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Sample records for solving linear algebra

  1. Some Applications of Algebraic System Solving

    Science.gov (United States)

    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…

  2. Linear Algebra and Smarandache Linear Algebra

    OpenAIRE

    Vasantha, Kandasamy

    2003-01-01

    The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and vector spaces over finite p...

  3. Linear algebra

    CERN Document Server

    Edwards, Harold M

    1995-01-01

    In his new undergraduate textbook, Harold M Edwards proposes a radically new and thoroughly algorithmic approach to linear algebra Originally inspired by the constructive philosophy of mathematics championed in the 19th century by Leopold Kronecker, the approach is well suited to students in the computer-dominated late 20th century Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples, making the study of linear algebra a truly interactive experience Designed for a one-semester course, this text adopts an algorithmic approach to linear algebra giving the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject Students at all levels will find much interactive instruction in this text while teachers will find stimulating examples and methods of approach to the subject

  4. Essential linear algebra with applications a problem-solving approach

    CERN Document Server

    Andreescu, Titu

    2014-01-01

    This textbook provides a rigorous introduction to linear algebra in addition to material suitable for a more advanced course while emphasizing the subject’s interactions with other topics in mathematics such as calculus and geometry. A problem-based approach is used to develop the theoretical foundations of vector spaces, linear equations, matrix algebra, eigenvectors, and orthogonality. Key features include: • a thorough presentation of the main results in linear algebra along with numerous examples to illustrate the theory;  • over 500 problems (half with complete solutions) carefully selected for their elegance and theoretical significance; • an interleaved discussion of geometry and linear algebra, giving readers a solid understanding of both topics and the relationship between them.   Numerous exercises and well-chosen examples make this text suitable for advanced courses at the junior or senior levels. It can also serve as a source of supplementary problems for a sophomore-level course.    ...

  5. Schwarz maps of algebraic linear ordinary differential equations

    Science.gov (United States)

    Sanabria Malagón, Camilo

    2017-12-01

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

  6. An Application of Linear Algebra over Lattices

    OpenAIRE

    M. Hosseinyazdi

    2008-01-01

    In this paper, first we consider L n as a semimodule over a complete bounded distributive lattice L. Then we define the basic concepts of module theory for L n. After that, we proved many similar theorems in linear algebra for the space L n. An application of linear algebra over lattices for solving linear systems, was given

  7. Numerical linear algebra with applications using Matlab

    CERN Document Server

    Ford, William

    2014-01-01

    Designed for those who want to gain a practical knowledge of modern computational techniques for the numerical solution of linear algebra problems, Numerical Linear Algebra with Applications contains all the material necessary for a first year graduate or advanced undergraduate course on numerical linear algebra with numerous applications to engineering and science. With a unified presentation of computation, basic algorithm analysis, and numerical methods to compute solutions, this book is ideal for solving real-world problems. It provides necessary mathematical background information for

  8. An Application of Linear Algebra over Lattices

    Directory of Open Access Journals (Sweden)

    M. Hosseinyazdi

    2008-03-01

    Full Text Available In this paper, first we consider L n as a semimodule over a complete bounded distributive lattice L. Then we define the basic concepts of module theory for L n. After that, we proved many similar theorems in linear algebra for the space L n. An application of linear algebra over lattices for solving linear systems, was given

  9. A novel algebraic procedure for solving non-linear evolution equations of higher order

    International Nuclear Information System (INIS)

    Huber, Alfred

    2007-01-01

    We report here a systematic approach that can easily be used for solving non-linear partial differential equations (nPDE), especially of higher order. We restrict the analysis to the so called evolution equations describing any wave propagation. The proposed new algebraic approach leads us to traveling wave solutions and moreover, new class of solution can be obtained. The crucial step of our method is the basic assumption that the solutions satisfy an ordinary differential equation (ODE) of first order that can be easily integrated. The validity and reliability of the method is tested by its application to some non-linear evolution equations. The important aspect of this paper however is the fact that we are able to calculate distinctive class of solutions which cannot be found in the current literature. In other words, using this new algebraic method the solution manifold is augmented to new class of solution functions. Simultaneously we would like to stress the necessity of such sophisticated methods since a general theory of nPDE does not exist. Otherwise, for practical use the algebraic construction of new class of solutions is of fundamental interest

  10. Relation of deformed nonlinear algebras with linear ones

    International Nuclear Information System (INIS)

    Nowicki, A; Tkachuk, V M

    2014-01-01

    The relation between nonlinear algebras and linear ones is established. For a one-dimensional nonlinear deformed Heisenberg algebra with two operators we find the function of deformation for which this nonlinear algebra can be transformed to a linear one with three operators. We also establish the relation between the Lie algebra of total angular momentum and corresponding nonlinear one. This relation gives a possibility to simplify and to solve the eigenvalue problem for the Hamiltonian in a nonlinear case using the reduction of this problem to the case of linear algebra. It is demonstrated in an example of a harmonic oscillator. (paper)

  11. Applied linear algebra and matrix analysis

    CERN Document Server

    Shores, Thomas S

    2018-01-01

    In its second edition, this textbook offers a fresh approach to matrix and linear algebra. Its blend of theory, computational exercises, and analytical writing projects is designed to highlight the interplay between these aspects of an application. This approach places special emphasis on linear algebra as an experimental science that provides tools for solving concrete problems. The second edition’s revised text discusses applications of linear algebra like graph theory and network modeling methods used in Google’s PageRank algorithm. Other new materials include modeling examples of diffusive processes, linear programming, image processing, digital signal processing, and Fourier analysis. These topics are woven into the core material of Gaussian elimination and other matrix operations; eigenvalues, eigenvectors, and discrete dynamical systems; and the geometrical aspects of vector spaces. Intended for a one-semester undergraduate course without a strict calculus prerequisite, Applied Linear Algebra and M...

  12. Introduction to computational linear algebra

    CERN Document Server

    Nassif, Nabil; Erhel, Jocelyne

    2015-01-01

    Introduction to Computational Linear Algebra introduces the reader with a background in basic mathematics and computer programming to the fundamentals of dense and sparse matrix computations with illustrating examples. The textbook is a synthesis of conceptual and practical topics in ""Matrix Computations."" The book's learning outcomes are twofold: to understand state-of-the-art computational tools to solve matrix computations problems (BLAS primitives, MATLAB® programming) as well as essential mathematical concepts needed to master the topics of numerical linear algebra. It is suitable for s

  13. Special set linear algebra and special set fuzzy linear algebra

    OpenAIRE

    Kandasamy, W. B. Vasantha; Smarandache, Florentin; Ilanthenral, K.

    2009-01-01

    The authors in this book introduce the notion of special set linear algebra and special set fuzzy Linear algebra, which is an extension of the notion set linear algebra and set fuzzy linear algebra. These concepts are best suited in the application of multi expert models and cryptology. This book has five chapters. In chapter one the basic concepts about set linear algebra is given in order to make this book a self contained one. The notion of special set linear algebra and their fuzzy analog...

  14. Modeling digital switching circuits with linear algebra

    CERN Document Server

    Thornton, Mitchell A

    2014-01-01

    Modeling Digital Switching Circuits with Linear Algebra describes an approach for modeling digital information and circuitry that is an alternative to Boolean algebra. While the Boolean algebraic model has been wildly successful and is responsible for many advances in modern information technology, the approach described in this book offers new insight and different ways of solving problems. Modeling the bit as a vector instead of a scalar value in the set {0, 1} allows digital circuits to be characterized with transfer functions in the form of a linear transformation matrix. The use of transf

  15. Computational linear and commutative algebra

    CERN Document Server

    Kreuzer, Martin

    2016-01-01

    This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems. This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is not available at all. The authors continue to prese...

  16. Linear {GLP}-algebras and their elementary theories

    Science.gov (United States)

    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.

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

    Science.gov (United States)

    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…

  18. Solution of systems of linear algebraic equations by the method of summation of divergent series

    International Nuclear Information System (INIS)

    Kirichenko, G.A.; Korovin, Ya.S.; Khisamutdinov, M.V.; Shmojlov, V.I.

    2015-01-01

    A method for solving systems of linear algebraic equations has been proposed on the basis on the summation of the corresponding continued fractions. The proposed algorithm for solving systems of linear algebraic equations is classified as direct algorithms providing an exact solution in a finite number of operations. Examples of solving systems of linear algebraic equations have been presented and the effectiveness of the algorithm has been estimated [ru

  19. Assessing Algebraic Solving Ability: A Theoretical Framework

    Science.gov (United States)

    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…

  20. An approach for solving linear fractional programming problems ...

    African Journals Online (AJOL)

    The paper presents a new approach for solving a fractional linear programming problem in which the objective function is a linear fractional function, while the constraint functions are in the form of linear inequalities. The approach adopted is based mainly upon solving the problem algebraically using the concept of duality ...

  1. Gauss Elimination: Workhorse of Linear Algebra.

    Science.gov (United States)

    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

  2. Linearizing W-algebras

    International Nuclear Information System (INIS)

    Krivonos, S.O.; Sorin, A.S.

    1994-06-01

    We show that the Zamolodchikov's and Polyakov-Bershadsky nonlinear algebras W 3 and W (2) 3 can be embedded as subalgebras into some linear algebras with finite set of currents. Using these linear algebras we find new field realizations of W (2) 3 and W 3 which could be a starting point for constructing new versions of W-string theories. We also reveal a number of hidden relationships between W 3 and W (2) 3 . We conjecture that similar linear algebras can exist for other W-algebra as well. (author). 10 refs

  3. Fundamentals of linear algebra

    CERN Document Server

    Dash, Rajani Ballav

    2008-01-01

    FUNDAMENTALS OF LINEAR ALGEBRA is a comprehensive Text Book, which can be used by students and teachers of All Indian Universities. The Text has easy, understandable form and covers all topics of UGC Curriculum. There are lots of worked out examples which helps the students in solving the problems without anybody's help. The Problem sets have been designed keeping in view of the questions asked in different examinations.

  4. Linear algebraic groups

    CERN Document Server

    Springer, T A

    1998-01-01

    "[The first] ten chapters...are an efficient, accessible, and self-contained introduction to affine algebraic groups over an algebraically closed field. The author includes exercises and the book is certainly usable by graduate students as a text or for self-study...the author [has a] student-friendly style… [The following] seven chapters... would also be a good introduction to rationality issues for algebraic groups. A number of results from the literature…appear for the first time in a text." –Mathematical Reviews (Review of the Second Edition) "This book is a completely new version of the first edition. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. Reading that book, many people entered the research field of linear algebraic groups. The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of t...

  5. Embodied, Symbolic and Formal Thinking in Linear Algebra

    Science.gov (United States)

    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…

  6. Linear algebra

    CERN Document Server

    Liesen, Jörg

    2015-01-01

    This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations. The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exerc...

  7. Linear algebra

    CERN Document Server

    Stoll, R R

    1968-01-01

    Linear Algebra is intended to be used as a text for a one-semester course in linear algebra at the undergraduate level. The treatment of the subject will be both useful to students of mathematics and those interested primarily in applications of the theory. The major prerequisite for mastering the material is the readiness of the student to reason abstractly. Specifically, this calls for an understanding of the fact that axioms are assumptions and that theorems are logical consequences of one or more axioms. Familiarity with calculus and linear differential equations is required for understand

  8. Linear-Algebra Programs

    Science.gov (United States)

    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.

  9. Linear algebra meets Lie algebra: the Kostant-Wallach theory

    OpenAIRE

    Shomron, Noam; Parlett, Beresford N.

    2008-01-01

    In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.

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

    Science.gov (United States)

    Gonzalez-Vega, Laureano

    1999-01-01

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

  11. Principles of linear algebra with Mathematica

    CERN Document Server

    Shiskowski, Kenneth M

    2013-01-01

    A hands-on introduction to the theoretical and computational aspects of linear algebra using Mathematica® Many topics in linear algebra are simple, yet computationally intensive, and computer algebra systems such as Mathematica® are essential not only for learning to apply the concepts to computationally challenging problems, but also for visualizing many of the geometric aspects within this field of study. Principles of Linear Algebra with Mathematica uniquely bridges the gap between beginning linear algebra and computational linear algebra that is often encountered in applied settings,

  12. Basic linear algebra

    CERN Document Server

    Blyth, T S

    2002-01-01

    Basic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises. More exercises (of the kind a student may expect in examination papers) are grouped at the end of each section. The book covers the most important basics of any first course on linear algebra, explaining the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations and complex numbers. Linear equations are treated via Hermite normal forms which provides a successful and concrete explanation of the notion of linear independence. Another important highlight is the connection between linear mappings and matrices leading to the change of basis theorem which opens the door to the notion of similarity. This new and revised edition features additional exercises and coverage of Cramer's rule (omitted from the first edition). However, it is the new, extra chapter on computer assistance that will be of particular interest to readers:...

  13. W-algebra for solving problems with fuzzy parameters

    Science.gov (United States)

    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.

  14. Convergence of hybrid methods for solving non-linear partial ...

    African Journals Online (AJOL)

    This paper is concerned with the numerical solution and convergence analysis of non-linear partial differential equations using a hybrid method. The solution technique involves discretizing the non-linear system of PDE to obtain a corresponding non-linear system of algebraic difference equations to be solved at each time ...

  15. Linear algebra done right

    CERN Document Server

    Axler, Sheldon

    2015-01-01

    This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the ...

  16. Matrices and linear algebra

    CERN Document Server

    Schneider, Hans

    1989-01-01

    Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related t

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

    Energy Technology Data Exchange (ETDEWEB)

    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 people 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.

  18. Further linear algebra

    CERN Document Server

    Blyth, T S

    2002-01-01

    Most of the introductory courses on linear algebra develop the basic theory of finite­ dimensional vector spaces, and in so doing relate the notion of a linear mapping to that of a matrix. Generally speaking, such courses culminate in the diagonalisation of certain matrices and the application of this process to various situations. Such is the case, for example, in our previous SUMS volume Basic Linear Algebra. The present text is a continuation of that volume, and has the objective of introducing the reader to more advanced properties of vector spaces and linear mappings, and consequently of matrices. For readers who are not familiar with the contents of Basic Linear Algebra we provide an introductory chapter that consists of a compact summary of the prerequisites for the present volume. In order to consolidate the student's understanding we have included a large num­ ber of illustrative and worked examples, as well as many exercises that are strategi­ cally placed throughout the text. Solutions to the ex...

  19. Non-linear singular problems in p-adic analysis: associative algebras of p-adic distributions

    International Nuclear Information System (INIS)

    Albeverio, S; Khrennikov, A Yu; Shelkovich, V M

    2005-01-01

    We propose an algebraic theory which can be used for solving both linear and non-linear singular problems of p-adic analysis related to p-adic distributions (generalized functions). We construct the p-adic Colombeau-Egorov algebra of generalized functions, in which Vladimirov's pseudo-differential operator plays the role of differentiation. This algebra is closed under Fourier transformation and associative convolution. Pointvalues of generalized functions are defined, and it turns out that any generalized function is uniquely determined by its pointvalues. We also construct an associative algebra of asymptotic distributions, which is generated by the linear span of the set of associated homogeneous p-adic distributions. This algebra is embedded in the Colombeau-Egorov algebra as a subalgebra. In addition, a new technique for constructing weak asymptotics is developed

  20. SUPPORTING STUDENTS’ UNDERSTANDING OF LINEAR EQUATIONS WITH ONE VARIABLE USING ALGEBRA TILES

    Directory of Open Access Journals (Sweden)

    Sari Saraswati

    2016-01-01

    Full Text Available 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 use the algebra tiles to find a method to solve linear equations with one variable. Design research was used as an approach in this study. It consists of three phases, namely preliminary design, teaching experiment and retrospective analysis. Video registrations, students’ written works, pre-test, post-test, field notes, and interview are technic to collect data. The data were analyzed by comparing the hypothetical learning trajectory (HLT and the actual learning process. The result shows that algebra tiles could supports students’ understanding to find the formal solution of linear equation with one variable.

  1. Templates for Linear Algebra Problems

    NARCIS (Netherlands)

    Bai, Z.; Day, D.; Demmel, J.; Dongarra, J.; Gu, M.; Ruhe, A.; Vorst, H.A. van der

    1995-01-01

    The increasing availability of advanced-architecture computers is having a very signicant eect on all spheres of scientic computation, including algorithm research and software development in numerical linear algebra. Linear algebra {in particular, the solution of linear systems of equations and

  2. Students’ difficulties in solving linear equation problems

    Science.gov (United States)

    Wati, S.; Fitriana, L.; Mardiyana

    2018-03-01

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

  3. SUPPORTING STUDENTS’ UNDERSTANDING OF LINEAR EQUATIONS WITH ONE VARIABLE USING ALGEBRA TILES

    Directory of Open Access Journals (Sweden)

    Sari Saraswati

    2016-01-01

    Full Text Available 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 use the algebra tiles to find a method to solve linear equations with one variable. Design research was used as an approach in this study. It consists of three phases, namely preliminary design, teaching experiment and retrospective analysis. Video registrations, students’ written works, pre-test, post-test, field notes, and interview are technic to collect data. The data were analyzed by comparing the hypothetical learning trajectory (HLT and the actual learning process. The result shows that algebra tiles could supports students’ understanding to find the formal solution of linear equation with one variable.Keywords: linear equation with one variable, algebra tiles, design research, balancing method, HLT DOI: http://dx.doi.org/10.22342/jme.7.1.2814.19-30

  4. Computer Program For Linear Algebra

    Science.gov (United States)

    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.

  5. Nonlinear evolution equations and solving algebraic systems: the importance of computer algebra

    International Nuclear Information System (INIS)

    Gerdt, V.P.; Kostov, N.A.

    1989-01-01

    In the present paper we study the application of computer algebra to solve the nonlinear polynomial systems which arise in investigation of nonlinear evolution equations. We consider several systems which are obtained in classification of integrable nonlinear evolution equations with uniform rank. Other polynomial systems are related with the finding of algebraic curves for finite-gap elliptic potentials of Lame type and generalizations. All systems under consideration are solved using the method based on construction of the Groebner basis for corresponding polynomial ideals. The computations have been carried out using computer algebra systems. 20 refs

  6. Applied linear algebra

    CERN Document Server

    Olver, Peter J

    2018-01-01

    This textbook develops the essential tools of linear algebra, with the goal of imparting technique alongside contextual understanding. Applications go hand-in-hand with theory, each reinforcing and explaining the other. This approach encourages students to develop not only the technical proficiency needed to go on to further study, but an appreciation for when, why, and how the tools of linear algebra can be used across modern applied mathematics. Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra. In this second edition, applications have been updated and expanded to include numerical methods, dynamical systems, data analysis, and signal processing, while the pedagogical flow of the core material has been improved. Throughout, the text emphasizes the conceptual connections between each application and the un...

  7. Undergraduate Mathematics Students' Emotional Experiences in Linear Algebra Courses

    Science.gov (United States)

    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…

  8. Dynamical systems and linear algebra

    OpenAIRE

    Colonius, Fritz (Prof.)

    2007-01-01

    Dynamical systems and linear algebra / F. Colonius, W. Kliemann. - In: Handbook of linear algebra / ed. by Leslie Hogben. - Boca Raton : Chapman & Hall/CRC, 2007. - S. 56,1-56,22. - (Discrete mathematics and its applications)

  9. Gender differences in algebraic thinking ability to solve mathematics problems

    Science.gov (United States)

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

    2018-05-01

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

  10. Linear Algebraic Method for Non-Linear Map Analysis

    International Nuclear Information System (INIS)

    Yu, L.; Nash, B.

    2009-01-01

    We present a newly developed method to analyze some non-linear dynamics problems such as the Henon map using a matrix analysis method from linear algebra. Choosing the Henon map as an example, we analyze the spectral structure, the tune-amplitude dependence, the variation of tune and amplitude during the particle motion, etc., using the method of Jordan decomposition which is widely used in conventional linear algebra.

  11. Sensitivity theory for general non-linear algebraic equations with constraints

    International Nuclear Information System (INIS)

    Oblow, E.M.

    1977-04-01

    Sensitivity theory has been developed to a high state of sophistication for applications involving solutions of the linear Boltzmann equation or approximations to it. The success of this theory in the field of radiation transport has prompted study of possible extensions of the method to more general systems of non-linear equations. Initial work in the U.S. and in Europe on the reactor fuel cycle shows that the sensitivity methodology works equally well for those non-linear problems studied to date. The general non-linear theory for algebraic equations is summarized and applied to a class of problems whose solutions are characterized by constrained extrema. Such equations form the basis of much work on energy systems modelling and the econometrics of power production and distribution. It is valuable to have a sensitivity theory available for these problem areas since it is difficult to repeatedly solve complex non-linear equations to find out the effects of alternative input assumptions or the uncertainties associated with predictions of system behavior. The sensitivity theory for a linear system of algebraic equations with constraints which can be solved using linear programming techniques is discussed. The role of the constraints in simplifying the problem so that sensitivity methodology can be applied is highlighted. The general non-linear method is summarized and applied to a non-linear programming problem in particular. Conclusions are drawn in about the applicability of the method for practical problems

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

    International Nuclear Information System (INIS)

    Winicour, Jeffrey

    2017-01-01

    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. (note)

  13. AN ADA LINEAR ALGEBRA PACKAGE MODELED AFTER HAL/S

    Science.gov (United States)

    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.

  14. Synthesis of models for order-sorted first-order theories using linear algebra and constraint solving

    Directory of Open Access Journals (Sweden)

    Salvador Lucas

    2015-12-01

    Full Text Available Recent developments in termination analysis for declarative programs emphasize the use of appropriate models for the logical theory representing the program at stake as a generic approach to prove termination of declarative programs. In this setting, Order-Sorted First-Order Logic provides a powerful framework to represent declarative programs. It also provides a target logic to obtain models for other logics via transformations. We investigate the automatic generation of numerical models for order-sorted first-order logics and its use in program analysis, in particular in termination analysis of declarative programs. We use convex domains to give domains to the different sorts of an order-sorted signature; we interpret the ranked symbols of sorted signatures by means of appropriately adapted convex matrix interpretations. Such numerical interpretations permit the use of existing algorithms and tools from linear algebra and arithmetic constraint solving to synthesize the models.

  15. Students’ Algebraic Reasonsing In Solving Mathematical Problems With Adversity Quotient

    Science.gov (United States)

    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.

  16. Numerical linear algebra theory and applications

    CERN Document Server

    Beilina, Larisa; Karchevskii, Mikhail

    2017-01-01

    This book combines a solid theoretical background in linear algebra with practical algorithms for numerical solution of linear algebra problems. Developed from a number of courses taught repeatedly by the authors, the material covers topics like matrix algebra, theory for linear systems of equations, spectral theory, vector and matrix norms combined with main direct and iterative numerical methods, least squares problems, and eigen problems. Numerical algorithms illustrated by computer programs written in MATLAB® are also provided as supplementary material on SpringerLink to give the reader a better understanding of professional numerical software for the solution of real-life problems. Perfect for a one- or two-semester course on numerical linear algebra, matrix computation, and large sparse matrices, this text will interest students at the advanced undergraduate or graduate level.

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

    Science.gov (United States)

    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…

  18. Linear Algebra and Image Processing

    Science.gov (United States)

    Allali, Mohamed

    2010-01-01

    We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)

  19. More on the linearization of W-algebras

    International Nuclear Information System (INIS)

    Krivonos, S.; Sorin, A.

    1995-01-01

    We show that a wide class of W-(super)algebras, including W N (N-1) , U(N)-superconformal as well as W N nonlinear algebras, can be linearized by embedding them as subalgebras into some linear (super)conformal algebras with finite sets of currents. The general construction is illustrated by the example of W 4 algebra. 16 refs

  20. Generalization of the linear algebraic method to three dimensions

    International Nuclear Information System (INIS)

    Lynch, D.L.; Schneider, B.I.

    1991-01-01

    We present a numerical method for the solution of the Lippmann-Schwinger equation for electron-molecule collisions. By performing a three-dimensional numerical quadrature, this approach avoids both a basis-set representation of the wave function and a partial-wave expansion of the scattering potential. The resulting linear equations, analogous in form to the one-dimensional linear algebraic method, are solved with the direct iteration-variation method. Several numerical examples are presented. The prospect for using this numerical quadrature scheme for electron-polyatomic molecules is discussed

  1. Student Obstacles in Solving Algebraic Thinking Problems

    Science.gov (United States)

    Andini, W.; Suryadi, D.

    2017-09-01

    The aim of this research is to analize the student obstacles on solving algebraic thinking problems in low grades elementary school. This research is a preliminary qualitative research, and involved 66 students of grade 3 elementary school. From the analysis student test results, most of student experience difficulty in solving algebraic thinking problems. The main obstacle is the student’s difficulty in understanding the problem of generalizing the pattern because the students are not accustomed to see the rules that exist in generalize the pattern.

  2. Linear Algebra Thoroughly Explained

    CERN Document Server

    Vujičić, Milan

    2008-01-01

    Linear Algebra Thoroughly Explained provides a comprehensive introduction to the subject suitable for adoption as a self-contained text for courses at undergraduate and postgraduate level. The clear and comprehensive presentation of the basic theory is illustrated throughout with an abundance of worked examples. The book is written for teachers and students of linear algebra at all levels and across mathematics and the applied sciences, particularly physics and engineering. It will also be an invaluable addition to research libraries as a comprehensive resource book for the subject.

  3. Solving Absolute Value Equations Algebraically and Geometrically

    Science.gov (United States)

    Shiyuan, Wei

    2005-01-01

    The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.

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

    Science.gov (United States)

    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…

  5. Handbook of linear algebra

    CERN Document Server

    Hogben, Leslie

    2013-01-01

    With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and updates throughout, the second edition of this bestseller includes 20 new chapters.New to the Second EditionSeparate chapters on Schur complements, additional types of canonical forms, tensors, matrix polynomials, matrix equations, special types of

  6. The Growing Importance of Linear Algebra in Undergraduate Mathematics.

    Science.gov (United States)

    Tucker, Alan

    1993-01-01

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

  7. An Inquiry-Based Linear Algebra Class

    Science.gov (United States)

    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…

  8. On Associative Conformal Algebras of Linear Growth

    OpenAIRE

    Retakh, Alexander

    2000-01-01

    Lie conformal algebras appear in the theory of vertex algebras. Their relation is similar to that of Lie algebras and their universal enveloping algebras. Associative conformal algebras play a role in conformal representation theory. We introduce the notions of conformal identity and unital associative conformal algebras and classify finitely generated simple unital associative conformal algebras of linear growth. These are precisely the complete algebras of conformal endomorphisms of finite ...

  9. A linear algebraic approach to electron-molecule collisions

    International Nuclear Information System (INIS)

    Collins, L.A.; Schnieder, B.I.

    1982-01-01

    The linear algebraic approach to electron-molecule collisions is examined by firstly deriving the general set of coupled integrodifferential equations that describe electron collisional processes and then describing the linear algebraic approach for obtaining a solution to the coupled equations. Application of the linear algebraic method to static-exchange, separable exchange and effective optical potential, is examined. (U.K.)

  10. Topics in quaternion linear algebra

    CERN Document Server

    Rodman, Leiba

    2014-01-01

    Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses...

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

    Science.gov (United States)

    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.

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

    Science.gov (United States)

    Çelik, Derya

    2015-01-01

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

  13. Mathematical methods linear algebra normed spaces distributions integration

    CERN Document Server

    Korevaar, Jacob

    1968-01-01

    Mathematical Methods, Volume I: Linear Algebra, Normed Spaces, Distributions, Integration focuses on advanced mathematical tools used in applications and the basic concepts of algebra, normed spaces, integration, and distributions.The publication first offers information on algebraic theory of vector spaces and introduction to functional analysis. Discussions focus on linear transformations and functionals, rectangular matrices, systems of linear equations, eigenvalue problems, use of eigenvectors and generalized eigenvectors in the representation of linear operators, metric and normed vector

  14. Insights into the School Mathematics Tradition from Solving Linear Equations

    Science.gov (United States)

    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…

  15. On the economical solution method for a system of linear algebraic equations

    Directory of Open Access Journals (Sweden)

    Jan Awrejcewicz

    2004-01-01

    Full Text Available The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ3 is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O(hx12+hx22. The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.

  16. Study of solving a Toda dynamic system with loop algebra

    International Nuclear Information System (INIS)

    Zhu Qiao; Yang Zhanying; Shi Kangjie; Wen Junqing

    2006-01-01

    The authors construct a Toda system with Loop algebra, and prove that the Lax equation L=[L,M] can be solved by means of solving a regular Riemann-Hilbert problem. In our system, M in Lax pair is an antisymmetrical matrix, while L=L + + M, and L + is a quasi-upper triangular matrix of loop algebra. In order to check our result, the authors exactly solve an R-H problem under a given initial condition as an example. (authors)

  17. Advanced linear algebra for engineers with Matlab

    CERN Document Server

    Dianat, Sohail A

    2009-01-01

    Matrices, Matrix Algebra, and Elementary Matrix OperationsBasic Concepts and NotationMatrix AlgebraElementary Row OperationsSolution of System of Linear EquationsMatrix PartitionsBlock MultiplicationInner, Outer, and Kronecker ProductsDeterminants, Matrix Inversion and Solutions to Systems of Linear EquationsDeterminant of a MatrixMatrix InversionSolution of Simultaneous Linear EquationsApplications: Circuit AnalysisHomogeneous Coordinates SystemRank, Nu

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

    Science.gov (United States)

    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.

  19. Linear algebra and analytic geometry for physical sciences

    CERN Document Server

    Landi, Giovanni

    2018-01-01

    A self-contained introduction to finite dimensional vector spaces, matrices, systems of linear equations, spectral analysis on euclidean and hermitian spaces, affine euclidean geometry, quadratic forms and conic sections. The mathematical formalism is motivated and introduced by problems from physics, notably mechanics (including celestial) and electro-magnetism, with more than two hundreds examples and solved exercises. Topics include: The group of orthogonal transformations on euclidean spaces, in particular rotations, with Euler angles and angular velocity. The rigid body with its inertia matrix. The unitary group. Lie algebras and exponential map. The Dirac’s bra-ket formalism. Spectral theory for self-adjoint endomorphisms on euclidean and hermitian spaces. The Minkowski spacetime from special relativity and the Maxwell equations. Conic sections with the use of eccentricity and Keplerian motions. An appendix collects basic algebraic notions like group, ring and field; and complex numbers and integers m...

  20. Linear algebra

    CERN Document Server

    Said-Houari, Belkacem

    2017-01-01

    This self-contained, clearly written textbook on linear algebra is easily accessible for students. It begins with the simple linear equation and generalizes several notions from this equation for the system of linear equations and introduces the main ideas using matrices. It then offers a detailed chapter on determinants and introduces the main ideas with detailed proofs. The third chapter introduces the Euclidean spaces using very simple geometric ideas and discusses various major inequalities and identities. These ideas offer a solid basis for understanding general Hilbert spaces in functional analysis. The following two chapters address general vector spaces, including some rigorous proofs to all the main results, and linear transformation: areas that are ignored or are poorly explained in many textbooks. Chapter 6 introduces the idea of matrices using linear transformation, which is easier to understand than the usual theory of matrices approach. The final two chapters are more advanced, introducing t...

  1. Solving the Unknown with Algebra: Poster/Teaching Guide for Pre-Algebra Students. Expect the Unexpected with Math[R

    Science.gov (United States)

    Actuarial Foundation, 2013

    2013-01-01

    "Solving the Unknown with Algebra" is a new math program aligned with the National Council of Teachers of Mathematics (NCTM) standards and designed to help students practice pre-algebra skills including using formulas, solving for unknowns, and manipulating equations. Developed by The Actuarial Foundation with Scholastic, this program provides…

  2. Ten-Year-Old Students Solving Linear Equations

    Science.gov (United States)

    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…

  3. Linear operators in Clifford algebras

    International Nuclear Information System (INIS)

    Laoues, M.

    1991-01-01

    We consider the real vector space structure of the algebra of linear endomorphisms of a finite-dimensional real Clifford algebra (2, 4, 5, 6, 7, 8). A basis of that space is constructed in terms of the operators M eI,eJ defined by x→e I .x.e J , where the e I are the generators of the Clifford algebra and I is a multi-index (3, 7). In particular, it is shown that the family (M eI,eJ ) is exactly a basis in the even case. (orig.)

  4. Linear algebra a first course with applications

    CERN Document Server

    Knop, Larry E

    2008-01-01

    Linear Algebra: A First Course with Applications explores the fundamental ideas of linear algebra, including vector spaces, subspaces, basis, span, linear independence, linear transformation, eigenvalues, and eigenvectors, as well as a variety of applications, from inventories to graphics to Google's PageRank. Unlike other texts on the subject, this classroom-tested book gives students enough time to absorb the material by focusing on vector spaces early on and using computational sections as numerical interludes. It offers introductions to Maple™, MATLAB®, and TI-83 Plus for calculating matri

  5. Inhomogeneous linear equation in Rota-Baxter algebra

    OpenAIRE

    Pietrzkowski, Gabriel

    2014-01-01

    We consider a complete filtered Rota-Baxter algebra of weight $\\lambda$ over a commutative ring. Finding the unique solution of a non-homogeneous linear algebraic equation in this algebra, we generalize Spitzer's identity in both commutative and non-commutative cases. As an application, considering the Rota-Baxter algebra of power series in one variable with q-integral as the Rota-Baxter operator, we show certain Eulerian identities.

  6. Linear algebra as an alternative approach to the synthesis of digital devices of automation and control systems

    Directory of Open Access Journals (Sweden)

    Nikolay Chernov

    2018-01-01

    Full Text Available The article considers linear algebra as an alternative mathematical tool of logic synthesis of digital structures to Boolean algebra and synthesis methods of digital electronic component base (ECB on its ground. The methods of solving the applied problems of logic synthesis are shown, including the expansion of an arbitrary logic function by means of monotonic functions. The proposed mathematical apparatus actually provides the creation of digital structures on the principles of analog circuitry. It can find application in the design of multivalued digital ECB, specialized system-on-chip and analog-digital sensors with current output. The examples of synthesis of the combinational and sequential two-valued and multivalued digital devices are given. In conclusion, the advantages of linear algebra in comparison with Boolean algebra are formulated.

  7. The linear algebra survival guide illustrated with Mathematica

    CERN Document Server

    Szabo, Fred

    2015-01-01

    The Linear Algebra Survival Guide is a reference book with a free downloadable Mathematica notebook containing all of interactive code to make the content of the book playable in Mathematica and the Mathematica Player. It offers a concise introduction to the core topics of linear algebra which includes numerous exercises that will accompany a first or second course in linear algebra. This book will guide you through the powerful graphic displays and visualization of Mathematica that make the most abstract theories seem simple-- allowing you to tackle realistic problems using simple mathematic

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

    Science.gov (United States)

    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 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.

  9. Ada Linear-Algebra Program

    Science.gov (United States)

    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.

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

    Science.gov (United States)

    Hoover, Jerome D; Healy, Alice F

    2017-12-01

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

  11. Differential Equation over Banach Algebra

    OpenAIRE

    Kleyn, Aleks

    2018-01-01

    In the book, I considered differential equations of order $1$ over Banach $D$-algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. In noncommutative Banach algebra, initial value problem for linear homogeneous equation has infinitely many solutions.

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

    Science.gov (United States)

    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.

  13. BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS

    Science.gov (United States)

    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.

  14. INPUT-OUTPUT STRUCTURE OF LINEAR-DIFFERENTIAL ALGEBRAIC SYSTEMS

    NARCIS (Netherlands)

    KUIJPER, M; SCHUMACHER, JM

    Systems of linear differential and algebraic equations occur in various ways, for instance, as a result of automated modeling procedures and in problems involving algebraic constraints, such as zero dynamics and exact model matching. Differential/algebraic systems may represent an input-output

  15. Solving Linear Differential Equations

    NARCIS (Netherlands)

    Nguyen, K.A.; Put, M. van der

    2010-01-01

    The theme of this paper is to 'solve' an absolutely irreducible differential module explicitly in terms of modules of lower dimension and finite extensions of the differential field K. Representations of semi-simple Lie algebras and differential Galo is theory are the main tools. The results extend

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

    Science.gov (United States)

    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.

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

    Science.gov (United States)

    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.

  18. Emphasizing Language and Visualization in Teaching Linear Algebra

    Science.gov (United States)

    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…

  19. A Systematic Approach for Solving the Great Circle Track Problems based on Vector Algebra

    Directory of Open Access Journals (Sweden)

    Chen Chih-Li

    2016-04-01

    Full Text Available A systematic approach, based on multiple products of the vector algebra (S-VA, is proposed to derive the spherical triangle formulae for solving the great circle track (GCT problems. Because the mathematical properties of the geometry and algebra are both embedded in the S-VA approach, derivations of the spherical triangle formulae become more understandable and more straightforward as compared with those approaches which use the complex linear combination of a vector basis. In addition, the S-VA approach can handle all given initial conditions for solving the GCT problems simpler, clearer and avoid redundant formulae existing in the conventional approaches. With the technique of transforming the Earth coordinates system of latitudes and longitudes into the Cartesian one and adopting the relative longitude concept, the concise governing equations of the S-VA approach can be easily and directly derived. Owing to the advantage of the S-VA approach, it makes the practical navigator quickly adjust to solve the GCT problems. Based on the S-VA approach, a program namely GCTPro_VA is developed for friendly use of the navigator. Several validation examples are provided to show the S-VA approach is simple and versatile to solve the GCT problems.

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

    Science.gov (United States)

    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.

  1. Computer programs for the solution of systems of linear algebraic equations

    Science.gov (United States)

    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.

  2. Stability of Linear Equations--Algebraic Approach

    Science.gov (United States)

    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…

  3. A modified linear algebraic approach to electron scattering using cubic splines

    International Nuclear Information System (INIS)

    Kinney, R.A.

    1986-01-01

    A modified linear algebraic approach to the solution of the Schrodiner equation for low-energy electron scattering is presented. The method uses a piecewise cubic-spline approximation of the wavefunction. Results in the static-potential and the static-exchange approximations for e - +H s-wave scattering are compared with unmodified linear algebraic and variational linear algebraic methods. (author)

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

    Science.gov (United States)

    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…

  5. IDEALS GENERATED BY LINEAR FORMS AND SYMMETRIC ALGEBRAS

    Directory of Open Access Journals (Sweden)

    Gaetana Restuccia

    2016-01-01

    Full Text Available We consider ideals generated by linear forms in the variables X1 : : : ;Xn in the polynomial ring R[X1; : : : ;Xn], being R a commutative, Noetherian ring with identity. We investigate when a sequence a1; a2; : : : ; am of linear forms is an ssequence, in order to compute algebraic invariants of the symmetric algebra of the ideal I = (a1; a2; : : : ; am.

  6. Teaching Linear Algebra: Must the Fog Always Roll In?

    Science.gov (United States)

    Carlson, David

    1993-01-01

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

  7. Lie algebras and linear differential equations.

    Science.gov (United States)

    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.

  8. Solving linear inequalities in a least squares sense

    Energy Technology Data Exchange (ETDEWEB)

    Bramley, R.; Winnicka, B. [Indiana Univ., Bloomington, IN (United States)

    1994-12-31

    Let A {element_of} {Re}{sup mxn} be an arbitrary real matrix, and let b {element_of} {Re}{sup m} a given vector. A familiar problem in computational linear algebra is to solve the system Ax = b in a least squares sense; that is, to find an x* minimizing {parallel}Ax {minus} b{parallel}, where {parallel} {center_dot} {parallel} refers to the vector two-norm. Such an x* solves the normal equations A{sup T}(Ax {minus} b) = 0, and the optimal residual r* = b {minus} Ax* is unique (although x* need not be). The least squares problem is usually interpreted as corresponding to multiple observations, represented by the rows of A and b, on a vector of data x. The observations may be inconsistent, and in this case a solution is sought that minimizes the norm of the residuals. A less familiar problem to numerical linear algebraists is the solution of systems of linear inequalities Ax {le} b in a least squares sense, but the motivation is similar: if a set of observations places upper or lower bounds on linear combinations of variables, the authors want to find x* minimizing {parallel} (Ax {minus} b){sub +} {parallel}, where the i{sup th} component of the vector v{sub +} is the maximum of zero and the i{sup th} component of v.

  9. Solving algebraic computational problems in geodesy and geoinformatics the answer to modern challenges

    CERN Document Server

    Awange, Joseph L

    2004-01-01

    While preparing and teaching 'Introduction to Geodesy I and II' to - dergraduate students at Stuttgart University, we noticed a gap which motivated the writing of the present book: Almost every topic that we taughtrequiredsomeskillsinalgebra,andinparticular,computeral- bra! From positioning to transformation problems inherent in geodesy and geoinformatics, knowledge of algebra and application of computer algebra software were required. In preparing this book therefore, we haveattemptedtoputtogetherbasicconceptsofabstractalgebra which underpin the techniques for solving algebraic problems. Algebraic c- putational algorithms useful for solving problems which require exact solutions to nonlinear systems of equations are presented and tested on various problems. Though the present book focuses mainly on the two ?elds,theconceptsand techniquespresented hereinarenonetheless- plicable to other ?elds where algebraic computational problems might be encountered. In Engineering for example, network densi?cation and robo...

  10. Resources for Teaching Linear Algebra. MAA Notes Volume 42.

    Science.gov (United States)

    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…

  11. Data Compression with Linear Algebra

    OpenAIRE

    Etler, David

    2015-01-01

    A presentation on the applications of linear algebra to image compression. Covers entropy, the discrete cosine transform, thresholding, quantization, and examples of images compressed with DCT. Given in Spring 2015 at Ocean County College as part of the honors program.

  12. A linear process-algebraic format for probabilistic systems with data

    NARCIS (Netherlands)

    Katoen, Joost P.; van de Pol, Jan Cornelis; Stoelinga, Mariëlle Ida Antoinette; Timmer, Mark; Gomes, L.; Khomenko, V.; Fernandes, J.M.

    This paper presents a novel linear process algebraic format for probabilistic automata. The key ingredient is a symbolic transformation of probabilistic process algebra terms that incorporate data into this linear format while preserving strong probabilistic bisimulation. This generalises similar

  13. A Computer Algebra Approach to Solving Chemical Equilibria in General Chemistry

    Science.gov (United States)

    Kalainoff, Melinda; Lachance, Russ; Riegner, Dawn; Biaglow, Andrew

    2012-01-01

    In this article, we report on a semester-long study of the incorporation into our general chemistry course, of advanced algebraic and computer algebra techniques for solving chemical equilibrium problems. The method presented here is an alternative to the commonly used concentration table method for describing chemical equilibria in general…

  14. Linear algebra a first course with applications to differential equations

    CERN Document Server

    Apostol, Tom M

    2014-01-01

    Developed from the author's successful two-volume Calculus text this book presents Linear Algebra without emphasis on abstraction or formalization. To accommodate a variety of backgrounds, the text begins with a review of prerequisites divided into precalculus and calculus prerequisites. It continues to cover vector algebra, analytic geometry, linear spaces, determinants, linear differential equations and more.

  15. Quadratic algebras

    CERN Document Server

    Polishchuk, Alexander

    2005-01-01

    Quadratic algebras, i.e., algebras defined by quadratic relations, often occur in various areas of mathematics. One of the main problems in the study of these (and similarly defined) algebras is how to control their size. A central notion in solving this problem is the notion of a Koszul algebra, which was introduced in 1970 by S. Priddy and then appeared in many areas of mathematics, such as algebraic geometry, representation theory, noncommutative geometry, K-theory, number theory, and noncommutative linear algebra. The book offers a coherent exposition of the theory of quadratic and Koszul algebras, including various definitions of Koszulness, duality theory, Poincar�-Birkhoff-Witt-type theorems for Koszul algebras, and the Koszul deformation principle. In the concluding chapter of the book, they explain a surprising connection between Koszul algebras and one-dependent discrete-time stochastic processes.

  16. Variational linear algebraic equations method

    International Nuclear Information System (INIS)

    Moiseiwitsch, B.L.

    1982-01-01

    A modification of the linear algebraic equations method is described which ensures a variational bound on the phaseshifts for potentials having a definite sign at all points. The method is illustrated by the elastic scattering of s-wave electrons by the static field of atomic hydrogen. (author)

  17. Primary school students’ strategies in early algebra problem solving supported by an online game

    NARCIS (Netherlands)

    van den Heuvel-Panhuizen, M.H.A.M; Kolovou, A.; Robitzsch, A.

    2013-01-01

    In this study we investigated the role of a dynamic online game on students’ early algebra problem solving. In total 253 students from grades 4, 5, and 6 (10–12 years old) used the game at home to solve a sequence of early algebra problems consisting of contextual problems addressing covarying

  18. Using Computer Symbolic Algebra to Solve Differential Equations.

    Science.gov (United States)

    Mathews, John H.

    1989-01-01

    This article illustrates that mathematical theory can be incorporated into the process to solve differential equations by a computer algebra system, muMATH. After an introduction to functions of muMATH, several short programs for enhancing the capabilities of the system are discussed. Listed are six references. (YP)

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

    Science.gov (United States)

    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…

  20. Thirty-three miniatures mathematical and algorithmic applications of linear algebra

    CERN Document Server

    Matousek, Jiří

    2010-01-01

    This volume contains a collection of clever mathematical applications of linear algebra, mainly in combinatorics, geometry, and algorithms. Each chapter covers a single main result with motivation and full proof in at most ten pages and can be read independently of all other chapters (with minor exceptions), assuming only a modest background in linear algebra. The topics include a number of well-known mathematical gems, such as Hamming codes, the matrix-tree theorem, the Lov�sz bound on the Shannon capacity, and a counterexample to Borsuk's conjecture, as well as other, perhaps less popular but similarly beautiful results, e.g., fast associativity testing, a lemma of Steinitz on ordering vectors, a monotonicity result for integer partitions, or a bound for set pairs via exterior products. The simpler results in the first part of the book provide ample material to liven up an undergraduate course of linear algebra. The more advanced parts can be used for a graduate course of linear-algebraic methods or for s...

  1. Numerical stability in problems of linear algebra.

    Science.gov (United States)

    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.

  2. Matlab linear algebra

    CERN Document Server

    Lopez, Cesar

    2014-01-01

    MATLAB is a high-level language and environment for numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java. MATLAB Linear Algebra introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. In addition to giving an introduction to

  3. Current algebra of classical non-linear sigma models

    International Nuclear Information System (INIS)

    Forger, M.; Laartz, J.; Schaeper, U.

    1992-01-01

    The current algebra of classical non-linear sigma models on arbitrary Riemannian manifolds is analyzed. It is found that introducing, in addition to the Noether current j μ associated with the global symmetry of the theory, a composite scalar field j, the algebra closes under Poisson brackets. (orig.)

  4. Algebraic Reasoning in Solving Mathematical Problem Based on Learning Style

    Science.gov (United States)

    Indraswari, N. F.; Budayasa, I. K.; Ekawati, R.

    2018-01-01

    This study aimed to describe algebraic reasoning of secondary school’s pupils with different learning styles in solving mathematical problem. This study begins by giving the questionnaire to find out the learning styles and followed by mathematical ability test to get three subjects of 8th-grade whereas the learning styles of each pupil is visual, auditory, kinesthetic and had similar mathematical abilities. Then it continued with given algebraic problems and interviews. The data is validated using triangulation of time. The result showed that in the pattern of seeking indicator, subjects identified the things that were known and asked based on them observations. The visual and kinesthetic learners represented the known information in a chart, whereas the auditory learner in a table. In addition, they found the elements which makes the pattern and made a relationship between two quantities. In the pattern recognition indicator, they created conjectures on the relationship between two quantities and proved it. In the generalization indicator, they were determining the general rule of pattern found on each element of pattern using algebraic symbols and created a mathematical model. Visual and kinesthetic learners determined the general rule of equations which was used to solve problems using algebraic symbols, but auditory learner in a sentence.

  5. Linear algebra and group theory for physicists

    CERN Document Server

    Rao, K N Srinivasa

    2006-01-01

    Professor Srinivasa Rao's text on Linear Algebra and Group Theory is directed to undergraduate and graduate students who wish to acquire a solid theoretical foundation in these mathematical topics which find extensive use in physics. Based on courses delivered during Professor Srinivasa Rao's long career at the University of Mysore, this text is remarkable for its clear exposition of the subject. Advanced students will find a range of topics such as the Representation theory of Linear Associative Algebras, a complete analysis of Dirac and Kemmer algebras, Representations of the Symmetric group via Young Tableaux, a systematic derivation of the Crystallographic point groups, a comprehensive and unified discussion of the Rotation and Lorentz groups and their representations, and an introduction to Dynkin diagrams in the classification of Lie groups. In addition, the first few chapters on Elementary Group Theory and Vector Spaces also provide useful instructional material even at an introductory level. An author...

  6. Linearized dynamical approach to current algebra

    International Nuclear Information System (INIS)

    Scadron, M.D.

    1995-07-01

    We study the original motivations searching for a nonlinear chiral Lagrangian to replace the linear sigma model while manifesting all the successful properties of current algebra and partial conservation of axial currents (PCAC). (author). 26 refs

  7. Linear algebra and matrices topics for a second course

    CERN Document Server

    Shapiro, Helene

    2015-01-01

    Linear algebra and matrix theory are fundamental tools for almost every area of mathematics, both pure and applied. This book combines coverage of core topics with an introduction to some areas in which linear algebra plays a key role, for example, block designs, directed graphs, error correcting codes, and linear dynamical systems. Notable features include a discussion of the Weyr characteristic and Weyr canonical forms, and their relationship to the better-known Jordan canonical form; the use of block cyclic matrices and directed graphs to prove Frobenius's theorem on the structure of the eigenvalues of a nonnegative, irreducible matrix; and the inclusion of such combinatorial topics as BIBDs, Hadamard matrices, and strongly regular graphs. Also included are McCoy's theorem about matrices with property P, the Bruck-Ryser-Chowla theorem on the existence of block designs, and an introduction to Markov chains. This book is intended for those who are familiar with the linear algebra covered in a typical first c...

  8. The algebra of non-local charges in non-linear sigma models

    International Nuclear Information System (INIS)

    Abdalla, E.; Abdalla, M.C.B.; Brunelli, J.C.; Zadra, A.

    1993-07-01

    We obtain the exact Dirac algebra obeyed by the conserved non-local charges in bosonic non-linear sigma models. Part of the computation is specialized for a symmetry group O(N). As it turns out the algebra corresponds to a cubic deformation of the Kac-Moody algebra. The non-linear terms are computed in closed form. In each Dirac bracket we only find highest order terms (as explained in the paper), defining a saturated algebra. We generalize the results for the presence of a Wess-Zumino term. The algebra is very similar to the previous one, containing now a calculable correction of order one unit lower. (author). 22 refs, 5 figs

  9. Equivalency of two-dimensional algebras

    International Nuclear Information System (INIS)

    Santos, Gildemar Carneiro dos; Pomponet Filho, Balbino Jose S.

    2011-01-01

    Full text: Let us consider a vector z = xi + yj over the field of real numbers, whose basis (i,j) satisfy a given algebra. Any property of this algebra will be reflected in any function of z, so we can state that the knowledge of the properties of an algebra leads to more general conclusions than the knowledge of the properties of a function. However structural properties of an algebra do not change when this algebra suffers a linear transformation, though the structural constants defining this algebra do change. We say that two algebras are equivalent to each other whenever they are related by a linear transformation. In this case, we have found that some relations between the structural constants are sufficient to recognize whether or not an algebra is equivalent to another. In spite that the basis transform linearly, the structural constants change like a third order tensor, but some combinations of these tensors result in a linear transformation, allowing to write the entries of the transformation matrix as function of the structural constants. Eventually, a systematic way to find the transformation matrix between these equivalent algebras is obtained. In this sense, we have performed the thorough classification of associative commutative two-dimensional algebras, and find that even non-division algebra may be helpful in solving non-linear dynamic systems. The Mandelbrot set was used to have a pictorial view of each algebra, since equivalent algebras result in the same pattern. Presently we have succeeded in classifying some non-associative two-dimensional algebras, a task more difficult than for associative one. (author)

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

    Science.gov (United States)

    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…

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

    Science.gov (United States)

    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…

  12. Expert Strategies in Solving Algebraic Structure Sense Problems: The Case of Quadratic Equations

    Science.gov (United States)

    Jupri, Al; Sispiyati, R.

    2017-02-01

    Structure sense, an intuitive ability towards symbolic expressions, including skills to interpret, to manipulate, and to perceive symbols in different roles, is considered as a key success in learning algebra. In this article, we report results of three phases of a case study on solving algebraic structure sense problems aiming at testing the appropriateness of algebraic structure sense tasks and at investigating expert strategies dealing with the tasks. First, we developed three tasks on quadratic equations based on the characteristics of structure sense for high school algebra. Next, we validated the tasks to seven experts. In the validation process, we requested these experts to solve each task using two different strategies. Finally, we analyzing expert solution strategies in the light of structure sense characteristics. We found that even if eventual expert strategies are in line with the characteristics of structure sense; some of their initial solution strategies used standard procedures which might pay less attention to algebraic structures. This finding suggests that experts have reconsidered their procedural work and have provided more efficient solution strategies. For further investigation, we consider to test the tasks to high school algebra students and to see whether they produce similar results as experts.

  13. Constructive Learning in Undergraduate Linear Algebra

    Science.gov (United States)

    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.

  14. Hamiltonian structure of linearly extended Virasoro algebra

    International Nuclear Information System (INIS)

    Arakelyan, T.A.; Savvidi, G.K.

    1991-01-01

    The Hamiltonian structure of linearly extended Virasoro algebra which admits free bosonic field representation is described. An example of a non-trivial extension is found. The hierarchy of integrable non-linear equations corresponding to this Hamiltonian structure is constructed. This hierarchy admits the Lax representation by matrix Lax operator of second order

  15. Formalized Linear Algebra over Elementary Divisor Rings in Coq

    OpenAIRE

    Cano , Guillaume; Cohen , Cyril; Dénès , Maxime; Mörtberg , Anders; Siles , Vincent

    2016-01-01

    International audience; This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely pre-sented modules over such rings and the uniqueness of the Smith normal form up to multiplication by units. We present formally verified algorithms comput-in...

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

    Science.gov (United States)

    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.

  17. Numerical linear algebra a concise introduction with Matlab and Julia

    CERN Document Server

    Bornemann, Folkmar

    2018-01-01

    This book offers an introduction to the algorithmic-numerical thinking using basic problems of linear algebra. By focusing on linear algebra, it ensures a stronger thematic coherence than is otherwise found in introductory lectures on numerics. The book highlights the usefulness of matrix partitioning compared to a component view, leading not only to a clearer notation and shorter algorithms, but also to significant runtime gains in modern computer architectures. The algorithms and accompanying numerical examples are given in the programming environment MATLAB, and additionally – in an appendix – in the future-oriented, freely accessible programming language Julia. This book is suitable for a two-hour lecture on numerical linear algebra from the second semester of a bachelor's degree in mathematics.

  18. 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

    Science.gov (United States)

    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…

  19. Operational matrices with respect to Hermite polynomials and their applications in solving linear dierential equations with variable coecients

    Directory of Open Access Journals (Sweden)

    A. Aminataei

    2014-05-01

    Full Text Available In this paper, a new and ecient approach is applied for numerical approximation of the linear dierential equations with variable coecients based on operational matrices with respect to Hermite polynomials. Explicit formulae which express the Hermite expansioncoecients for the moments of derivatives of any dierentiable function in terms of the original expansion coecients of the function itself are given in the matrix form. The mainimportance of this scheme is that using this approach reduces solving the linear dierentialequations to solve a system of linear algebraic equations, thus greatly simplifying the problem. In addition, two experiments are given to demonstrate the validity and applicability of the method

  20. Primary School Students' Strategies in Early Algebra Problem Solving Supported by an Online Game

    Science.gov (United States)

    van den Heuvel-Panhuizen, Marja; Kolovou, Angeliki; Robitzsch, Alexander

    2013-01-01

    In this study we investigated the role of a dynamic online game on students' early algebra problem solving. In total 253 students from grades 4, 5, and 6 (10-12 years old) used the game at home to solve a sequence of early algebra problems consisting of contextual problems addressing covarying quantities. Special software monitored the…

  1. Fast and Elegant Numerical Linear Algebra Using the RcppEigen Package

    Directory of Open Access Journals (Sweden)

    Douglas Bates

    2013-01-01

    Full Text Available The RcppEigen package provides access from R (R Core Team 2012a to the Eigen (Guennebaud, Jacob, and others 2012 C++ template library for numerical linear algebra. Rcpp (Eddelbuettel and François 2011, 2012 classes and specializations of the C++ templated functions as and wrap from Rcpp provide the "glue" for passing objects from R to C++ and back. Several introductory examples are presented. This is followed by an in-depth discussion of various available approaches for solving least-squares problems, including rank-revealing methods, concluding with an empirical run-time comparison. Last but not least, sparse matrix methods are discussed.

  2. Multicore Performance of Block Algebraic Iterative Reconstruction Methods

    DEFF Research Database (Denmark)

    Sørensen, Hans Henrik B.; Hansen, Per Christian

    2014-01-01

    Algebraic iterative methods are routinely used for solving the ill-posed sparse linear systems arising in tomographic image reconstruction. Here we consider the algebraic reconstruction technique (ART) and the simultaneous iterative reconstruction techniques (SIRT), both of which rely on semiconv......Algebraic iterative methods are routinely used for solving the ill-posed sparse linear systems arising in tomographic image reconstruction. Here we consider the algebraic reconstruction technique (ART) and the simultaneous iterative reconstruction techniques (SIRT), both of which rely...... on semiconvergence. Block versions of these methods, based on a partitioning of the linear system, are able to combine the fast semiconvergence of ART with the better multicore properties of SIRT. These block methods separate into two classes: those that, in each iteration, access the blocks in a sequential manner...... a fixed relaxation parameter in each method, namely, the one that leads to the fastest semiconvergence. Computational results show that for multicore computers, the sequential approach is preferable....

  3. Positive projections of symmetric matrices and Jordan algebras

    DEFF Research Database (Denmark)

    Fuglede, Bent; Jensen, Søren Tolver

    2013-01-01

    An elementary proof is given that the projection from the space of all symmetric p×p matrices onto a linear subspace is positive if and only if the subspace is a Jordan algebra. This solves a problem in a statistical model.......An elementary proof is given that the projection from the space of all symmetric p×p matrices onto a linear subspace is positive if and only if the subspace is a Jordan algebra. This solves a problem in a statistical model....

  4. Journal Writing: Enlivening Elementary Linear Algebra.

    Science.gov (United States)

    Meel, David E.

    1999-01-01

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

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

    Science.gov (United States)

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

    2017-10-01

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

  6. Matrix form of Legendre polynomials for solving linear integro-differential equations of high order

    Science.gov (United States)

    Kammuji, M.; Eshkuvatov, Z. K.; Yunus, Arif A. M.

    2017-04-01

    This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods.

  7. Linear algebraic methods applied to intensity modulated radiation therapy.

    Science.gov (United States)

    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.

  8. A linear process-algebraic format for probabilistic systems with data (extended version)

    NARCIS (Netherlands)

    Katoen, Joost P.; van de Pol, Jan Cornelis; Stoelinga, Mariëlle Ida Antoinette; Timmer, Mark

    2010-01-01

    This paper presents a novel linear process-algebraic format for probabilistic automata. The key ingredient is a symbolic transformation of probabilistic process algebra terms that incorporate data into this linear format while preserving strong probabilistic bisimulation. This generalises similar

  9. MODELING IN MAPLE AS THE RESEARCHING MEANS OF FUNDAMENTAL CONCEPTS AND PROCEDURES IN LINEAR ALGEBRA

    Directory of Open Access Journals (Sweden)

    Vasil Kushnir

    2016-05-01

    Full Text Available The article is devoted to binary technology and "fundamental training technology." Binary training refers to the simultaneous teaching of mathematics and computer science, for example differential equations and Maple, linear algebra and Maple. Moreover the system of traditional course of Maple is not performed. The use of the opportunities of Maple-technology in teaching mathematics is based on the following fundamental concepts of computer science as an algorithm, program, a linear program, cycle, branching, relative operators, etc. That’s why only a certain system of command operators in Maple is considered. They are necessary for fundamental concepts of linear algebra and differential equations studying in Maple-environment. Relative name - "the technology of fundamental training" reflects the study of fundamental mathematical concepts and procedures that express the properties of these concepts in Maple-environment. This article deals with the study of complex fundamental concepts of linear algebra (determinant of the matrix and algorithm of its calculation, the characteristic polynomial of the matrix and the eigenvalues of matrix, canonical form of characteristic matrix, eigenvectors of matrix, elementary divisors of the characteristic matrix, etc., which are discussed in the appropriate courses briefly enough, and sometimes are not considered at all, but they are important in linear systems of differential equations, asymptotic methods for solving differential equations, systems of linear equations. Herewith complex and voluminous procedures of finding of these linear algebra concepts embedded in Maple can be performed as a result of a simple command-operator. Especially important issue is building matrix to canonical form. In fact matrix functions are effectively reduced to the functions of the diagonal matrix or matrix in Jordan canonical form. These matrices are used to rise a square matrix to a power, to extract the roots of the n

  10. On Graph C*-Algebras with a Linear Ideal Lattice

    DEFF Research Database (Denmark)

    Eilers, Søren; Restorff, Gunnar; Ruiz, Efren

    2010-01-01

    At the cost of restricting the nature of the involved K-groups, we prove a classication result for a hitherto unexplored class of graph C-algebras, allowing us to classify all graph C-algebras on nitely many vertices with a nite linear ideal lattice if all pair of vertices are connected by innitely...

  11. Linear algebraic approach to electron-molecule collisions

    International Nuclear Information System (INIS)

    Schneider, B.I.; Collins, L.A.

    1983-01-01

    The various levels of sophistication of the linear algebraic method are discussed and its application to electron-molecule collisions of H 2 , N 2 LiH, LiF and HCl is described. 13 references, 2 tables

  12. Mathematical modelling in engineering: A proposal to introduce linear algebra concepts

    Directory of Open Access Journals (Sweden)

    Andrea Dorila Cárcamo

    2016-03-01

    Full Text Available The modern dynamic world requires that basic science courses for engineering, including linear algebra, emphasize the development of mathematical abilities primarily associated with modelling and interpreting, which aren´t limited only to calculus abilities. Considering this, an instructional design was elaborated based on mathematic modelling and emerging heuristic models for the construction of specific linear algebra concepts:  span and spanning set. This was applied to first year engineering students. Results suggest that this type of instructional design contributes to the construction of these mathematical concepts and can also favour first year engineering students understanding of key linear algebra concepts and potentiate the development of higher order skills.

  13. The Linear Span of Projections in AH Algebras and for Inclusions of C*-Algebras

    Directory of Open Access Journals (Sweden)

    Dinh Trung Hoa

    2013-01-01

    Full Text Available In the first part of this paper, we show that an AH algebra A=lim→(Ai,ϕi has the LP property if and only if every element of the centre of Ai belongs to the closure of the linear span of projections in A. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation in the sense of Bratteli and Elliott. The second contribution of this paper is that for an inclusion of unital C*-algebras P⊂A with a finite Watatani index, if a faithful conditional expectation E:A→P has the Rokhlin property in the sense of Kodaka et al., then P has the LP property under the condition thatA has the LP property. As an application, let A be a simple unital C*-algebra with the LP property, α an action of a finite group G onto Aut(A. If α has the Rokhlin property in the sense of Izumi, then the fixed point algebra AG and the crossed product algebra A ⋊α G have the LP property. We also point out that there is a symmetry on the CAR algebra such that its fixed point algebra does not have the LP property.

  14. GPU Linear algebra extensions for GNU/Octave

    International Nuclear Information System (INIS)

    Bosi, L B; Mariotti, M; Santocchia, A

    2012-01-01

    Octave is one of the most widely used open source tools for numerical analysis and liner algebra. Our project aims to improve Octave by introducing support for GPU computing in order to speed up some linear algebra operations. The core of our work is a C library that executes some BLAS operations concerning vector- vector, vector matrix and matrix-matrix functions on the GPU. OpenCL functions are used to program GPU kernels, which are bound within the GNU/octave framework. We report the project implementation design and some preliminary results about performance.

  15. Non-linear realization of the Virasoro-Kac-Moody algebra and the anomalies

    International Nuclear Information System (INIS)

    Aoyama, S.

    1988-01-01

    The non-linear realization of the Virasoro algebra x Kac-Moody algebra will be studied. We will calculate the Ricci tensor of the relevant Kaehler manifold to show a new vacuum structure for this coupled algebra. (orig.)

  16. Solving the linear inviscid shallow water equations in one dimension, with variable depth, using a recursion formula

    Science.gov (United States)

    Hernandez-Walls, R.; Martín-Atienza, B.; Salinas-Matus, M.; Castillo, J.

    2017-11-01

    When solving the linear inviscid shallow water equations with variable depth in one dimension using finite differences, a tridiagonal system of equations must be solved. Here we present an approach, which is more efficient than the commonly used numerical method, to solve this tridiagonal system of equations using a recursion formula. We illustrate this approach with an example in which we solve for a rectangular channel to find the resonance modes. Our numerical solution agrees very well with the analytical solution. This new method is easy to use and understand by undergraduate students, so it can be implemented in undergraduate courses such as Numerical Methods, Lineal Algebra or Differential Equations.

  17. Solving the linear inviscid shallow water equations in one dimension, with variable depth, using a recursion formula

    International Nuclear Information System (INIS)

    Hernandez-Walls, R; Martín-Atienza, B; Salinas-Matus, M; Castillo, J

    2017-01-01

    When solving the linear inviscid shallow water equations with variable depth in one dimension using finite differences, a tridiagonal system of equations must be solved. Here we present an approach, which is more efficient than the commonly used numerical method, to solve this tridiagonal system of equations using a recursion formula. We illustrate this approach with an example in which we solve for a rectangular channel to find the resonance modes. Our numerical solution agrees very well with the analytical solution. This new method is easy to use and understand by undergraduate students, so it can be implemented in undergraduate courses such as Numerical Methods, Lineal Algebra or Differential Equations. (paper)

  18. Non-linear realizations of superconformal and W-algebras as embeddings of strings

    International Nuclear Information System (INIS)

    Bellucci, S.

    1998-01-01

    We propose a simple method for constructing representations of (super)conformal and non-linear W-type algebras in terms of their subalgebras and corresponding Nambu-Goldstone fields. We apply it to N=2 and N=1 superconformal algebras and describe in this way various embeddings of strings and superstrings for which these algebras and their subalgebras define world-sheet symmetries. Besides reproducing the known examples, we present some new ones, in particular an embedding of the bosonic string with additional U(1) affine symmetry into N=2 superstring. We also apply our method to the non-linear W 3 (2) algebra and demonstrate that the linearization procedure worked out for it some time ago gets a natural interpretation as a kind of string embedding. All these embeddings include the critical ones as particular cases. (orig.)

  19. Solving differential–algebraic equation systems by means of index reduction methodology

    DEFF Research Database (Denmark)

    Sørensen, Kim; Houbak, Niels; Condra, Thomas

    2006-01-01

    of a number of differential equations and algebraic equations — a so called DAE system. Two of the DAE systems are of index 1 and they can be solved by means of standard DAE-solvers. For the actual application, the equation systems are integrated by means of MATLAB’s solver: ode23t, that solves moderately...... stiff ODEs and index 1 DAEs by means of the trapezoidal rule. The last sub-model that models the boilers steam drum consist of two differential and three algebraic equations. The index of this model is greater than 1, which means that ode23t cannot integrate this equation system. In this paper......, it is shown how the equation system, by means of an index reduction methodology, can be reduced to a system of ordinary differential equations — ODEs....

  20. Tracking control of concentration profiles in a fed-batch bioreactor using a linear algebra methodology.

    Science.gov (United States)

    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.

  1. A Simple and Practical Linear Algebra Library Interface with Static Size Checking

    Directory of Open Access Journals (Sweden)

    Akinori Abe

    2015-12-01

    Full Text Available Linear algebra is a major field of numerical computation and is widely applied. Most linear algebra libraries (in most programming languages do not statically guarantee consistency of the dimensions of vectors and matrices, causing runtime errors. While advanced type systems—specifically, dependent types on natural numbers—can ensure consistency among the sizes of collections such as lists and arrays, such type systems generally require non-trivial changes to existing languages and application programs, or tricky type-level programming. We have developed a linear algebra library interface that verifies the consistency (with respect to dimensions of matrix operations by means of generative phantom types, implemented via fairly standard ML types and module system. To evaluate its usability, we ported to it a practical machine learning library from a traditional linear algebra library. We found that most of the changes required for the porting could be made mechanically, and changes that needed human thought are minor.

  2. Adjamagbo Determinant and Serre conjecture for linear groups over Weyl algebras

    OpenAIRE

    Adjamagbo, Kossivi

    2008-01-01

    Thanks to the theory of determinants over an Ore domain, also called Adjamagbo determinant by the Russian school of non commutative algebra, we extend to any Weyl algebra over a field of characteristic zero Suslin theorem solving what Suslin himself called the $K_1$-analogue of the well-known Serre Conjecture and asserting that for any integer $n$ greater than 2, any $n$ by $n$ matrix with coefficients in any algebra of polynomials over a field and with determinant one is the product of eleme...

  3. Answers to selected problems in multivariable calculus with linear algebra and series

    CERN Document Server

    Trench, William F

    1972-01-01

    Answers to Selected Problems in Multivariable Calculus with Linear Algebra and Series contains the answers to selected problems in linear algebra, the calculus of several variables, and series. Topics covered range from vectors and vector spaces to linear matrices and analytic geometry, as well as differential calculus of real-valued functions. Theorems and definitions are included, most of which are followed by worked-out illustrative examples.The problems and corresponding solutions deal with linear equations and matrices, including determinants; vector spaces and linear transformations; eig

  4. Quasi exactly solvable operators and abstract associative algebras

    International Nuclear Information System (INIS)

    Brihaye, Y.; Kosinski, P.

    1998-01-01

    We consider the vector spaces consisting of direct sums of polynomials of given degrees and we show how to classify the linear differential operators preserving these spaces. The families of operators so obtained are identified as the envelopping algebras of particular abstract associative algebras. Some of these operators can be transformed into quasi exactly solvable Schroedinger operators which, having a hidden algebra, can be partially solved algebraically; we exhibit however a series of Schoedinger equations which, while completely solvable algebraically, do not possess a hidden algebra

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

    Science.gov (United States)

    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.

  6. Solving differential-algebraic equation systems by means of index reduction methodology

    DEFF Research Database (Denmark)

    Sørensen, Kim; Houbak, Niels; Condra, Thomas Joseph

    2006-01-01

    of a number of differential equations and algebraic equations - a so called DAE system. Two of the DAE systems are of index 1 and they can be solved by means of standard DAE-solvers. For the actual application, the equation systems are integrated by means of MATLAB’s solver: ode23t, that solves moderately...... stiff ODE’s and index 1 DAE’s by means of the trapezoidal rule. The last sub-model that models the boilers steam drum consist of two differential and three algebraic equations. The index of this model is greater than 1, which means that ode23t cannot integrate this equation system. In this paper......, it is shown how the equation system, by means of an index reduction methodology, can be reduced to a system of Ordinary- Differential-Equations - ODE’s....

  7. The algebra of non-local charges in non-linear sigma models

    International Nuclear Information System (INIS)

    Abdalla, E.; Abdalla, M.C.B.; Brunelli, J.C.; Zadra, A.

    1994-01-01

    It is derived the complete Dirac algebra satisfied by non-local charges conserved in non-linear sigma models. Some examples of calculation are given for the O(N) symmetry group. The resulting algebra corresponds to a saturated cubic deformation (with only maximum order terms) of the Kac-Moody algebra. The results are generalized for when a Wess-Zumino term be present. In that case the algebra contains a minor order correction (sub-saturation). (author). 1 ref

  8. Quasi-Linear Algebras and Integrability (the Heisenberg Picture

    Directory of Open Access Journals (Sweden)

    Alexei Zhedanov

    2008-02-01

    Full Text Available We study Poisson and operator algebras with the ''quasi-linear property'' from the Heisenberg picture point of view. This means that there exists a set of one-parameter groups yielding an explicit expression of dynamical variables (operators as functions of ''time'' t. We show that many algebras with nonlinear commutation relations such as the Askey-Wilson, q-Dolan-Grady and others satisfy this property. This provides one more (explicit Heisenberg evolution interpretation of the corresponding integrable systems.

  9. Numerical linear algebra on emerging architectures: The PLASMA and MAGMA projects

    International Nuclear Information System (INIS)

    Agullo, Emmanuel; Demmel, Jim; Dongarra, Jack; Hadri, Bilel; Kurzak, Jakub; Langou, Julien; Ltaief, Hatem; Luszczek, Piotr; Tomov, Stanimire

    2009-01-01

    The emergence and continuing use of multi-core architectures and graphics processing units require changes in the existing software and sometimes even a redesign of the established algorithms in order to take advantage of now prevailing parallelism. Parallel Linear Algebra for Scalable Multi-core Architectures (PLASMA) and Matrix Algebra on GPU and Multics Architectures (MAGMA) are two projects that aims to achieve high performance and portability across a wide range of multi-core architectures and hybrid systems respectively. We present in this document a comparative study of PLASMA's performance against established linear algebra packages and some preliminary results of MAGMA on hybrid multi-core and GPU systems.

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

    Science.gov (United States)

    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…

  11. Linear-algebraic approach to electron-molecule collisions: General formulation

    International Nuclear Information System (INIS)

    Collins, L.A.; Schneider, B.I.

    1981-01-01

    We present a linear-algebraic approach to electron-molecule collisions based on an integral equations form with either logarithmic or asymptotic boundary conditions. The introduction of exchange effects does not alter the basic form or order of the linear-algebraic equations for a local potential. In addition to the standard procedure of directly evaluating the exchange integrals by numerical quadrature, we also incorporate exchange effects through a separable-potential approximation. Efficient schemes are developed for reducing the number of points and channels that must be included. The method is applied at the static-exchange level to a number of molecular systems including H 2 , N 2 , LiH, and CO 2

  12. Symmetric linear systems - An application of algebraic systems theory

    Science.gov (United States)

    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.

  13. Problems in abstract algebra

    CERN Document Server

    Wadsworth, A R

    2017-01-01

    This is a book of problems in abstract algebra for strong undergraduates or beginning graduate students. It can be used as a supplement to a course or for self-study. The book provides more variety and more challenging problems than are found in most algebra textbooks. It is intended for students wanting to enrich their learning of mathematics by tackling problems that take some thought and effort to solve. The book contains problems on groups (including the Sylow Theorems, solvable groups, presentation of groups by generators and relations, and structure and duality for finite abelian groups); rings (including basic ideal theory and factorization in integral domains and Gauss's Theorem); linear algebra (emphasizing linear transformations, including canonical forms); and fields (including Galois theory). Hints to many problems are also included.

  14. Many-core graph analytics using accelerated sparse linear algebra routines

    Science.gov (United States)

    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.

  15. Hardware Tailored Linear Algebra for Implicit Integrators in Embedded NMPC

    DEFF Research Database (Denmark)

    Frison, Gianluca; Quirynen, Rien; Zanelli, Andrea

    2017-01-01

    . In the case of stiff or implicitly defined dynamics, implicit integration schemes are typically preferred. This paper proposes a tailored implementation of the necessary linear algebra routines (LU factorization and triangular solutions), in order to allow for a considerable computational speedup...... of such integrators. In particular, the open-source BLASFEO framework is presented as a library of efficient linear algebra routines for small to medium-scale embedded optimization applications. Its performance is illustrated on the nonlinear optimal control example of a chain of masses. The proposed library allows...

  16. Accelerating Dense Linear Algebra on the GPU

    DEFF Research Database (Denmark)

    Sørensen, Hans Henrik Brandenborg

    and matrix-vector operations on GPUs. Such operations form the backbone of level 1 and level 2 routines in the Basic Linear Algebra Subroutines (BLAS) library and are therefore of great importance in many scientific applications. The target hardware is the most recent NVIDIA Tesla 20-series (Fermi...

  17. Noise limitations in optical linear algebra processors.

    Science.gov (United States)

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

    1990-05-10

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

  18. Modules as Learning Tools in Linear Algebra

    Science.gov (United States)

    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…

  19. Simplified neural networks for solving linear least squares and total least squares problems in real time.

    Science.gov (United States)

    Cichocki, A; Unbehauen, R

    1994-01-01

    In this paper a new class of simplified low-cost analog artificial neural networks with on chip adaptive learning algorithms are proposed for solving linear systems of algebraic equations in real time. The proposed learning algorithms for linear least squares (LS), total least squares (TLS) and data least squares (DLS) problems can be considered as modifications and extensions of well known algorithms: the row-action projection-Kaczmarz algorithm and/or the LMS (Adaline) Widrow-Hoff algorithms. The algorithms can be applied to any problem which can be formulated as a linear regression problem. The correctness and high performance of the proposed neural networks are illustrated by extensive computer simulation results.

  20. High performance linear algebra algorithms: An introduction

    DEFF Research Database (Denmark)

    Gustavson, F.G.; Wasniewski, Jerzy

    2006-01-01

    his Mini-Symposium consisted of two back to back sessions, each consisting of five presentations, held on the afternoon of Monday, June 21, 2004. A major theme of both sessions was novel data structures for the matrices of dense linear algebra, DLA. Talks one to four of session one all centered...

  1. Finding the radical of an algebra of linear transformations

    NARCIS (Netherlands)

    Cohen, A.M.; Ivanyos, G.; Wales, D.B.

    1997-01-01

    We present a method that reduces the problem of computing the radical of a matrix algebra over an arbitrary field to solving systems of semilinear equations. The complexity of the algorithm, measured in the number of arithmetic operations and the total number of the coefficients passed to an oracle

  2. Groups, matrices, and vector spaces a group theoretic approach to linear algebra

    CERN Document Server

    Carrell, James B

    2017-01-01

    This unique text provides a geometric approach to group theory and linear algebra, bringing to light the interesting ways in which these subjects interact. Requiring few prerequisites beyond understanding the notion of a proof, the text aims to give students a strong foundation in both geometry and algebra. Starting with preliminaries (relations, elementary combinatorics, and induction), the book then proceeds to the core topics: the elements of the theory of groups and fields (Lagrange's Theorem, cosets, the complex numbers and the prime fields), matrix theory and matrix groups, determinants, vector spaces, linear mappings, eigentheory and diagonalization, Jordan decomposition and normal form, normal matrices, and quadratic forms. The final two chapters consist of a more intensive look at group theory, emphasizing orbit stabilizer methods, and an introduction to linear algebraic groups, which enriches the notion of a matrix group. Applications involving symm etry groups, determinants, linear coding theory ...

  3. A Linear Algebra Measure of Cluster Quality.

    Science.gov (United States)

    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)

  4. Representations of Lie algebras and partial differential equations

    CERN Document Server

    Xu, Xiaoping

    2017-01-01

    This book provides explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic codes, combinatorics and algebraic varieties, summarizing the author’s works and his joint works with his former students.  Further, it presents various oscillator generalizations of the classical representation theorem on harmonic polynomials, and highlights new functors from the representation category of a simple Lie algebra to that of another simple Lie algebra. Partial differential equations play a key role in solving certain representation problems. The weight matrices of the minimal and adjoint representations over the simple Lie algebras of types E and F are proved to generate ternary orthogonal linear codes with large minimal distances. New multi-variable hypergeometric functions related to the root systems of simple Lie algebras are introduced in connection with quantum many-body systems in one dimension. In addition, the book identifies certai...

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

    Science.gov (United States)

    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…

  6. Matrix algebra for linear models

    CERN Document Server

    Gruber, Marvin H J

    2013-01-01

    Matrix methods have evolved from a tool for expressing statistical problems to an indispensable part of the development, understanding, and use of various types of complex statistical analyses. This evolution has made matrix methods a vital part of statistical education. Traditionally, matrix methods are taught in courses on everything from regression analysis to stochastic processes, thus creating a fractured view of the topic. Matrix Algebra for Linear Models offers readers a unique, unified view of matrix analysis theory (where and when necessary), methods, and their applications. Written f

  7. Study on infrared multiphoton excitation of the linear triatomic molecule by the Lie-algebra approach

    International Nuclear Information System (INIS)

    Feng, H.; Zheng, Y.; Ding, S.

    2007-01-01

    Infrared multiphoton vibrational excitation of the linear triatomic molecule has been studied using the quadratic anharmonic Lie-algebra model, unitary transformations, and Magnus approximation. An explicit Lie-algebra expression for the vibrational transition probability is obtained by using a Lie-algebra approach. This explicit Lie-algebra expressions for time-evolution operator and vibrational transition probabilities make the computation clearer and easier. The infrared multiphoton vibrational excitation of the DCN linear tri-atomic molecule is discussed as an example

  8. Topics in computational linear optimization

    DEFF Research Database (Denmark)

    Hultberg, Tim Helge

    2000-01-01

    Linear optimization has been an active area of research ever since the pioneering work of G. Dantzig more than 50 years ago. This research has produced a long sequence of practical as well as theoretical improvements of the solution techniques avilable for solving linear optimization problems...... of high quality solvers and the use of algebraic modelling systems to handle the communication between the modeller and the solver. This dissertation features four topics in computational linear optimization: A) automatic reformulation of mixed 0/1 linear programs, B) direct solution of sparse unsymmetric...... systems of linear equations, C) reduction of linear programs and D) integration of algebraic modelling of linear optimization problems in C++. Each of these topics is treated in a separate paper included in this dissertation. The efficiency of solving mixed 0-1 linear programs by linear programming based...

  9. An introduction to the history of algebra solving equations from Mesopotamian times to the Renaissance

    CERN Document Server

    Sesiano, Jacques

    2009-01-01

    This text should not be viewed as a comprehensive history of algebra before 1600, but as a basic introduction to the types of problems that illustrate the earliest forms of algebra. It would be particularly useful for an instructor who is looking for examples to help enliven a course on elementary algebra with problems drawn from actual historical texts. -Warren Van Egmond about the French edition for MathSciNet This book does not aim to give an exhaustive survey of the history of algebra up to early modern times but merely to present some significant steps in solving equations and, wherever

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

    Science.gov (United States)

    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.

  11. A generalized variational algebra and conserved densities for linear evolution equations

    International Nuclear Information System (INIS)

    Abellanas, L.; Galindo, A.

    1978-01-01

    The symbolic algebra of Gel'fand and Dikii is generalized to the case of n variables. Using this algebraic approach a rigorous characterization of the polynomial kernel of the variational derivative is given. This is applied to classify all the conservation laws for linear polynomial evolution equations of arbitrary order. (Auth.)

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

    Science.gov (United States)

    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.

  13. Algebraic Theory of Linear Viscoelastic Nematodynamics

    International Nuclear Information System (INIS)

    Leonov, Arkady I.

    2008-01-01

    This paper consists of two parts. The first one develops algebraic theory of linear anisotropic nematic 'N-operators' build up on the additive group of traceless second rank 3D tensors. These operators have been implicitly used in continual theories of nematic liquid crystals and weakly elastic nematic elastomers. It is shown that there exists a non-commutative, multiplicative group N 6 of N-operators build up on a manifold in 6D space of parameters. Positive N-operators, which in physical applications hold thermodynamic stability constraints, do not generally form a subgroup of group N 6 . A three-parametric, commutative transversal-isotropic subgroup S 3 subset of N 6 of positive symmetric nematic operators is also briefly discussed. The special case of singular, non-negative symmetric N-operators reveals the algebraic structure of nematic soft deformation modes. The second part of the paper develops a theory of linear viscoelastic nematodynamics applicable to liquid crystalline polymer. The viscous and elastic nematic components in theory are described by using the Leslie-Ericksen-Parodi (LEP) approach for viscous nematics and de Gennes free energy for weakly elastic nematic elastomers. The case of applied external magnetic field exemplifies the occurrence of non-symmetric stresses. In spite of multi-(10) parametric character of the theory, the use of nematic operators presents it in a transparent form. When the magnetic field is absent, the theory is simplified for symmetric case with six parameters, and takes an extremely simple, two-parametric form for viscoelastic nematodynamics with possible soft deformation modes. It is shown that the linear nematodynamics is always reducible to the LEP-like equations where the coefficients are changed for linear memory functionals whose parameters are calculated from original viscosities and moduli

  14. su(1,2) Algebraic Structure of XYZ Antiferromagnetic Model in Linear Spin-Wave Frame

    International Nuclear Information System (INIS)

    Jin Shuo; Xie Binghao; Yu Zhaoxian; Hou Jingmin

    2008-01-01

    The XYZ antiferromagnetic model in linear spin-wave frame is shown explicitly to have an su(1,2) algebraic structure: the Hamiltonian can be written as a linear function of the su(1,2) algebra generators. Based on it, the energy eigenvalues are obtained by making use of the similar transformations, and the algebraic diagonalization method is investigated. Some numerical solutions are given, and the results indicate that only one group solution could be accepted in physics

  15. A linear programming manual

    Science.gov (United States)

    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.

  16. Meromorphic functions and linear algebra

    CERN Document Server

    Nevanlinna, Olavi

    2003-01-01

    This volume describes for the first time in monograph form important applications in numerical methods of linear algebra. The author presents new material and extended results from recent papers in a very readable style. The main goal of the book is to study the behavior of the resolvent of a matrix under the perturbation by low rank matrices. Whereas the eigenvalues (the poles of the resolvent) and the pseudospectra (the sets where the resolvent takes large values) can move dramatically under such perturbations, the growth of the resolvent as a matrix-valued meromorphic function remains essen

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

    Science.gov (United States)

    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)

  18. Inverse Modelling Problems in Linear Algebra Undergraduate Courses

    Science.gov (United States)

    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…

  19. Near-infrared reflectance analysis by Gauss-Jordan linear algebra

    International Nuclear Information System (INIS)

    Honigs, D.E.; Freelin, J.M.; Hieftje, G.M.; Hirschfeld, T.B.

    1983-01-01

    Near-infrared reflectance analysis is an analytical technique that uses the near-infrared diffuse reflectance of a sample at several discrete wavelengths to predict the concentration of one or more of the chemical species in that sample. However, because near-infrared bands from solid samples are both abundant and broad, the reflectance at a given wavelength usually contains contributions from several sample components, requiring extensive calculations on overlapped bands. In the present study, these calculations have been performed using an approach similar to that employed in multi-component spectrophotometry, but with Gauss-Jordan linear algebra serving as the computational vehicle. Using this approach, correlations for percent protein in wheat flour and percent benzene in hydrocarbons have been obtained and are evaluated. The advantages of a linear-algebra approach over the common one employing stepwise regression are explored

  20. Matrix Operations for Engineers and Scientists An Essential Guide in Linear Algebra

    CERN Document Server

    Jeffrey, Alan

    2010-01-01

    Engineers and scientists need to have an introduction to the basics of linear algebra in a context they understand. Computer algebra systems make the manipulation of matrices and the determination of their properties a simple matter, and in practical applications such software is often essential. However, using this tool when learning about matrices, without first gaining a proper understanding of the underlying theory, limits the ability to use matrices and to apply them to new problems. This book explains matrices in the detail required by engineering or science students, and it discusses linear systems of ordinary differential equations. These students require a straightforward introduction to linear algebra illustrated by applications to which they can relate. It caters of the needs of undergraduate engineers in all disciplines, and provides considerable detail where it is likely to be helpful. According to the author the best way to understand the theory of matrices is by working simple exercises designe...

  1. Application of laser speckle to randomized numerical linear algebra

    Science.gov (United States)

    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.

  2. Minimal deformation of the commutative algebra and the linear group GL(n)

    International Nuclear Information System (INIS)

    Zupnik, B.M.

    1993-01-01

    We consider the relations of generalized commutativity in the algebra of formal series M q (x i ), which conserve a tensor I q -graduation and depend on parameters q(i,k). We choose the I q -invariant version of differential calculus on M q . A new construction of the symmetrized tensor product for M q -type algebras and the corresponding definition of minimally deformed linear group QGL(n) and Lie algebra qgl(n) are proposed. We study the connection of QGL(n) and qgl(n) with the special matrix algebra Mat(n, Q) containing matrices with noncommutative elements. A definition of the deformed determinant in the algebra Mat(n, Q) is given. The exponential parametrization in the algebra Mat(n, Q) is considered on the basis of Campbell-Hausdorf formula

  3. Partially Flipped Linear Algebra: A Team-Based Approach

    Science.gov (United States)

    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.…

  4. Parallel algorithms for numerical linear algebra

    CERN Document Server

    van der Vorst, H

    1990-01-01

    This is the first in a new series of books presenting research results and developments concerning the theory and applications of parallel computers, including vector, pipeline, array, fifth/future generation computers, and neural computers.All aspects of high-speed computing fall within the scope of the series, e.g. algorithm design, applications, software engineering, networking, taxonomy, models and architectural trends, performance, peripheral devices.Papers in Volume One cover the main streams of parallel linear algebra: systolic array algorithms, message-passing systems, algorithms for p

  5. Definitions Are Important: The Case of Linear Algebra

    Science.gov (United States)

    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…

  6. Structuring students’ analogical reasoning in solving algebra problem

    Science.gov (United States)

    Lailiyah, S.; Nusantara, T.; Sa'dijah, C.; Irawan, E. B.; Kusaeri; Asyhar, A. H.

    2018-01-01

    The average achievement of Indonesian students’ mathematics skills according to Benchmark International Trends in Mathematics and Science Study (TIMSS) is ranked at the 38th out of 42 countries and according to the survey result in Program for International Student Assessment (PISA) is ranked at the 64th out of 65 countries. The low mathematics skill of Indonesian student has become an important reason to research more deeply about reasoning and algebra in mathematics. Analogical reasoning is a very important component in mathematics because it is the key to creativity and it can make the learning process in the classroom become effective. The major part of the analogical reasoning is about structuring including the processes of inferencing and decision-making happens. Those processes involve base domain and target domain. Methodologically, the subjects of this research were 42 students from class XII. The sources of data were derived from the results of thinks aloud, the transcribed interviews, and the videos taken while the subject working on the instruments and interviews. The collected data were analyzed using qualitative techniques. The result of this study described the structuring characteristics of students’ analogical reasoning in solving algebra problems from all the research subjects.

  7. Causal structure and algebraic classification of non-dissipative linear optical media

    International Nuclear Information System (INIS)

    Schuller, Frederic P.; Witte, Christof; Wohlfarth, Mattias N.R.

    2010-01-01

    In crystal optics and quantum electrodynamics in gravitational vacua, the propagation of light is not described by a metric, but an area metric geometry. In this article, this prompts us to study conditions for linear electrodynamics on area metric manifolds to be well-posed. This includes an identification of the timelike future cones and their duals associated to an area metric geometry, and thus paves the ground for a discussion of the related local and global causal structures in standard fashion. In order to provide simple algebraic criteria for an area metric manifold to present a consistent spacetime structure, we develop a complete algebraic classification of area metric tensors up to general transformations of frame. This classification, valuable in its own right, is then employed to prove a theorem excluding the majority of algebraic classes of area metrics as viable spacetimes. Physically, these results classify and drastically restrict the viable constitutive tensors of non-dissipative linear optical media.

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

    Science.gov (United States)

    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.

  9. Creating Discussions with Classroom Voting in Linear Algebra

    Science.gov (United States)

    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…

  10. Linear Algebra and the Experiences of a "Flipper"

    Science.gov (United States)

    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…

  11. A linear algebra course with PC-MATLAB : some experiences

    NARCIS (Netherlands)

    Smits, J.G.M.M.; Rijpkema, J.J.M.

    1992-01-01

    The authors present their views on the impact that the use of computers and software packages should have on the contents of a first service course on linear algebra. Furthermore they report on their experiences using the software package PC-MATLAB in such a course.

  12. From Rota-Baxter algebras to pre-Lie algebras

    International Nuclear Information System (INIS)

    An Huihui; Ba, Chengming

    2008-01-01

    Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras. In this paper, we give all Rota-Baxter operators of weight 1 on complex associative algebras in dimension ≤3 and their corresponding pre-Lie algebras

  13. Solving applied mathematical problems with Matlab

    CERN Document Server

    Xue, Dingyu

    2008-01-01

    Computer Mathematics Language-An Overview. Fundamentals of MATLAB Programming. Calculus Problems. MATLAB Computations of Linear Algebra Problems. Integral Transforms and Complex Variable Functions. Solutions to Nonlinear Equations and Optimization Problems. MATLAB Solutions to Differential Equation Problems. Solving Interpolations and Approximations Problems. Solving Probability and Mathematical Statistics Problems. Nontraditional Solution Methods for Mathematical Problems.

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

    Science.gov (United States)

    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…

  15. A parallel algorithm for solving linear equations arising from one-dimensional network problems

    International Nuclear Information System (INIS)

    Mesina, G.L.

    1991-01-01

    One-dimensional (1-D) network problems, such as those arising from 1- D fluid simulations and electrical circuitry, produce systems of sparse linear equations which are nearly tridiagonal and contain a few non-zero entries outside the tridiagonal. Most direct solution techniques for such problems either do not take advantage of the special structure of the matrix or do not fully utilize parallel computer architectures. We describe a new parallel direct linear equation solution algorithm, called TRBR, which is especially designed to take advantage of this structure on MIMD shared memory machines. The new method belongs to a family of methods which split the coefficient matrix into the sum of a tridiagonal matrix T and a matrix comprised of the remaining coefficients R. Efficient tridiagonal methods are used to algebraically simplify the linear system. A smaller auxiliary subsystem is created and solved and its solution is used to calculate the solution of the original system. The newly devised BR method solves the subsystem. The serial and parallel operation counts are given for the new method and related earlier methods. TRBR is shown to have the smallest operation count in this class of direct methods. Numerical results are given. Although the algorithm is designed for one-dimensional networks, it has been applied successfully to three-dimensional problems as well. 20 refs., 2 figs., 4 tabs

  16. A Newton method for solving continuous multiple material minimum compliance problems

    DEFF Research Database (Denmark)

    Stolpe, M; Stegmann, Jan

    method, one or two linear saddle point systems are solved. These systems involve the Hessian of the objective function, which is both expensive to compute and completely dense. Therefore, the linear algebra is arranged such that the Hessian is not explicitly formed. The main concern is to solve...

  17. A Newton method for solving continuous multiple material minimum compliance problems

    DEFF Research Database (Denmark)

    Stolpe, Mathias; Stegmann, Jan

    2007-01-01

    method, one or two linear saddle point systems are solved. These systems involve the Hessian of the objective function, which is both expensive to compute and completely dense. Therefore, the linear algebra is arranged such that the Hessian is not explicitly formed. The main concern is to solve...

  18. Advanced topics in linear algebra weaving matrix problems through the Weyr form

    CERN Document Server

    O'Meara, Kevin; Vinsonhaler, Charles

    2011-01-01

    The Weyr matrix canonical form is a largely unknown cousin of the Jordan canonical form. Discovered by Eduard Weyr in 1885, the Weyr form outperforms the Jordan form in a number of mathematical situations, yet it remains somewhat of a mystery, even to many who are skilled in linear algebra. Written in an engaging style, this book presents various advanced topics in linear algebra linked through the Weyr form. Kevin O'Meara, John Clark, and Charles Vinsonhaler develop the Weyr form from scratch and include an algorithm for computing it. A fascinating duality exists between the Weyr form and the

  19. Lie Algebraic Treatment of Linear and Nonlinear Beam Dynamics

    Energy Technology Data Exchange (ETDEWEB)

    Alex J. Dragt; Filippo Neri; Govindan Rangarajan; David Douglas; Liam M. Healy; Robert D. Ryne

    1988-12-01

    The purpose of this paper is to present a summary of new methods, employing Lie algebraic tools, for characterizing beam dynamics in charged-particle optical systems. These methods are applicable to accelerator design, charged-particle beam transport, electron microscopes, and also light optics. The new methods represent the action of each separate element of a compound optical system, including all departures from paraxial optics, by a certain operator. The operators for the various elements can then be concatenated, following well-defined rules, to obtain a resultant operator that characterizes the entire system. This paper deals mostly with accelerator design and charged-particle beam transport. The application of Lie algebraic methods to light optics and electron microscopes is described elsewhere (1, see also 44). To keep its scope within reasonable bounds, they restrict their treatment of accelerator design and charged-particle beam transport primarily to the use of Lie algebraic methods for the description of particle orbits in terms of transfer maps. There are other Lie algebraic or related approaches to accelerator problems that the reader may find of interest (2). For a general discussion of linear and nonlinear problems in accelerator physics see (3).

  20. Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry

    Directory of Open Access Journals (Sweden)

    K. S. Mahomed

    2013-01-01

    Full Text Available By use of the Lie symmetry group methods we analyze the relationship between the first integrals of the simplest linear third-order ordinary differential equations (ODEs and their point symmetries. It is well known that there are three classes of linear third-order ODEs for maximal cases of point symmetries which are 4, 5, and 7. The simplest scalar linear third-order equation has seven-point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equation y′′′=0 is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals.

  1. Application of symbolic and algebraic manipulation software in solving applied mechanics problems

    Science.gov (United States)

    Tsai, Wen-Lang; Kikuchi, Noboru

    1993-01-01

    As its name implies, symbolic and algebraic manipulation is an operational tool which not only can retain symbols throughout computations but also can express results in terms of symbols. This report starts with a history of symbolic and algebraic manipulators and a review of the literatures. With the help of selected examples, the capabilities of symbolic and algebraic manipulators are demonstrated. These applications to problems of applied mechanics are then presented. They are the application of automatic formulation to applied mechanics problems, application to a materially nonlinear problem (rigid-plastic ring compression) by finite element method (FEM) and application to plate problems by FEM. The advantages and difficulties, contributions, education, and perspectives of symbolic and algebraic manipulation are discussed. It is well known that there exist some fundamental difficulties in symbolic and algebraic manipulation, such as internal swelling and mathematical limitation. A remedy for these difficulties is proposed, and the three applications mentioned are solved successfully. For example, the closed from solution of stiffness matrix of four-node isoparametrical quadrilateral element for 2-D elasticity problem was not available before. Due to the work presented, the automatic construction of it becomes feasible. In addition, a new advantage of the application of symbolic and algebraic manipulation found is believed to be crucial in improving the efficiency of program execution in the future. This will substantially shorten the response time of a system. It is very significant for certain systems, such as missile and high speed aircraft systems, in which time plays an important role.

  2. Electronic excitation of atoms and molecules by electron impact in a linear algebraic, separable potential approach

    International Nuclear Information System (INIS)

    Collins, L.A.; Schneider, B.I.

    1984-01-01

    The linear algebraic, separable potential approach is applied to the electronic excitation of atoms and molecules by electron impact. By representing the exchange and off-diagonal direct terms on a basis, the standard set of coupled inelastic equations is reduced to a set of elastic inhomogeneous equations. The procedure greatly simplifies the formulation by allowing a large portion of the problem to be handled by standard bound-state techniques and by greatly reducing the order of the scattering equations that must be solved. Application is made to the excitation of atomic hydrogen in the three-state close-coupling (1s, 2s, 2p) approximation. (author)

  3. Optical linear algebra processors - Noise and error-source modeling

    Science.gov (United States)

    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.

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

    Science.gov (United States)

    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.

  5. Communication Avoiding and Overlapping for Numerical Linear Algebra

    Science.gov (United States)

    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

  6. Linear representation of algebras with non-associative operations which are satisfy in the balanced functional equations

    International Nuclear Information System (INIS)

    Ehsani, Amir

    2015-01-01

    Algebras with a pair of non-associative binary operations (f, g) which are satisfy in the balanced quadratic functional equations with four object variables considered. First, we obtain a linear representation for the operations, of this kind of binary algebras (A,f,g), over an abelian group (A, +) and then we generalize the linear representation of operations, to an algebra (A,F) with non-associative binary operations which are satisfy in the balanced quadratic functional equations with four object variables. (paper)

  7. GPU TECHNOLOGIES EMBODIED IN PARALLEL SOLVERS OF LINEAR ALGEBRAIC EQUATION SYSTEMS

    Directory of Open Access Journals (Sweden)

    Sidorov Alexander Vladimirovich

    2012-10-01

    Full Text Available The author reviews existing shareware solvers that are operated by graphical computer devices. The purpose of this review is to explore the opportunities and limitations of the above parallel solvers applicable for resolution of linear algebraic problems that arise at Research and Educational Centre of Computer Modeling at MSUCE, and Research and Engineering Centre STADYO. The author has explored new applications of the GPU in the PETSc suite and compared them with the results generated absent of the GPU. The research is performed within the CUSP library developed to resolve the problems of linear algebra through the application of GPU. The author has also reviewed the new MAGMA project which is analogous to LAPACK for the GPU.

  8. The Analysis of the Grade of the Students' Understanding in "Linear Algebra" in National College of Technology

    OpenAIRE

    中沢, 喜昌

    1989-01-01

    We gave linear algebra lessons to the fifth grade students as an elective subject and analyzed that to what extent students understood the linear algebra, judging from the result of questionaires and tests. It showed that they are good at the problems accompanied by calculations such as inverse matrix, simultaneous linear equation, and proper value problem and that, on the contrary, it is difficult to understand the abstract notion like linear space and linear map.

  9. Monte Carlo method for solving a parabolic problem

    Directory of Open Access Journals (Sweden)

    Tian Yi

    2016-01-01

    Full Text Available In this paper, we present a numerical method based on random sampling for a parabolic problem. This method combines use of the Crank-Nicolson method and Monte Carlo method. In the numerical algorithm, we first discretize governing equations by Crank-Nicolson method, and obtain a large sparse system of linear algebraic equations, then use Monte Carlo method to solve the linear algebraic equations. To illustrate the usefulness of this technique, we apply it to some test problems.

  10. Developing ontological model of computational linear algebra - preliminary considerations

    Science.gov (United States)

    Wasielewska, K.; Ganzha, M.; Paprzycki, M.; Lirkov, I.

    2013-10-01

    The aim of this paper is to propose a method for application of ontologically represented domain knowledge to support Grid users. The work is presented in the context provided by the Agents in Grid system, which aims at development of an agent-semantic infrastructure for efficient resource management in the Grid. Decision support within the system should provide functionality beyond the existing Grid middleware, specifically, help the user to choose optimal algorithm and/or resource to solve a problem from a given domain. The system assists the user in at least two situations. First, for users without in-depth knowledge about the domain, it should help them to select the method and the resource that (together) would best fit the problem to be solved (and match the available resources). Second, if the user explicitly indicates the method and the resource configuration, it should "verify" if her choice is consistent with the expert recommendations (encapsulated in the knowledge base). Furthermore, one of the goals is to simplify the use of the selected resource to execute the job; i.e., provide a user-friendly method of submitting jobs, without required technical knowledge about the Grid middleware. To achieve the mentioned goals, an adaptable method of expert knowledge representation for the decision support system has to be implemented. The selected approach is to utilize ontologies and semantic data processing, supported by multicriterial decision making. As a starting point, an area of computational linear algebra was selected to be modeled, however, the paper presents a general approach that shall be easily extendable to other domains.

  11. The Model Method: Singapore Children's Tool for Representing and Solving Algebraic Word Problems

    Science.gov (United States)

    Ng, Swee Fong; Lee, Kerry

    2009-01-01

    Solving arithmetic and algebraic word problems is a key component of the Singapore elementary mathematics curriculum. One heuristic taught, the model method, involves drawing a diagram to represent key information in the problem. We describe the model method and a three-phase theoretical framework supporting its use. We conducted 2 studies to…

  12. Exact solution of some linear matrix equations using algebraic methods

    Science.gov (United States)

    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.

  13. Optical linear algebra processors - Architectures and algorithms

    Science.gov (United States)

    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.

  14. Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables

    OpenAIRE

    Alesker, Semyon

    2003-01-01

    We recall known and establish new properties of the Dieudonn\\'e and Moore determinants of quaternionic matrices.Using these linear algebraic results we develop a basic theory of plurisubharmonic functions of quaternionic variables. Then we introduce and briefly discuss quaternionic Monge-Amp\\'ere equations.

  15. On Solving Linear Recurrences

    Science.gov (United States)

    Dobbs, David E.

    2013-01-01

    A direct method is given for solving first-order linear recurrences with constant coefficients. The limiting value of that solution is studied as "n to infinity." This classroom note could serve as enrichment material for the typical introductory course on discrete mathematics that follows a calculus course.

  16. An Approach for Solving Linear Fractional Programming Problems

    OpenAIRE

    Andrew Oyakhobo Odior

    2012-01-01

    Linear fractional programming problems are useful tools in production planning, financial and corporate planning, health care and hospital planning and as such have attracted considerable research interest. The paper presents a new approach for solving a fractional linear programming problem in which the objective function is a linear fractional function, while the constraint functions are in the form of linear inequalities. The approach adopted is based mainly upon solving the problem algebr...

  17. Algebras of holomorphic functions and control theory

    CERN Document Server

    Sasane, Amol

    2009-01-01

    This accessible, undergraduate-level text illustrates the role of algebras of holomorphic functions in the solution of an important engineering problem: the stabilization of a linear control system. Its concise and self-contained treatment avoids the use of higher mathematics and forms a bridge to more advanced treatments. The treatment consists of two components: the algebraic framework, which serves as the abstract language for posing and solving the problem of stabilization; and the analysis component, which examines properties of specific rings of holomorphic functions. Elementary, self-co

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

    Science.gov (United States)

    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.

  19. Basic linear algebra subprograms for FORTRAN usage

    Science.gov (United States)

    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.

  20. Linear response theory an analytic-algebraic approach

    CERN Document Server

    De Nittis, Giuseppe

    2017-01-01

    This book presents a modern and systematic approach to Linear Response Theory (LRT) by combining analytic and algebraic ideas. LRT is a tool to study systems that are driven out of equilibrium by external perturbations. In particular the reader is provided with a new and robust tool to implement LRT for a wide array of systems. The proposed formalism in fact applies to periodic and random systems in the discrete and the continuum. After a short introduction describing the structure of the book, its aim and motivation, the basic elements of the theory are presented in chapter 2. The mathematical framework of the theory is outlined in chapters 3–5: the relevant von Neumann algebras, noncommutative $L^p$- and Sobolev spaces are introduced; their construction is then made explicit for common physical systems; the notion of isopectral perturbations and the associated dynamics are studied. Chapter 6 is dedicated to the main results, proofs of the Kubo and Kubo-Streda formulas. The book closes with a chapter about...

  1. Investigating Integer Restrictions in Linear Programming

    Science.gov (United States)

    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…

  2. On Numerical Stability in Large Scale Linear Algebraic Computations

    Czech Academy of Sciences Publication Activity Database

    Strakoš, Zdeněk; Liesen, J.

    2005-01-01

    Roč. 85, č. 5 (2005), s. 307-325 ISSN 0044-2267 R&D Projects: GA AV ČR 1ET400300415 Institutional research plan: CEZ:AV0Z10300504 Keywords : linear algebraic systems * eigenvalue problems * convergence * numerical stability * backward error * accuracy * Lanczos method * conjugate gradient method * GMRES method Subject RIV: BA - General Mathematics Impact factor: 0.351, year: 2005

  3. Energy footprint of advanced dense numerical linear algebra using tile algorithms on multicore architectures

    KAUST Repository

    Dongarra, Jack

    2012-11-01

    We propose to study the impact on the energy footprint of two advanced algorithmic strategies in the context of high performance dense linear algebra libraries: (1) mixed precision algorithms with iterative refinement allow to run at the peak performance of single precision floating-point arithmetic while achieving double precision accuracy and (2) tree reduction technique exposes more parallelism when factorizing tall and skinny matrices for solving over determined systems of linear equations or calculating the singular value decomposition. Integrated within the PLASMA library using tile algorithms, which will eventually supersede the block algorithms from LAPACK, both strategies further excel in performance in the presence of a dynamic task scheduler while targeting multicore architecture. Energy consumption measurements are reported along with parallel performance numbers on a dual-socket quad-core Intel Xeon as well as a quad-socket quad-core Intel Sandy Bridge chip, both providing component-based energy monitoring at all levels of the system, through the Power Pack framework and the Running Average Power Limit model, respectively. © 2012 IEEE.

  4. Energy footprint of advanced dense numerical linear algebra using tile algorithms on multicore architectures

    KAUST Repository

    Dongarra, Jack; Ltaief, Hatem; Luszczek, Piotr R.; Weaver, Vincent M.

    2012-01-01

    We propose to study the impact on the energy footprint of two advanced algorithmic strategies in the context of high performance dense linear algebra libraries: (1) mixed precision algorithms with iterative refinement allow to run at the peak performance of single precision floating-point arithmetic while achieving double precision accuracy and (2) tree reduction technique exposes more parallelism when factorizing tall and skinny matrices for solving over determined systems of linear equations or calculating the singular value decomposition. Integrated within the PLASMA library using tile algorithms, which will eventually supersede the block algorithms from LAPACK, both strategies further excel in performance in the presence of a dynamic task scheduler while targeting multicore architecture. Energy consumption measurements are reported along with parallel performance numbers on a dual-socket quad-core Intel Xeon as well as a quad-socket quad-core Intel Sandy Bridge chip, both providing component-based energy monitoring at all levels of the system, through the Power Pack framework and the Running Average Power Limit model, respectively. © 2012 IEEE.

  5. Real forms of non-linear superconformal and quasi-superconformal algebras and their unified realization

    International Nuclear Information System (INIS)

    Bina, B.; Guenaydin, M.

    1997-01-01

    We give a complete classification of the real forms of simple non-linear superconformal algebras (SCA) and quasi-superconformal algebras (QSCA) and present a unified realization of these algebras with simple symmetry groups. This classification is achieved by establishing a correspondence between simple non-linear QSCA's and SCA's and quaternionic and super-quaternionic symmetric spaces of simple Lie groups and Lie supergroups, respectively. The unified realization we present involves a dimension zero scalar field (dilaton), dimension-1 symmetry currents, and dimension-1/2 free bosons for QSCA's and dimension-1/2 free fermions for SCA's. The free bosons and fermions are associated with the quaternionic and super-quaternionic symmetric spaces of corresponding Lie groups and Lie supergroups, respectively. We conclude with a discussion of possible applications of our results. (orig.)

  6. Ghost field realizations of the spinor $W_{2,s}$ strings based on the linear W(1,2,s) algebras

    OpenAIRE

    Liu, Yu-Xiao; Zhang, Li-Jie; Ren, Ji-Rong

    2005-01-01

    It has been shown that certain W algebras can be linearized by the inclusion of a spin-1 current. This Provides a way of obtaining new realizations of the W algebras. In this paper, we investigate the new ghost field realizations of the W(2,s)(s=3,4) algebras, making use of the fact that these two algebras can be linearized. We then construct the nilpotent BRST charges of the spinor non-critical W(2,s) strings with these new realizations.

  7. Ghost field realizations of the spinor W2,s strings based on the linear W1,2,s algebras

    International Nuclear Information System (INIS)

    Liu Yuxiao; Ren Jirong; Zhang Lijie

    2005-01-01

    It has been shown that certain W algebras can be linearized by the inclusion of a spin-1 current. This provides a way of obtaining new realizations of the W algebras. In this paper, we investigate the new ghost field realizations of the W 2,s (s=3,4) algebras, making use of the fact that these two algebras can be linearized. We then construct the nilpotent BRST charges of the spinor non-critical W 2,s strings with these new realizations. (author)

  8. Interior Point Method for Solving Fuzzy Number Linear Programming Problems Using Linear Ranking Function

    Directory of Open Access Journals (Sweden)

    Yi-hua Zhong

    2013-01-01

    Full Text Available Recently, various methods have been developed for solving linear programming problems with fuzzy number, such as simplex method and dual simplex method. But their computational complexities are exponential, which is not satisfactory for solving large-scale fuzzy linear programming problems, especially in the engineering field. A new method which can solve large-scale fuzzy number linear programming problems is presented in this paper, which is named a revised interior point method. Its idea is similar to that of interior point method used for solving linear programming problems in crisp environment before, but its feasible direction and step size are chosen by using trapezoidal fuzzy numbers, linear ranking function, fuzzy vector, and their operations, and its end condition is involved in linear ranking function. Their correctness and rationality are proved. Moreover, choice of the initial interior point and some factors influencing the results of this method are also discussed and analyzed. The result of algorithm analysis and example study that shows proper safety factor parameter, accuracy parameter, and initial interior point of this method may reduce iterations and they can be selected easily according to the actual needs. Finally, the method proposed in this paper is an alternative method for solving fuzzy number linear programming problems.

  9. On the linearization of nonlinear supersymmetry based on the commutator algebra

    Energy Technology Data Exchange (ETDEWEB)

    Tsuda, Motomu, E-mail: tsuda@sit.ac.jp

    2017-01-10

    We discuss a linearization procedure of nonlinear supersymmetry (NLSUSY) based on the closure of the commutator algebra for variations of functionals of Nambu–Goldstone fermions and their derivative terms under NLSUSY transformations in Volkov–Akulov NLSUSY theory. In the case of a set of bosonic and fermionic functionals, which leads to (massless) vector linear supermultiplets, we explicitly show that general linear SUSY transformations of basic components defined from those functionals are uniquely determined by examining the commutation relation in the NLSUSY theory.

  10. Hom-Novikov algebras

    International Nuclear Information System (INIS)

    Yau, Donald

    2011-01-01

    We study a twisted generalization of Novikov algebras, called Hom-Novikov algebras, in which the two defining identities are twisted by a linear map. It is shown that Hom-Novikov algebras can be obtained from Novikov algebras by twisting along any algebra endomorphism. All algebra endomorphisms on complex Novikov algebras of dimensions 2 or 3 are computed, and their associated Hom-Novikov algebras are described explicitly. Another class of Hom-Novikov algebras is constructed from Hom-commutative algebras together with a derivation, generalizing a construction due to Dorfman and Gel'fand. Two other classes of Hom-Novikov algebras are constructed from Hom-Lie algebras together with a suitable linear endomorphism, generalizing a construction due to Bai and Meng.

  11. Fiber-wise linear Poisson structures related to W∗-algebras

    Science.gov (United States)

    Odzijewicz, Anatol; Jakimowicz, Grzegorz; Sliżewska, Aneta

    2018-01-01

    In the framework of Banach differential geometry we investigate the fiber-wise linear Poisson structures as well as the Lie groupoid and Lie algebroid structures which are defined in the canonical way by the structure of a W∗-algebra (von Neumann algebra) M. The main role in this theory is played by the complex Banach-Lie groupoid G(M) ⇉ L(M) of partially invertible elements of M over the lattice L(M) of orthogonal projections of M. The Atiyah sequence and the predual Atiyah sequence corresponding to this groupoid are investigated from the point of view of Banach Poisson geometry. In particular we show that the predual Atiyah sequence fits in a short exact sequence of complex Banach sub-Poisson V B-groupoids with G(M) ⇉ L(M) as the side groupoid.

  12. Using Cognitive Tutor Software in Learning Linear Algebra Word Concept

    Science.gov (United States)

    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…

  13. A Framework for Mathematical Thinking: The Case of Linear Algebra

    Science.gov (United States)

    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…

  14. The Hilbert polynomial and linear forms in the logarithms of algebraic numbers

    International Nuclear Information System (INIS)

    Aleksentsev, Yu M

    2008-01-01

    We prove a new estimate for homogeneous linear forms with integer coefficients in the logarithms of algebraic numbers. We obtain a qualitative improvement of the estimate depending on the coefficients of the linear form and the best value of the constant in the estimate in the case when the number of logarithms is not too large

  15. Isomorphic Operators and Functional Equations for the Skew-Circulant Algebra

    Directory of Open Access Journals (Sweden)

    Zhaolin Jiang

    2014-01-01

    Full Text Available The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants with complex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on the skew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra of n×n complex skew-circulant matrices are displayed in this paper.

  16. The Algebra Teacher's Activity-a-Day, Grades 6-12 Over 180 Quick Challenges for Developing Math and Problem-Solving Skills

    CERN Document Server

    Thompson, Frances McBroom

    2010-01-01

    Fun-filled math problems that put the emphasis on problem-solving strategies and reasoning. The Algebra Teacher's Activity-a-Day offers activities for test prep, warm-ups, down time, homework, or just for fun. These unique activities are correlated with national math education standards and emphasize problem-solving strategies and logical reasoning skills. In many of the activities, students are encouraged to communicate their different approaches to other students in the class.: Filled with dozens of quick and fun algebra activities that can be used inside and outside the classroom; Designed

  17. Head First Algebra A Learner's Guide to Algebra I

    CERN Document Server

    Pilone, Tracey

    2008-01-01

    Having trouble understanding algebra? Do algebraic concepts, equations, and logic just make your head spin? We have great news: Head First Algebra is designed for you. Full of engaging stories and practical, real-world explanations, this book will help you learn everything from natural numbers and exponents to solving systems of equations and graphing polynomials. Along the way, you'll go beyond solving hundreds of repetitive problems, and actually use what you learn to make real-life decisions. Does it make sense to buy two years of insurance on a car that depreciates as soon as you drive i

  18. A Practical Approach to Inquiry-Based Learning in Linear Algebra

    Science.gov (United States)

    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…

  19. A convex optimization approach for solving large scale linear systems

    Directory of Open Access Journals (Sweden)

    Debora Cores

    2017-01-01

    Full Text Available The well-known Conjugate Gradient (CG method minimizes a strictly convex quadratic function for solving large-scale linear system of equations when the coefficient matrix is symmetric and positive definite. In this work we present and analyze a non-quadratic convex function for solving any large-scale linear system of equations regardless of the characteristics of the coefficient matrix. For finding the global minimizers, of this new convex function, any low-cost iterative optimization technique could be applied. In particular, we propose to use the low-cost globally convergent Spectral Projected Gradient (SPG method, which allow us to extend this optimization approach for solving consistent square and rectangular linear system, as well as linear feasibility problem, with and without convex constraints and with and without preconditioning strategies. Our numerical results indicate that the new scheme outperforms state-of-the-art iterative techniques for solving linear systems when the symmetric part of the coefficient matrix is indefinite, and also for solving linear feasibility problems.

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

    Science.gov (United States)

    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…

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

    Science.gov (United States)

    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…

  2. Linear algebra applications using Matlab software

    Directory of Open Access Journals (Sweden)

    Cornelia Victoria Anghel

    2005-10-01

    Full Text Available The paper presents two ways of special matrix generating using some functions included in the MatLab software package. The MatLab software package contains a set of functions that generate special matrixes used in the linear algebra applications and the signal processing from different activity fields. The paper presents two tipes of special matrixes that can be generated using written sintaxes in the dialog window of the MatLab software and for the command validity we need to press the Enter task. The applications presented in the paper represent eamples of numerical calculus using the MatLab software and belong to the scientific field „Computer Assisted Mathematics” thus creating the symbiosis between mathematics and informatics.

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

    Science.gov (United States)

    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

  4. Fast Solvers for Dense Linear Systems

    Energy Technology Data Exchange (ETDEWEB)

    Kauers, Manuel [Research Institute for Symbolic Computation (RISC), Altenbergerstrasse 69, A4040 Linz (Austria)

    2008-10-15

    It appears that large scale calculations in particle physics often require to solve systems of linear equations with rational number coefficients exactly. If classical Gaussian elimination is applied to a dense system, the time needed to solve such a system grows exponentially in the size of the system. In this tutorial paper, we present a standard technique from computer algebra that avoids this exponential growth: homomorphic images. Using this technique, big dense linear systems can be solved in a much more reasonable time than using Gaussian elimination over the rationals.

  5. Hopf-algebraic renormalization of QED in the linear covariant gauge

    Energy Technology Data Exchange (ETDEWEB)

    Kißler, Henry, E-mail: kissler@physik.hu-berlin.de

    2016-09-15

    In the context of massless quantum electrodynamics (QED) with a linear covariant gauge fixing, the connection between the counterterm and the Hopf-algebraic approach to renormalization is examined. The coproduct formula of Green’s functions contains two invariant charges, which give rise to different renormalization group functions. All formulas are tested by explicit computations to third loop order. The possibility of a finite electron self-energy by fixing a generalized linear covariant gauge is discussed. An analysis of subdivergences leads to the conclusion that such a gauge only exists in quenched QED.

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

    Science.gov (United States)

    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…

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

    Science.gov (United States)

    Beaver, Scott

    2015-01-01

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

  8. Solving the generalized Langevin equation with the algebraically correlated noise

    International Nuclear Information System (INIS)

    Srokowski, T.; Ploszajczak, M.

    1997-01-01

    The Langevin equation with the memory kernel is solved. The stochastic force possesses algebraic correlations, proportional to 1/t. The velocity autocorrelation function and related quantities characterizing transport properties are calculated at the assumption that the system is in the thermal equilibrium. Stochastic trajectories are simulated numerically, using the kangaroo process as a noise generator. Results of this simulation resemble Levy walks with divergent moments of the velocity distribution. The motion of a Brownian particle is considered both without any external potential and in the harmonic oscillator field, in particular the escape from a potential well. The results are compared with memory-free calculations for the Brownian particle. (author)

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

    Science.gov (United States)

    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…

  10. Solving of Clock Problems Using An Algebraic Approach And Developing An Application For Automatic Conversion

    Science.gov (United States)

    Lakshmi Devaraj, Shanmuga

    2018-04-01

    The recent trend in learning Mathematics is through android apps like Byju’s. The clock problems asked in aptitude tests could be learnt using such computer applications. The Clock problems are of four categories namely: 1. What is the angle between the hands of a clock at a particular time 2. When the hands of a clock will meet after a particular time 3. When the hands of a clock will be at right angle after a particular time 4. When the hands of a clock will be in a straight line but not together after a particular time The aim of this article is to convert the clock problems which were solved using the traditional approach to algebraic equations and solve them. Shortcuts are arrived which help in solving the questions in just a few seconds. Any aptitude problem could be converted to an algebraic equation by tracing the way the problem proceeds by applying our analytical skills. Solving of equations would be the easiest part in coming up with the solution. Also a computer application could be developed by using the equations that were arrived at in the analysis part. The computer application aims at solving the four different problems in Clocks. The application helps the learners of aptitude for CAT and other competitive exams to know the approach of the problem. Learning Mathematics with a gaming tool like this would be interesting to the learners. This paper provides a path to creating gaming apps to learn Mathematics.

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

    Science.gov (United States)

    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.…

  12. PC-BLAS, PC Linear Algebra Subroutines

    International Nuclear Information System (INIS)

    Hanson, R.J.

    1989-01-01

    1 - Description of program or function: PC-BLAS is a highly optimized version of the Basic Linear Algebra Subprograms (BLAS), a standardized set of 38 routines that perform low-level operations on vectors of numbers in single- and double-precision real and complex arithmetic. Routines are included to find the index of the largest component of a vector, apply a Givens or modified Givens rotation, multiply a vector by a constant, determine the Euclidean length, perform a dot product, swap and copy vectors, and find the norm of a vector. 2 - Restrictions on the complexity of the problem: The number of components in any vector and the spacing or stride between their entries must not exceed 32,767 (2 15 -1). PC-BLAS will not work with an 80286 CPU operating in 'protected' mode

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

    Science.gov (United States)

    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.)…

  14. First order linear ordinary differential equations in associative algebras

    Directory of Open Access Journals (Sweden)

    Gordon Erlebacher

    2004-01-01

    Full Text Available In this paper, we study the linear differential equation $$ frac{dx}{dt}=sum_{i=1}^n a_i(t x b_i(t + f(t $$ in an associative but non-commutative algebra $mathcal{A}$, where the $b_i(t$ form a set of commuting $mathcal{A}$-valued functions expressed in a time-independent spectral basis consisting of mutually annihilating idempotents and nilpotents. Explicit new closed solutions are derived, and examples are presented to illustrate the theory.

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

    Science.gov (United States)

    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.

  16. Green's functions through so(2,1) lie algebra in nonrelativistic quantum mechanics

    International Nuclear Information System (INIS)

    Boschi-Filho, H.; Vaidya, A.N.

    1991-01-01

    The authors discuss an algebraic technique to construct the Green's function for systems described by the noncompact so(2,1) Lie algebra. They show that this technique solves the one-dimensional linear oscillator and Coulomb potentials and also generates particular solutions for other one-dimensional potentials. Then they construct explicitly the Green's function for the three-dimensional oscillator and the three-dimensional Coulomb potential, which are generalizations of the one-dimensional cases, and the Coulomb plus an Aharanov-Bohm, potential. They discuss the dynamical algebra involved in each case and also find their wave functions and bound state spectra. Finally they introduce in each case and also find their wave functions and bound state spectra. Finally they introduce a point canonical transformation in the generators of so(2,10) Lie algebra, show that this procedure permits us to solve the one-dimensional Morse potential in addition to the previous cases, and construct its Green's function and find its energy spectrum and wave functions

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

    Science.gov (United States)

    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,…

  18. A new fuzzy Monte Carlo method for solving SLAE with ergodic fuzzy Markov chains

    Directory of Open Access Journals (Sweden)

    Maryam Gharehdaghi

    2015-05-01

    Full Text Available In this paper we introduce a new fuzzy Monte Carlo method for solving system of linear algebraic equations (SLAE over the possibility theory and max-min algebra. To solve the SLAE, we first define a fuzzy estimator and prove that this is an unbiased estimator of the solution. To prove unbiasedness, we apply the ergodic fuzzy Markov chains. This new approach works even for cases with coefficients matrix with a norm greater than one.

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

    Science.gov (United States)

    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.

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

    Science.gov (United States)

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

    2016-07-01

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

  1. A Type System for the Vectorial Aspect of the Linear-Algebraic Lambda-Calculus

    Directory of Open Access Journals (Sweden)

    Pablo Arrighi

    2012-07-01

    Full Text Available We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We show that the resulting typed lambda-calculus is strongly normalizing and features a weak subject-reduction.

  2. Algebra

    CERN Document Server

    Flanders, Harley

    1975-01-01

    Algebra presents the essentials of algebra with some applications. The emphasis is on practical skills, problem solving, and computational techniques. Topics covered range from equations and inequalities to functions and graphs, polynomial and rational functions, and exponentials and logarithms. Trigonometric functions and complex numbers are also considered, together with exponentials and logarithms.Comprised of eight chapters, this book begins with a discussion on the fundamentals of algebra, each topic explained, illustrated, and accompanied by an ample set of exercises. The proper use of a

  3. Relating Reasoning Methodologies in Linear Logic and Process Algebra

    Directory of Open Access Journals (Sweden)

    Yuxin Deng

    2012-11-01

    Full Text Available We show that the proof-theoretic notion of logical preorder coincides with the process-theoretic notion of contextual preorder for a CCS-like calculus obtained from the formula-as-process interpretation of a fragment of linear logic. The argument makes use of other standard notions in process algebra, namely a labeled transition system and a coinductively defined simulation relation. This result establishes a connection between an approach to reason about process specifications and a method to reason about logic specifications.

  4. Experimental quantum computing to solve systems of linear equations.

    Science.gov (United States)

    Cai, X-D; Weedbrook, C; Su, Z-E; Chen, M-C; Gu, Mile; Zhu, M-J; Li, Li; Liu, Nai-Le; Lu, Chao-Yang; Pan, Jian-Wei

    2013-06-07

    Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time proportional to the number of variables N. A recently proposed quantum algorithm shows that quantum computers could solve linear systems in a time scale of order log(N), giving an exponential speedup over classical computers. Here we realize the simplest instance of this algorithm, solving 2×2 linear equations for various input vectors on a quantum computer. We use four quantum bits and four controlled logic gates to implement every subroutine required, demonstrating the working principle of this algorithm.

  5. Negative base encoding in optical linear algebra processors

    Science.gov (United States)

    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.

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

    Science.gov (United States)

    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…

  7. Linear operator pencils on Lie algebras and Laurent biorthogonal polynomials

    International Nuclear Information System (INIS)

    Gruenbaum, F A; Vinet, Luc; Zhedanov, Alexei

    2004-01-01

    We study operator pencils on generators of the Lie algebras sl 2 and the oscillator algebra. These pencils are linear in a spectral parameter λ. The corresponding generalized eigenvalue problem gives rise to some sets of orthogonal polynomials and Laurent biorthogonal polynomials (LBP) expressed in terms of the Gauss 2 F 1 and degenerate 1 F 1 hypergeometric functions. For special choices of the parameters of the pencils, we identify the resulting polynomials with the Hendriksen-van Rossum LBP which are widely believed to be the biorthogonal analogues of the classical orthogonal polynomials. This places these examples under the umbrella of the generalized bispectral problem which is considered here. Other (non-bispectral) cases give rise to some 'nonclassical' orthogonal polynomials including Tricomi-Carlitz and random-walk polynomials. An application to solutions of relativistic Toda chain is considered

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

    Science.gov (United States)

    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.

  9. Design and Implementation of Numerical Linear Algebra Algorithms on Fixed Point DSPs

    Directory of Open Access Journals (Sweden)

    Gene Frantz

    2007-01-01

    Full Text Available Numerical linear algebra algorithms use the inherent elegance of matrix formulations and are usually implemented using C/C++ floating point representation. The system implementation is faced with practical constraints because these algorithms usually need to run in real time on fixed point digital signal processors (DSPs to reduce total hardware costs. Converting the simulation model to fixed point arithmetic and then porting it to a target DSP device is a difficult and time-consuming process. In this paper, we analyze the conversion process. We transformed selected linear algebra algorithms from floating point to fixed point arithmetic, and compared real-time requirements and performance between the fixed point DSP and floating point DSP algorithm implementations. We also introduce an advanced code optimization and an implementation by DSP-specific, fixed point C code generation. By using the techniques described in the paper, speed can be increased by a factor of up to 10 compared to floating point emulation on fixed point hardware.

  10. Asymptotic aspect of derivations in Banach algebras

    Directory of Open Access Journals (Sweden)

    Jaiok Roh

    2017-02-01

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

  11. Kac-Moody algebras derived from linearization systems using Zsub(N) reduction and extended to supersymmetry

    International Nuclear Information System (INIS)

    Bohr, H.; Roy Chowdhury, A.

    1984-10-01

    The hidden symmetries in various integrable models are derived by applying a newly developed method that uses the Riemann-Hilbert transform in a Zsub(N)-reduction of the linearization systems. The method is extended to linearization systems with higher algebras and with supersymmetry. (author)

  12. On MV-algebras of non-linear functions

    Directory of Open Access Journals (Sweden)

    Antonio Di Nola

    2017-01-01

    Full Text Available In this paper, the main results are:a study of the finitely generated MV-algebras of continuous functions from the n-th power of the unit real interval I to I;a study of Hopfian MV-algebras; anda category-theoretic study of the map sending an MV-algebra as above to the range of its generators (up to a suitable form of homeomorphism.

  13. On MV-algebras of non-linear functions

    Directory of Open Access Journals (Sweden)

    Antonio Di Nola

    2017-01-01

    Full Text Available In this paper, the main results are: a study of the finitely generated MV-algebras of continuous functions from the n-th power of the unit real interval I to I; a study of Hopfian MV-algebras; and a category-theoretic study of the map sending an MV-algebra as above to the range of its generators (up to a suitable form of homeomorphism.

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

    Science.gov (United States)

    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…

  15. New approach to solve symmetric fully fuzzy linear systems

    Indian Academy of Sciences (India)

    concepts of fuzzy set theory and then define a fully fuzzy linear system of equations. .... To represent the above problem as fully fuzzy linear system, we represent x .... Fully fuzzy linear systems can be solved by Linear programming approach, ...

  16. The bubble algebra: structure of a two-colour Temperley-Lieb Algebra

    International Nuclear Information System (INIS)

    Grimm, Uwe; Martin, Paul P

    2003-01-01

    We define new diagram algebras providing a sequence of multiparameter generalizations of the Temperley-Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional statistical mechanics. These algebras give a rigorous foundation to the various 'multi-colour algebras' of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in statistical mechanics. We demonstrate how these algebras may be used to solve the Yang-Baxter equations

  17. On the Liouvillian solution of second-order linear differential equations and algebraic invariant curves

    International Nuclear Information System (INIS)

    Man, Yiu-Kwong

    2010-01-01

    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. (fast track communication)

  18. Monomial algebras

    CERN Document Server

    Villarreal, Rafael

    2015-01-01

    The book stresses the interplay between several areas of pure and applied mathematics, emphasizing the central role of monomial algebras. It unifies the classical results of commutative algebra with central results and notions from graph theory, combinatorics, linear algebra, integer programming, and combinatorial optimization. The book introduces various methods to study monomial algebras and their presentation ideals, including Stanley-Reisner rings, subrings and blowup algebra-emphasizing square free quadratics, hypergraph clutters, and effective computational methods.

  19. High Productivity Programming of Dense Linear Algebra on Heterogeneous NUMA Architectures

    KAUST Repository

    Alomairy, Rabab M.

    2013-07-01

    High-end multicore systems with GPU-based accelerators are now ubiquitous in the hardware landscape. Besides dealing with the nontrivial heterogeneous environ- ment, end users should often take into consideration the underlying memory architec- ture to decrease the overhead of data motion, especially when running on non-uniform memory access (NUMA) platforms. We propose the OmpSs parallel programming model approach using its Nanos++ dynamic runtime system to solve the two challeng- ing problems aforementioned, through 1) an innovative NUMA node-aware scheduling policy to reduce data movement between NUMA nodes and 2) a nested parallelism feature to concurrently exploit the resources available from the GPU devices as well as the CPU host, without compromising the overall performance. Our approach fea- tures separation of concerns by abstracting the complexity of the hardware from the end users so that high productivity can be achieved. The Cholesky factorization is used as a benchmark representative of dense numerical linear algebra algorithms. Superior performance is also demonstrated on the symmetric matrix inversion based on Cholesky factorization, commonly used in co-variance computations in statistics. Performance on a NUMA system with Kepler-based GPUs exceeds that of existing implementations, while the OmpSs-enabled code remains very similar to its original sequential version.

  20. Non-geometric flux vacua, S-duality and algebraic geometry

    International Nuclear Information System (INIS)

    Guarino, Adolfo; Weatherill, George James

    2009-01-01

    The four dimensional gauged supergravities descending from non-geometric string compactifications involve a wide class of flux objects which are needed to make the theory invariant under duality transformations at the effective level. Additionally, complex algebraic conditions involving these fluxes arise from Bianchi identities and tadpole cancellations in the effective theory. In this work we study a simple T and S-duality invariant gauged supergravity, that of a type IIB string compactified on a T 6 /Z 2 x Z 2 orientifold with O3/O7-planes. We build upon the results of recent works and develop a systematic method for solving all the flux constraints based on the algebra structure underlying the fluxes. Starting with the T-duality invariant supergravity, we find that the fluxes needed to restore S-duality can be simply implemented as linear deformations of the gauge subalgebra by an element of its second cohomology class. Algebraic geometry techniques are extensively used to solve these constraints and supersymmetric vacua, centering our attention on Minkowski solutions, become systematically computable and are also provided to clarify the methods.

  1. Explicit field realizations of W algebras

    OpenAIRE

    Wei, Shao-Wen; Liu, Yu-Xiao; Zhang, Li-Jie; Ren, Ji-Rong

    2009-01-01

    The fact that certain non-linear $W_{2,s}$ algebras can be linearized by the inclusion of a spin-1 current can provide a simple way to realize $W_{2,s}$ algebras from linear $W_{1,2,s}$ algebras. In this paper, we first construct the explicit field realizations of linear $W_{1,2,s}$ algebras with double-scalar and double-spinor, respectively. Then, after a change of basis, the realizations of $W_{2,s}$ algebras are presented. The results show that all these realizations are Romans-type realiz...

  2. Relativistic algebraic spinors and quantum motions in phase space

    International Nuclear Information System (INIS)

    Holland, P.R.

    1986-01-01

    Following suggestions of Schonberg and Bohm, we study the tensorial phase space representation of the Dirac and Feynman-Gell-Mann equations in terms of the complex Dirac algebra C 4 , a Jordan-Wigner algebra G 4 , and Wigner transformations. To do this we solve the problem of the conditions under which elements in C 4 generate minimal ideals, and extend this to G 4 . This yields the linear theory of Dirac spin spaces and tensor representations of Dirac spinors, and the spin-1/2 wave equations are represented through fermionic state vectors in a higher space as a set of interconnected tensor relations

  3. Stopping test of iterative methods for solving PDE

    International Nuclear Information System (INIS)

    Wang Bangrong

    1991-01-01

    In order to assure the accuracy of the numerical solution of the iterative method for solving PDE (partial differential equation), the stopping test is very important. If the coefficient matrix of the system of linear algebraic equations is strictly diagonal dominant or irreducible weakly diagonal dominant, the stopping test formulas of the iterative method for solving PDE is proposed. Several numerical examples are given to illustrate the applications of the stopping test formulas

  4. LAPACKrc: Fast linear algebra kernels/solvers for FPGA accelerators

    International Nuclear Information System (INIS)

    Gonzalez, Juan; Nunez, Rafael C

    2009-01-01

    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.

  5. Ab initio optical potentials applied to low-energy e-H2 and e-N2 collisions in the linear-algebraic approach

    International Nuclear Information System (INIS)

    Schneider, B.I.; Collins, L.A.

    1983-01-01

    We propose a method for constructing an effective optical potential through which correlation effects can be introduced into the electron-molecule scattering formulation. The optical potential is based on a nonperturbative, Feshbach projection-operator procedure and is evaluated on an L 2 basis. The optical potential is incorporated into the scattering equations by means of a separable expansion, and the resulting scattering equations are solved by a linear-algebraic method based on the integral-equation formulation. We report the results of scattering calculations, which include polarization effects, for low-energy e-H 2 and e-N 2 collisions. The agreement with other theoretical and with experimental results is quite good

  6. Analytical exact solution of the non-linear Schroedinger equation

    International Nuclear Information System (INIS)

    Martins, Alisson Xavier; Rocha Filho, Tarcisio Marciano da

    2011-01-01

    Full text: In this work we present how to classify and obtain analytical solutions of the Schroedinger equation with a generic non-linearity in 1+1 dimensions. Our approach is based on the determination of Lie symmetry transformation mapping solutions into solutions, and non-classical symmetry transformations, mapping a given solution into itself. From these symmetries it is then possible to reduce the equation to a system of ordinary differential equations which can then be solved using standard methods. The generic non-linearity is handled by considering it as an additional unknown in the determining equations for the symmetry transformations. This results in an over-determined system of non-linear partial differential equations. Its solution can then be determined in some cases by reducing it to the so called involutive (triangular) form, and then solved. This reduction is very tedious and can only performed using a computer algebra system. Once the determining system is solved, we obtain the explicit form for the non-linearity admitting a Lie or non-classical symmetry. The analytical solutions are then derived by solving the reduced ordinary differential equations. The non-linear determining system for the non-classical symmetry transformations and Lie symmetry generators are obtaining using the computer algebra package SADE (symmetry analysis of differential equations), developed at our group. (author)

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

    Science.gov (United States)

    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.

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

    Science.gov (United States)

    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.

  9. Direct estimation of elements of quantum states algebra and entanglement detection via linear contractions

    International Nuclear Information System (INIS)

    Horodecki, Pawel

    2003-01-01

    Possibility of some nonlinear-like operations in quantum mechanics are studied. Some general formula for real linear maps are derived. With the results we show how to perform physically separability tests based on any linear contraction (on product states) that either is real or Hermitian. We also show how to estimate either product or linear combinations of quantum states without knowledge about the states themselves. This can be viewed as a sort of quantum computing on quantum states algebra

  10. New approach to solve fully fuzzy system of linear equations using ...

    Indian Academy of Sciences (India)

    Known example problems are solved to illustrate the efficacy and ... The concept of fuzzy set and fuzzy number were first introduced by Zadeh .... (iii) Fully fuzzy linear systems can be solved by linear programming approach, Gauss elim-.

  11. The algebras of higher order currents of the fermionic Gross-Neveu model

    International Nuclear Information System (INIS)

    Saltini, Luis Eduardo

    1996-01-01

    Results are reported from our studies on the following 2-dimensional field theories: the supersymmetric non-linear sigma model and the fermionic Gross-Neveu model. About the supersymmetric non-linear sigma model, an attempt is made to solve the the algebraic problem of finding the non-local conserved charges and the corresponding algebra, extending the methods described in a previous article for the case of the purely bosonic non linear sigma model. For the fermionic Gross-Neveu model, we intend to construct the conserved currents and the respective charges, related to the abelian U(1) symmetry and non-abelian SU(n) symmetry, at the conformal point and calculate the correlation functions between them. From these results at the conformal point, we want to study the effects of perturbation to get a massive but integral theory

  12. Batched Triangular Dense Linear Algebra Kernels for Very Small Matrix Sizes on GPUs

    KAUST Repository

    Charara, Ali; Keyes, David E.; Ltaief, Hatem

    2017-01-01

    Batched dense linear algebra kernels are becoming ubiquitous in scientific applications, ranging from tensor contractions in deep learning to data compression in hierarchical low-rank matrix approximation. Within a single API call, these kernels are capable of simultaneously launching up to thousands of similar matrix computations, removing the expensive overhead of multiple API calls while increasing the occupancy of the underlying hardware. A challenge is that for the existing hardware landscape (x86, GPUs, etc.), only a subset of the required batched operations is implemented by the vendors, with limited support for very small problem sizes. We describe the design and performance of a new class of batched triangular dense linear algebra kernels on very small data sizes using single and multiple GPUs. By deploying two-sided recursive formulations, stressing the register usage, maintaining data locality, reducing threads synchronization and fusing successive kernel calls, the new batched kernels outperform existing state-of-the-art implementations.

  13. Batched Triangular Dense Linear Algebra Kernels for Very Small Matrix Sizes on GPUs

    KAUST Repository

    Charara, Ali

    2017-03-06

    Batched dense linear algebra kernels are becoming ubiquitous in scientific applications, ranging from tensor contractions in deep learning to data compression in hierarchical low-rank matrix approximation. Within a single API call, these kernels are capable of simultaneously launching up to thousands of similar matrix computations, removing the expensive overhead of multiple API calls while increasing the occupancy of the underlying hardware. A challenge is that for the existing hardware landscape (x86, GPUs, etc.), only a subset of the required batched operations is implemented by the vendors, with limited support for very small problem sizes. We describe the design and performance of a new class of batched triangular dense linear algebra kernels on very small data sizes using single and multiple GPUs. By deploying two-sided recursive formulations, stressing the register usage, maintaining data locality, reducing threads synchronization and fusing successive kernel calls, the new batched kernels outperform existing state-of-the-art implementations.

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

    Science.gov (United States)

    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.

  15. Classical versus Computer Algebra Methods in Elementary Geometry

    Science.gov (United States)

    Pech, Pavel

    2005-01-01

    Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic…

  16. Elementary linear programming with applications

    CERN Document Server

    Kolman, Bernard

    1995-01-01

    Linear programming finds the least expensive way to meet given needs with available resources. Its results are used in every area of engineering and commerce: agriculture, oil refining, banking, and air transport. Authors Kolman and Beck present the basic notions of linear programming and illustrate how they are used to solve important common problems. The software on the included disk leads students step-by-step through the calculations. The Second Edition is completely revised and provides additional review material on linear algebra as well as complete coverage of elementary linear program

  17. ADM For Solving Linear Second-Order Fredholm Integro-Differential Equations

    Science.gov (United States)

    Karim, Mohd F.; Mohamad, Mahathir; Saifullah Rusiman, Mohd; Che-Him, Norziha; Roslan, Rozaini; Khalid, Kamil

    2018-04-01

    In this paper, we apply Adomian Decomposition Method (ADM) as numerically analyse linear second-order Fredholm Integro-differential Equations. The approximate solutions of the problems are calculated by Maple package. Some numerical examples have been considered to illustrate the ADM for solving this equation. The results are compared with the existing exact solution. Thus, the Adomian decomposition method can be the best alternative method for solving linear second-order Fredholm Integro-Differential equation. It converges to the exact solution quickly and in the same time reduces computational work for solving the equation. The result obtained by ADM shows the ability and efficiency for solving these equations.

  18. Explicit field realizations of W algebras

    International Nuclear Information System (INIS)

    Wei Shaowen; Liu Yuxiao; Ren Jirong; Zhang Lijie

    2009-01-01

    The fact that certain nonlinear W 2,s algebras can be linearized by the inclusion of a spin-1 current can provide a simple way to realize W 2,s algebras from linear W 1,2,s algebras. In this paper, we first construct the explicit field realizations of linear W 1,2,s algebras with double scalar and double spinor, respectively. Then, after a change of basis, the realizations of W 2,s algebras are presented. The results show that all these realizations are Romans-type realizations.

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

    Science.gov (United States)

    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…

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

    Science.gov (United States)

    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…

  1. Linear algebra and linear operators in engineering with applications in Mathematica

    CERN Document Server

    Davis, H Ted

    2000-01-01

    Designed for advanced engineering, physical science, and applied mathematics students, this innovative textbook is an introduction to both the theory and practical application of linear algebra and functional analysis. The book is self-contained, beginning with elementary principles, basic concepts, and definitions. The important theorems of the subject are covered and effective application tools are developed, working up to a thorough treatment of eigenanalysis and the spectral resolution theorem. Building on a fundamental understanding of finite vector spaces, infinite dimensional Hilbert spaces are introduced from analogy. Wherever possible, theorems and definitions from matrix theory are called upon to drive the analogy home. The result is a clear and intuitive segue to functional analysis, culminating in a practical introduction to the functional theory of integral and differential operators. Numerous examples, problems, and illustrations highlight applications from all over engineering and the physical ...

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

    Science.gov (United States)

    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…

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

    Science.gov (United States)

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

    2015-08-01

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

  4. Characterization of the order relation on the set of completely n-positive linear maps between C*-algebras

    Directory of Open Access Journals (Sweden)

    Maria Joita

    2007-12-01

    Full Text Available In this paper we characterize the order relation on the set of all nondegenerate completely n-positive linear maps between C*-algebras in terms of a self-dual Hilbert module induced by each completely n-positive linear map.

  5. Regular Riemann-Hilbert transforms, Baecklund transformations and hidden symmetry algebra for some linearization systems

    International Nuclear Information System (INIS)

    Chau Ling-Lie; Ge Mo-Lin; Teh, Rosy.

    1984-09-01

    The Baecklund Transformations and the hidden symmetry algebra for Self-Dual Yang-Mills Equations, Landau-Lifshitz equations and the Extended Super Yang-Mills fields (N>2) are discussed on the base of the Regular Riemann-Hilbert Transform and the linearization equations. (author)

  6. Linear algebra

    CERN Document Server

    Shilov, Georgi E

    1977-01-01

    Covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems with hints and answers.

  7. Planning under uncertainty solving large-scale stochastic linear programs

    Energy Technology Data Exchange (ETDEWEB)

    Infanger, G. [Stanford Univ., CA (United States). Dept. of Operations Research]|[Technische Univ., Vienna (Austria). Inst. fuer Energiewirtschaft

    1992-12-01

    For many practical problems, solutions obtained from deterministic models are unsatisfactory because they fail to hedge against certain contingencies that may occur in the future. Stochastic models address this shortcoming, but up to recently seemed to be intractable due to their size. Recent advances both in solution algorithms and in computer technology now allow us to solve important and general classes of practical stochastic problems. We show how large-scale stochastic linear programs can be efficiently solved by combining classical decomposition and Monte Carlo (importance) sampling techniques. We discuss the methodology for solving two-stage stochastic linear programs with recourse, present numerical results of large problems with numerous stochastic parameters, show how to efficiently implement the methodology on a parallel multi-computer and derive the theory for solving a general class of multi-stage problems with dependency of the stochastic parameters within a stage and between different stages.

  8. Bounds on achievable accuracy in analog optical linear-algebra processors

    Science.gov (United States)

    Batsell, Stephen G.; Walkup, John F.; Krile, Thomas F.

    1990-07-01

    Upper arid lower bounds on the number of bits of accuracy achievable are determined by applying a seconth-ortler statistical model to the linear algebra processor. The use of bounds was found necessary due to the strong signal-dependence of the noise at the output of the optical linear algebra processor (OLAP). 1 1. ACCURACY BOUNDS One of the limiting factors in applying OLAPs to real world problems has been the poor achievable accuracy of these processors. Little previous research has been done on determining noise sources from a systems perspective which would include noise generated in the multiplication ard addition operations spatial variations across arrays and crosstalk. We have previously examined these noise sources and determined a general model for the output noise mean and variance. The model demonstrates a strony signaldependency in the noise at the output of the processor which has been confirmed by our experiments. 1 We define accuracy similar to its definition for an analog signal input to an analog-to-digital (ND) converter. The number of bits of accuracy achievable is related to the log (base 2) of the number of separable levels at the P/D converter output. The number of separable levels is fouri by dividing the dynamic range by m times the standard deviation of the signal a. 2 Here m determines the error rate in the P/D conversion. The dynamic range can be expressed as the

  9. Graded associative conformal algebras of finite type

    OpenAIRE

    Kolesnikov, Pavel

    2011-01-01

    In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group $\\Gamma $ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group $G$ such that the identity component $G^0$ is the affine line and $G/G^0\\simeq \\Gamma $. A classification of simple...

  10. Effects of Modified Schema-Based Instruction on Real-World Algebra Problem Solving of Students with Autism Spectrum Disorder and Moderate Intellectual Disability

    Science.gov (United States)

    Root, Jenny Rose

    2016-01-01

    The current study evaluated the effects of modified schema-based instruction (SBI) on the algebra problem solving skills of three middle school students with autism spectrum disorder and moderate intellectual disability (ASD/ID). Participants learned to solve two types of group word problems: missing-whole and missing-part. The themes of the word…

  11. Linear algebra for dense matrices on a hypercube

    International Nuclear Information System (INIS)

    Sears, M.P.

    1990-01-01

    A set of routines has been written for dense matrix operations optimized for the NCUBE/6400 parallel processor. This paper was motivated by a Sandia effort to parallelize certain electronic structure calculations. Routines are included for matrix transpose, multiply, Cholesky decomposition, triangular inversion, and Householder tridiagonalization. The library is written in C and is callable from Fortran. Matrices up to order 1600 can be handled on 128 processors. For each operation, the algorithm used is presented along with typical timings and estimates of performance. Performance for order 1600 on 128 processors varies from 42 MFLOPs (House-holder tridiagonalization, triangular inverse) up to 126 MFLOPs (matrix multiply). The authors also present performance results for communications and basic linear algebra operations (saxpy and dot products)

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

    Science.gov (United States)

    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.

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

    Science.gov (United States)

    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…

  14. On the PR-algebras

    International Nuclear Information System (INIS)

    Lebedenko, V.M.

    1978-01-01

    The PR-algebras, i.e. the Lie algebras with commutation relations of [Hsub(i),Hsub(j)]=rsub(ij)Hsub(i)(i< j) type are investigated. On the basis of former results a criterion for the membership of 2-solvable Lie algebras to the PR-algebra class is given. The conditions imposed by the criterion are formulated in the linear algebra language

  15. Solving Differential Equations in R: Package deSolve

    Science.gov (United States)

    In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines appr...

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

    Science.gov (United States)

    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…

  17. ADART: an adaptive algebraic reconstruction algorithm for discrete tomography.

    Science.gov (United States)

    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.

  18. The universal C*-algebra of the electromagnetic field II. Topological charges and spacelike linear fields

    Science.gov (United States)

    Buchholz, Detlev; Ciolli, Fabio; Ruzzi, Giuseppe; Vasselli, Ezio

    2017-02-01

    Conditions for the appearance of topological charges are studied in the framework of the universal C*-algebra of the electromagnetic field, which is represented in any theory describing electromagnetism. It is shown that non-trivial topological charges, described by pairs of fields localised in certain topologically non-trivial spacelike separated regions, can appear in regular representations of the algebra only if the fields depend non-linearly on the mollifying test functions. On the other hand, examples of regular vacuum representations with non-trivial topological charges are constructed, where the underlying field still satisfies a weakened form of "spacelike linearity". Such representations also appear in the presence of electric currents. The status of topological charges in theories with several types of electromagnetic fields, which appear in the short distance (scaling) limit of asymptotically free non-abelian gauge theories, is also briefly discussed.

  19. Wavelets and quantum algebras

    International Nuclear Information System (INIS)

    Ludu, A.; Greiner, M.

    1995-09-01

    A non-linear associative algebra is realized in terms of translation and dilation operators, and a wavelet structure generating algebra is obtained. We show that this algebra is a q-deformation of the Fourier series generating algebra, and reduces to this for certain value of the deformation parameter. This algebra is also homeomorphic with the q-deformed su q (2) algebra and some of its extensions. Through this algebraic approach new methods for obtaining the wavelets are introduced. (author). 20 refs

  20. Investigation of the Practical Possibility of Solving Problems on Generalized Cellular Automata Associated with Cryptanalysis by Mean Algebraic Methods

    Directory of Open Access Journals (Sweden)

    P. G. Klyucharev

    2017-01-01

    Full Text Available A number of previous author’s papers proposed methods for constructing various cryptographic algorithms, including block ciphers and cryptographic hash functions, based on generalized cellular automata. This one is aimed at studying a possibility to use the algebraic cryptanalysis methods related to the construction of Gröbner bases for the generalized cellular automata to be applied in cryptography, i.e. this paper studies the possibility for using algebraic cryptanalysis methods to solve the problems of inversion of a generalized cellular automaton and recovering the key of such an automaton.If the cryptographic algorithm is represented as a system of polynomial equations over a certain finite field, then its breach is reduced to solving this system with respect to the key. Although the problem of solving a system of polynomial equations in a finite field is NP-difficult in the general case, the solution of a particular system can have low computational cost.Cryptanalysis based on the construction of a system of polynomial equations that links plain text, cipher-text and key, and its solution by algebraic methods, is usually called algebraic cryptanalysis. Among the main methods to solve systems of polynomial equations are those to construct Gröbner bases.Cryptanalysis of ciphers and hash functions based on generalized cellular automata can be reduced to various problems. We will consider two such problems: the problem of inversion of a generalized cellular automaton, which, in case we know the values of the cells after k iterations, enables us to find the initial values. And the task of recovering the key, which is to find the initial values of the remaining cells, using the cell values after k steps and the initial values of a part of the cells.A computational experiment was carried out to solve the two problems above stated in order to determine the maximum size of a generalized cellular automaton for which the solution of these

  1. Solving Differential Equations in R: Package deSolve

    NARCIS (Netherlands)

    Soetaert, K.E.R.; Petzoldt, T.; Setzer, R.W.

    2010-01-01

    In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines approach. The

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

    Science.gov (United States)

    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…

  3. Implementing the Standards: Teaching Informal Algebra.

    Science.gov (United States)

    Schultz, James E.

    1991-01-01

    Presents suggestions for developing algebraic concepts beginning in the early grades to develop a gradual building from informal to formal algebraic concepts that progresses over the K-12 curriculum. Includes suggestions for representing relationships, solving equations, employing meaningful applications of algebra, and using of technology. (MDH)

  4. Profiling high performance dense linear algebra algorithms on multicore architectures for power and energy efficiency

    KAUST Repository

    Ltaief, Hatem; Luszczek, Piotr R.; Dongarra, Jack

    2011-01-01

    This paper presents the power profile of two high performance dense linear algebra libraries i.e., LAPACK and PLASMA. The former is based on block algorithms that use the fork-join paradigm to achieve parallel performance. The latter uses fine

  5. Algebra II workbook for dummies

    CERN Document Server

    Sterling, Mary Jane

    2014-01-01

    To succeed in Algebra II, start practicing now Algebra II builds on your Algebra I skills to prepare you for trigonometry, calculus, and a of myriad STEM topics. Working through practice problems helps students better ingest and retain lesson content, creating a solid foundation to build on for future success. Algebra II Workbook For Dummies, 2nd Edition helps you learn Algebra II by doing Algebra II. Author and math professor Mary Jane Sterling walks you through the entire course, showing you how to approach and solve the problems you encounter in class. You'll begin by refreshing your Algebr

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

    Science.gov (United States)

    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.

  7. Extended conformal algebras

    International Nuclear Information System (INIS)

    Goddard, Peter

    1990-01-01

    The algebra of the group of conformal transformations in two dimensions consists of two commuting copies of the Virasoro algebra. In many mathematical and physical contexts, the representations of ν which are relevant satisfy two conditions: they are unitary and they have the ''positive energy'' property that L o is bounded below. In an irreducible unitary representation the central element c takes a fixed real value. In physical contexts, the value of c is a characteristic of a theory. If c < 1, it turns out that the conformal algebra is sufficient to ''solve'' the theory, in the sense of relating the calculation of the infinite set of physically interesting quantities to a finite subset which can be handled in principle. For c ≥ 1, this is no longer the case for the algebra alone and one needs some sort of extended conformal algebra, such as the superconformal algebra. It is these algebras that this paper aims at addressing. (author)

  8. Algebraic properties of generalized inverses

    CERN Document Server

    Cvetković‐Ilić, Dragana S

    2017-01-01

    This book addresses selected topics in the theory of generalized inverses. Following a discussion of the “reverse order law” problem and certain problems involving completions of operator matrices, it subsequently presents a specific approach to solving the problem of the reverse order law for {1} -generalized inverses. Particular emphasis is placed on the existence of Drazin invertible completions of an upper triangular operator matrix; on the invertibility and different types of generalized invertibility of a linear combination of operators on Hilbert spaces and Banach algebra elements; on the problem of finding representations of the Drazin inverse of a 2x2 block matrix; and on selected additive results and algebraic properties for the Drazin inverse. In addition to the clarity of its content, the book discusses the relevant open problems for each topic discussed. Comments on the latest references on generalized inverses are also included. Accordingly, the book will be useful for graduate students, Ph...

  9. The algebra of the energy-momentum tensor and the Noether currents in classical non-linear sigma models

    International Nuclear Information System (INIS)

    Forger, M.; Mannheim Univ.; Laartz, J.; Schaeper, U.

    1994-01-01

    The recently derived current algrbra of classical non-linear sigma models on arbitrary Riemannian manifolds is extended to include the energy-momentum tensor. It is found that in two dimensions the energy-momentum tensor θ μv , the Noether current j μ associated with the global symmetry of the theory and the composite field j appearing as the coefficient of the Schwinger term in the current algebra, together with the derivatives of j μ and j, generte a closed algebra. The subalgebra generated by the light-cone components of the energy-momentum tensor consists of two commuting copies of the Virasoro algebra, with central charge c=0, reflecting the classical conformal invariance of the theory, but the current algebra part and the semidirect product structure are quite different from the usual Kac-Moody/Sugawara type contruction. (orig.)

  10. Multi-Threaded Dense Linear Algebra Libraries for Low-Power Asymmetric Multicore Processors

    OpenAIRE

    Catalán, Sandra; Herrero, José R.; Igual, Francisco D.; Rodríguez-Sánchez, Rafael; Quintana-Ortí, Enrique S.

    2015-01-01

    Dense linear algebra libraries, such as BLAS and LAPACK, provide a relevant collection of numerical tools for many scientific and engineering applications. While there exist high performance implementations of the BLAS (and LAPACK) functionality for many current multi-threaded architectures,the adaption of these libraries for asymmetric multicore processors (AMPs)is still pending. In this paper we address this challenge by developing an asymmetry-aware implementation of the BLAS, based on the...

  11. Representations of fundamental groups of algebraic varieties

    CERN Document Server

    Zuo, Kang

    1999-01-01

    Using harmonic maps, non-linear PDE and techniques from algebraic geometry this book enables the reader to study the relation between fundamental groups and algebraic geometry invariants of algebraic varieties. The reader should have a basic knowledge of algebraic geometry and non-linear analysis. This book can form the basis for graduate level seminars in the area of topology of algebraic varieties. It also contains present new techniques for researchers working in this area.

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

    Science.gov (United States)

    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…

  13. Non-linear analysis of skew thin plate by finite difference method

    International Nuclear Information System (INIS)

    Kim, Chi Kyung; Hwang, Myung Hwan

    2012-01-01

    This paper deals with a discrete analysis capability for predicting the geometrically nonlinear behavior of skew thin plate subjected to uniform pressure. The differential equations are discretized by means of the finite difference method which are used to determine the deflections and the in-plane stress functions of plates and reduced to several sets of linear algebraic simultaneous equations. For the geometrically non-linear, large deflection behavior of the plate, the non-linear plate theory is used for the analysis. An iterative scheme is employed to solve these quasi-linear algebraic equations. Several problems are solved which illustrate the potential of the method for predicting the finite deflection and stress. For increasing lateral pressures, the maximum principal tensile stress occurs at the center of the plate and migrates toward the corners as the load increases. It was deemed important to describe the locations of the maximum principal tensile stress as it occurs. The load-deflection relations and the maximum bending and membrane stresses for each case are presented and discussed

  14. Galerkin projection methods for solving multiple related linear systems

    Energy Technology Data Exchange (ETDEWEB)

    Chan, T.F.; Ng, M.; Wan, W.L.

    1996-12-31

    We consider using Galerkin projection methods for solving multiple related linear systems A{sup (i)}x{sup (i)} = b{sup (i)} for 1 {le} i {le} s, where A{sup (i)} and b{sup (i)} are different in general. We start with the special case where A{sup (i)} = A and A is symmetric positive definite. The method generates a Krylov subspace from a set of direction vectors obtained by solving one of the systems, called the seed system, by the CG method and then projects the residuals of other systems orthogonally onto the generated Krylov subspace to get the approximate solutions. The whole process is repeated with another unsolved system as a seed until all the systems are solved. We observe in practice a super-convergence behaviour of the CG process of the seed system when compared with the usual CG process. We also observe that only a small number of restarts is required to solve all the systems if the right-hand sides are close to each other. These two features together make the method particularly effective. In this talk, we give theoretical proof to justify these observations. Furthermore, we combine the advantages of this method and the block CG method and propose a block extension of this single seed method. The above procedure can actually be modified for solving multiple linear systems A{sup (i)}x{sup (i)} = b{sup (i)}, where A{sup (i)} are now different. We can also extend the previous analytical results to this more general case. Applications of this method to multiple related linear systems arising from image restoration and recursive least squares computations are considered as examples.

  15. A control volume based finite difference method for solving the equilibrium equations in terms of displacements

    DEFF Research Database (Denmark)

    Hattel, Jesper; Hansen, Preben

    1995-01-01

    This paper presents a novel control volume based FD method for solving the equilibrium equations in terms of displacements, i.e. the generalized Navier equations. The method is based on the widely used cv-FDM solution of heat conduction and fluid flow problems involving a staggered grid formulati....... The resulting linear algebraic equations are solved by line-Gauss-Seidel....

  16. Lectures on algebraic statistics

    CERN Document Server

    Drton, Mathias; Sullivant, Seth

    2009-01-01

    How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.

  17. Students' errors in solving linear equation word problems: Case ...

    African Journals Online (AJOL)

    The study examined errors students make in solving linear equation word problems with a view to expose the nature of these errors and to make suggestions for classroom teaching. A diagnostic test comprising 10 linear equation word problems, was administered to a sample (n=130) of senior high school first year Home ...

  18. A logic circuit for solving linear function by digital method

    International Nuclear Information System (INIS)

    Ma Yonghe

    1986-01-01

    A mathematical method for determining the linear relation of physical quantity with rediation intensity is described. A logic circuit has been designed for solving linear function by digital method. Some applications and the circuit function are discussed

  19. Galois Theory of Differential Equations, Algebraic Groups and Lie Algebras

    NARCIS (Netherlands)

    Put, Marius van der

    1999-01-01

    The Galois theory of linear differential equations is presented, including full proofs. The connection with algebraic groups and their Lie algebras is given. As an application the inverse problem of differential Galois theory is discussed. There are many exercises in the text.

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

    International Nuclear Information System (INIS)

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

    2012-01-01

    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. (paper)

  1. Projection of angular momentum via linear algebra

    Science.gov (United States)

    Johnson, Calvin W.; O'Mara, Kevin D.

    2017-12-01

    Projection of many-body states with good angular momentum from an initial state is usually accomplished by a three-dimensional integral. We show how projection can instead be done by solving a straightforward system of linear equations. We demonstrate the method and give sample applications to 48Cr and 60Fe in the p f shell. This new projection scheme, which is competitive against the standard numerical quadrature, should also be applicable to other quantum numbers such as isospin and particle number.

  2. JTpack90: A parallel, object-based, Fortran 90 linear algebra package

    Energy Technology Data Exchange (ETDEWEB)

    Turner, J.A.; Kothe, D.B. [Los Alamos National Lab., NM (United States); Ferrell, R.C. [Cambridge Power Computing Associates, Ltd., Brookline, MA (United States)

    1997-03-01

    The authors have developed an object-based linear algebra package, currently with emphasis on sparse Krylov methods, driven primarily by needs of the Los Alamos National Laboratory parallel unstructured-mesh casting simulation tool Telluride. Support for a number of sparse storage formats, methods, and preconditioners have been implemented, driven primarily by application needs. They describe the object-based Fortran 90 approach, which enhances maintainability, performance, and extensibility, the parallelization approach using a new portable gather/scatter library (PGSLib), current capabilities and future plans, and present preliminary performance results on a variety of platforms.

  3. "Real-Time Optical Laboratory Linear Algebra Solution Of Partial Differential Equations"

    Science.gov (United States)

    Casasent, David; Jackson, James

    1986-03-01

    A Space Integrating (SI) Optical Linear Algebra Processor (OLAP) employing space and frequency-multiplexing, new partitioning and data flow, and achieving high accuracy performance with a non base-2 number system is described. Laboratory data on the performance of this system and the solution of parabolic Partial Differential Equations (PDEs) is provided. A multi-processor OLAP system is also described for the first time. It use in the solution of multiple banded matrices that frequently arise is then discussed. The utility and flexibility of this processor compared to digital systolic architectures should be apparent.

  4. Biderivations of finite dimensional complex simple Lie algebras

    OpenAIRE

    Tang, Xiaomin

    2016-01-01

    In this paper, we prove that a biderivation of a finite dimensional complex simple Lie algebra without the restriction of skewsymmetric is inner. As an application, the biderivation of a general linear Lie algebra is presented. In particular, we find a class of a non-inner and non-skewsymmetric biderivations. Furthermore, we also get the forms of linear commuting maps on the finite dimensional complex simple Lie algebra or general linear Lie algebra.

  5. Lectures on W algebras and W gravity

    International Nuclear Information System (INIS)

    Pope, C.N.

    1992-01-01

    We give a review of the extended conformal algebras, known as W algebras, which contain currents of spins higher than 2 in addition to the energy-momentum tensor. These include the non-linear W N algebras; the linear W ∞ and W 1+∞ algebras; and their super-extensions. We discuss their applications to the construction of W-gravity and W-string theories. (author). 46 refs

  6. The simplex method of linear programming

    CERN Document Server

    Ficken, Frederick A

    1961-01-01

    This concise but detailed and thorough treatment discusses the rudiments of the well-known simplex method for solving optimization problems in linear programming. Geared toward undergraduate students, the approach offers sufficient material for readers without a strong background in linear algebra. Many different kinds of problems further enrich the presentation. The text begins with examinations of the allocation problem, matrix notation for dual problems, feasibility, and theorems on duality and existence. Subsequent chapters address convex sets and boundedness, the prepared problem and boun

  7. An Optimal Linear Coding for Index Coding Problem

    OpenAIRE

    Pezeshkpour, Pouya

    2015-01-01

    An optimal linear coding solution for index coding problem is established. Instead of network coding approach by focus on graph theoric and algebraic methods a linear coding program for solving both unicast and groupcast index coding problem is presented. The coding is proved to be the optimal solution from the linear perspective and can be easily utilize for any number of messages. The importance of this work is lying mostly on the usage of the presented coding in the groupcast index coding ...

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

    Science.gov (United States)

    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.

  9. Killing scalar of non-linear σ-model on G/H realizing the classical exchange algebra

    International Nuclear Information System (INIS)

    Aoyama, Shogo

    2014-01-01

    The Poisson brackets for non-linear σ-models on G/H are set up on the light-like plane. A quantity which transforms irreducibly by the Killing vectors, called Killing scalar, is constructed in an arbitrary representation of G. It is shown to satisfy the classical exchange algebra

  10. Waterloo Workshop on Computer Algebra

    CERN Document Server

    Zima, Eugene; WWCA-2016; Advances in computer algebra : in honour of Sergei Abramov's' 70th birthday

    2018-01-01

    This book discusses the latest advances in algorithms for symbolic summation, factorization, symbolic-numeric linear algebra and linear functional equations. It presents a collection of papers on original research topics from the Waterloo Workshop on Computer Algebra (WWCA-2016), a satellite workshop of the International Symposium on Symbolic and Algebraic Computation (ISSAC’2016), which was held at Wilfrid Laurier University (Waterloo, Ontario, Canada) on July 23–24, 2016.   This workshop and the resulting book celebrate the 70th birthday of Sergei Abramov (Dorodnicyn Computing Centre of the Russian Academy of Sciences, Moscow), whose highly regarded and inspirational contributions to symbolic methods have become a crucial benchmark of computer algebra and have been broadly adopted by many Computer Algebra systems.

  11. Acoustooptic linear algebra processors - Architectures, algorithms, and applications

    Science.gov (United States)

    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.

  12. Chosen interval methods for solving linear interval systems with special type of matrix

    Science.gov (United States)

    Szyszka, Barbara

    2013-10-01

    The paper is devoted to chosen direct interval methods for solving linear interval systems with special type of matrix. This kind of matrix: band matrix with a parameter, from finite difference problem is obtained. Such linear systems occur while solving one dimensional wave equation (Partial Differential Equations of hyperbolic type) by using the central difference interval method of the second order. Interval methods are constructed so as the errors of method are enclosed in obtained results, therefore presented linear interval systems contain elements that determining the errors of difference method. The chosen direct algorithms have been applied for solving linear systems because they have no errors of method. All calculations were performed in floating-point interval arithmetic.

  13. Algebraic solution for the vector potential in the Dirac equation

    Energy Technology Data Exchange (ETDEWEB)

    Booth, H.S. [School of Mathematics and Physics, University of Tasmania, Hobart Tas (Australia); Centre for Mathematics and its Applications, Australian National University (Australia)]. E-mail: hbooth@wintermute.anu.edu.au; Legg, G.; Jarvis, P.D. [School of Mathematics and Physics, University of Tasmania, Hobart Tas (Australia)

    2001-07-20

    The Dirac equation for an electron in an external electromagnetic field can be regarded as a singular set of linear equations for the vector potential. Radford's method of algebraically solving for the vector potential is reviewed, with attention to the additional constraints arising from non-maximality of the rank. The extension of the method to general spacetimes is illustrated by examples in diverse dimensions with both c- and a-number wavefunctions. (author)

  14. Abstract algebra

    CERN Document Server

    Garrett, Paul B

    2007-01-01

    Designed for an advanced undergraduate- or graduate-level course, Abstract Algebra provides an example-oriented, less heavily symbolic approach to abstract algebra. The text emphasizes specifics such as basic number theory, polynomials, finite fields, as well as linear and multilinear algebra. This classroom-tested, how-to manual takes a more narrative approach than the stiff formalism of many other textbooks, presenting coherent storylines to convey crucial ideas in a student-friendly, accessible manner. An unusual feature of the text is the systematic characterization of objects by universal

  15. Solution of generalized shifted linear systems with complex symmetric matrices

    International Nuclear Information System (INIS)

    Sogabe, Tomohiro; Hoshi, Takeo; Zhang, Shao-Liang; Fujiwara, Takeo

    2012-01-01

    We develop the shifted COCG method [R. Takayama, T. Hoshi, T. Sogabe, S.-L. Zhang, T. Fujiwara, Linear algebraic calculation of Green’s function for large-scale electronic structure theory, Phys. Rev. B 73 (165108) (2006) 1–9] and the shifted WQMR method [T. Sogabe, T. Hoshi, S.-L. Zhang, T. Fujiwara, On a weighted quasi-residual minimization strategy of the QMR method for solving complex symmetric shifted linear systems, Electron. Trans. Numer. Anal. 31 (2008) 126–140] for solving generalized shifted linear systems with complex symmetric matrices that arise from the electronic structure theory. The complex symmetric Lanczos process with a suitable bilinear form plays an important role in the development of the methods. The numerical examples indicate that the methods are highly attractive when the inner linear systems can efficiently be solved.

  16. Boundary Lax pairs from non-ultra-local Poisson algebras

    International Nuclear Information System (INIS)

    Avan, Jean; Doikou, Anastasia

    2009-01-01

    We consider non-ultra-local linear Poisson algebras on a continuous line. Suitable combinations of representations of these algebras yield representations of novel generalized linear Poisson algebras or 'boundary' extensions. They are parametrized by a boundary scalar matrix and depend, in addition, on the choice of an antiautomorphism. The new algebras are the classical-linear counterparts of the known quadratic quantum boundary algebras. For any choice of parameters, the non-ultra-local contribution of the original Poisson algebra disappears. We also systematically construct the associated classical Lax pair. The classical boundary principal chiral model is examined as a physical example.

  17. Quadratic algebras and noncommutative integration of Klein-Gordon equations in non-steckel Riemann spaces

    International Nuclear Information System (INIS)

    Varaksin, O.L.; Firstov, V.V.; Shapovalov, A.V.; Shirokov, I.V.

    1995-01-01

    The method of noncommutative integration of linear partial differential equations is used to solve the Klein-Gordon equations in Riemann space, in the case when the set of noncommutating symmetry operators of this equation for a quadratic algebra consists of one second-order operator and several first-order operators. Solutions that do not permit variable separation are presented

  18. Biderivations of W-algebra $W(2,2)$ and Virasoro algebra without skewsymmetric condition and their applications

    OpenAIRE

    Tang, Xiaomin

    2016-01-01

    In this paper, we characterize the biderivations of W-algebra $W(2,2)$ and Virasoro algebra $Vir$ without skewsymmetric condition. We get two classes of non-inner biderivations. As applications, we also get the forms of linear commuting maps on W-algebra $W(2,2)$ and Virasoro algebra $Vir$.

  19. Bicovariant quantum algebras and quantum Lie algebras

    International Nuclear Information System (INIS)

    Schupp, P.; Watts, P.; Zumino, B.

    1993-01-01

    A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun(G q ) to U q g, given by elements of the pure braid group. These operators - the 'reflection matrix' Y= triple bond L + SL - being a special case - generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for Y in SO q (N). (orig.)

  20. Choosing processor array configuration by performance modeling for a highly parallel linear algebra algorithm

    International Nuclear Information System (INIS)

    Littlefield, R.J.; Maschhoff, K.J.

    1991-04-01

    Many linear algebra algorithms utilize an array of processors across which matrices are distributed. Given a particular matrix size and a maximum number of processors, what configuration of processors, i.e., what size and shape array, will execute the fastest? The answer to this question depends on tradeoffs between load balancing, communication startup and transfer costs, and computational overhead. In this paper we analyze in detail one algorithm: the blocked factored Jacobi method for solving dense eigensystems. A performance model is developed to predict execution time as a function of the processor array and matrix sizes, plus the basic computation and communication speeds of the underlying computer system. In experiments on a large hypercube (up to 512 processors), this model has been found to be highly accurate (mean error ∼ 2%) over a wide range of matrix sizes (10 x 10 through 200 x 200) and processor counts (1 to 512). The model reveals, and direct experiment confirms, that the tradeoffs mentioned above can be surprisingly complex and counterintuitive. We propose decision procedures based directly on the performance model to choose configurations for fastest execution. The model-based decision procedures are compared to a heuristic strategy and shown to be significantly better. 7 refs., 8 figs., 1 tab

  1. Numerical algebra, matrix theory, differential-algebraic equations and control theory festschrift in honor of Volker Mehrmann

    CERN Document Server

    Bollhöfer, Matthias; Kressner, Daniel; Mehl, Christian; Stykel, Tatjana

    2015-01-01

    This edited volume highlights the scientific contributions of Volker Mehrmann, a leading expert in the area of numerical (linear) algebra, matrix theory, differential-algebraic equations and control theory. These mathematical research areas are strongly related and often occur in the same real-world applications. The main areas where such applications emerge are computational engineering and sciences, but increasingly also social sciences and economics. This book also reflects some of Volker Mehrmann's major career stages. Starting out working in the areas of numerical linear algebra (his first full professorship at TU Chemnitz was in "Numerical Algebra," hence the title of the book) and matrix theory, Volker Mehrmann has made significant contributions to these areas ever since. The highlights of these are discussed in Parts I and II of the present book. Often the development of new algorithms in numerical linear algebra is motivated by problems in system and control theory. These and his later major work on ...

  2. The large N=4 superconformal W∞ algebra

    International Nuclear Information System (INIS)

    Beccaria, Matteo; Candu, Constantin; Gaberdiel, Matthias R.

    2014-01-01

    The most general large N=4 superconformal W ∞ algebra, containing in addition to the superconformal algebra one supermultiplet for each integer spin, is analysed in detail. It is found that the W ∞ algebra is uniquely determined by the levels of the two su(2) algebras, a conclusion that holds both for the linear and the non-linear case. We also perform various cross-checks of our analysis, and exhibit two different types of truncations in some detail.

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

    Science.gov (United States)

    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…

  4. The intelligence of dual simplex method to solve linear fractional fuzzy transportation problem.

    Science.gov (United States)

    Narayanamoorthy, S; Kalyani, S

    2015-01-01

    An approach is presented to solve a fuzzy transportation problem with linear fractional fuzzy objective function. In this proposed approach the fractional fuzzy transportation problem is decomposed into two linear fuzzy transportation problems. The optimal solution of the two linear fuzzy transportations is solved by dual simplex method and the optimal solution of the fractional fuzzy transportation problem is obtained. The proposed method is explained in detail with an example.

  5. Algebraic groups and their birational invariants

    CERN Document Server

    Voskresenskiĭ, V E

    2011-01-01

    Since the late 1960s, methods of birational geometry have been used successfully in the theory of linear algebraic groups, especially in arithmetic problems. This book--which can be viewed as a significant revision of the author's book, Algebraic Tori (Nauka, Moscow, 1977)--studies birational properties of linear algebraic groups focusing on arithmetic applications. The main topics are forms and Galois cohomology, the Picard group and the Brauer group, birational geometry of algebraic tori, arithmetic of algebraic groups, Tamagawa numbers, R-equivalence, projective toric varieties, invariants of finite transformation groups, and index-formulas. Results and applications are recent. There is an extensive bibliography with additional comments that can serve as a guide for further reading.

  6. N=2 current algebra and coset models

    International Nuclear Information System (INIS)

    Hull, C.M.; Spence, B.

    1990-01-01

    The N=2 supersymmetric extension of the Kac-Moody algebra and the corresponding Sugawara construction of the N=2 superconformal algebra are discussed both in components and in N=1 superspace. A formulation of the Kac-Moody algebra and Sugawara construction is given in N=2 superspace in terms of supercurrents satisfying a non-linear chiral constraint. The operator product of two supercurrents includes terms that are non-linear in the supercurrents. The N=2 generalization of the GKO coset construction is then given and the conditions found by Kazama and Suzuki are seen to arise from the non-linearity of the algebra. (orig.)

  7. Homogeneous Buchberger algorithms and Sullivant's computational commutative algebra challenge

    DEFF Research Database (Denmark)

    Lauritzen, Niels

    2005-01-01

    We give a variant of the homogeneous Buchberger algorithm for positively graded lattice ideals. Using this algorithm we solve the Sullivant computational commutative algebra challenge.......We give a variant of the homogeneous Buchberger algorithm for positively graded lattice ideals. Using this algorithm we solve the Sullivant computational commutative algebra challenge....

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

    Science.gov (United States)

    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…

  9. The Intelligence of Dual Simplex Method to Solve Linear Fractional Fuzzy Transportation Problem

    Directory of Open Access Journals (Sweden)

    S. Narayanamoorthy

    2015-01-01

    Full Text Available An approach is presented to solve a fuzzy transportation problem with linear fractional fuzzy objective function. In this proposed approach the fractional fuzzy transportation problem is decomposed into two linear fuzzy transportation problems. The optimal solution of the two linear fuzzy transportations is solved by dual simplex method and the optimal solution of the fractional fuzzy transportation problem is obtained. The proposed method is explained in detail with an example.

  10. Computers in nonassociative rings and algebras

    CERN Document Server

    Beck, Robert E

    1977-01-01

    Computers in Nonassociative Rings and Algebras provides information pertinent to the computational aspects of nonassociative rings and algebras. This book describes the algorithmic approaches for solving problems using a computer.Organized into 10 chapters, this book begins with an overview of the concept of a symmetrized power of a group representation. This text then presents data structures and other computational methods that may be useful in the field of computational algebra. Other chapters consider several mathematical ideas, including identity processing in nonassociative algebras, str

  11. Multilinear Computing and Multilinear Algebraic Geometry

    Science.gov (United States)

    2016-08-10

    algebra : linear systems, least squares, eigevalue problems, singular value problems, determinant evaluation, low-rank approximations, etc — problems...intractability to move beyond linear algebra , substantiating what the PI had proposed. High-resolution MRI with tensors: In another piece of work... applications . One reason is that we found out that many statistical estimation problems ( linear regression, errors-in-variables regression, principal components

  12. A goal programming procedure for solving fuzzy multiobjective fractional linear programming problems

    Directory of Open Access Journals (Sweden)

    Tunjo Perić

    2014-12-01

    Full Text Available This paper presents a modification of Pal, Moitra and Maulik's goal programming procedure for fuzzy multiobjective linear fractional programming problem solving. The proposed modification of the method allows simpler solving of economic multiple objective fractional linear programming (MOFLP problems, enabling the obtained solutions to express the preferences of the decision maker defined by the objective function weights. The proposed method is tested on the production planning example.

  13. Clifford Algebras and Spinorial Representation of Linear Canonical Transformations in Quantum Theory

    International Nuclear Information System (INIS)

    Raoelina Andriambololona; Ranaivoson, R.T.R.; Rakotoson, H.

    2017-11-01

    This work is a continuation of previous works that we have done concerning linear canonical transformations and a phase space representation of quantum theory. It is mainly focused on the description of an approach which permits to establish spinorial representation of linear canonical transformations. It begins with an introduction section in which the reason and context of the content are discussed. The introduction section is followed by a brief recall about Clifford algebra and spin group. The description of the approach is started with the presentation of an adequate parameterization of linear canonical transformations which permits to represent them with special pseudo-orthogonal transformations in an operators space. The establishment of the spinorial representation is deduced using relation between special pseudo-orthogonal groups and spin groups. The cases of one dimension quantum mechanics and general multidimensional theory are both studied. The case of linear canonical transformation related to Minkowski space is particularly studied and it is shown that Lorentz transformation may be considered as particular case of linear canonical transformation. Some results from the spinorial representation are also exploited to define operators which may be used to establish equations for fields if one considers the possibility of envisaging a field theory which admits as main symmetry group the group constituted by linear canonical transformations.

  14. Confluence via strong normalisation in an algebraic λ-calculus with rewriting

    Directory of Open Access Journals (Sweden)

    Pablo Buiras

    2012-03-01

    Full Text Available The linear-algebraic lambda-calculus and the algebraic lambda-calculus are untyped lambda-calculi extended with arbitrary linear combinations of terms. The former presents the axioms of linear algebra in the form of a rewrite system, while the latter uses equalities. When given by rewrites, algebraic lambda-calculi are not confluent unless further restrictions are added. We provide a type system for the linear-algebraic lambda-calculus enforcing strong normalisation, which gives back confluence. The type system allows an abstract interpretation in System F.

  15. Using linear algebra for protein structural comparison and classification.

    Science.gov (United States)

    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.

  16. Using linear algebra for protein structural comparison and classification

    Directory of Open Access Journals (Sweden)

    Janaína Gomide

    2009-01-01

    Full Text Available 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.

  17. Polynomials in algebraic analysis

    OpenAIRE

    Multarzyński, Piotr

    2012-01-01

    The concept of polynomials in the sense of algebraic analysis, for a single right invertible linear operator, was introduced and studied originally by D. Przeworska-Rolewicz \\cite{DPR}. One of the elegant results corresponding with that notion is a purely algebraic version of the Taylor formula, being a generalization of its usual counterpart, well known for functions of one variable. In quantum calculus there are some specific discrete derivations analyzed, which are right invertible linear ...

  18. Clifford algebras and the minimal representations of the 1D N-extended supersymmetry algebra

    International Nuclear Information System (INIS)

    Toppan, Francesco

    2008-01-01

    The Atiyah-Bott-Shapiro classification of the irreducible Clifford algebra is used to derive general properties of the minimal representations of the 1D N-Extended Supersymmetry algebra (the Z 2 -graded symmetry algebra of the Supersymmetric Quantum Mechanics) linearly realized on a finite number of fields depending on a real parameter t, the time. (author)

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

    Science.gov (United States)

    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…

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

    Science.gov (United States)

    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.)

  1. SD-CAS: Spin Dynamics by Computer Algebra System.

    Science.gov (United States)

    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.

  2. Difference sets connecting algebra, combinatorics, and geometry

    CERN Document Server

    Moore, Emily H

    2013-01-01

    Difference sets belong both to group theory and to combinatorics. Studying them requires tools from geometry, number theory, and representation theory. This book lays a foundation for these topics, including a primer on representations and characters of finite groups. It makes the research literature on difference sets accessible to students who have studied linear algebra and abstract algebra, and it prepares them to do their own research. This text is suitable for an undergraduate capstone course, since it illuminates the many links among topics that the students have already studied. To this end, almost every chapter ends with a coda highlighting the main ideas and emphasizing mathematical connections. This book can also be used for self-study by anyone interested in these connections and concrete examples. An abundance of exercises, varying from straightforward to challenging, invites the reader to solve puzzles, construct proofs, and investigate problems--by hand or on a computer. Hints and solutions are...

  3. A comparison of iterative methods to solve complex valued linear algebraic systems

    Czech Academy of Sciences Publication Activity Database

    Axelsson, Owe; Neytcheva, M.; Ahmad, B.

    2013-01-01

    Roč. 66, č. 4 (2013), s. 811-841 ISSN 1017-1398 R&D Projects: GA MŠk ED1.1.00/02.0070 Institutional support: RVO:68145535 Keywords : linear systems * complex symmetric * real valued form * preconditioning Subject RIV: BA - General Mathematics Impact factor: 1.005, year: 2013 http://www.it.uu.se/research/publications/reports/2013-005/2013-005-nc.pdf

  4. Handbook of algebra Vol. 1

    CERN Document Server

    1996-01-01

    Handbook of Algebra defines algebra as consisting of many different ideas, concepts and results. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. Each chapter of the book combines some of the features of both a graduate-level textbook and a research-level survey. This book is divided into eight sections. Section 1A focuses on linear algebra and discusses such concepts as matrix functions and equations and random matrices. Section 1B cover linear d

  5. A Proposed Method for Solving Fuzzy System of Linear Equations

    Directory of Open Access Journals (Sweden)

    Reza Kargar

    2014-01-01

    Full Text Available This paper proposes a new method for solving fuzzy system of linear equations with crisp coefficients matrix and fuzzy or interval right hand side. Some conditions for the existence of a fuzzy or interval solution of m×n linear system are derived and also a practical algorithm is introduced in detail. The method is based on linear programming problem. Finally the applicability of the proposed method is illustrated by some numerical examples.

  6. Solving polynomial differential equations by transforming them to linear functional-differential equations

    OpenAIRE

    Nahay, John Michael

    2008-01-01

    We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order Abel differential equation with two nonlinear terms in order to demonstrate in as much detail as possible the computations necessary for a complete solution. We mention in our section on further developments that the basic transformation idea can be generali...

  7. Continuum analogues of contragredient Lie algebras

    International Nuclear Information System (INIS)

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

    1989-03-01

    We present an axiomatic formulation of a new class of infinite-dimensional 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. (author). 9 refs

  8. Study of the 'non-Abelian' current algebra of a non-linear σ-model

    International Nuclear Information System (INIS)

    Ghosh, Subir

    2006-01-01

    A particular form of non-linear σ-model, having a global gauge invariance, is studied. The detailed discussion on current algebra structures reveals the non-Abelian nature of the invariance, with field dependent structure functions. Reduction of the field theory to a point particle framework yields a non-linear harmonic oscillator, which is a special case of similar models studied before in [J.F. Carinena et al., Nonlinearity 17 (2004) 1941, math-ph/0406002; J.F. Carinena et al., in: Proceedings of 10th International Conference in Modern Group Analysis, Larnaca, Cyprus, 2004, p. 39, math-ph/0505028; J.F. Carinena et al., Rep. Math. Phys. 54 (2004) 285, hep-th/0501106]. The connection with non-commutative geometry is also established

  9. Linear algebraic analyses of structures with one predominant type of anomalous scatterer

    International Nuclear Information System (INIS)

    Karle, J.

    1989-01-01

    Further studies have been made of the information content of the exact linear equations for analyzing anomalous dispersion data in one-wavelength experiments. The case of interest concerns structures containing atoms that essentially do not scatter anomalously and one type of anomalously scattering atoms. For this case, there are three alternative ways of writing the equations. The alternative sets of equations and the transformations for transforming one set into the other are given explicitly. Comparison calculations were made with different sets of equations. Isomorphous replacement information is readily introduced into the calculations and the advantage of doing so is clearly illustrated by the results. Another aspect of the potential of the exact linear algebraic theory is its application to multiple-wavelength experiments. Successful applications of the latter have been made by several collaborative groups of investigators. (orig.)

  10. Observable algebras for the rational and trigonometric Euler-Calogero-Moser Models

    International Nuclear Information System (INIS)

    Avan, J.; Billey, E.

    1995-01-01

    We construct polynomial Poisson algebras of observables for the classical Euler-Calogero-Moser (ECM) models. Their structure connects them to flavour-indexed non-linear W ∞ algebras, albeit with qualitative differences. The conserved Hamiltonians and symmetry algebras derived in a previous work are subsets of these algebra. We define their linear, N →∞ limits, realizing W ∞ type algebras coupled to current algebras. ((orig.))

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

    Science.gov (United States)

    Sialaros, Michalis; Christianidis, Jean

    2016-06-01

    Argument The aim of this paper is to employ the newly contextualized historiographical category of "premodern algebra" in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on "geometrical algebra." Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called "semi-algebraic" alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing "premodern algebra," and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.

  12. Algebra for Gifted Third Graders.

    Science.gov (United States)

    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)

  13. An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation

    KAUST Repository

    Liu, Da-Yan; Tian, Yang; Boutat, Driss; Laleg-Kirati, Taous-Meriem

    2015-01-01

    This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann-Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.

  14. Discrete Fourier and wavelet transforms an introduction through linear algebra with applications to signal processing

    CERN Document Server

    Goodman, Roe W

    2016-01-01

    This textbook for undergraduate mathematics, science, and engineering students introduces the theory and applications of discrete Fourier and wavelet transforms using elementary linear algebra, without assuming prior knowledge of signal processing or advanced analysis.It explains how to use the Fourier matrix to extract frequency information from a digital signal and how to use circulant matrices to emphasize selected frequency ranges. It introduces discrete wavelet transforms for digital signals through the lifting method and illustrates through examples and computer explorations how these transforms are used in signal and image processing. Then the general theory of discrete wavelet transforms is developed via the matrix algebra of two-channel filter banks. Finally, wavelet transforms for analog signals are constructed based on filter bank results already presented, and the mathematical framework of multiresolution analysis is examined.

  15. An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation

    KAUST Repository

    Liu, Da-Yan

    2015-04-30

    This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann-Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.

  16. Non-linear partial differential equations an algebraic view of generalized solutions

    CERN Document Server

    Rosinger, Elemer E

    1990-01-01

    A massive transition of interest from solving linear partial differential equations to solving nonlinear ones has taken place during the last two or three decades. The availability of better computers has often made numerical experimentations progress faster than the theoretical understanding of nonlinear partial differential equations. The three most important nonlinear phenomena observed so far both experimentally and numerically, and studied theoretically in connection with such equations have been the solitons, shock waves and turbulence or chaotical processes. In many ways, these phenomen

  17. Construction Example for Algebra System Using Harmony Search Algorithm

    Directory of Open Access Journals (Sweden)

    FangAn Deng

    2015-01-01

    Full Text Available The construction example of algebra system is to verify the existence of a complex algebra system, and it is a NP-hard problem. In this paper, to solve this kind of problems, firstly, a mathematical optimization model for construction example of algebra system is established. Secondly, an improved harmony search algorithm based on NGHS algorithm (INGHS is proposed to find as more solutions as possible for the optimization model; in the proposed INGHS algorithm, to achieve the balance between exploration power and exploitation power in the search process, a global best strategy and parameters dynamic adjustment method are present. Finally, nine construction examples of algebra system are used to evaluate the optimization model and performance of INGHS. The experimental results show that the proposed algorithm has strong performance for solving complex construction example problems of algebra system.

  18. Linear-algebraic approach to electronic excitation of atoms and molecules by electron impact

    International Nuclear Information System (INIS)

    Collins, L.A.; Schneider, B.I.

    1983-01-01

    A linear-algebraic method, based on an integral equations formulation, is applied to the excitation of atoms and molecules by electron impact. Various schemes are devised for treating the one-electron terms that sometimes cause instabilities when directly incorporated into the solution matrix. These include introducing Lagrange undetermined multipliers and correlation terms. Good agreement between the method and other computational techniques is obtained for electron scattering for hydrogenic and Li-like atomic ions and for H 2 + in two- to five-state close-coupling calculations

  19. Diagrams benefit symbolic problem-solving.

    Science.gov (United States)

    Chu, Junyi; Rittle-Johnson, Bethany; Fyfe, Emily R

    2017-06-01

    The format of a mathematics problem often influences students' problem-solving performance. For example, providing diagrams in conjunction with story problems can benefit students' understanding, choice of strategy, and accuracy on story problems. However, it remains unclear whether providing diagrams in conjunction with symbolic equations can benefit problem-solving performance as well. We tested the impact of diagram presence on students' performance on algebra equation problems to determine whether diagrams increase problem-solving success. We also examined the influence of item- and student-level factors to test the robustness of the diagram effect. We worked with 61 seventh-grade students who had received 2 months of pre-algebra instruction. Students participated in an experimenter-led classroom session. Using a within-subjects design, students solved algebra problems in two matched formats (equation and equation-with-diagram). The presence of diagrams increased equation-solving accuracy and the use of informal strategies. This diagram benefit was independent of student ability and item complexity. The benefits of diagrams found previously for story problems generalized to symbolic problems. The findings are consistent with cognitive models of problem-solving and suggest that diagrams may be a useful additional representation of symbolic problems. © 2017 The British Psychological Society.

  20. Toward robust scalable algebraic multigrid solvers

    International Nuclear Information System (INIS)

    Waisman, Haim; Schroder, Jacob; Olson, Luke; Hiriyur, Badri; Gaidamour, Jeremie; Siefert, Christopher; Hu, Jonathan Joseph; Tuminaro, Raymond Stephen

    2010-01-01

    This talk highlights some multigrid challenges that arise from several application areas including structural dynamics, fluid flow, and electromagnetics. A general framework is presented to help introduce and understand algebraic multigrid methods based on energy minimization concepts. Connections between algebraic multigrid prolongators and finite element basis functions are made to explored. It is shown how the general algebraic multigrid framework allows one to adapt multigrid ideas to a number of different situations. Examples are given corresponding to linear elasticity and specifically in the solution of linear systems associated with extended finite elements for fracture problems.

  1. Finite-dimensional linear algebra

    CERN Document Server

    Gockenbach, Mark S

    2010-01-01

    Some Problems Posed on Vector SpacesLinear equationsBest approximationDiagonalizationSummaryFields and Vector SpacesFields Vector spaces Subspaces Linear combinations and spanning sets Linear independence Basis and dimension Properties of bases Polynomial interpolation and the Lagrange basis Continuous piecewise polynomial functionsLinear OperatorsLinear operatorsMore properties of linear operatorsIsomorphic vector spaces Linear operator equations Existence and uniqueness of solutions The fundamental theorem; inverse operatorsGaussian elimination Newton's method Linear ordinary differential eq

  2. Strategies for Solving Fraction Tasks and Their Link to Algebraic Thinking

    Science.gov (United States)

    Pearn, Catherine; Stephens, Max

    2015-01-01

    Many researchers argue that a deep understanding of fractions is important for a successful transition to algebra. Teaching, especially in the middle years, needs to focus specifically on those areas of fraction knowledge and operations that support subsequent solution processes for algebraic equations. This paper focuses on the results of Year 6…

  3. Method of mechanical quadratures for solving singular integral equations of various types

    Science.gov (United States)

    Sahakyan, A. V.; Amirjanyan, H. A.

    2018-04-01

    The method of mechanical quadratures is proposed as a common approach intended for solving the integral equations defined on finite intervals and containing Cauchy-type singular integrals. This method can be used to solve singular integral equations of the first and second kind, equations with generalized kernel, weakly singular equations, and integro-differential equations. The quadrature rules for several different integrals represented through the same coefficients are presented. This allows one to reduce the integral equations containing integrals of different types to a system of linear algebraic equations.

  4. Prime alternative algebras that are nearly commutative

    International Nuclear Information System (INIS)

    Pchelintsev, S V

    2004-01-01

    We prove that by deforming the multiplication in a prime commutative alternative algebra using a C-operation we obtain a prime non-commutative alternative algebra. Under certain restrictions on non-commutative algebras this relation between algebras is reversible. Isotopes are special cases of deformations. We introduce and study a linear space generated by the Bruck C-operations. We prove that the Bruck space is generated by operations of rank 1 and 2 and that 'general' Bruck operations of rank 2 are independent in the following sense: a sum of n operations of rank 2 cannot be written as a linear combination of (n-1) operations of rank 2 and an arbitrary operation of rank 1. We describe infinite series of non-isomorphic prime non-commutative algebras of bounded degree that are deformations of a concrete prime commutative algebra

  5. Quantum algebras as quantizations of dual Poisson–Lie groups

    International Nuclear Information System (INIS)

    Ballesteros, Ángel; Musso, Fabio

    2013-01-01

    A systematic computational approach for the explicit construction of any quantum Hopf algebra (U z (g), Δ z ) starting from the Lie bialgebra (g, δ) that gives the first-order deformation of the coproduct map Δ z is presented. The procedure is based on the well-known ‘quantum duality principle’, namely the fact that any quantum algebra can be viewed as the quantization of the unique Poisson–Lie structure (G*, Λ g ) on the dual group G*, which is obtained by exponentiating the Lie algebra g* defined by the dual map δ*. From this perspective, the coproduct for U z (g) is just the pull-back of the group law for G*, and the Poisson analogues of the quantum commutation rules for U z (g) are given by the unique Poisson–Lie structure Λ g on G* whose linearization is the Poisson analogue of the initial Lie algebra g. This approach is shown to be a very useful technical tool in order to solve the Lie bialgebra quantization problem explicitly since, once a Lie bialgebra (g, δ) is given, the full dual Poisson–Lie group (G*, Λ) can be obtained either by applying standard Poisson–Lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs. As a consequence, the quantization of (G*, Λ) will give rise to the full U z (g) quantum algebra, provided that ordering problems are appropriately fixed through the choice of certain local coordinates on G* whose coproduct fulfils a precise ‘quantum symmetry’ property. The applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant Lie algebras such as sl(2,R), the (2+1) anti-de Sitter algebra so(2, 2) and the Poincaré algebra in (3+1) dimensions. (paper)

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

    Science.gov (United States)

    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.

  7. Algebra & trigonometry super review

    CERN Document Server

    2012-01-01

    Get all you need to know with Super Reviews! Each Super Review is packed with in-depth, student-friendly topic reviews that fully explain everything about the subject. The Algebra and Trigonometry Super Review includes sets and set operations, number systems and fundamental algebraic laws and operations, exponents and radicals, polynomials and rational expressions, equations, linear equations and systems of linear equations, inequalities, relations and functions, quadratic equations, equations of higher order, ratios, proportions, and variations. Take the Super Review quizzes to see how much y

  8. Elementary Algebra Connections to Precalculus

    Science.gov (United States)

    Lopez-Boada, Roberto; Daire, Sandra Arguelles

    2013-01-01

    This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…

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

    Science.gov (United States)

    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…

  10. Using graph theory for automated electric circuit solving

    International Nuclear Information System (INIS)

    Toscano, L; Stella, S; Milotti, E

    2015-01-01

    Graph theory plays many important roles in modern physics and in many different contexts, spanning diverse topics such as the description of scale-free networks and the structure of the universe as a complex directed graph in causal set theory. Graph theory is also ideally suited to describe many concepts in computer science. Therefore it is increasingly important for physics students to master the basic concepts of graph theory. Here we describe a student project where we develop a computational approach to electric circuit solving which is based on graph theoretic concepts. This highly multidisciplinary approach combines abstract mathematics, linear algebra, the physics of circuits, and computer programming to reach the ambitious goal of implementing automated circuit solving. (paper)

  11. Solution of linear and nonlinear matrix systems. Application to a nonlinear diffusion equation

    International Nuclear Information System (INIS)

    Bonnet, M.; Meurant, G.

    1978-01-01

    The object of this study is to compare different methods of solving linear and nonlinear algebraic systems and to apply them to the nonlinear system obtained by discretizing a nonlinear diffusion equation. For linear systems the conventional methods of alternating direction type or Gauss Seidel's methods are compared to more recent ones of the type of generalized conjugate gradient; the superiority of the latter is shown by numerical examples. For nonlinear systems, a method of nonlinear conjugate gradient is studied together with Newton's method and some of its variants. It should be noted, however, that Newton's method is found to be more efficient when coupled with a good method for solving the linear system. As a conclusion, these methods are used to solve a nonlinear diffusion problem and the numerical results obtained are compared [fr

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

    Science.gov (United States)

    Schonberger, Ann K.

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

  13. The development of an algebraic multigrid algorithm for symmetric positive definite linear systems

    Energy Technology Data Exchange (ETDEWEB)

    Vanek, P.; Mandel, J.; Brezina, M. [Univ. of Colorado, Denver, CO (United States)

    1996-12-31

    An algebraic multigrid algorithm for symmetric, positive definite linear systems is developed based on the concept of prolongation by smoothed aggregation. Coarse levels are generated automatically. We present a set of requirements motivated heuristically by a convergence theory. The algorithm then attempts to satisfy the requirements. Input to the method are the coefficient matrix and zero energy modes, which are determined from nodal coordinates and knowledge of the differential equation. Efficiency of the resulting algorithm is demonstrated by computational results on real world problems from solid elasticity, plate blending, and shells.

  14. New approach to solve symmetric fully fuzzy linear systems

    Indian Academy of Sciences (India)

    In this paper, we present a method to solve fully fuzzy linear systems with symmetric coefficient matrix. The symmetric coefficient matrix is decomposed into two systems of equations by using Cholesky method and then a solution can be obtained. Numerical examples are given to illustrate our method.

  15. Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

    Directory of Open Access Journals (Sweden)

    S. Narayanamoorthy

    2015-01-01

    Full Text Available We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

  16. Linear algebra

    CERN Document Server

    Berberian, Sterling K

    2014-01-01

    Introductory treatment covers basic theory of vector spaces and linear maps - dimension, determinants, eigenvalues, and eigenvectors - plus more advanced topics such as the study of canonical forms for matrices. 1992 edition.

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

    Science.gov (United States)

    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…

  18. Algebraic Thinking: Conceptions of Elementary School Teachers

    OpenAIRE

    Rodrigues Rézio, Ana Sofia

    2015-01-01

    Students’ algebraic reasoning, at the beginning of their schooling years, includes the development and promotion of functional thinking and the understanding of mathematical properties, which can be stimulated by solving problems. In the latest Portuguese Program for Mathematics Elementary Education, we do not see the topic Algebra in the first year of school although some other topics include objectives of algebraic nature. This fact showed the importance of research about the introduction o...

  19. New approach to solve fully fuzzy system of linear equations using ...

    Indian Academy of Sciences (India)

    This paper proposes two new methods to solve fully fuzzy system of linear equations. The fuzzy system has been converted to a crisp system of linear equations by using single and double parametric form of fuzzy numbers to obtain the non-negative solution. Double parametric form of fuzzy numbers is defined and applied ...

  20. Anyons, deformed oscillator algebras and projectors

    International Nuclear Information System (INIS)

    Engquist, Johan

    2009-01-01

    We initiate an algebraic approach to the many-anyon problem based on deformed oscillator algebras. The formalism utilizes a generalization of the deformed Heisenberg algebras underlying the operator solution of the Calogero problem. We define a many-body Hamiltonian and an angular momentum operator which are relevant for a linearized analysis in the statistical parameter ν. There exists a unique ground state and, in spite of the presence of defect lines, the anyonic weight lattices are completely connected by the application of the oscillators of the algebra. This is achieved by supplementing the oscillator algebra with a certain projector algebra.

  1. Solving Fully Fuzzy Linear System of Equations in General Form

    Directory of Open Access Journals (Sweden)

    A. Yousefzadeh

    2012-06-01

    Full Text Available In this work, we propose an approach for computing the positive solution of a fully fuzzy linear system where the coefficient matrix is a fuzzy $nimes n$ matrix. To do this, we use arithmetic operations on fuzzy numbers that introduced by Kaffman in and convert the fully fuzzy linear system into two $nimes n$ and $2nimes 2n$ crisp linear systems. If the solutions of these linear systems don't satisfy in positive fuzzy solution condition, we introduce the constrained least squares problem to obtain optimal fuzzy vector solution by applying the ranking function in given fully fuzzy linear system. Using our proposed method, the fully fuzzy linear system of equations always has a solution. Finally, we illustrate the efficiency of proposed method by solving some numerical examples.

  2. Spectral theory of linear operators and spectral systems in Banach algebras

    CERN Document Server

    Müller, Vladimir

    2003-01-01

    This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach algebras. It presents a survey of results concerning various types of spectra, both of single and n-tuples of elements. Typical examples are the one-sided spectra, the approximate point, essential, local and Taylor spectrum, and their variants. The theory is presented in a unified, axiomatic and elementary way. Many results appear here for the first time in a monograph. The material is self-contained. Only a basic knowledge of functional analysis, topology, and complex analysis is assumed. The monograph should appeal both to students who would like to learn about spectral theory and to experts in the field. It can also serve as a reference book. The present second edition contains a number of new results, in particular, concerning orbits and their relations to the invariant subspace problem. This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach alg...

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

    Science.gov (United States)

    Downie, John D.; Goodman, Joseph W.

    1989-10-01

    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.

  4. Linear-algebraic bath transformation for simulating complex open quantum systems

    International Nuclear Information System (INIS)

    Huh, Joonsuk; Mostame, Sarah; Fujita, Takatoshi; Aspuru-Guzik, Alán; Yung, Man-Hong

    2014-01-01

    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 parallel chains. 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. (paper)

  5. The structure of algebraic problem in high schools

    OpenAIRE

    Chio, José Angel; Álvarez, Aida; Estrada, Pablo

    2010-01-01

    The paper is aimed at discussing the importance of pupil’s knowledge of algebraic problem structure. The research started by diagnosing pupil’s actual command of algebraic problem structure. Finally suggestions to teachers of mathematics for facing difficulties in solving problems are given.

  6. Lie-Algebras. Pt. 1

    International Nuclear Information System (INIS)

    Baeuerle, G.G.A.; Kerf, E.A. de

    1990-01-01

    The structure of the laws in physics is largely based on symmetries. This book is on Lie algebras, the mathematics of symmetry. It gives a thorough mathematical treatment of finite dimensional Lie algebras and Kac-Moody algebras. Concepts such as Cartan matrix, root system, Serre's construction are carefully introduced. Although the book can be read by an undergraduate with only an elementary knowledge of linear algebra, the book will also be of use to the experienced researcher. Experience has shown that students who followed the lectures are well-prepared to take on research in the realms of string-theory, conformal field-theory and integrable systems. 48 refs.; 66 figs.; 3 tabs

  7. Solving large-scale sparse eigenvalue problems and linear systems of equations for accelerator modeling

    International Nuclear Information System (INIS)

    Gene Golub; Kwok Ko

    2009-01-01

    The solutions of sparse eigenvalue problems and linear systems constitute one of the key computational kernels in the discretization of partial differential equations for the modeling of linear accelerators. The computational challenges faced by existing techniques for solving those sparse eigenvalue problems and linear systems call for continuing research to improve on the algorithms so that ever increasing problem size as required by the physics application can be tackled. Under the support of this award, the filter algorithm for solving large sparse eigenvalue problems was developed at Stanford to address the computational difficulties in the previous methods with the goal to enable accelerator simulations on then the world largest unclassified supercomputer at NERSC for this class of problems. Specifically, a new method, the Hemitian skew-Hemitian splitting method, was proposed and researched as an improved method for solving linear systems with non-Hermitian positive definite and semidefinite matrices.

  8. Classification and identification of Lie algebras

    CERN Document Server

    Snobl, Libor

    2014-01-01

    The purpose of this book is to serve as a tool for researchers and practitioners who apply Lie algebras and Lie groups to solve problems arising in science and engineering. The authors address the problem of expressing a Lie algebra obtained in some arbitrary basis in a more suitable basis in which all essential features of the Lie algebra are directly visible. This includes algorithms accomplishing decomposition into a direct sum, identification of the radical and the Levi decomposition, and the computation of the nilradical and of the Casimir invariants. Examples are given for each algorithm. For low-dimensional Lie algebras this makes it possible to identify the given Lie algebra completely. The authors provide a representative list of all Lie algebras of dimension less or equal to 6 together with their important properties, including their Casimir invariants. The list is ordered in a way to make identification easy, using only basis independent properties of the Lie algebras. They also describe certain cl...

  9. An algebraic approach to linear-optical schemes for deterministic quantum computing

    International Nuclear Information System (INIS)

    Aniello, Paolo; Cagli, Ruben Coen

    2005-01-01

    Linear-optical passive (LOP) devices and photon counters are sufficient to implement universal quantum computation with single photons, and particular schemes have already been proposed. In this paper we discuss the link between the algebraic structure of LOP transformations and quantum computing. We first show how to decompose the Fock space of N optical modes in finite-dimensional subspaces that are suitable for encoding strings of qubits and invariant under LOP transformations (these subspaces are related to the spaces of irreducible unitary representations of U (N). Next we show how to design in algorithmic fashion LOP circuits which implement any quantum circuit deterministically. We also present some simple examples, such as the circuits implementing a cNOT gate and a Bell state generator/analyser

  10. A q-deformed Lorentz algebra

    International Nuclear Information System (INIS)

    Schmidke, W.B.; Wess, J.; Muenchen Univ.; Zumino, B.; Lawrence Berkeley Lab., CA

    1991-01-01

    We derive a q-deformed version of the Lorentz algebra by deformating the algebra SL(2, C). The method is based on linear representations of the algebra on the complex quantum spinor space. We find that the generators usually identified with SL q (2, C) generate SU q (2) only. Four additional generators are added which generate Lorentz boosts. The full algebra of all seven generators and their coproduct is presented. We show that in the limit q→1 the generators are those of the classical Lorentz algebra plus an additional U(1). Thus we have a deformation of SL(2, C)xU(1). (orig.)

  11. Students' errors in solving linear equation word problems: Case ...

    African Journals Online (AJOL)

    kofi.mereku

    Development in most areas of life is based on effective knowledge of science and ... Problem solving, as used in mathematics education literature, refers ... word problems, on the other hand, are those linear equation tasks or ... taught LEWPs in the junior high school, many of them reach the senior high school without a.

  12. Fuzzy logic of quasi-truth an algebraic treatment

    CERN Document Server

    Di Nola, Antonio; Turunen, Esko

    2016-01-01

    This book presents the first algebraic treatment of quasi-truth fuzzy logic and covers the algebraic foundations of many-valued logic. It offers a comprehensive account of basic techniques and reports on important results showing the pivotal role played by perfect many-valued algebras (MV-algebras). It is well known that the first-order predicate Łukasiewicz logic is not complete with respect to the canonical set of truth values. However, it is complete with respect to all linearly ordered MV –algebras. As there are no simple linearly ordered MV-algebras in this case, infinitesimal elements of an MV-algebra are allowed to be truth values. The book presents perfect algebras as an interesting subclass of local MV-algebras and provides readers with the necessary knowledge and tools for formalizing the fuzzy concept of quasi true and quasi false. All basic concepts are introduced in detail to promote a better understanding of the more complex ones. It is an advanced and inspiring reference-guide for graduate s...

  13. Mathematics Literacy of Secondary Students in Solving Simultanenous Linear Equations

    Science.gov (United States)

    Sitompul, R. S. I.; Budayasa, I. K.; Masriyah

    2018-01-01

    This study examines the profile of secondary students’ mathematical literacy in solving simultanenous linear equations problems in terms of cognitive style of visualizer and verbalizer. This research is a descriptive research with qualitative approach. The subjects in this research consist of one student with cognitive style of visualizer and one student with cognitive style of verbalizer. The main instrument in this research is the researcher herself and supporting instruments are cognitive style tests, mathematics skills tests, problem-solving tests and interview guidelines. Research was begun by determining the cognitive style test and mathematics skill test. The subjects chosen were given problem-solving test about simultaneous linear equations and continued with interview. To ensure the validity of the data, the researcher conducted data triangulation; the steps of data reduction, data presentation, data interpretation, and conclusion drawing. The results show that there is a similarity of visualizer and verbalizer-cognitive style in identifying and understanding the mathematical structure in the process of formulating. There are differences in how to represent problems in the process of implementing, there are differences in designing strategies and in the process of interpreting, and there are differences in explaining the logical reasons.

  14. CHEBYSHEV ACCELERATION TECHNIQUE FOR SOLVING FUZZY LINEAR SYSTEM

    Directory of Open Access Journals (Sweden)

    S.H. Nasseri

    2011-07-01

    Full Text Available In this paper, Chebyshev acceleration technique is used to solve the fuzzy linear system (FLS. This method is discussed in details and followed by summary of some other acceleration techniques. Moreover, we show that in some situations that the methods such as Jacobi, Gauss-Sidel, SOR and conjugate gradient is divergent, our proposed method is applicable and the acquired results are illustrated by some numerical examples.

  15. CHEBYSHEV ACCELERATION TECHNIQUE FOR SOLVING FUZZY LINEAR SYSTEM

    Directory of Open Access Journals (Sweden)

    S.H. Nasseri

    2009-10-01

    Full Text Available In this paper, Chebyshev acceleration technique is used to solve the fuzzy linear system (FLS. This method is discussed in details and followed by summary of some other acceleration techniques. Moreover, we show that in some situations that the methods such as Jacobi, Gauss-Sidel, SOR and conjugate gradient is divergent, our proposed method is applicable and the acquired results are illustrated by some numerical examples.

  16. Algebraic complexities and algebraic curves over finite fields.

    Science.gov (United States)

    Chudnovsky, D V; Chudnovsky, G V

    1987-04-01

    We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, 169-180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields.

  17. Simple Lie algebras and Dynkin diagrams

    International Nuclear Information System (INIS)

    Buccella, F.

    1983-01-01

    The following theorem is studied: in a simple Lie algebra of rank p there are p positive roots such that all the other n-3p/2 positive roots are linear combinations of them with integer non negative coefficients. Dykin diagrams are built by representing the simple roots with circles and drawing a junction between the roots. Five exceptional algebras are studied, focusing on triple junction algebra, angular momentum algebra, weights of the representation, antisymmetric tensors, and subalgebras

  18. Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

    Institute of Scientific and Technical Information of China (English)

    2008-01-01

    Using functional derivative technique in quantum field theory, the algebraic dy-namics approach for solution of ordinary differential evolution equations was gen-eralized to treat partial differential evolution equations. The partial differential evo-lution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynam-ics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new nu-merical algorithm—algebraic dynamics algorithm was proposed for partial differ-ential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experi-ments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.

  19. UCSMP Algebra. What Works Clearinghouse Intervention Report

    Science.gov (United States)

    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…

  20. The algebras of large N matrix mechanics

    Energy Technology Data Exchange (ETDEWEB)

    Halpern, M.B.; Schwartz, C.

    1999-09-16

    Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.

  1. The structure of algebraic problem in high schools

    Directory of Open Access Journals (Sweden)

    Chio, José Angel

    2010-01-01

    Full Text Available The paper is aimed at discussing the importance of pupil’s knowledge of algebraic problem structure. The research started by diagnosing pupil’s actual command of algebraic problem structure. Finally suggestions to teachers of mathematics for facing difficulties in solving problems are given.

  2. On unitary representations of the exceptional non-linear N=7 and N=8 superconformal algebras in terms of free fields

    International Nuclear Information System (INIS)

    Ketov, S.V.

    1996-01-01

    The simplest free-field realizations of the exceptional non-linear (quadratically generated, or W-type) N=8 and N=7 superconformal algebras with Spin(7) and G 2 affine currents, respectively, are investigated. Both the N=8 and N=7 algebras are found to admit unitary and highest-weight irreducible representations in terms of a single free boson and free fermions in 8 of Spin(7) or 7 of G 2 , respectively, at level k=1 and the corresponding central charges c 8 =26/5 and c 7 =5. (orig.)

  3. On the stability, the periodic solutions and the resolution of certain types of non linear equations, and of non linearly coupled systems of these equations, appearing in betatronic oscillations

    International Nuclear Information System (INIS)

    Valat, J.

    1960-12-01

    Universal stability diagrams have been calculated and experimentally checked for Hill-Meissner type equations with square-wave coefficients. The study of these equations in the phase-plane has then made it possible to extend the periodic solution calculations to the case of non-linear differential equations with periodic square-wave coefficients. This theory has been checked experimentally. For non-linear coupled systems with constant coefficients, a search was first made for solutions giving an algebraic motion. The elliptical and Fuchs's functions solve such motions. The study of non-algebraic motions is more delicate, apart from the study of nonlinear Lissajous's motions. A functional analysis shows that it is possible however in certain cases to decouple the system and to find general solutions. For non-linear coupled systems with periodic square-wave coefficients it is then possible to calculate the conditions leading to periodic solutions, if the two non-linear associated systems with constant coefficients fall into one of the categories of the above paragraph. (author) [fr

  4. Basic algebra

    CERN Document Server

    Jacobson, Nathan

    2009-01-01

    A classic text and standard reference for a generation, this volume and its companion are the work of an expert algebraist who taught at Yale for two decades. Nathan Jacobson's books possess a conceptual and theoretical orientation, and in addition to their value as classroom texts, they serve as valuable references.Volume I explores all of the topics typically covered in undergraduate courses, including the rudiments of set theory, group theory, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra. Its comprehensive treatment extends to such rigorous topics as L

  5. PENGEMBANGAN BAHAN AJAR MATA KULIAH ALJABAR LINEAR BERBASIS OPEN ENDED

    Directory of Open Access Journals (Sweden)

    Nurul Farida

    2017-01-01

    Full Text Available Teaching materials is one important component in learning because teaching materials can be used for independent study. On the other hand, the lack of teaching materials are on open-ended problems leads to an underdevelopment of creativity in learning. With the open-ended problems, students are expected to have more than one solution to solve problems in everyday life, especially in mathematical problem solving. The purpose of this study is to develop teaching materials based on open-ended valildly, practically and effectively in the subject of linear algebra. This research is the development by using Plomp models which consists of three stages: preliminary research, prototyping, and assessment. Data collection technique through observation, questionnaires and tests. The instruments used were observation sheet, questionnaire sheet and a test sheet. Based on result of the research that has been done shows that the teaching materials is based on open-ended in the subject linear algebra that have been developed otherwise valid with a value of 3.43, practice with a value 3,11 or in the percentage reached 77,75%, and effective with value of learning outcomes experiment class (use teaching materials developed higher than control class.

  6. Computer Algebra Systems in Undergraduate Instruction.

    Science.gov (United States)

    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)

  7. Linear homotopy solution of nonlinear systems of equations in geodesy

    Science.gov (United States)

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

    2010-01-01

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

  8. On a new iterative method for solving linear systems and comparison results

    Science.gov (United States)

    Jing, Yan-Fei; Huang, Ting-Zhu

    2008-10-01

    In Ujevic [A new iterative method for solving linear systems, Appl. Math. Comput. 179 (2006) 725-730], the author obtained a new iterative method for solving linear systems, which can be considered as a modification of the Gauss-Seidel method. In this paper, we show that this is a special case from a point of view of projection techniques. And a different approach is established, which is both theoretically and numerically proven to be better than (at least the same as) Ujevic's. As the presented numerical examples show, in most cases, the convergence rate is more than one and a half that of Ujevic.

  9. Method for solving fully fuzzy linear programming problems using deviation degree measure

    Institute of Scientific and Technical Information of China (English)

    Haifang Cheng; Weilai Huang; Jianhu Cai

    2013-01-01

    A new ful y fuzzy linear programming (FFLP) prob-lem with fuzzy equality constraints is discussed. Using deviation degree measures, the FFLP problem is transformed into a crispδ-parametric linear programming (LP) problem. Giving the value of deviation degree in each constraint, the δ-fuzzy optimal so-lution of the FFLP problem can be obtained by solving this LP problem. An algorithm is also proposed to find a balance-fuzzy optimal solution between two goals in conflict: to improve the va-lues of the objective function and to decrease the values of the deviation degrees. A numerical example is solved to il ustrate the proposed method.

  10. The impact of fraction magnitude knowledge on algebra performance and learning.

    Science.gov (United States)

    Booth, Julie L; Newton, Kristie J; Twiss-Garrity, Laura K

    2014-02-01

    Knowledge of fractions is thought to be crucial for success with algebra, but empirical evidence supporting this conjecture is just beginning to emerge. In the current study, Algebra 1 students completed magnitude estimation tasks on three scales (0-1 [fractions], 0-1,000,000, and 0-62,571) just before beginning their unit on equation solving. Results indicated that fraction magnitude knowledge, and not whole number knowledge, was especially related to students' pretest knowledge of equation solving and encoding of equation features. Pretest fraction knowledge was also predictive of students' improvement in equation solving and equation encoding skills. Students' placement of unit fractions (e.g., those with a numerator of 1) was not especially useful for predicting algebra performance and learning in this population. Placement of non-unit fractions was more predictive, suggesting that proportional reasoning skills might be an important link between fraction knowledge and learning algebra. Copyright © 2013 Elsevier Inc. All rights reserved.

  11. Stability and Linear Quadratic Differential Games of Discrete-Time Markovian Jump Linear Systems with State-Dependent Noise

    Directory of Open Access Journals (Sweden)

    Huiying Sun

    2014-01-01

    Full Text Available We mainly consider the stability of discrete-time Markovian jump linear systems with state-dependent noise as well as its linear quadratic (LQ differential games. A necessary and sufficient condition involved with the connection between stochastic Tn-stability of Markovian jump linear systems with state-dependent noise and Lyapunov equation is proposed. And using the theory of stochastic Tn-stability, we give the optimal strategies and the optimal cost values for infinite horizon LQ stochastic differential games. It is demonstrated that the solutions of infinite horizon LQ stochastic differential games are concerned with four coupled generalized algebraic Riccati equations (GAREs. Finally, an iterative algorithm is presented to solve the four coupled GAREs and a simulation example is given to illustrate the effectiveness of it.

  12. Algebraic coarsening methods for linear and nonlinear PDE and systems

    International Nuclear Information System (INIS)

    McWilliams, J C

    2000-01-01

    In [l] Brandt describes a general approach for algebraic coarsening. Given fine-grid equations and a prescribed relaxation method, an approach is presented for defining both the coarse-grid variables and the coarse-grid equations corresponding to these variables. Although, these two tasks are not necessarily related (and, indeed, are often performed independently and with distinct techniques) in the approaches of [1] both revolve around the same underlying observation. To determine whether a given set of coarse-grid variables is appropriate it is suggested that one should employ compatible relaxation. This is a generalization of so-called F-relaxation (e.g., [2]). Suppose that the coarse-grid variables are defined as a subset of the fine-grid variables. Then, F-relaxation simply means relaxing only the F-variables (i.e., fine-grid variables that do not correspond to coarse-grid variables), while leaving the remaining fine-grid variables (C-variables) unchanged. The generalization of compatible relaxation is in allowing the coarse-grid variables to be defined differently, say as linear combinations of fine-grid variables, or even nondeterministically (see examples in [1]). For the present summary it suffices to consider the simple case. The central observation regarding the set of coarse-grid variables is the following [1]: Observation 1--A general measure for the quality of the set of coarse-grid variables is the convergence rate of compatible relaxation. The conclusion is that a necessary condition for efficient multigrid solution (e.g., with convergence rates independent of problem size) is that the compatible-relaxation convergence be bounded away from 1, independently of the number of variables. This is often a sufficient condition, provided that the coarse-grid equations are sufficiently accurate. Therefore, it is suggested in [1] that the convergence rate of compatible relaxation should be used as a criterion for choosing and evaluating the set of coarse

  13. Solving the nuclear shell model with an algebraic method

    International Nuclear Information System (INIS)

    Feng, D.H.; Pan, X.W.; Guidry, M.

    1997-01-01

    We illustrate algebraic methods in the nuclear shell model through a concrete example, the fermion dynamical symmetry model (FDSM). We use this model to introduce important concepts such as dynamical symmetry, symmetry breaking, effective symmetry, and diagonalization within a higher-symmetry basis. (orig.)

  14. Stability Analysis for Fractional-Order Linear Singular Delay Differential Systems

    Directory of Open Access Journals (Sweden)

    Hai Zhang

    2014-01-01

    Full Text Available We investigate the delay-independently asymptotic stability of fractional-order linear singular delay differential systems. Based on the algebraic approach, the sufficient conditions are presented to ensure the asymptotic stability for any delay parameter. By applying the stability criteria, one can avoid solving the roots of transcendental equations. An example is also provided to illustrate the effectiveness and applicability of the theoretical results.

  15. Algebraic special functions and SO(3,2)

    International Nuclear Information System (INIS)

    Celeghini, E.; Olmo, M.A. del

    2013-01-01

    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 2 functions defined on (−1,1)×Z and on the sphere S 2 , respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining in 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 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 2 functions

  16. The multilevel fast multipole algorithm (MLFMA) for solving large-scale computational electromagnetics problems

    CERN Document Server

    Ergul, Ozgur

    2014-01-01

    The Multilevel Fast Multipole Algorithm (MLFMA) for Solving Large-Scale Computational Electromagnetic Problems provides a detailed and instructional overview of implementing MLFMA. The book: Presents a comprehensive treatment of the MLFMA algorithm, including basic linear algebra concepts, recent developments on the parallel computation, and a number of application examplesCovers solutions of electromagnetic problems involving dielectric objects and perfectly-conducting objectsDiscusses applications including scattering from airborne targets, scattering from red

  17. EZLP: An Interactive Computer Program for Solving Linear Programming Problems. Final Report.

    Science.gov (United States)

    Jarvis, John J.; And Others

    Designed for student use in solving linear programming problems, the interactive computer program described (EZLP) permits the student to input the linear programming model in exactly the same manner in which it would be written on paper. This report includes a brief review of the development of EZLP; narrative descriptions of program features,…

  18. Basic math and pre-algebra practice problems for dummies

    CERN Document Server

    Zegarelli, Mark

    2013-01-01

    1001 Basic Math & Pre- Algebra Practice Problems For  Dummies   Practice makes perfect-and helps deepen your understanding of basic math and pre-algebra 1001 Basic Math & Pre-Algebra Practice Problems For Dummies, with free access to online practice problems, takes you beyond the instruction and guidance offered in Basic Math & Pre-Algebra For Dummies, giving you 1,001 opportunities to practice solving problems from the major topics in your math course. You begin with some basic arithmetic practice, move on to fractions, decimals, and per

  19. Applied matrix algebra in the statistical sciences

    CERN Document Server

    Basilevsky, Alexander

    2005-01-01

    This comprehensive text offers teachings relevant to both applied and theoretical branches of matrix algebra and provides a bridge between linear algebra and statistical models. Appropriate for advanced undergraduate and graduate students. 1983 edition.

  20. The Universal Askey-Wilson Algebra

    Directory of Open Access Journals (Sweden)

    Paul Terwilliger

    2011-07-01

    Full Text Available In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3 and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension Δ of AW, obtained from AW by reinterpreting certain parameters as central elements in the algebra. We call Δ the universal Askey-Wilson algebra. We give a faithful action of the modular group PSL_2(Z on Δ as a group of automorphisms. We give a linear basis for Δ. We describe the center of Δ and the 2-sided ideal Δ[Δ,Δ]Δ. We discuss how Δ is related to the q-Onsager algebra.

  1. Accuracy Limitations in Optical Linear Algebra Processors

    Science.gov (United States)

    Batsell, Stephen Gordon

    1990-01-01

    One of the limiting factors in applying optical linear algebra processors (OLAPs) to real-world problems has been the poor achievable accuracy of these processors. Little previous research has been done on determining noise sources from a systems perspective which would include noise generated in the multiplication and addition operations, noise from spatial variations across arrays, and from crosstalk. In this dissertation, we propose a second-order statistical model for an OLAP which incorporates all these system noise sources. We now apply this knowledge to determining upper and lower bounds on the achievable accuracy. This is accomplished by first translating the standard definition of accuracy used in electronic digital processors to analog optical processors. We then employ our second-order statistical model. Having determined a general accuracy equation, we consider limiting cases such as for ideal and noisy components. From the ideal case, we find the fundamental limitations on improving analog processor accuracy. From the noisy case, we determine the practical limitations based on both device and system noise sources. These bounds allow system trade-offs to be made both in the choice of architecture and in individual components in such a way as to maximize the accuracy of the processor. Finally, by determining the fundamental limitations, we show the system engineer when the accuracy desired can be achieved from hardware or architecture improvements and when it must come from signal pre-processing and/or post-processing techniques.

  2. Topological characterizations of S-Linearity

    Directory of Open Access Journals (Sweden)

    Carfi', David

    2007-10-01

    Full Text Available We give several characterizations of basic concepts of S-linear algebra in terms of weak duality on topological vector spaces. On the way, some classic results of Functional Analysis are reinterpreted in terms of S-linear algebra, by an application-oriented fashion. The results are required in the S-linear algebra formulation of infinite dimensional Decision Theory and in the study of abstract evolution equations in economical and physical Theories.

  3. Comments on N=4 superconformal algebras

    International Nuclear Information System (INIS)

    Rasmussen, J.

    2001-01-01

    We present a new and asymmetric N=4 superconformal algebra for arbitrary central charge, thus completing our recent work on its classical analogue with vanishing central charge. Besides the Virasoro generator and 4 supercurrents, the algebra consists of an internal SU(2)xU(1) Kac-Moody algebra in addition to two spin 1/2 fermions and a bosonic scalar. The algebra is shown to be invariant under a linear twist of the generators, except for a unique value of the continuous twist parameter. At this value, the invariance is broken and the algebra collapses to the small N=4 superconformal algebra. The asymmetric N=4 superconformal algebra may be seen as induced by an affine SL(2 vertical bar 2) current superalgebra. Replacing SL(2 vertical bar 2) with the coset SL(2 vertical bar 2)/U(1), results directly in the small N=4 superconformal algebra

  4. A Linear Algebra Framework for Static High Performance Fortran Code Distribution

    Directory of Open Access Journals (Sweden)

    Corinne Ancourt

    1997-01-01

    Full Text Available High Performance Fortran (HPF was developed to support data parallel programming for single-instruction multiple-data (SIMD and multiple-instruction multiple-data (MIMD machines with distributed memory. The programmer is provided a familiar uniform logical address space and specifies the data distribution by directives. The compiler then exploits these directives to allocate arrays in the local memories, to assign computations to elementary processors, and to migrate data between processors when required. We show here that linear algebra is a powerful framework to encode HPF directives and to synthesize distributed code with space-efficient array allocation, tight loop bounds, and vectorized communications for INDEPENDENT loops. The generated code includes traditional optimizations such as guard elimination, message vectorization and aggregation, and overlap analysis. The systematic use of an affine framework makes it possible to prove the compilation scheme correct.

  5. A Quantitative Reasoning Approach to Algebra Using Inquiry-Based Learning

    Directory of Open Access Journals (Sweden)

    Victor I. Piercey

    2017-07-01

    Full Text Available In this paper, I share a hybrid quantitative reasoning/algebra two-course sequence that challenges the common assumption that quantitative literacy and reasoning are less rigorous mathematics alternatives to algebra and illustrates that a quantitative reasoning framework can be used to teach traditional algebra. The presentation is made in two parts. In the first part, which is somewhat philosophical and theoretical, I explain my personal perspective of what I mean by “algebra” and “doing algebra.” I contend that algebra is a form of communication whose value is precision, which allows us to perform algebraic manipulations in the form of simplification and solving moves. A quantitative reasoning approach to traditional algebraic manipulations rests on intentional and purposeful use of simplification and solving moves within contextual situations. In part 2, I describe a 6-week instructional module intended for undergraduate business students that was delivered to students who had placed into beginning algebra. The perspective described in part 1 heavily informed the design of this module. The course materials, which involve the use of Excel in multiple authentic contexts, are built around the use of inquiry-based learning. Upon completion of this module, the percentage of students who successfully complete model problems in an assessment is in the same range as surveyed students in precalculus and calculus, approximately two “grade levels” ahead of their placement.

  6. Discrete event systems in dioid algebra and conventional algebra

    CERN Document Server

    Declerck, Philippe

    2013-01-01

    This book concerns the use of dioid algebra as (max, +) algebra to treat the synchronization of tasks expressed by the maximum of the ends of the tasks conditioning the beginning of another task - a criterion of linear programming. A classical example is the departure time of a train which should wait for the arrival of other trains in order to allow for the changeover of passengers.The content focuses on the modeling of a class of dynamic systems usually called "discrete event systems" where the timing of the events is crucial. Events are viewed as sudden changes in a process which i

  7. Development of abstract mathematical reasoning: the case of algebra.

    Science.gov (United States)

    Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja

    2014-01-01

    Algebra typically represents the students' first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students' ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16-17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15-16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students' transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition.

  8. Comments on new iterative methods for solving linear systems

    Directory of Open Access Journals (Sweden)

    Wang Ke

    2017-06-01

    Full Text Available Some new iterative methods were presented by Du, Zheng and Wang for solving linear systems in [3], where it is shown that the new methods, comparing to the classical Jacobi or Gauss-Seidel method, can be applied to more systems and have faster convergence. This note shows that their methods are suitable for more matrices than positive matrices which the authors suggested through further analysis and numerical examples.

  9. Solving Linear Equations by Classical Jacobi-SR Based Hybrid Evolutionary Algorithm with Uniform Adaptation Technique

    OpenAIRE

    Jamali, R. M. Jalal Uddin; Hashem, M. M. A.; Hasan, M. Mahfuz; Rahman, Md. Bazlar

    2013-01-01

    Solving a set of simultaneous linear equations is probably the most important topic in numerical methods. For solving linear equations, iterative methods are preferred over the direct methods especially when the coefficient matrix is sparse. The rate of convergence of iteration method is increased by using Successive Relaxation (SR) technique. But SR technique is very much sensitive to relaxation factor, {\\omega}. Recently, hybridization of classical Gauss-Seidel based successive relaxation t...

  10. Optimal Homotopy Asymptotic Method for Solving the Linear Fredholm Integral Equations of the First Kind

    Directory of Open Access Journals (Sweden)

    Mohammad Almousa

    2013-01-01

    Full Text Available The aim of this study is to present the use of a semi analytical method called the optimal homotopy asymptotic method (OHAM for solving the linear Fredholm integral equations of the first kind. Three examples are discussed to show the ability of the method to solve the linear Fredholm integral equations of the first kind. The results indicated that the method is very effective and simple.

  11. A type of loop algebra and the associated loop algebras

    International Nuclear Information System (INIS)

    Tam Honwah; Zhang Yufeng

    2008-01-01

    A higher-dimensional twisted loop algebra is constructed. As its application, a new Lax pair is presented, whose compatibility gives rise to a Liouville integrable hierarchy of evolution equations by making use of Tu scheme. One of the reduction cases of the hierarchy is an analogous of the well-known AKNS system. Next, the twisted loop algebra, furthermore, is extended to another higher dimensional loop algebra, from which a hierarchy of evolution equations with 11-potential component functions is obtained, whose reduction is just standard AKNS system. Especially, we prove that an arbitrary linear combination of the four Hamiltonian operators directly obtained from the recurrence relations is still a Hamiltonian operator. Therefore, the hierarchy with 11-potential functions possesses 4-Hamiltonian structures. Finally, an integrable coupling of the hierarchy is worked out

  12. A complex linear least-squares method to derive relative and absolute orientations of seismic sensors

    OpenAIRE

    F. Grigoli; Simone Cesca; Torsten Dahm; L. Krieger

    2012-01-01

    Determining the relative orientation of the horizontal components of seismic sensors is a common problem that limits data analysis and interpretation for several acquisition setups, including linear arrays of geophones deployed in borehole installations or ocean bottom seismometers deployed at the seafloor. To solve this problem we propose a new inversion method based on a complex linear algebra approach. Relative orientation angles are retrieved by minimizing, in a least-squares sense, the l...

  13. Lie n-derivations on 7 -subspace lattice algebras

    Indian Academy of Sciences (India)

    all x ∈ K and all A ∈ Alg L. Based on this result, a complete characterization of linear n-Lie derivations on Alg L is obtained. Keywords. J -subspace lattice algebras; Lie derivations; Lie n-derivations; derivations. 2010 Mathematics Subject Classification. 47B47, 47L35. 1. Introduction. Let A be an algebra. Recall that a linear ...

  14. A difference quotient-numerical integration method for solving radiative transfer problems

    International Nuclear Information System (INIS)

    Ding Peizhu

    1992-01-01

    A difference quotient-numerical integration method is adopted to solve radiative transfer problems in an anisotropic scattering slab medium. By using the method, the radiative transfer problem is separated into a system of linear algebraic equations and the coefficient matrix of the system is a band matrix, so the method is very simple to evaluate on computer and to deduce formulae and easy to master for experimentalists. An example is evaluated and it is shown that the method is precise

  15. Un-equivalency theorem between deformed and undeformed Heisenberg-Weyl's algebras

    International Nuclear Information System (INIS)

    Zhang Jianzu

    2006-01-01

    Two fundamental issues about the relation between the deformed Heisenberg-Weyl algebra in noncommutative space and the undeformed one in commutative space are elucidated. First the un-equivalency theorem between two algebras is proved: the deformed algebra related to the undeformed one by a non-orthogonal similarity transformation is explored; furthermore, non-existence of a unitary similarity transformation which transforms the deformed algebra to the undeformed one is demonstrated. Secondly the uniqueness of realizing the deformed phase space variables via the undeformed ones is elucidated: both the deformed Heisenberg-Weyl algebra and the deformed bosonic algebra should be maintained under a linear transformation between two sets of phase space variables which fixes that such a linear transformation is unique. Elucidation of this un-equivalency theorem has basic meaning both in theory and experiment

  16. Algebraic and stochastic coding theory

    CERN Document Server

    Kythe, Dave K

    2012-01-01

    Using a simple yet rigorous approach, Algebraic and Stochastic Coding Theory makes the subject of coding theory easy to understand for readers with a thorough knowledge of digital arithmetic, Boolean and modern algebra, and probability theory. It explains the underlying principles of coding theory and offers a clear, detailed description of each code. More advanced readers will appreciate its coverage of recent developments in coding theory and stochastic processes. After a brief review of coding history and Boolean algebra, the book introduces linear codes, including Hamming and Golay codes.

  17. Error-Detecting Identification Codes for Algebra Students.

    Science.gov (United States)

    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)

  18. A new neural network model for solving random interval linear programming problems.

    Science.gov (United States)

    Arjmandzadeh, Ziba; Safi, Mohammadreza; Nazemi, Alireza

    2017-05-01

    This paper presents a neural network model for solving random interval linear programming problems. The original problem involving random interval variable coefficients is first transformed into an equivalent convex second order cone programming problem. A neural network model is then constructed for solving the obtained convex second order cone problem. Employing Lyapunov function approach, it is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact satisfactory solution of the original problem. Several illustrative examples are solved in support of this technique. Copyright © 2017 Elsevier Ltd. All rights reserved.

  19. Matrix algebra theory, computations and applications in statistics

    CERN Document Server

    Gentle, James E

    2017-01-01

    This textbook for graduate and advanced undergraduate students presents the theory of matrix algebra for statistical applications, explores various types of matrices encountered in statistics, and covers numerical linear algebra. Matrix algebra is one of the most important areas of mathematics in data science and in statistical theory, and the second edition of this very popular textbook provides essential updates and comprehensive coverage on critical topics in mathematics in data science and in statistical theory. Part I offers a self-contained description of relevant aspects of the theory of matrix algebra for applications in statistics. It begins with fundamental concepts of vectors and vector spaces; covers basic algebraic properties of matrices and analytic properties of vectors and matrices in multivariate calculus; and concludes with a discussion on operations on matrices in solutions of linear systems and in eigenanalysis. Part II considers various types of matrices encountered in statistics, such as...

  20. Parallel algorithms for network routing problems and recurrences

    International Nuclear Information System (INIS)

    Wisniewski, J.A.; Sameh, A.H.

    1982-01-01

    In this paper, we consider the parallel solution of recurrences, and linear systems in the regular algebra of Carre. These problems are equivalent to solving the shortest path problem in graph theory, and they also arise in the analysis of Fortran programs. Our methods for solving linear systems in the regular algebra are analogues of well-known methods for solving systems of linear algebraic equations. A parallel version of Dijkstra's method, which has no linear algebraic analogue, is presented. Considerations for choosing an algorithm when the problem is large and sparse are also discussed