DOCSLIB.ORG

Cocoa 4.7 Manual

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Cocoa 4.7 Manual CoCoA 4.7 Manual March 29, 2007 2 Contents I 17 I-1 Preamble 19 I-1.1 Version ............................................. 19 I-1.2 Preface ............................................. 19 I-1.3 System Distribution ...................................... 19 I-1.4 System Requirements ..................................... 20 I-1.5 Copyright and Trademarks .................................. 20 I-1.6 Acknowledgments ....................................... 20 II Introduction to CoCoA 21 II-1 The CoCoA System 23 II-1.1 An Overview of the System .................................. 23 II-1.2 System Structure ........................................ 23 II-1.3 Contributions .......................................... 24 II-1.4 CoCoA and Macaulay ..................................... 24 II-1.5 Pointers to the Literature ................................... 24 II-2 Tutorial 27 II-2.1 A Tutorial Introduction to CoCoA .............................. 27 II-2.2 Setting Up CoCoA for the Tutorial .............................. 27 II-2.3 Entering Commands ...................................... 27 II-2.4 Examples of Entering Commands ............................... 28 II-2.5 More on Entering Commands ................................. 28 II-2.6 After the Tutorial ....................................... 28 II-2.7 Arithmetic ........................................... 29 II-2.8 Variables ............................................ 29 II-2.9 The Variable “It” ........................................ 29 II-2.10 Making Lists .......................................... 30 II-2.11 Setting Up a Ring ....................................... 31 II-2.12 A Groebner Basis Example .................................. 31 II-2.13 Eliminating Variables ..................................... 32 II-2.14 Using More Than One Ring .................................. 33 II-2.15 Substitutions .......................................... 33 II-2.16 First Functions ......................................... 34 II-2.17 More First Functions ...................................... 35 II-2.18 Rings Inside User-Defined Functions ............................. 35 II-2.19 Rational Normal Curve .................................... 37 II-2.20 Generic Minors ......................................... 38 II-2.21 Leading Term (Initial) Ideals, Generic Polynomials ..................... 39 II-2.22 Ring Mapping Example .................................... 39 II-2.23 Output to a File ........................................ 39 II-2.24 Finite Point Sets: Buchberger-Moeller ............................ 40 II-2.25 Syzygies and Resolution Example ............................... 40 3 4 Contents II-2.26 Factoring Polynomials ..................................... 41 III The CoCoA Programming Language 43 III-1 Introduction to CoCoA Programming 45 III-1.1 An Overview of CoCoA Programming ............................ 45 III-2 Language Elements 47 III-2.1 Character Set and Special Symbols .............................. 47 III-2.2 Identifiers ............................................ 47 III-2.3 Names of Indeterminates ................................... 47 III-2.4 Reserved Names ........................................ 48 III-2.5 Comments ........................................... 48 III-2.6 Data Types ........................................... 48 III-2.7 Commands and Functions for Data Types .......................... 49 III-3 Operators 51 III-3.1 CoCoA Operators ....................................... 51 III-3.2 Algebraic Operators ...................................... 51 III-3.3 Relational Operators ...................................... 52 III-3.4 Boolean Operators ....................................... 53 III-3.5 Selection Operators ...................................... 53 III-3.6 Range Operator ........................................ 53 III-4 Evaluation and Assignment 55 III-4.1 Evaluation ........................................... 55 III-4.2 Assignment ........................................... 55 III-5 User-Defined Functions 57 III-5.1 Introduction to User-Defined Functions ........................... 57 III-5.2 Commands and Functions for User-Defined Functions ................... 57 III-6 Flow Control: Conditional Statements and Loops 59 III-6.1 Commands and Functions for Branching ........................... 59 III-6.2 Commands and Functions for Loops ............................. 59 III-7 Input/Output 61 III-7.1 Introduction to IO ....................................... 61 III-7.2 Standard IO .......................................... 61 III-7.3 File IO ............................................. 61 III-7.4 String IO ............................................ 62 III-7.5 Commands and Functions for IO ............................... 62 III-7.6 Tagged Printing ........................................ 63 III-7.7 Tagging an Object ....................................... 63 III-7.8 Printing a Tagged Object ................................... 64 III-7.9 Describing a Tagged Object .................................. 64 III-7.10 Another Example Using Tags ................................. 65 III-7.11 Commands and Functions for Tags .............................. 65 III-8 Memory Management 67 III-8.1 Introduction to Memory .................................... 67 III-8.2 Working Memory ........................................ 67 III-8.3 Global Memory ......................................... 68 III-8.4 Ring-Bound Memory ...................................... 69 III-8.5 Commands and Functions for Memory ............................ 70 Contents 5 III-9 CoCoA Packages 71 III-9.1 Introduction to Packages ................................... 71 III-9.2 First Example of a Package .................................. 71 III-9.3 Package Essentials ....................................... 72 III-9.4 Package Sourcing and Autoloading .............................. 72 III-9.5 Global Aliases ......................................... 73 III-9.6 Local Aliases .......................................... 74 III-9.7 More Examples of Packages .................................. 74 III-9.8 Package Initialization ..................................... 75 III-9.9 Sharing Your Package ..................................... 76 III-9.10 Commands and Functions for Packages ........................... 76 III-9.11 Supported Packages ...................................... 76 III-9.12 K-Algebra Homomorphisms .................................. 77 III-9.13 Galois Package ......................................... 77 III-9.14 Integer Programming ..................................... 77 III-9.15 Algebra of Invariants ...................................... 78 III-9.16 Primary Ideals ......................................... 78 III-9.17 Special Varieties ........................................ 78 III-9.18 Statistics ............................................ 79 III-9.19 Geometrical Theorem-Proving ................................ 79 III-9.20 Typevectors ........................................... 79 III-9.21 Conductor ............................................ 80 III-9.22 Matrix Normal Form ...................................... 80 III-9.23 CantStop ............................................ 80 III-9.24 Control ............................................. 80 IV Doing Mathematics with CoCoA 83 IV-1 Booleans 85 IV-1.1 Introduction to Booleans ................................... 85 IV-1.2 Commands and Functions for Booleans ........................... 85 IV-2 Numbers 87 IV-2.1 Introduction to Numbers ................................... 87 IV-2.2 Rationals ............................................ 87 IV-2.3 Numerators and Denominators for Rational Numbers ................... 87 IV-2.4 Modular Integers ........................................ 88 IV-2.5 Commands and Functions for Numbers ........................... 88 IV-3 Strings 91 IV-3.1 Introduction to Strings .................................... 91 IV-3.2 Concatenation ......................................... 91 IV-3.3 Substrings ............................................ 91 IV-3.4 Quotes Within Strings ..................................... 92 IV-3.5 Commands and Functions for Strings ............................ 92 IV-4 Lists 93 IV-4.1 Introduction to Lists ...................................... 93 IV-4.2 Commands and Functions for Lists .............................. 93 IV-5 Records 97 IV-5.1 Introduction to Records .................................... 97 IV-5.2 Commands and Functions for Records ............................ 98 IV-6 Vectors 99 IV-6.1 Introduction to Vectors .................................... 99 IV-6.2 Commands and Functions for Vectors ............................ 99 6 Contents IV-7 Matrices 101 IV-7.1 Introduction to Matrices .................................... 101 IV-7.2 Commands and Functions for Matrices ............................ 102 IV-8 Rings 103 IV-8.1 Introduction to Rings ..................................... 103 IV-8.2 New Rings ........................................... 103 IV-8.3 Coefficient Rings ........................................ 104 IV-8.4 Indeterminates ......................................... 104 IV-8.5 Weights Modifier ........................................ 105 IV-8.6 Orderings ............................................ 106 IV-8.7 Predefined Term-Orderings .................................. 106 IV-8.8 Temporary Term-Orderings .................................. 107 IV-8.9 Custom Term-Orderings .................................... 107 IV-8.10 Module Orderings ....................................... 108 IV-8.11 Accessing Other Rings
Load more

Recommended publications
  • Using Macaulay2 Effectively in Practice
    Using Macaulay2 effectively in practice Mike Stillman ([email protected]) Department of Mathematics Cornell 22 July 2019 / IMA Sage/M2 Macaulay2: at a glance Project started in 1993, Dan Grayson and Mike Stillman. Open source. Key computations: Gr¨obnerbases, free resolutions, Hilbert functions and applications of these. Rings, Modules and Chain Complexes are first class objects. Language which is comfortable for mathematicians, yet powerful, expressive, and fun to program in. Now a community project Journal of Software for Algebra and Geometry (started in 2009. Now we handle: Macaulay2, Singular, Gap, Cocoa) (original editors: Greg Smith, Amelia Taylor). Strong community: including about 2 workshops per year. User contributed packages (about 200 so far). Each has doc and tests, is tested every night, and is distributed with M2. Lots of activity Over 2000 math papers refer to Macaulay2. History: 1976-1978 (My undergrad years at Urbana) E. Graham Evans: asked me to write a program to compute syzygies, from Hilbert's algorithm from 1890. Really didn't work on computers of the day (probably might still be an issue!). Instead: Did computation degree by degree, no finishing condition. Used Buchsbaum-Eisenbud \What makes a complex exact" (by hand!) to see if the resulting complex was exact. Winfried Bruns was there too. Very exciting time. History: 1978-1983 (My grad years, with Dave Bayer, at Harvard) History: 1978-1983 (My grad years, with Dave Bayer, at Harvard) I tried to do \real mathematics" but Dave Bayer (basically) rediscovered Groebner bases, and saw that they gave an algorithm for computing all syzygies. I got excited, dropped what I was doing, and we programmed (in Pascal), in less than one week, the first version of what would be Macaulay.
    [Show full text]
  • Luis David Garcıa Puente
    Luis David Garc´ıa Puente Department of Mathematics and Statistics (936) 294-1581 Sam Houston State University [email protected] Huntsville, TX 77341–2206 http://www.shsu.edu/ldg005/ Professional Preparation Universidad Nacional Autonoma´ de Mexico´ (UNAM) Mexico City, Mexico´ B.S. Mathematics (with Honors) 1999 Virginia Polytechnic Institute and State University Blacksburg, VA Ph.D. Mathematics 2004 – Advisor: Reinhard Laubenbacher – Dissertation: Algebraic Geometry of Bayesian Networks University of California, Berkeley Berkeley, CA Postdoctoral Fellow Summer 2004 – Mentor: Lior Pachter Mathematical Sciences Research Institute (MSRI) Berkeley, CA Postdoctoral Fellow Fall 2004 – Mentor: Bernd Sturmfels Texas A&M University College Station, TX Visiting Assistant Professor 2005 – 2007 – Mentor: Frank Sottile Appointments Colorado College Colorado Springs, CO Professor of Mathematics and Computer Science 2021 – Sam Houston State University Huntsville, TX Professor of Mathematics 2019 – 2021 Sam Houston State University Huntsville, TX Associate Department Chair Fall 2017 – 2021 Sam Houston State University Huntsville, TX Associate Professor of Mathematics 2013 – 2019 Statistical and Applied Mathematical Sciences Institute Research Triangle Park, NC SAMSI New Researcher fellowship Spring 2009 Sam Houston State University Huntsville, TX Assistant Professor of Mathematics 2007 – 2013 Virginia Bioinformatics Institute (Virginia Tech) Blacksburg, VA Graduate Research Assistant Spring 2004 Virginia Polytechnic Institute and State University Blacksburg,
    [Show full text]
  • An Alternative Algorithm for Computing the Betti Table of a Monomial Ideal 3
    AN ALTERNATIVE ALGORITHM FOR COMPUTING THE BETTI TABLE OF A MONOMIAL IDEAL MARIA-LAURA TORRENTE AND MATTEO VARBARO Abstract. In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform numerical experiments suggesting a very promising efficiency. On the way of describing the method, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal. 1. Introduction Since many years syzygies, and more generally free resolutions, are central in purely theo- retical aspects of algebraic geometry; more recently, after the connection between algebra and statistics have been initiated by Diaconis and Sturmfels in [DS98], free resolutions have also become an important tool in statistics (for instance, see [D11, SW09]). As a consequence, it is fundamental to have efficient algorithms to compute them. The usual approach uses Gr¨obner bases and exploits a result of Schreyer (for more details see [Sc80, Sc91] or [Ei95, Chapter 15, Section 5]). The packages for free resolutions of the most used computer algebra systems, like [Macaulay2, Singular, CoCoA], are based on these techniques. In this paper, we introduce a new algorithmic method to compute the minimal graded free resolution of any finitely generated graded module over a polynomial ring such that some (possibly non- minimal) graded free resolution is known a priori. We describe this method and we present the resulting algorithm in the case of monomial ideals in a polynomial ring, in which situ- ation we always have a starting nonminimal graded free resolution.
    [Show full text]
  • Computations in Algebraic Geometry with Macaulay 2
    Computations in algebraic geometry with Macaulay 2 Editors: D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels Preface Systems of polynomial equations arise throughout mathematics, science, and engineering. Algebraic geometry provides powerful theoretical techniques for studying the qualitative and quantitative features of their solution sets. Re- cently developed algorithms have made theoretical aspects of the subject accessible to a broad range of mathematicians and scientists. The algorith- mic approach to the subject has two principal aims: developing new tools for research within mathematics, and providing new tools for modeling and solv- ing problems that arise in the sciences and engineering. A healthy synergy emerges, as new theorems yield new algorithms and emerging applications lead to new theoretical questions. This book presents algorithmic tools for algebraic geometry and experi- mental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. A wide range of mathematical scientists should find these expositions valuable. This includes both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all.
    [Show full text]
  • A Simplified Introduction to Virus Propagation Using Maple's Turtle Graphics Package
    E. Roanes-Lozano, C. Solano-Macías & E. Roanes-Macías.: A simplified introduction to virus propagation using Maple's Turtle Graphics package A simplified introduction to virus propagation using Maple's Turtle Graphics package Eugenio Roanes-Lozano Instituto de Matemática Interdisciplinar & Departamento de Didáctica de las Ciencias Experimentales, Sociales y Matemáticas Facultad de Educación, Universidad Complutense de Madrid, Spain Carmen Solano-Macías Departamento de Información y Comunicación Facultad de CC. de la Documentación y Comunicación, Universidad de Extremadura, Spain Eugenio Roanes-Macías Departamento de Álgebra, Universidad Complutense de Madrid, Spain [email protected] ; [email protected] ; [email protected] Partially funded by the research project PGC2018-096509-B-100 (Government of Spain) 1 E. Roanes-Lozano, C. Solano-Macías & E. Roanes-Macías.: A simplified introduction to virus propagation using Maple's Turtle Graphics package 1. INTRODUCTION: TURTLE GEOMETRY AND LOGO • Logo language: developed at the end of the ‘60s • Characterized by the use of Turtle Geometry (a.k.a. as Turtle Graphics). • Oriented to introduce kids to programming (Papert, 1980). • Basic movements of the turtle (graphic cursor): FD, BK RT, LT. • It is not based on a Cartesian Coordinate system. 2 E. Roanes-Lozano, C. Solano-Macías & E. Roanes-Macías.: A simplified introduction to virus propagation using Maple's Turtle Graphics package • Initially robots were used to plot the trail of the turtle. http://cyberneticzoo.com/cyberneticanimals/1969-the-logo-turtle-seymour-papert-marvin-minsky-et-al-american/ 3 E. Roanes-Lozano, C. Solano-Macías & E. Roanes-Macías.: A simplified introduction to virus propagation using Maple's Turtle Graphics package • IBM Logo / LCSI Logo (’80) 4 E.
    [Show full text]
  • What Can Computer Algebraic Geometry Do Today?
    What can computer Algebraic Geometry do today? Gregory G. Smith Wolfram Decker Mike Stillman 14 July 2015 Essential Questions ̭ What can be computed? ̭ What software is currently available? ̭ What would you like to compute? ̭ How should software advance your research? Basic Mathematical Types ̭ Polynomial Rings, Ideals, Modules, ̭ Varieties (affine, projective, toric, abstract), ̭ Sheaves, Divisors, Intersection Rings, ̭ Maps, Chain Complexes, Homology, ̭ Polyhedra, Graphs, Matroids, ̯ Established Geometric Tools ̭ Elimination, Blowups, Normalization, ̭ Rational maps, Working with divisors, ̭ Components, Parametrizing curves, ̭ Sheaf Cohomology, ঠ-modules, ̯ Emerging Geometric Tools ̭ Classification of singularities, ̭ Numerical algebraic geometry, ̭ ैक़௴Ь, Derived equivalences, ̭ Deformation theory,Positivity, ̯ Some Geometric Successes ̭ GEOGRAPHY OF SURFACES: exhibiting surfaces with given invariants ̭ BOIJ-SÖDERBERG: examples lead to new conjectures and theorems ̭ MODULI SPACES: computer aided proofs of unirationality Some Existing Software ̭ GAP,Macaulay2, SINGULAR, ̭ CoCoA, Magma, Sage, PARI, RISA/ASIR, ̭ Gfan, Polymake, Normaliz, 4ti2, ̭ Bertini, PHCpack, Schubert, Bergman, an idiosyncratic and incomplete list Effective Software ̭ USEABLE: documented examples ̭ MAINTAINABLE: includes tests, part of a larger distribution ̭ PUBLISHABLE: Journal of Software for Algebra and Geometry; www.j-sag.org ̭ CITATIONS: reference software Recent Developments in Singular Wolfram Decker Janko B¨ohm, Hans Sch¨onemann, Mathias Schulze Mohamed Barakat TU Kaiserslautern July 14, 2015 Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 1 / 24 commutative and non-commutative algebra, singularity theory, and with packages for convex and tropical geometry. It is free and open-source under the GNU General Public Licence.
    [Show full text]
  • Computational Algebraic Geometry and Quantum Mechanics: an Initiative Toward Post Contemporary Quantum Chemistry
    J Multidis Res Rev 2019 Inno Journal of Multidisciplinary Volume 1: 2 Research and Reviews Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post Contemporary Quantum Chemistry Ichio Kikuchi International Research Center for Quantum Technology, Tokyo, Japan Akihito Kikuchi* Abstract Article Information A new framework in quantum chemistry has been proposed recently Article Type: Review Article Number: JMRR118 symbolic-numeric computation” by A. Kikuchi). It is based on the Received Date: 30 September, 2019 modern(“An approach technique to first of principlescomputational electronic algebraic structure geometry, calculation viz. theby Accepted Date: 18 October, 2019 symbolic computation of polynomial systems. Although this framework Published Date: 25 October, 2019 belongs to molecular orbital theory, it fully adopts the symbolic method. The analytic integrals in the secular equations are approximated by the *Corresponding Author: Akihito Kikuchi, Independent polynomials. The indeterminate variables of polynomials represent Researcher, Member of International Research Center for the wave-functions and other parameters for the optimization, such as Quantum Technology, 2-17-7 Yaguchi, Ota-ku, Tokyo, Japan. Tel: +81-3-3759-6810; Email: akihito_kikuchi(at)gakushikai.jp the symbolic computation digests and decomposes the polynomials into aatomic tame positionsform of the and set contraction of equations, coefficients to which ofnumerical atomic orbitals.computations Then Citation: Kikuchi A, Kikuchi I (2019) Computational are easily applied. The key technique is Gröbner basis theory, by Algebraic Geometry and Quantum Mechanics: An Initiative which one can investigate the electronic structure by unraveling the toward Post Contemporary Quantum Chemistry. J Multidis Res Rev Vol: 1, Issu: 2 (47-79). demonstrate the featured result of this new theory.
    [Show full text]
  • Modeling and Analysis of Hybrid Systems
    Building Bridges between Symbolic Computation and Satisfiability Checking Erika Abrah´ am´ RWTH Aachen University, Germany in cooperation with Florian Corzilius, Gereon Kremer, Stefan Schupp and others ISSAC’15, 7 July 2015 Photo: Prior Park, Bath / flickr Liam Gladdy What is this talk about? Satisfiability problem The satisfiability problem is the problem of deciding whether a logical formula is satisfiable. We focus on the automated solution of the satisfiability problem for first-order logic over arithmetic theories, especially on similarities and differences in symbolic computation and SAT and SMT solving. Erika Abrah´ am´ - SMT solving and Symbolic Computation 2 / 39 CAS SAT SMT (propositional logic) (SAT modulo theories) Enumeration Computer algebra DP (resolution) systems [Davis, Putnam’60] DPLL (propagation) [Davis,Putnam,Logemann,Loveland’62] Decision procedures NP-completeness [Cook’71] for combined theories CAD Conflict-directed [Shostak’79] [Nelson, Oppen’79] backjumping Partial CAD Virtual CDCL [GRASP’97] [zChaff’04] DPLL(T) substitution Watched literals Equalities and uninterpreted Clause learning/forgetting functions Variable ordering heuristics Bit-vectors Restarts Array theory Arithmetic Decision procedures for first-order logic over arithmetic theories in mathematical logic 1940 Computer architecture development 1960 1970 1980 2000 2010 Erika Abrah´ am´ - SMT solving and Symbolic Computation 3 / 39 SAT SMT (propositional logic) (SAT modulo theories) Enumeration DP (resolution) [Davis, Putnam’60] DPLL (propagation) [Davis,Putnam,Logemann,Loveland’62]
    [Show full text]
  • Combinatorics of Rank Jumps in Simplicial Hypergeometric Systems
    COMBINATORICS OF RANK JUMPS IN SIMPLICIAL HYPERGEOMETRIC SYSTEMS LAURA FELICIA MATUSEVICH AND EZRA MILLER Abstract. Let A be an integer d × n matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d − 1. It is known that the semigroup ring C[NA] is Cohen–Macaulay if and only if the rank of the GKZ hypergeometric system HA(β) equals the normalized volume of conv(A) for all complex parameters β ∈ Cd [Sai02]. Our refinement here shows that HA(β) has rank strictly larger than the volume of conv(A) if and only if β lies in the Zariski closure (in Cd) of all Zd-graded degrees where the local L i cohomology i<d Hm(C[NA]) is nonzero. References [Ado94] Alan Adolphson, Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), no. 2, 269–290. [BH93] Winfried Bruns and J¨urgenHerzog, Cohen-Macaulay rings, Cambridge University Press, Cam- bridge, 1993. [CDD99] Eduardo Cattani, Carlos D’Andrea, and Alicia Dickenstein, The A-hypergeometric system asso- ciated with a monomial curve, Duke Math. J. 99 (1999), no. 2, 179–207. [CoC] CoCoATeam, CoCoA: a system for doing computations in commutative algebra, available at http://cocoa.dima.unige.it/. [GGZ87] I. M. Gelfand, M. I. Graev, and A. V. Zelevinsky, Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR 295 (1987), no. 1, 14–19. [GPS01] G.-M. Greuel, G. Pfister, and H. Sch¨onemann, Singular 2.0, A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern, 2001, http://www.singular.uni-kl.de/.
    [Show full text]
  • Vidya to Host AICTE Sponsored STTP
    Vidya to host AICTE sponsored STTP The Department of Mechanical Engineering and the Department of Applied Sciences are jointly organising an AICTE Sponsored online Short Term Training Program (STTP) on “Software Packages for Mathematics in Engineering” during 16 November – 28 December 2020. The STTP is of five days’ duration and it will be repeated in five different slots with identical content. The various slots are 16 – 21 November 2020, 23 – 28 November 2020, 14 – 19 December 2020 and 21 – 28 December 2020. The organisers are planning to give intensive training to the participants in various software packages like MATLAB, SPSS, R, Demetra, SageMath, CFD, Axiom, MAXIMA, GAP, Cadabra, CoCoA, Xcas, PARI/GP, and Sympy. Mathematical techniques provide a scientific base for engineering and a good mathematical tool is a stepping stone for engineering education. Software packages for mathematics improve the understanding of concepts with visualizations and explanations. The traditional teaching methodologies are limited to solve problems manually involving vectors in a three-dimensional space, matrices of order three by three, and third-degree ordinary differential equations. Ultimately software packages provide the solution to problems of higher dimension, help in geometrical interpretation, and also efficient programs and thus reduce time complexity. Incorporating these software packages for problem-solving techniques significantly supports the existing teaching methodologies. There is a wide range of mathematical software available worldwide of which some are open to all. These packages help in solving simple to advanced problems representing various real-life mathematical models not limited to engineering problems. Awareness of these packages to our teaching community, students, research scholars, and industrialists is made through this programme.
    [Show full text]
  • 2 Computers in Education Journal Benchmarking
    BENCHMARKING SOFTWARE FOR SOLVING MULTI-POLYNOMIAL SYSTEMS Richard V. Schmidt and Mark J. DeBonis Department of Mathematics Manhattan College Abstract the runtime and accuracy of the new algorithm versus the computer algebraic systems Singular, Benchmarking different software which CoCoA [3] and Macaulay2 [4]. performs the same task gives students in computer science and computer engineering an Method opportunity to develop important skills. It also demonstrates how effective benchmarking can Solving systems of nonlinear equations is a be applied to computer application packages well-known NP-complete problem, and no designed for solving systems of multi- polynomial-time algorithm is known for solving polynomial equations versus a new proposed any NP-complete problem. Some known method by the second author methods for solving multi-polynomial systems employ Gröbner bases and resultants [5]. The Introduction run time for the Gröbner basis algorithm is exponential in 2d, or doubly exponential [6] [7], The second author has proposed a new way where d is the number of unknowns. With to solve multi-polynomial equations all of the resultants, the dimension of the determinant same total degree in which the number of which needs to be computed in order to solve equations equals the number of unknowns. the system grows at an alarming rate. For Below is a simple example of such a system instance, d multi-polynomial equations in d with three equations in three unknowns. variables each of degree n yields a determinant d-1 of dimension 2d-1n2 -1 [8]. 2 2 2 4x + 6xy + 6xz + 2y + 8yz + 3z = 1 2 2 2 2x + 3xy + 7xz - 3y - 3yz - 3z = 0 The new algorithm proposed by the second 2 2 2 4x + 7xy + 2xz + y - 7yz - 2z = -4 author is a generalization of a standard technique for solving systems of linear Such systems are found in many applied equations by multiplying by the inverse of the mathematical and scientific fields.
    [Show full text]
  • Appendix a Some Concepts from Algebra
    Appendix A Some Concepts from Algebra This appendix contains precise statements of various algebraic facts and definitions used in the text. For students who have had a course in abstract algebra, much of this material will be familiar. For students seeing these terms for the first time, keep in mind that the abstract concepts defined here are used in the text in very concrete situations. §1 Fields and Rings We first give a precise definition of a field. Definition 1. A field consists of a set k and two binary operations “+” and “·” de- fined on k for which the following conditions are satisfied: (i) (a+b)+c = a+(b+c) and (a·b)·c = a·(b·c) for all a, b, c ∈ k (associativity). (ii) a + b = b + a and a · b = b · a for all a, b ∈ k (commutativity). (iii) a · (b + c)=a · b + a · c for all a, b, c ∈ k (distributivity). (iv) There are 0, 1 ∈ k such that a + 0 = a · 1 = a for all a ∈ k (identities). (v) Given a ∈ k, there is b ∈ k such that a + b = 0 (additive inverses). (vi) Given a ∈ k, a = 0, there is c ∈ k such that a · c = 1 (multiplicative inverses). The fields most commonly used in the text are Q, R, and C. In the exercises to §1 of Chapter 1, we mention the field F2 which consists of the two elements 0 and 1. Some more complicated fields are discussed in the text. For example, in §3 of Chapter 1, we define the field k(t1,...,tm) of rational functions in t1,...,tm with coefficients in k.
    [Show full text]