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…

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…

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…

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

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

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.

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.

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…

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.

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.

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

NASA Astrophysics Data System (ADS)

Seo, Jae Hong

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

An Inquiry-Based Linear Algebra Class

ERIC Educational Resources Information Center

Wang, Haohao; Posey, Lisa

2011-01-01

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

Numerical Linear Algebra.

DTIC Science & Technology

1980-09-08

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

Linear {GLP}-algebras and their elementary theories

NASA Astrophysics Data System (ADS)

Pakhomov, F. N.

2016-12-01

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

The Algebra of Sleepiness: Investigating the Interaction of Homeostatic (S) and Circadian (C) Processes in Sleepiness Using Linear Metrics"

ERIC Educational Resources Information Center

Mairesse, Olivier; Hofmans, Joeri; Neu, Daniel; Dinis Monica de Oliveira, Armando Luis; Cluydts, Raymond; Theuns, Peter

2010-01-01

The present studies were conducted to contribute to the debate on the interaction between circadian (C) and homeostatic (S) processes in models of sleep regulation. The Two-Process Model of Sleep Regulation assumes a linear relationship between processes S and C. However, recent elaborations of the model, based on data from forced desynchrony…

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

NASA Astrophysics Data System (ADS)

Caglayan, Günhan

2018-05-01

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

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.

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…

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.

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…

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…

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.

Numerical stability in problems of linear algebra.

NASA Technical Reports Server (NTRS)

Babuska, I.

1972-01-01

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

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.

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.

Parallel Algorithms for Least Squares and Related Computations.

DTIC Science & Technology

1991-03-22

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

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

ERIC Educational Resources Information Center

What Works Clearinghouse, 2009

2009-01-01

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

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

PubMed

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

2002-08-01

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

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.

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.

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

Commentary on A General Curriculum in Mathematics for Colleges.

ERIC Educational Resources Information Center

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

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

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

ERIC Educational Resources Information Center

Yildiz Ulus, Aysegul

2013-01-01

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

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

ERIC Educational Resources Information Center

Uhlig, Frank

2002-01-01

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

Constitutive relations in optics in terms of geometric algebra

NASA Astrophysics Data System (ADS)

Dargys, A.

2015-11-01

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

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

DTIC Science & Technology

1979-09-01

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

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

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…

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

An algebraic equation solution process formulated in anticipation of banded linear equations.

DOT National Transportation Integrated Search

1971-01-01

A general method for the solution of large, sparsely banded, positive-definite, coefficient matrices is presented. The goal in developing the method was to produce an efficient and reliable solution process and to provide the user-programmer with a p...

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.

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.

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.

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

Science.gov Websites

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

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

ERIC Educational Resources Information Center

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

2010-01-01

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

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.

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…

Ada Linear-Algebra Program

NASA Technical Reports Server (NTRS)

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

1988-01-01

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

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.

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…

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.

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

NASA Technical Reports Server (NTRS)

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

2008-01-01

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

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.

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…

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 wavelet-based ECG delineation algorithm for 32-bit integer online processing

PubMed Central

2011-01-01

Background Since the first well-known electrocardiogram (ECG) delineator based on Wavelet Transform (WT) presented by Li et al. in 1995, a significant research effort has been devoted to the exploitation of this promising method. Its ability to reliably delineate the major waveform components (mono- or bi-phasic P wave, QRS, and mono- or bi-phasic T wave) would make it a suitable candidate for efficient online processing of ambulatory ECG signals. Unfortunately, previous implementations of this method adopt non-linear operators such as root mean square (RMS) or floating point algebra, which are computationally demanding. Methods This paper presents a 32-bit integer, linear algebra advanced approach to online QRS detection and P-QRS-T waves delineation of a single lead ECG signal, based on WT. Results The QRS detector performance was validated on the MIT-BIH Arrhythmia Database (sensitivity Se = 99.77%, positive predictive value P+ = 99.86%, on 109010 annotated beats) and on the European ST-T Database (Se = 99.81%, P+ = 99.56%, on 788050 annotated beats). The ECG delineator was validated on the QT Database, showing a mean error between manual and automatic annotation below 1.5 samples for all fiducial points: P-onset, P-peak, P-offset, QRS-onset, QRS-offset, T-peak, T-offset, and a mean standard deviation comparable to other established methods. Conclusions The proposed algorithm exhibits reliable QRS detection as well as accurate ECG delineation, in spite of a simple structure built on integer linear algebra. PMID:21457580

A wavelet-based ECG delineation algorithm for 32-bit integer online processing.

PubMed

Di Marco, Luigi Y; Chiari, Lorenzo

2011-04-03

Since the first well-known electrocardiogram (ECG) delineator based on Wavelet Transform (WT) presented by Li et al. in 1995, a significant research effort has been devoted to the exploitation of this promising method. Its ability to reliably delineate the major waveform components (mono- or bi-phasic P wave, QRS, and mono- or bi-phasic T wave) would make it a suitable candidate for efficient online processing of ambulatory ECG signals. Unfortunately, previous implementations of this method adopt non-linear operators such as root mean square (RMS) or floating point algebra, which are computationally demanding. This paper presents a 32-bit integer, linear algebra advanced approach to online QRS detection and P-QRS-T waves delineation of a single lead ECG signal, based on WT. The QRS detector performance was validated on the MIT-BIH Arrhythmia Database (sensitivity Se = 99.77%, positive predictive value P+ = 99.86%, on 109010 annotated beats) and on the European ST-T Database (Se = 99.81%, P+ = 99.56%, on 788050 annotated beats). The ECG delineator was validated on the QT Database, showing a mean error between manual and automatic annotation below 1.5 samples for all fiducial points: P-onset, P-peak, P-offset, QRS-onset, QRS-offset, T-peak, T-offset, and a mean standard deviation comparable to other established methods. The proposed algorithm exhibits reliable QRS detection as well as accurate ECG delineation, in spite of a simple structure built on integer linear algebra.

Dual-scale topology optoelectronic processor.

PubMed

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

1991-12-15

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

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

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)

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…

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 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…

Motivating the Concept of Eigenvectors via Cryptography

ERIC Educational Resources Information Center

Siap, Irfan

2008-01-01

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

A Linear Algebra Measure of Cluster Quality.

ERIC Educational Resources Information Center

Mather, Laura A.

2000-01-01

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

The Transformation App Redux: The Notion of Linearity

ERIC Educational Resources Information Center

Domenick, Anthony

2015-01-01

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

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

PubMed

Casasent, D; Ghosh, A

1985-06-01

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

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.

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.

Numerical methods on some structured matrix algebra problems

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

Jessup, E.R.

1996-06-01

This proposal concerned the design, analysis, and implementation of serial and parallel algorithms for certain structured matrix algebra problems. It emphasized large order problems and so focused on methods that can be implemented efficiently on distributed-memory MIMD multiprocessors. Such machines supply the computing power and extensive memory demanded by the large order problems. We proposed to examine three classes of matrix algebra problems: the symmetric and nonsymmetric eigenvalue problems (especially the tridiagonal cases) and the solution of linear systems with specially structured coefficient matrices. As all of these are of practical interest, a major goal of this work was tomore » translate our research in linear algebra into useful tools for use by the computational scientists interested in these and related applications. Thus, in addition to software specific to the linear algebra problems, we proposed to produce a programming paradigm and library to aid in the design and implementation of programs for distributed-memory MIMD computers. We now report on our progress on each of the problems and on the programming tools.« less

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)…

QuBiLS-MIDAS: a parallel free-software for molecular descriptors computation based on multilinear algebraic maps.

PubMed

García-Jacas, César R; Marrero-Ponce, Yovani; Acevedo-Martínez, Liesner; Barigye, Stephen J; Valdés-Martiní, José R; Contreras-Torres, Ernesto

2014-07-05

The present report introduces the QuBiLS-MIDAS software belonging to the ToMoCoMD-CARDD suite for the calculation of three-dimensional molecular descriptors (MDs) based on the two-linear (bilinear), three-linear, and four-linear (multilinear or N-linear) algebraic forms. Thus, it is unique software that computes these tensor-based indices. These descriptors, establish relations for two, three, and four atoms by using several (dis-)similarity metrics or multimetrics, matrix transformations, cutoffs, local calculations and aggregation operators. The theoretical background of these N-linear indices is also presented. The QuBiLS-MIDAS software was developed in the Java programming language and employs the Chemical Development Kit library for the manipulation of the chemical structures and the calculation of the atomic properties. This software is composed by a desktop user-friendly interface and an Abstract Programming Interface library. The former was created to simplify the configuration of the different options of the MDs, whereas the library was designed to allow its easy integration to other software for chemoinformatics applications. This program provides functionalities for data cleaning tasks and for batch processing of the molecular indices. In addition, it offers parallel calculation of the MDs through the use of all available processors in current computers. The studies of complexity of the main algorithms demonstrate that these were efficiently implemented with respect to their trivial implementation. Lastly, the performance tests reveal that this software has a suitable behavior when the amount of processors is increased. Therefore, the QuBiLS-MIDAS software constitutes a useful application for the computation of the molecular indices based on N-linear algebraic maps and it can be used freely to perform chemoinformatics studies. Copyright © 2014 Wiley Periodicals, Inc.

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…

libdrdc: software standards library

NASA Astrophysics Data System (ADS)

Erickson, David; Peng, Tie

2008-04-01

This paper presents the libdrdc software standards library including internal nomenclature, definitions, units of measure, coordinate reference frames, and representations for use in autonomous systems research. This library is a configurable, portable C-function wrapped C++ / Object Oriented C library developed to be independent of software middleware, system architecture, processor, or operating system. It is designed to use the automatically-tuned linear algebra suite (ATLAS) and Basic Linear Algebra Suite (BLAS) and port to firmware and software. The library goal is to unify data collection and representation for various microcontrollers and Central Processing Unit (CPU) cores and to provide a common Application Binary Interface (ABI) for research projects at all scales. The library supports multi-platform development and currently works on Windows, Unix, GNU/Linux, and Real-Time Executive for Multiprocessor Systems (RTEMS). This library is made available under LGPL version 2.1 license.

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…

Using Cognitive Tutor Software in Learning Linear Algebra Word Concept

ERIC Educational Resources Information Center

Yang, Kai-Ju

2015-01-01

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

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

ERIC Educational Resources Information Center

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

2016-01-01

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

A Framework for Mathematical Thinking: The Case of Linear Algebra

ERIC Educational Resources Information Center

Stewart, Sepideh; Thomas, Michael O. J.

2009-01-01

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

Partially Flipped Linear Algebra: A Team-Based Approach

ERIC Educational Resources Information Center

Carney, Debra; Ormes, Nicholas; Swanson, Rebecca

2015-01-01

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

Definitions Are Important: The Case of Linear Algebra

ERIC Educational Resources Information Center

Berman, Abraham; Shvartsman, Ludmila

2016-01-01

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

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.)…

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

NASA Astrophysics Data System (ADS)

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

2018-01-01

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

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

NASA Astrophysics Data System (ADS)

Syarifuddin, H.

2018-04-01

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

Some aspects of mathematical and chemical modeling of complex chemical processes

NASA Technical Reports Server (NTRS)

Nemes, I.; Botar, L.; Danoczy, E.; Vidoczy, T.; Gal, D.

1983-01-01

Some theoretical questions involved in the mathematical modeling of the kinetics of complex chemical process are discussed. The analysis is carried out for the homogeneous oxidation of ethylbenzene in the liquid phase. Particular attention is given to the determination of the general characteristics of chemical systems from an analysis of mathematical models developed on the basis of linear algebra.

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…

Mardi Gras Math

ERIC Educational Resources Information Center

Eubanks-Turner, Christina; Hajj, Najat

2015-01-01

In this article, Christina Eubanks-Turner and Najat Hajj describe a planning process that they used to create a fun-filled eighth-grade math activity that focused on parade planning. The activity was designed to enhance and supplement the eighth-grade algebra curriculum on linear equations and functions, help students use mathematical habits of…

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.

The Processing and Mechanical Properties of High Temperature/ High Performance Composites. Fatigue and Creep. Book 3

DTIC Science & Technology

1994-03-01

bilinear forms of their rates. Setting the partial derivatives of fl with respect to the rates to be zero, one obtains simultaneous linear algebraic ...Figure 3 shows the variation in 8/P with P for one such test. In this case, the degree of linearity is high, with a correlation coefficient, r...each cycle is shown.) The linearity of the data suggests that the traction law can be represented by a power law, with the power law exponent, d log

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

NASA Astrophysics Data System (ADS)

Campoamor-Stursberg, R.

2018-03-01

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

Some Applications of Algebraic System Solving

ERIC Educational Resources Information Center

Roanes-Lozano, Eugenio

2011-01-01

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

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

ERIC Educational Resources Information Center

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

2015-01-01

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

Ten-Year-Old Students Solving Linear Equations

ERIC Educational Resources Information Center

Brizuela, Barbara; Schliemann, Analucia

2004-01-01

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

Measuring the Readability of Elementary Algebra Using the Cloze Technique.

ERIC Educational Resources Information Center

Kulm, Gerald

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

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 .

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

Acoustooptic linear algebra processors - Architectures, algorithms, and applications

NASA Technical Reports Server (NTRS)

Casasent, D.

1984-01-01

Architectures, algorithms, and applications for systolic processors are described with attention to the realization of parallel algorithms on various optical systolic array processors. Systolic processors for matrices with special structure and matrices of general structure, and the realization of matrix-vector, matrix-matrix, and triple-matrix products and such architectures are described. Parallel algorithms for direct and indirect solutions to systems of linear algebraic equations and their implementation on optical systolic processors are detailed with attention to the pipelining and flow of data and operations. Parallel algorithms and their optical realization for LU and QR matrix decomposition are specifically detailed. These represent the fundamental operations necessary in the implementation of least squares, eigenvalue, and SVD solutions. Specific applications (e.g., the solution of partial differential equations, adaptive noise cancellation, and optimal control) are described to typify the use of matrix processors in modern advanced signal processing.

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.

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.

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

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).

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.

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…

Optical pattern recognition algorithms on neural-logic equivalent models and demonstration of their prospects and possible implementations

NASA Astrophysics Data System (ADS)

Krasilenko, Vladimir G.; Nikolsky, Alexander I.; Zaitsev, Alexandr V.; Voloshin, Victor M.

2001-03-01

Historic information regarding the appearance and creation of fundamentals of algebra-logical apparatus-`equivalental algebra' for description of neuro-nets paradigms and algorithms is considered which is unification of theory of neuron nets (NN), linear algebra and the most generalized neuro-biology extended for matrix case. A survey is given of `equivalental models' of neuron nets and associative memory is suggested new, modified matrix-tenzor neurological equivalental models (MTNLEMS) are offered with double adaptive-equivalental weighing (DAEW) for spatial-non- invariant recognition (SNIR) and space-invariant recognition (SIR) of 2D images (patterns). It is shown, that MTNLEMS DAEW are the most generalized, they can describe the processes in NN both within the frames of known paradigms and within new `equivalental' paradigm of non-interaction type, and the computing process in NN under using the offered MTNLEMs DAEW is reduced to two-step and multi-step algorithms and step-by-step matrix-tenzor procedures (for SNIR) and procedures of defining of space-dependent equivalental functions from two images (for SIR).

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.

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.

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.

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.

A Mathematics Software Database Update.

ERIC Educational Resources Information Center

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

1987-01-01

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

College Algebra I.

ERIC Educational Resources Information Center

Benjamin, Carl; And Others

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

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

The Use of E-Portfolio in a Linear Algebra Course

ERIC Educational Resources Information Center

Torres, Judit Taberna; García-Planas, María Isabel; Domínguez-García, Santiago

2016-01-01

The use of e-portfolio becomes a standard tool when it comes to learning and student's assessment. This is due to the teachers need for enhancing their students' autonomy. The use of e-portfolio helps students to focus on their own learning process. Lectures should not be limited only to classes, but must foster active learning, and in this…

Introduction to the Discrete Fourier Series Considering Both Mathematical and Engineering Aspects--A Linear Algebra Approach

ERIC Educational Resources Information Center

Kohaupt, Ludwig

2015-01-01

The discrete Fourier series is a valuable tool developed and used by mathematicians and engineers alike. One of the most prominent applications is signal processing. Usually, it is important that the signals be transmitted fast, for example, when transmitting images over large distances such as between the moon and the earth or when generating…

Phylogenetic mixtures and linear invariants for equal input models.

PubMed

Casanellas, Marta; Steel, Mike

2017-04-01

The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the 'equal input model'. This model generalizes the 'Felsenstein 1981' model (and thereby the Jukes-Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a 'random cluster' process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees-the so called 'model invariants'), on any number n of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of [Formula: see text] leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167-191, 1987).

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

DOE PAGES

Song, Fengguang; Dongarra, Jack

2014-10-01

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

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

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

Song, Fengguang; Dongarra, Jack

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

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

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.

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

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

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

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

Simultaneous quantification of protein phosphorylation sites using liquid chromatography-tandem mass spectrometry-based targeted proteomics: a linear algebra approach for isobaric phosphopeptides.

PubMed

Xu, Feifei; Yang, Ting; Sheng, Yuan; Zhong, Ting; Yang, Mi; Chen, Yun

2014-12-05

As one of the most studied post-translational modifications (PTM), protein phosphorylation plays an essential role in almost all cellular processes. Current methods are able to predict and determine thousands of phosphorylation sites, whereas stoichiometric quantification of these sites is still challenging. Liquid chromatography coupled with tandem mass spectrometry (LC-MS/MS)-based targeted proteomics is emerging as a promising technique for site-specific quantification of protein phosphorylation using proteolytic peptides as surrogates of proteins. However, several issues may limit its application, one of which relates to the phosphopeptides with different phosphorylation sites and the same mass (i.e., isobaric phosphopeptides). While employment of site-specific product ions allows for these isobaric phosphopeptides to be distinguished and quantified, site-specific product ions are often absent or weak in tandem mass spectra. In this study, linear algebra algorithms were employed as an add-on to targeted proteomics to retrieve information on individual phosphopeptides from their common spectra. To achieve this simultaneous quantification, a LC-MS/MS-based targeted proteomics assay was first developed and validated for each phosphopeptide. Given the slope and intercept of calibration curves of phosphopeptides in each transition, linear algebraic equations were developed. Using a series of mock mixtures prepared with varying concentrations of each phosphopeptide, the reliability of the approach to quantify isobaric phosphopeptides containing multiple phosphorylation sites (≥ 2) was discussed. Finally, we applied this approach to determine the phosphorylation stoichiometry of heat shock protein 27 (HSP27) at Ser78 and Ser82 in breast cancer cells and tissue samples.

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

DOE PAGES

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

2018-02-06

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

Choosing order of operations to accelerate strip structure analysis in parameter range

NASA Astrophysics Data System (ADS)

Kuksenko, S. P.; Akhunov, R. R.; Gazizov, T. R.

2018-05-01

The paper considers the issue of using iteration methods in solving the sequence of linear algebraic systems obtained in quasistatic analysis of strip structures with the method of moments. Using the analysis of 4 strip structures, the authors have proved that additional acceleration (up to 2.21 times) of the iterative process can be obtained during the process of solving linear systems repeatedly by means of choosing a proper order of operations and a preconditioner. The obtained results can be used to accelerate the process of computer-aided design of various strip structures. The choice of the order of operations to accelerate the process is quite simple, universal and could be used not only for strip structure analysis but also for a wide range of computational problems.

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.

Numerical Problem Solving Using Mathcad in Undergraduate Reaction Engineering

ERIC Educational Resources Information Center

Parulekar, Satish J.

2006-01-01

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

On iterative processes in the Krylov-Sonneveld subspaces

NASA Astrophysics Data System (ADS)

Ilin, Valery P.

2016-10-01

The iterative Induced Dimension Reduction (IDR) methods are considered for solving large systems of linear algebraic equations (SLAEs) with nonsingular nonsymmetric matrices. These approaches are investigated by many authors and are charachterized sometimes as the alternative to the classical processes of Krylov type. The key moments of the IDR algorithms consist in the construction of the embedded Sonneveld subspaces, which have the decreasing dimensions and use the orthogonalization to some fixed subspace. Other independent approaches for research and optimization of the iterations are based on the augmented and modified Krylov subspaces by using the aggregation and deflation procedures with present various low rank approximations of the original matrices. The goal of this paper is to show, that IDR method in Sonneveld subspaces present an original interpretation of the modified algorithms in the Krylov subspaces. In particular, such description is given for the multi-preconditioned semi-conjugate direction methods which are actual for the parallel algebraic domain decomposition approaches.

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

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

Symbolic-numeric interface: A review

NASA Technical Reports Server (NTRS)

Ng, E. W.

1980-01-01

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

Quasi- and pseudo-maximum likelihood estimators for discretely observed continuous-time Markov branching processes

PubMed Central

Chen, Rui; Hyrien, Ollivier

2011-01-01

This article deals with quasi- and pseudo-likelihood estimation in a class of continuous-time multi-type Markov branching processes observed at discrete points in time. “Conventional” and conditional estimation are discussed for both approaches. We compare their properties and identify situations where they lead to asymptotically equivalent estimators. Both approaches possess robustness properties, and coincide with maximum likelihood estimation in some cases. Quasi-likelihood functions involving only linear combinations of the data may be unable to estimate all model parameters. Remedial measures exist, including the resort either to non-linear functions of the data or to conditioning the moments on appropriate sigma-algebras. The method of pseudo-likelihood may also resolve this issue. We investigate the properties of these approaches in three examples: the pure birth process, the linear birth-and-death process, and a two-type process that generalizes the previous two examples. Simulations studies are conducted to evaluate performance in finite samples. PMID:21552356

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

NASA Astrophysics Data System (ADS)

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

2018-05-01

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

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.

Proceedings of the Tenth Annual National Conference on Ada Technology. Held in Arlington, VA, on February 24-28, 1992

DTIC Science & Technology

1992-02-01

Newsletter, Vol. 5, No. 1, January 1983 be translated from HAL’S. 4. Klumpp, Allan R., An Ada Linear Algebra Software development costs for using the...a linear algebra approach to As noted above, the concept of the problem and address the problem of unitdimensional analysis extends beyond problems...you will join us again next year. The 11th Annual Conference on Ada Technology (1993) will be held here at the Hyatt Regency - Crystal City

An algebra of reversible computation.

PubMed

Wang, Yong

2016-01-01

We design an axiomatization for reversible computation called reversible ACP (RACP). It has four extendible modules: basic reversible processes algebra, algebra of reversible communicating processes, recursion and abstraction. Just like process algebra ACP in classical computing, RACP can be treated as an axiomatization foundation for reversible computation.

Modeling Stochastic Complexity in Complex Adaptive Systems: Non-Kolmogorov Probability and the Process Algebra Approach.

PubMed

Sulis, William H

2017-10-01

Walter Freeman III pioneered the application of nonlinear dynamical systems theories and methodologies in his work on mesoscopic brain dynamics.Sadly, mainstream psychology and psychiatry still cling to linear correlation based data analysis techniques, which threaten to subvert the process of experimentation and theory building. In order to progress, it is necessary to develop tools capable of managing the stochastic complexity of complex biopsychosocial systems, which includes multilevel feedback relationships, nonlinear interactions, chaotic dynamics and adaptability. In addition, however, these systems exhibit intrinsic randomness, non-Gaussian probability distributions, non-stationarity, contextuality, and non-Kolmogorov probabilities, as well as the absence of mean and/or variance and conditional probabilities. These properties and their implications for statistical analysis are discussed. An alternative approach, the Process Algebra approach, is described. It is a generative model, capable of generating non-Kolmogorov probabilities. It has proven useful in addressing fundamental problems in quantum mechanics and in the modeling of developing psychosocial systems.

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)

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.

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.

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…

AN ADA LINEAR ALGEBRA PACKAGE MODELED AFTER HAL/S

NASA Technical Reports Server (NTRS)

Klumpp, A. R.

1994-01-01

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

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.

LAPACKrc: Fast linear algebra kernels/solvers for FPGA accelerators

NASA Astrophysics Data System (ADS)

Gonzalez, Juan; Núñez, Rafael C.

2009-07-01

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

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.

Structural identifiability analyses of candidate models for in vitro Pitavastatin hepatic uptake.

PubMed

Grandjean, Thomas R B; Chappell, Michael J; Yates, James W T; Evans, Neil D

2014-05-01

In this paper a review of the application of four different techniques (a version of the similarity transformation approach for autonomous uncontrolled systems, a non-differential input/output observable normal form approach, the characteristic set differential algebra and a recent algebraic input/output relationship approach) to determine the structural identifiability of certain in vitro nonlinear pharmacokinetic models is provided. The Organic Anion Transporting Polypeptide (OATP) substrate, Pitavastatin, is used as a probe on freshly isolated animal and human hepatocytes. Candidate pharmacokinetic non-linear compartmental models have been derived to characterise the uptake process of Pitavastatin. As a prerequisite to parameter estimation, structural identifiability analyses are performed to establish that all unknown parameters can be identified from the experimental observations available. Copyright © 2013. Published by Elsevier Ireland Ltd.

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

Kitanidis, Peter

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

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.

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.

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

PubMed

Li, Ming; Zhao, Wei

2012-01-01

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

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

PubMed Central

Li, Ming; Zhao, Wei

2012-01-01

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

The 6th International Conference on Computer Science and Computational Mathematics (ICCSCM 2017)

NASA Astrophysics Data System (ADS)

2017-09-01

The ICCSCM 2017 (The 6th International Conference on Computer Science and Computational Mathematics) has aimed to provide a platform to discuss computer science and mathematics related issues including Algebraic Geometry, Algebraic Topology, Approximation Theory, Calculus of Variations, Category Theory; Homological Algebra, Coding Theory, Combinatorics, Control Theory, Cryptology, Geometry, Difference and Functional Equations, Discrete Mathematics, Dynamical Systems and Ergodic Theory, Field Theory and Polynomials, Fluid Mechanics and Solid Mechanics, Fourier Analysis, Functional Analysis, Functions of a Complex Variable, Fuzzy Mathematics, Game Theory, General Algebraic Systems, Graph Theory, Group Theory and Generalizations, Image Processing, Signal Processing and Tomography, Information Fusion, Integral Equations, Lattices, Algebraic Structures, Linear and Multilinear Algebra; Matrix Theory, Mathematical Biology and Other Natural Sciences, Mathematical Economics and Financial Mathematics, Mathematical Physics, Measure Theory and Integration, Neutrosophic Mathematics, Number Theory, Numerical Analysis, Operations Research, Optimization, Operator Theory, Ordinary and Partial Differential Equations, Potential Theory, Real Functions, Rings and Algebras, Statistical Mechanics, Structure Of Matter, Topological Groups, Wavelets and Wavelet Transforms, 3G/4G Network Evolutions, Ad-Hoc, Mobile, Wireless Networks and Mobile Computing, Agent Computing & Multi-Agents Systems, All topics related Image/Signal Processing, Any topics related Computer Networks, Any topics related ISO SC-27 and SC- 17 standards, Any topics related PKI(Public Key Intrastructures), Artifial Intelligences(A.I.) & Pattern/Image Recognitions, Authentication/Authorization Issues, Biometric authentication and algorithms, CDMA/GSM Communication Protocols, Combinatorics, Graph Theory, and Analysis of Algorithms, Cryptography and Foundation of Computer Security, Data Base(D.B.) Management & Information Retrievals, Data Mining, Web Image Mining, & Applications, Defining Spectrum Rights and Open Spectrum Solutions, E-Comerce, Ubiquitous, RFID, Applications, Fingerprint/Hand/Biometrics Recognitions and Technologies, Foundations of High-performance Computing, IC-card Security, OTP, and Key Management Issues, IDS/Firewall, Anti-Spam mail, Anti-virus issues, Mobile Computing for E-Commerce, Network Security Applications, Neural Networks and Biomedical Simulations, Quality of Services and Communication Protocols, Quantum Computing, Coding, and Error Controls, Satellite and Optical Communication Systems, Theory of Parallel Processing and Distributed Computing, Virtual Visions, 3-D Object Retrievals, & Virtual Simulations, Wireless Access Security, etc. The success of ICCSCM 2017 is reflected in the received papers from authors around the world from several countries which allows a highly multinational and multicultural idea and experience exchange. The accepted papers of ICCSCM 2017 are published in this Book. Please check http://www.iccscm.com for further news. A conference such as ICCSCM 2017 can only become successful using a team effort, so herewith we want to thank the International Technical Committee and the Reviewers for their efforts in the review process as well as their valuable advices. We are thankful to all those who contributed to the success of ICCSCM 2017. The Secretary

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

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.

Rank-based decompositions of morphological templates.

PubMed

Sussner, P; Ritter, G X

2000-01-01

Methods for matrix decomposition have found numerous applications in image processing, in particular for the problem of template decomposition. Since existing matrix decomposition techniques are mainly concerned with the linear domain, we consider it timely to investigate matrix decomposition techniques in the nonlinear domain with applications in image processing. The mathematical basis for these investigations is the new theory of rank within minimax algebra. Thus far, only minimax decompositions of rank 1 and rank 2 matrices into outer product expansions are known to the image processing community. We derive a heuristic algorithm for the decomposition of matrices having arbitrary rank.

Algebraic approach to solve ttbar dilepton equations

NASA Astrophysics Data System (ADS)

Sonnenschein, Lars

2006-01-01

The set of non-linear equations describing the Standard Model kinematics of the top quark an- tiqark production system in the dilepton decay channel has at most a four-fold ambiguity due to two not fully reconstructed neutrinos. Its most precise and robust solution is of major importance for measurements of top quark properties like the top quark mass and t t spin correlations. Simple algebraic operations allow to transform the non-linear equations into a system of two polynomial equations with two unknowns. These two polynomials of multidegree eight can in turn be an- alytically reduced to one polynomial with one unknown by means of resultants. The obtained univariate polynomial is of degree sixteen and the coefficients are free of any singularity. The number of its real solutions is determined analytically by means of Sturm’s theorem, which is as well used to isolate each real solution into a unique pairwise disjoint interval. The solutions are polished by seeking the sign change of the polynomial in a given interval through binary brack- eting. Further a new Ansatz - exploiting an accidental cancelation in the process of transforming the equations - is presented. It permits to transform the initial system of equations into two poly- nomial equations with two unknowns. These two polynomials of multidegree two can be reduced to one univariate polynomial of degree four by means of resultants. The obtained quartic equation can be solved analytically. The analytical solution has singularities which can be circumvented by the algebraic approach described above.

Efficient computer algebra algorithms for polynomial matrices in control design

NASA Technical Reports Server (NTRS)

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

1989-01-01

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

Communication: A reduced scaling J-engine based reformulation of SOS-MP2 using graphics processing units.

PubMed

Maurer, S A; Kussmann, J; Ochsenfeld, C

2014-08-07

We present a low-prefactor, cubically scaling scaled-opposite-spin second-order Møller-Plesset perturbation theory (SOS-MP2) method which is highly suitable for massively parallel architectures like graphics processing units (GPU). The scaling is reduced from O(N⁵) to O(N³) by a reformulation of the MP2-expression in the atomic orbital basis via Laplace transformation and the resolution-of-the-identity (RI) approximation of the integrals in combination with efficient sparse algebra for the 3-center integral transformation. In contrast to previous works that employ GPUs for post Hartree-Fock calculations, we do not simply employ GPU-based linear algebra libraries to accelerate the conventional algorithm. Instead, our reformulation allows to replace the rate-determining contraction step with a modified J-engine algorithm, that has been proven to be highly efficient on GPUs. Thus, our SOS-MP2 scheme enables us to treat large molecular systems in an accurate and efficient manner on a single GPU-server.

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

Descriptions of Free and Freeware Software in the Mathematics Teaching

NASA Astrophysics Data System (ADS)

Antunes de Macedo, Josue; Neves de Almeida, Samara; Voelzke, Marcos Rincon

2016-05-01

This paper presents the analysis and the cataloging of free and freeware mathematical software available on the internet, a brief explanation of them, and types of licenses for use in teaching and learning. The methodology is based on the qualitative research. Among the different types of software found, it stands out in algebra, the Winmat, that works with linear algebra, matrices and linear systems. In geometry, the GeoGebra, which can be used in the study of functions, plan and spatial geometry, algebra and calculus. For graphing, can quote the Graph and Graphequation. With Graphmatica software, it is possible to build various graphs of mathematical equations on the same screen, representing cartesian equations, inequalities, parametric among other functions. The Winplot allows the user to build graphics in two and three dimensions functions and mathematical equations. Thus, this work aims to present the teachers some free math software able to be used in the classroom.

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.

An Ada Linear-Algebra Software Package Modeled After HAL/S

NASA Technical Reports Server (NTRS)

Klumpp, Allan R.; Lawson, Charles L.

1990-01-01

New avionics software written more easily. Software package extends Ada programming language to include linear-algebra capabilities similar to those of HAL/S programming language. Designed for such avionics applications as Space Station flight software. In addition to built-in functions of HAL/S, package incorporates quaternion functions used in Space Shuttle and Galileo projects and routines from LINPAK solving systems of equations involving general square matrices. Contains two generic programs: one for floating-point computations and one for integer computations. Written on IBM/AT personal computer running under PC DOS, v.3.1.

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…

Assessing Algebraic Solving Ability: A Theoretical Framework

ERIC Educational Resources Information Center

Lian, Lim Hooi; Yew, Wun Thiam

2012-01-01

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

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.

Capitalizing on Basic Brain Processes in Developmental Algebra--Part 2

ERIC Educational Resources Information Center

Laughbaum, Edward D.

2011-01-01

Basic brain function is not a mystery. Given that neuroscientists understand its basic functioning processes, one wonders what their research suggests to teachers of developmental algebra. What if we knew how to teach so as to improve understanding of the algebra taught to developmental algebra students? What if we knew how the brain processes…

Capitalizing on Basic Brain Processes in Developmental Algebra--Part One

ERIC Educational Resources Information Center

Laughbaum, Edward D.

2011-01-01

Basic brain function is not a mystery. Given that neuroscientists understand the brain's basic functioning processes, one wonders what their research suggests to teachers of developmental algebra. What if we knew how to teach so as to improve understanding of the algebra taught to developmental algebra students? What if we knew how the brain…

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

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.

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.

Optical Data Processing for Missile Guidance.

DTIC Science & Technology

1984-11-21

and architectures for back -substitution and the solution of triangular systems of LAEs (linear algebraic equations). Most recently, a parallel QR...Calculation of I1 is quite difficult since the o T exact Z matrix is quite ill-conditioned. The two VC choices considered in our system are E - I and E I - 0...shown in fig. 1. It These operations are most commonly referred to as shows the ship in water with a sky and shoreline back - segmentation and also

Explorations in fuzzy physics and non-commutative geometry

NASA Astrophysics Data System (ADS)

Kurkcuoglu, Seckin

Fuzzy spaces arise as discrete approximations to continuum manifolds. They are usually obtained through quantizing coadjoint orbits of compact Lie groups and they can be described in terms of finite-dimensional matrix algebras, which for large matrix sizes approximate the algebra of functions of the limiting continuum manifold. Their ability to exactly preserve the symmetries of their parent manifolds is especially appealing for physical applications. Quantum Field Theories are built over them as finite-dimensional matrix models preserving almost all the symmetries of their respective continuum models. In this dissertation, we first focus our attention to the study of fuzzy supersymmetric spaces. In this regard, we obtain the fuzzy supersphere S2,2F through quantizing the supersphere, and demonstrate that it has exact supersymmetry. We derive a finite series formula for the *-product of functions over S2,2F and analyze the differential geometric information encoded in this formula. Subsequently, we show that quantum field theories on S2,2F are realized as finite-dimensional supermatrix models, and in particular we obtain the non-linear sigma model over the fuzzy supersphere by constructing the fuzzy supersymmetric extensions of a certain class of projectors. We show that this model too, is realized as a finite-dimensional supermatrix model with exact supersymmetry. Next, we show that fuzzy spaces have a generalized Hopf algebra structure. By focusing on the fuzzy sphere, we establish that there is a *-homomorphism from the group algebra SU(2)* of SU(2) to the fuzzy sphere. Using this and the canonical Hopf algebra structure of SU(2)* we show that both the fuzzy sphere and their direct sum are Hopf algebras. Using these results, we discuss processes in which a fuzzy sphere with angular momenta J splits into fuzzy spheres with angular momenta K and L. Finally, we study the formulation of Chern-Simons (CS) theory on an infinite strip of the non-commutative plane. We develop a finite-dimensional matrix model, whose large size limit approximates the CS theory on the infinite strip, and show that there are edge observables in this model obeying a finite-dimensional Lie algebra, that resembles the Kac-Moody algebra.

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.

Improving Machining Accuracy of CNC Machines with Innovative Design Methods

NASA Astrophysics Data System (ADS)

Yemelyanov, N. V.; Yemelyanova, I. V.; Zubenko, V. L.

2018-03-01

The article considers achieving the machining accuracy of CNC machines by applying innovative methods in modelling and design of machining systems, drives and machine processes. The topological method of analysis involves visualizing the system as matrices of block graphs with a varying degree of detail between the upper and lower hierarchy levels. This approach combines the advantages of graph theory and the efficiency of decomposition methods, it also has visual clarity, which is inherent in both topological models and structural matrices, as well as the resiliency of linear algebra as part of the matrix-based research. The focus of the study is on the design of automated machine workstations, systems, machines and units, which can be broken into interrelated parts and presented as algebraic, topological and set-theoretical models. Every model can be transformed into a model of another type, and, as a result, can be interpreted as a system of linear and non-linear equations which solutions determine the system parameters. This paper analyses the dynamic parameters of the 1716PF4 machine at the stages of design and exploitation. Having researched the impact of the system dynamics on the component quality, the authors have developed a range of practical recommendations which have enabled one to reduce considerably the amplitude of relative motion, exclude some resonance zones within the spindle speed range of 0...6000 min-1 and improve machining accuracy.

Efficient parallel linear scaling construction of the density matrix for Born-Oppenheimer molecular dynamics.

PubMed

Mniszewski, S M; Cawkwell, M J; Wall, M E; Mohd-Yusof, J; Bock, N; Germann, T C; Niklasson, A M N

2015-10-13

We present an algorithm for the calculation of the density matrix that for insulators scales linearly with system size and parallelizes efficiently on multicore, shared memory platforms with small and controllable numerical errors. The algorithm is based on an implementation of the second-order spectral projection (SP2) algorithm [ Niklasson, A. M. N. Phys. Rev. B 2002 , 66 , 155115 ] in sparse matrix algebra with the ELLPACK-R data format. We illustrate the performance of the algorithm within self-consistent tight binding theory by total energy calculations of gas phase poly(ethylene) molecules and periodic liquid water systems containing up to 15,000 atoms on up to 16 CPU cores. We consider algorithm-specific performance aspects, such as local vs nonlocal memory access and the degree of matrix sparsity. Comparisons to sparse matrix algebra implementations using off-the-shelf libraries on multicore CPUs, graphics processing units (GPUs), and the Intel many integrated core (MIC) architecture are also presented. The accuracy and stability of the algorithm are illustrated with long duration Born-Oppenheimer molecular dynamics simulations of 1000 water molecules and a 303 atom Trp cage protein solvated by 2682 water molecules.

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

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…

The exact fundamental solution for the Benes tracking problem

NASA Astrophysics Data System (ADS)

Balaji, Bhashyam

2009-05-01

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

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…

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.

Optical Linear Algebra for Computational Light Transport

NASA Astrophysics Data System (ADS)

O'Toole, Matthew

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

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.

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.

Communication: A reduced scaling J-engine based reformulation of SOS-MP2 using graphics processing units

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

Maurer, S. A.; Kussmann, J.; Ochsenfeld, C., E-mail: Christian.Ochsenfeld@cup.uni-muenchen.de

2014-08-07

We present a low-prefactor, cubically scaling scaled-opposite-spin second-order Møller-Plesset perturbation theory (SOS-MP2) method which is highly suitable for massively parallel architectures like graphics processing units (GPU). The scaling is reduced from O(N{sup 5}) to O(N{sup 3}) by a reformulation of the MP2-expression in the atomic orbital basis via Laplace transformation and the resolution-of-the-identity (RI) approximation of the integrals in combination with efficient sparse algebra for the 3-center integral transformation. In contrast to previous works that employ GPUs for post Hartree-Fock calculations, we do not simply employ GPU-based linear algebra libraries to accelerate the conventional algorithm. Instead, our reformulation allows tomore » replace the rate-determining contraction step with a modified J-engine algorithm, that has been proven to be highly efficient on GPUs. Thus, our SOS-MP2 scheme enables us to treat large molecular systems in an accurate and efficient manner on a single GPU-server.« less

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…

An Arithmetic-Algebraic Work Space for the Promotion of Arithmetic and Algebraic Thinking: Triangular Numbers

ERIC Educational Resources Information Center

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

2016-01-01

This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…

Students’ Algebraic Thinking Process in Context of Point and Line Properties

NASA Astrophysics Data System (ADS)

Nurrahmi, H.; Suryadi, D.; Fatimah, S.

2017-09-01

Learning of schools algebra is limited to symbols and operating procedures, so students are able to work on problems that only require the ability to operate symbols but unable to generalize a pattern as one of part of algebraic thinking. The purpose of this study is to create a didactic design that facilitates students to do algebraic thinking process through the generalization of patterns, especially in the context of the property of point and line. This study used qualitative method and includes Didactical Design Research (DDR). The result is students are able to make factual, contextual, and symbolic generalization. This happen because the generalization arises based on facts on local terms, then the generalization produced an algebraic formula that was described in the context and perspective of each student. After that, the formula uses the algebraic letter symbol from the symbol t hat uses the students’ language. It can be concluded that the design has facilitated students to do algebraic thinking process through the generalization of patterns, especially in the context of property of the point and line. The impact of this study is this design can use as one of material teaching alternative in learning of school algebra.

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.

A Electro-Optical Image Algebra Processing System for Automatic Target Recognition

NASA Astrophysics Data System (ADS)

Coffield, Patrick Cyrus

The proposed electro-optical image algebra processing system is designed specifically for image processing and other related computations. The design is a hybridization of an optical correlator and a massively paralleled, single instruction multiple data processor. The architecture of the design consists of three tightly coupled components: a spatial configuration processor (the optical analog portion), a weighting processor (digital), and an accumulation processor (digital). The systolic flow of data and image processing operations are directed by a control buffer and pipelined to each of the three processing components. The image processing operations are defined in terms of basic operations of an image algebra developed by the University of Florida. The algebra is capable of describing all common image-to-image transformations. The merit of this architectural design is how it implements the natural decomposition of algebraic functions into spatially distributed, point use operations. The effect of this particular decomposition allows convolution type operations to be computed strictly as a function of the number of elements in the template (mask, filter, etc.) instead of the number of picture elements in the image. Thus, a substantial increase in throughput is realized. The implementation of the proposed design may be accomplished in many ways. While a hybrid electro-optical implementation is of primary interest, the benefits and design issues of an all digital implementation are also discussed. The potential utility of this architectural design lies in its ability to control a large variety of the arithmetic and logic operations of the image algebra's generalized matrix product. The generalized matrix product is the most powerful fundamental operation in the algebra, thus allowing a wide range of applications. No other known device or design has made this claim of processing speed and general implementation of a heterogeneous image algebra.

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.

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

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.

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.

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

ERIC Educational Resources Information Center

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

2017-01-01

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

Quantum stochastic calculus associated with quadratic quantum noises

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

Ji, Un Cig, E-mail: uncigji@chungbuk.ac.kr; Sinha, Kalyan B., E-mail: kbs-jaya@yahoo.co.in

2016-02-15

We first study a class of fundamental quantum stochastic processes induced by the generators of a six dimensional non-solvable Lie †-algebra consisting of all linear combinations of the generalized Gross Laplacian and its adjoint, annihilation operator, creation operator, conservation, and time, and then we study the quantum stochastic integrals associated with the class of fundamental quantum stochastic processes, and the quantum Itô formula is revisited. The existence and uniqueness of solution of a quantum stochastic differential equation is proved. The unitarity conditions of solutions of quantum stochastic differential equations associated with the fundamental processes are examined. The quantum stochastic calculusmore » extends the Hudson-Parthasarathy quantum stochastic calculus.« less

Visual Salience of Algebraic Transformations

ERIC Educational Resources Information Center

Kirshner, David; Awtry, Thomas

2004-01-01

Information processing researchers have assumed that algebra symbol skills depend on mastery of the abstract rules presented in the curriculum (Matz, 1980; Sleeman, 1986). Thus, students' ubiquitous algebra errors have been taken as indicating the need to embed algebra in rich contextual settings (Kaput, 1995; National Council of Teachers of…

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

NASA Astrophysics Data System (ADS)

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

2018-04-01

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

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.

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

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

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.

Invariants of polarization transformations.

PubMed

Sadjadi, Firooz A

2007-05-20

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

Principles of Stagewise Separation Process Calculations: A Simple Algebraic Approach Using Solvent Extraction.

ERIC Educational Resources Information Center

Crittenden, Barry D.

1991-01-01

A simple liquid-liquid equilibrium (LLE) system involving a constant partition coefficient based on solute ratios is used to develop an algebraic understanding of multistage contacting in a first-year separation processes course. This algebraic approach to the LLE system is shown to be operable for the introduction of graphical techniques…

Highly Productive Application Development with ViennaCL for Accelerators

NASA Astrophysics Data System (ADS)

Rupp, K.; Weinbub, J.; Rudolf, F.

2012-12-01

The use of graphics processing units (GPUs) for the acceleration of general purpose computations has become very attractive over the last years, and accelerators based on many integrated CPU cores are about to hit the market. However, there are discussions about the benefit of GPU computing when comparing the reduction of execution times with the increased development effort [1]. To counter these concerns, our open-source linear algebra library ViennaCL [2,3] uses modern programming techniques such as generic programming in order to provide a convenient access layer for accelerator and GPU computing. Other GPU-accelerated libraries are primarily tuned for performance, but less tailored to productivity and portability: MAGMA [4] provides dense linear algebra operations via a LAPACK-comparable interface, but no dedicated matrix and vector types. Cusp [5] is closest in functionality to ViennaCL for sparse matrices, but is based on CUDA and thus restricted to devices from NVIDIA. However, no convenience layer for dense linear algebra is provided with Cusp. ViennaCL is written in C++ and uses OpenCL to access the resources of accelerators, GPUs and multi-core CPUs in a unified way. On the one hand, the library provides iterative solvers from the family of Krylov methods, including various preconditioners, for the solution of linear systems typically obtained from the discretization of partial differential equations. On the other hand, dense linear algebra operations are supported, including algorithms such as QR factorization and singular value decomposition. The user application interface of ViennaCL is compatible to uBLAS [6], which is part of the peer-reviewed Boost C++ libraries [7]. This allows to port existing applications based on uBLAS with a minimum of effort to ViennaCL. Conversely, the interface compatibility allows to use the iterative solvers from ViennaCL with uBLAS types directly, thus enabling code reuse beyond CPU-GPU boundaries. Out-of-the-box support for types from the Eigen library [8] and MTL 4 [9] are provided as well, enabling a seamless transition from single-core CPU to GPU and multi-core CPU computations. Case studies from the numerical solution of PDEs are given and isolated performance benchmarks are discussed. Also, pitfalls in scientific computing with GPUs and accelerators are addressed, allowing for a first evaluation of whether these novel devices can be mapped well to certain applications. References: [1] R. Bordawekar et al., Technical Report, IBM, 2010 [2] ViennaCL library. Online: http://viennacl.sourceforge.net/ [3] K. Rupp et al., GPUScA, 2010 [4] MAGMA library. Online: http://icl.cs.utk.edu/magma/ [5] Cusp library. Online: http://code.google.com/p/cusp-library/ [6] uBLAS library. Online: http://www.boost.org/libs/numeric/ublas/ [7] Boost C++ Libraries. Online: http://www.boost.org/ [8] Eigen library. Online: http://eigen.tuxfamily.org/ [9] MTL 4 Library. Online: http://www.mtl4.org/

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.

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.

Meanings Given to Algebraic Symbolism in Problem-Posing

ERIC Educational Resources Information Center

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

2018-01-01

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

Students’ Algebraic Reasonsing In Solving Mathematical Problems With Adversity Quotient

NASA Astrophysics Data System (ADS)

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

2018-01-01

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

Mastering algebra retrains the visual system to perceive hierarchical structure in equations.

PubMed

Marghetis, Tyler; Landy, David; Goldstone, Robert L

2016-01-01

Formal mathematics is a paragon of abstractness. It thus seems natural to assume that the mathematical expert should rely more on symbolic or conceptual processes, and less on perception and action. We argue instead that mathematical proficiency relies on perceptual systems that have been retrained to implement mathematical skills. Specifically, we investigated whether the visual system-in particular, object-based attention-is retrained so that parsing algebraic expressions and evaluating algebraic validity are accomplished by visual processing. Object-based attention occurs when the visual system organizes the world into discrete objects, which then guide the deployment of attention. One classic signature of object-based attention is better perceptual discrimination within, rather than between, visual objects. The current study reports that object-based attention occurs not only for simple shapes but also for symbolic mathematical elements within algebraic expressions-but only among individuals who have mastered the hierarchical syntax of algebra. Moreover, among these individuals, increased object-based attention within algebraic expressions is associated with a better ability to evaluate algebraic validity. These results suggest that, in mastering the rules of algebra, people retrain their visual system to represent and evaluate abstract mathematical structure. We thus argue that algebraic expertise involves the regimentation and reuse of evolutionarily ancient perceptual processes. Our findings implicate the visual system as central to learning and reasoning in mathematics, leading us to favor educational approaches to mathematics and related STEM fields that encourage students to adapt, not abandon, their use of perception.

A joint asymmetric watermarking and image encryption scheme

NASA Astrophysics Data System (ADS)

Boato, G.; Conotter, V.; De Natale, F. G. B.; Fontanari, C.

2008-02-01

Here we introduce a novel watermarking paradigm designed to be both asymmetric, i.e., involving a private key for embedding and a public key for detection, and commutative with a suitable encryption scheme, allowing both to cipher watermarked data and to mark encrypted data without interphering with the detection process. In order to demonstrate the effectiveness of the above principles, we present an explicit example where the watermarking part, based on elementary linear algebra, and the encryption part, exploiting a secret random permutation, are integrated in a commutative scheme.

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

Linear non-normality as the cause of nonlinear instability in LAPD

NASA Astrophysics Data System (ADS)

Friedman, Brett; Carter, Troy; Umansky, Maxim

2013-10-01

A BOUT + + simulation using a Braginskii fluid model reproduces drift-wave turbulence in LAPD with high qualitative and quantitative agreement. The turbulent fluctuations in the simulation sustain themselves through a nonlinear instability mechanism that injects energy into k|| = 0 fluctuations despite the fact that all of the linear eigenmodes at k|| = 0 are stable. The reason for this is the high non-orthogonality of the eigenmodes caused by the non-normality of the linear operator, which is common in fluid and plasma models that contain equilibrium gradients. While individual stable eigenmodes must decay when acted upon by their linear operator, the sum of the eigenmodes may grow transiently with initial algebraic time dependence. This transient growth can inject energy into the system, and the nonlinearities can remix the eigenmode amplitudes to self-sustain the growth. Such a mechanism also acts in subcritical neutral fluid turbulence, and the self-sustainment process is quite similar, indicating the universality of this nonlinear instability.

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.

A novel encryption scheme for high-contrast image data in the Fresnelet domain

PubMed Central

Bibi, Nargis; Farwa, Shabieh; Jahngir, Adnan; Usman, Muhammad

2018-01-01

In this paper, a unique and more distinctive encryption algorithm is proposed. This is based on the complexity of highly nonlinear S box in Flesnelet domain. The nonlinear pattern is transformed further to enhance the confusion in the dummy data using Fresnelet technique. The security level of the encrypted image boosts using the algebra of Galois field in Fresnelet domain. At first level, the Fresnelet transform is used to propagate the given information with desired wavelength at specified distance. It decomposes given secret data into four complex subbands. These complex sub-bands are separated into two components of real subband data and imaginary subband data. At second level, the net subband data, produced at the first level, is deteriorated to non-linear diffused pattern using the unique S-box defined on the Galois field F28. In the diffusion process, the permuted image is substituted via dynamic algebraic S-box substitution. We prove through various analysis techniques that the proposed scheme enhances the cipher security level, extensively. PMID:29608609

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

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

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

2014-05-30

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

A Comparison of Solver Performance for Complex Gastric Electrophysiology Models

PubMed Central

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

2016-01-01

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

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.

Development of a Computerized Adaptive Testing for Diagnosing the Cognitive Process of Grade 7 Students in Learning Algebra, Using Multidimensional Item Response Theory

ERIC Educational Resources Information Center

Senarat, Somprasong; Tayraukham, Sombat; Piyapimonsit, Chatsiri; Tongkhambanjong, Sakesan

2013-01-01

The purpose of this research is to develop a multidimensional computerized adaptive test for diagnosing the cognitive process of grade 7 students in learning algebra by applying multidimensional item response theory. The research is divided into 4 steps: 1) the development of item bank of algebra, 2) the development of the multidimensional…

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)

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.

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.

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

GPU-accelerated computation of electron transfer.

PubMed

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

2012-11-05

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

A Process Algebra Approach to Quantum Electrodynamics

NASA Astrophysics Data System (ADS)

Sulis, William

2017-12-01

The process algebra program is directed towards developing a realist model of quantum mechanics free of paradoxes, divergences and conceptual confusions. From this perspective, fundamental phenomena are viewed as emerging from primitive informational elements generated by processes. The process algebra has been shown to successfully reproduce scalar non-relativistic quantum mechanics (NRQM) without the usual paradoxes and dualities. NRQM appears as an effective theory which emerges under specific asymptotic limits. Space-time, scalar particle wave functions and the Born rule are all emergent in this framework. In this paper, the process algebra model is reviewed, extended to the relativistic setting, and then applied to the problem of electrodynamics. A semiclassical version is presented in which a Minkowski-like space-time emerges as well as a vector potential that is discrete and photon-like at small scales and near-continuous and wave-like at large scales. QED is viewed as an effective theory at small scales while Maxwell theory becomes an effective theory at large scales. The process algebra version of quantum electrodynamics is intuitive and realist, free from divergences and eliminates the distinction between particle, field and wave. Computations are carried out using the configuration space process covering map, although the connection to second quantization has not been fully explored.

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.

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…

The Effects of Multiple Linked Representations on Students' Learning of Linear Relationships

ERIC Educational Resources Information Center

Ozgun-Koca, S. Asli

2004-01-01

The focus of this study was on comparing three groups of Algebra I 9th-year students: one group using linked representation software, the second group using similar software but with semi-linked representations, and the control group in order to examine the effects on students' understanding of linear relationships. Data collection methods…

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…

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.

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.

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

Blooms' separation of the final exam of Engineering Mathematics II: Item reliability using Rasch measurement model

NASA Astrophysics Data System (ADS)

Fuaad, Norain Farhana Ahmad; Nopiah, Zulkifli Mohd; Tawil, Norgainy Mohd; Othman, Haliza; Asshaari, Izamarlina; Osman, Mohd Hanif; Ismail, Nur Arzilah

2014-06-01

In engineering studies and researches, Mathematics is one of the main elements which express physical, chemical and engineering laws. Therefore, it is essential for engineering students to have a strong knowledge in the fundamental of mathematics in order to apply the knowledge to real life issues. However, based on the previous results of Mathematics Pre-Test, it shows that the engineering students lack the fundamental knowledge in certain topics in mathematics. Due to this, apart from making improvements in the methods of teaching and learning, studies on the construction of questions (items) should also be emphasized. The purpose of this study is to assist lecturers in the process of item development and to monitor the separation of items based on Blooms' Taxonomy and to measure the reliability of the items itself usingRasch Measurement Model as a tool. By using Rasch Measurement Model, the final exam questions of Engineering Mathematics II (Linear Algebra) for semester 2 sessions 2012/2013 were analysed and the results will provide the details onthe extent to which the content of the item providesuseful information about students' ability. This study reveals that the items used in Engineering Mathematics II (Linear Algebra) final exam are well constructed but the separation of the items raises concern as it is argued that it needs further attention, as there is abig gap between items at several levels of Blooms' cognitive skill.

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

Carpenter, J.A.

This report is a sequel to ORNL/CSD-96 in the ongoing supplements to Professor A.S. Householder's KWIC Index for Numerical Algebra. With this supplement, the coverage has been restricted to Numerical Linear Algebra and is now roughly characterized by the American Mathematical Society's classification section 15 and 65F but with little coverage of inifinite matrices, matrices over fields of characteristics other than zero, operator theory, optimization and those parts of matrix theory primarily combinatorial in nature. Some recognition is made of the uses of graph theory in Numerical Linear Algebra, particularly as regards their use in algorithms for sparse matrix computations.more » The period covered by this report is roughly the calendar year 1981 as measured by the appearance of the articles in the American Mathematical Society's Contents of Mathematical Publications. The review citations are limited to the Mathematical Reviews (MR) and Das Zentralblatt fur Mathematik und Ihre Grenzgebiete (ZBL). Future reports will be made more timely by closer ovservation of the few journals which supply the bulk of the listings rather than what appears to be too much reliance on secondary sources. Some thought is being given to the physical appearance of these reports and the author welcomes comments concerning both their appearance and contents.« less

Algebraic solutions of shape-invariant position-dependent effective mass systems

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

Amir, Naila, E-mail: naila.amir@live.com, E-mail: naila.amir@seecs.edu.pk; Iqbal, Shahid, E-mail: sic80@hotmail.com, E-mail: siqbal@sns.nust.edu.pk

2016-06-15

Keeping in view the ordering ambiguity that arises due to the presence of position-dependent effective mass in the kinetic energy term of the Hamiltonian, a general scheme for obtaining algebraic solutions of quantum mechanical systems with position-dependent effective mass is discussed. We quantize the Hamiltonian of the pertaining system by using symmetric ordering of the operators concerning momentum and the spatially varying mass, initially proposed by von Roos and Lévy-Leblond. The algebraic method, used to obtain the solutions, is based on the concepts of supersymmetric quantum mechanics and shape invariance. In order to exemplify the general formalism a class ofmore » non-linear oscillators has been considered. This class includes the particular example of a one-dimensional oscillator with different position-dependent effective mass profiles. Explicit expressions for the eigenenergies and eigenfunctions in terms of generalized Hermite polynomials are presented. Moreover, properties of these modified Hermite polynomials, like existence of generating function and recurrence relations among the polynomials have also been studied. Furthermore, it has been shown that in the harmonic limit, all the results for the linear harmonic oscillator are recovered.« less

Algebraic Approaches for Scalable End-to-End Monitoring and Diagnosis

NASA Astrophysics Data System (ADS)

Zhao, Yao; Chen, Yan

The rigidity of the Internet architecture led to flourish in the research of end-to-end based systems. In this chapter, we describe a linear algebra-based end-to-end monitoring and diagnosis system. We first propose a tomography-based overlay monitoring system (TOM). Given n end hosts, TOM selectively monitors a basis set of O(nlogn) paths out of all n(n - 1) end-to-end paths. Any end-to-end path can be written as a unique linear combination of paths in the basis set. Consequently, by monitoring loss rates for the paths in the basis set, TOM infers loss rates for all end-to-end paths. Furthermore, leveraging on the scalable measurements from the TOM system, we propose the Least-biased End-to-End Network Diagnosis (in short, LEND) system. We define a minimal identifiable link sequence (MILS) as a link sequence of minimal length whose properties can be uniquely identified from end-to-end measurements. LEND applies an algebraic approach to find out the MILSes and infers the properties of the MILSes efficiently. This also means LEND system achieves the finest diagnosis granularity under the least biased statistical assumptions.

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

PubMed Central

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

2012-01-01

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

Computer Classification of Triangles and Quadrilaterals--A Challenging Application

ERIC Educational Resources Information Center

Dennis, J. Richard

1978-01-01

Two computer exercises involving the classification of geometric figures are given. The mathematics required is relatively simple but comes from several areas--synthetic geometry, analytic geometry, and linear algebra. (MN)

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

PubMed

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

2011-03-01

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

Diagnosing students' misconceptions in algebra: results from an experimental pilot study.

PubMed

Russell, Michael; O'Dwyer, Laura M; Miranda, Helena

2009-05-01

Computer-based diagnostic assessment systems hold potential to help teachers identify sources of poor performance and to connect teachers and students to learning activities designed to help advance students' conceptual understandings. The present article presents findings from a study that examined how students' performance in algebra and their overcoming of common algebraic misconceptions were affected by the use of a diagnostic assessment system that focused on important algebra concepts. This study used a four-group randomized cluster trial design in which teachers were assigned randomly to one of four groups: a "business as usual" control group, a partial intervention group that was provided with access to diagnostic tests results, a partial intervention group that was provided with access to the learning activities, and a full intervention group that was given access to the test results and learning activities. Data were collected from 905 students (6th-12th grade) nested within 44 teachers. We used hierarchical linear modeling techniques to compare the effects of full, partial, and no (control) intervention on students' algebraic ability and misconceptions. The analyses indicate that full intervention had a net positive effect on ability and misconception measures.

Literal algebra for satellite dynamics. [perturbation analysis

NASA Technical Reports Server (NTRS)

Gaposchkin, E. M.

1975-01-01

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

Artificial Neural Networks: an overview and their use in the analysis of the AMPHORA-3 dataset.

PubMed

Buscema, Paolo Massimo; Massini, Giulia; Maurelli, Guido

2014-10-01

The Artificial Adaptive Systems (AAS) are theories with which generative algebras are able to create artificial models simulating natural phenomenon. Artificial Neural Networks (ANNs) are the more diffused and best-known learning system models in the AAS. This article describes an overview of ANNs, noting its advantages and limitations for analyzing dynamic, complex, non-linear, multidimensional processes. An example of a specific ANN application to alcohol consumption in Spain, as part of the EU AMPHORA-3 project, during 1961-2006 is presented. Study's limitations are noted and future needed research using ANN methodologies are suggested.

Localization and stationary phase approximation on supermanifolds

NASA Astrophysics Data System (ADS)

Zakharevich, Valentin

2017-08-01

Given an odd vector field Q on a supermanifold M and a Q-invariant density μ on M, under certain compactness conditions on Q, the value of the integral ∫Mμ is determined by the value of μ on any neighborhood of the vanishing locus N of Q. We present a formula for the integral in the case where N is a subsupermanifold which is appropriately non-degenerate with respect to Q. In the process, we discuss the linear algebra necessary to express our result in a coordinate independent way. We also extend the stationary phase approximation and the Morse-Bott lemma to supermanifolds.

Edge-based nonlinear diffusion for finite element approximations of convection-diffusion equations and its relation to algebraic flux-correction schemes.

PubMed

Barrenechea, Gabriel R; Burman, Erik; Karakatsani, Fotini

2017-01-01

For the case of approximation of convection-diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions.

Yangians and Yang-Baxter R-operators for ortho-symplectic superalgebras

NASA Astrophysics Data System (ADS)

Fuksa, J.; Isaev, A. P.; Karakhanyan, D.; Kirschner, R.

2017-04-01

Yang-Baxter relations symmetric with respect to the ortho-symplectic superalgebras are studied. We start with the formulation of graded algebras and the linear superspace carrying the vector (fundamental) representation of the ortho-symplectic supergroup. On this basis we study the analogy of the Yang-Baxter operators considered earlier for the cases of orthogonal and symplectic symmetries: the vector (fundamental) R-matrix, the L-operator defining the Yangian algebra and its first and second order evaluations. We investigate the condition for L (u) in the case of the truncated expansion in inverse powers of u and give examples of Lie algebra representations obeying these conditions. We construct the R-operator intertwining two superspinor representations and study the fusion of L-operators involving the tensor product of such representations.

A spatial operator algebra for manipulator modeling and control

NASA Technical Reports Server (NTRS)

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

1989-01-01

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

Lumped Model Generation and Evaluation: Sensitivity and Lie Algebraic Techniques with Applications to Combustion

DTIC Science & Technology

1989-03-03

address global parameter space mapping issues for first order differential equations. The rigorous criteria for the existence of exact lumping by linear projective transformations was also established.

A computing method for sound propagation through a nonuniform jet stream

NASA Technical Reports Server (NTRS)

Padula, S. L.; Liu, C. H.

1974-01-01

The classical formulation of sound propagation through a jet flow was found to be inadequate for computer solutions. Previous investigations selected the phase and amplitude of the acoustic pressure as dependent variables requiring the solution of a system of nonlinear algebraic equations. The nonlinearities complicated both the analysis and the computation. A reformulation of the convective wave equation in terms of a new set of dependent variables is developed with a special emphasis on its suitability for numerical solutions on fast computers. The technique is very attractive because the resulting equations are linear in nonwaving variables. The computer solution to such a linear system of algebraic equations may be obtained by well-defined and direct means which are conservative of computer time and storage space. Typical examples are illustrated and computational results are compared with available numerical and experimental data.

G-sequentially connectedness for topological groups with operations

NASA Astrophysics Data System (ADS)

Mucuk, Osman; Cakalli, Huseyin

2016-08-01

It is a well-known fact that for a Hausdorff topological group X, the limits of convergent sequences in X define a function denoted by lim from the set of all convergent sequences in X to X. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing lim with an arbitrary linear functional G defined on a linear subspace of the vector space of all real sequences. Recently some authors have extended the concept to the topological group setting and introduced the concepts of G-sequential continuity, G-sequential compactness and G-sequential connectedness. In this work, we present some results about G-sequentially closures, G-sequentially connectedness and fundamental system of G-sequentially open neighbourhoods for topological group with operations which include topological groups, topological rings without identity, R-modules, Lie algebras, Jordan algebras, and many others.

Deformed coset models from gauged WZW actions

NASA Astrophysics Data System (ADS)

Park, Q.-Han

1994-06-01

A general Lagrangian formulation of integrably deformed G/H-coset models is given. We consider the G/H-coset model in terms of the gauged Wess-Zumino-Witten action and obtain an integrable deformation by adding a potential energy term Tr(gTg -1overlineT) , where algebra elements T, overlineT belong to the center of the algebra h associated with the subgroup H. We show that the classical equation of motion of the deformed coset model can be identified with the integrability condition of certain linear equations which makes the use of the inverse scattering method possible. Using the linear equation, we give a systematic way to construct infinitely many conserved currents as well as soliton solutions. In the case of the parafermionic SU(2)/U(1)-coset model, we derive n-solitons and conserved currents explicitly.

Using linear algebra for protein structural comparison and classification

PubMed Central

2009-01-01

In this article, we describe a novel methodology to extract semantic characteristics from protein structures using linear algebra in order to compose structural signature vectors which may be used efficiently to compare and classify protein structures into fold families. These signatures are built from the pattern of hydrophobic intrachain interactions using Singular Value Decomposition (SVD) and Latent Semantic Indexing (LSI) techniques. Considering proteins as documents and contacts as terms, we have built a retrieval system which is able to find conserved contacts in samples of myoglobin fold family and to retrieve these proteins among proteins of varied folds with precision of up to 80%. The classifier is a web tool available at our laboratory website. Users can search for similar chains from a specific PDB, view and compare their contact maps and browse their structures using a JMol plug-in. PMID:21637532

Using linear algebra for protein structural comparison and classification.

PubMed

Gomide, Janaína; Melo-Minardi, Raquel; Dos Santos, Marcos Augusto; Neshich, Goran; Meira, Wagner; Lopes, Júlio César; Santoro, Marcelo

2009-07-01

In this article, we describe a novel methodology to extract semantic characteristics from protein structures using linear algebra in order to compose structural signature vectors which may be used efficiently to compare and classify protein structures into fold families. These signatures are built from the pattern of hydrophobic intrachain interactions using Singular Value Decomposition (SVD) and Latent Semantic Indexing (LSI) techniques. Considering proteins as documents and contacts as terms, we have built a retrieval system which is able to find conserved contacts in samples of myoglobin fold family and to retrieve these proteins among proteins of varied folds with precision of up to 80%. The classifier is a web tool available at our laboratory website. Users can search for similar chains from a specific PDB, view and compare their contact maps and browse their structures using a JMol plug-in.

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.

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

NASA Technical Reports Server (NTRS)

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

1990-01-01

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

Algebras Generated by Geometric Scalar Forms and their Applications in Physics and Social Sciences

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

Keller, Jaime

2008-09-17

The present paper analyzes the consequences of defining that the geometric scalar form is not necessarily quadratic, but in general K-atic, that is obtained from the K{sup th} power of the linear form, requiring {l_brace}e{sub i};i = 1,...,N;(e{sub i}){sup K} = 1{r_brace} and d-vector {sigma}{sub i}x{sub i}e{sub i}. We consider the algebras which are thus generated, for positive integer K, a generalization of the geometric algebras we know under the names of Clifford or Grassmann algebras. We then obtain a set of geometric K-algebras. We also consider the generalization of special functions of geometry which corresponds to the K-order scalarmore » forms (as trigonometric functions and other related geometric functions which are based on the use of quadratic forms). We present an overview of the use of quadratic forms in physics as in our general theory, we have called START. And, in order to give an introduction to the use of the more general K-algebras and to the possible application to sciences other than physics, the application to social sciences is considered.For the applications to physics we show that quadratic spaces are a fundamental clue to understand the structure of theoretical physics (see, for example, Keller in ICNAAM 2005 and 2006)« less

Individual Differences in Algebraic Cognition: Relation to the Approximate Number and Sematic Memory Systems

PubMed Central

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

2015-01-01

The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 (92 girls) 9th graders, controlling parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x,y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation, but not schema memory. Frequency of fact-retrieval errors was related to schema memory but not coordinate plane or expression evaluation accuracy. The results suggest the ANS may contribute to or is influenced by spatial-numerical and numerical only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest different brain and cognitive systems are engaged during the learning of different components of algebraic competence, controlling demographic and domain general abilities. PMID:26255604

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

ERIC Educational Resources Information Center

Strickland, Tricia K.; Maccini, Paula

2013-01-01

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

Applications of a Sequence of Points in Teaching Linear Algebra, Numerical Methods and Discrete Mathematics

ERIC Educational Resources Information Center

Shi, Yixun

2009-01-01

Based on a sequence of points and a particular linear transformation generalized from this sequence, two recent papers (E. Mauch and Y. Shi, "Using a sequence of number pairs as an example in teaching mathematics". Math. Comput. Educ., 39 (2005), pp. 198-205; Y. Shi, "Case study projects for college mathematics courses based on a particular…

How Intuition and Language Use Relate to Students' Understanding of Span and Linear Independence in an Elementary Linear Algebra Class

ERIC Educational Resources Information Center

Parker, Catherine Frieda

2010-01-01

A possible contributing factor to students' difficulty in learning advanced mathematics is the conflict between students' "natural" learning styles and the formal structure of mathematics, which is based on definitions, theorems, and proofs. Students' natural learning styles may be a function of their intuition and language skills. The purpose of…

Roots of Linear Algebra: An Historical Exploration of Linear Systems

ERIC Educational Resources Information Center

Andrews-Larson, Christine

2015-01-01

There is a long-standing tradition in mathematics education to look to history to inform instruction. An historical analysis of the genesis of a mathematical idea offers insight into: (i) the contexts that give rise to a need for a mathematical construct; (ii) the ways in which available tools might shape the development of that mathematical idea;…

An Extended Microcomputer-Based Network Optimization Package.

DTIC Science & Technology

1982-10-01

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

On Generalizations of Cochran’s Theorem and Projection Matrices.

DTIC Science & Technology

1980-08-01

Definiteness of the Estimated Dispersion Matrix in a Multivariate Linear Model ," F. Pukelsheim and George P.H. Styan, May 1978. TECHNICAL REPORTS...with applications to the analysis of covariance," Proc. Cambridge Philos. Soc., 30, pp. 178-191. Graybill , F. A. and Marsaglia, G. (1957...34Idempotent matrices and quad- ratic forms in the general linear hypothesis," Ann. Math. Statist., 28, pp. 678-686. Greub, W. (1975). Linear Algebra (4th ed

Analysis of Information Content in High-Spectral Resolution Sounders using Subset Selection Analysis

NASA Technical Reports Server (NTRS)

Velez-Reyes, Miguel; Joiner, Joanna

1998-01-01

In this paper, we summarize the results of the sensitivity analysis and data reduction carried out to determine the information content of AIRS and IASI channels. The analysis and data reduction was based on the use of subset selection techniques developed in the linear algebra and statistical community to study linear dependencies in high dimensional data sets. We applied the subset selection method to study dependency among channels by studying the dependency among their weighting functions. Also, we applied the technique to study the information provided by the different levels in which the atmosphere is discretized for retrievals and analysis. Results from the method correlate well with intuition in many respects and point out to possible modifications for band selection in sensor design and number and location of levels in the analysis process.

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

PubMed Central

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

2014-01-01

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

Technical report. The application of probability-generating functions to linear-quadratic radiation survival curves.

PubMed

Kendal, W S

2000-04-01

To illustrate how probability-generating functions (PGFs) can be employed to derive a simple probabilistic model for clonogenic survival after exposure to ionizing irradiation. Both repairable and irreparable radiation damage to DNA were assumed to occur by independent (Poisson) processes, at intensities proportional to the irradiation dose. Also, repairable damage was assumed to be either repaired or further (lethally) injured according to a third (Bernoulli) process, with the probability of lethal conversion being directly proportional to dose. Using the algebra of PGFs, these three processes were combined to yield a composite PGF that described the distribution of lethal DNA lesions in irradiated cells. The composite PGF characterized a Poisson distribution with mean, chiD+betaD2, where D was dose and alpha and beta were radiobiological constants. This distribution yielded the conventional linear-quadratic survival equation. To test the composite model, the derived distribution was used to predict the frequencies of multiple chromosomal aberrations in irradiated human lymphocytes. The predictions agreed well with observation. This probabilistic model was consistent with single-hit mechanisms, but it was not consistent with binary misrepair mechanisms. A stochastic model for radiation survival has been constructed from elementary PGFs that exactly yields the linear-quadratic relationship. This approach can be used to investigate other simple probabilistic survival models.

Linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

NASA Astrophysics Data System (ADS)

Camporesi, Roberto

2011-06-01

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 the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and the variation of constants method. The approach presented here can be used in a first course on differential equations for science and engineering majors.

Complete characterization of fourth-order symplectic integrators with extended-linear coefficients.

PubMed

Chin, Siu A

2006-02-01

The structure of symplectic integrators up to fourth order can be completely and analytically understood when the factorization (split) coefficients are related linearly but with a uniform nonlinear proportional factor. The analytic form of these extended-linear symplectic integrators greatly simplified proofs of their general properties and allowed easy construction of both forward and nonforward fourth-order algorithms with an arbitrary number of operators. Most fourth-order forward integrators can now be derived analytically from this extended-linear formulation without the use of symbolic algebra.

Development process of in-service training intended for teachers to perform teaching of mathematics with computer algebra systems

NASA Astrophysics Data System (ADS)

Ardıç, Mehmet Alper; Işleyen, Tevfik

2018-01-01

In this study, we deal with the development process of in-service training activities designed in order for mathematics teachers of secondary education to realize teaching of mathematics, utilizing computer algebra systems. In addition, the results obtained from the researches carried out during and after the in-service training were summarized. Last section focuses on suggestions any teacher can use to carry out activities aimed at using computer algebra systems in teaching environments.

Development of a GNSS water vapour tomography system using algebraic reconstruction techniques

NASA Astrophysics Data System (ADS)

Bender, Michael; Dick, Galina; Ge, Maorong; Deng, Zhiguo; Wickert, Jens; Kahle, Hans-Gert; Raabe, Armin; Tetzlaff, Gerd

2011-05-01

A GNSS water vapour tomography system developed to reconstruct spatially resolved humidity fields in the troposphere is described. The tomography system was designed to process the slant path delays of about 270 German GNSS stations in near real-time with a temporal resolution of 30 min, a horizontal resolution of 40 km and a vertical resolution of 500 m or better. After a short introduction to the GPS slant delay processing the framework of the GNSS tomography is described in detail. Different implementations of the iterative algebraic reconstruction techniques (ART) used to invert the linear inverse problem are discussed. It was found that the multiplicative techniques (MART) provide the best results with least processing time, i.e., a tomographic reconstruction of about 26,000 slant delays on a 8280 cell grid can be obtained in less than 10 min. Different iterative reconstruction techniques are compared with respect to their convergence behaviour and some numerical parameters. The inversion can be considerably stabilized by using additional non-GNSS observations and implementing various constraints. Different strategies for initialising the tomography and utilizing extra information are discussed. At last an example of a reconstructed field of the wet refractivity is presented and compared to the corresponding distribution of the integrated water vapour, an analysis of a numerical weather model (COSMO-DE) and some radiosonde profiles.

Chaos, Fractals, and Polynomials.

ERIC Educational Resources Information Center

Tylee, J. Louis; Tylee, Thomas B.

1996-01-01

Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)

Mathematics Conceptual Visualization with HyperCard.

ERIC Educational Resources Information Center

Haws, LaDawn

1992-01-01

Hypermedia provides an easy-to-use option for adding visualization, via the computer, to the classroom. Some examples of this medium are presented, including applications in basic linear algebra and calculus, and a tutorial in electromagnetism. (Author)

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

NASA Astrophysics Data System (ADS)

Taqiyuddin, Muhammad; Sumiaty, Encum; Jupri, Al

2017-08-01

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

Linear or linearizable first-order delay ordinary differential equations and their Lie point symmetries

NASA Astrophysics Data System (ADS)

Dorodnitsyn, Vladimir A.; Kozlov, Roman; Meleshko, Sergey V.; Winternitz, Pavel

2018-05-01

A recent article was devoted to an analysis of the symmetry properties of a class of first-order delay ordinary differential systems (DODSs). Here we concentrate on linear DODSs, which have infinite-dimensional Lie point symmetry groups due to the linear superposition principle. Their symmetry algebra always contains a two-dimensional subalgebra realized by linearly connected vector fields. We identify all classes of linear first-order DODSs that have additional symmetries, not due to linearity alone, and we present representatives of each class. These additional symmetries are then used to construct exact analytical particular solutions using symmetry reduction.

General purpose graphic processing unit implementation of adaptive pulse compression algorithms

NASA Astrophysics Data System (ADS)

Cai, Jingxiao; Zhang, Yan

2017-07-01

This study introduces a practical approach to implement real-time signal processing algorithms for general surveillance radar based on NVIDIA graphical processing units (GPUs). The pulse compression algorithms are implemented using compute unified device architecture (CUDA) libraries such as CUDA basic linear algebra subroutines and CUDA fast Fourier transform library, which are adopted from open source libraries and optimized for the NVIDIA GPUs. For more advanced, adaptive processing algorithms such as adaptive pulse compression, customized kernel optimization is needed and investigated. A statistical optimization approach is developed for this purpose without needing much knowledge of the physical configurations of the kernels. It was found that the kernel optimization approach can significantly improve the performance. Benchmark performance is compared with the CPU performance in terms of processing accelerations. The proposed implementation framework can be used in various radar systems including ground-based phased array radar, airborne sense and avoid radar, and aerospace surveillance radar.

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.

Measurement theory in local quantum physics

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

Okamura, Kazuya, E-mail: okamura@math.cm.is.nagoya-u.ac.jp; Ozawa, Masanao, E-mail: ozawa@is.nagoya-u.ac.jp

In this paper, we aim to establish foundations of measurement theory in local quantum physics. For this purpose, we discuss a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras. We introduce a condition called the normal extension property (NEP) and establish a one-to-one correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes. We show that every CP instrument on an atomic von Neumann algebra has the NEP, extending the well-known result for type I factors. Moreover, we show that every CP instrument on an injective von Neumann algebra is approximated bymore » CP instruments with the NEP. The concept of posterior states is also discussed to show that the NEP is equivalent to the existence of a strongly measurable family of posterior states for every normal state. Two examples of CP instruments without the NEP are obtained from this result. It is thus concluded that in local quantum physics not every CP instrument represents a measuring process, but in most of physically relevant cases every CP instrument can be realized by a measuring process within arbitrary error limits, as every approximately finite dimensional von Neumann algebra on a separable Hilbert space is injective. To conclude the paper, the concept of local measurement in algebraic quantum field theory is examined in our framework. In the setting of the Doplicher-Haag-Roberts and Doplicher-Roberts theory describing local excitations, we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a CP instrument with the NEP, provided that the split property holds for the net of local algebras.« less

Capitalizing on Basic Brain Processes in Developmental Algebra--Part 3

ERIC Educational Resources Information Center

Laughbaum, Edward D.

2011-01-01

In Part Three, the author reviews the basic ideas presented in Parts One and Two while arguing why the traditional equation-solving developmental algebra curricula is not a good choice for implementing neural response strategies presented in the first two parts. He continues by showing that the developmental algebra student audience is simply…

Calif. Laws Shift Gears on Algebra, Textbooks

ERIC Educational Resources Information Center

Robelen, Erik W.

2012-01-01

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

A modified discrete algebraic reconstruction technique for multiple grey image reconstruction for limited angle range tomography.

PubMed

Liang, Zhiting; Guan, Yong; Liu, Gang; Chen, Xiangyu; Li, Fahu; Guo, Pengfei; Tian, Yangchao

2016-03-01

The `missing wedge', which is due to a restricted rotation range, is a major challenge for quantitative analysis of an object using tomography. With prior knowledge of the grey levels, the discrete algebraic reconstruction technique (DART) is able to reconstruct objects accurately with projections in a limited angle range. However, the quality of the reconstructions declines as the number of grey levels increases. In this paper, a modified DART (MDART) was proposed, in which each independent region of homogeneous material was chosen as a research object, instead of the grey values. The grey values of each discrete region were estimated according to the solution of the linear projection equations. The iterative process of boundary pixels updating and correcting the grey values of each region was executed alternately. Simulation experiments of binary phantoms as well as multiple grey phantoms show that MDART is capable of achieving high-quality reconstructions with projections in a limited angle range. The interesting advancement of MDART is that neither prior knowledge of the grey values nor the number of grey levels is necessary.

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

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

Dechant, Lawrence J.

Wave packet analysis provides a connection between linear small disturbance theory and subsequent nonlinear turbulent spot flow behavior. The traditional association between linear stability analysis and nonlinear wave form is developed via the method of stationary phase whereby asymptotic (simplified) mean flow solutions are used to estimate dispersion behavior and stationary phase approximation are used to invert the associated Fourier transform. The resulting process typically requires nonlinear algebraic equations inversions that can be best performed numerically, which partially mitigates the value of the approximation as compared to a more complete, e.g. DNS or linear/nonlinear adjoint methods. To obtain a simpler,more » closed-form analytical result, the complete packet solution is modeled via approximate amplitude (linear convected kinematic wave initial value problem) and local sinusoidal (wave equation) expressions. Significantly, the initial value for the kinematic wave transport expression follows from a separable variable coefficient approximation to the linearized pressure fluctuation Poisson expression. The resulting amplitude solution, while approximate in nature, nonetheless, appears to mimic many of the global features, e.g. transitional flow intermittency and pressure fluctuation magnitude behavior. A low wave number wave packet models also recover meaningful auto-correlation and low frequency spectral behaviors.« less

Some applications of mathematics in theoretical physics - A review

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

Bora, Kalpana

2016-06-21

Mathematics is a very beautiful subject−very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like−differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical toolsmore » are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.« less

Some applications of mathematics in theoretical physics - A review

NASA Astrophysics Data System (ADS)

Bora, Kalpana

2016-06-01

Mathematics is a very beautiful subject-very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like-differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical tools are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.

Frequency analysis via the method of moment functionals

NASA Technical Reports Server (NTRS)

Pearson, A. E.; Pan, J. Q.

1990-01-01

Several variants are presented of a linear-in-parameters least squares formulation for determining the transfer function of a stable linear system at specified frequencies given a finite set of Fourier series coefficients calculated from transient nonstationary input-output data. The basis of the technique is Shinbrot's classical method of moment functionals using complex Fourier based modulating functions to convert a differential equation model on a finite time interval into an algebraic equation which depends linearly on frequency-related parameters.

A Search for Mountain Waves in MLS Stratospheric Limb Radiances from the Winter Northern Hemisphere: Data Analysis and Global Mountain Wave Modeling

DTIC Science & Technology

2004-02-11

the general circulation of the middle atmosphere, Philos. Trans. R. Soc. London, Ser. A, 323, 693–705. Anton , H. (2000), Elementary Linear Algebra ...Because the saturated radiances may depend slightly on tangent height as the limb path length decreases, a linear trend (described by parameters a and b...track days and interpolated onto the same limb-track orbits. The color bar scale for radiance variance is linear . (b) Digital elevations of northern

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

DTIC Science & Technology

1992-03-01

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

An elementary proof of a criterion for linear disjointness

NASA Astrophysics Data System (ADS)

Dobbs, David E.

2013-06-01

An elementary proof using matrix theory is given for the following criterion: if F/K and L/K are field extensions, with F and L both contained in a common extension field, then F and L are linearly disjoint over K if (and only if) some K-vector space basis of F is linearly independent over L. The material in this note could serve as enrichment material for the unit on fields in a first course on abstract algebra.

A mechanized process algebra for verification of device synchronization protocols

NASA Technical Reports Server (NTRS)

Schubert, E. Thomas

1992-01-01

We describe the formalization of a process algebra based on CCS within the Higher Order Logic (HOL) theorem-proving system. The representation of four types of device interactions and a correctness proof of the communication between a microprocessor and MMU is presented.

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

PubMed Central

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

2016-01-01

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

Algebraic signal processing theory: 2-D spatial hexagonal lattice.

PubMed

Pünschel, Markus; Rötteler, Martin

2007-06-01

We develop the framework for signal processing on a spatial, or undirected, 2-D hexagonal lattice for both an infinite and a finite array of signal samples. This framework includes the proper notions of z-transform, boundary conditions, filtering or convolution, spectrum, frequency response, and Fourier transform. In the finite case, the Fourier transform is called discrete triangle transform. Like the hexagonal lattice, this transform is nonseparable. The derivation of the framework makes it a natural extension of the algebraic signal processing theory that we recently introduced. Namely, we construct the proper signal models, given by polynomial algebras, bottom-up from a suitable definition of hexagonal space shifts using a procedure provided by the algebraic theory. These signal models, in turn, then provide all the basic signal processing concepts. The framework developed in this paper is related to Mersereau's early work on hexagonal lattices in the same way as the discrete cosine and sine transforms are related to the discrete Fourier transform-a fact that will be made rigorous in this paper.

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

NASA Astrophysics Data System (ADS)

Mortensen, Mikael; Valen-Sendstad, Kristian

2015-03-01

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

Image-analysis library

NASA Technical Reports Server (NTRS)

1980-01-01

MATHPAC image-analysis library is collection of general-purpose mathematical and statistical routines and special-purpose data-analysis and pattern-recognition routines for image analysis. MATHPAC library consists of Linear Algebra, Optimization, Statistical-Summary, Densities and Distribution, Regression, and Statistical-Test packages.

Projective formulation of Maggi's method for nonholonomic systems analysis

NASA Astrophysics Data System (ADS)

Blajer, Wojciech

1992-04-01

A projective interpretation of Maggi'a approach to dynamic analysis of nonholonomic systems is presented. Both linear and nonlinear constraint cases are treatment in unified fashion, using the language of vector spaces and tensor algebra analysis.

The History of Mathematics and Mathematical Education

ERIC Educational Resources Information Center

Grattan-Guinness, I.

1977-01-01

Answers to questions which were asked after the author's various lectures in Australia are gathered here. Topics touched upon include "new" mathematics, unknown constants and free variables, propositional functions, linear algebra, arithmetic and geometry, and student assessment. (MN)

A new localization set for generalized eigenvalues.

PubMed

Gao, Jing; Li, Chaoqian

2017-01-01

A new localization set for generalized eigenvalues is obtained. It is shown that the new set is tighter than that in (Numer. Linear Algebra Appl. 16:883-898, 2009). Numerical examples are given to verify the corresponding results.

Thought beyond language: neural dissociation of algebra and natural language.

PubMed

Monti, Martin M; Parsons, Lawrence M; Osherson, Daniel N

2012-08-01

A central question in cognitive science is whether natural language provides combinatorial operations that are essential to diverse domains of thought. In the study reported here, we addressed this issue by examining the role of linguistic mechanisms in forging the hierarchical structures of algebra. In a 3-T functional MRI experiment, we showed that processing of the syntax-like operations of algebra does not rely on the neural mechanisms of natural language. Our findings indicate that processing the syntax of language elicits the known substrate of linguistic competence, whereas algebraic operations recruit bilateral parietal brain regions previously implicated in the representation of magnitude. This double dissociation argues against the view that language provides the structure of thought across all cognitive domains.

DEGAS: Dynamic Exascale Global Address Space Programming Environments

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

Demmel, James

The Dynamic, Exascale Global Address Space programming environment (DEGAS) project will develop the next generation of programming models and runtime systems to meet the challenges of Exascale computing. The Berkeley part of the project concentrated on communication-optimal code generation to optimize speed and energy efficiency by reducing data movement. Our work developed communication lower bounds, and/or communication avoiding algorithms (that either meet the lower bound, or do much less communication than their conventional counterparts) for a variety of algorithms, including linear algebra, machine learning and genomics. The Berkeley part of the project concentrated on communication-optimal code generation to optimize speedmore » and energy efficiency by reducing data movement. Our work developed communication lower bounds, and/or communication avoiding algorithms (that either meet the lower bound, or do much less communication than their conventional counterparts) for a variety of algorithms, including linear algebra, machine learning and genomics.« less

Linear-algebraic bath transformation for simulating complex open quantum systems

DOE PAGES

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

2014-12-02

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

Families of vector-like deformations of relativistic quantum phase spaces, twists and symmetries

NASA Astrophysics Data System (ADS)

Meljanac, Daniel; Meljanac, Stjepan; Pikutić, Danijel

2017-12-01

Families of vector-like deformed relativistic quantum phase spaces and corresponding realizations are analyzed. A method for a general construction of the star product is presented. The corresponding twist, expressed in terms of phase space coordinates, in the Hopf algebroid sense is presented. General linear realizations are considered and corresponding twists, in terms of momenta and Poincaré-Weyl generators or gl(n) generators are constructed and R-matrix is discussed. A classification of linear realizations leading to vector-like deformed phase spaces is given. There are three types of spaces: (i) commutative spaces, (ii) κ -Minkowski spaces and (iii) κ -Snyder spaces. The corresponding star products are (i) associative and commutative (but non-local), (ii) associative and non-commutative and (iii) non-associative and non-commutative, respectively. Twisted symmetry algebras are considered. Transposed twists and left-right dual algebras are presented. Finally, some physical applications are discussed.

Simulation of the Flow Through Porous Layers Composed of Converging-Diverging Capillary Fissures or Tubes

NASA Astrophysics Data System (ADS)

Walicka, A.

2018-02-01

In this paper, a porous medium is modelled by a network of converging-diverging capillaries which may be considered as fissures or tubes. This model makes it necessary to consider flows through capillary fissures or tubes. Therefore an analytical method for deriving the relationships between pressure drops, volumetric flow rates and velocities for the following fluids: Newtonian, polar, power-law, pseudoplastic (DeHaven and Sisko types) and Shulmanian, was developed. Next, considerations on the models of pore network for Newtonian and non-Newtonian fluids were presented. The models, similar to the schemes of central finite differences may provide a good basis for transforming the governing equations of a flow through the porous medium into a set of linear or quasi-linear algebraic equations. It was shown that the some coefficients in these algebraic equations depend on the kind of the capillary convergence.

Prediction of High-Lift Flows using Turbulent Closure Models

NASA Technical Reports Server (NTRS)

Rumsey, Christopher L.; Gatski, Thomas B.; Ying, Susan X.; Bertelrud, Arild

1997-01-01

The flow over two different multi-element airfoil configurations is computed using linear eddy viscosity turbulence models and a nonlinear explicit algebraic stress model. A subset of recently-measured transition locations using hot film on a McDonnell Douglas configuration is presented, and the effect of transition location on the computed solutions is explored. Deficiencies in wake profile computations are found to be attributable in large part to poor boundary layer prediction on the generating element, and not necessarily inadequate turbulence modeling in the wake. Using measured transition locations for the main element improves the prediction of its boundary layer thickness, skin friction, and wake profile shape. However, using measured transition locations on the slat still yields poor slat wake predictions. The computation of the slat flow field represents a key roadblock to successful predictions of multi-element flows. In general, the nonlinear explicit algebraic stress turbulence model gives very similar results to the linear eddy viscosity models.

Joint Services Electronics Program.

DTIC Science & Technology

1983-09-30

environment. The research is under three interrelated heads: (1) algebraic Methodologies for Control Systems design , both linear and non -linear, (2) robust...properties of the device. After study of these experimental results, we plan to design a millimeter- wave version of the Gunn device. This will...appropriate dose discretization level for an adju- stable width beam. 2) Experimental Device Fabrication In a collaborative effort with the IC design group