WorldWideScience

Sample records for linear algebra algorithms

  1. 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)

  2. 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

  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. 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...

  5. 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.

  6. 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.

  7. 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

  8. 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...

  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. 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.

  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. 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

  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. 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.

  15. 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...

  16. 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.

  17. 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

  18. 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...

  19. 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.

  20. Robust Algebraic Multilevel Methods and Algorithms

    CERN Document Server

    Kraus, Johannes

    2009-01-01

    This book deals with algorithms for the solution of linear systems of algebraic equations with large-scale sparse matrices, with a focus on problems that are obtained after discretization of partial differential equations using finite element methods. Provides a systematic presentation of the recent advances in robust algebraic multilevel methods. Can be used for advanced courses on the topic.

  1. 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...

  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. 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

  4. Algorithms in Algebraic Geometry

    CERN Document Server

    Dickenstein, Alicia; Sommese, Andrew J

    2008-01-01

    In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric computation. Some of these algorithms were originally designed for abstract algebraic geometry, but now are of interest for use in applications and some of these algorithms were originally designed for applications, but now are of interest for use in abstract algebraic geometry. The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its

  5. 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

  6. 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...

  7. 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.

  8. 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.

  9. 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...

  10. 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

  11. 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

  12. 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...

  13. 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.

  14. 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.

  15. 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,

  16. 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:...

  17. 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 ...

  18. 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

  19. 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)

  20. 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

  1. 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...

  2. 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.

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

    KAUST Repository

    Ltaief, Hatem

    2011-08-31

    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-grained task parallelism that recasts the computation to operate on submatrices called tiles. In this way tile algorithms are formed. We show results from the power profiling of the most common routines, which permits us to clearly identify the different phases of the computations. This allows us to isolate the bottlenecks in terms of energy efficiency. Our results show that PLASMA surpasses LAPACK not only in terms of performance but also in terms of energy efficiency. © 2011 Springer-Verlag.

  4. 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.

  5. 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.

  6. 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.

  7. 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...

  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. An algorithm to construct the basic algebra of a skew group algebra

    NARCIS (Netherlands)

    Horobeţ, E.

    2016-01-01

    We give an algorithm for the computation of the basic algebra Morita equivalent to a skew group algebra of a path algebra by obtaining formulas for the number of vertices and arrows of the new quiver Qb. We apply this algorithm to compute the basic algebra corresponding to all simple quaternion

  10. 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.

  11. 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.

  12. 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.)

  13. 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

  14. 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.

  15. 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

  16. 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

  17. 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.

  18. 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…

  19. 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.

  20. Algebraic dynamics algorithm: Numerical comparison with Runge-Kutta algorithm and symplectic geometric algorithm

    Institute of Scientific and Technical Information of China (English)

    WANG ShunJin; ZHANG Hua

    2007-01-01

    Based on the exact analytical solution of ordinary differential equations,a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm.A detailed numerical comparison is presented with Runge-Kutta algorithm and symplectic geometric algorithm for 12 test models.The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision,and it can solve the problem of algorithm-induced dissipation for the Runge-Kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm.

  1. Algebraic dynamics algorithm:Numerical comparison with Runge-Kutta algorithm and symplectic geometric algorithm

    Institute of Scientific and Technical Information of China (English)

    2007-01-01

    Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numerical comparison is presented with Runge-Kutta algorithm and symplectic geometric algorithm for 12 test models. The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision, and it can solve the problem of algorithm-induced dissipation for the Runge-Kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm.

  2. 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)

  3. 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…

  4. 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)

  5. 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 ...

  6. 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.)

  7. Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear ordinary differential equations

    Institute of Scientific and Technical Information of China (English)

    WANG; Shunjin; ZHANG; Hua

    2006-01-01

    The problem of preserving fidelity in numerical computation of nonlinear ordinary differential equations is studied in terms of preserving local differential structure and approximating global integration structure of the dynamical system.The ordinary differential equations are lifted to the corresponding partial differential equations in the framework of algebraic dynamics,and a new algorithm-algebraic dynamics algorithm is proposed based on the exact analytical solutions of the ordinary differential equations by the algebraic dynamics method.In the new algorithm,the time evolution of the ordinary differential system is described locally by the time translation operator and globally by the time evolution operator.The exact analytical piece-like solution of the ordinary differential equations is expressd in terms of Taylor series with a local convergent radius,and its finite order truncation leads to the new numerical algorithm with a controllable precision better than Runge Kutta Algorithm and Symplectic Geometric Algorithm.

  8. 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...

  9. 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.

  10. 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…

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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.

  17. 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...

  18. 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.

  19. 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

  20. 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.

  1. Algebraic Algorithm Design and Local Search

    National Research Council Canada - National Science Library

    Graham, Robert

    1996-01-01

    .... Algebraic techniques have been applied successfully to algorithm synthesis by the use of algorithm theories and design tactics, an approach pioneered in the Kestrel Interactive Development System (KIDS...

  2. 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.)

  3. 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....

  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. 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

  6. 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.

  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. 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.

  9. 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.

  10. 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.

  11. 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.

  12. 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

  13. 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…

  14. El desempeño del docente en el proceso de desarrollo de habilidades de trabajo con algoritmos en la disciplina Álgebra Lineal / Teachers' performance and the process of developing skills to work with algorithms in Linear Algebra

    Directory of Open Access Journals (Sweden)

    Ivonne Burguet Lago

    2018-05-01

    Full Text Available ABSTRACT The paper describes a proposal of professional pedagogical performance tests to assess teachers’ role in the process of developing the skill of working with algorithms in Linear Algebra. It aims at devising a testing tool to assess teachers’ performance in the skill-developing process. This tool is a finding of Cuba theory of Advanced Education, systematically used in recent years. The findings include the test design and the illustration of its use in a sample of 22 Linear Algebra teachers during the first term of the 2017-2018 academic year at Informatics Sciences Engineering major. Keywords: ABSTRACT The paper describes a proposal of professional pedagogical performance tests to assess teachers’ role in the process of developing the skill of working with algorithms in Linear Algebra. It aims at devising a testing tool to assess teachers’ performance in the skill-developing process. This tool is a finding of Cuba theory of Advanced Education, systematically used in recent years. The findings include the test design and the illustration of its use in a sample of 22 Linear Algebra teachers during the first term of the 2017-2018 academic year at Informatics Sciences Engineering major.

  15. A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric

    DEFF Research Database (Denmark)

    Zimmermann, Ralf

    2017-01-01

    We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the optimization-based approach known from the literature, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm...... converges locally and exhibits a linear rate of convergence....

  16. A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric

    OpenAIRE

    Zimmermann, Ralf

    2016-01-01

    We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the optimization-based approach known from the literature, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm converges locally and exhibits a linear rate of convergence.

  17. 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…

  18. 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)

  19. 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.    ...

  20. 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.

  1. 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…

  2. 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

  3. Linear programming algorithms and applications

    CERN Document Server

    Vajda, S

    1981-01-01

    This text is based on a course of about 16 hours lectures to students of mathematics, statistics, and/or operational research. It is intended to introduce readers to the very wide range of applicability of linear programming, covering problems of manage­ ment, administration, transportation and a number of other uses which are mentioned in their context. The emphasis is on numerical algorithms, which are illustrated by examples of such modest size that the solutions can be obtained using pen and paper. It is clear that these methods, if applied to larger problems, can also be carried out on automatic (electronic) computers. Commercially available computer packages are, in fact, mainly based on algorithms explained in this book. The author is convinced that the user of these algorithms ought to be knowledgeable about the underlying theory. Therefore this volume is not merely addressed to the practitioner, but also to the mathematician who is interested in relatively new developments in algebraic theory and in...

  4. 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.

  5. 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…

  6. 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.

  7. 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

  8. 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.

  9. 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)

  10. 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…

  11. 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.

  12. 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.)

  13. 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)

  14. 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.

  15. 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...

  16. 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

  17. 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...

  18. 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

  19. 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…

  20. 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…

  1. 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.

  2. 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

  3. 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.

  4. 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…

  5. Performance Analysis of a Decoding Algorithm for Algebraic Geometry Codes

    DEFF Research Database (Denmark)

    Jensen, Helge Elbrønd; Nielsen, Rasmus Refslund; Høholdt, Tom

    1998-01-01

    We analyse the known decoding algorithms for algebraic geometry codes in the case where the number of errors is greater than or equal to [(dFR-1)/2]+1, where dFR is the Feng-Rao distance......We analyse the known decoding algorithms for algebraic geometry codes in the case where the number of errors is greater than or equal to [(dFR-1)/2]+1, where dFR is the Feng-Rao distance...

  6. 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.…

  7. Quantum complexity of graph and algebraic problems

    International Nuclear Information System (INIS)

    Doern, Sebastian

    2008-01-01

    This thesis is organized as follows: In Chapter 2 we give some basic notations, definitions and facts from linear algebra, graph theory, group theory and quantum computation. In Chapter 3 we describe three important methods for the construction of quantum algorithms. We present the quantum search algorithm by Grover, the quantum amplitude amplification and the quantum walk search technique by Magniez et al. These three tools are the basis for the development of our new quantum algorithms for graph and algebra problems. In Chapter 4 we present two tools for proving quantum query lower bounds. We present the quantum adversary method by Ambainis and the polynomial method introduced by Beals et al. The quantum adversary tool is very useful to prove good lower bounds for many graph and algebra problems. The part of the thesis containing the original results is organized in two parts. In the first part we consider the graph problems. In Chapter 5 we give a short summary of known quantum graph algorithms. In Chapter 6 to 8 we study the complexity of our new algorithms for matching problems, graph traversal and independent set problems on quantum computers. In the second part of our thesis we present new quantum algorithms for algebraic problems. In Chapter 9 to 10 we consider group testing problems and prove quantum complexity bounds for important problems from linear algebra. (orig.)

  8. Quantum complexity of graph and algebraic problems

    Energy Technology Data Exchange (ETDEWEB)

    Doern, Sebastian

    2008-02-04

    This thesis is organized as follows: In Chapter 2 we give some basic notations, definitions and facts from linear algebra, graph theory, group theory and quantum computation. In Chapter 3 we describe three important methods for the construction of quantum algorithms. We present the quantum search algorithm by Grover, the quantum amplitude amplification and the quantum walk search technique by Magniez et al. These three tools are the basis for the development of our new quantum algorithms for graph and algebra problems. In Chapter 4 we present two tools for proving quantum query lower bounds. We present the quantum adversary method by Ambainis and the polynomial method introduced by Beals et al. The quantum adversary tool is very useful to prove good lower bounds for many graph and algebra problems. The part of the thesis containing the original results is organized in two parts. In the first part we consider the graph problems. In Chapter 5 we give a short summary of known quantum graph algorithms. In Chapter 6 to 8 we study the complexity of our new algorithms for matching problems, graph traversal and independent set problems on quantum computers. In the second part of our thesis we present new quantum algorithms for algebraic problems. In Chapter 9 to 10 we consider group testing problems and prove quantum complexity bounds for important problems from linear algebra. (orig.)

  9. 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.

  10. 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

  11. 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

  12. 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.

  13. Algorithmic and experimental methods in algebra, geometry, and number theory

    CERN Document Server

    Decker, Wolfram; Malle, Gunter

    2017-01-01

    This book presents state-of-the-art research and survey articles that highlight work done within the Priority Program SPP 1489 “Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory”, which was established and generously supported by the German Research Foundation (DFG) from 2010 to 2016. The goal of the program was to substantially advance algorithmic and experimental methods in the aforementioned disciplines, to combine the different methods where necessary, and to apply them to central questions in theory and practice. Of particular concern was the further development of freely available open source computer algebra systems and their interaction in order to create powerful new computational tools that transcend the boundaries of the individual disciplines involved.  The book covers a broad range of topics addressing the design and theoretical foundations, implementation and the successful application of algebraic algorithms in order to solve mathematical research problems. It off...

  14. 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...

  15. 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

  16. 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.

  17. 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.

  18. 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.

  19. 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.

  20. 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…

  1. 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.)

  2. An algorithm for analysis of the structure of finitely presented Lie algebras

    Directory of Open Access Journals (Sweden)

    Vladimir P. Gerdt

    1997-12-01

    Full Text Available We consider the following problem: what is the most general Lie algebra satisfying a given set of Lie polynomial equations? The presentation of Lie algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance, covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are constructionof prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie algebras arising in different physical models. The finite presentations also indicate a way to q-quantize Lie algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason, in practice one needs to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie algebra and its commutator table, and its implementation in the C language. Some computer results illustrating our algorithmand its actual implementation are also presented.

  3. 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.)

  4. 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.

  5. Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes

    Science.gov (United States)

    Lin, Shu

    1998-01-01

    A code trellis is a graphical representation of a code, block or convolutional, in which every path represents a codeword (or a code sequence for a convolutional code). This representation makes it possible to implement Maximum Likelihood Decoding (MLD) of a code with reduced decoding complexity. The most well known trellis-based MLD algorithm is the Viterbi algorithm. The trellis representation was first introduced and used for convolutional codes [23]. This representation, together with the Viterbi decoding algorithm, has resulted in a wide range of applications of convolutional codes for error control in digital communications over the last two decades. There are two major reasons for this inactive period of research in this area. First, most coding theorists at that time believed that block codes did not have simple trellis structure like convolutional codes and maximum likelihood decoding of linear block codes using the Viterbi algorithm was practically impossible, except for very short block codes. Second, since almost all of the linear block codes are constructed algebraically or based on finite geometries, it was the belief of many coding theorists that algebraic decoding was the only way to decode these codes. These two reasons seriously hindered the development of efficient soft-decision decoding methods for linear block codes and their applications to error control in digital communications. This led to a general belief that block codes are inferior to convolutional codes and hence, that they were not useful. Chapter 2 gives a brief review of linear block codes. The goal is to provide the essential background material for the development of trellis structure and trellis-based decoding algorithms for linear block codes in the later chapters. Chapters 3 through 6 present the fundamental concepts, finite-state machine model, state space formulation, basic structural properties, state labeling, construction procedures, complexity, minimality, and

  6. 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

  7. 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

  8. An Algorithm for Isolating the Real Solutions of Piecewise Algebraic Curves

    Directory of Open Access Journals (Sweden)

    Jinming Wu

    2011-01-01

    to compute the real solutions of two piecewise algebraic curves. It is primarily based on the Krawczyk-Moore iterative algorithm and good initial iterative interval searching algorithm. The proposed algorithm is relatively easy to implement.

  9. 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.

  10. Algorithmic algebraic geometry and flux vacua

    International Nuclear Information System (INIS)

    Gray, James; He Yanghui; Lukas, Andre

    2006-01-01

    We develop a new and efficient method to systematically analyse four dimensional effective supergravities which descend from flux compactifications. The issue of finding vacua of such systems, both supersymmetric and non-supersymmetric, is mapped into a problem in computational algebraic geometry. Using recent developments in computer algebra, the problem can then be rapidly dealt with in a completely algorithmic fashion. Two main results are (1) a procedure for calculating constraints which the flux parameters must satisfy in these models if any given type of vacuum is to exist; (2) a stepwise process for finding all of the isolated vacua of such systems and their physical properties. We illustrate our discussion with several concrete examples, some of which have eluded conventional methods so far

  11. 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

  12. 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.

  13. 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…

  14. 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

  15. 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.

  16. 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 ...

  17. 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...

  18. 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...

  19. 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.

  20. 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…

  1. 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 ...

  2. Elements of algebraic coding systems

    CERN Document Server

    Cardoso da Rocha, Jr, Valdemar

    2014-01-01

    Elements of Algebraic Coding Systems is an introductory text to algebraic coding theory. In the first chapter, you'll gain inside knowledge of coding fundamentals, which is essential for a deeper understanding of state-of-the-art coding systems. This book is a quick reference for those who are unfamiliar with this topic, as well as for use with specific applications such as cryptography and communication. Linear error-correcting block codes through elementary principles span eleven chapters of the text. Cyclic codes, some finite field algebra, Goppa codes, algebraic decoding algorithms, and applications in public-key cryptography and secret-key cryptography are discussed, including problems and solutions at the end of each chapter. Three appendices cover the Gilbert bound and some related derivations, a derivation of the Mac- Williams' identities based on the probability of undetected error, and two important tools for algebraic decoding-namely, the finite field Fourier transform and the Euclidean algorithm f...

  3. 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…

  4. A differential algebraic integration algorithm for symplectic mappings in systems with three-dimensional magnetic field

    International Nuclear Information System (INIS)

    Chang, P.; Lee, S.Y.; Yan, Y.T.

    2006-01-01

    A differential algebraic integration algorithm is developed for symplectic mapping through a three-dimensional (3-D) magnetic field. The self-consistent reference orbit in phase space is obtained by making a canonical transformation to eliminate the linear part of the Hamiltonian. Transfer maps from the entrance to the exit of any 3-D magnetic field are then obtained through slice-by-slice symplectic integration. The particle phase-space coordinates are advanced by using the integrable polynomial procedure. This algorithm is a powerful tool to attain nonlinear maps for insertion devices in synchrotron light source or complicated magnetic field in the interaction region in high energy colliders

  5. A Differential Algebraic Integration Algorithm for Symplectic Mappings in Systems with Three-Dimensional Magnetic Field

    International Nuclear Information System (INIS)

    Chang, P

    2004-01-01

    A differential algebraic integration algorithm is developed for symplectic mapping through a three-dimensional (3-D) magnetic field. The self-consistent reference orbit in phase space is obtained by making a canonical transformation to eliminate the linear part of the Hamiltonian. Transfer maps from the entrance to the exit of any 3-D magnetic field are then obtained through slice-by-slice symplectic integration. The particle phase-space coordinates are advanced by using the integrable polynomial procedure. This algorithm is a powerful tool to attain nonlinear maps for insertion devices in synchrotron light source or complicated magnetic field in the interaction region in high energy colliders

  6. Performance analysis of a decoding algorithm for algebraic-geometry codes

    DEFF Research Database (Denmark)

    Høholdt, Tom; Jensen, Helge Elbrønd; Nielsen, Rasmus Refslund

    1999-01-01

    The fast decoding algorithm for one point algebraic-geometry codes of Sakata, Elbrond Jensen, and Hoholdt corrects all error patterns of weight less than half the Feng-Rao minimum distance. In this correspondence we analyze the performance of the algorithm for heavier error patterns. It turns out...

  7. 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

  8. 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

  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. 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. Searching dependency between algebraic equations: An algorithm applied to automated reasoning

    International Nuclear Information System (INIS)

    Yang Lu; Zhang Jingzhong

    1990-01-01

    An efficient computer algorithm is given to decide how many branches of the solution to a system of algebraic also solve another equation. As one of the applications, this can be used in practice to verify a conjecture with hypotheses and conclusion expressed by algebraic equations, despite the variety of reducible or irreducible. (author). 10 refs

  13. New matrix bounds and iterative algorithms for the discrete coupled algebraic Riccati equation

    Science.gov (United States)

    Liu, Jianzhou; Wang, Li; Zhang, Juan

    2017-11-01

    The discrete coupled algebraic Riccati equation (DCARE) has wide applications in control theory and linear system. In general, for the DCARE, one discusses every term of the coupled term, respectively. In this paper, we consider the coupled term as a whole, which is different from the recent results. When applying eigenvalue inequalities to discuss the coupled term, our method has less error. In terms of the properties of special matrices and eigenvalue inequalities, we propose several upper and lower matrix bounds for the solution of DCARE. Further, we discuss the iterative algorithms for the solution of the DCARE. In the fixed point iterative algorithms, the scope of Lipschitz factor is wider than the recent results. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived results.

  14. 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

  15. 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

  16. MultiAspect Graphs: Algebraic Representation and Algorithms

    Directory of Open Access Journals (Sweden)

    Klaus Wehmuth

    2016-12-01

    Full Text Available We present the algebraic representation and basic algorithms for MultiAspect Graphs (MAGs. A MAG is a structure capable of representing multilayer and time-varying networks, as well as higher-order networks, while also having the property of being isomorphic to a directed graph. In particular, we show that, as a consequence of the properties associated with the MAG structure, a MAG can be represented in matrix form. Moreover, we also show that any possible MAG function (algorithm can be obtained from this matrix-based representation. This is an important theoretical result since it paves the way for adapting well-known graph algorithms for application in MAGs. We present a set of basic MAG algorithms, constructed from well-known graph algorithms, such as degree computing, Breadth First Search (BFS, and Depth First Search (DFS. These algorithms adapted to the MAG context can be used as primitives for building other more sophisticated MAG algorithms. Therefore, such examples can be seen as guidelines on how to properly derive MAG algorithms from basic algorithms on directed graphs. We also make available Python implementations of all the algorithms presented in this paper.

  17. Algorithmic Algebraic Combinatorics and Gröbner Bases

    CERN Document Server

    Klin, Mikhail; Jurisic, Aleksandar

    2009-01-01

    This collection of tutorial and research papers introduces readers to diverse areas of modern pure and applied algebraic combinatorics and finite geometries with a special emphasis on algorithmic aspects and the use of the theory of Grobner bases. Topics covered include coherent configurations, association schemes, permutation groups, Latin squares, the Jacobian conjecture, mathematical chemistry, extremal combinatorics, coding theory, designs, etc. Special attention is paid to the description of innovative practical algorithms and their implementation in software packages such as GAP and MAGM

  18. 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

  19. 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

  20. 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)

  1. 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…

  2. 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

  3. 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...

  4. 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.

  5. 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

  6. 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.…

  7. An algebraic approach to the scattering equations

    Energy Technology Data Exchange (ETDEWEB)

    Huang, Rijun; Rao, Junjie [Zhejiang Institute of Modern Physics, Zhejiang University,Hangzhou, 310027 (China); Feng, Bo [Zhejiang Institute of Modern Physics, Zhejiang University,Hangzhou, 310027 (China); Center of Mathematical Science, Zhejiang University,Hangzhou, 310027 (China); He, Yang-Hui [School of Physics, NanKai University,Tianjin, 300071 (China); Department of Mathematics, City University,London, EC1V 0HB (United Kingdom); Merton College, University of Oxford,Oxford, OX14JD (United Kingdom)

    2015-12-10

    We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

  8. An algebraic approach to the scattering equations

    International Nuclear Information System (INIS)

    Huang, Rijun; Rao, Junjie; Feng, Bo; He, Yang-Hui

    2015-01-01

    We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

  9. 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.

  10. Linearization Method and Linear Complexity

    Science.gov (United States)

    Tanaka, Hidema

    We focus on the relationship between the linearization method and linear complexity and show that the linearization method is another effective technique for calculating linear complexity. We analyze its effectiveness by comparing with the logic circuit method. We compare the relevant conditions and necessary computational cost with those of the Berlekamp-Massey algorithm and the Games-Chan algorithm. The significant property of a linearization method is that it needs no output sequence from a pseudo-random number generator (PRNG) because it calculates linear complexity using the algebraic expression of its algorithm. When a PRNG has n [bit] stages (registers or internal states), the necessary computational cost is smaller than O(2n). On the other hand, the Berlekamp-Massey algorithm needs O(N2) where N(≅2n) denotes period. Since existing methods calculate using the output sequence, an initial value of PRNG influences a resultant value of linear complexity. Therefore, a linear complexity is generally given as an estimate value. On the other hand, a linearization method calculates from an algorithm of PRNG, it can determine the lower bound of linear complexity.

  11. 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…

  12. 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.

  13. 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.

  14. 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.

  15. 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…

  16. 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…

  17. 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.

  18. 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

  19. An Improved Algorithm for Generating Database Transactions from Relational Algebra Specifications

    Directory of Open Access Journals (Sweden)

    Daniel J. Dougherty

    2010-03-01

    Full Text Available Alloy is a lightweight modeling formalism based on relational algebra. In prior work with Fisler, Giannakopoulos, Krishnamurthi, and Yoo, we have presented a tool, Alchemy, that compiles Alloy specifications into implementations that execute against persistent databases. The foundation of Alchemy is an algorithm for rewriting relational algebra formulas into code for database transactions. In this paper we report on recent progress in improving the robustness and efficiency of this transformation.

  20. 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…

  1. 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

  2. Additional operations in algebra of structural numbers for control algorithm development

    Directory of Open Access Journals (Sweden)

    Morhun A.V.

    2016-12-01

    Full Text Available The structural numbers and the algebra of the structural numbers due to the simplicity of representation, flexibility and current algebraic operations are the powerful tool for a wide range of applications. In autonomous power supply systems and systems with distributed generation (Micro Grid mathematical apparatus of structural numbers can be effectively used for the calculation of the parameters of the operating modes of consumption of electric energy. The purpose of the article is the representation of the additional algebra of structural numbers. The standard algebra was proposed to be extended by the additional operations and modification current in order to expand the scope of their use, namely to construct a flexible, adaptive algorithms of control systems. It is achieved due to the possibility to consider each individual component of the system with its parameters and provide easy management of entire system and each individual component. Thus, structural numbers and extended algebra are the perspective line of research and further studying is required.

  3. 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).

  4. Commutative algebra constructive methods finite projective modules

    CERN Document Server

    Lombardi, Henri

    2015-01-01

    Translated from the popular French edition, this book offers a detailed introduction to various basic concepts, methods, principles, and results of commutative algebra. It takes a constructive viewpoint in commutative algebra and studies algorithmic approaches alongside several abstract classical theories. Indeed, it revisits these traditional topics with a new and simplifying manner, making the subject both accessible and innovative. The algorithmic aspects of such naturally abstract topics as Galois theory, Dedekind rings, Prüfer rings, finitely generated projective modules, dimension theory of commutative rings, and others in the current treatise, are all analysed in the spirit of the great developers of constructive algebra in the nineteenth century. This updated and revised edition contains over 350 well-arranged exercises, together with their helpful hints for solution. A basic knowledge of linear algebra, group theory, elementary number theory as well as the fundamentals of ring and module theory is r...

  5. 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.

  6. ALGEBRA: ALgorithm for the heterogeneous dosimetry based on GEANT4 for BRAchytherapy.

    Science.gov (United States)

    Afsharpour, H; Landry, G; D'Amours, M; Enger, S; Reniers, B; Poon, E; Carrier, J-F; Verhaegen, F; Beaulieu, L

    2012-06-07

    Task group 43 (TG43)-based dosimetry algorithms are efficient for brachytherapy dose calculation in water. However, human tissues have chemical compositions and densities different than water. Moreover, the mutual shielding effect of seeds on each other (interseed attenuation) is neglected in the TG43-based dosimetry platforms. The scientific community has expressed the need for an accurate dosimetry platform in brachytherapy. The purpose of this paper is to present ALGEBRA, a Monte Carlo platform for dosimetry in brachytherapy which is sufficiently fast and accurate for clinical and research purposes. ALGEBRA is based on the GEANT4 Monte Carlo code and is capable of handling the DICOM RT standard to recreate a virtual model of the treated site. Here, the performance of ALGEBRA is presented for the special case of LDR brachytherapy in permanent prostate and breast seed implants. However, the algorithm is also capable of handling other treatments such as HDR brachytherapy.

  7. 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.

  8. 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…

  9. 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.

  10. 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)

  11. 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.

  12. 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.

  13. Modular Regularization Algorithms

    DEFF Research Database (Denmark)

    Jacobsen, Michael

    2004-01-01

    The class of linear ill-posed problems is introduced along with a range of standard numerical tools and basic concepts from linear algebra, statistics and optimization. Known algorithms for solving linear inverse ill-posed problems are analyzed to determine how they can be decomposed into indepen...

  14. 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

  15. Application of Symplectic Algebraic Dynamics Algorithm to Circular Restricted Three-Body Problem

    International Nuclear Information System (INIS)

    Wei-Tao, Lu; Hua, Zhang; Shun-Jin, Wang

    2008-01-01

    Symplectic algebraic dynamics algorithm (SADA) for ordinary differential equations is applied to solve numerically the circular restricted three-body problem (CR3BP) in dynamical astronomy for both stable motion and chaotic motion. The result is compared with those of Runge–Kutta algorithm and symplectic algorithm under the fourth order, which shows that SADA has higher accuracy than the others in the long-term calculations of the CR3BP. (general)

  16. 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.

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

    Science.gov (United States)

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

    2017-05-01

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

  18. A polynomial time algorithm for checking regularity of totally normed process algebra

    NARCIS (Netherlands)

    Yang, F.; Huang, H.

    2015-01-01

    A polynomial algorithm for the regularity problem of weak and branching bisimilarity on totally normed process algebra (PA) processes is given. Its time complexity is O(n 3 +mn) O(n3+mn), where n is the number of transition rules and m is the maximal length of the rules. The algorithm works for

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

    Science.gov (United States)

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

    2017-11-01

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

  20. 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.

  1. 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.

  2. 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...

  3. 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...

  4. 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

  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. 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.

  9. 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.

  10. 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.

  11. Algorithms and Methods for High-Performance Model Predictive Control

    DEFF Research Database (Denmark)

    Frison, Gianluca

    routines employed in the numerical tests. The main focus of this thesis is on linear MPC problems. In this thesis, both the algorithms and their implementation are equally important. About the implementation, a novel implementation strategy for the dense linear algebra routines in embedded optimization...... is proposed, aiming at improving the computational performance in case of small matrices. About the algorithms, they are built on top of the proposed linear algebra, and they are tailored to exploit the high-level structure of the MPC problems, with special care on reducing the computational complexity....

  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. A Global algorithm for linear radiosity

    OpenAIRE

    Sbert Cassasayas, Mateu; Pueyo Sánchez, Xavier

    1993-01-01

    A linear algorithm for radiosity is presented, linear both in time and storage. The new algorithm is based on previous work by the authors and on the well known algorithms for progressive radiosity and Monte Carlo particle transport.

  16. Algorithm for solving polynomial algebraic Riccati equations and its application

    Czech Academy of Sciences Publication Activity Database

    Augusta, Petr; Augustová, Petra

    2012-01-01

    Roč. 1, č. 4 (2012), s. 237-242 ISSN 2223-7038 R&D Projects: GA ČR GPP103/12/P494 Institutional support: RVO:67985556 Keywords : Numerical algorithms * algebraic Riccati equation * spatially distributed systems * optimal control Subject RIV: BC - Control Systems Theory http://lib.physcon.ru/doc?id=8b4876d6a57d

  17. GENERAL: Application of Symplectic Algebraic Dynamics Algorithm to Circular Restricted Three-Body Problem

    Science.gov (United States)

    Lu, Wei-Tao; Zhang, Hua; Wang, Shun-Jin

    2008-07-01

    Symplectic algebraic dynamics algorithm (SADA) for ordinary differential equations is applied to solve numerically the circular restricted three-body problem (CR3BP) in dynamical astronomy for both stable motion and chaotic motion. The result is compared with those of Runge-Kutta algorithm and symplectic algorithm under the fourth order, which shows that SADA has higher accuracy than the others in the long-term calculations of the CR3BP.

  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. 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…

  20. 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.

  1. 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…

  2. 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.

  3. 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…

  4. 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.

  5. 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…

  6. 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…

  7. 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.

  8. 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.…

  9. 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)

  10. 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

  11. 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.)…

  12. Algorithmic Verification of Linearizability for Ordinary Differential Equations

    KAUST Repository

    Lyakhov, Dmitry A.

    2017-07-19

    For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a linear one by a point transformation of the dependent and independent variables. The first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra. The second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. The implementation of both algorithms is discussed and their application is illustrated using several examples.

  13. 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.

  14. [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.

  15. 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,…

  16. 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.

  17. Linear feature detection algorithm for astronomical surveys - I. Algorithm description

    Science.gov (United States)

    Bektešević, Dino; Vinković, Dejan

    2017-11-01

    Computer vision algorithms are powerful tools in astronomical image analyses, especially when automation of object detection and extraction is required. Modern object detection algorithms in astronomy are oriented towards detection of stars and galaxies, ignoring completely the detection of existing linear features. With the emergence of wide-field sky surveys, linear features attract scientific interest as possible trails of fast flybys of near-Earth asteroids and meteors. In this work, we describe a new linear feature detection algorithm designed specifically for implementation in big data astronomy. The algorithm combines a series of algorithmic steps that first remove other objects (stars and galaxies) from the image and then enhance the line to enable more efficient line detection with the Hough algorithm. The rate of false positives is greatly reduced thanks to a step that replaces possible line segments with rectangles and then compares lines fitted to the rectangles with the lines obtained directly from the image. The speed of the algorithm and its applicability in astronomical surveys are also discussed.

  18. 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

  19. Algebraic Verification Method for SEREs Properties via Groebner Bases Approaches

    Directory of Open Access Journals (Sweden)

    Ning Zhou

    2013-01-01

    Full Text Available This work presents an efficient solution using computer algebra system to perform linear temporal properties verification for synchronous digital systems. The method is essentially based on both Groebner bases approaches and symbolic simulation. A mechanism for constructing canonical polynomial set based symbolic representations for both circuit descriptions and assertions is studied. We then present a complete checking algorithm framework based on these algebraic representations by using Groebner bases. The computational experience result in this work shows that the algebraic approach is a quite competitive checking method and will be a useful supplement to the existent verification methods based on simulation.

  20. 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.

  1. Generalization of Risch's algorithm to special functions

    International Nuclear Information System (INIS)

    Raab, Clemens G.

    2013-05-01

    Symbolic integration deals with the evaluation of integrals in closed form. We present an overview of Risch's algorithm including recent developments. The algorithms discussed are suited for both indefinite and definite integration. They can also be used to compute linear relations among integrals and to find identities for special functions given by parameter integrals. The aim of this presentation is twofold: to introduce the reader to some basic ideas of differential algebra in the context of integration and to raise awareness in the physics community of computer algebra algorithms for indefinite and definite integration.

  2. 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…

  3. 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

  4. 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.

  5. 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)

  6. 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.

  7. 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)

  8. 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.

  9. 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.

  10. Seismic noise attenuation using an online subspace tracking algorithm

    NARCIS (Netherlands)

    Zhou, Yatong; Li, Shuhua; Zhang, D.; Chen, Yangkang

    2018-01-01

    We propose a new low-rank based noise attenuation method using an efficient algorithm for tracking subspaces from highly corrupted seismic observations. The subspace tracking algorithm requires only basic linear algebraic manipulations. The algorithm is derived by analysing incremental gradient

  11. 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…

  12. Combined genetic algorithm and multiple linear regression (GA-MLR) optimizer: Application to multi-exponential fluorescence decay surface.

    Science.gov (United States)

    Fisz, Jacek J

    2006-12-07

    The optimization approach based on the genetic algorithm (GA) combined with multiple linear regression (MLR) method, is discussed. The GA-MLR optimizer is designed for the nonlinear least-squares problems in which the model functions are linear combinations of nonlinear functions. GA optimizes the nonlinear parameters, and the linear parameters are calculated from MLR. GA-MLR is an intuitive optimization approach and it exploits all advantages of the genetic algorithm technique. This optimization method results from an appropriate combination of two well-known optimization methods. The MLR method is embedded in the GA optimizer and linear and nonlinear model parameters are optimized in parallel. The MLR method is the only one strictly mathematical "tool" involved in GA-MLR. The GA-MLR approach simplifies and accelerates considerably the optimization process because the linear parameters are not the fitted ones. Its properties are exemplified by the analysis of the kinetic biexponential fluorescence decay surface corresponding to a two-excited-state interconversion process. A short discussion of the variable projection (VP) algorithm, designed for the same class of the optimization problems, is presented. VP is a very advanced mathematical formalism that involves the methods of nonlinear functionals, algebra of linear projectors, and the formalism of Fréchet derivatives and pseudo-inverses. Additional explanatory comments are added on the application of recently introduced the GA-NR optimizer to simultaneous recovery of linear and weakly nonlinear parameters occurring in the same optimization problem together with nonlinear parameters. The GA-NR optimizer combines the GA method with the NR method, in which the minimum-value condition for the quadratic approximation to chi(2), obtained from the Taylor series expansion of chi(2), is recovered by means of the Newton-Raphson algorithm. The application of the GA-NR optimizer to model functions which are multi-linear

  13. 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

  14. 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)

  15. 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.

  16. 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...

  17. Computations in finite-dimensional Lie algebras

    Directory of Open Access Journals (Sweden)

    A. M. Cohen

    1997-12-01

    Full Text Available This paper describes progress made in context with the construction of a general library of Lie algebra algorithms, called ELIAS (Eindhoven Lie Algebra System, within the computer algebra package GAP. A first sketch of the package can be found in Cohen and de Graaf[1]. Since then, in a collaborative effort with G. Ivanyos, the authors have continued to develop algorithms which were implemented in ELIAS by the second author. These activities are part of a bigger project, called ACELA and financed by STW, the Dutch Technology Foundation, which aims at an interactive book on Lie algebras (cf. Cohen and Meertens [2]. This paper gives a global description of the main ways in which to present Lie algebras on a computer. We focus on the transition from a Lie algebra abstractly given by an array of structure constants to a Lie algebra presented as a subalgebra of the Lie algebra of n×n matrices. We describe an algorithm typical of the structure analysis of a finite-dimensional Lie algebra: finding a Levi subalgebra of a Lie algebra.

  18. Linear programming using Matlab

    CERN Document Server

    Ploskas, Nikolaos

    2017-01-01

    This book offers a theoretical and computational presentation of a variety of linear programming algorithms and methods with an emphasis on the revised simplex method and its components. A theoretical background and mathematical formulation is included for each algorithm as well as comprehensive numerical examples and corresponding MATLAB® code. The MATLAB® implementations presented in this book  are sophisticated and allow users to find solutions to large-scale benchmark linear programs. Each algorithm is followed by a computational study on benchmark problems that analyze the computational behavior of the presented algorithms. As a solid companion to existing algorithmic-specific literature, this book will be useful to researchers, scientists, mathematical programmers, and students with a basic knowledge of linear algebra and calculus.  The clear presentation enables the reader to understand and utilize all components of simplex-type methods, such as presolve techniques, scaling techniques, pivoting ru...

  19. 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.

  20. Linear scaling of density functional algorithms

    International Nuclear Information System (INIS)

    Stechel, E.B.; Feibelman, P.J.; Williams, A.R.

    1993-01-01

    An efficient density functional algorithm (DFA) that scales linearly with system size will revolutionize electronic structure calculations. Density functional calculations are reliable and accurate in determining many condensed matter and molecular ground-state properties. However, because current DFA's, including methods related to that of Car and Parrinello, scale with the cube of the system size, density functional studies are not routinely applied to large systems. Linear scaling is achieved by constructing functions that are both localized and fully occupied, thereby eliminating the need to calculate global eigenfunctions. It is, however, widely believed that exponential localization requires the existence of an energy gap between the occupied and unoccupied states. Despite this, the authors demonstrate that linear scaling can still be achieved for metals. Using a linear scaling algorithm, they have explicitly constructed localized, almost fully occupied orbitals for the quintessential metallic system, jellium. The algorithm is readily generalizable to any system geometry and Hamiltonian. They will discuss the conceptual issues involved, convergence properties and scaling for their new algorithm

  1. 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.

  2. Recursive Algorithm For Linear Regression

    Science.gov (United States)

    Varanasi, S. V.

    1988-01-01

    Order of model determined easily. Linear-regression algorithhm includes recursive equations for coefficients of model of increased order. Algorithm eliminates duplicative calculations, facilitates search for minimum order of linear-regression model fitting set of data satisfactory.

  3. 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

  4. Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra

    CERN Document Server

    Cox, David A; O'Shea, Donal

    2015-01-01

    This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem, and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D). The book may serve as a first or second course in undergraduate abstract algebra and, with some supplementation perhaps, for beginning graduate level courses in algebraic geom...

  5. 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.

  6. Quantum algorithm for linear regression

    Science.gov (United States)

    Wang, Guoming

    2017-07-01

    We present a quantum algorithm for fitting a linear regression model to a given data set using the least-squares approach. Differently from previous algorithms which yield a quantum state encoding the optimal parameters, our algorithm outputs these numbers in the classical form. So by running it once, one completely determines the fitted model and then can use it to make predictions on new data at little cost. Moreover, our algorithm works in the standard oracle model, and can handle data sets with nonsparse design matrices. It runs in time poly( log2(N ) ,d ,κ ,1 /ɛ ) , where N is the size of the data set, d is the number of adjustable parameters, κ is the condition number of the design matrix, and ɛ is the desired precision in the output. We also show that the polynomial dependence on d and κ is necessary. Thus, our algorithm cannot be significantly improved. Furthermore, we also give a quantum algorithm that estimates the quality of the least-squares fit (without computing its parameters explicitly). This algorithm runs faster than the one for finding this fit, and can be used to check whether the given data set qualifies for linear regression in the first place.

  7. 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.

  8. 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.

  9. 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.

  10. Linear programming mathematics, theory and algorithms

    CERN Document Server

    1996-01-01

    Linear Programming provides an in-depth look at simplex based as well as the more recent interior point techniques for solving linear programming problems. Starting with a review of the mathematical underpinnings of these approaches, the text provides details of the primal and dual simplex methods with the primal-dual, composite, and steepest edge simplex algorithms. This then is followed by a discussion of interior point techniques, including projective and affine potential reduction, primal and dual affine scaling, and path following algorithms. Also covered is the theory and solution of the linear complementarity problem using both the complementary pivot algorithm and interior point routines. A feature of the book is its early and extensive development and use of duality theory. Audience: The book is written for students in the areas of mathematics, economics, engineering and management science, and professionals who need a sound foundation in the important and dynamic discipline of linear programming.

  11. 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.

  12. 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…

  13. 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…

  14. 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 ...

  15. The relation between quantum W algebras and Lie algebras

    International Nuclear Information System (INIS)

    Boer, J. de; Tjin, T.

    1994-01-01

    By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrary sl 2 embeddings we show that a large set W of quantum W algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set W contains many known W algebras such as W N and W 3 (2) . Our formalism yields a completely algorithmic method for calculating the W algebra generators and their operator product expansions, replacing the cumbersome construction of W algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that any W algebra in W can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Therefore any realization of this semisimple affine Lie algebra leads to a realization of the W algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolutions for all algebras in W. Some examples are explicitly worked out. (orig.)

  16. 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.

  17. An Ordering Linear Unification Algorithm

    Institute of Scientific and Technical Information of China (English)

    胡运发

    1989-01-01

    In this paper,we present an ordering linear unification algorithm(OLU).A new idea on substituteion of the binding terms is introduced to the algorithm,which is able to overcome some drawbacks of other algorithms,e.g.,MM algorithm[1],RG1 and RG2 algorithms[2],Particularly,if we use the directed eyclie graphs,the algoritm needs not check the binding order,then the OLU algorithm can also be aplied to the infinite tree data struceture,and a higher efficiency can be expected.The paper focuses upon the discussion of OLU algorithm and a partial order structure with respect to the unification algorithm.This algorithm has been implemented in the GKD-PROLOG/VAX 780 interpreting system.Experimental results have shown that the algorithm is very simple and efficient.

  18. 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…

  19. Observability of linear control systems on Lie groups

    International Nuclear Information System (INIS)

    Ayala, V.; Hacibekiroglu, A.K.

    1995-01-01

    In this paper, we study the observability problem for a linear control system Σ on a Lie group G. The drift vector field of Σ is an infinitesimal automorphism of G and the control vectors are elements in the Lie algebra of G. We establish algebraic conditions to characterize locally and globally observability for Σ. As in the linear case on R n , these conditions are independent of the control vector. We give an algorithm on the co-tangent bundle of G to calculate the equivalence class of the neutral element. (author). 6 refs

  20. 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.

  1. 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)

  2. 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.

  3. 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.

  4. 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

  5. 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...

  6. The Automation of Stochastization Algorithm with Use of SymPy Computer Algebra Library

    Science.gov (United States)

    Demidova, Anastasya; Gevorkyan, Migran; Kulyabov, Dmitry; Korolkova, Anna; Sevastianov, Leonid

    2018-02-01

    SymPy computer algebra library is used for automatic generation of ordinary and stochastic systems of differential equations from the schemes of kinetic interaction. Schemes of this type are used not only in chemical kinetics but also in biological, ecological and technical models. This paper describes the automatic generation algorithm with an emphasis on application details.

  7. 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.

  8. 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.

  9. 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.

  10. 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.

  11. 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.

  12. Arc-Search Infeasible Interior-Point Algorithm for Linear Programming

    OpenAIRE

    Yang, Yaguang

    2014-01-01

    Mehrotra's algorithm has been the most successful infeasible interior-point algorithm for linear programming since 1990. Most popular interior-point software packages for linear programming are based on Mehrotra's algorithm. This paper proposes an alternative algorithm, arc-search infeasible interior-point algorithm. We will demonstrate, by testing Netlib problems and comparing the test results obtained by arc-search infeasible interior-point algorithm and Mehrotra's algorithm, that the propo...

  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. 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…

  16. 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.

  17. 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

  18. Performance of direct and iterative algorithms on an optical systolic processor

    Science.gov (United States)

    Ghosh, A. K.; Casasent, D.; Neuman, C. P.

    1985-11-01

    The frequency-multiplexed optical linear algebra processor (OLAP) is treated in detail with attention to its performance in the solution of systems of linear algebraic equations (LAEs). General guidelines suitable for most OLAPs, including digital-optical processors, are advanced concerning system and component error source models, guidelines for appropriate use of direct and iterative algorithms, the dominant error sources, and the effect of multiple simultaneous error sources. Specific results are advanced on the quantitative performance of both direct and iterative algorithms in the solution of systems of LAEs and in the solution of nonlinear matrix equations. Acoustic attenuation is found to dominate iterative algorithms and detector noise to dominate direct algorithms. The effect of multiple spatial errors is found to be additive. A theoretical expression for the amount of acoustic attenuation allowed is advanced and verified. Simulations and experimental data are included.

  19. Algorithmic mathematics

    CERN Document Server

    Hougardy, Stefan

    2016-01-01

    Algorithms play an increasingly important role in nearly all fields of mathematics. This book allows readers to develop basic mathematical abilities, in particular those concerning the design and analysis of algorithms as well as their implementation. It presents not only fundamental algorithms like the sieve of Eratosthenes, the Euclidean algorithm, sorting algorithms, algorithms on graphs, and Gaussian elimination, but also discusses elementary data structures, basic graph theory, and numerical questions. In addition, it provides an introduction to programming and demonstrates in detail how to implement algorithms in C++. This textbook is suitable for students who are new to the subject and covers a basic mathematical lecture course, complementing traditional courses on analysis and linear algebra. Both authors have given this "Algorithmic Mathematics" course at the University of Bonn several times in recent years.

  20. 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…

  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. 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.

  3. Computer algebra and operators

    Science.gov (United States)

    Fateman, Richard; Grossman, Robert

    1989-01-01

    The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.

  4. Practical algorithms for linear boolean-width

    NARCIS (Netherlands)

    ten Brinke, C.B.; van Houten, F.J.P.; Bodlaender, H.L.

    2015-01-01

    In this paper, we give a number of new exact algorithms and heuristics to compute linear boolean decompositions, and experimentally evaluate these algorithms. The experimental evaluation shows that significant improvements can be made with respect to running time without increasing the width of the

  5. Practical algorithms for linear Boolean-width

    NARCIS (Netherlands)

    ten Brinke, C.B.; van Houten, F.J.P.; Bodlaender, H.L.

    2015-01-01

    In this paper, we give a number of new exact algorithms and heuristics to compute linear boolean decompositions, and experimentally evaluate these algorithms. The experimental evaluation shows that significant improvements can be made with respect to running time without increasing the width of the

  6. 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.)

  7. 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...

  8. 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.

  9. 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…

  10. A Primal-Dual Interior Point-Linear Programming Algorithm for MPC

    DEFF Research Database (Denmark)

    Edlund, Kristian; Sokoler, Leo Emil; Jørgensen, John Bagterp

    2009-01-01

    Constrained optimal control problems for linear systems with linear constraints and an objective function consisting of linear and l1-norm terms can be expressed as linear programs. We develop an efficient primal-dual interior point algorithm for solution of such linear programs. The algorithm...

  11. A Numerical Algorithm to find All Scalar Feedback Nash Equilibria

    NARCIS (Netherlands)

    Engwerda, J.C.

    2013-01-01

    Abstract: In this note we generalize a numerical algorithm presented in [9] to calculate all solutions of the scalar algebraic Riccati equations that play an important role in finding feedback Nash equilibria of the scalar N-player linear affine-quadratic differential game. The algorithm is based on

  12. The Sum-Product Algorithm for Degree-2 Check Nodes and Trapping Sets

    OpenAIRE

    Brevik, John O.; O'Sullivan, Michael E.

    2014-01-01

    The sum-product algorithm for decoding of binary codes is analyzed for bipartite graphs in which the check nodes all have degree $2$. The algorithm simplifies dramatically and may be expressed using linear algebra. Exact results about the convergence of the algorithm are derived and applied to trapping sets.

  13. A New Finite Continuation Algorithm for Linear Programming

    DEFF Research Database (Denmark)

    Madsen, Kaj; Nielsen, Hans Bruun; Pinar, Mustafa

    1996-01-01

    We describe a new finite continuation algorithm for linear programming. The dual of the linear programming problem with unit lower and upper bounds is formulated as an $\\ell_1$ minimization problem augmented with the addition of a linear term. This nondifferentiable problem is approximated...... by a smooth problem. It is shown that the minimizers of the smooth problem define a family of piecewise-linear paths as a function of a smoothing parameter. Based on this property, a finite algorithm that traces these paths to arrive at an optimal solution of the linear program is developed. The smooth...

  14. 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.

  15. 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.

  16. 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.

  17. An implementation of the Heaviside algorithm

    International Nuclear Information System (INIS)

    Dimovski, I.H.; Spiridonova, M.N.

    2011-01-01

    The so-called Heaviside algorithm based on the operational calculus approach is intended for solving initial value problems for linear ordinary differential equations with constant coefficients. We use it in the framework of Mikusinski's operational calculus. A description and implementation of the Heaviside algorithm using a computer algebra system are considered. Special attention is paid to the features making this implementation efficient. Illustrative examples are included

  18. ORACLS- OPTIMAL REGULATOR ALGORITHMS FOR THE CONTROL OF LINEAR SYSTEMS (CDC VERSION)

    Science.gov (United States)

    Armstrong, E. S.

    1994-01-01

    This control theory design package, called Optimal Regulator Algorithms for the Control of Linear Systems (ORACLS), was developed to aid in the design of controllers and optimal filters for systems which can be modeled by linear, time-invariant differential and difference equations. Optimal linear quadratic regulator theory, currently referred to as the Linear-Quadratic-Gaussian (LQG) problem, has become the most widely accepted method of determining optimal control policy. Within this theory, the infinite duration time-invariant problems, which lead to constant gain feedback control laws and constant Kalman-Bucy filter gains for reconstruction of the system state, exhibit high tractability and potential ease of implementation. A variety of new and efficient methods in the field of numerical linear algebra have been combined into the ORACLS program, which provides for the solution to time-invariant continuous or discrete LQG problems. The ORACLS package is particularly attractive to the control system designer because it provides a rigorous tool for dealing with multi-input and multi-output dynamic systems in both continuous and discrete form. The ORACLS programming system is a collection of subroutines which can be used to formulate, manipulate, and solve various LQG design problems. The ORACLS program is constructed in a manner which permits the user to maintain considerable flexibility at each operational state. This flexibility is accomplished by providing primary operations, analysis of linear time-invariant systems, and control synthesis based on LQG methodology. The input-output routines handle the reading and writing of numerical matrices, printing heading information, and accumulating output information. The basic vector-matrix operations include addition, subtraction, multiplication, equation, norm construction, tracing, transposition, scaling, juxtaposition, and construction of null and identity matrices. The analysis routines provide for the following

  19. 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...

  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. Algorithms for non-linear M-estimation

    DEFF Research Database (Denmark)

    Madsen, Kaj; Edlund, O; Ekblom, H

    1997-01-01

    In non-linear regression, the least squares method is most often used. Since this estimator is highly sensitive to outliers in the data, alternatives have became increasingly popular during the last decades. We present algorithms for non-linear M-estimation. A trust region approach is used, where...

  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. Computer algebra applications

    International Nuclear Information System (INIS)

    Calmet, J.

    1982-01-01

    A survey of applications based either on fundamental algorithms in computer algebra or on the use of a computer algebra system is presented. Recent work in biology, chemistry, physics, mathematics and computer science is discussed. In particular, applications in high energy physics (quantum electrodynamics), celestial mechanics and general relativity are reviewed. (Auth.)

  5. 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.

  6. Error-source effects on the performance of direct and iterative algorithms on an optical matrix-vector processor

    Science.gov (United States)

    Perlee, Caroline J.; Casasent, David P.

    1990-09-01

    Error sources in an optical matrix-vector processor are analyzed in terms of their effect on the performance of the algorithms used to solve a set of nonlinear and linear algebraic equations. A direct and an iterative algorithm are used to solve a nonlinear time-dependent case-study from computational fluid dynamics. A simulator which emulates the data flow and number representation of the OLAP is used to studs? these error effects. The ability of each algorithm to tolerate or correct the error sources is quantified. These results are extended to the general case of solving nonlinear and linear algebraic equations on the optical system.

  7. 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

  8. Algorithms for a parallel implementation of Hidden Markov Models with a small state space

    DEFF Research Database (Denmark)

    Nielsen, Jesper; Sand, Andreas

    2011-01-01

    Two of the most important algorithms for Hidden Markov Models are the forward and the Viterbi algorithms. We show how formulating these using linear algebra naturally lends itself to parallelization. Although the obtained algorithms are slow for Hidden Markov Models with large state spaces...

  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. Graphs and matroids weighted in a bounded incline algebra.

    Science.gov (United States)

    Lu, Ling-Xia; Zhang, Bei

    2014-01-01

    Firstly, for a graph weighted in a bounded incline algebra (or called a dioid), a longest path problem (LPP, for short) is presented, which can be considered the uniform approach to the famous shortest path problem, the widest path problem, and the most reliable path problem. The solutions for LPP and related algorithms are given. Secondly, for a matroid weighted in a linear matroid, the maximum independent set problem is studied.

  11. 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

  12. PARALLEL ALGORITHM FOR THREE-DIMENSIONAL STOKES FLOW SIMULATION USING BOUNDARY ELEMENT METHOD

    Directory of Open Access Journals (Sweden)

    D. G. Pribytok

    2016-01-01

    Full Text Available Parallel computing technique for modeling three-dimensional viscous flow (Stokes flow using direct boundary element method is presented. The problem is solved in three phases: sampling and construction of system of linear algebraic equations (SLAE, its decision and finding the velocity of liquid at predetermined points. For construction of the system and finding the velocity, the parallel algorithms using graphics CUDA cards programming technology have been developed and implemented. To solve the system of linear algebraic equations the implemented software libraries are used. A comparison of time consumption for three main algorithms on the example of calculation of viscous fluid motion in three-dimensional cavity is performed.

  13. 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.

  14. Boolean Operations with Prism Algebraic Patches

    Science.gov (United States)

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

    2009-01-01

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

  15. 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 $Vir$.

  16. 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.)

  17. Algebraic Side-Channel Attack on Twofish

    Directory of Open Access Journals (Sweden)

    Chujiao Ma

    2017-05-01

    Full Text Available While algebraic side-channel attack (ASCA has been successful in breaking simple cryptographic algorithms, it has never been done on larger or more complex algorithms such as Twofish. Compared to other algorithms that ASCA has been used on, Twofish is more difficult to attack due to the key-dependent S-boxes as well as the complex key scheduling. In this paper, we propose the first algebraic side-channel attack on Twofish, and examine the importance of side-channel information in getting past the key-dependent S-boxes and the complex key scheduling. The cryptographic algorithm and side-channel information are both expressed as boolean equations and a SAT solver is used to recover the key. While algebraic attack by itself is not sufficient to break the algorithm, with the help of side-channel information such as Hamming weights, we are able to correctly solve for 96 bits of the 128 bits key in under 2 hours with known plaintext/ciphertext.

  18. 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.

  19. A Global Optimization Algorithm for Sum of Linear Ratios Problem

    Directory of Open Access Journals (Sweden)

    Yuelin Gao

    2013-01-01

    Full Text Available We equivalently transform the sum of linear ratios programming problem into bilinear programming problem, then by using the linear characteristics of convex envelope and concave envelope of double variables product function, linear relaxation programming of the bilinear programming problem is given, which can determine the lower bound of the optimal value of original problem. Therefore, a branch and bound algorithm for solving sum of linear ratios programming problem is put forward, and the convergence of the algorithm is proved. Numerical experiments are reported to show the effectiveness of the proposed algorithm.

  20. 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.

  1. Iterative algorithms for large sparse linear systems on parallel computers

    Science.gov (United States)

    Adams, L. M.

    1982-01-01

    Algorithms for assembling in parallel the sparse system of linear equations that result from finite difference or finite element discretizations of elliptic partial differential equations, such as those that arise in structural engineering are developed. Parallel linear stationary iterative algorithms and parallel preconditioned conjugate gradient algorithms are developed for solving these systems. In addition, a model for comparing parallel algorithms on array architectures is developed and results of this model for the algorithms are given.

  2. 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.

  3. 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.)

  4. ORACLS- OPTIMAL REGULATOR ALGORITHMS FOR THE CONTROL OF LINEAR SYSTEMS (DEC VAX VERSION)

    Science.gov (United States)

    Frisch, H.

    1994-01-01

    This control theory design package, called Optimal Regulator Algorithms for the Control of Linear Systems (ORACLS), was developed to aid in the design of controllers and optimal filters for systems which can be modeled by linear, time-invariant differential and difference equations. Optimal linear quadratic regulator theory, currently referred to as the Linear-Quadratic-Gaussian (LQG) problem, has become the most widely accepted method of determining optimal control policy. Within this theory, the infinite duration time-invariant problems, which lead to constant gain feedback control laws and constant Kalman-Bucy filter gains for reconstruction of the system state, exhibit high tractability and potential ease of implementation. A variety of new and efficient methods in the field of numerical linear algebra have been combined into the ORACLS program, which provides for the solution to time-invariant continuous or discrete LQG problems. The ORACLS package is particularly attractive to the control system designer because it provides a rigorous tool for dealing with multi-input and multi-output dynamic systems in both continuous and discrete form. The ORACLS programming system is a collection of subroutines which can be used to formulate, manipulate, and solve various LQG design problems. The ORACLS program is constructed in a manner which permits the user to maintain considerable flexibility at each operational state. This flexibility is accomplished by providing primary operations, analysis of linear time-invariant systems, and control synthesis based on LQG methodology. The input-output routines handle the reading and writing of numerical matrices, printing heading information, and accumulating output information. The basic vector-matrix operations include addition, subtraction, multiplication, equation, norm construction, tracing, transposition, scaling, juxtaposition, and construction of null and identity matrices. The analysis routines provide for the following

  5. Synthesizing Dynamic Programming Algorithms from Linear Temporal Logic Formulae

    Science.gov (United States)

    Rosu, Grigore; Havelund, Klaus

    2001-01-01

    The problem of testing a linear temporal logic (LTL) formula on a finite execution trace of events, generated by an executing program, occurs naturally in runtime analysis of software. We present an algorithm which takes an LTL formula and generates an efficient dynamic programming algorithm. The generated algorithm tests whether the LTL formula is satisfied by a finite trace of events given as input. The generated algorithm runs in linear time, its constant depending on the size of the LTL formula. The memory needed is constant, also depending on the size of the formula.

  6. 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…

  7. Implementation of software-based sensor linearization algorithms on low-cost microcontrollers.

    Science.gov (United States)

    Erdem, Hamit

    2010-10-01

    Nonlinear sensors and microcontrollers are used in many embedded system designs. As the input-output characteristic of most sensors is nonlinear in nature, obtaining data from a nonlinear sensor by using an integer microcontroller has always been a design challenge. This paper discusses the implementation of six software-based sensor linearization algorithms for low-cost microcontrollers. The comparative study of the linearization algorithms is performed by using a nonlinear optical distance-measuring sensor. The performance of the algorithms is examined with respect to memory space usage, linearization accuracy and algorithm execution time. The implementation and comparison results can be used for selection of a linearization algorithm based on the sensor transfer function, expected linearization accuracy and microcontroller capacity. Copyright © 2010 ISA. Published by Elsevier Ltd. All rights reserved.

  8. 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

  9. 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.

  10. 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.

  11. Algorithms of estimation for nonlinear systems a differential and algebraic viewpoint

    CERN Document Server

    Martínez-Guerra, Rafael

    2017-01-01

    This book acquaints readers with recent developments in dynamical systems theory and its applications, with a strong focus on the control and estimation of nonlinear systems. Several algorithms are proposed and worked out for a set of model systems, in particular so-called input-affine or bilinear systems, which can serve to approximate a wide class of nonlinear control systems. These can either take the form of state space models or be represented by an input-output equation. The approach taken here further highlights the role of modern mathematical and conceptual tools, including differential algebraic theory, observer design for nonlinear systems and generalized canonical forms.

  12. 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.

  13. 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.

  14. 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 ...

  15. Portfolio optimization by using linear programing models based on genetic algorithm

    Science.gov (United States)

    Sukono; Hidayat, Y.; Lesmana, E.; Putra, A. S.; Napitupulu, H.; Supian, S.

    2018-01-01

    In this paper, we discussed the investment portfolio optimization using linear programming model based on genetic algorithms. It is assumed that the portfolio risk is measured by absolute standard deviation, and each investor has a risk tolerance on the investment portfolio. To complete the investment portfolio optimization problem, the issue is arranged into a linear programming model. Furthermore, determination of the optimum solution for linear programming is done by using a genetic algorithm. As a numerical illustration, we analyze some of the stocks traded on the capital market in Indonesia. Based on the analysis, it is shown that the portfolio optimization performed by genetic algorithm approach produces more optimal efficient portfolio, compared to the portfolio optimization performed by a linear programming algorithm approach. Therefore, genetic algorithms can be considered as an alternative on determining the investment portfolio optimization, particularly using linear programming models.

  16. 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)

  17. 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…

  18. 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.)

  19. 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.

  20. Efficient Implementation of Nested-Loop Multimedia Algorithms

    Directory of Open Access Journals (Sweden)

    Kittitornkun Surin

    2001-01-01

    Full Text Available A novel dependence graph representation called the multiple-order dependence graph for nested-loop formulated multimedia signal processing algorithms is proposed. It allows a concise representation of an entire family of dependence graphs. This powerful representation facilitates the development of innovative implementation approach for nested-loop formulated multimedia algorithms such as motion estimation, matrix-matrix product, 2D linear transform, and others. In particular, algebraic linear mapping (assignment and scheduling methodology can be applied to implement such algorithms on an array of simple-processing elements. The feasibility of this new approach is demonstrated in three major target architectures: application-specific integrated circuit (ASIC, field programmable gate array (FPGA, and a programmable clustered VLIW processor.

  1. Non-freely generated W-algebras and construction of N=2 super W-algebras

    International Nuclear Information System (INIS)

    Blumenhagen, R.

    1994-07-01

    Firstly, we investigate the origin of the bosonic W-algebras W(2, 3, 4, 5), W(2, 4, 6) and W(2, 4, 6) found earlier by direct construction. We present a coset construction for all three examples leading to a new type of finitely, non-freely generated quantum W-algebras, which we call unifying W-algebras. Secondly, we develop a manifest covariant formalism to construct N = 2 super W-algebras explicitly on a computer. Applying this algorithm enables us to construct the first four examples of N = 2 super W-algebras with two generators and the N = 2 super W 4 algebra involving three generators. The representation theory of the former ones shows that all examples could be divided into four classes, the largest one containing the N = 2 special type of spectral flow algebras. Besides the W-algebra of the CP(3) Kazama-Suzuki coset model, the latter example with three generators discloses a second solution which could also be explained as a unifying W-algebra for the CP(n) models. (orig.)

  2. Scalable algorithms for contact problems

    CERN Document Server

    Dostál, Zdeněk; Sadowská, Marie; Vondrák, Vít

    2016-01-01

    This book presents a comprehensive and self-contained treatment of the authors’ newly developed scalable algorithms for the solutions of multibody contact problems of linear elasticity. The brand new feature of these algorithms is theoretically supported numerical scalability and parallel scalability demonstrated on problems discretized by billions of degrees of freedom. The theory supports solving multibody frictionless contact problems, contact problems with possibly orthotropic Tresca’s friction, and transient contact problems. It covers BEM discretization, jumping coefficients, floating bodies, mortar non-penetration conditions, etc. The exposition is divided into four parts, the first of which reviews appropriate facets of linear algebra, optimization, and analysis. The most important algorithms and optimality results are presented in the third part of the volume. The presentation is complete, including continuous formulation, discretization, decomposition, optimality results, and numerical experimen...

  3. Epsilon topological accelerating algorithms for difference method for initial-value problems

    International Nuclear Information System (INIS)

    Hristea, V.; Posirca, M.

    1992-01-01

    Linear and nonlinear parabolic equations can be solved by discretization methods which lead to linear and nonlinear algebraic systems. The iterative methods (e.g. Gauss - Seidel) show a very slow convergence and instability in the case of nonlinear equations. This paper proposes an ε topological algorithm for accelerating slow iterative methods used in the thermohydraulic code COBRA and the dynamic code ADEP. The results show an executing time approximately ten times lower than original algorithms. (Author)

  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. Construction and decoding of a class of algebraic geometry codes

    DEFF Research Database (Denmark)

    Justesen, Jørn; Larsen, Knud J.; Jensen, Helge Elbrønd

    1989-01-01

    A class of codes derived from algebraic plane curves is constructed. The concepts and results from algebraic geometry that were used are explained in detail; no further knowledge of algebraic geometry is needed. Parameters, generator and parity-check matrices are given. The main result is a decod...... is a decoding algorithm which turns out to be a generalization of the Peterson algorithm for decoding BCH decoder codes......A class of codes derived from algebraic plane curves is constructed. The concepts and results from algebraic geometry that were used are explained in detail; no further knowledge of algebraic geometry is needed. Parameters, generator and parity-check matrices are given. The main result...

  6. 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

  7. A note on the linear memory Baum-Welch algorithm

    DEFF Research Database (Denmark)

    Jensen, Jens Ledet

    2009-01-01

    We demonstrate the simplicity and generality of the recently introduced linear space Baum-Welch algorithm for hidden Markov models. We also point to previous literature on the subject.......We demonstrate the simplicity and generality of the recently introduced linear space Baum-Welch algorithm for hidden Markov models. We also point to previous literature on the subject....

  8. Algebraic curves and cryptography

    CERN Document Server

    Murty, V Kumar

    2010-01-01

    It is by now a well-known paradigm that public-key cryptosystems can be built using finite Abelian groups and that algebraic geometry provides a supply of such groups through Abelian varieties over finite fields. Of special interest are the Abelian varieties that are Jacobians of algebraic curves. All of the articles in this volume are centered on the theme of point counting and explicit arithmetic on the Jacobians of curves over finite fields. The topics covered include Schoof's \\ell-adic point counting algorithm, the p-adic algorithms of Kedlaya and Denef-Vercauteren, explicit arithmetic on

  9. Efficient high-precision matrix algebra on parallel architectures for nonlinear combinatorial optimization

    KAUST Repository

    Gunnels, John; Lee, Jon; Margulies, Susan

    2010-01-01

    We provide a first demonstration of the idea that matrix-based algorithms for nonlinear combinatorial optimization problems can be efficiently implemented. Such algorithms were mainly conceived by theoretical computer scientists for proving efficiency. We are able to demonstrate the practicality of our approach by developing an implementation on a massively parallel architecture, and exploiting scalable and efficient parallel implementations of algorithms for ultra high-precision linear algebra. Additionally, we have delineated and implemented the necessary algorithmic and coding changes required in order to address problems several orders of magnitude larger, dealing with the limits of scalability from memory footprint, computational efficiency, reliability, and interconnect perspectives. © Springer and Mathematical Programming Society 2010.

  10. Efficient high-precision matrix algebra on parallel architectures for nonlinear combinatorial optimization

    KAUST Repository

    Gunnels, John

    2010-06-01

    We provide a first demonstration of the idea that matrix-based algorithms for nonlinear combinatorial optimization problems can be efficiently implemented. Such algorithms were mainly conceived by theoretical computer scientists for proving efficiency. We are able to demonstrate the practicality of our approach by developing an implementation on a massively parallel architecture, and exploiting scalable and efficient parallel implementations of algorithms for ultra high-precision linear algebra. Additionally, we have delineated and implemented the necessary algorithmic and coding changes required in order to address problems several orders of magnitude larger, dealing with the limits of scalability from memory footprint, computational efficiency, reliability, and interconnect perspectives. © Springer and Mathematical Programming Society 2010.

  11. "Accelerated Perceptron": A Self-Learning Linear Decision Algorithm

    OpenAIRE

    Zuev, Yu. A.

    2003-01-01

    The class of linear decision rules is studied. A new algorithm for weight correction, called an "accelerated perceptron", is proposed. In contrast to classical Rosenblatt's perceptron this algorithm modifies the weight vector at each step. The algorithm may be employed both in learning and in self-learning modes. The theoretical aspects of the behaviour of the algorithm are studied when the algorithm is used for the purpose of increasing the decision reliability by means of weighted voting. I...

  12. Quantum associative memory with linear and non-linear algorithms for the diagnosis of some tropical diseases.

    Science.gov (United States)

    Tchapet Njafa, J-P; Nana Engo, S G

    2018-01-01

    This paper presents the QAMDiagnos, a model of Quantum Associative Memory (QAM) that can be a helpful tool for medical staff without experience or laboratory facilities, for the diagnosis of four tropical diseases (malaria, typhoid fever, yellow fever and dengue) which have several similar signs and symptoms. The memory can distinguish a single infection from a polyinfection. Our model is a combination of the improved versions of the original linear quantum retrieving algorithm proposed by Ventura and the non-linear quantum search algorithm of Abrams and Lloyd. From the given simulation results, it appears that the efficiency of recognition is good when particular signs and symptoms of a disease are inserted given that the linear algorithm is the main algorithm. The non-linear algorithm helps confirm or correct the diagnosis or give some advice to the medical staff for the treatment. So, our QAMDiagnos that has a friendly graphical user interface for desktop and smart-phone is a sensitive and a low-cost diagnostic tool that enables rapid and accurate diagnosis of four tropical diseases. Copyright © 2017 Elsevier Ltd. All rights reserved.

  13. 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

  14. 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.)

  15. 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.))

  16. 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)

  17. 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.

  18. 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.

  19. 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.

  20. 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.

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

    Science.gov (United States)

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

    2014-12-05

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

  2. Fast numerical algorithm for the linear canonical transform.

    Science.gov (United States)

    Hennelly, Bryan M; Sheridan, John T

    2005-05-01

    The linear canonical transform (LCT) describes the effect of any quadratic phase system (QPS) on an input optical wave field. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT), and the Fresnel transform (FST) describing free-space propagation. Currently there are numerous efficient algorithms used (for purposes of numerical simulation in the area of optical signal processing) to calculate the discrete FT, FRT, and FST. All of these algorithms are based on the use of the fast Fourier transform (FFT). In this paper we develop theory for the discrete linear canonical transform (DLCT), which is to the LCT what the discrete Fourier transform (DFT) is to the FT. We then derive the fast linear canonical transform (FLCT), an N log N algorithm for its numerical implementation by an approach similar to that used in deriving the FFT from the DFT. Our algorithm is significantly different from the FFT, is based purely on the properties of the LCT, and can be used for FFT, FRT, and FST calculations and, in the most general case, for the rapid calculation of the effect of any QPS.

  3. 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

  4. 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…

  5. 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.

  6. 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

  7. Tracing a planar algebraic curve

    International Nuclear Information System (INIS)

    Chen Falai; Kozak, J.

    1994-09-01

    In this paper, an algorithm that determines a real algebraic curve is outlined. Its basic step is to divide the plane into subdomains that include only simple branches of the algebraic curve without singular points. Each of the branches is then stably and efficiently traced in the particular subdomain. Except for the tracing, the algorithm requires only a couple of simple operations on polynomials that can be carried out exactly if the coefficients are rational, and the determination of zeros of several polynomials of one variable. (author). 5 refs, 4 figs

  8. A Warm-Started Homogeneous and Self-Dual Interior-Point Method for Linear Economic Model Predictive Control

    DEFF Research Database (Denmark)

    Sokoler, Leo Emil; Skajaa, Anders; Frison, Gianluca

    2013-01-01

    algorithm in MATLAB and its performance is analyzed based on a smart grid power management case study. Closed loop simulations show that 1) our algorithm is significantly faster than state-of-the-art IPMs based on sparse linear algebra routines, and 2) warm-starting reduces the number of iterations...

  9. 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

  10. Error Estimation for the Linearized Auto-Localization Algorithm

    Directory of Open Access Journals (Sweden)

    Fernando Seco

    2012-02-01

    Full Text Available The Linearized Auto-Localization (LAL algorithm estimates the position of beacon nodes in Local Positioning Systems (LPSs, using only the distance measurements to a mobile node whose position is also unknown. The LAL algorithm calculates the inter-beacon distances, used for the estimation of the beacons’ positions, from the linearized trilateration equations. In this paper we propose a method to estimate the propagation of the errors of the inter-beacon distances obtained with the LAL algorithm, based on a first order Taylor approximation of the equations. Since the method depends on such approximation, a confidence parameter τ is defined to measure the reliability of the estimated error. Field evaluations showed that by applying this information to an improved weighted-based auto-localization algorithm (WLAL, the standard deviation of the inter-beacon distances can be improved by more than 30% on average with respect to the original LAL method.

  11. 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

  12. 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…

  13. Applied algebra codes, ciphers and discrete algorithms

    CERN Document Server

    Hardy, Darel W; Walker, Carol L

    2009-01-01

    This book attempts to show the power of algebra in a relatively simple setting.-Mathematical Reviews, 2010… The book supports learning by doing. In each section we can find many examples which clarify the mathematics introduced in the section and each section is followed by a series of exercises of which approximately half are solved in the end of the book. Additional the book comes with a CD-ROM containing an interactive version of the book powered by the computer algebra system Scientific Notebook. … the mathematics in the book are developed as needed and the focus of the book lies clearly o

  14. Pattern Nulling of Linear Antenna Arrays Using Backtracking Search Optimization Algorithm

    Directory of Open Access Journals (Sweden)

    Kerim Guney

    2015-01-01

    Full Text Available An evolutionary method based on backtracking search optimization algorithm (BSA is proposed for linear antenna array pattern synthesis with prescribed nulls at interference directions. Pattern nulling is obtained by controlling only the amplitude, position, and phase of the antenna array elements. BSA is an innovative metaheuristic technique based on an iterative process. Various numerical examples of linear array patterns with the prescribed single, multiple, and wide nulls are given to illustrate the performance and flexibility of BSA. The results obtained by BSA are compared with the results of the following seventeen algorithms: particle swarm optimization (PSO, genetic algorithm (GA, modified touring ant colony algorithm (MTACO, quadratic programming method (QPM, bacterial foraging algorithm (BFA, bees algorithm (BA, clonal selection algorithm (CLONALG, plant growth simulation algorithm (PGSA, tabu search algorithm (TSA, memetic algorithm (MA, nondominated sorting GA-2 (NSGA-2, multiobjective differential evolution (MODE, decomposition with differential evolution (MOEA/D-DE, comprehensive learning PSO (CLPSO, harmony search algorithm (HSA, seeker optimization algorithm (SOA, and mean variance mapping optimization (MVMO. The simulation results show that the linear antenna array synthesis using BSA provides low side-lobe levels and deep null levels.

  15. 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

  16. Linear Algorithms for Radioelectric Spectrum Forecast

    Directory of Open Access Journals (Sweden)

    Luis F. Pedraza

    2016-12-01

    Full Text Available This paper presents the development and evaluation of two linear algorithms for forecasting reception power for different channels at an assigned spectrum band of global systems for mobile communications (GSM, in order to analyze the spatial opportunity for reuse of frequencies by secondary users (SUs in a cognitive radio (CR network. The algorithms employed correspond to seasonal autoregressive integrated moving average (SARIMA and generalized autoregressive conditional heteroskedasticity (GARCH, which allow for a forecast of channel occupancy status. Results are evaluated using the following criteria: availability and occupancy time for channels, different types of mean absolute error, and observation time. The contributions of this work include a more integral forecast as the algorithm not only forecasts reception power but also the occupancy and availability time of a channel to determine its precision percentage during the use by primary users (PUs and SUs within a CR system. Algorithm analyses demonstrate a better performance for SARIMA over GARCH algorithm in most of the evaluated variables.

  17. 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.

  18. 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.

  19. Linear Time Local Approximation Algorithm for Maximum Stable Marriage

    Directory of Open Access Journals (Sweden)

    Zoltán Király

    2013-08-01

    Full Text Available We consider a two-sided market under incomplete preference lists with ties, where the goal is to find a maximum size stable matching. The problem is APX-hard, and a 3/2-approximation was given by McDermid [1]. This algorithm has a non-linear running time, and, more importantly needs global knowledge of all preference lists. We present a very natural, economically reasonable, local, linear time algorithm with the same ratio, using some ideas of Paluch [2]. In this algorithm every person make decisions using only their own list, and some information asked from members of these lists (as in the case of the famous algorithm of Gale and Shapley. Some consequences to the Hospitals/Residents problem are also discussed.

  20. Rationality problem for algebraic tori

    CERN Document Server

    Hoshi, Akinari

    2017-01-01

    The authors give the complete stably rational classification of algebraic tori of dimensions 4 and 5 over a field k. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank 4 and 5 is given. The authors show that there exist exactly 487 (resp. 7, resp. 216) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 4, and there exist exactly 3051 (resp. 25, resp. 3003) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 5. The authors make a procedure to compute a flabby resolution of a G-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a G-lattice is invertible (resp. zero) or not. Using the algorithms, the suthors determine all the flabby and coflabby G-lattices of rank up to 6 and verify that they are stably permutation. The authors also show that the Krull-Schmidt theorem for G-...

  1. Invariants of triangular Lie algebras

    International Nuclear Information System (INIS)

    Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman

    2007-01-01

    Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of Boyko et al (2006 J. Phys. A: Math. Gen.39 5749 (Preprint math-ph/0602046)), developed further in Boyko et al (2007 J. Phys. A: Math. Theor.40 113 (Preprint math-ph/0606045)), is used to determine the invariants. A conjecture of Tremblay and Winternitz (2001 J. Phys. A: Math. Gen.34 9085), concerning the number of independent invariants and their form, is corroborated

  2. Advanced computer algebra algorithms for the expansion of Feynman integrals

    International Nuclear Information System (INIS)

    Ablinger, Jakob; Round, Mark; Schneider, Carsten

    2012-10-01

    Two-point Feynman parameter integrals, with at most one mass and containing local operator insertions in 4+ε-dimensional Minkowski space, can be transformed to multi-integrals or multi-sums over hyperexponential and/or hypergeometric functions depending on a discrete parameter n. Given such a specific representation, we utilize an enhanced version of the multivariate Almkvist-Zeilberger algorithm (for multi-integrals) and a common summation framework of the holonomic and difference field approach (for multi-sums) to calculate recurrence relations in n. Finally, solving the recurrence we can decide efficiently if the first coefficients of the Laurent series expansion of a given Feynman integral can be expressed in terms of indefinite nested sums and products; if yes, the all n solution is returned in compact representations, i.e., no algebraic relations exist among the occurring sums and products.

  3. Advanced computer algebra algorithms for the expansion of Feynman integrals

    Energy Technology Data Exchange (ETDEWEB)

    Ablinger, Jakob; Round, Mark; Schneider, Carsten [Johannes Kepler Univ., Linz (Austria). Research Inst. for Symbolic Computation; Bluemlein, Johannes [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany)

    2012-10-15

    Two-point Feynman parameter integrals, with at most one mass and containing local operator insertions in 4+{epsilon}-dimensional Minkowski space, can be transformed to multi-integrals or multi-sums over hyperexponential and/or hypergeometric functions depending on a discrete parameter n. Given such a specific representation, we utilize an enhanced version of the multivariate Almkvist-Zeilberger algorithm (for multi-integrals) and a common summation framework of the holonomic and difference field approach (for multi-sums) to calculate recurrence relations in n. Finally, solving the recurrence we can decide efficiently if the first coefficients of the Laurent series expansion of a given Feynman integral can be expressed in terms of indefinite nested sums and products; if yes, the all n solution is returned in compact representations, i.e., no algebraic relations exist among the occurring sums and products.

  4. 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.

  5. 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...

  6. Scheduling Two-Sided Transformations Using Tile Algorithms on Multicore Architectures

    Directory of Open Access Journals (Sweden)

    Hatem Ltaief

    2010-01-01

    Full Text Available The objective of this paper is to describe, in the context of multicore architectures, three different scheduler implementations for the two-sided linear algebra transformations, in particular the Hessenberg and Bidiagonal reductions which are the first steps for the standard eigenvalue problems and the singular value decompositions respectively. State-of-the-art dense linear algebra softwares, such as the LAPACK and ScaLAPACK libraries, suffer performance losses on multicore processors due to their inability to fully exploit thread-level parallelism. At the same time the fine-grain dataflow model gains popularity as a paradigm for programming multicore architectures. Buttari et al. (Parellel Comput. Syst. Appl. 35 (2009, 38–53 introduced the concept of tile algorithms in which parallelism is no longer hidden inside Basic Linear Algebra Subprograms but is brought to the fore to yield much better performance. Along with efficient scheduling mechanisms for data-driven execution, these tile two-sided reductions achieve high performance computing by reaching up to 75% of the DGEMM peak on a 12000×12000 matrix with 16 Intel Tigerton 2.4 GHz processors. The main drawback of the tile algorithms approach for two-sided transformations is that the full reduction cannot be obtained in one stage. Other methods have to be considered to further reduce the band matrices to the required forms.

  7. 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.

  8. 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)

  9. 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

  10. Algorithms in Singular

    Directory of Open Access Journals (Sweden)

    Hans Schonemann

    1996-12-01

    Full Text Available Some algorithms for singularity theory and algebraic geometry The use of Grobner basis computations for treating systems of polynomial equations has become an important tool in many areas. This paper introduces of the concept of standard bases (a generalization of Grobner bases and the application to some problems from algebraic geometry. The examples are presented as SINGULAR commands. A general introduction to Grobner bases can be found in the textbook [CLO], an introduction to syzygies in [E] and [St1]. SINGULAR is a computer algebra system for computing information about singularities, for use in algebraic geometry. The basic algorithms in SINGULAR are several variants of a general standard basis algorithm for general monomial orderings (see [GG]. This includes wellorderings (Buchberger algorithm ([B1], [B2] and tangent cone orderings (Mora algorithm ([M1], [MPT] as special cases: It is able to work with non-homogeneous and homogeneous input and also to compute in the localization of the polynomial ring in 0. Recent versions include algorithms to factorize polynomials and a factorizing Grobner basis algorithm. For a complete description of SINGULAR see [Si].

  11. 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.)

  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. 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...

  14. 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.

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

    Energy Technology Data Exchange (ETDEWEB)

    Matone, Marco [Universita di Padova, Dipartimento di Fisica e Astronomia ' ' G. Galilei' ' , Padua (Italy); Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Padua (Italy)

    2016-11-15

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

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

    International Nuclear Information System (INIS)

    Matone, Marco

    2016-01-01

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

  17. A primal-dual exterior point algorithm for linear programming problems

    Directory of Open Access Journals (Sweden)

    Samaras Nikolaos

    2009-01-01

    Full Text Available The aim of this paper is to present a new simplex type algorithm for the Linear Programming Problem. The Primal - Dual method is a Simplex - type pivoting algorithm that generates two paths in order to converge to the optimal solution. The first path is primal feasible while the second one is dual feasible for the original problem. Specifically, we use a three-phase-implementation. The first two phases construct the required primal and dual feasible solutions, using the Primal Simplex algorithm. Finally, in the third phase the Primal - Dual algorithm is applied. Moreover, a computational study has been carried out, using randomly generated sparse optimal linear problems, to compare its computational efficiency with the Primal Simplex algorithm and also with MATLAB's Interior Point Method implementation. The algorithm appears to be very promising since it clearly shows its superiority to the Primal Simplex algorithm as well as its robustness over the IPM algorithm.

  18. 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

  19. 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)

  20. Algorithms for finding Chomsky and Greibach normal forms for a fuzzy context-free grammar using an algebraic approach

    Energy Technology Data Exchange (ETDEWEB)

    Lee, E.T.

    1983-01-01

    Algorithms for the construction of the Chomsky and Greibach normal forms for a fuzzy context-free grammar using the algebraic approach are presented and illustrated by examples. The results obtained in this paper may have useful applications in fuzzy languages, pattern recognition, information storage and retrieval, artificial intelligence, database and pictorial information systems. 16 references.

  1. 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…

  2. Linear parallel processing machines I

    Energy Technology Data Exchange (ETDEWEB)

    Von Kunze, M

    1984-01-01

    As is well-known, non-context-free grammars for generating formal languages happen to be of a certain intrinsic computational power that presents serious difficulties to efficient parsing algorithms as well as for the development of an algebraic theory of contextsensitive languages. In this paper a framework is given for the investigation of the computational power of formal grammars, in order to start a thorough analysis of grammars consisting of derivation rules of the form aB ..-->.. A/sub 1/ ... A /sub n/ b/sub 1/...b /sub m/ . These grammars may be thought of as automata by means of parallel processing, if one considers the variables as operators acting on the terminals while reading them right-to-left. This kind of automata and their 2-dimensional programming language prove to be useful by allowing a concise linear-time algorithm for integer multiplication. Linear parallel processing machines (LP-machines) which are, in their general form, equivalent to Turing machines, include finite automata and pushdown automata (with states encoded) as special cases. Bounded LP-machines yield deterministic accepting automata for nondeterministic contextfree languages, and they define an interesting class of contextsensitive languages. A characterization of this class in terms of generating grammars is established by using derivation trees with crossings as a helpful tool. From the algebraic point of view, deterministic LP-machines are effectively represented semigroups with distinguished subsets. Concerning the dualism between generating and accepting devices of formal languages within the algebraic setting, the concept of accepting automata turns out to reduce essentially to embeddability in an effectively represented extension monoid, even in the classical cases.

  3. Circle Maps and C*-algebras

    DEFF Research Database (Denmark)

    Schmidt, Thomas Lundsgaard; Thomsen, Klaus

    2015-01-01

    We consider a construction of $C^*$-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map...... is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the $C^*$-algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the K-theory groups and turn them into an algorithmic method for Markov maps....

  4. A Linearized Relaxing Algorithm for the Specific Nonlinear Optimization Problem

    Directory of Open Access Journals (Sweden)

    Mio Horai

    2016-01-01

    Full Text Available We propose a new method for the specific nonlinear and nonconvex global optimization problem by using a linear relaxation technique. To simplify the specific nonlinear and nonconvex optimization problem, we transform the problem to the lower linear relaxation form, and we solve the linear relaxation optimization problem by the Branch and Bound Algorithm. Under some reasonable assumptions, the global convergence of the algorithm is certified for the problem. Numerical results show that this method is more efficient than the previous methods.

  5. Algorithms for sorting unsigned linear genomes by the DCJ operations.

    Science.gov (United States)

    Jiang, Haitao; Zhu, Binhai; Zhu, Daming

    2011-02-01

    The double cut and join operation (abbreviated as DCJ) has been extensively used for genomic rearrangement. Although the DCJ distance between signed genomes with both linear and circular (uni- and multi-) chromosomes is well studied, the only known result for the NP-complete unsigned DCJ distance problem is an approximation algorithm for unsigned linear unichromosomal genomes. In this article, we study the problem of computing the DCJ distance on two unsigned linear multichromosomal genomes (abbreviated as UDCJ). We devise a 1.5-approximation algorithm for UDCJ by exploiting the distance formula for signed genomes. In addition, we show that UDCJ admits a weak kernel of size 2k and hence an FPT algorithm running in O(2(2k)n) time.

  6. 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.)

  7. APPLICATION OF NEURAL NETWORK ALGORITHMS FOR BPM LINEARIZATION

    Energy Technology Data Exchange (ETDEWEB)

    Musson, John C. [JLAB; Seaton, Chad [JLAB; Spata, Mike F. [JLAB; Yan, Jianxun [JLAB

    2012-11-01

    Stripline BPM sensors contain inherent non-linearities, as a result of field distortions from the pickup elements. Many methods have been devised to facilitate corrections, often employing polynomial fitting. The cost of computation makes real-time correction difficult, particulalry when integer math is utilized. The application of neural-network technology, particularly the multi-layer perceptron algorithm, is proposed as an efficient alternative for electrode linearization. A process of supervised learning is initially used to determine the weighting coefficients, which are subsequently applied to the incoming electrode data. A non-linear layer, known as an activation layer, is responsible for the removal of saturation effects. Implementation of a perceptron in an FPGA-based software-defined radio (SDR) is presented, along with performance comparisons. In addition, efficient calculation of the sigmoidal activation function via the CORDIC algorithm is presented.

  8. Linear array implementation of the EM algorithm for PET image reconstruction

    International Nuclear Information System (INIS)

    Rajan, K.; Patnaik, L.M.; Ramakrishna, J.

    1995-01-01

    The PET image reconstruction based on the EM algorithm has several attractive advantages over the conventional convolution back projection algorithms. However, the PET image reconstruction based on the EM algorithm is computationally burdensome for today's single processor systems. In addition, a large memory is required for the storage of the image, projection data, and the probability matrix. Since the computations are easily divided into tasks executable in parallel, multiprocessor configurations are the ideal choice for fast execution of the EM algorithms. In tis study, the authors attempt to overcome these two problems by parallelizing the EM algorithm on a multiprocessor systems. The parallel EM algorithm on a linear array topology using the commercially available fast floating point digital signal processor (DSP) chips as the processing elements (PE's) has been implemented. The performance of the EM algorithm on a 386/387 machine, IBM 6000 RISC workstation, and on the linear array system is discussed and compared. The results show that the computational speed performance of a linear array using 8 DSP chips as PE's executing the EM image reconstruction algorithm is about 15.5 times better than that of the IBM 6000 RISC workstation. The novelty of the scheme is its simplicity. The linear array topology is expandable with a larger number of PE's. The architecture is not dependant on the DSP chip chosen, and the substitution of the latest DSP chip is straightforward and could yield better speed performance

  9. 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

  10. Comparison of l₁-Norm SVR and Sparse Coding Algorithms for Linear Regression.

    Science.gov (United States)

    Zhang, Qingtian; Hu, Xiaolin; Zhang, Bo

    2015-08-01

    Support vector regression (SVR) is a popular function estimation technique based on Vapnik's concept of support vector machine. Among many variants, the l1-norm SVR is known to be good at selecting useful features when the features are redundant. Sparse coding (SC) is a technique widely used in many areas and a number of efficient algorithms are available. Both l1-norm SVR and SC can be used for linear regression. In this brief, the close connection between the l1-norm SVR and SC is revealed and some typical algorithms are compared for linear regression. The results show that the SC algorithms outperform the Newton linear programming algorithm, an efficient l1-norm SVR algorithm, in efficiency. The algorithms are then used to design the radial basis function (RBF) neural networks. Experiments on some benchmark data sets demonstrate the high efficiency of the SC algorithms. In particular, one of the SC algorithms, the orthogonal matching pursuit is two orders of magnitude faster than a well-known RBF network designing algorithm, the orthogonal least squares algorithm.

  11. 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)

  12. Acceleration of the direct reconstruction of linear parametric images using nested algorithms

    International Nuclear Information System (INIS)

    Wang Guobao; Qi Jinyi

    2010-01-01

    Parametric imaging using dynamic positron emission tomography (PET) provides important information for biological research and clinical diagnosis. Indirect and direct methods have been developed for reconstructing linear parametric images from dynamic PET data. Indirect methods are relatively simple and easy to implement because the image reconstruction and kinetic modeling are performed in two separate steps. Direct methods estimate parametric images directly from raw PET data and are statistically more efficient. However, the convergence rate of direct algorithms can be slow due to the coupling between the reconstruction and kinetic modeling. Here we present two fast gradient-type algorithms for direct reconstruction of linear parametric images. The new algorithms decouple the reconstruction and linear parametric modeling at each iteration by employing the principle of optimization transfer. Convergence speed is accelerated by running more sub-iterations of linear parametric estimation because the computation cost of the linear parametric modeling is much less than that of the image reconstruction. Computer simulation studies demonstrated that the new algorithms converge much faster than the traditional expectation maximization (EM) and the preconditioned conjugate gradient algorithms for dynamic PET.

  13. High-order quantum algorithm for solving linear differential equations

    International Nuclear Information System (INIS)

    Berry, Dominic W

    2014-01-01

    Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms to general inhomogeneous sparse linear differential equations, which describe many classical physical systems. We examine the use of high-order methods (where the error over a time step is a high power of the size of the time step) to improve the efficiency. These provide scaling close to Δt 2 in the evolution time Δt. As with other algorithms of this type, the solution is encoded in amplitudes of the quantum state, and it is possible to extract global features of the solution. (paper)

  14. Optimizing relational algebra operations using generic equivalence discriminators and lazy products

    DEFF Research Database (Denmark)

    Henglein, Fritz

    2010-01-01

    We show how to efficiently evaluate generic map-filter-product queries, generalizations of select-project-join (SPJ) queries in re- lational algebra, based on a combination of two novel techniques: generic discrimination-based joins and lazy (formal) products. Discrimination-based joins are based...... that discriminators can be constructed generically (by structural recursion on equivalence expressions), purely func- tionally, and efficiently (worst-case linear time). The array-based basic multiset discrimination algorithm of Cai and Paige (1995) provides a base discriminator that is both asymptotically and prac...... on relational algebra equalities, without need for a query preprocessing phase. They require no indexes and behave purely functionally. They can be considered a form of symbolic execution of set expressions that automate and encapsulate dynamic program transformation of such expressions and lead to asymptotic...

  15. 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...

  16. Fast algorithms for computing defects and their derivatives in the Regge calculus

    International Nuclear Information System (INIS)

    Brewin, Leo

    2011-01-01

    Any practical attempt to solve the Regge equations, these being a large system of non-linear algebraic equations, will almost certainly employ a Newton-Raphson-like scheme. In such cases, it is essential that efficient algorithms be used when computing the defect angles and their derivatives with respect to the leg lengths. The purpose of this paper is to present details of such an algorithm.

  17. Novel Supercomputing Approaches for High Performance Linear Algebra Using FPGAs, Phase I

    Data.gov (United States)

    National Aeronautics and Space Administration — We propose to develop novel FPGA-based algorithmic technology that will enable unprecedented computational power for the solution of large sparse linear equation...

  18. Contribution to the resolution of algebraic differential equations. Application to electronic circuits and nuclear reactors

    International Nuclear Information System (INIS)

    Monsef, Youssef.

    1977-05-01

    This note deals with the resolution of large algebraic differential systems involved in the physical sciences, with special reference to electronics and nuclear physics. The theoretical aspect of the algorithms established and developed for this purpose is discussed in detail. A decomposition algorithm based on the graph theory is developed in detail and the regressive analysis of the error involved in the decomposition is carried out. The specific application of these algorithms on the analyses of non-linear electronic circuits and to the integration of algebraic differential equations simulating the general operation of nuclear reactors coupled to heat exchangers is discussed in detail. To conclude, it is shown that the development of efficient digital resolution techniques dealing with the elements in order is sub-optimal for large systems and calls for the revision of conventional formulation methods. Thus for a high-order physical system, the larger, the number of auxiliary unknowns introduced, the easier the formulation and resolution, owing to the elimination of any form of complex matricial calculation such as those given by the state variables method [fr

  19. Algorithms for Zero-Dimensional Ideals Using Linear Recurrent Sequences

    DEFF Research Database (Denmark)

    Neiger, Vincent; Rahkooy, Hamid; Schost, Éric

    2017-01-01

    Inspired by Faugére and Mou´s sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing of the annihilator of one or several such sequences.......Inspired by Faugére and Mou´s sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing of the annihilator of one or several such sequences....

  20. 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

  1. 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

  2. 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.

  3. 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.

  4. 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.

  5. Recent results in the decoding of Algebraic geometry codes

    DEFF Research Database (Denmark)

    Høholdt, Tom; Jensen, Helge Elbrønd; Nielsen, Rasmus Refslund

    1998-01-01

    We analyse the known decoding algorithms for algebraic geometry codes in the case where the number of errors is [(dFR-1)/2]+1, where dFR is the Feng-Rao distance......We analyse the known decoding algorithms for algebraic geometry codes in the case where the number of errors is [(dFR-1)/2]+1, where dFR is the Feng-Rao distance...

  6. 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.

  7. 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

  8. Numerical algorithms for contact problems in linear elastostatics

    International Nuclear Information System (INIS)

    Barbosa, H.J.C.; Feijoo, R.A.

    1984-01-01

    In this work contact problems in linear elasticity are analysed by means of Finite Elements and Mathematical Programming Techniques. The principle of virtual work leads in this case to a variational inequality which in turn is equivalent, for Hookean materials and infinitesimal strains, to the minimization of the total potential energy over the set of all admissible virtual displacements. The use of Gauss-Seidel algorithm with relaxation and projection and also Lemke's algorithm and Uzawa's algorithm for solving the minimization problem is discussed. Finally numerical examples are presented. (Author) [pt

  9. 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.

  10. Modern algorithms for large sparse eigenvalue problems

    International Nuclear Information System (INIS)

    Meyer, A.

    1987-01-01

    The volume is written for mathematicians interested in (numerical) linear algebra and in the solution of large sparse eigenvalue problems, as well as for specialists in engineering, who use the considered algorithms in the investigation of eigenoscillations of structures, in reactor physics, etc. Some variants of the algorithms based on the idea of a gradient-type direction of movement are presented and their convergence properties are discussed. From this, a general strategy for the direct use of preconditionings for the eigenvalue problem is derived. In this new approach the necessity of the solution of large linear systems is entirely avoided. Hence, these methods represent a new alternative to some other modern eigenvalue algorithms, as they show a slightly slower convergence on the one hand but essentially lower numerical and data processing problems on the other hand. A brief description and comparison of some well-known methods (i.e. simultaneous iteration, Lanczos algorithm) completes this volume. (author)

  11. 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

  12. 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....

  13. Fast Bitwise Implementation of the Algebraic Normal Form Transform

    OpenAIRE

    Bakoev, Valentin

    2017-01-01

    The representation of Boolean functions by their algebraic normal forms (ANFs) is very important for cryptography, coding theory and other scientific areas. The ANFs are used in computing the algebraic degree of S-boxes, some other cryptographic criteria and parameters of errorcorrecting codes. Their applications require these criteria and parameters to be computed by fast algorithms. Hence the corresponding ANFs should also be obtained by fast algorithms. Here we continue o...

  14. 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

  15. A scalable parallel algorithm for multiple objective linear programs

    Science.gov (United States)

    Wiecek, Malgorzata M.; Zhang, Hong

    1994-01-01

    This paper presents an ADBASE-based parallel algorithm for solving multiple objective linear programs (MOLP's). Job balance, speedup and scalability are of primary interest in evaluating efficiency of the new algorithm. Implementation results on Intel iPSC/2 and Paragon multiprocessors show that the algorithm significantly speeds up the process of solving MOLP's, which is understood as generating all or some efficient extreme points and unbounded efficient edges. The algorithm gives specially good results for large and very large problems. Motivation and justification for solving such large MOLP's are also included.

  16. 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 ...

  17. Synthesis of Greedy Algorithms Using Dominance Relations

    Science.gov (United States)

    Nedunuri, Srinivas; Smith, Douglas R.; Cook, William R.

    2010-01-01

    Greedy algorithms exploit problem structure and constraints to achieve linear-time performance. Yet there is still no completely satisfactory way of constructing greedy algorithms. For example, the Greedy Algorithm of Edmonds depends upon translating a problem into an algebraic structure called a matroid, but the existence of such a translation can be as hard to determine as the existence of a greedy algorithm itself. An alternative characterization of greedy algorithms is in terms of dominance relations, a well-known algorithmic technique used to prune search spaces. We demonstrate a process by which dominance relations can be methodically derived for a number of greedy algorithms, including activity selection, and prefix-free codes. By incorporating our approach into an existing framework for algorithm synthesis, we demonstrate that it could be the basis for an effective engineering method for greedy algorithms. We also compare our approach with other characterizations of greedy algorithms.

  18. 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

  19. 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.

  20. 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)

  1. 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...

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

    Science.gov (United States)

    Dunham, Benjamin Z.

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

  3. Non-local matrix generalizations of W-algebras

    International Nuclear Information System (INIS)

    Bilal, A.

    1995-01-01

    There is a standard way to define two symplectic (hamiltonian) structures, the first and second Gelfand-Dikii brackets, on the space of ordinary m th -order linear differential operators L=-d m +U 1 d m-1 +U 2 d m-2 +..+U m . In this paper, I consider in detail the case where the U k are nxn-matrix-valued functions, with particular emphasis on the (more interesting) second Gelfand-Dikii bracket. Of particular interest is the reduction to the symplectic submanifold U 1 =0. This reduction gives rise to matrix generalizations of (the classical version of) the non-linear W m -algebras, called V n,m -algebras. The non-commutativity of the matrices leads to non-local terms in these V n,m -algebras. I show that these algebras contain a conformal Virasoro subalgebra and that combinations W k of the U k can be formed that are nxn-matrices of conformally primary fields of spin k, in analogy with the scalar case n=1. In general however, the V m,n -algebras have a much richer structure than the W m -algebras as can be seen on the examples of the non-linear and non-local Poisson brackets {(U 2 ) ab (σ),(U 2 ) cd (σ')}, {(U 2 ) ab (σ),(W 3 ) cd (σ')} and {(W 3 ) ab (σ),(W 3 ) cd (σ')} which I work out explicitly for all m and n. A matrix Miura transformation is derived, mapping these complicated (second Gelfand-Dikii) brackets of the U k to a set of much simpler Poisson brackets, providing the analogue of the free-field representation of the W m -algebras. (orig.)

  4. Probing the Locality of Excited States with Linear Algebra.

    Science.gov (United States)

    Etienne, Thibaud

    2015-04-14

    This article reports a novel theoretical approach related to the analysis of molecular excited states. The strategy introduced here involves gathering two pieces of physical information, coming from Hilbert and direct space operations, into a general, unique quantum mechanical descriptor of electronic transitions' locality. Moreover, the projection of Hilbert and direct space-derived indices in an Argand plane delivers a straightforward way to visually probe the ability of a dye to undergo a long- or short-range charge-transfer. This information can be applied, for instance, to the analysis of the electronic response of families of dyes to light absorption by unveiling the trend of a given push-pull chromophore to increase the electronic cloud polarization magnitude of its main transition with respect to the size extension of its conjugated spacer. We finally demonstrate that all the quantities reported in this article can be reliably approximated by a linear algebraic derivation, based on the contraction of detachment/attachment density matrices from canonical to atomic space. This alternative derivation has the remarkable advantage of a very low computational cost with respect to the previously used numerical integrations, making fast and accurate characterization of large molecular systems' excited states easily affordable.

  5. Constructing canonical bases of quantized enveloping algebras

    OpenAIRE

    Graaf, W.A. de

    2001-01-01

    An algorithm for computing the elements of a given weight of the canonical basis of a quantized enveloping algebra is described. Subsequently, a similar algorithm is presented for computing the canonical basis of a finite-dimensional module.

  6. Intermediate algebra & analytic geometry

    CERN Document Server

    Gondin, William R

    1967-01-01

    Intermediate Algebra & Analytic Geometry Made Simple focuses on the principles, processes, calculations, and methodologies involved in intermediate algebra and analytic geometry. The publication first offers information on linear equations in two unknowns and variables, functions, and graphs. Discussions focus on graphic interpretations, explicit and implicit functions, first quadrant graphs, variables and functions, determinate and indeterminate systems, independent and dependent equations, and defective and redundant systems. The text then examines quadratic equations in one variable, system

  7. An algorithm for the solution of dynamic linear programs

    Science.gov (United States)

    Psiaki, Mark L.

    1989-01-01

    The algorithm's objective is to efficiently solve Dynamic Linear Programs (DLP) by taking advantage of their special staircase structure. This algorithm constitutes a stepping stone to an improved algorithm for solving Dynamic Quadratic Programs, which, in turn, would make the nonlinear programming method of Successive Quadratic Programs more practical for solving trajectory optimization problems. The ultimate goal is to being trajectory optimization solution speeds into the realm of real-time control. The algorithm exploits the staircase nature of the large constraint matrix of the equality-constrained DLPs encountered when solving inequality-constrained DLPs by an active set approach. A numerically-stable, staircase QL factorization of the staircase constraint matrix is carried out starting from its last rows and columns. The resulting recursion is like the time-varying Riccati equation from multi-stage LQR theory. The resulting factorization increases the efficiency of all of the typical LP solution operations over that of a dense matrix LP code. At the same time numerical stability is ensured. The algorithm also takes advantage of dynamic programming ideas about the cost-to-go by relaxing active pseudo constraints in a backwards sweeping process. This further decreases the cost per update of the LP rank-1 updating procedure, although it may result in more changes of the active set that if pseudo constraints were relaxed in a non-stagewise fashion. The usual stability of closed-loop Linear/Quadratic optimally-controlled systems, if it carries over to strictly linear cost functions, implies that the saving due to reduced factor update effort may outweigh the cost of an increased number of updates. An aerospace example is presented in which a ground-to-ground rocket's distance is maximized. This example demonstrates the applicability of this class of algorithms to aerospace guidance. It also sheds light on the efficacy of the proposed pseudo constraint relaxation

  8. 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…

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

    Science.gov (United States)

    Payton, Spencer D.

    2017-01-01

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

  10. 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.

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

    Science.gov (United States)

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

    2017-10-01

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

  12. Elementary matrix algebra

    CERN Document Server

    Hohn, Franz E

    2012-01-01

    This complete and coherent exposition, complemented by numerous illustrative examples, offers readers a text that can teach by itself. Fully rigorous in its treatment, it offers a mathematically sound sequencing of topics. The work starts with the most basic laws of matrix algebra and progresses to the sweep-out process for obtaining the complete solution of any given system of linear equations - homogeneous or nonhomogeneous - and the role of matrix algebra in the presentation of useful geometric ideas, techniques, and terminology.Other subjects include the complete treatment of the structur

  13. 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

  14. A Partitioning and Bounded Variable Algorithm for Linear Programming

    Science.gov (United States)

    Sheskin, Theodore J.

    2006-01-01

    An interesting new partitioning and bounded variable algorithm (PBVA) is proposed for solving linear programming problems. The PBVA is a variant of the simplex algorithm which uses a modified form of the simplex method followed by the dual simplex method for bounded variables. In contrast to the two-phase method and the big M method, the PBVA does…

  15. The N=2 super-W3 algebra

    International Nuclear Information System (INIS)

    Romans, L.J.

    1992-01-01

    We present the complete structure of the N=2 super-W 3 algebra, a non-linear extended conformal algebra containing the usual N=2 superconformal algebra (with generators of spins 1, 3/2, 3/2 and 2) and a higher-spin multiplet of generators with spins 2, 5/2, 5/2 and 3. We investigate various sub-algebras and related algebras, and find necessary conditions upon possible unitary representations of the algebra. In particular, the central charge c is restricted to two discrete series, one ascending and one descending to a common accumulation point c=6. The results suggest that the algebra is realised in certain (compact or non-compact) Kazama-Suzuki coset models, including a c=9 model proposed by Bars based on SU(2, 1)/U(2). (orig.)

  16. Expansion of the Lie algebra and its applications

    International Nuclear Information System (INIS)

    Guo Fukui; Zhang Yufeng

    2006-01-01

    We take the Lie algebra A1 as an example to illustrate a detail approach for expanding a finite dimensional Lie algebra into a higher-dimensional one. By making use of the late and its resulting loop algebra, a few linear isospectral problems with multi-component potential functions are established. It follows from them that some new integrable hierarchies of soliton equations are worked out. In addition, various Lie algebras may be constructed for which the integrable couplings of soliton equations are obtained by employing the expanding technique of the the Lie algebras

  17. 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…

  18. Realizations of κ-Minkowski space, Drinfeld twists, and related symmetry algebras

    Energy Technology Data Exchange (ETDEWEB)

    Juric, Tajron; Meljanac, Stjepan; Pikutic, Danijel [Ruder Boskovic Institute, Theoretical Physics Division, Zagreb (Croatia)

    2015-11-15

    Realizations of κ-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of the gl(n) generators. There are three one-parameter families of linear realizations for timelike and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between a deformed Heisenberg algebra, the star product, the coproduct of momenta, and the twist operator is presented. It is proved that for each linear realization there exists a Drinfeld twist satisfying normalization and cocycle conditions. κ-Deformed igl(n)-Hopf algebras are presented for all cases. The κ-Poincare-Weyl and κ-Poincare-Hopf algebras are discussed. The left-right dual κ-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All Drinfeld twists related to κ-Minkowski space are obtained from our construction. Finally, some physical applications are discussed. (orig.)

  19. Realizations of κ-Minkowski space, Drinfeld twists, and related symmetry algebras

    International Nuclear Information System (INIS)

    Juric, Tajron; Meljanac, Stjepan; Pikutic, Danijel

    2015-01-01

    Realizations of κ-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of the gl(n) generators. There are three one-parameter families of linear realizations for timelike and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between a deformed Heisenberg algebra, the star product, the coproduct of momenta, and the twist operator is presented. It is proved that for each linear realization there exists a Drinfeld twist satisfying normalization and cocycle conditions. κ-Deformed igl(n)-Hopf algebras are presented for all cases. The κ-Poincare-Weyl and κ-Poincare-Hopf algebras are discussed. The left-right dual κ-Minkowski algebra is constructed from the transposed twists. The corresponding realizations are nonlinear. All Drinfeld twists related to κ-Minkowski space are obtained from our construction. Finally, some physical applications are discussed. (orig.)

  20. A faithful functor among algebras and graphs

    OpenAIRE

    Falcón Ganfornina, Óscar Jesús; Falcón Ganfornina, Raúl Manuel; Núñez Valdés, Juan; Pacheco Martínez, Ana María; Villar Liñán, María Trinidad; Vigo Aguiar, Jesús (Coordinador)

    2016-01-01

    The problem of identifying a functor between the categories of algebras and graphs is currently open. Based on a known algorithm that identifies isomorphisms of Latin squares with isomorphism of vertex-colored graphs, we describe here a pair of graphs that enable us to find a faithful functor between finite-dimensional algebras over finite fields and these graphs.