Plotting Direction Fields in Matlab and Maxima – a Short Tutorial
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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. -
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. -
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). -
Sage Tutorial (Pdf)
Sage Tutorial Release 9.4 The Sage Development Team Aug 24, 2021 CONTENTS 1 Introduction 3 1.1 Installation................................................4 1.2 Ways to Use Sage.............................................4 1.3 Longterm Goals for Sage.........................................5 2 A Guided Tour 7 2.1 Assignment, Equality, and Arithmetic..................................7 2.2 Getting Help...............................................9 2.3 Functions, Indentation, and Counting.................................. 10 2.4 Basic Algebra and Calculus....................................... 14 2.5 Plotting.................................................. 20 2.6 Some Common Issues with Functions.................................. 23 2.7 Basic Rings................................................ 26 2.8 Linear Algebra.............................................. 28 2.9 Polynomials............................................... 32 2.10 Parents, Conversion and Coercion.................................... 36 2.11 Finite Groups, Abelian Groups...................................... 42 2.12 Number Theory............................................. 43 2.13 Some More Advanced Mathematics................................... 46 3 The Interactive Shell 55 3.1 Your Sage Session............................................ 55 3.2 Logging Input and Output........................................ 57 3.3 Paste Ignores Prompts.......................................... 58 3.4 Timing Commands............................................ 58 3.5 Other IPython -
Lisp: Program Is Data
LISP: PROGRAM IS DATA A HISTORICAL PERSPECTIVE ON MACLISP Jon L White Laboratory for Computer Science, M.I.T.* ABSTRACT For over 10 years, MACLISP has supported a variety of projects at M.I.T.'s Artificial Intelligence Laboratory, and the Laboratory for Computer Science (formerly Project MAC). During this time, there has been a continuing development of the MACLISP system, spurred in great measure by the needs of MACSYMAdevelopment. Herein are reported, in amosiac, historical style, the major features of the system. For each feature discussed, an attempt will be made to mention the year of initial development, andthe names of persons or projectsprimarily responsible for requiring, needing, or suggestingsuch features. INTRODUCTION In 1964,Greenblatt and others participated in thecheck-out phase of DigitalEquipment Corporation's new computer, the PDP-6. This machine had a number of innovative features that were thought to be ideal for the development of a list processing system, and thus it was very appropriate that thefirst working program actually run on thePDP-6 was anancestor of thecurrent MACLISP. This earlyLISP was patterned after the existing PDP-1 LISP (see reference l), and was produced by using the text editor and a mini-assembler on the PDP-1. That first PDP-6 finally found its way into M.I.T.'s ProjectMAC for use by theArtificial lntelligence group (the A.1. grouplater became the M.I.T. Artificial Intelligence Laboratory, and Project MAC became the Laboratory for Computer Science). By 1968, the PDP-6 wasrunning the Incompatible Time-sharing system, and was soon supplanted by the PDP-IO.Today, the KL-I 0, anadvanced version of thePDP-10, supports a variety of time sharing systems, most of which are capable of running a MACLISP. -
Highlighting Wxmaxima in Calculus
mathematics Article Not Another Computer Algebra System: Highlighting wxMaxima in Calculus Natanael Karjanto 1,* and Husty Serviana Husain 2 1 Department of Mathematics, University College, Natural Science Campus, Sungkyunkwan University Suwon 16419, Korea 2 Department of Mathematics Education, Faculty of Mathematics and Natural Science Education, Indonesia University of Education, Bandung 40154, Indonesia; [email protected] * Correspondence: [email protected] Abstract: This article introduces and explains a computer algebra system (CAS) wxMaxima for Calculus teaching and learning at the tertiary level. The didactic reasoning behind this approach is the need to implement an element of technology into classrooms to enhance students’ understanding of Calculus concepts. For many mathematics educators who have been using CAS, this material is of great interest, particularly for secondary teachers and university instructors who plan to introduce an alternative CAS into their classrooms. By highlighting both the strengths and limitations of the software, we hope that it will stimulate further debate not only among mathematics educators and software users but also also among symbolic computation and software developers. Keywords: computer algebra system; wxMaxima; Calculus; symbolic computation Citation: Karjanto, N.; Husain, H.S. 1. Introduction Not Another Computer Algebra A computer algebra system (CAS) is a program that can solve mathematical problems System: Highlighting wxMaxima in by rearranging formulas and finding a formula that solves the problem, as opposed to Calculus. Mathematics 2021, 9, 1317. just outputting the numerical value of the result. Maxima is a full-featured open-source https://doi.org/10.3390/ CAS: the software can serve as a calculator, provide analytical expressions, and perform math9121317 symbolic manipulations. -
General Relativity Computations with Sagemanifolds
General relativity computations with SageManifolds Eric´ Gourgoulhon Laboratoire Univers et Th´eories (LUTH) CNRS / Observatoire de Paris / Universit´eParis Diderot 92190 Meudon, France http://luth.obspm.fr/~luthier/gourgoulhon/ NewCompStar School 2016 Neutron stars: gravitational physics theory and observations Coimbra (Portugal) 5-9 September 2016 Eric´ Gourgoulhon (LUTH) GR computations with SageManifolds NewCompStar, Coimbra, 6 Sept. 2016 1 / 29 Outline 1 Computer differential geometry and tensor calculus 2 The SageManifolds project 3 Let us practice! 4 Other examples 5 Conclusion and perspectives Eric´ Gourgoulhon (LUTH) GR computations with SageManifolds NewCompStar, Coimbra, 6 Sept. 2016 2 / 29 Computer differential geometry and tensor calculus Outline 1 Computer differential geometry and tensor calculus 2 The SageManifolds project 3 Let us practice! 4 Other examples 5 Conclusion and perspectives Eric´ Gourgoulhon (LUTH) GR computations with SageManifolds NewCompStar, Coimbra, 6 Sept. 2016 3 / 29 In 1965, J.G. Fletcher developed the GEOM program, to compute the Riemann tensor of a given metric In 1969, during his PhD under Pirani supervision, Ray d'Inverno wrote ALAM (Atlas Lisp Algebraic Manipulator) and used it to compute the Riemann tensor of Bondi metric. The original calculations took Bondi and his collaborators 6 months to go. The computation with ALAM took 4 minutes and yielded to the discovery of 6 errors in the original paper [J.E.F. Skea, Applications of SHEEP (1994)] Since then, many softwares for tensor calculus have been developed... Computer differential geometry and tensor calculus Introduction Computer algebra system (CAS) started to be developed in the 1960's; for instance Macsyma (to become Maxima in 1998) was initiated in 1968 at MIT Eric´ Gourgoulhon (LUTH) GR computations with SageManifolds NewCompStar, Coimbra, 6 Sept. -
Programming for Computations – Python
15 Svein Linge · Hans Petter Langtangen Programming for Computations – Python Editorial Board T. J.Barth M.Griebel D.E.Keyes R.M.Nieminen D.Roose T.Schlick Texts in Computational 15 Science and Engineering Editors Timothy J. Barth Michael Griebel David E. Keyes Risto M. Nieminen Dirk Roose Tamar Schlick More information about this series at http://www.springer.com/series/5151 Svein Linge Hans Petter Langtangen Programming for Computations – Python A Gentle Introduction to Numerical Simulations with Python Svein Linge Hans Petter Langtangen Department of Process, Energy and Simula Research Laboratory Environmental Technology Lysaker, Norway University College of Southeast Norway Porsgrunn, Norway On leave from: Department of Informatics University of Oslo Oslo, Norway ISSN 1611-0994 Texts in Computational Science and Engineering ISBN 978-3-319-32427-2 ISBN 978-3-319-32428-9 (eBook) DOI 10.1007/978-3-319-32428-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2016945368 Mathematic Subject Classification (2010): 26-01, 34A05, 34A30, 34A34, 39-01, 40-01, 65D15, 65D25, 65D30, 68-01, 68N01, 68N19, 68N30, 70-01, 92D25, 97-04, 97U50 © The Editor(s) (if applicable) and the Author(s) 2016 This book is published open access. Open Access This book is distributed under the terms of the Creative Commons Attribution-Non- Commercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, a link is provided to the Creative Commons license and any changes made are indicated. -
They Have Very Good Docs At
Intro to the Julia programming language Brendan O’Connor CMU, Dec 2013 They have very good docs at: http://julialang.org/ I’m borrowing some slides from: http://julialang.org/blog/2013/03/julia-tutorial-MIT/ 1 Tuesday, December 17, 13 Julia • A relatively new, open-source numeric programming language that’s both convenient and fast • Version 0.2. Still in flux, especially libraries. But the basics are very usable. • Lots of development momentum 2 Tuesday, December 17, 13 Why Julia? Dynamic languages are extremely popular for numerical work: ‣ Matlab, R, NumPy/SciPy, Mathematica, etc. ‣ very simple to learn and easy to do research in However, all have a “split language” approach: ‣ high-level dynamic language for scripting low-level operations ‣ C/C++/Fortran for implementing fast low-level operations Libraries in C — no productivity boost for library writers Forces vectorization — sometimes a scalar loop is just better slide from ?? 2012 3 Bezanson, Karpinski, Shah, Edelman Tuesday, December 17, 13 “Gang of Forty” Matlab Maple Mathematica SciPy SciLab IDL R Octave S-PLUS SAS J APL Maxima Mathcad Axiom Sage Lush Ch LabView O-Matrix PV-WAVE Igor Pro OriginLab FreeMat Yorick GAUSS MuPad Genius SciRuby Ox Stata JLab Magma Euler Rlab Speakeasy GDL Nickle gretl ana Torch7 slide from March 2013 4 Bezanson, Karpinski, Shah, Edelman Tuesday, December 17, 13 Numeric programming environments Core properties Dynamic Fast? and math-y? C/C++/ Fortran/Java − + Matlab + − Num/SciPy + − R + − Older table: http://brenocon.com/blog/2009/02/comparison-of-data-analysis-packages-r-matlab-scipy-excel-sas-spss-stata/ Tuesday, December 17, 13 - Dynamic vs Fast: the usual tradeof - PL quality: more subjective. -
Teaching Dynamical Systems with Maxima 1. Mechanics For
125 Teaching Dynamical Systems with Maxima Jaime E. Viliate University of Porto, Faculty of Engineering Rua Dr. Roberto Frias, Porto 42OO-~65, Portugal, villatec~fe.up.pt WWW home page: http://viliate.org/ We report on our experience in teaching a dynamical systems course to second year engineering students. The course has been taught with a hands-on ap proach, using a Computer Algebra System: Maxima. We have developed some additional programs to help students explore the dynamics of the systems, in a more intuitiva way and without losing too much time with computer program ming. The result has been very encouraging; students are better motivated using this approach, as compared to the more traditional methods that we used in the past. 1. Mechanics for Engineering students Our engineering students in the University of Porto have to take two courses in Physics during their first. or second year of studies. The first course is a course on Newtonian mechanics. When they attend that course, they have already taken one or two semesters of calculus, and during their high-school studies they must have already studied particle kinematics and dynamics. The main goal of the first course in physics is then to solve the equations of motion for some mechanical systems; for example, for a one-dimensional system the equations of motion are: (1) .F(x,v) (2) where x is the position, v the speed, m the mass and F is the total external force, which might depend on x and v. In general, equations 1 and 2 lead to a non-linear system. -
A Survey of User Interfaces for Computer Algebra Systems
J. Symbolic Computation (1998) 25, 127–159 A Survey of User Interfaces for Computer Algebra Systems NORBERT KAJLER† AND NEIL SOIFFER‡§ †Ecole des Mines de Paris, 60 Bd. St-Michel, 75006 Paris, France ‡Wolfram Research, Inc., 100 Trade Center Drive, Champaign, IL 61820, U.S.A. This paper surveys work within the Computer Algebra community (and elsewhere) di- rected towards improving user interfaces for scientific computation during the period 1963–1994. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. Work related to improving human interaction with computer algebra systems is the main focus of the paper. However, the paper includes additional materials on some closely related issues such as structured document editing, graphics, and communication protocols. c 1998 Academic Press Limited 1. Introduction There are several problems with current computer algebra systems (CASs) that are interface-related. These problems include: the use of an unnatural linear notation to enter and edit expressions, the inherent difficulty of selecting and modifying subexpressions with commands, and the display of large expressions that run off the screen. These problems may intimidate novice users and frustrate experienced users. The more natural and intuitive the interface (the closer it corresponds to pencil and paper manipulations), the more likely it is that people will want to take advantage of the CAS for its ability to do tedious computations and to verify derivations. -
Maxima by Example: Ch. 1, Getting Started ∗
Maxima by Example: Ch. 1, Getting Started ¤ Edwin L. Woollett Contents 1 Getting Started 2 1.1 Introduction . 2 1.1.1 What is Maxima? . 2 1.2 Using the XMaxima Interface . 2 1.2.1 Using Console - Command Line Maxima . 3 1.2.2 Text Editors . 3 1.2.3 The Maxima Manual . 3 1.3 Expressions in Maxima . 4 1.3.1 Operators in Maxima . 4 1.3.2 Numbers in Maxima . 5 1.3.3 Variables in Maxima . 6 1.3.4 Constants in Maxima . 7 1.3.5 Rational Expressions in Maxima, rat(...) . 8 1.4 List Creation and Manipulation . 9 1.4.1 The Maxima List and the Lisp List . 10 1.4.2 cons, endcons, append . 10 1.4.3 makelist and map . 11 1.4.4 Anonymous Unnamed Function lamda . 14 1.4.5 Accessing List Elements Using part(...) or Bracket Pairs . 15 1.4.6 Creating a List of Fractional Values . 15 1.5 Data Files: Read, Write, Fit, and Plot . 15 1.5.1 le search . 15 1.5.2 le search maxima . 16 1.5.3 maxima-init.mac and maxima userdir . 17 1.5.4 Viewing a File with printle . 18 1.5.5 Creating a Data File using with stdout . 18 1.5.6 Creating a List from a Data File: read nested list . 18 1.5.7 Write List to Data File with write data . 19 1.5.8 Using Random Numbers to Create Noisy Data . 19 1.5.9 Using lsquares estimates for a Fit . 21 1.5.10 Nested List $ Matrix using apply and substpart .