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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. -
Utilizing MATHEMATICA Software to Improve Students' Problem Solving
International Journal of Education and Research Vol. 7 No. 11 November 2019 Utilizing MATHEMATICA Software to Improve Students’ Problem Solving Skills of Derivative and its Applications Hiyam, Bataineh , Jordan University of Science and Technology, Jordan Ali Zoubi, Yarmouk University, Jordan Abdalla Khataybeh, Yarmouk University, Jordan Abstract Traditional methods of teaching calculus (1) at most Jordanian universities are usually used. This paper attempts to introduce an innovative approach for teaching Calculus (1) using Computerized Algebra Systems (CAS). This paper examined utilizing Mathematica as a supporting tool for the teaching learning process of Calculus (1), especially for derivative and its applications. The research created computerized educational materials using Mathematica, which was used as an approach for teaching a (25) students as experimental group and another (25) students were taught traditionally as control group. The understandings of both samples were tested using problem solving test of 6 questions. The results revealed the experimental group outscored the control group significantly on the problem solving test. The use of Mathematica not just improved students’ abilities to interpret graphs and make connection between the graph of a function and its derivative, but also motivate students’ thinking in different ways to come up with innovative solutions for unusual and non routine problems involving the derivative and its applications. This research suggests that CAS tools should be integrated to teaching calculus (1) courses. Keywords: Mathematica, Problem solving, Derivative, Derivative’s applivations. INTRODUCTION The major technological advancements assisted decision makers and educators to pay greater attention on making the learner as the focus of the learning-teaching process. The use of computer algebra systems (CAS) in teaching mathematics has proved to be more effective compared to traditional teaching methods (Irving & Bell, 2004; Dhimar & Petal, 2013; Abdul Majid, huneiti, Al- Nafa & Balachander, 2012). -
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, -
The Yacas Book of Algorithms
The Yacas Book of Algorithms by the Yacas team 1 Yacas version: 1.3.6 generated on November 25, 2014 This book is a detailed description of the algorithms used in the Yacas system for exact symbolic and arbitrary-precision numerical computations. Very few of these algorithms are new, and most are well-known. The goal of this book is to become a compendium of all relevant issues of design and implementation of these algorithms. 1This text is part of the Yacas software package. Copyright 2000{2002. Principal documentation authors: Ayal Zwi Pinkus, Serge Winitzki, Jitse Niesen. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts and no Back-Cover Texts. A copy of the license is included in the section entitled \GNU Free Documentation License". Contents 1 Symbolic algebra algorithms 3 1.1 Sparse representations . 3 1.2 Implementation of multivariate polynomials . 4 1.3 Integration . 5 1.4 Transforms . 6 1.5 Finding real roots of polynomials . 7 2 Number theory algorithms 10 2.1 Euclidean GCD algorithms . 10 2.2 Prime numbers: the Miller-Rabin test and its improvements . 10 2.3 Factorization of integers . 11 2.4 The Jacobi symbol . 12 2.5 Integer partitions . 12 2.6 Miscellaneous functions . 13 2.7 Gaussian integers . 13 3 A simple factorization algorithm for univariate polynomials 15 3.1 Modular arithmetic . 15 3.2 Factoring using modular arithmetic . -
Rapid Research with Computer Algebra Systems
doi: 10.21495/71-0-109 25th International Conference ENGINEERING MECHANICS 2019 Svratka, Czech Republic, 13 – 16 May 2019 RAPID RESEARCH WITH COMPUTER ALGEBRA SYSTEMS C. Fischer* Abstract: Computer algebra systems (CAS) are gaining popularity not only among young students and schol- ars but also as a tool for serious work. These highly complicated software systems, which used to just be regarded as toys for computer enthusiasts, have reached maturity. Nowadays such systems are available on a variety of computer platforms, starting from freely-available on-line services up to complex and expensive software packages. The aim of this review paper is to show some selected capabilities of CAS and point out some problems with their usage from the point of view of 25 years of experience. Keywords: Computer algebra system, Methodology, Wolfram Mathematica 1. Introduction The Wikipedia page (Wikipedia contributors, 2019a) defines CAS as a package comprising a set of algo- rithms for performing symbolic manipulations on algebraic objects, a language to implement them, and an environment in which to use the language. There are 35 different systems listed on the page, four of them discontinued. The oldest one, Reduce, was publicly released in 1968 (Hearn, 2005) and is still available as an open-source project. Maple (2019a) is among the most popular CAS. It was first publicly released in 1984 (Maple, 2019b) and is still popular, also among users in the Czech Republic. PTC Mathcad (2019) was published in 1986 in DOS as an engineering calculation solution, and gained popularity for its ability to work with typeset mathematical notation in combination with automatic computations. -
CAS (Computer Algebra System) Mathematica
CAS (Computer Algebra System) Mathematica- UML students can download a copy for free as part of the UML site license; see the course website for details From: Wikipedia 2/9/2014 A computer algebra system (CAS) is a software program that allows [one] to compute with mathematical expressions in a way which is similar to the traditional handwritten computations of the mathematicians and other scientists. The main ones are Axiom, Magma, Maple, Mathematica and Sage (the latter includes several computer algebras systems, such as Macsyma and SymPy). Computer algebra systems began to appear in the 1960s, and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence. A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics Martin Veltman, who designed a program for symbolic mathematics, especially High Energy Physics, called Schoonschip (Dutch for "clean ship") in 1963. Using LISP as the programming basis, Carl Engelman created MATHLAB in 1964 at MITRE within an artificial intelligence research environment. Later MATHLAB was made available to users on PDP-6 and PDP-10 Systems running TOPS-10 or TENEX in universities. Today it can still be used on SIMH-Emulations of the PDP-10. MATHLAB ("mathematical laboratory") should not be confused with MATLAB ("matrix laboratory") which is a system for numerical computation built 15 years later at the University of New Mexico, accidentally named rather similarly. The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma; a popular copyleft version of Macsyma called Maxima is actively being maintained. -
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. -
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. -
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. -
About the Polynomial System Solve Facility of Axiom, Macsyma, Maple
Ab out the Polynomial System Solve Facility of Axiom, Macsyma, Maple, Mathematica, MuPAD, and Reduce Hans-Gert Grab e Institut fur Informatik, Universitat Leipzig, Germany February 23, 1998 In memoriam to Renate. Abstract We rep ort on some exp eriences with the general purp ose Computer Algebra Systems (CAS) Axiom, Macsyma, Maple, Mathematica, MuPAD, and Reduce solving systems of p olynomial equations and the way they present their solutions. This snapshot (taken in the spring 1996) of the current power of the di erent systems in a sp ecial area concentrates b oth on CPU-times and the quality of the output. 1 Intro duction Let S := k [x ;::: ;x ] be the p olynomial ring in the variables x ;::: ;x over the eld 1 n 1 n k and B := ff ;::: ;f g S be a nite system of p olynomials. Denote by I (B ) the 1 m ideal generated by these p olynomials. One of the ma jor tasks of constructive commutative n algebra is the derivation of information ab out the structure of Z (B ) := fa 2 k : 8 f 2 B such that f (a)=0g, the set of common zero es of the system B over the algebraic closure k of k . Splitting the system into smaller ones, solving them separately, and patching all solu- tions together is often a go o d guess for a quick solution of even highly nontrivial problems. This can b e done by several techniques, e.g. characteristic sets, resultants, the Grobner fac- torizer or some ad ho c metho ds. -
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. -
GNU Texmacs User Manual Joris Van Der Hoeven
GNU TeXmacs User Manual Joris van der Hoeven To cite this version: Joris van der Hoeven. GNU TeXmacs User Manual. 2013. hal-00785535 HAL Id: hal-00785535 https://hal.archives-ouvertes.fr/hal-00785535 Preprint submitted on 6 Feb 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. GNU TEXMACS user manual Joris van der Hoeven & others Table of contents 1. Getting started ...................................... 11 1.1. Conventionsforthismanual . .......... 11 Menuentries ..................................... 11 Keyboardmodifiers ................................. 11 Keyboardshortcuts ................................ 11 Specialkeys ..................................... 11 1.2. Configuring TEXMACS ..................................... 12 1.3. Creating, saving and loading documents . ............ 12 1.4. Printingdocuments .............................. ........ 13 2. Writing simple documents ............................. 15 2.1. Generalities for typing text . ........... 15 2.2. Typingstructuredtext ........................... ......... 15 2.3. Content-basedtags