Cognitive Tutor"®, published by Carnegie Learning, is a secondary math curricula that offers textbooks and interactive software to provide individualized, self-paced instruction based on student needs. The program includes pre-Algebra, Algebra I, Algebra II, and Geometry, as well as a three-course series…

ACTS: from ATLAS software towards a common track reconstruction software

NASA Astrophysics Data System (ADS)

Gumpert, C.; Salzburger, A.; Kiehn, M.; Hrdinka, J.; Calace, N.; ATLAS Collaboration

2017-10-01

Reconstruction of charged particles’ trajectories is a crucial task for most particle physics experiments. The high instantaneous luminosity achieved at the LHC leads to a high number of proton-proton collisions per bunch crossing, which has put the track reconstruction software of the LHC experiments through a thorough test. Preserving track reconstruction performance under increasingly difficult experimental conditions, while keeping the usage of computational resources at a reasonable level, is an inherent problem for many HEP experiments. Exploiting concurrent algorithms and using multivariate techniques for track identification are the primary strategies to achieve that goal. Starting from current ATLAS software, the ACTS project aims to encapsulate track reconstruction software into a generic, framework- and experiment-independent software package. It provides a set of high-level algorithms and data structures for performing track reconstruction tasks as well as fast track simulation. The software is developed with special emphasis on thread-safety to support parallel execution of the code and data structures are optimised for vectorisation to speed up linear algebra operations. The implementation is agnostic to the details of the detection technologies and magnetic field configuration which makes it applicable to many different experiments.

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.

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

Selecting reusable components using algebraic specifications

NASA Technical Reports Server (NTRS)

Eichmann, David A.

1992-01-01

A significant hurdle confronts the software reuser attempting to select candidate components from a software repository - discriminating between those components without resorting to inspection of the implementation(s). We outline a mixed classification/axiomatic approach to this problem based upon our lattice-based faceted classification technique and Guttag and Horning's algebraic specification techniques. This approach selects candidates by natural language-derived classification, by their interfaces, using signatures, and by their behavior, using axioms. We briefly outline our problem domain and related work. Lattice-based faceted classifications are described; the reader is referred to surveys of the extensive literature for algebraic specification techniques. Behavioral support for reuse queries is presented, followed by the conclusions.

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)

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

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.

The Impact of Technology-Enhanced Curriculum on Learning Advanced Algebra in US High School Classrooms

ERIC Educational Resources Information Center

Hegedus, Stephen J.; Dalton, Sara; Tapper, John R.

2015-01-01

We report on two large studies conducted in advanced algebra classrooms in the US, which evaluated the effect of replacing traditional algebra 2 curriculum with an integrated suite of dynamic interactive software, wireless networks and technology-enhanced curriculum on student learning. The first study was a cluster randomized trial and the second…

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…

ATLAS offline software performance monitoring and optimization

NASA Astrophysics Data System (ADS)

Chauhan, N.; Kabra, G.; Kittelmann, T.; Langenberg, R.; Mandrysch, R.; Salzburger, A.; Seuster, R.; Ritsch, E.; Stewart, G.; van Eldik, N.; Vitillo, R.; Atlas Collaboration

2014-06-01

In a complex multi-developer, multi-package software environment, such as the ATLAS offline framework Athena, tracking the performance of the code can be a non-trivial task in itself. In this paper we describe improvements in the instrumentation of ATLAS offline software that have given considerable insight into the performance of the code and helped to guide the optimization work. The first tool we used to instrument the code is PAPI, which is a programing interface for accessing hardware performance counters. PAPI events can count floating point operations, cycles, instructions and cache accesses. Triggering PAPI to start/stop counting for each algorithm and processed event results in a good understanding of the algorithm level performance of ATLAS code. Further data can be obtained using Pin, a dynamic binary instrumentation tool. Pin tools can be used to obtain similar statistics as PAPI, but advantageously without requiring recompilation of the code. Fine grained routine and instruction level instrumentation is also possible. Pin tools can additionally interrogate the arguments to functions, like those in linear algebra libraries, so that a detailed usage profile can be obtained. These tools have characterized the extensive use of vector and matrix operations in ATLAS tracking. Currently, CLHEP is used here, which is not an optimal choice. To help evaluate replacement libraries a testbed has been setup allowing comparison of the performance of different linear algebra libraries (including CLHEP, Eigen and SMatrix/SVector). Results are then presented via the ATLAS Performance Management Board framework, which runs daily with the current development branch of the code and monitors reconstruction and Monte-Carlo jobs. This framework analyses the CPU and memory performance of algorithms and an overview of results are presented on a web page. These tools have provided the insight necessary to plan and implement performance enhancements in ATLAS code by identifying the most common operations, with the call parameters well understood, and allowing improvements to be quantified in detail.

The Matrix Pencil and its Applications to Speech Processing

DTIC Science & Technology

2007-03-01

Elementary Linear Algebra ” 8th edition, pp. 278, 2000 John Wiley & Sons, Inc., New York [37] Wai C. Chu, “Speech Coding Algorithms”, New Jeresy: John...Ben; Daniel, James W.; “Applied Linear Algebra ”, pp. 342-345, 1988 Prentice Hall, Englewood Cliffs, NJ [35] Haykin, Simon “Applied Linear Adaptive...ABSTRACT Matrix Pencils facilitate the study of differential equations resulting from oscillating systems. Certain problems in linear ordinary

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.

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.

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)

High-performance computing on GPUs for resistivity logging of oil and gas wells

NASA Astrophysics Data System (ADS)

Glinskikh, V.; Dudaev, A.; Nechaev, O.; Surodina, I.

2017-10-01

We developed and implemented into software an algorithm for high-performance simulation of electrical logs from oil and gas wells using high-performance heterogeneous computing. The numerical solution of the 2D forward problem is based on the finite-element method and the Cholesky decomposition for solving a system of linear algebraic equations (SLAE). Software implementations of the algorithm used the NVIDIA CUDA technology and computing libraries are made, allowing us to perform decomposition of SLAE and find its solution on central processor unit (CPU) and graphics processor unit (GPU). The calculation time is analyzed depending on the matrix size and number of its non-zero elements. We estimated the computing speed on CPU and GPU, including high-performance heterogeneous CPU-GPU computing. Using the developed algorithm, we simulated resistivity data in realistic models.

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…

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.

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

A high-speed linear algebra library with automatic parallelism

NASA Technical Reports Server (NTRS)

Boucher, Michael L.

1994-01-01

Parallel or distributed processing is key to getting highest performance workstations. However, designing and implementing efficient parallel algorithms is difficult and error-prone. It is even more difficult to write code that is both portable to and efficient on many different computers. Finally, it is harder still to satisfy the above requirements and include the reliability and ease of use required of commercial software intended for use in a production environment. As a result, the application of parallel processing technology to commercial software has been extremely small even though there are numerous computationally demanding programs that would significantly benefit from application of parallel processing. This paper describes DSSLIB, which is a library of subroutines that perform many of the time-consuming computations in engineering and scientific software. DSSLIB combines the high efficiency and speed of parallel computation with a serial programming model that eliminates many undesirable side-effects of typical parallel code. The result is a simple way to incorporate the power of parallel processing into commercial software without compromising maintainability, reliability, or ease of use. This gives significant advantages over less powerful non-parallel entries in the market.

Developing Information Power Grid Based Algorithms and Software

NASA Technical Reports Server (NTRS)

Dongarra, Jack

1998-01-01

This was an exploratory study to enhance our understanding of problems involved in developing large scale applications in a heterogeneous distributed environment. It is likely that the large scale applications of the future will be built by coupling specialized computational modules together. For example, efforts now exist to couple ocean and atmospheric prediction codes to simulate a more complete climate system. These two applications differ in many respects. They have different grids, the data is in different unit systems and the algorithms for inte,-rating in time are different. In addition the code for each application is likely to have been developed on different architectures and tend to have poor performance when run on an architecture for which the code was not designed, if it runs at all. Architectural differences may also induce differences in data representation which effect precision and convergence criteria as well as data transfer issues. In order to couple such dissimilar codes some form of translation must be present. This translation should be able to handle interpolation from one grid to another as well as construction of the correct data field in the correct units from available data. Even if a code is to be developed from scratch, a modular approach will likely be followed in that standard scientific packages will be used to do the more mundane tasks such as linear algebra or Fourier transform operations. This approach allows the developers to concentrate on their science rather than becoming experts in linear algebra or signal processing. Problems associated with this development approach include difficulties associated with data extraction and translation from one module to another, module performance on different nodal architectures, and others. In addition to these data and software issues there exists operational issues such as platform stability and resource management.

A Linear Algebraic Approach to Teaching Interpolation

ERIC Educational Resources Information Center

Tassa, Tamir

2007-01-01

A novel approach for teaching interpolation in the introductory course in numerical analysis is presented. The interpolation problem is viewed as a problem in linear algebra, whence the various forms of interpolating polynomial are seen as different choices of a basis to the subspace of polynomials of the corresponding degree. This approach…

Teaching Linear Algebra: Proceeding More Efficiently by Staying Comfortably within Z

ERIC Educational Resources Information Center

Beaver, Scott

2015-01-01

For efficiency in a linear algebra course the instructor may wish to avoid the undue arithmetical distractions of rational arithmetic. In this paper we explore how to write fraction-free problems of various types including elimination, matrix inverses, orthogonality, and the (non-normalizing) Gram-Schmidt process.

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.

Inverse Modelling Problems in Linear Algebra Undergraduate Courses

ERIC Educational Resources Information Center

Martinez-Luaces, Victor E.

2013-01-01

This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…

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…

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…

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.

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)…

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…

Pre Service Teachers' Usage of Dynamic Mathematics Software

ERIC Educational Resources Information Center

Bulut, Mehmet; Bulut, Neslihan

2011-01-01

Aim of this study is about mathematics education and dynamic mathematics software. Dynamic mathematics software provides new opportunities for using both computer algebra system and dynamic geometry software. GeoGebra selected as dynamic mathematics software in this research. In this study, it is investigated that what is the usage of pre service…

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…

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…

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…

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…

Visual, Algebraic and Mixed Strategies in Visually Presented Linear Programming Problems.

ERIC Educational Resources Information Center

Shama, Gilli; Dreyfus, Tommy

1994-01-01

Identified and classified solution strategies of (n=49) 10th-grade students who were presented with linear programming problems in a predominantly visual setting in the form of a computerized game. Visual strategies were developed more frequently than either algebraic or mixed strategies. Appendix includes questionnaires. (Contains 11 references.)…

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.

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.

Suboptimal LQR-based spacecraft full motion control: Theory and experimentation

NASA Astrophysics Data System (ADS)

Guarnaccia, Leone; Bevilacqua, Riccardo; Pastorelli, Stefano P.

2016-05-01

This work introduces a real time suboptimal control algorithm for six-degree-of-freedom spacecraft maneuvering based on a State-Dependent-Algebraic-Riccati-Equation (SDARE) approach and real-time linearization of the equations of motion. The control strategy is sub-optimal since the gains of the linear quadratic regulator (LQR) are re-computed at each sample time. The cost function of the proposed controller has been compared with the one obtained via a general purpose optimal control software, showing, on average, an increase in control effort of approximately 15%, compensated by real-time implementability. Lastly, the paper presents experimental tests on a hardware-in-the-loop six-degree-of-freedom spacecraft simulator, designed for testing new guidance, navigation, and control algorithms for nano-satellites in a one-g laboratory environment. The tests show the real-time feasibility of the proposed approach.

Software Reviews.

ERIC Educational Resources Information Center

Davis, Shelly J., Ed.; Knaupp, Jon, Ed.

1984-01-01

Reviewed is computer software on: (1) classification of living things, a tutorial program for grades 5-10; and (2) polynomial practice using tiles, a drill-and-practice program for algebra students. (MNS)

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.

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.

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

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…

Carnegie Learning Curricula and Cognitive Tutor[R] Software. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2010

2010-01-01

The combination of "Carnegie Learning Curricula and Cognitive Tutor[R] Software" merges algebra textbooks with interactive software developed around an artificial intelligence model that identifies strengths and weaknesses in an individual student's mastery of mathematical concepts. The software customizes prompts to focus on areas in…

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.

Description and User Instructions for the Quaternion_to_Orbit_v3 Software

NASA Technical Reports Server (NTRS)

Strekalov, Dmitry V.; Kruizinga, Gerhard L.; Paik, Meegyeong; Yuan, Dah-Ning; Asmar, Sami W.

2012-01-01

For a given inertial frame of reference, the software combines the spacecraft orbits with the spacecraft attitude quaternions, and rotates the body-fixed reference frame of a particular spacecraft to the inertial reference frame. The conversion assumes that the two spacecraft are aligned with respect to the mutual line of sight, with a parameterized time tag. The software is implemented in Python and is completely open source. It is very versatile, and may be applied under various circumstances and for other related purposes. Based on the solid linear algebra analysis, it has an extra option for compensating the linear pitch. This software has been designed for simulation of the calibration maneuvers performed by the two spacecraft comprising the GRAIL mission to the Moon, but has potential use for other applications. In simulations of formation flights, one needs to coordinate the spacecraft orbits represented in an appropriate inertial reference frame and the spacecraft attitudes. The latter are usually given as the time series of quaternions rotating the body-fixed reference frame of a particular spacecraft to the inertial reference frame. It is often desirable to simulate the same maneuver for different segments of the orbit. It is also useful to study various maneuvers that could be performed at the same orbit segment. These two lines of study are more timeand labor-efficient if the attitude and orbit data are generated independently, so that the part of the data that has not been changed can be recycled in the course of multiple simulations.

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…

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.

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

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

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.

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.

Titration Calculations with Computer Algebra Software

ERIC Educational Resources Information Center

Lachance, Russ; Biaglow, Andrew

2012-01-01

This article examines the symbolic algebraic solution of the titration equations for a diprotic acid, as obtained using "Mathematica," "Maple," and "Mathcad." The equilibrium and conservation equations are solved symbolically by the programs to eliminate the approximations that normally would be performed by the student. Of the three programs,…

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…

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.

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.

Kinetic modeling and fitting software for interconnected reaction schemes: VisKin.

PubMed

Zhang, Xuan; Andrews, Jared N; Pedersen, Steen E

2007-02-15

Reaction kinetics for complex, highly interconnected kinetic schemes are modeled using analytical solutions to a system of ordinary differential equations. The algorithm employs standard linear algebra methods that are implemented using MatLab functions in a Visual Basic interface. A graphical user interface for simple entry of reaction schemes facilitates comparison of a variety of reaction schemes. To ensure microscopic balance, graph theory algorithms are used to determine violations of thermodynamic cycle constraints. Analytical solutions based on linear differential equations result in fast comparisons of first order kinetic rates and amplitudes as a function of changing ligand concentrations. For analysis of higher order kinetics, we also implemented a solution using numerical integration. To determine rate constants from experimental data, fitting algorithms that adjust rate constants to fit the model to imported data were implemented using the Levenberg-Marquardt algorithm or using Broyden-Fletcher-Goldfarb-Shanno methods. We have included the ability to carry out global fitting of data sets obtained at varying ligand concentrations. These tools are combined in a single package, which we have dubbed VisKin, to guide and analyze kinetic experiments. The software is available online for use on PCs.

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

Software Reviews.

ERIC Educational Resources Information Center

Bitter, Gary G., Ed.

1989-01-01

Describes three software packages: (1) "MacMendeleev"--database/graphic display for chemistry, grades 10-12, Macintosh; (2) "Geometry One: Foundations"--geometry tutorial, grades 7-12, IBM; (3) "Mathematics Exploration Toolkit"--algebra and calculus tutorial, grades 8-12, IBM. (MVL)

Software Reviews.

ERIC Educational Resources Information Center

Miller, Anne, Ed.; Radziemski, Cathy, Ed.

1988-01-01

Three pieces of computer software are described and reviewed: HyperCard, to build and use varied applications; Iggy's Gnees, for problem solving with shapes in grades kindergarten-two; and Algebra Shop, for practicing skills and problem solving. (MNS)

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).

Specter: linear deconvolution for targeted analysis of data-independent acquisition mass spectrometry proteomics.

PubMed

Peckner, Ryan; Myers, Samuel A; Jacome, Alvaro Sebastian Vaca; Egertson, Jarrett D; Abelin, Jennifer G; MacCoss, Michael J; Carr, Steven A; Jaffe, Jacob D

2018-05-01

Mass spectrometry with data-independent acquisition (DIA) is a promising method to improve the comprehensiveness and reproducibility of targeted and discovery proteomics, in theory by systematically measuring all peptide precursors in a biological sample. However, the analytical challenges involved in discriminating between peptides with similar sequences in convoluted spectra have limited its applicability in important cases, such as the detection of single-nucleotide polymorphisms (SNPs) and alternative site localizations in phosphoproteomics data. We report Specter (https://github.com/rpeckner-broad/Specter), an open-source software tool that uses linear algebra to deconvolute DIA mixture spectra directly through comparison to a spectral library, thus circumventing the problems associated with typical fragment-correlation-based approaches. We validate the sensitivity of Specter and its performance relative to that of other methods, and show that Specter is able to successfully analyze cases involving highly similar peptides that are typically challenging for DIA analysis methods.

High-Performance Mixed Models Based Genome-Wide Association Analysis with omicABEL software

PubMed Central

Fabregat-Traver, Diego; Sharapov, Sodbo Zh.; Hayward, Caroline; Rudan, Igor; Campbell, Harry; Aulchenko, Yurii; Bientinesi, Paolo

2014-01-01

To raise the power of genome-wide association studies (GWAS) and avoid false-positive results in structured populations, one can rely on mixed model based tests. When large samples are used, and when multiple traits are to be studied in the ’omics’ context, this approach becomes computationally challenging. Here we consider the problem of mixed-model based GWAS for arbitrary number of traits, and demonstrate that for the analysis of single-trait and multiple-trait scenarios different computational algorithms are optimal. We implement these optimal algorithms in a high-performance computing framework that uses state-of-the-art linear algebra kernels, incorporates optimizations, and avoids redundant computations, increasing throughput while reducing memory usage and energy consumption. We show that, compared to existing libraries, our algorithms and software achieve considerable speed-ups. The OmicABEL software described in this manuscript is available under the GNU GPL v. 3 license as part of the GenABEL project for statistical genomics at http: //www.genabel.org/packages/OmicABEL. PMID:25717363

High-Performance Mixed Models Based Genome-Wide Association Analysis with omicABEL software.

PubMed

Fabregat-Traver, Diego; Sharapov, Sodbo Zh; Hayward, Caroline; Rudan, Igor; Campbell, Harry; Aulchenko, Yurii; Bientinesi, Paolo

2014-01-01

To raise the power of genome-wide association studies (GWAS) and avoid false-positive results in structured populations, one can rely on mixed model based tests. When large samples are used, and when multiple traits are to be studied in the 'omics' context, this approach becomes computationally challenging. Here we consider the problem of mixed-model based GWAS for arbitrary number of traits, and demonstrate that for the analysis of single-trait and multiple-trait scenarios different computational algorithms are optimal. We implement these optimal algorithms in a high-performance computing framework that uses state-of-the-art linear algebra kernels, incorporates optimizations, and avoids redundant computations, increasing throughput while reducing memory usage and energy consumption. We show that, compared to existing libraries, our algorithms and software achieve considerable speed-ups. The OmicABEL software described in this manuscript is available under the GNU GPL v. 3 license as part of the GenABEL project for statistical genomics at http: //www.genabel.org/packages/OmicABEL.

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

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…

Some Issues about the Introduction of First Concepts in Linear Algebra during Tutorial Sessions at the Beginning of University

ERIC Educational Resources Information Center

Grenier-Boley, Nicolas

2014-01-01

Certain mathematical concepts were not introduced to solve a specific open problem but rather to solve different problems with the same tools in an economic formal way or to unify several approaches: such concepts, as some of those of linear algebra, are presumably difficult to introduce to students as they are potentially interwoven with many…

Introduction to COFFE: The Next-Generation HPCMP CREATE-AV CFD Solver

NASA Technical Reports Server (NTRS)

Glasby, Ryan S.; Erwin, J. Taylor; Stefanski, Douglas L.; Allmaras, Steven R.; Galbraith, Marshall C.; Anderson, W. Kyle; Nichols, Robert H.

2016-01-01

HPCMP CREATE-AV Conservative Field Finite Element (COFFE) is a modular, extensible, robust numerical solver for the Navier-Stokes equations that invokes modularity and extensibility from its first principles. COFFE implores a flexible, class-based hierarchy that provides a modular approach consisting of discretization, physics, parallelization, and linear algebra components. These components are developed with modern software engineering principles to ensure ease of uptake from a user's or developer's perspective. The Streamwise Upwind/Petrov-Galerkin (SU/PG) method is utilized to discretize the compressible Reynolds-Averaged Navier-Stokes (RANS) equations tightly coupled with a variety of turbulence models. The mathematics and the philosophy of the methodology that makes up COFFE are presented.

Computer Activities for College Algebra and Precalculus.

ERIC Educational Resources Information Center

White, Jacci Wozniak; Norwich, Vicki Howard

Mathematics software can be a great aid in understanding difficult mathematics concepts at all levels. This paper presents nine exercises on calculus concepts by using different software used in mathematics education. Each exercise includes instruction on how to use software in order to highlight a specific concept in mathematics. This paper also…

Computer-Aided College Algebra: Learning Components that Students Find Beneficial

ERIC Educational Resources Information Center

Aichele, Douglas B.; Francisco, Cynthia; Utley, Juliana; Wescoatt, Benjamin

2011-01-01

A mixed-method study was conducted during the Fall 2008 semester to better understand the experiences of students participating in computer-aided instruction of College Algebra using the software MyMathLab. The learning environment included a computer learning system for the majority of the instruction, a support system via focus groups (weekly…

Math/Measurement Literacy for Upgrading Skills of Industrial Hourly Workers. Math Manual.

ERIC Educational Resources Information Center

McMahon, Joan L.

This manual contains materials for a numeracy course for adult industrial workers. In addition to assessment tests, seven units are provided. Unit topics are whole numbers; fractions; decimals; percents, median, and range; measurement and signed numbers; ratio/proportion and introduction to algebra; and computer literacy using algebra software.…

Some Unexpected Results Using Computer Algebra Systems.

ERIC Educational Resources Information Center

Alonso, Felix; Garcia, Alfonsa; Garcia, Francisco; Hoya, Sara; Rodriguez, Gerardo; de la Villa, Agustin

2001-01-01

Shows how teachers can often use unexpected outputs from Computer Algebra Systems (CAS) to reinforce concepts and to show students the importance of thinking about how they use the software and reflecting on their results. Presents different examples where DERIVE, MAPLE, or Mathematica does not work as expected and suggests how to use them as a…

Using Group Explorer in Teaching Abstract Algebra

ERIC Educational Resources Information Center

Schubert, Claus; Gfeller, Mary; Donohue, Christopher

2013-01-01

This study explores the use of Group Explorer in an undergraduate mathematics course in abstract algebra. The visual nature of Group Explorer in representing concepts in group theory is an attractive incentive to use this software in the classroom. However, little is known about students' perceptions on this technology in learning concepts in…

Software for Training in Pre-College Mathematics

NASA Technical Reports Server (NTRS)

Shelton, Robert O.; Moebes, Travis A.; VanAlstine, Scot

2003-01-01

The Intelligent Math Tutor (IMT) is a computer program for training students in pre-college and college-level mathematics courses, including fundamentals, intermediate algebra, college algebra, and trigonometry. The IMT can be executed on a server computer for access by students via the Internet; alternatively, it can be executed on students computers equipped with compact- disk/read-only-memory (CD-ROM) drives. The IMT provides interactive exercises, assessment, tracking, and an on-line graphing calculator with algebraic-manipulation capabilities. The IMT provides an innovative combination of content, delivery mechanism, and artificial intelligence. Careful organization and presentation of the content make it possible to provide intelligent feedback to the student based on performance on exercises and tests. The tracking and feedback mechanisms are implemented within the capabilities of a commercial off-the-shelf development software tool and are written in the Unified Modeling Language to maximize reuse and minimize development cost. The graphical calculator is a standard feature of most college and pre-college algebra and trigonometry courses. Placing this functionality in a Java applet decreases the cost, provides greater capabilities, and provides an opportunity to integrate the calculator with the lessons.

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.

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.

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.

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.

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.

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.

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)

Finite-dimensional integrable systems: A collection of research problems

NASA Astrophysics Data System (ADS)

Bolsinov, A. V.; Izosimov, A. M.; Tsonev, D. M.

2017-05-01

This article suggests a series of problems related to various algebraic and geometric aspects of integrability. They reflect some recent developments in the theory of finite-dimensional integrable systems such as bi-Poisson linear algebra, Jordan-Kronecker invariants of finite dimensional Lie algebras, the interplay between singularities of Lagrangian fibrations and compatible Poisson brackets, and new techniques in projective geometry.

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

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.

Tensor Algebra Library for NVidia Graphics Processing Units

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

Liakh, Dmitry

This is a general purpose math library implementing basic tensor algebra operations on NVidia GPU accelerators. This software is a tensor algebra library that can perform basic tensor algebra operations, including tensor contractions, tensor products, tensor additions, etc., on NVidia GPU accelerators, asynchronously with respect to the CPU host. It supports a simultaneous use of multiple NVidia GPUs. Each asynchronous API function returns a handle which can later be used for querying the completion of the corresponding tensor algebra operation on a specific GPU. The tensors participating in a particular tensor operation are assumed to be stored in local RAMmore » of a node or GPU RAM. The main research area where this library can be utilized is the quantum many-body theory (e.g., in electronic structure theory).« less

Apprentissage dans un Environnement Informatique: Possibilite, Nature, Transfert des Acquis (Learning within an Information Environment: Possibilities, Native, and Transfer of Ideas).

ERIC Educational Resources Information Center

Dagher, Antoine

1996-01-01

Examines possibilities for learning offered by a piece of software, Fonctuse, likely to encourage the linking of algebraic and graphical representations of functions. Studied the influence of prior algebraic knowledge on the cognitive processes and constructions of knowledge at play in this environment. (Author/MKR)

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.

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.

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…

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 Unified Algebraic and Logic-Based Framework Towards Safe Routing Implementations

DTIC Science & Technology

2015-08-13

Software - defined Networks ( SDN ). We developed a declarative platform for implementing SDN protocols using declarative...and debugging several SDN applications. Example-based SDN synthesis. Recent emergence of software - defined networks offers an opportunity to design...domain of Software - defined Networks ( SDN ). We developed a declarative platform for implementing SDN protocols using declarative networking

R&D Project on Algebra Software Seen to Show Promise

ERIC Educational Resources Information Center

Trotter, Andrew

2007-01-01

Computer software that shows students visual models of mathematical concepts--and lets them manipulate those models by doing math--has a certain intuitive appeal. Now, recent research on SimCalc Mathworlds, one of the pioneering examples of such software, is providing some of the best evidence so far that the approach can lead to gains in student…

A Symbolic Dance: The Interplay between Movement, Notation, and Mathematics on a Journey toward Solving Equations

ERIC Educational Resources Information Center

Hewitt, Dave

2014-01-01

This article analyzes the use of the software Grid Algebra with a mixed ability class of 21 nine-to-ten-year-old students who worked with complex formal notation involving all four arithmetic operations. Unlike many other models to support learning, Grid Algebra has formal notation ever present and allows students to "look through" that…

Introducing Computer Algebra to School Teachers of Mathematics

ERIC Educational Resources Information Center

Man, Yiu-Kwong

2007-01-01

Since the last decade, the use of computer algebra systems at the Hong Kong school level is still very limited. Among various reasons behind, the lack of exposure of this kind of software to local school teachers should be taken into account. In this article, we describe how to introduce MAPLE in a BEd module of a local teacher-training programme.…

Solving a System of Nonlinear Algebraic Equations You Only Get Error Messages--What to Do Next?

ERIC Educational Resources Information Center

Shacham, Mordechai; Brauner, Neima

2017-01-01

Chemical engineering problems often involve the solution of systems of nonlinear algebraic equations (NLE). There are several software packages that can be used for solving NLE systems, but they may occasionally fail, especially in cases where the mathematical model contains discontinuities and/or regions where some of the functions are undefined.…

Exploring the concept of interaction computing through the discrete algebraic analysis of the Belousov-Zhabotinsky reaction.

PubMed

Dini, Paolo; Nehaniv, Chrystopher L; Egri-Nagy, Attila; Schilstra, Maria J

2013-05-01

Interaction computing (IC) aims to map the properties of integrable low-dimensional non-linear dynamical systems to the discrete domain of finite-state automata in an attempt to reproduce in software the self-organizing and dynamically stable properties of sub-cellular biochemical systems. As the work reported in this paper is still at the early stages of theory development it focuses on the analysis of a particularly simple chemical oscillator, the Belousov-Zhabotinsky (BZ) reaction. After retracing the rationale for IC developed over the past several years from the physical, biological, mathematical, and computer science points of view, the paper presents an elementary discussion of the Krohn-Rhodes decomposition of finite-state automata, including the holonomy decomposition of a simple automaton, and of its interpretation as an abstract positional number system. The method is then applied to the analysis of the algebraic properties of discrete finite-state automata derived from a simplified Petri net model of the BZ reaction. In the simplest possible and symmetrical case the corresponding automaton is, not surprisingly, found to contain exclusively cyclic groups. In a second, asymmetrical case, the decomposition is much more complex and includes five different simple non-abelian groups whose potential relevance arises from their ability to encode functionally complete algebras. The possible computational relevance of these findings is discussed and possible conclusions are drawn. Copyright © 2013 Elsevier Ireland Ltd. All rights reserved.

Using Spherical-Harmonics Expansions for Optics Surface Reconstruction from Gradients.

PubMed

Solano-Altamirano, Juan Manuel; Vázquez-Otero, Alejandro; Khikhlukha, Danila; Dormido, Raquel; Duro, Natividad

2017-11-30

In this paper, we propose a new algorithm to reconstruct optics surfaces (aka wavefronts) from gradients, defined on a circular domain, by means of the Spherical Harmonics. The experimental results indicate that this algorithm renders the same accuracy, compared to the reconstruction based on classical Zernike polynomials, using a smaller number of polynomial terms, which potentially speeds up the wavefront reconstruction. Additionally, we provide an open-source C++ library, released under the terms of the GNU General Public License version 2 (GPLv2), wherein several polynomial sets are coded. Therefore, this library constitutes a robust software alternative for wavefront reconstruction in a high energy laser field, optical surface reconstruction, and, more generally, in surface reconstruction from gradients. The library is a candidate for being integrated in control systems for optical devices, or similarly to be used in ad hoc simulations. Moreover, it has been developed with flexibility in mind, and, as such, the implementation includes the following features: (i) a mock-up generator of various incident wavefronts, intended to simulate the wavefronts commonly encountered in the field of high-energy lasers production; (ii) runtime selection of the library in charge of performing the algebraic computations; (iii) a profiling mechanism to measure and compare the performance of different steps of the algorithms and/or third-party linear algebra libraries. Finally, the library can be easily extended to include additional dependencies, such as porting the algebraic operations to specific architectures, in order to exploit hardware acceleration features.

Using Spherical-Harmonics Expansions for Optics Surface Reconstruction from Gradients

PubMed Central

Solano-Altamirano, Juan Manuel; Khikhlukha, Danila

2017-01-01

In this paper, we propose a new algorithm to reconstruct optics surfaces (aka wavefronts) from gradients, defined on a circular domain, by means of the Spherical Harmonics. The experimental results indicate that this algorithm renders the same accuracy, compared to the reconstruction based on classical Zernike polynomials, using a smaller number of polynomial terms, which potentially speeds up the wavefront reconstruction. Additionally, we provide an open-source C++ library, released under the terms of the GNU General Public License version 2 (GPLv2), wherein several polynomial sets are coded. Therefore, this library constitutes a robust software alternative for wavefront reconstruction in a high energy laser field, optical surface reconstruction, and, more generally, in surface reconstruction from gradients. The library is a candidate for being integrated in control systems for optical devices, or similarly to be used in ad hoc simulations. Moreover, it has been developed with flexibility in mind, and, as such, the implementation includes the following features: (i) a mock-up generator of various incident wavefronts, intended to simulate the wavefronts commonly encountered in the field of high-energy lasers production; (ii) runtime selection of the library in charge of performing the algebraic computations; (iii) a profiling mechanism to measure and compare the performance of different steps of the algorithms and/or third-party linear algebra libraries. Finally, the library can be easily extended to include additional dependencies, such as porting the algebraic operations to specific architectures, in order to exploit hardware acceleration features. PMID:29189722

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.

A linear programming manual

NASA Technical Reports Server (NTRS)

Tuey, R. C.

1972-01-01

Computer solutions of linear programming problems are outlined. Information covers vector spaces, convex sets, and matrix algebra elements for solving simultaneous linear equations. Dual problems, reduced cost analysis, ranges, and error analysis are illustrated.

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

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

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.

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.

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.

Three-dimensional polarization algebra for all polarization sensitive optical systems.

PubMed

Li, Yahong; Fu, Yuegang; Liu, Zhiying; Zhou, Jianhong; Bryanston-Cross, P J; Li, Yan; He, Wenjun

2018-05-28

Using three-dimensional (3D) coherency vector (9 × 1), we develop a new 3D polarization algebra to calculate the polarization properties of all polarization sensitive optical systems, especially when the incident optical field is partially polarized or un-polarized. The polarization properties of a high numerical aperture (NA) microscope objective (NA = 1.25 immersed in oil) are analyzed based on the proposed 3D polarization algebra. Correspondingly, the polarization simulation of this high NA optical system is performed by the commercial software VirtualLAB Fusion. By comparing the theoretical calculations with polarization simulations, a perfect matching relation is obtained, which demonstrates that this 3D polarization algebra is valid to quantify the 3D polarization properties for all polarization sensitive optical systems.

Symbolic Algebra Development for Higher-Order Electron Propagator Formulation and Implementation.

PubMed

Tamayo-Mendoza, Teresa; Flores-Moreno, Roberto

2014-06-10

Through the use of symbolic algebra, implemented in a program, the algebraic expression of the elements of the self-energy matrix for the electron propagator to different orders were obtained. In addition, a module for the software package Lowdin was automatically generated. Second- and third-order electron propagator results have been calculated to test the correct operation of the program. It was found that the Fortran 90 modules obtained automatically with our algorithm succeeded in calculating ionization energies with the second- and third-order electron propagator in the diagonal approximation. The strategy for the development of this symbolic algebra program is described in detail. This represents a solid starting point for the automatic derivation and implementation of higher-order electron propagator methods.

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.

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

Methodology and application of high performance electrostatic field simulation in the KATRIN experiment

NASA Astrophysics Data System (ADS)

Corona, Thomas

The Karlsruhe Tritium Neutrino (KATRIN) experiment is a tritium beta decay experiment designed to make a direct, model independent measurement of the electron neutrino mass. The experimental apparatus employs strong ( O[T]) magnetostatic and (O[10 5 V/m]) electrostatic fields in regions of ultra high (O[10-11 mbar]) vacuum in order to obtain precise measurements of the electron energy spectrum near the endpoint of tritium beta-decay. The electrostatic fields in KATRIN are formed by multiscale electrode geometries, necessitating the development of high performance field simulation software. To this end, we present a Boundary Element Method (BEM) with analytic boundary integral terms in conjunction with the Robin Hood linear algebraic solver, a nonstationary successive subspace correction (SSC) method. We describe an implementation of these techniques for high performance computing environments in the software KEMField, along with the geometry modeling and discretization software KGeoBag. We detail the application of KEMField and KGeoBag to KATRIN's spectrometer and detector sections, and demonstrate its use in furthering several of KATRIN's scientific goals. Finally, we present the results of a measurement designed to probe the electrostatic profile of KATRIN's main spectrometer in comparison to simulated results.

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.

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

Addressing the United States Navy Need for Software Engineering Education

DTIC Science & Technology

1999-09-01

taught in MA 1996 (5 - 0). Precalculus review, complex numbers and algebra, complex plane, DeMovire’s Theorem, matrix algebra, LU decomposition...This course was designed for the METOC and Combat Systems curricula. PREREQUISITE: Precalculus mathematics. MA1996 MATHEMATICS FOR SCIENTISTS AND...description for MAI995 (5 - 0). This course was designed for the METOC and Combat Systems curricula. PREREQUISITE: Precalculus mathematics. PHYSICS/SYSTEMS

Development and Initial Evaluation of the ClearSpeak Style for Automated Speaking of Algebra. Research Report No. RR-16-23

ERIC Educational Resources Information Center

Frankel, Lois; Brownstein, Beth; Soiffer, Neil; Hansen, Eric

2016-01-01

The work described in this report is the first phase of a project to provide easy-to-use tools for authoring and rendering secondary-school algebra-level math expressions in synthesized speech that is useful for students with blindness or low vision. This report describes the initial development, software implementation, and evaluation of the…

Development and Initial Evaluation of the ClearSpeak Style for Automated Speaking of Algebra. Research Report. ETS RR-16-23

ERIC Educational Resources Information Center

Frankel, Lois; Brownstein, Beth; Soiffer, Neil; Hansen, Eric

2016-01-01

The work described in this report is the first phase of a project to provide easy-to-use tools for authoring and rendering secondary-school algebra-level math expressions in synthesized speech that is useful for students with blindness or low vision. This report describes the initial development, software implementation, and evaluation of the…

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.

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…

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.

Problems Relating Mathematics and Science in the High School.

ERIC Educational Resources Information Center

Morrow, Richard; Beard, Earl

This document contains various science problems which require a mathematical solution. The problems are arranged under two general areas. The first (algebra I) contains biology, chemistry, and physics problems which require solutions related to linear equations, exponentials, and nonlinear equations. The second (algebra II) contains physics…

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.

Exciting Normal Distribution

ERIC Educational Resources Information Center

Fuchs, Karl Josef; Simonovits, Reinhard; Thaller, Bernd

2008-01-01

This paper describes a high school project where the mathematics teaching and learning software M@th Desktop (MD) based on the Computer Algebra System Mathematica was used for symbolical and numerical calculations and for visualisation. The mathematics teaching and learning software M@th Desktop 2.0 (MD) contains the modules Basics including tools…

A Simple Interactive Software Package for Plotting, Animating, and Calculating

ERIC Educational Resources Information Center

Engelhardt, Larry

2012-01-01

We introduce a new open source (free) software package that provides a simple, highly interactive interface for carrying out certain mathematical tasks that are commonly encountered in physics. These tasks include plotting and animating functions, solving systems of coupled algebraic equations, and basic calculus (differentiating and integrating…

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.

The SoRReL papers: Recent publications of the Software Reuse Repository Lab

NASA Technical Reports Server (NTRS)

Eichmann, David A. (Editor)

1992-01-01

The entire publication is presented of some of the papers recently published by the SoRReL. Some typical titles are as follows: Design of a Lattice-Based Faceted Classification System; A Hybrid Approach to Software Reuse Repository Retrieval; Selecting Reusable Components Using Algebraic Specifications; Neural Network-Based Retrieval from Reuse Repositories; and A Neural Net-Based Approach to Software Metrics.

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…

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.

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…

Further studies on stability analysis of nonlinear Roesser-type two-dimensional systems

NASA Astrophysics Data System (ADS)

Dai, Xiao-Lin

2014-04-01

This paper is concerned with further relaxations of the stability analysis of nonlinear Roesser-type two-dimensional (2D) systems in the Takagi-Sugeno fuzzy form. To achieve the goal, a novel slack matrix variable technique, which is homogenous polynomially parameter-dependent on the normalized fuzzy weighting functions with arbitrary degree, is developed and the algebraic properties of the normalized fuzzy weighting functions are collected into a set of augmented matrices. Consequently, more information about the normalized fuzzy weighting functions is involved and the relaxation quality of the stability analysis is significantly improved. Moreover, the obtained result is formulated in the form of linear matrix inequalities, which can be easily solved via standard numerical software. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed result.

DCA_Tools

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

Summers, Michael S

2017-11-08

HPC software for ab-initio, condensed-matter physics, quantum mechanics calculations needs to be built on top of well tested libraries some of which address requirements unique to the programming domain. During the development of the DCA++ code, that we use in our research, we have developed a collection of libraries that may be of use to other computational scientists working in the same or similar domains. The libraries include: a) a pythonic input-language system, b) tensors whose shape is constructed from generalized dimension objects such at time domains. frequency domains, momentum domains, vertex domains et. al. and c) linear algebra operationsmore » that resolve to BLA/LAPACK operations when possible. This supports the implementation of Greens functions and operations on them such as are used in condensed matter physics.« less

The Seismic Tool-Kit (STK): an open source software for seismology and signal processing.

NASA Astrophysics Data System (ADS)

Reymond, Dominique

2016-04-01

We present an open source software project (GNU public license), named STK: Seismic ToolKit, that is dedicated mainly for seismology and signal processing. The STK project that started in 2007, is hosted by SourceForge.net, and count more than 19 500 downloads at the date of writing. The STK project is composed of two main branches: First, a graphical interface dedicated to signal processing (in the SAC format (SAC_ASCII and SAC_BIN): where the signal can be plotted, zoomed, filtered, integrated, derivated, ... etc. (a large variety of IFR and FIR filter is proposed). The estimation of spectral density of the signal are performed via the Fourier transform, with visualization of the Power Spectral Density (PSD) in linear or log scale, and also the evolutive time-frequency representation (or sonagram). The 3-components signals can be also processed for estimating their polarization properties, either for a given window, or either for evolutive windows along the time. This polarization analysis is useful for extracting the polarized noises, differentiating P waves, Rayleigh waves, Love waves, ... etc. Secondly, a panel of Utilities-Program are proposed for working in a terminal mode, with basic programs for computing azimuth and distance in spherical geometry, inter/auto-correlation, spectral density, time-frequency for an entire directory of signals, focal planes, and main components axis, radiation pattern of P waves, Polarization analysis of different waves (including noize), under/over-sampling the signals, cubic-spline smoothing, and linear/non linear regression analysis of data set. A MINimum library of Linear AlGebra (MIN-LINAG) is also provided for computing the main matrix process like: QR/QL decomposition, Cholesky solve of linear system, finding eigen value/eigen vectors, QR-solve/Eigen-solve of linear equations systems ... etc. STK is developed in C/C++, mainly under Linux OS, and it has been also partially implemented under MS-Windows. Usefull links: http://sourceforge.net/projects/seismic-toolkit/ http://sourceforge.net/p/seismic-toolkit/wiki/browse_pages/

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

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.

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.

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.

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.

Reusable Software Component Retrieval via Normalized Algebraic Specifications

DTIC Science & Technology

1991-12-01

outputs. In fact, this method of query is simpler for matching since it relieves the system from the burden of generating a test set. Eichmann [Eich9l...September 1991. [Eich9l] Eichmann , David A., "Selecting Reusable Components Using Algebraic Specifications", Proceedings of the Second International...Technology Atlanta, Georgia 30332-0800 12. Dr. David Eichmann 1 Department of Statistics and Computer Science Knapp Hall West Virginia University Morgantown, West Virginia 26506 226

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…

Differential Geometry and Lie Groups for Physicists

NASA Astrophysics Data System (ADS)

Fecko, Marián.

2006-10-01

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

Differential Geometry and Lie Groups for Physicists

NASA Astrophysics Data System (ADS)

Fecko, Marián.

2011-03-01

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

Global identifiability of linear compartmental models--a computer algebra algorithm.

PubMed

Audoly, S; D'Angiò, L; Saccomani, M P; Cobelli, C

1998-01-01

A priori global identifiability deals with the uniqueness of the solution for the unknown parameters of a model and is, thus, a prerequisite for parameter estimation of biological dynamic models. Global identifiability is however difficult to test, since it requires solving a system of algebraic nonlinear equations which increases both in nonlinearity degree and number of terms and unknowns with increasing model order. In this paper, a computer algebra tool, GLOBI (GLOBal Identifiability) is presented, which combines the topological transfer function method with the Buchberger algorithm, to test global identifiability of linear compartmental models. GLOBI allows for the automatic testing of a priori global identifiability of general structure compartmental models from general multi input-multi output experiments. Examples of usage of GLOBI to analyze a priori global identifiability of some complex biological compartmental models are provided.

Linear-scaling density-functional simulations of charged point defects in Al2O3 using hierarchical sparse matrix algebra.

PubMed

Hine, N D M; Haynes, P D; Mostofi, A A; Payne, M C

2010-09-21

We present calculations of formation energies of defects in an ionic solid (Al(2)O(3)) extrapolated to the dilute limit, corresponding to a simulation cell of infinite size. The large-scale calculations required for this extrapolation are enabled by developments in the approach to parallel sparse matrix algebra operations, which are central to linear-scaling density-functional theory calculations. The computational cost of manipulating sparse matrices, whose sizes are determined by the large number of basis functions present, is greatly improved with this new approach. We present details of the sparse algebra scheme implemented in the ONETEP code using hierarchical sparsity patterns, and demonstrate its use in calculations on a wide range of systems, involving thousands of atoms on hundreds to thousands of parallel processes.

Problem Solving in Calculus with Symbolic Geometry and CAS

ERIC Educational Resources Information Center

Todd, Philip; Wiechmann, James

2008-01-01

Computer algebra systems (CAS) have been around for a number of years, as has dynamic geometry. Symbolic geometry software is new. It bears a superficial similarity to dynamic geometry software, but differs in that problems may be set up involving symbolic variables and constants, and measurements are given as symbolic expressions. Mathematical…

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

Vertex Algebras W(p)Am and W(p)Dm and Constant Term Identities

NASA Astrophysics Data System (ADS)

Adamović, Dražen; Lin, Xianzu; Milas, Antun

2015-03-01

We consider AD-type orbifolds of the triplet vertex algebras W(p) extending the well-known c=1 orbifolds of lattice vertex algebras. We study the structure of Zhu's algebras A(W(p)^{A_m}) and A(W(p)^{D_m}), where A_m and D_m are cyclic and dihedral groups, respectively. A combinatorial algorithm for classification of irreducible W(p)^Γ-modules is developed, which relies on a family of constant term identities and properties of certain polynomials based on constant terms. All these properties can be checked for small values of m and p with a computer software. As a result, we argue that if certain constant term properties hold, the irreducible modules constructed in [Commun. Contemp. Math. 15 (2013), 1350028, 30 pages; Internat. J. Math. 25 (2014), 1450001, 34 pages] provide a complete list of irreducible W(p)^{A_m} and W(p)^{D_m}-modules. This paper is a continuation of our previous work on the ADE subalgebras of the triplet vertex algebra W(p).

Line defect Schur indices, Verlinde algebras and U(1) r fixed points

NASA Astrophysics Data System (ADS)

Neitzke, Andrew; Yan, Fei

2017-11-01

Given an N=2 superconformal field theory, we reconsider the Schur index ℐ L ( q) in the presence of a half line defect L. Recently Cordova-Gaiotto-Shao found that ℐ L ( q) admits an expansion in terms of characters of the chiral algebra A introduced by Beem et al., with simple coefficients υ L, β ( q). We report a puzzling new feature of this expansion: the q → 1 limit of the coefficients υ L, β ( q) is linearly related to the vacuum expectation values 〈 L〉 in U(1) r -invariant vacua of the theory compactified on S 1. This relation can be expressed algebraically as a commutative diagram involving three algebras: the algebra generated by line defects, the algebra of functions on U(1) r -invariant vacua, and a Verlindelike algebra associated to A . Our evidence is experimental, by direct computation in the Argyres-Douglas theories of type ( A 1, A 2), ( A 1, A 4), ( A 1, A 6), ( A 1, D 3) and ( A 1, D 5). In the latter two theories, which have flavor symmetries, the Verlinde-like algebra which appears is a new deformation of algebras previously considered.

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…

A Third Grader's Way of Thinking about Linear Function Tables

ERIC Educational Resources Information Center

Martinez, Mara; Brizuela, Barbara M.

2006-01-01

This paper is inscribed within the research effort to produce evidence regarding primary school students' learning of algebra. Given the results obtained so far in the research community, we are convinced that young elementary school students can successfully learn algebra. Moreover, children this young can make use of different representational…

Unification of the general non-linear sigma model and the Virasoro master equation

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

Boer, J. de; Halpern, M.B.

1997-06-01

The Virasoro master equation describes a large set of conformal field theories known as the affine-Virasoro constructions, in the operator algebra (affinie Lie algebra) of the WZW model, while the einstein equations of the general non-linear sigma model describe another large set of conformal field theories. This talk summarizes recent work which unifies these two sets of conformal field theories, together with a presumable large class of new conformal field theories. The basic idea is to consider spin-two operators of the form L{sub ij}{partial_derivative}x{sup i}{partial_derivative}x{sup j} in the background of a general sigma model. The requirement that these operators satisfymore » the Virasoro algebra leads to a set of equations called the unified Einstein-Virasoro master equation, in which the spin-two spacetime field L{sub ij} cuples to the usual spacetime fields of the sigma model. The one-loop form of this unified system is presented, and some of its algebraic and geometric properties are discussed.« less

Developing integrated parametric planning models for budgeting and managing complex projects

NASA Technical Reports Server (NTRS)

Etnyre, Vance A.; Black, Ken U.

1988-01-01

The applicability of integrated parametric models for the budgeting and management of complex projects is investigated. Methods for building a very flexible, interactive prototype for a project planning system, and software resources available for this purpose, are discussed and evaluated. The prototype is required to be sensitive to changing objectives, changing target dates, changing costs relationships, and changing budget constraints. To achieve the integration of costs and project and task durations, parametric cost functions are defined by a process of trapezoidal segmentation, where the total cost for the project is the sum of the various project cost segments, and each project cost segment is the integral of a linearly segmented cost loading function over a specific interval. The cost can thus be expressed algebraically. The prototype was designed using Lotus-123 as the primary software tool. This prototype implements a methodology for interactive project scheduling that provides a model of a system that meets most of the goals for the first phase of the study and some of the goals for the second phase.

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

NASA Astrophysics Data System (ADS)

Abdelaziz, Y.; Maillard, J.-M.

2017-05-01

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same {}_2F1 hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus {}_2F1 hypergeometric function example. We then focus on identities relating the same {}_2F1 hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale’s paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to {}_3F2 , hypergeometric functions, and show that one just reduces to the previous {}_2F1 cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a _2F1 hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or {}_2F1 hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

Compositional Specification of Software Architecture

NASA Technical Reports Server (NTRS)

Penix, John; Lau, Sonie (Technical Monitor)

1998-01-01

This paper describes our experience using parameterized algebraic specifications to model properties of software architectures. The goal is to model the decomposition of requirements independent of the style used to implement the architecture. We begin by providing an overview of the role of architecture specification in software development. We then describe how architecture specifications are build up from component and connector specifications and give an overview of insights gained from a case study used to validate the method.

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.

Curve Fitting via the Criterion of Least Squares. Applications of Algebra and Elementary Calculus to Curve Fitting. [and] Linear Programming in Two Dimensions: I. Applications of High School Algebra to Operations Research. Modules and Monographs in Undergraduate Mathematics and Its Applications Project. UMAP Units 321, 453.

ERIC Educational Resources Information Center

Alexander, John W., Jr.; Rosenberg, Nancy S.

This document consists of two modules. The first of these views applications of algebra and elementary calculus to curve fitting. The user is provided with information on how to: 1) construct scatter diagrams; 2) choose an appropriate function to fit specific data; 3) understand the underlying theory of least squares; 4) use a computer program to…

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

Carpenter, J.A.

This report is a sequel to ORNL/CSD-106 in the ongoing supplements to Professor A.S. Householder's KWIC Index for Numerical Algebra. Beginning with the previous supplement, the subject has been restricted to Numerical Linear Algebra, roughly characterized by the American Mathematical Society's classification sections 15 and 65F but with little coverage of infinite matrices, matrices over fields of characteristics other than zero, operator theory, optimization and those parts of matrix theory primarily combinatorial in nature. Some consideration is given to the uses of graph theory in Numerical Linear Algebra, particularly with respect to algorithms for sparse matrix computations. The period coveredmore » by this report is roughly the calendar year 1982 as measured by the appearance of the articles in the American Mathematical Society's Contents of Mathematical Publications lagging actual appearance dates by up to nearly half a year. The review citations are limited to the Mathematical Reviews (MR).« less

ADART: an adaptive algebraic reconstruction algorithm for discrete tomography.

PubMed

Maestre-Deusto, F Javier; Scavello, Giovanni; Pizarro, Joaquín; Galindo, Pedro L

2011-08-01

In this paper we suggest an algorithm based on the Discrete Algebraic Reconstruction Technique (DART) which is capable of computing high quality reconstructions from substantially fewer projections than required for conventional continuous tomography. Adaptive DART (ADART) goes a step further than DART on the reduction of the number of unknowns of the associated linear system achieving a significant reduction in the pixel error rate of reconstructed objects. The proposed methodology automatically adapts the border definition criterion at each iteration, resulting in a reduction of the number of pixels belonging to the border, and consequently of the number of unknowns in the general algebraic reconstruction linear system to be solved, being this reduction specially important at the final stage of the iterative process. Experimental results show that reconstruction errors are considerably reduced using ADART when compared to original DART, both in clean and noisy environments.

Accelerate quasi Monte Carlo method for solving systems of linear algebraic equations through shared memory

NASA Astrophysics Data System (ADS)

Lai, Siyan; Xu, Ying; Shao, Bo; Guo, Menghan; Lin, Xiaola

2017-04-01

In this paper we study on Monte Carlo method for solving systems of linear algebraic equations (SLAE) based on shared memory. Former research demostrated that GPU can effectively speed up the computations of this issue. Our purpose is to optimize Monte Carlo method simulation on GPUmemoryachritecture specifically. Random numbers are organized to storein shared memory, which aims to accelerate the parallel algorithm. Bank conflicts can be avoided by our Collaborative Thread Arrays(CTA)scheme. The results of experiments show that the shared memory based strategy can speed up the computaions over than 3X at most.

Negative base encoding in optical linear algebra processors

NASA Technical Reports Server (NTRS)

Perlee, C.; Casasent, D.

1986-01-01

In the digital multiplication by analog convolution algorithm, the bits of two encoded numbers are convolved to form the product of the two numbers in mixed binary representation; this output can be easily converted to binary. Attention is presently given to negative base encoding, treating base -2 initially, and then showing that the negative base system can be readily extended to any radix. In general, negative base encoding in optical linear algebra processors represents a more efficient technique than either sign magnitude or 2's complement encoding, when the additions of digitally encoded products are performed in parallel.

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…

Functional Thinking Ways in Relation to Linear Function Tables of Elementary School Students

ERIC Educational Resources Information Center

Tanisli, Dilek

2011-01-01

One of the basic components of algebraic thinking is functional thinking. Functional thinking involves focusing on the relationship between two (or more) varying quantities and such thinking facilitates the studies on both algebra and the notion of function. The development of functional thinking of students should start in the early grades and it…

Diagonalization and Jordan Normal Form--Motivation through "Maple"[R

ERIC Educational Resources Information Center

Glaister, P.

2009-01-01

Following an introduction to the diagonalization of matrices, one of the more difficult topics for students to grasp in linear algebra is the concept of Jordan normal form. In this note, we show how the important notions of diagonalization and Jordan normal form can be introduced and developed through the use of the computer algebra package…

An Introduction to the Profound Potential of Connected Algebra Activities: Issues of Representation, Engagement and Pedagogy

ERIC Educational Resources Information Center

Hegedus, Stephen J.; Kaput, James J.

2004-01-01

We present two vignettes of classroom episodes that exemplify new activity structures for introducing core algebra ideas such as linear functions, slope as rate and parametric variation within a new educational technology environment that combines two kinds of classroom technology affordances, one based in dynamic representation and the other…

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

Hart, William; Laird, Carl; Siirola, John

Pyomo provides a rich software environment for formulating and analyzing optimization applications. Pyomo supports the algebraic specification of complex sets of objectives and constraints, which enables optimization solvers to exploit problem structure to efficiently perform optimization.

Improved parallel data partitioning by nested dissection with applications to information retrieval.

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

Wolf, Michael M.; Chevalier, Cedric; Boman, Erik Gunnar

The computational work in many information retrieval and analysis algorithms is based on sparse linear algebra. Sparse matrix-vector multiplication is a common kernel in many of these computations. Thus, an important related combinatorial problem in parallel computing is how to distribute the matrix and the vectors among processors so as to minimize the communication cost. We focus on minimizing the total communication volume while keeping the computation balanced across processes. In [1], the first two authors presented a new 2D partitioning method, the nested dissection partitioning algorithm. In this paper, we improve on that algorithm and show that it ismore » a good option for data partitioning in information retrieval. We also show partitioning time can be substantially reduced by using the SCOTCH software, and quality improves in some cases, too.« less

Automated speech understanding: the next generation

NASA Astrophysics Data System (ADS)

Picone, J.; Ebel, W. J.; Deshmukh, N.

1995-04-01

Modern speech understanding systems merge interdisciplinary technologies from Signal Processing, Pattern Recognition, Natural Language, and Linguistics into a unified statistical framework. These systems, which have applications in a wide range of signal processing problems, represent a revolution in Digital Signal Processing (DSP). Once a field dominated by vector-oriented processors and linear algebra-based mathematics, the current generation of DSP-based systems rely on sophisticated statistical models implemented using a complex software paradigm. Such systems are now capable of understanding continuous speech input for vocabularies of several thousand words in operational environments. The current generation of deployed systems, based on small vocabularies of isolated words, will soon be replaced by a new technology offering natural language access to vast information resources such as the Internet, and provide completely automated voice interfaces for mundane tasks such as travel planning and directory assistance.

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.

Using computer algebra and SMT-solvers to analyze a mathematical model of cholera propagation

NASA Astrophysics Data System (ADS)

Trujillo Arredondo, Mariana

2014-06-01

We analyze a mathematical model for the transmission of cholera. The model is already defined and involves variables such as the pathogen agent, which in this case is the bacterium Vibrio cholera, and the human population. The human population is divided into three classes: susceptible, infectious and removed. Using Computer Algebra, specifically Maple we obtain two equilibrium states: the disease free state and the endemic state. Using Maple it is possible to prove that the disease free state is locally asymptotically stable if and only if R0 < 1. Using Maple it is possible to prove that the endemic equilibrium state is locally stable when it exists, it is to say when R0 > 1. Using the package Red-Log of the Computer algebra system Reduce and the SMT-Solver Z3Py it is possible to obtain numerical conditions for the model. The formula for the basic reproductive number makes a synthesis with all epidemic parameters in the model. Also it is possible to make numerical simulations which are very illustrative about the epidemic patters that are expected to be observed in real situations. We claim that these kinds of software are very useful in the analysis of epidemic models given that the symbolic computation provides algebraic formulas for the basic reproductive number and such algebraic formulas are very useful to derive control measures. For other side, computer algebra software is a powerful tool to make the stability analysis for epidemic models given that the all steps in the stability analysis can be made automatically: finding the equilibrium points, computing the jacobian, computing the characteristic polynomial for the jacobian, and applying the Routh-Hurwitz theorem to the characteristic polynomial. Finally, using SMT-Solvers is possible to make automatically checks of satisfiability, validity and quantifiers elimination being these computations very useful to analyse complicated epidemic models.

Proceedings of the International Conference on Algebraic Methodology and Software Technology (2nd) Held in Iowa City, Iowa on May 22-25, 1991.

DTIC Science & Technology

1992-05-25

d’Etat, University of Paris-Sud. [Boudol 84] G. Boudol, An asynchronous calculus MEIJE, in NATO summer school, La - Colle - sur - Loup , France (1984). [Da...on the propositional p-calculus. Theoretical Comput. Sci., 27:333-354, 1983. [12] M. Nivat. Sur la synchronisation des processus. Revue Technique...Meulen, E.A. Deriving In- Traynor, 0., de la Cruz, P., Uniform (Meta- ) De- cremental Implementations from Algebraic Specifica- velopment in the PROSPECTRA

Security analysis of boolean algebra based on Zhang-Wang digital signature scheme

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

Zheng, Jinbin, E-mail: jbzheng518@163.com

2014-10-06

In 2005, Zhang and Wang proposed an improvement signature scheme without using one-way hash function and message redundancy. In this paper, we show that this scheme exits potential safety concerns through the analysis of boolean algebra, such as bitwise exclusive-or, and point out that mapping is not one to one between assembly instructions and machine code actually by means of the analysis of the result of the assembly program segment, and which possibly causes safety problems unknown to the software.

Adaptive Identification by Systolic Arrays.

DTIC Science & Technology

1987-12-01

BIBLIOGRIAPHY Anton , Howard, Elementary Linear Algebra , John Wiley & Sons, 19S4. Cristi, Roberto, A Parallel Structure Jor Adaptive Pole Placement...10 11. SYSTEM IDENTIFICATION M*YETHODS ....................... 12 A. LINEAR SYSTEM MODELING ......................... 12 B. SOLUTION OF SYSTEMS OF... LINEAR EQUATIONS ......... 13 C. QR DECOMPOSITION ................................ 14 D. RECURSIVE LEAST SQUARES ......................... 16 E. BLOCK

PONS2train: tool for testing the MLP architecture and local traning methods for runoff forecast

NASA Astrophysics Data System (ADS)

Maca, P.; Pavlasek, J.; Pech, P.

2012-04-01

The purpose of presented poster is to introduce the PONS2train developed for runoff prediction via multilayer perceptron - MLP. The software application enables the implementation of 12 different MLP's transfer functions, comparison of 9 local training algorithms and finally the evaluation the MLP performance via 17 selected model evaluation metrics. The PONS2train software is written in C++ programing language. Its implementation consists of 4 classes. The NEURAL_NET and NEURON classes implement the MLP, the CRITERIA class estimates model evaluation metrics and for model performance evaluation via testing and validation datasets. The DATA_PATTERN class prepares the validation, testing and calibration datasets. The software application uses the LAPACK, BLAS and ARMADILLO C++ linear algebra libraries. The PONS2train implements the first order local optimization algorithms: standard on-line and batch back-propagation with learning rate combined with momentum and its variants with the regularization term, Rprop and standard batch back-propagation with variable momentum and learning rate. The second order local training algorithms represents: the Levenberg-Marquardt algorithm with and without regularization and four variants of scaled conjugate gradients. The other important PONS2train features are: the multi-run, the weight saturation control, early stopping of trainings, and the MLP weights analysis. The weights initialization is done via two different methods: random sampling from uniform distribution on open interval or Nguyen Widrow method. The data patterns can be transformed via linear and nonlinear transformation. The runoff forecast case study focuses on PONS2train implementation and shows the different aspects of the MLP training, the MLP architecture estimation, the neural network weights analysis and model uncertainty estimation.

Linearized gravity in terms of differential forms

NASA Astrophysics Data System (ADS)

Baykal, Ahmet; Dereli, Tekin

2017-01-01

A technique to linearize gravitational field equations is developed in which the perturbation metric coefficients are treated as second rank, symmetric, 1-form fields belonging to the Minkowski background spacetime by using the exterior algebra of differential forms.

A Systematic Approach for Obtaining Performance on Matrix-Like Operations

NASA Astrophysics Data System (ADS)

Veras, Richard Michael

Scientific Computation provides a critical role in the scientific process because it allows us ask complex queries and test predictions that would otherwise be unfeasible to perform experimentally. Because of its power, Scientific Computing has helped drive advances in many fields ranging from Engineering and Physics to Biology and Sociology to Economics and Drug Development and even to Machine Learning and Artificial Intelligence. Common among these domains is the desire for timely computational results, thus a considerable amount of human expert effort is spent towards obtaining performance for these scientific codes. However, this is no easy task because each of these domains present their own unique set of challenges to software developers, such as domain specific operations, structurally complex data and ever-growing datasets. Compounding these problems are the myriads of constantly changing, complex and unique hardware platforms that an expert must target. Unfortunately, an expert is typically forced to reproduce their effort across multiple problem domains and hardware platforms. In this thesis, we demonstrate the automatic generation of expert level high-performance scientific codes for Dense Linear Algebra (DLA), Structured Mesh (Stencil), Sparse Linear Algebra and Graph Analytic. In particular, this thesis seeks to address the issue of obtaining performance on many complex platforms for a certain class of matrix-like operations that span across many scientific, engineering and social fields. We do this by automating a method used for obtaining high performance in DLA and extending it to structured, sparse and scale-free domains. We argue that it is through the use of the underlying structure found in the data from these domains that enables this process. Thus, obtaining performance for most operations does not occur in isolation of the data being operated on, but instead depends significantly on the structure of the data.

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.

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.

Extensions of algebraic image operators: An approach to model-based vision

NASA Technical Reports Server (NTRS)

Lerner, Bao-Ting; Morelli, Michael V.

1990-01-01

Researchers extend their previous research on a highly structured and compact algebraic representation of grey-level images which can be viewed as fuzzy sets. Addition and multiplication are defined for the set of all grey-level images, which can then be described as polynomials of two variables. Utilizing this new algebraic structure, researchers devised an innovative, efficient edge detection scheme. An accurate method for deriving gradient component information from this edge detector is presented. Based upon this new edge detection system researchers developed a robust method for linear feature extraction by combining the techniques of a Hough transform and a line follower. The major advantage of this feature extractor is its general, object-independent nature. Target attributes, such as line segment lengths, intersections, angles of intersection, and endpoints are derived by the feature extraction algorithm and employed during model matching. The algebraic operators are global operations which are easily reconfigured to operate on any size or shape region. This provides a natural platform from which to pursue dynamic scene analysis. A method for optimizing the linear feature extractor which capitalizes on the spatially reconfiguration nature of the edge detector/gradient component operator is discussed.

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.

Attitude control with realization of linear error dynamics

NASA Technical Reports Server (NTRS)

Paielli, Russell A.; Bach, Ralph E.

1993-01-01

An attitude control law is derived to realize linear unforced error dynamics with the attitude error defined in terms of rotation group algebra (rather than vector algebra). Euler parameters are used in the rotational dynamics model because they are globally nonsingular, but only the minimal three Euler parameters are used in the error dynamics model because they have no nonlinear mathematical constraints to prevent the realization of linear error dynamics. The control law is singular only when the attitude error angle is exactly pi rad about any eigenaxis, and a simple intuitive modification at the singularity allows the control law to be used globally. The forced error dynamics are nonlinear but stable. Numerical simulation tests show that the control law performs robustly for both initial attitude acquisition and attitude control.

Tissue characterization using electrical impedance spectroscopy data: a linear algebra approach.

PubMed

Laufer, Shlomi; Solomon, Stephen B; Rubinsky, Boris

2012-06-01

In this study, we use a new linear algebra manipulation on electrical impedance spectroscopy measurements to provide real-time information regarding the nature of the tissue surrounding the needle in minimal invasive procedures. Using a Comsol Multiphysics three-dimensional model, a phantom based on ex vivo animal tissue and in vivo animal data, we demonstrate how tissue inhomogeneity can be characterized without any previous knowledge of the electrical properties of the different tissues, except that they should not be linearly dependent on a certain frequency range. This method may have applications in needle biopsies, radiation seeds, or minimally invasive surgery and can reduce the number of computer tomography or magnetic resonance imaging images. We conclude by demonstrating how this mathematical approach can be useful in other applications.

Complementary Reliability-Based Decodings of Binary Linear Block Codes

NASA Technical Reports Server (NTRS)

Fossorier, Marc P. C.; Lin, Shu

1997-01-01

This correspondence presents a hybrid reliability-based decoding algorithm which combines the reprocessing method based on the most reliable basis and a generalized Chase-type algebraic decoder based on the least reliable positions. It is shown that reprocessing with a simple additional algebraic decoding effort achieves significant coding gain. For long codes, the order of reprocessing required to achieve asymptotic optimum error performance is reduced by approximately 1/3. This significantly reduces the computational complexity, especially for long codes. Also, a more efficient criterion for stopping the decoding process is derived based on the knowledge of the algebraic decoding solution.

Mathematical model for Dengue with three states of infection

NASA Astrophysics Data System (ADS)

Hincapie, Doracelly; Ospina, Juan

2012-06-01

A mathematical model for dengue with three states of infection is proposed and analyzed. The model consists in a system of differential equations. The three states of infection are respectively asymptomatic, partially asymptomatic and fully asymptomatic. The model is analyzed using computer algebra software, specifically Maple, and the corresponding basic reproductive number and the epidemic threshold are computed. The resulting basic reproductive number is an algebraic synthesis of all epidemic parameters and it makes clear the possible control measures. The microscopic structure of the epidemic parameters is established using the quantum theory of the interactions between the atoms and radiation. In such approximation, the human individual is represented by an atom and the mosquitoes are represented by radiation. The force of infection from the mosquitoes to the humans is considered as the transition probability from the fundamental state of atom to excited states. The combination of computer algebra software and quantum theory provides a very complete formula for the basic reproductive number and the possible control measures tending to stop the propagation of the disease. It is claimed that such result may be important in military medicine and the proposed method can be applied to other vector-borne diseases.

Matrix preconditioning: a robust operation for optical linear algebra processors.

PubMed

Ghosh, A; Paparao, P

1987-07-15

Analog electrooptical processors are best suited for applications demanding high computational throughput with tolerance for inaccuracies. Matrix preconditioning is one such application. Matrix preconditioning is a preprocessing step for reducing the condition number of a matrix and is used extensively with gradient algorithms for increasing the rate of convergence and improving the accuracy of the solution. In this paper, we describe a simple parallel algorithm for matrix preconditioning, which can be implemented efficiently on a pipelined optical linear algebra processor. From the results of our numerical experiments we show that the efficacy of the preconditioning algorithm is affected very little by the errors of the optical system.

Operator pencil passing through a given operator

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

Biggs, A., E-mail: khudian@manchester.ac.uk, E-mail: adam.biggs@student.manchester.ac.uk; Khudaverdian, H. M., E-mail: khudian@manchester.ac.uk, E-mail: adam.biggs@student.manchester.ac.uk

Let Δ be a linear differential operator acting on the space of densities of a given weight λ{sub 0} on a manifold M. One can consider a pencil of operators Π-circumflex(Δ)=(Δ{sub λ}) passing through the operator Δ such that any Δ{sub λ} is a linear differential operator acting on densities of weight λ. This pencil can be identified with a linear differential operator Δ-circumflex acting on the algebra of densities of all weights. The existence of an invariant scalar product in the algebra of densities implies a natural decomposition of operators, i.e., pencils of self-adjoint and anti-self-adjoint operators. We studymore » lifting maps that are on one hand equivariant with respect to divergenceless vector fields, and, on the other hand, with values in self-adjoint or anti-self-adjoint operators. In particular, we analyze the relation between these two concepts, and apply it to the study of diff (M)-equivariant liftings. Finally, we briefly consider the case of liftings equivariant with respect to the algebra of projective transformations and describe all regular self-adjoint and anti-self-adjoint liftings. Our constructions can be considered as a generalisation of equivariant quantisation.« less

Ground-based microwave radiometric remote sensing of the tropical atmosphere

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

Han, Yong.

1992-01-01

A partially developed 9-channel ground-based microwave radiometer for the Department of Meteorology at Penn State was completed and tested. Complementary units were added, corrections to both hardware and software were made, and system software was corrected and upgraded. Measurements from this radiometer were used to infer tropospheric temperature, water vapor and cloud liquid water. The various weighting functions at each of the 9 channels were calculated and analyzed to estimate the sensitivities of the brightness temperature to the desired atmospheric variables. The mathematical inversion problem, in a linear form, was viewed in terms of the theory of linear algebra. Severalmore » methods for solving the inversion problem were reviewed. Radiometric observations were conducted during the 1990 Tropical Cyclone Motion Experiment. The radiometer was installed on the island of Saipan in a tropical region. The radiometer was calibrated using tipping curve and radiosonde data as well as measurements of the radiation from a blackbody absorber. A linear statistical method was applied for the data inversion. The inversion coefficients in the equation were obtained using a large number of radiosonde profiles from Guam and a radiative transfer model. Retrievals were compared with those from local, Saipan, radiosonde measurements. Water vapor profiles, integrated water vapor, and integrated liquid water were retrieved successfully. For temperature profile retrievals, however, the radiometric measurements with experimental noises added no more profile information to the inversion than that they were determined mainly by the surface pressure measurements. A method was developed to derive the integrated water vapor and liquid water from combined radiometer and ceilometer measurements. Significant improvement on radiometric measurements of the integrated liquid water can be gained with this method.« less

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…

A Hypothetical Learning Trajectory for Conceptualizing Matrices as Linear Transformations

ERIC Educational Resources Information Center

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

2017-01-01

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…

Symmetry-preserving perturbations of the Bateman Lagrangian and dissipative systems

NASA Astrophysics Data System (ADS)

Campoamor-Stursberg, Rutwig

2017-03-01

Perturbations of the classical Bateman Lagrangian preserving a certain subalgebra of Noether symmetries are studied, and conservative perturbations are characterized by the Lie algebra sl(2, ℝ) ⊕ so(2). Non-conservative albeit integrable perturbations are determined by the simple Lie algebra sl(2,ℝ), showing further the relation of the corresponding non-linear systems with the notion of generalized Ermakov systems.

Gordan—Capelli series in superalgebras

PubMed Central

Brini, Andrea; Palareti, Aldopaolo; Teolis, Antonio G. B.

1988-01-01

We derive two Gordan—Capelli series for the supersymmetric algebra of the tensor product of two [unk]2-graded [unk]-vector spaces U and V, being [unk] a field of characteristic zero. These expansions yield complete decompositions of the supersymmetric algebra regarded as a pl(U)- and a pl(V)- module, where pl(U) and pl(V) are the general linear Lie superalgebras of U and V, respectively. PMID:16593911

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,…

Symmetry-preserving perturbations of the Bateman Lagrangian and dissipative systems

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

Campoamor-Stursberg, Rutwig, E-mail: rutwig@ucm.es

Perturbations of the classical Bateman Lagrangian preserving a certain subalgebra of Noether symmetries are studied, and conservative perturbations are characterized by the Lie algebra sl(2, ℝ) ⊕ so(2). Non-conservative albeit integrable perturbations are determined by the simple Lie algebra sl(2,ℝ), showing further the relation of the corresponding non-linear systems with the notion of generalized Ermakov systems.

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

ERIC Educational Resources Information Center

Schmidt, Karsten

2008-01-01

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

Signal Processing for Radar Target Tracking and Identification

DTIC Science & Technology

1996-12-01

Computes the likelihood for various potential jump moves. 12. matrix_mult.m: Parallel implementation of linear algebra ... Elementary Lineary Algebra with Applications, John Wiley k Sons, Inc., New York, 1987. [9] A. K. Bhattacharyya, and D. L. Sengupta, Radar Cross...Miller, ’Target Tracking and Recognition Using Jump-Diffusion Processes," ARO’s 11th Army Conf. on Applied Mathemat- ics and Computing, June 8-11

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.

Proceedings of the 1977 MACSYMA users' conference (NASA)

NASA Technical Reports Server (NTRS)

1977-01-01

The MACSYMA program for symbolic and algebraic manipulation enables exact, symbolic mathematical computations to be performed on a computer. This program is rather large, and various approaches to the hardware and software problems are examined.

Global exponential stability of octonion-valued neural networks with leakage delay and mixed delays.

PubMed

Popa, Călin-Adrian

2018-06-08

This paper discusses octonion-valued neural networks (OVNNs) with leakage delay, time-varying delays, and distributed delays, for which the states, weights, and activation functions belong to the normed division algebra of octonions. The octonion algebra is a nonassociative and noncommutative generalization of the complex and quaternion algebras, but does not belong to the category of Clifford algebras, which are associative. In order to avoid the nonassociativity of the octonion algebra and also the noncommutativity of the quaternion algebra, the Cayley-Dickson construction is used to decompose the OVNNs into 4 complex-valued systems. By using appropriate Lyapunov-Krasovskii functionals, with double and triple integral terms, the free weighting matrix method, and simple and double integral Jensen inequalities, delay-dependent criteria are established for the exponential stability of the considered OVNNs. The criteria are given in terms of complex-valued linear matrix inequalities, for two types of Lipschitz conditions which are assumed to be satisfied by the octonion-valued activation functions. Finally, two numerical examples illustrate the feasibility, effectiveness, and correctness of the theoretical results. Copyright © 2018 Elsevier Ltd. All rights reserved.

An algorithmic approach to solving polynomial equations associated with quantum circuits

NASA Astrophysics Data System (ADS)

Gerdt, V. P.; Zinin, M. V.

2009-12-01

In this paper we present two algorithms for reducing systems of multivariate polynomial equations over the finite field F 2 to the canonical triangular form called lexicographical Gröbner basis. This triangular form is the most appropriate for finding solutions of the system. On the other hand, the system of polynomials over F 2 whose variables also take values in F 2 (Boolean polynomials) completely describes the unitary matrix generated by a quantum circuit. In particular, the matrix itself can be computed by counting the number of solutions (roots) of the associated polynomial system. Thereby, efficient construction of the lexicographical Gröbner bases over F 2 associated with quantum circuits gives a method for computing their circuit matrices that is alternative to the direct numerical method based on linear algebra. We compare our implementation of both algorithms with some other software packages available for computing Gröbner bases over F 2.

Handling Big Data in Medical Imaging: Iterative Reconstruction with Large-Scale Automated Parallel Computation

PubMed Central

Lee, Jae H.; Yao, Yushu; Shrestha, Uttam; Gullberg, Grant T.; Seo, Youngho

2014-01-01

The primary goal of this project is to implement the iterative statistical image reconstruction algorithm, in this case maximum likelihood expectation maximum (MLEM) used for dynamic cardiac single photon emission computed tomography, on Spark/GraphX. This involves porting the algorithm to run on large-scale parallel computing systems. Spark is an easy-to- program software platform that can handle large amounts of data in parallel. GraphX is a graph analytic system running on top of Spark to handle graph and sparse linear algebra operations in parallel. The main advantage of implementing MLEM algorithm in Spark/GraphX is that it allows users to parallelize such computation without any expertise in parallel computing or prior knowledge in computer science. In this paper we demonstrate a successful implementation of MLEM in Spark/GraphX and present the performance gains with the goal to eventually make it useable in clinical setting. PMID:27081299

Handling Big Data in Medical Imaging: Iterative Reconstruction with Large-Scale Automated Parallel Computation.

PubMed

Lee, Jae H; Yao, Yushu; Shrestha, Uttam; Gullberg, Grant T; Seo, Youngho

2014-11-01

The primary goal of this project is to implement the iterative statistical image reconstruction algorithm, in this case maximum likelihood expectation maximum (MLEM) used for dynamic cardiac single photon emission computed tomography, on Spark/GraphX. This involves porting the algorithm to run on large-scale parallel computing systems. Spark is an easy-to- program software platform that can handle large amounts of data in parallel. GraphX is a graph analytic system running on top of Spark to handle graph and sparse linear algebra operations in parallel. The main advantage of implementing MLEM algorithm in Spark/GraphX is that it allows users to parallelize such computation without any expertise in parallel computing or prior knowledge in computer science. In this paper we demonstrate a successful implementation of MLEM in Spark/GraphX and present the performance gains with the goal to eventually make it useable in clinical setting.

Compartmental modelling of the pharmacokinetics of a breast cancer resistance protein.

PubMed

Grandjean, Thomas R B; Chappell, Mike J; Yates, James T W; Jones, Kevin; Wood, Gemma; Coleman, Tanya

2011-11-01

A mathematical model for the pharmacokinetics of Hoechst 33342 following administration into a culture medium containing a population of transfected cells (HEK293 hBCRP) with a potent breast cancer resistance protein inhibitor, Fumitremorgin C (FTC), present is described. FTC is reported to almost completely annul resistance mediated by BCRP in vitro. This non-linear compartmental model has seven macroscopic sub-units, with 14 rate parameters. It describes the relationship between the concentration of Hoechst 33342 and FTC, initially spiked in the medium, and the observed change in fluorescence due to Hoechst 33342 binding to DNA. Structural identifiability analysis has been performed using two methods, one based on the similarity transformation/exhaustive modelling approach and the other based on the differential algebra approach. The analyses demonstrated that all models derived are uniquely identifiable for the experiments/observations available. A kinetic modelling software package, namely FACSIMILE (MPCA Software, UK), was used for parameter fitting and to obtain numerical solutions for the system equations. Model fits gave very good agreement with in vitro data provided by AstraZeneca across a variety of experimental scenarios. Copyright © 2010 Elsevier Ireland Ltd. All rights reserved.

Computational algebraic geometry of epidemic models

NASA Astrophysics Data System (ADS)

Rodríguez Vega, Martín.

2014-06-01

Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.

Chemical Equation Balancing.

ERIC Educational Resources Information Center

Blakley, G. R.

1982-01-01

Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)

Linear time-invariant controller design for two-channel decentralized control systems

NASA Technical Reports Server (NTRS)

Desoer, Charles A.; Gundes, A. Nazli

1987-01-01

This paper analyzes a linear time-invariant two-channel decentralized control system with a 2 x 2 strictly proper plant. It presents an algorithm for the algebraic design of a class of decentralized compensators which stabilize the given plant.

Algebraic Construction of Exact Difference Equations from Symmetry of Equations

NASA Astrophysics Data System (ADS)

Itoh, Toshiaki

2009-09-01

Difference equations or exact numerical integrations, which have general solutions, are treated algebraically. Eliminating the symmetries of the equation, we can construct difference equations (DCE) or numerical integrations equivalent to some ODEs or PDEs that means both have the same solution functions. When arbitrary functions are given, whether we can construct numerical integrations that have solution functions equal to given function or not are treated in this work. Nowadays, Lie's symmetries solver for ODE and PDE has been implemented in many symbolic software. Using this solver we can construct algebraic DCEs or numerical integrations which are correspond to some ODEs or PDEs. In this work, we treated exact correspondence between ODE or PDE and DCE or numerical integration with Gröbner base and Janet base from the view of Lie's symmetries.

Linear Ordinary Differential Equations with Constant Coefficients. Revisiting the Impulsive Response Method Using Factorization

ERIC Educational Resources Information Center

Camporesi, Roberto

2011-01-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of…

Teaching Linear Functions in Context with Graphics Calculators: Students' Responses and the Impact of the Approach on Their Use of Algebraic Symbols

ERIC Educational Resources Information Center

Bardini, Caroline; Pierce, Robyn U.; Stacey, Kaye

2004-01-01

This study analyses some of the consequences of adopting a functional/modelling approach to the teaching of algebra. The teaching of one class of 17 students was observed over five weeks, with 15 students undertaking both pre- and post-tests and 6 students and the teacher being interviewed individually. Use of graphics calculators made the…

Algebraic Riccati equations in zero-sum differential games

NASA Technical Reports Server (NTRS)

Johnson, T. L.; Chao, A.

1974-01-01

The procedure for finding the closed-loop Nash equilibrium solution of two-player zero-sum linear time-invariant differential games with quadratic performance criteria and classical information pattern may be reduced in most cases to the solution of an algebraic Riccati equation. Based on the results obtained by Willems, necessary and sufficient conditions for existence of solutions to these equations are derived, and explicit conditions for a scalar example are given.

SYVA: A program to analyze symmetry of molecules based on vector algebra

NASA Astrophysics Data System (ADS)

Gyevi-Nagy, László; Tasi, Gyula

2017-06-01

Symmetry is a useful concept in physics and chemistry. It can be used to find out some simple properties of a molecule or simplify complex calculations. In this paper a simple vector algebraic method is described to determine all symmetry elements of an arbitrary molecule. To carry out the symmetry analysis, a program has been written, which is also capable of generating the framework group of the molecule, revealing the symmetry properties of normal modes of vibration and symmetrizing the structure. To demonstrate the capabilities of the program, it is compared to other common widely used stand-alone symmetry analyzer (SYMMOL, Symmetrizer) and molecular modeling (NWChem, ORCA, MRCC) software. SYVA can generate input files for molecular modeling programs, e.g. Gaussian, using precisely symmetrized molecular structures. Possible applications are also demonstrated by integrating SYVA with the GAMESS and MRCC software.

The elastic theory of shells using geometric algebra

PubMed Central

Lasenby, J.; Agarwal, A.

2017-01-01

We present a novel derivation of the elastic theory of shells. We use the language of geometric algebra, which allows us to express the fundamental laws in component-free form, thus aiding physical interpretation. It also provides the tools to express equations in an arbitrary coordinate system, which enhances their usefulness. The role of moments and angular velocity, and the apparent use by previous authors of an unphysical angular velocity, has been clarified through the use of a bivector representation. In the linearized theory, clarification of previous coordinate conventions which have been the cause of confusion is provided, and the introduction of prior strain into the linearized theory of shells is made possible. PMID:28405404

New infinite-dimensional hidden symmetries for heterotic string theory

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

Gao Yajun

The symmetry structures of two-dimensional heterotic string theory are studied further. A (2d+n)x(2d+n) matrix complex H-potential is constructed and the field equations are extended into a complex matrix formulation. A pair of Hauser-Ernst-type linear systems are established. Based on these linear systems, explicit formulations of new hidden symmetry transformations for the considered theory are given and then these symmetry transformations are verified to constitute infinite-dimensional Lie algebras: the semidirect product of the Kac-Moody o(d,d+n-circumflex) and Virasoro algebras (without center charges). These results demonstrate that the heterotic string theory under consideration possesses more and richer symmetry structures than previously expected.

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.

On recent advances and future research directions for computational fluid dynamics

NASA Technical Reports Server (NTRS)

Baker, A. J.; Soliman, M. O.; Manhardt, P. D.

1986-01-01

This paper highlights some recent accomplishments regarding CFD numerical algorithm constructions for generation of discrete approximate solutions to classes of Reynolds-averaged Navier-Stokes equations. Following an overview of turbulent closure modeling, and development of appropriate conservation law systems, a Taylor weak-statement semi-discrete approximate solution algorithm is developed. Various forms for completion to the final linear algebra statement are cited, as are a range of candidate numerical linear algebra solution procedures. This development sequence emphasizes the key building blocks of a CFD RNS algorithm, including solution trial and test spaces, integration procedure and added numerical stability mechanisms. A range of numerical results are discussed focusing on key topics guiding future research directions.

The elastic theory of shells using geometric algebra.

PubMed

Gregory, A L; Lasenby, J; Agarwal, A

2017-03-01

We present a novel derivation of the elastic theory of shells. We use the language of geometric algebra, which allows us to express the fundamental laws in component-free form, thus aiding physical interpretation. It also provides the tools to express equations in an arbitrary coordinate system, which enhances their usefulness. The role of moments and angular velocity, and the apparent use by previous authors of an unphysical angular velocity, has been clarified through the use of a bivector representation. In the linearized theory, clarification of previous coordinate conventions which have been the cause of confusion is provided, and the introduction of prior strain into the linearized theory of shells is made possible.

Accuracy requirements of optical linear algebra processors in adaptive optics imaging systems

NASA Technical Reports Server (NTRS)

Downie, John D.; Goodman, Joseph W.

1989-01-01

The accuracy requirements of optical processors in adaptive optics systems are determined by estimating the required accuracy in a general optical linear algebra processor (OLAP) that results in a smaller average residual aberration than that achieved with a conventional electronic digital processor with some specific computation speed. Special attention is given to an error analysis of a general OLAP with regard to the residual aberration that is created in an adaptive mirror system by the inaccuracies of the processor, and to the effect of computational speed of an electronic processor on the correction. Results are presented on the ability of an OLAP to compete with a digital processor in various situations.

Choreographing Patterns and Functions

ERIC Educational Resources Information Center

Hawes, Zachary; Moss, Joan; Finch, Heather; Katz, Jacques

2012-01-01

In this article, the authors begin with a description of an algebraic dance--the translation of composite linear growing patterns into choreographed movement--which was the last component of a research-based instructional unit that focused on fostering an understanding of linear functional rules through geometric growing patterns and…

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.

Investigating Years 7 to 12 students' knowledge of linear relationships through different contexts and representations

NASA Astrophysics Data System (ADS)

Wilkie, Karina J.; Ayalon, Michal

2018-02-01

A foundational component of developing algebraic thinking for meaningful calculus learning is the idea of "function" that focuses on the relationship between varying quantities. Students have demonstrated widespread difficulties in learning calculus, particularly interpreting and modeling dynamic events, when they have a poor understanding of relationships between variables. Yet, there are differing views on how to develop students' functional thinking over time. In the Australian curriculum context, linear relationships are introduced to lower secondary students with content that reflects a hybrid of traditional and reform algebra pedagogy. This article discusses an investigation into Australian secondary students' understanding of linear functional relationships from Years 7 to 12 (approximately 12 to 18 years old; n = 215) in their approaches to three tasks (finding rate of change, pattern generalisation and interpretation of gradient) involving four different representations (table, geometric growing pattern, equation and graph). From the findings, it appears that these students' knowledge of linear functions remains context-specific rather than becoming connected over time.

Validation of DNA-based identification software by computation of pedigree likelihood ratios.

PubMed

Slooten, K

2011-08-01

Disaster victim identification (DVI) can be aided by DNA-evidence, by comparing the DNA-profiles of unidentified individuals with those of surviving relatives. The DNA-evidence is used optimally when such a comparison is done by calculating the appropriate likelihood ratios. Though conceptually simple, the calculations can be quite involved, especially with large pedigrees, precise mutation models etc. In this article we describe a series of test cases designed to check if software designed to calculate such likelihood ratios computes them correctly. The cases include both simple and more complicated pedigrees, among which inbred ones. We show how to calculate the likelihood ratio numerically and algebraically, including a general mutation model and possibility of allelic dropout. In Appendix A we show how to derive such algebraic expressions mathematically. We have set up these cases to validate new software, called Bonaparte, which performs pedigree likelihood ratio calculations in a DVI context. Bonaparte has been developed by SNN Nijmegen (The Netherlands) for the Netherlands Forensic Institute (NFI). It is available free of charge for non-commercial purposes (see www.dnadvi.nl for details). Commercial licenses can also be obtained. The software uses Bayesian networks and the junction tree algorithm to perform its calculations. Copyright © 2010 Elsevier Ireland Ltd. All rights reserved.

Topologically massive gravity and galilean conformal algebra: a study of correlation functions

NASA Astrophysics Data System (ADS)

Bagchi, Arjun

2011-02-01

The Galilean Conformal Algebra (GCA) arises from the conformal algebra in the non-relativistic limit. In two dimensions, one can view it as a limit of linear combinations of the two copies Virasoro algebra. Recently, it has been argued that Topologically Massive Gravity (TMG) realizes the quantum 2d GCA in a particular scaling limit of the gravitational Chern-Simons term. To add strength to this claim, we demonstrate a matching of correlation functions on both sides of this correspondence. A priori looking for spatially dependent correlators seems to force us to deal with high spin operators in the bulk. We get around this difficulty by constructing the non-relativistic Energy-Momentum tensor and considering its correlation functions. On the gravity side, our analysis makes heavy use of recent results of Holographic Renormalization in Topologically Massive Gravity.

Measurements and mathematical formalism of quantum mechanics

NASA Astrophysics Data System (ADS)

Slavnov, D. A.

2007-03-01

A scheme for constructing quantum mechanics is given that does not have Hilbert space and linear operators as its basic elements. Instead, a version of algebraic approach is considered. Elements of a noncommutative algebra (observables) and functionals on this algebra (elementary states) associated with results of single measurements are used as primary components of the scheme. On the one hand, it is possible to use within the scheme the formalism of the standard (Kolmogorov) probability theory, and, on the other hand, it is possible to reproduce the mathematical formalism of standard quantum mechanics, and to study the limits of its applicability. A short outline is given of the necessary material from the theory of algebras and probability theory. It is described how the mathematical scheme of the paper agrees with the theory of quantum measurements, and avoids quantum paradoxes.

Colombeau algebra as a mathematical tool for investigating step load and step deformation of systems of nonlinear springs and dashpots

NASA Astrophysics Data System (ADS)

Průša, Vít; Řehoř, Martin; Tůma, Karel

2017-02-01

The response of mechanical systems composed of springs and dashpots to a step input is of eminent interest in the applications. If the system is formed by linear elements, then its response is governed by a system of linear ordinary differential equations. In the linear case, the mathematical method of choice for the analysis of the response is the classical theory of distributions. However, if the system contains nonlinear elements, then the classical theory of distributions is of no use, since it is strictly limited to the linear setting. Consequently, a question arises whether it is even possible or reasonable to study the response of nonlinear systems to step inputs. The answer is positive. A mathematical theory that can handle the challenge is the so-called Colombeau algebra. Building on the abstract result by Průša and Rajagopal (Int J Non-Linear Mech 81:207-221, 2016), we show how to use the theory in the analysis of response of nonlinear spring-dashpot and spring-dashpot-mass systems.

GeoGebra: A Global Platform for Teaching and Learning Math Together and Using the Synergy of Mathematicians

NASA Astrophysics Data System (ADS)

Kllogjeri, Pellumb

In present age we are witnesses and practioners of computer-based education which is highly speed progressing. The computer-based education allows educators and students to use educational programming language and e-tutors to teach and learn, to interact with one another and share together the results of their work. The computer-based education is done possible by special electronic tools among which the most important are the mathematical programmes. There are many mathematical programmes, but one which is being embraced and used by a daily increasing number of users throughout the world is GeoGebra. The recently published software GeoGebra by Markus Hohenwater (2004) explicitly links geometry and algebra. GeoGebra affords a bidirectional combination of geometry and algebra that differs from earlier software forms. The bidirectional combination means that, for instance, by typing in an equation in the algebra window, the graph of the equation will be shown in the dynamic and graphic window. This programme is so much preferred because of its three main features: the double representation of the mathematical object(geometric and algebraic), there are not strong requirements as to the age and the knowledge in using it(the students of the elementary school can use it as well) and, it is offered free of charge(simply by downloading it). In this paper we are concentrating in the double representation of the mathematical object and its advantages in explaining and forming mathematical concepts and performing operations, in the global opportunities for using GeoGebra and the benefits of using it by cooperating and sharing experiences.

General purpose nonlinear system solver based on Newton-Krylov method.

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

2013-12-01

KINSOL is part of a software family called SUNDIALS: SUite of Nonlinear and Differential/Algebraic equation Solvers [1]. KINSOL is a general-purpose nonlinear system solver based on Newton-Krylov and fixed-point solver technologies [2].

Autonomous robot software development using simple software components

NASA Astrophysics Data System (ADS)

Burke, Thomas M.; Chung, Chan-Jin

2004-10-01

Developing software to control a sophisticated lane-following, obstacle-avoiding, autonomous robot can be demanding and beyond the capabilities of novice programmers - but it doesn"t have to be. A creative software design utilizing only basic image processing and a little algebra, has been employed to control the LTU-AISSIG autonomous robot - a contestant in the 2004 Intelligent Ground Vehicle Competition (IGVC). This paper presents a software design equivalent to that used during the IGVC, but with much of the complexity removed. The result is an autonomous robot software design, that is robust, reliable, and can be implemented by programmers with a limited understanding of image processing. This design provides a solid basis for further work in autonomous robot software, as well as an interesting and achievable robotics project for students.

Mathematical Modeling of Chemical Stoichiometry

ERIC Educational Resources Information Center

Croteau, Joshua; Fox, William P.; Varazo, Kristofoland

2007-01-01

In beginning chemistry classes, students are taught a variety of techniques for balancing chemical equations. The most common method is inspection. This paper addresses using a system of linear mathematical equations to solve for the stoichiometric coefficients. Many linear algebra books carry the standard balancing of chemical equations as an…

Quantum monodromy and quantum phase transitions in floppy molecules

NASA Astrophysics Data System (ADS)

Larese, Danielle

2012-10-01

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 analyzing the spectroscopic 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 CH3NCO and GeH3NCO. Extraction of potential functions are completed for several molecules, resulting in predictions of barriers to linearity and equilibrium bond angles.

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

Luszczek, Piotr R; Tomov, Stanimire Z; Dongarra, Jack J

We present an efficient and scalable programming model for the development of linear algebra in heterogeneous multi-coprocessor environments. The model incorporates some of the current best design and implementation practices for the heterogeneous acceleration of dense linear algebra (DLA). Examples are given as the basis for solving linear systems' algorithms - the LU, QR, and Cholesky factorizations. To generate the extreme level of parallelism needed for the efficient use of coprocessors, algorithms of interest are redesigned and then split into well-chosen computational tasks. The tasks execution is scheduled over the computational components of a hybrid system of multi-core CPUs andmore » coprocessors using a light-weight runtime system. The use of lightweight runtime systems keeps scheduling overhead low, while enabling the expression of parallelism through otherwise sequential code. This simplifies the development efforts and allows the exploration of the unique strengths of the various hardware components.« less

A Fresh Look at Linear Ordinary Differential Equations with Constant Coefficients. Revisiting the Impulsive Response Method Using Factorization

ERIC Educational Resources Information Center

Camporesi, Roberto

2016-01-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as…

Automorphisms of Order Structures of Abelian Parts of Operator Algebras and Their Role in Quantum Theory

NASA Astrophysics Data System (ADS)

Hamhalter, Jan; Turilova, Ekaterina

2014-10-01

It is shown that any order isomorphism between the structures of unital associative JB subalgebras of JB algebras is given naturally by a partially linear Jordan isomorphism. The same holds for nonunital subalgebras and order isomorphisms preserving the unital subalgebra. Finally, we recover usual action of time evolution group on a von Neumann factor from group of automorphisms of the structure of Abelian subalgebras.

Electrokinetics Models for Micro and Nano Fluidic Impedance Sensors

DTIC Science & Technology

2010-11-01

primitive Differential-Algebraic Equations (DAEs), used to process and interpret the experimentally measured electrical impedance data (Sun and Morgan...field, and species respectively. A second-order scheme was used to calculate the ionic species distribution. The linearized algebraic equations were...is governed by the Poisson equation 2 0 0 r i i i F z cε ε φ∇ + =∑ where ε0 and εr are, respectively, the electrical permittivity in the vacuum

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.

Boolean Operations with Prism Algebraic Patches

PubMed Central

Bajaj, Chandrajit; Paoluzzi, Alberto; Portuesi, Simone; Lei, Na; Zhao, Wenqi

2009-01-01

In this paper we discuss a symbolic-numeric algorithm for Boolean operations, closed in the algebra of curved polyhedra whose boundary is triangulated with algebraic patches (A-patches). This approach uses a linear polyhedron as a first approximation of both the arguments and the result. On each triangle of a boundary representation of such linear approximation, a piecewise cubic algebraic interpolant is built, using a C1-continuous prism algebraic patch (prism A-patch) that interpolates the three triangle vertices, with given normal vectors. The boundary representation only stores the vertices of the initial triangulation and their external vertex normals. In order to represent also flat and/or sharp local features, the corresponding normal-per-face and/or normal-per-edge may be also given, respectively. The topology is described by storing, for each curved triangle, the two triples of pointers to incident vertices and to adjacent triangles. For each triangle, a scaffolding prism is built, produced by its extreme vertices and normals, which provides a containment volume for the curved interpolating A-patch. When looking for the result of a regularized Boolean operation, the 0-set of a tri-variate polynomial within each such prism is generated, and intersected with the analogous 0-sets of the other curved polyhedron, when two prisms have non-empty intersection. The intersection curves of the boundaries are traced and used to decompose each boundary into the 3 standard classes of subpatches, denoted in, out and on. While tracing the intersection curves, the locally refined triangulation of intersecting patches is produced, and added to the boundary representation. PMID:21516262

Integrand-level reduction of loop amplitudes by computational algebraic geometry methods

NASA Astrophysics Data System (ADS)

Zhang, Yang

2012-09-01

We present an algorithm for the integrand-level reduction of multi-loop amplitudes of renormalizable field theories, based on computational algebraic geometry. This algorithm uses (1) the Gröbner basis method to determine the basis for integrand-level reduction, (2) the primary decomposition of an ideal to classify all inequivalent solutions of unitarity cuts. The resulting basis and cut solutions can be used to reconstruct the integrand from unitarity cuts, via polynomial fitting techniques. The basis determination part of the algorithm has been implemented in the Mathematica package, BasisDet. The primary decomposition part can be readily carried out by algebraic geometry softwares, with the output of the package BasisDet. The algorithm works in both D = 4 and D = 4 - 2 ɛ dimensions, and we present some two and three-loop examples of applications of this algorithm.

Using MathCAD to Teach One-Dimensional Graphs

ERIC Educational Resources Information Center

Yushau, B.

2004-01-01

Topics such as linear and nonlinear equations and inequalities, compound inequalities, linear and nonlinear absolute value equations and inequalities, rational equations and inequality are commonly found in college algebra and precalculus textbooks. What is common about these topics is the fact that their solutions and graphs lie in the real line…

Discovering Linear Equations in Explicit Tables

ERIC Educational Resources Information Center

Burton, Lauren

2017-01-01

When teaching algebra concepts to middle school students, the author often hears questions that echo her own past confusion as a young student learning to write linear equations using data tables that show only input and output values. Students, expected to synthesize the relationship between these values in symbolic representation, grow…

From the Laboratory to the Classroom: A Technology-Intensive Curriculum for Functions and Graphs.

ERIC Educational Resources Information Center

Magidson, Susan

1992-01-01

Addresses the challenges, risks, and rewards of teaching about linear functions in a technology-rich environment from a constructivist perspective. Describes an algebra class designed for junior high school students that focuses of the representations and real-world applications of linear functions. (MDH)

Reading between the Lines: Teaching Linear Algebra

ERIC Educational Resources Information Center

Lewis, Jennifer M.; Blunk, Merrie L.

2012-01-01

This paper compares lessons on linear equations from the same curriculum materials taught by two teachers of different levels of mathematical knowledge for teaching (MKT). The analysis indicates that the mathematical quality of instruction in these two classrooms appears to be a function of differences in MKT. Although the two teachers were…

Wick Product for Commutation Relations Connected with Yang-Baxter Operators and New Constructions of Factors

NASA Astrophysics Data System (ADS)

Krsolarlak, Ilona

We analyze a certain class of von Neumann algebras generated by selfadjoint elements , for satisfying the general commutation relations: Such algebras can be continuously embedded into some closure of the set of finite linear combinations of vectors , where is an orthonormal basis of a Hilbert space . The operator which represents the vector is denoted by and called the ``Wick product'' of the operators . We describe explicitly the form of this product. Also, we estimate the operator norm of for . Finally we apply these two results and prove that under the assumption all the von Neumann algebras considered are II1 factors.

[Feature extraction for breast cancer data based on geometric algebra theory and feature selection using differential evolution].

PubMed

Li, Jing; Hong, Wenxue

2014-12-01

The feature extraction and feature selection are the important issues in pattern recognition. Based on the geometric algebra representation of vector, a new feature extraction method using blade coefficient of geometric algebra was proposed in this study. At the same time, an improved differential evolution (DE) feature selection method was proposed to solve the elevated high dimension issue. The simple linear discriminant analysis was used as the classifier. The result of the 10-fold cross-validation (10 CV) classification of public breast cancer biomedical dataset was more than 96% and proved superior to that of the original features and traditional feature extraction method.

Restoring Low Sidelobe Antenna Patterns with Failed Elements in a Phased Array Antenna

DTIC Science & Technology

2016-02-01

optimum low sidelobes are demonstrated in several examples. Index Terms — Array signal processing, beams, linear algebra , phased arrays, shaped...represented by a linear combination of low sidelobe beamformers with no failed elements, ’s, in a neighborhood around under the constraint that the linear ...would expect that linear combinations of them in a neighborhood around would also have low sidelobes. The algorithms in this paper exploit this

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

Natural differential operations on manifolds: an algebraic approach

NASA Astrophysics Data System (ADS)

Katsylo, P. I.; Timashev, D. A.

2008-10-01

Natural algebraic differential operations on geometric quantities on smooth manifolds are considered. A method for the investigation and classification of such operations is described, the method of IT-reduction. With it the investigation of natural operations reduces to the analysis of rational maps between k-jet spaces, which are equivariant with respect to certain algebraic groups. On the basis of the method of IT-reduction a finite generation theorem is proved: for tensor bundles \\mathscr{V},\\mathscr{W}\\to M all the natural differential operations D\\colon\\Gamma(\\mathscr{V})\\to\\Gamma(\\mathscr{W}) of degree at most d can be algebraically constructed from some finite set of such operations. Conceptual proofs of known results on the classification of natural linear operations on arbitrary and symplectic manifolds are presented. A non-existence theorem is proved for natural deformation quantizations on Poisson manifolds and symplectic manifolds.Bibliography: 21 titles.

Calculus domains modelled using an original bool algebra based on polygons

NASA Astrophysics Data System (ADS)

Oanta, E.; Panait, C.; Raicu, A.; Barhalescu, M.; Axinte, T.

2016-08-01

Analytical and numerical computer based models require analytical definitions of the calculus domains. The paper presents a method to model a calculus domain based on a bool algebra which uses solid and hollow polygons. The general calculus relations of the geometrical characteristics that are widely used in mechanical engineering are tested using several shapes of the calculus domain in order to draw conclusions regarding the most effective methods to discretize the domain. The paper also tests the results of several CAD commercial software applications which are able to compute the geometrical characteristics, being drawn interesting conclusions. The tests were also targeting the accuracy of the results vs. the number of nodes on the curved boundary of the cross section. The study required the development of an original software consisting of more than 1700 computer code lines. In comparison with other calculus methods, the discretization using convex polygons is a simpler approach. Moreover, this method doesn't lead to large numbers as the spline approximation did, in that case being required special software packages in order to offer multiple, arbitrary precision. The knowledge resulted from this study may be used to develop complex computer based models in engineering.

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.

Private algebras in quantum information and infinite-dimensional complementarity

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

Crann, Jason, E-mail: jason-crann@carleton.ca; Laboratoire de Mathématiques Paul Painlevé–UMR CNRS 8524, UFR de Mathématiques, Université Lille 1–Sciences et Technologies, 59655 Villeneuve d’Ascq Cédex; Kribs, David W., E-mail: dkribs@uoguelph.ca

We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.

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.

Closed form of the Baker-Campbell-Hausdorff formula for the generators of semisimple complex Lie algebras

NASA Astrophysics Data System (ADS)

Matone, Marco

2016-11-01

Recently it has been introduced an algorithm for the Baker-Campbell-Hausdorff (BCH) formula, which extends the Van-Brunt and Visser recent results, leading to new closed forms of BCH formula. More recently, it has been shown that there are 13 types of such commutator algebras. We show, by providing the explicit solutions, that these include the generators of the semisimple complex Lie algebras. More precisely, for any pair, X, Y of the Cartan-Weyl basis, we find W, linear combination of X, Y, such that exp (X) exp (Y)=exp (W). The derivation of such closed forms follows, in part, by using the above mentioned recent results. The complete derivation is provided by considering the structure of the root system. Furthermore, if X, Y, and Z are three generators of the Cartan-Weyl basis, we find, for a wide class of cases, W, a linear combination of X, Y and Z, such that exp (X) exp (Y) exp (Z)=exp (W). It turns out that the relevant commutator algebras are type 1c-i, type 4 and type 5. A key result concerns an iterative application of the algorithm leading to relevant extensions of the cases admitting closed forms of the BCH formula. Here we provide the main steps of such an iteration that will be developed in a forthcoming paper.

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

Control Strategies for Guided Collective Motion

DTIC Science & Technology

2015-01-30

Control, Atlanta, GA, USA, December 2010, pp. 5468-5473. [19] C. Rorres and H. Anton , “ Elementary linear algebra applications version,” 9th Edition...work addresses and analyses deviated linear cyclic pursuit in which an Distribution Code A: Approved for public release, distribution is unlimited...Pursuit 6. D. Mukherjee and D. Ghose: Deviated Linear Cyclic Pursuit 7. D. Mukherjee and D. Ghose; On Synchronous and Asynchronous Heterogeneous Cyclic

R&D 100, 2016: Pyomo 4.0 – Python Optimization Modeling Objects

ScienceCinema

Hart, William; Laird, Carl; Siirola, John

2018-06-13

Pyomo provides a rich software environment for formulating and analyzing optimization applications. Pyomo supports the algebraic specification of complex sets of objectives and constraints, which enables optimization solvers to exploit problem structure to efficiently perform optimization.

Turbulence Model Predictions of Strongly Curved Flow in a U-Duct

NASA Technical Reports Server (NTRS)

Rumsey, Christopher L.; Gatski, Thomas B.; Morrison, Joseph H.

2000-01-01

The ability of three types of turbulence models to accurately predict the effects of curvature on the flow in a U-duct is studied. An explicit algebraic stress model performs slightly better than one- or two-equation linear eddy viscosity models, although it is necessary to fully account for the variation of the production-to-dissipation-rate ratio in the algebraic stress model formulation. In their original formulations, none of these turbulence models fully captures the suppressed turbulence near the convex wall, whereas a full Reynolds stress model does. Some of the underlying assumptions used in the development of algebraic stress models are investigated and compared with the computed flowfield from the full Reynolds stress model. Through this analysis, the assumption of Reynolds stress anisotropy equilibrium used in the algebraic stress model formulation is found to be incorrect in regions of strong curvature. By the accounting for the local variation of the principal axes of the strain rate tensor, the explicit algebraic stress model correctly predicts the suppressed turbulence in the outer part of the boundary layer near the convex wall.

Application of Artificial Intelligence technology to the analysis and synthesis of reliable software systems

NASA Technical Reports Server (NTRS)

Wild, Christian; Eckhardt, Dave

1987-01-01

The development of a methodology for the production of highly reliable software is one of the greatest challenges facing the computer industry. Meeting this challenge will undoubtably involve the integration of many technologies. This paper describes the use of Artificial Intelligence technologies in the automated analysis of the formal algebraic specifications of abstract data types. These technologies include symbolic execution of specifications using techniques of automated deduction and machine learning through the use of examples. On-going research into the role of knowledge representation and problem solving in the process of developing software is also discussed.

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.

Invariant classification of second-order conformally flat superintegrable systems

NASA Astrophysics Data System (ADS)

Capel, J. J.; Kress, J. M.

2014-12-01

In this paper we continue the work of Kalnins et al in classifying all second-order conformally-superintegrable (Laplace-type) systems over conformally flat spaces, using tools from algebraic geometry and classical invariant theory. The results obtained show, through Stäckel equivalence, that the list of known nondegenerate superintegrable systems over three-dimensional conformally flat spaces is complete. In particular, a seven-dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems. This manifold is foliated by the nonlinear action of the conformal group in three dimensions. Two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation. This foliation is explicitly described using algebraic varieties formed from representations of the conformal group. The proof of these results rely heavily on Gröbner basis calculations using the computer algebra software packages Maple and Singular.

Lectures on algebraic system theory: Linear systems over rings

NASA Technical Reports Server (NTRS)

Kamen, E. W.

1978-01-01

The presentation centers on four classes of systems that can be treated as linear systems over a ring. These are: (1) discrete-time systems over a ring of scalars such as the integers; (2) continuous-time systems containing time delays; (3) large-scale discrete-time systems; and (4) time-varying discrete-time systems.

The Multifaceted Variable Approach: Selection of Method in Solving Simple Linear Equations

ERIC Educational Resources Information Center

Tahir, Salma; Cavanagh, Michael

2010-01-01

This paper presents a comparison of the solution strategies used by two groups of Year 8 students as they solved linear equations. The experimental group studied algebra following a multifaceted variable approach, while the comparison group used a traditional approach. Students in the experimental group employed different solution strategies,…

Solving rational matrix equations in the state space with applications to computer-aided control-system design

NASA Technical Reports Server (NTRS)

Packard, A. K.; Sastry, S. S.

1986-01-01

A method of solving a class of linear matrix equations over various rings is proposed, using results from linear geometric control theory. An algorithm, successfully implemented, is presented, along with non-trivial numerical examples. Applications of the method to the algebraic control system design methodology are discussed.

Three Interpretations of the Matrix Equation Ax = b

ERIC Educational Resources Information Center

Larson, Christine; Zandieh, Michelle

2013-01-01

Many of the central ideas in an introductory undergraduate linear algebra course are closely tied to a set of interpretations of the matrix equation Ax = b (A is a matrix, x and b are vectors): linear combination interpretations, systems interpretations, and transformation interpretations. We consider graphic and symbolic representations for each,…

Synthesizing Strategies Creatively: Solving Linear Equations

ERIC Educational Resources Information Center

Ponce, Gregorio A.; Tuba, Imre

2015-01-01

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

An Application of the Vandermonde Determinant

ERIC Educational Resources Information Center

Xu, Junqin; Zhao, Likuan

2006-01-01

Eigenvalue is an important concept in Linear Algebra. It is well known that the eigenvectors corresponding to different eigenvalues of a square matrix are linear independent. In most of the existing textbooks, this result is proven using mathematical induction. In this note, a new proof using Vandermonde determinant is given. It is shown that this…

True orbit simulation of piecewise linear and linear fractional maps of arbitrary dimension using algebraic numbers

NASA Astrophysics Data System (ADS)

Saito, Asaki; Yasutomi, Shin-ichi; Tamura, Jun-ichi; Ito, Shunji

2015-06-01

We introduce a true orbit generation method enabling exact simulations of dynamical systems defined by arbitrary-dimensional piecewise linear fractional maps, including piecewise linear maps, with rational coefficients. This method can generate sufficiently long true orbits which reproduce typical behaviors (inherent behaviors) of these systems, by properly selecting algebraic numbers in accordance with the dimension of the target system, and involving only integer arithmetic. By applying our method to three dynamical systems—that is, the baker's transformation, the map associated with a modified Jacobi-Perron algorithm, and an open flow system—we demonstrate that it can reproduce their typical behaviors that have been very difficult to reproduce with conventional simulation methods. In particular, for the first two maps, we show that we can generate true orbits displaying the same statistical properties as typical orbits, by estimating the marginal densities of their invariant measures. For the open flow system, we show that an obtained true orbit correctly converges to the stable period-1 orbit, which is inherently possessed by the system.

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.

Why Representations?

ERIC Educational Resources Information Center

Schultz, James E.; Waters, Michael S.

2000-01-01

Discusses representations in the context of solving a system of linear equations. Views representations (concrete, tables, graphs, algebraic, matrices) from perspectives of understanding, technology, generalization, exact versus approximate solution, and learning style. (KHR)

Technology Tips: Investigating Extrema with GeoGebra

ERIC Educational Resources Information Center

Cullen, Craig J.; Hertel, Joshua T.; John, Sheryl

2013-01-01

The NCTM Algebra Standard suggests that students use technology to explore the effects of varying the parameters in y = ax2 + bx + c. This article discusses an extension of this task that incorporates dynamic geometry software to engage students in generating, testing, and proving mathematical conjectures.

Mathematical Methods for Optical Physics and Engineering

NASA Astrophysics Data System (ADS)

Gbur, Gregory J.

2011-01-01

1. Vector algebra; 2. Vector calculus; 3. Vector calculus in curvilinear coordinate systems; 4. Matrices and linear algebra; 5. Advanced matrix techniques and tensors; 6. Distributions; 7. Infinite series; 8. Fourier series; 9. Complex analysis; 10. Advanced complex analysis; 11. Fourier transforms; 12. Other integral transforms; 13. Discrete transforms; 14. Ordinary differential equations; 15. Partial differential equations; 16. Bessel functions; 17. Legendre functions and spherical harmonics; 18. Orthogonal functions; 19. Green's functions; 20. The calculus of variations; 21. Asymptotic techniques; Appendices; References; Index.

Plethystic vertex operators and boson-fermion correspondences

NASA Astrophysics Data System (ADS)

Fauser, Bertfried; Jarvis, Peter D.; King, Ronald C.

2016-10-01

We study the algebraic properties of plethystic vertex operators, introduced in (2010 J. Phys. A: Math. Theor. 43 405202), underlying the structure of symmetric functions associated with certain generalized universal character rings of subgroups of the general linear group, defined to stabilize tensors of Young symmetry type characterized by a partition of arbitrary shape π. Here we establish an extension of the well-known boson-fermion correspondence involving Schur functions and their associated (Bernstein) vertex operators: for each π, the modes generated by the plethystic vertex operators and their suitably constructed duals, satisfy the anticommutation relations of a complex Clifford algebra. The combinatorial manipulations underlying the results involve exchange identities exploiting the Hopf-algebraic structure of certain symmetric function series and their plethysms.

Symbolic derivation of high-order Rayleigh-Schroedinger perturbation energies using computer algebra: Application to vibrational-rotational analysis of diatomic molecules

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

Herbert, John M.

1997-01-01

Rayleigh-Schroedinger perturbation theory is an effective and popular tool for describing low-lying vibrational and rotational states of molecules. This method, in conjunction with ab initio techniques for computation of electronic potential energy surfaces, can be used to calculate first-principles molecular vibrational-rotational energies to successive orders of approximation. Because of mathematical complexities, however, such perturbation calculations are rarely extended beyond the second order of approximation, although recent work by Herbert has provided a formula for the nth-order energy correction. This report extends that work and furnishes the remaining theoretical details (including a general formula for the Rayleigh-Schroedinger expansion coefficients) necessary formore » calculation of energy corrections to arbitrary order. The commercial computer algebra software Mathematica is employed to perform the prohibitively tedious symbolic manipulations necessary for derivation of generalized energy formulae in terms of universal constants, molecular constants, and quantum numbers. As a pedagogical example, a Hamiltonian operator tailored specifically to diatomic molecules is derived, and the perturbation formulae obtained from this Hamiltonian are evaluated for a number of such molecules. This work provides a foundation for future analyses of polyatomic molecules, since it demonstrates that arbitrary-order perturbation theory can successfully be applied with the aid of commercially available computer algebra software.« less

Directed Abelian algebras and their application to stochastic models.

PubMed

Alcaraz, F C; Rittenberg, V

2008-10-01

With each directed acyclic graph (this includes some D-dimensional lattices) one can associate some Abelian algebras that we call directed Abelian algebras (DAAs). On each site of the graph one attaches a generator of the algebra. These algebras depend on several parameters and are semisimple. Using any DAA, one can define a family of Hamiltonians which give the continuous time evolution of a stochastic process. The calculation of the spectra and ground-state wave functions (stationary state probability distributions) is an easy algebraic exercise. If one considers D-dimensional lattices and chooses Hamiltonians linear in the generators, in finite-size scaling the Hamiltonian spectrum is gapless with a critical dynamic exponent z=D. One possible application of the DAA is to sandpile models. In the paper we present this application, considering one- and two-dimensional lattices. In the one-dimensional case, when the DAA conserves the number of particles, the avalanches belong to the random walker universality class (critical exponent sigma_(tau)=32 ). We study the local density of particles inside large avalanches, showing a depletion of particles at the source of the avalanche and an enrichment at its end. In two dimensions we did extensive Monte-Carlo simulations and found sigma_(tau)=1.780+/-0.005 .

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.

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.

Prediction of Transonic Vortex Flows Using Linear and Nonlinear Turbulent Eddy Viscosity Models

NASA Technical Reports Server (NTRS)

Bartels, Robert E.; Gatski, Thomas B.

2000-01-01

Three-dimensional transonic flow over a delta wing is investigated with a focus on the effect of transition and influence of turbulence stress anisotropies. The performance of linear eddy viscosity models and an explicit algebraic stress model is assessed at the start of vortex flow, and the results compared with experimental data. To assess the effect of transition location, computations that either fix transition or are fully turbulent are performed. To assess the effect of the turbulent stress anisotropy, comparisons are made between predictions from the algebraic stress model and the linear eddy viscosity models. Both transition location and turbulent stress anisotropy significantly affect the 3D flow field. The most significant effect is found to be the modeling of transition location. At a Mach number of 0.90, the computed solution changes character from steady to unsteady depending on transition onset. Accounting for the anisotropies in the turbulent stresses also considerably impacts the flow, most notably in the outboard region of flow separation.

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.

CAS as Environments for Implementing Mathematical Microworlds.

ERIC Educational Resources Information Center

Alpers, Burkhard

2002-01-01

Investigates whether computer algebra systems (CAS) are suitable environments for implementing mathematical microworlds. Recalls what constitutes a microworld and explores how CAS can be used for implementation, stating potentials as well as limitations. Provides as an example the microworld "Formula 1", implemented in Maple Software. (Author/KHR)

Generalizing a Categorization of Students' Interpretations of Linear Kinematics Graphs

ERIC Educational Resources Information Center

Bollen, Laurens; De Cock, Mieke; Zuza, Kristina; Guisasola, Jenaro; van Kampen, Paul

2016-01-01

We have investigated whether and how a categorization of responses to questions on linear distance-time graphs, based on a study of Irish students enrolled in an algebra-based course, could be adopted and adapted to responses from students enrolled in calculus-based physics courses at universities in Flanders, Belgium (KU Leuven) and the Basque…

Insights into the School Mathematics Tradition from Solving Linear Equations

ERIC Educational Resources Information Center

Buchbinder, Orly; Chazan, Daniel; Fleming, Elizabeth

2015-01-01

In this article, we explore how the solving of linear equations is represented in English­-language algebra text books from the early nineteenth century when schooling was becoming institutionalized, and then survey contemporary teachers. In the text books, we identify the increasing presence of a prescribed order of steps (a canonical method) for…

Biomedical Mathematics, Unit I: Measurement, Linear Functions and Dimensional Algebra. Student Text. Revised Version, 1975.

ERIC Educational Resources Information Center

Biomedical Interdisciplinary Curriculum Project, Berkeley, CA.

This text presents lessons relating specific mathematical concepts to the ideas, skills, and tasks pertinent to the health care field. Among other concepts covered are linear functions, vectors, trigonometry, and statistics. Many of the lessons use data acquired during science experiments as the basis for exercises in mathematics. Lessons present…

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…

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…

On Rank and Nullity

ERIC Educational Resources Information Center

Dobbs, David E.

2012-01-01

This note explains how Emil Artin's proof that row rank equals column rank for a matrix with entries in a field leads naturally to the formula for the nullity of a matrix and also to an algorithm for solving any system of linear equations in any number of variables. This material could be used in any course on matrix theory or linear algebra.

Opening the Door on Triangular Numbers

ERIC Educational Resources Information Center

McMartin, Kimberley; McMaster, Heather

2016-01-01

As an alternative to looking solely at linear functions, a three-lesson learning progression developed for Year 6 students that incorporates triangular numbers to develop children's algebraic thinking is described and evaluated.

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.

Finding Coefficients of the Full Array of Motion-Independent N-Body Potentials of Metric Gravity from Gravity's Exterior and Interior Effacement Algebra

NASA Astrophysics Data System (ADS)

Nordtvedt, Kenneth

2018-01-01

In the author's previous publications, a recursive linear algebraic method was introduced for obtaining (without gravitational radiation) the full potential expansions for the gravitational metric field components and the Lagrangian for a general N-body system. Two apparent properties of gravity— Exterior Effacement and Interior Effacement—were defined and fully enforced to obtain the recursive algebra, especially for the motion-independent potential expansions of the general N-body situation. The linear algebraic equations of this method determine the potential coefficients at any order n of the expansions in terms of the lower-order coefficients. Then, enforcing Exterior and Interior Effacement on a selecedt few potential series of the full motion-independent potential expansions, the complete exterior metric field for a single, spherically-symmetric mass source was obtained, producing the Schwarzschild metric field of general relativity. In this fourth paper of this series, the complete spatial metric's motion-independent potentials for N bodies are obtained using enforcement of Interior Effacement and knowledge of the Schwarzschild potentials. From the full spatial metric, the complete set of temporal metric potentials and Lagrangian potentials in the motion-independent case can then be found by transfer equations among the coefficients κ( n, α) → λ( n, ɛ) → ξ( n, α) with κ( n, α), λ( n, ɛ), ξ( n, α) being the numerical coefficients in the spatial metric, the Lagrangian, and the temporal metric potential expansions, respectively.

Pu239 Cross-Section Variations Based on Experimental Uncertainties and Covariances

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

Sigeti, David Edward; Williams, Brian J.; Parsons, D. Kent

2016-10-18

Algorithms and software have been developed for producing variations in plutonium-239 neutron cross sections based on experimental uncertainties and covariances. The varied cross-section sets may be produced as random samples from the multi-variate normal distribution defined by an experimental mean vector and covariance matrix, or they may be produced as Latin-Hypercube/Orthogonal-Array samples (based on the same means and covariances) for use in parametrized studies. The variations obey two classes of constraints that are obligatory for cross-section sets and which put related constraints on the mean vector and covariance matrix that detemine the sampling. Because the experimental means and covariances domore » not obey some of these constraints to sufficient precision, imposing the constraints requires modifying the experimental mean vector and covariance matrix. Modification is done with an algorithm based on linear algebra that minimizes changes to the means and covariances while insuring that the operations that impose the different constraints do not conflict with each other.« less

Use of Massive Parallel Computing Libraries in the Context of Global Gravity Field Determination from Satellite Data

NASA Astrophysics Data System (ADS)

Brockmann, J. M.; Schuh, W.-D.

2011-07-01

The estimation of the global Earth's gravity field parametrized as a finite spherical harmonic series is computationally demanding. The computational effort depends on the one hand on the maximal resolution of the spherical harmonic expansion (i.e. the number of parameters to be estimated) and on the other hand on the number of observations (which are several millions for e.g. observations from the GOCE satellite missions). To circumvent these restrictions, a massive parallel software based on high-performance computing (HPC) libraries as ScaLAPACK, PBLAS and BLACS was designed in the context of GOCE HPF WP6000 and the GOCO consortium. A prerequisite for the use of these libraries is that all matrices are block-cyclic distributed on a processor grid comprised by a large number of (distributed memory) computers. Using this set of standard HPC libraries has the benefit that once the matrices are distributed across the computer cluster, a huge set of efficient and highly scalable linear algebra operations can be used.

Experiments with conjugate gradient algorithms for homotopy curve tracking

NASA Technical Reports Server (NTRS)

Irani, Kashmira M.; Ribbens, Calvin J.; Watson, Layne T.; Kamat, Manohar P.; Walker, Homer F.

1991-01-01

There are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve, and has separate routines for dense and sparse Jacobian matrices. The HOMPACK algorithms for sparse Jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernel of the homotopy Jacobian matrix, a required linear algebra step for homotopy curve tracking. Here, variants of the conjugate gradient algorithm are implemented in the context of homotopy curve tracking and compared with Craig's preconditioned conjugate gradient method used in HOMPACK. The test problems used include actual large scale, sparse structural mechanics problems.

Control Strategies for Guided Collective Motion

DTIC Science & Technology

2015-02-27

Rorres and H. Anton , “ Elementary linear algebra applications version,” 9th Edition, Wiley India Pvt. Ltd., 2011. [20] S.H. Strogatz, “From Kuramoto to... linear cyclic pursuit in which an agent pursues its leader with an angle of deviation. The sufficient conditions for the stability of such systems are...Generalized Hierarchical Cyclic Pursuit 6. D. Mukherjee and D. Ghose: Deviated Linear Cyclic Pursuit 7. D. Mukherjee and D. Ghose; On Synchronous and

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

Pulse Height Analyzer Interfacing and Computer Programming in the Environmental Laser Propagation Project

DTIC Science & Technology

1976-06-01

United States Naval Postgraduate School, Monterey , California, 1974. 6. Anton , H., Elementary Linear Algebra , John Wiley & Sons, 1973. 7. Parrat, L. G...CONVERTER ln(laser & bias) PULSE HEIGHT ANALYZER © LINEAR AMPLIFIER SAMPLE TRIGGER OSCILLATOR early ln(laser & bias) SCINTILLOMETERS recent BACKGROUND...DEMODULATOR LASER CALIBRATION BOX LASER OR CAL VOLTAGE LOG CONVERTER LN (LASER OR CAL VOLT) LINEAR AMPLIFIER uLN (LASER OR CAL VOLT) PULSE HEIGHTEN ANALYZER V

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.

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.

Predicting NonInertial Effects with Algebraic Stress Models which Account for Dissipation Rate Anisotropies

NASA Technical Reports Server (NTRS)

Jongen, T.; Machiels, L.; Gatski, T. B.

1997-01-01

Three types of turbulence models which account for rotational effects in noninertial frames of reference are evaluated for the case of incompressible, fully developed rotating turbulent channel flow. The different types of models are a Coriolis-modified eddy-viscosity model, a realizable algebraic stress model, and an algebraic stress model which accounts for dissipation rate anisotropies. A direct numerical simulation of a rotating channel flow is used for the turbulent model validation. This simulation differs from previous studies in that significantly higher rotation numbers are investigated. Flows at these higher rotation numbers are characterized by a relaminarization on the cyclonic or suction side of the channel, and a linear velocity profile on the anticyclonic or pressure side of the channel. The predictive performance of the three types of models are examined in detail, and formulation deficiencies are identified which cause poor predictive performance for some of the models. Criteria are identified which allow for accurate prediction of such flows by algebraic stress models and their corresponding Reynolds stress formulations.

Rational solutions of CYBE for simple compact real Lie algebras

NASA Astrophysics Data System (ADS)

Pop, Iulia; Stolin, Alexander

2007-04-01

In [A.A. Stolin, On rational solutions of Yang-Baxter equation for sl(n), Math. Scand. 69 (1991) 57-80; A.A. Stolin, On rational solutions of Yang-Baxter equation. Maximal orders in loop algebra, Comm. Math. Phys. 141 (1991) 533-548; A. Stolin, A geometrical approach to rational solutions of the classical Yang-Baxter equation. Part I, in: Walter de Gruyter & Co. (Ed.), Symposia Gaussiana, Conf. Alg., Berlin, New York, 1995, pp. 347-357] a theory of rational solutions of the classical Yang-Baxter equation for a simple complex Lie algebra g was presented. We discuss this theory for simple compact real Lie algebras g. We prove that up to gauge equivalence all rational solutions have the form X(u,v)={Ω}/{u-v}+t1∧t2+⋯+t∧t2n, where Ω denotes the quadratic Casimir element of g and {ti} are linearly independent elements in a maximal torus t of g. The quantization of these solutions is also emphasized.

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.

Final Report: Subcontract B623868 Algebraic Multigrid solvers for coupled PDE systems

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

Brannick, J.

The Pennsylvania State University (“Subcontractor”) continued to work on the design of algebraic multigrid solvers for coupled systems of partial differential equations (PDEs) arising in numerical modeling of various applications, with a main focus on solving the Dirac equation arising in Quantum Chromodynamics (QCD). The goal of the proposed work was to develop combined geometric and algebraic multilevel solvers that are robust and lend themselves to efficient implementation on massively parallel heterogeneous computers for these QCD systems. The research in these areas built on previous works, focusing on the following three topics: (1) the development of parallel full-multigrid (PFMG) andmore » non-Galerkin coarsening techniques in this frame work for solving the Wilson Dirac system; (2) the use of these same Wilson MG solvers for preconditioning the Overlap and Domain Wall formulations of the Dirac equation; and (3) the design and analysis of algebraic coarsening algorithms for coupled PDE systems including Stokes equation, Maxwell equation and linear elasticity.« less