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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. -
Performance Analysis of Effective Symbolic Methods for Solving Band Matrix Slaes
Performance Analysis of Effective Symbolic Methods for Solving Band Matrix SLAEs Milena Veneva ∗1 and Alexander Ayriyan y1 1Joint Institute for Nuclear Research, Laboratory of Information Technologies, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia Abstract This paper presents an experimental performance study of implementations of three symbolic algorithms for solving band matrix systems of linear algebraic equations with heptadiagonal, pen- tadiagonal, and tridiagonal coefficient matrices. The only assumption on the coefficient matrix in order for the algorithms to be stable is nonsingularity. These algorithms are implemented using the GiNaC library of C++ and the SymPy library of Python, considering five different data storing classes. Performance analysis of the implementations is done using the high-performance computing (HPC) platforms “HybriLIT” and “Avitohol”. The experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed. An analysis of the three algorithms is performed. 1 Introduction Systems of linear algebraic equations (SLAEs) with heptadiagonal (HD), pentadiagonal (PD) and tridiagonal (TD) coefficient matrices may arise after many different scientific and engineering prob- lems, as well as problems of the computational linear algebra where finding the solution of a SLAE is considered to be one of the most important problems. On the other hand, special matrix’s char- acteristics like diagonal dominance, positive definiteness, etc. are not always feasible. The latter two points explain why there is a need of methods for solving of SLAEs which take into account the band structure of the matrices and do not have any other special requirements to them. One possible approach to this problem is the symbolic algorithms. -
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. -
SNC: a Cloud Service Platform for Symbolic-Numeric Computation Using Just-In-Time Compilation
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCC.2017.2656088, IEEE Transactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING 1 SNC: A Cloud Service Platform for Symbolic-Numeric Computation using Just-In-Time Compilation Peng Zhang1, Member, IEEE, Yueming Liu1, and Meikang Qiu, Senior Member, IEEE and SQL. Other types of Cloud services include: Software as a Abstract— Cloud services have been widely employed in IT Service (SaaS) in which the software is licensed and hosted in industry and scientific research. By using Cloud services users can Clouds; and Database as a Service (DBaaS) in which managed move computing tasks and data away from local computers to database services are hosted in Clouds. While enjoying these remote datacenters. By accessing Internet-based services over lightweight and mobile devices, users deploy diversified Cloud great services, we must face the challenges raised up by these applications on powerful machines. The key drivers towards this unexploited opportunities within a Cloud environment. paradigm for the scientific computing field include the substantial Complex scientific computing applications are being widely computing capacity, on-demand provisioning and cross-platform interoperability. To fully harness the Cloud services for scientific studied within the emerging Cloud-based services environment computing, however, we need to design an application-specific [4-12]. Traditionally, the focus is given to the parallel scientific platform to help the users efficiently migrate their applications. In HPC (high performance computing) applications [6, 12, 13], this, we propose a Cloud service platform for symbolic-numeric where substantial effort has been given to the integration of computation– SNC. -
Axiom / Fricas
Axiom / FriCAS Christoph Koutschan Research Institute for Symbolic Computation Johannes Kepler Universit¨atLinz, Austria Computer Algebra Systems 15.11.2010 Master's Thesis: The ISAC project • initiative at Graz University of Technology • Institute for Software Technology • Institute for Information Systems and Computer Media • experimental software assembling open source components with as little glue code as possible • feasibility study for a novel kind of transparent single-stepping software for applied mathematics • experimenting with concepts and technologies from • computer mathematics (theorem proving, symbolic computation, model based reasoning, etc.) • e-learning (knowledge space theory, usability engineering, computer-supported collaboration, etc.) • The development employs academic expertise from several disciplines. The challenge for research is interdisciplinary cooperation. Rewriting, a basic CAS technique This technique is used in simplification, equation solving, and many other CAS functions, and it is intuitively comprehensible. This would make rewriting useful for educational systems|if one copes with the problem, that even elementary simplifications involve hundreds of rewrites. As an example see: http://www.ist.tugraz.at/projects/isac/www/content/ publications.html#DA-M02-main \Reverse rewriting" for comprehensible justification Many CAS functions can not be done by rewriting, for instance cancelling multivariate polynomials, factoring or integration. However, respective inverse problems can be done by rewriting and produce human readable derivations. As an example see: http://www.ist.tugraz.at/projects/isac/www/content/ publications.html#GGTs-von-Polynomen Equation solving made transparent Re-engineering equation solvers in \transparent single-stepping systems" leads to types of equations, arranged in a tree. ISAC's tree of equations are to be compared with what is produced by tracing facilities of Mathematica and/or Maple. -
Estimado Profesor
México, D.F., a 15 de abril de 2016 CURRICULUM VITAE Nombre: RICARDO LÓPEZ BAUTISTA Departamento de Adscripción: CIENCIAS BASICAS, Profesor titular "A", tiempo Completo, definitivo en UAM-Azcapotzalco Correo electrónico: [email protected] Formación Profesional: DOCTOR EN CIENCIAS en la especialidad de Matemáticas por el Centro de Investigación y de Estudios Avanzados del I.P.N. (CINVESTAV) con Mención Honorífica. ESTANCIA POSTDOCTORAL Departamento de Matemáticas de la Ohio StateUniversity (USA). 2000 Líneas de investigación: 1) EVALUACIÓN EN LINEA y GENERACIÓN AUTOMATIZADA DE MILLONES DE REACTIVOS CON RETROALIMENTACION AUTOMATICA, PARA BANCOS DE EXÁMENES EN MATEMÁTICAS. a. Experto en plataforma MOODLE- LCMS (Learning Content Management System) b. Programador en diversos CAS (Computer Algebraic System) Axiom, Mathematica, Maple, Mathlab, Geogebra, Magma, SageMath, Maxima, SymPy, Xcas, Yacas, Pari-GP, Octave, GAP, CoCoA, Algebrator, MathCad 2) TEORIA ALGEBRAICA DE NUMEROS, FACTORIZACION DE ENTEROS, SEGURIDAD ELECTRONICA 3) SISTEMAS ALGEBRAICOS COMPUTACIONALES (CAS). a. INVESTIGACIÓN Y APLICACIONES DE CAS EN TEORÍA DE NÚMEROS. b. INVESTIGACIÓN Y APLICACIONES DE CAS EN MODELOS MIXTOS PARA LA EDUCACIÓN Y EVALUACIÓN EN MATEMÁTICAS 4) TECNOMATEMÁTICA EDUCATIVA EN MODELOS PRESENCIALES, NO PRESENCIALES Y MIXTOS. SISTEMAS ALGEBRAICOS COMPUTACIONALES EN LA EDUCACION Y EVALUACION EN MATEMÁTICAS. a. Portales activos para la comunidad en UAM-A: http://galois.azc.uam.mx http://evaluacionenlinea.azc.uam.mx http://matematicas.azc.uam.mx Proyecto CONACyT: Evaluación del aprendizaje en Matemáticas y propuesta de evaluaciones automáticas en línea. Aprobado en 2012, por concluir. 1 Formación de recursos humanos: Tesis de alumna de maestría: Jurado en examen doctoral. Uam Proyectos Terminales en Maestría en Ciencias de la Iztapalapa. -
Arxiv:2102.02679V1 [Cs.LO] 4 Feb 2021 HOL-ODE [10], Which Supports Reasoning for Systems of Ordinary Differential Equations (Sodes)
Certifying Differential Equation Solutions from Computer Algebra Systems in Isabelle/HOL Thomas Hickman, Christian Pardillo Laursen, and Simon Foster University of York Abstract. The Isabelle/HOL proof assistant has a powerful library for continuous analysis, which provides the foundation for verification of hybrid systems. However, Isabelle lacks automated proof support for continuous artifacts, which means that verification is often manual. In contrast, Computer Algebra Systems (CAS), such as Mathematica and SageMath, contain a wealth of efficient algorithms for matrices, differen- tial equations, and other related artifacts. Nevertheless, these algorithms are not verified, and thus their outputs cannot, of themselves, be trusted for use in a safety critical system. In this paper we integrate two CAS systems into Isabelle, with the aim of certifying symbolic solutions to or- dinary differential equations. This supports a verification technique that is both automated and trustworthy. 1 Introduction Verification of Cyber-Physical and Autonomous Systems requires that we can verify both discrete control, and continuous evolution, as envisaged by the hy- brid systems domain [1]. Whilst powerful bespoke verification tools exist, such as the KeYmaera X [2] proof assistant, software engineering requires a gen- eral framework, which can support a variety of notations and paradigms [3]. Isabelle/HOL [4] is a such a framework. Its combination of an extensible fron- tend for syntax processing, and a plug-in oriented backend, based in ML, which supports a wealth of heterogeneous semantic models and proof tools, supports a flexible platform for software development, verification, and assurance [5,6,7]. Verification of hybrid systems in Isabelle is supported by several detailed li- braries of Analysis, including Multivariate Analysis [8], Affine Arithmetic [9], and arXiv:2102.02679v1 [cs.LO] 4 Feb 2021 HOL-ODE [10], which supports reasoning for Systems of Ordinary Differential Equations (SODEs). -
Using Computer Programming As an Effective Complement To
Using Computer Programming as an Effective Complement to Mathematics Education: Experimenting with the Standards for Mathematics Practice in a Multidisciplinary Environment for Teaching and Learning with Technology in the 21st Century By Pavel Solin1 and Eugenio Roanes-Lozano2 1University of Nevada, Reno, 1664 N Virginia St, Reno, NV 89557, USA. Founder and Director of NCLab (http://nclab.com). 2Instituto de Matemática Interdisciplinar & Departamento de Didáctica de las Ciencias Experimentales, Sociales y Matemáticas, Facultad de Educación, Universidad Complutense de Madrid, c/ Rector Royo Villanova s/n, 28040 – Madrid, Spain. [email protected], [email protected] Received: 30 September 2018 Revised: 12 February 2019 DOI: 10.1564/tme_v27.3.03 Many mathematics educators are not aware of a strong 2. KAREL THE ROBOT connection that exists between the education of computer programming and mathematics. The reason may be that they Karel the Robot is a widely used educational have not been exposed to computer programming. This programming language which was introduced by Richard E. connection is worth exploring, given the current trends of Pattis in his 1981 textbook Karel the Robot: A Gentle automation and Industry 4.0. Therefore, in this paper we Introduction to the Art of Computer Programming (Pattis, take a closer look at the Common Core's eight Mathematical 1995). Let us note that Karel the Robot constitutes an Practice Standards. We show how each one of them can be environment related to Turtle Geometry (Abbelson and reinforced through computer programming. The following diSessa, 1981), but is not yet another implementation, as will discussion is virtually independent of the choice of a be detailed below. -
Addition and Subtraction
A Addition and Subtraction SUMMARY (475– 221 BCE), when arithmetic operations were per- formed by manipulating rods on a flat surface that was Addition and subtraction can be thought of as a pro- partitioned by vertical and horizontal lines. The num- cess of accumulation. For example, if a flock of 3 sheep bers were represented by a positional base- 10 system. is joined with a flock of 4 sheep, the combined flock Some scholars believe that this system— after moving will have 7 sheep. Thus, 7 is the sum that results from westward through India and the Islamic Empire— the addition of the numbers 3 and 4. This can be writ- became the modern system of representing numbers. ten as 3 + 4 = 7 where the sign “+” is read “plus” and The Greeks in the fifth century BCE, in addition the sign “=” is read “equals.” Both 3 and 4 are called to using a complex ciphered system for representing addends. Addition is commutative; that is, the order numbers, used a system that is very similar to Roman of the addends is irrelevant to how the sum is formed. numerals. It is possible that the Greeks performed Subtraction finds the remainder after a quantity is arithmetic operations by manipulating small stones diminished by a certain amount. If from a flock con- on a large, flat surface partitioned by lines. A simi- taining 5 sheep, 3 sheep are removed, then 2 sheep lar stone tablet was found on the island of Salamis remain. In this example, 5 is the minuend, 3 is the in the 1800s and is believed to date from the fourth subtrahend, and 2 is the remainder or difference. -
Performance Analysis of Effective Symbolic Methods for Solving Band Matrix Slaes
EPJ Web of Conferences 214, 05004 (2019) https://doi.org/10.1051/epjconf/201921405004 CHEP 2018 Performance Analysis of Effective Symbolic Methods for Solving Band Matrix SLAEs 1, 1, Milena Veneva ∗ and Alexander Ayriyan ∗∗ 1Joint Institute for Nuclear Research, Laboratory of Information Technologies, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia Abstract. This paper presents an experimental performance study of imple- mentations of three symbolic algorithms for solving band matrix systems of linear algebraic equations with heptadiagonal, pentadiagonal, and tridiagonal coefficient matrices. The only assumption on the coefficient matrix in order for the algorithms to be stable is nonsingularity. These algorithms are implemented using the GiNaC library of C++ and the SymPy library of Python, considering five different data storing classes. Performance analysis of the implementations is done using the high-performance computing (HPC) platforms “HybriLIT” and “Avitohol”. The experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed. An analysis of the three algorithms is performed. 1 Introduction Systems of linear algebraic equations (SLAEs) with heptadiagonal (HD), pentadiagonal (PD) and tridiagonal (TD) coefficient matrices may arise after many different scientific and engi- neering problems, as well as problems of the computational linear algebra where finding the solution of a SLAE is considered to be one of the most important problems. On the other hand, special matrix’s characteristics like diagonal dominance, positive definiteness, etc. are not always feasible. The latter two points explain why there is a need of methods for solving of SLAEs which take into account the band structure of the matrices and do not have any other special requirements to them. -
Ipad Educational Apps This List of Apps Was Compiled by the Following Individuals on Behalf of Innovative Educator Consulting: Naomi Harm Jenna Linskens Tim Nielsen
iPad Educational Apps This list of apps was compiled by the following individuals on behalf of Innovative Educator Consulting: Naomi Harm Jenna Linskens Tim Nielsen INNOVATIVE 295 South Marina Drive Brownsville, MN 55919 Home: (507) 750-0506 Cell: (608) 386-2018 EDUCATOR Email: [email protected] Website: http://naomiharm.org CONSULTING Inspired Technology Leadership to Transform Teaching & Learning CONTENTS Art ............................................................................................................... 3 Creativity and Digital Production ................................................................. 5 eBook Applications .................................................................................... 13 Foreign Language ....................................................................................... 22 Music ........................................................................................................ 25 PE / Health ................................................................................................ 27 Special Needs ............................................................................................ 29 STEM - General .......................................................................................... 47 STEM - Science ........................................................................................... 48 STEM - Technology ..................................................................................... 51 STEM - Engineering ................................................................................... -
MATH2010-1 Logiciels Mathématiques
MATH2010-1 Logiciels mathématiques Émilie Charlier Département de Mathématique Université de Liège 10 février 2020 Calculatrices électroniques I La HP-35, commercialisée en janvier 1972 par Hewlett-Packard. I Première calculatrice scientifique. I Devient célèbre sous le nom de “règle à calcul électronique”. I 395 dollars (moitié du salaire mensuel d’un enseignant de l’époque). https://fr.wikipedia.org/wiki/HP-35 Calculatrices graphiques I La TI-89, commercialisée par Texas Instruments en 1998. I Calcul formel. I Possibilités de programmation. http://fr.wikipedia.org/wiki/Calculatrice (Une multitude de) logiciels mathématiques Depuis 1960, au moins 45 logiciels mathématiques : Axiom FORM Magnus MuPAD SyMAT Cadabra FriCAS Maple OpenAxiom SymbolicC++ Calcinator FxSolver Mathcad PARI/GP Symbolism CoCoA-4 GAP Mathematica Reduce Symengine CoCoA-5 GiNaC MathHandbook Scilab SymPy Derive KANT/KASH Mathics SageMath TI-Nspire DataMelt Macaulay2 Mathomatic SINGULAR Wolfram Alpha Erable Macsyma Maxima SMath Xcas/Giac Fermat Magma MuMATH Symbolic Yacas http://en.wikipedia.org/wiki/List_of_computer_algebra_systems Logiciels de mathématiques Quelques logiciels commerciaux : I Maple, Waterloo Maple Inc., Maplesoft, depuis 1985. I Mathematica, Wolfram Research, depuis 1988. I Matlab, MathWorks, depuis 1989 I Magma, University of Sydney, depuis 1990 Quelques logiciels libres : I Maxima, W. Schelter et coll., depuis 1967 : Calcul symbolique I Singular, U. Kaiserslautern, depuis 1984 : Polynômes I PARI/GP, U. Bordeaux 1, depuis 1985 : Théorie des nombres I GAP, GAP Group, depuis 1986 : Théorie des groupes I R, U. Auckland, New Zealand, depuis 1993 : Statistiques Deux distinctions importantes I Calcul numérique vs calcul symbolique I Logiciel libre vs logiciel commercial Dans ce cours, présentation de 4 logiciels : I Mathematica : commercial, calcul symbolique I SymPy : libre, calcul symbolique I Geogebra : gratuit, géométrie et algèbre I Calc : gratuit, tableurs Remarque : gratuit 6= libre.