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Plotting Direction Fields in Matlab and Maxima – a Short Tutorial
Plotting direction fields in Matlab and Maxima – a short tutorial Luis Carvalho Introduction A first order differential equation can be expressed as dx x0(t) = = f(t; x) (1) dt where t is the independent variable and x is the dependent variable (on t). Note that the equation above is in normal form. The difficulty in solving equation (1) depends clearly on f(t; x). One way to gain geometric insight on how the solution might look like is to observe that any solution to (1) should be such that the slope at any point P (t0; x0) is f(t0; x0), since this is the value of the derivative at P . With this motivation in mind, if you select enough points and plot the slopes in each of these points, you will obtain a direction field for ODE (1). However, it is not easy to plot such a direction field for any function f(t; x), and so we must resort to some computational help. There are many computational packages to help us in this task. This is a short tutorial on how to plot direction fields for first order ODE’s in Matlab and Maxima. Matlab Matlab is a powerful “computing environment that combines numeric computation, advanced graphics and visualization” 1. A nice package for plotting direction field in Matlab (although resourceful, Matlab does not provide such facility natively) can be found at http://math.rice.edu/»dfield, contributed by John C. Polking from the Dept. of Mathematics at Rice University. To install this add-on, simply browse to the files for your version of Matlab and download them. -
A Note on Integral Points on Elliptic Curves 3
Journal de Th´eorie des Nombres de Bordeaux 00 (XXXX), 000–000 A note on integral points on elliptic curves par Mark WATKINS Abstract. We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writ- ing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have non- degenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of Macsyma, with there being a unique rational nondegenerate solution. For the second case we found that resultants and/or Gr¨obner bases were not very efficacious. Instead, at the suggestion of Elkies, we used multidimensional p-adic Newton iteration, and were able to find a nondegenerate solution, albeit over a quartic number field. Due to our methodol- ogy, we do not have much hope of proving that there are no other solutions. For the third case we found a solution in a nonic number field, but we were unable to make much progress with the fourth case. We make a few concluding comments and include an ap- pendix from Elkies regarding his calculations and correspondence with Zagier. Resum´ e.´ A` la suite de Zagier et Elkies, nous recherchons de grands points entiers sur des courbes elliptiques. En ´ecrivant une solution polynomiale g´en´erique et en ´egalisant des coefficients, nous obtenons quatre cas extr´emaux susceptibles d’avoir des so- lutions non d´eg´en´er´ees. Chacun de ces cas conduit `aun syst`eme d’´equations polynomiales, le premier ´etant r´esolu par Elkies en 1988 en utilisant les r´esultants de Macsyma; il admet une unique solution rationnelle non d´eg´en´er´ee. -
Some Reflections About the Success and Impact of the Computer Algebra System DERIVE with a 10-Year Time Perspective
Noname manuscript No. (will be inserted by the editor) Some reflections about the success and impact of the computer algebra system DERIVE with a 10-year time perspective Eugenio Roanes-Lozano · Jose Luis Gal´an-Garc´ıa · Carmen Solano-Mac´ıas Received: date / Accepted: date Abstract The computer algebra system DERIVE had a very important im- pact in teaching mathematics with technology, mainly in the 1990's. The au- thors analyze the possible reasons for its success and impact and give personal conclusions based on the facts collected. More than 10 years after it was discon- tinued it is still used for teaching and several scientific papers (most devoted to educational issues) still refer to it. A summary of the history, journals and conferences together with a brief bibliographic analysis are included. Keywords Computer algebra systems · DERIVE · educational software · symbolic computation · technology in mathematics education Eugenio Roanes-Lozano Instituto de Matem´aticaInterdisciplinar & Departamento de Did´actica de las Ciencias Experimentales, Sociales y Matem´aticas, Facultad de Educaci´on,Universidad Complutense de Madrid, c/ Rector Royo Villanova s/n, 28040-Madrid, Spain Tel.: +34-91-3946248 Fax: +34-91-3946133 E-mail: [email protected] Jose Luis Gal´an-Garc´ıa Departamento de Matem´atica Aplicada, Universidad de M´alaga, M´alaga, c/ Dr. Ortiz Ramos s/n. Campus de Teatinos, E{29071 M´alaga,Spain Tel.: +34-952-132873 Fax: +34-952-132766 E-mail: [email protected] Carmen Solano-Mac´ıas Departamento de Informaci´ony Comunicaci´on,Universidad de Extremadura, Plaza de Ibn Marwan s/n, 06001-Badajoz, Spain Tel.: +34-924-286400 Fax: +34-924-286401 E-mail: [email protected] 2 Eugenio Roanes-Lozano et al. -
Redberry: a Computer Algebra System Designed for Tensor Manipulation
Redberry: a computer algebra system designed for tensor manipulation Stanislav Poslavsky Institute for High Energy Physics, Protvino, Russia SRRC RF ITEP of NRC Kurchatov Institute, Moscow, Russia E-mail: [email protected] Dmitry Bolotin Institute of Bioorganic Chemistry of RAS, Moscow, Russia E-mail: [email protected] Abstract. In this paper we focus on the main aspects of computer-aided calculations with tensors and present a new computer algebra system Redberry which was specifically designed for algebraic tensor manipulation. We touch upon distinctive features of tensor software in comparison with pure scalar systems, discuss the main approaches used to handle tensorial expressions and present the comparison of Redberry performance with other relevant tools. 1. Introduction General-purpose computer algebra systems (CASs) have become an essential part of many scientific calculations. Focusing on the area of theoretical physics and particularly high energy physics, one can note that there is a wide area of problems that deal with tensors (or more generally | objects with indices). This work is devoted to the algebraic manipulations with abstract indexed expressions which forms a substantial part of computer aided calculations with tensors in this field of science. Today, there are many packages both on top of general-purpose systems (Maple Physics [1], xAct [2], Tensorial etc.) and standalone tools (Cadabra [3,4], SymPy [5], Reduce [6] etc.) that cover different topics in symbolic tensor calculus. However, it cannot be said that current demand on such a software is fully satisfied [7]. The main difference of tensorial expressions (in comparison with ordinary indexless) lies in the presence of contractions between indices. -
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. -
Getting Started Computing at the Al Lab by Christopher C. Stacy Abstract
MASSACHUSETTS INSTITUTE OF TECHNOLOGY ARTIFICIAL INTELLI..IGENCE LABORATORY WORKING PAPER 235 7 September 1982 Getting Started Computing at the Al Lab by Christopher C. Stacy Abstract This document describes the computing facilities at the M.I.T. Artificial Intelligence Laboratory, and explains how to get started using them. It is intended as an orientation document for newcomers to the lab, and will be updated by the author from time to time. A.I. Laboratory Working Papers are produced for internal circulation. and may contain information that is, for example, too preliminary or too detailed for formal publication. It is not intended that they should be considered papers to which reference can be made in the literature. a MASACHUSETS INSTITUTE OF TECHNOLOGY 1982 Getting Started Table of Contents Page i Table of Contents 1. Introduction 1 1.1. Lisp Machines 2 1.2. Timesharing 3 1.3. Other Computers 3 1.3.1. Field Engineering 3 1.3.2. Vision and Robotics 3 1.3.3. Music 4 1,3.4. Altos 4 1.4. Output Peripherals 4 1.5. Other Machines 5 1.6. Terminals 5 2. Networks 7 2.1. The ARPAnet 7 2.2. The Chaosnet 7 2.3. Services 8 2.3.1. TELNET/SUPDUP 8 2.3.2. FTP 8 2.4. Mail 9 2.4.1. Processing Mail 9 2.4.2. Ettiquette 9 2.5. Mailing Lists 10 2.5.1. BBoards 11 2.6. Finger/Inquire 11 2.7. TIPs and TACs 12 2.7.1. ARPAnet TAC 12 2.7.2. Chaosnet TIP 13 3. -
Computer Algebra Tems
short and expensive. Consequently, computer centre managers were not easily persuaded to install such sys Computer Algebra tems. This is another reason why many specialized CAS have been developed in many different institutions. From the Jacques Calmet, Grenoble * very beginning it was felt that the (UFIA / INPG) breakthrough of Computer Algebra is linked to the availability of adequate computers: several megabytes of main As soon as computers became avai handling of very large pieces of algebra memory and very large capacity disks. lable the wish to manipulate symbols which would be hopeless without com Since the advent of the VAX's and the rather than numbers was considered. puter aid, the possibility to get insight personal workstations, this era is open This resulted in what is called symbolic into a calculation, to experiment ideas ed and indeed the use of CAS is computing. The sub-field of symbolic interactively and to save time for less spreading very quickly. computing which deals with the sym technical problems. When compared to What are the main CAS of possible bolic solution of mathematical problems numerical computing, it is also much interest to physicists? A fairly balanced is known today as Computer Algebra. closer to human reasoning. answer is to quote MACSYMA, RE The discipline was rapidly organized, in DUCE, SMP, MAPLE among the general 1962, by a special interest group on Computer Algebra Systems purpose ones and SCHOONSHIP, symbolic and algebraic manipulation, Computer Algebra systems may be SHEEP and CAYLEY among the specia SIGSAM, of the Association for Com classified into two different categories: lized ones. -
The Evolution of Lisp
1 The Evolution of Lisp Guy L. Steele Jr. Richard P. Gabriel Thinking Machines Corporation Lucid, Inc. 245 First Street 707 Laurel Street Cambridge, Massachusetts 02142 Menlo Park, California 94025 Phone: (617) 234-2860 Phone: (415) 329-8400 FAX: (617) 243-4444 FAX: (415) 329-8480 E-mail: [email protected] E-mail: [email protected] Abstract Lisp is the world’s greatest programming language—or so its proponents think. The structure of Lisp makes it easy to extend the language or even to implement entirely new dialects without starting from scratch. Overall, the evolution of Lisp has been guided more by institutional rivalry, one-upsmanship, and the glee born of technical cleverness that is characteristic of the “hacker culture” than by sober assessments of technical requirements. Nevertheless this process has eventually produced both an industrial- strength programming language, messy but powerful, and a technically pure dialect, small but powerful, that is suitable for use by programming-language theoreticians. We pick up where McCarthy’s paper in the first HOPL conference left off. We trace the development chronologically from the era of the PDP-6, through the heyday of Interlisp and MacLisp, past the ascension and decline of special purpose Lisp machines, to the present era of standardization activities. We then examine the technical evolution of a few representative language features, including both some notable successes and some notable failures, that illuminate design issues that distinguish Lisp from other programming languages. We also discuss the use of Lisp as a laboratory for designing other programming languages. We conclude with some reflections on the forces that have driven the evolution of Lisp. -
Mathematical Modeling with Technology PURNIMA RAI
International Journal of Innovative Research in Computer Science & Technology (IJIRCST) ISSN: 2347-5552, Volume-3, Issue-2, March 2015 Mathematical Modeling with Technology PURNIMA RAI Abstract— Mathematics is very important for an Computer Algebraic System (CAS) increasing number of technical jobs in a quantitative A Computer Algebra system is a type of software package world. Over recent years technology has progressed to that is used in manipulation of mathematical formulae. The the point where many computer algebraic systems primary goal of a Computer Algebra system is to automate (CAS) are widely available. These technologies can tedious and sometimes difficult algebraic manipulation perform a wide variety of algebraic tasks. Computer Algebra systems often include facilities for manipulations.Computer algebra systems (CAS) have graphing equations and provide a programming language become powerful and flexible tools to support a variety for the user to define their own procedures. Computer of mathematical activities. This includes learning and Algebra systems have not only changed how mathematics is teaching, research and assessment. MATLAB, an taught at many universities, but have provided a flexible acronym for “Matrix Laboratory”, is the most tool for mathematicians worldwide. Examples of popular extensively used mathematical software in the general systems include Maple, Mathematica, and MathCAD. sciences. MATLAB is an interactive system for Computer Algebra systems can be used to simplify rational numerical computation and a programmable system. functions, factor polynomials, find the solutions to a system This paper attempts to investigate the impact of of equation, and various other manipulations. In Calculus, integrating mathematical modeling and applications on they can be used to find the limit of, symbolically integrate, higher level mathematics in India. -
Arxiv:Cs/0608005V2 [Cs.SC] 12 Jun 2007
AEI-2006-037 cs.SC/0608005 A field-theory motivated approach to symbolic computer algebra Kasper Peeters Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut Am M¨uhlenberg 1, 14476 Golm, GERMANY Abstract Field theory is an area in physics with a deceptively compact notation. Although general pur- pose computer algebra systems, built around generic list-based data structures, can be used to represent and manipulate field-theory expressions, this often leads to cumbersome input formats, unexpected side-effects, or the need for a lot of special-purpose code. This makes a direct trans- lation of problems from paper to computer and back needlessly time-consuming and error-prone. A prototype computer algebra system is presented which features TEX-like input, graph data structures, lists with Young-tableaux symmetries and a multiple-inheritance property system. The usefulness of this approach is illustrated with a number of explicit field-theory problems. 1. Field theory versus general-purpose computer algebra For good reasons, the area of general-purpose computer algebra programs has histor- ically been dominated by what one could call “list-based” systems. These are systems which are centred on the idea that, at the lowest level, mathematical expressions are nothing else but nested lists (or equivalently: nested functions, trees, directed acyclic graphs, . ). There is no doubt that a lot of mathematics indeed maps elegantly to problems concerning the manipulation of nested lists, as the success of a large class of LISP-based computer algebra systems illustrates (either implemented in LISP itself or arXiv:cs/0608005v2 [cs.SC] 12 Jun 2007 in another language with appropriate list data structures). -
Using Xcas in Calculus Curricula: a Plan of Lectures and Laboratory Projects
Computational and Applied Mathematics Journal 2015; 1(3): 131-138 Published online April 30, 2015 (http://www.aascit.org/journal/camj) Using Xcas in Calculus Curricula: a Plan of Lectures and Laboratory Projects George E. Halkos, Kyriaki D. Tsilika Laboratory of Operations Research, Department of Economics, University of Thessaly, Volos, Greece Email address [email protected] (K. D. Tsilika) Citation George E. Halkos, Kyriaki D. Tsilika. Using Xcas in Calculus Curricula: a Plan of Lectures and Keywords Laboratory Projects. Computational and Applied Mathematics Journal. Symbolic Computations, Vol. 1, No. 3, 2015, pp. 131-138. Computer-Based Education, Abstract Xcas Computer Software We introduce a topic in the intersection of symbolic mathematics and computation, concerning topics in multivariable Optimization and Dynamic Analysis. Our computational approach gives emphasis to mathematical methodology and aims at both symbolic and numerical results as implemented by a powerful digital mathematical tool, CAS software Received: March 31, 2015 Xcas. This work could be used as guidance to develop course contents in advanced Revised: April 20, 2015 calculus curricula, to conduct individual or collaborative projects for programming related Accepted: April 21, 2015 objectives, as Xcas is freely available to users and institutions. Furthermore, it could assist educators to reproduce calculus methodologies by generating automatically, in one entry, abstract calculus formulations. 1. Introduction Educational institutions are equipped with computer labs while modern teaching methods give emphasis in computer based learning, with curricula that include courses supported by the appropriate computer software. It has also been established that, software tools integrate successfully into Mathematics education and are considered essential in teaching Geometry, Statistics or Calculus (see indicatively [1], [2]). -
2.10. Sympy : Symbolic Mathematics in Python
2.10. Sympy : Symbolic Mathematics in Python author: Fabian Pedregosa Objectives 1. Evaluate expressions with arbitrary precision. 2. Perform algebraic manipulations on symbolic expressions. 3. Perform basic calculus tasks (limits, differentiation and integration) with symbolic expressions. 4. Solve polynomial and transcendental equations. 5. Solve some differential equations. What is SymPy? SymPy is a Python library for symbolic mathematics. It aims become a full featured computer algebra system that can compete directly with commercial alternatives (Mathematica, Maple) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python and does not require any external libraries. Sympy documentation and packages for installation can be found on http://sympy.org/ Chapters contents First Steps with SymPy Using SymPy as a calculator Exercises Symbols Algebraic manipulations Expand Simplify Exercises Calculus Limits Differentiation Series expansion Exercises Integration Exercises Equation solving Exercises Linear Algebra Matrices Differential Equations Exercises 2.10.1. First Steps with SymPy 2.10.1.1. Using SymPy as a calculator SymPy defines three numerical types: Real, Rational and Integer. The Rational class represents a rational number as a pair of two Integers: the numerator and the denomi‐ nator, so Rational(1,2) represents 1/2, Rational(5,2) 5/2 and so on: >>> from sympy import * >>> >>> a = Rational(1,2) >>> a 1/2 >>> a*2 1 SymPy uses mpmath in the background, which makes it possible to perform computations using arbitrary- precision arithmetic. That way, some special constants, like e, pi, oo (Infinity), are treated as symbols and can be evaluated with arbitrary precision: >>> pi**2 >>> pi**2 >>> pi.evalf() 3.14159265358979 >>> (pi+exp(1)).evalf() 5.85987448204884 as you see, evalf evaluates the expression to a floating-point number.