DOCSLIB.ORG

Noindent Mr $ Lj $ Ubica $ D−I $ Kovi $ Tsoft $ \Centerline

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Noindent Mr $ Lj $ Ubica $ D−I $ Kovi $ Tsoft $ \Centerline NASTAVA RAQUNARSTVA n centerlineMr Lj ubicaf D-iNASTAVA kovi tsoft RAQUNARSTVA g MATEMATIQKI SOFTVERSKI ALATI TIPA FOSS nnoindent1 period ..Mr Matemati $Lj$ q-k i ubicasoftvers i-k $D alati−i $ kovi $ tsoft $ Matematiqki softverski alati zauzimaj u znaqaj no mesto u procesu uqe nj a n centerlinei po d-u qavafMATEMATIQKI nj a period .... PrimarnaSOFTVERSKI upotreba ALATI kompj TIPA uterskih FOSS alatag za matematiqke namene j e kvalitetna reprezentacij a i verifikacij a rezultataNASTAVA period RAQUNARSTVA .... Matematiqki softver hyphen n centerline f1 . nquad Matemati $ q−k$ i softvers $ i−k $ a l a t i g ski alati su nameMr njLj eniubica za inovativnoD − i kovi commatsoft interaktivno i dinamiqko po d-u qava nj e iz raznih oblasti matematike openMATEMATIQKI square bracket 2 SOFTVERSKI closing square ALATIbracket TIPA periodFOSS n hspace ∗fn f i l l g Matematiqki softverski alati zauzimaj u znaqaj no mesto u procesu uqe Postoj i niz razliqitih programskih paketa1 . name Matemati nj enihq − zak radi softvers sa matematiqi − k alati hyphen $ nj $ a kim sadr zhe aj ima commaMatematiqki kao xto softverski su Mathematica alati zauzimaj comma u Maple znaqaj comma no mesto Scientific u procesu WorkPlace uqe nj a comma Manipula i po d − u qava nj a . Primarna upotreba kompj uterskih alata za matematiqke nnoindent i po $ d−u$ qava $nj$ a. n h f i l l Primarna upotreba kompj uterskih alata za matematiqke namene Math with Javanamene comma Derive and Calculus T slash L II comma Algebra i drugi period Ve tsoft ina omogu tsoft ava j e kvalitetna reprezentacij a i verifikacij a rezultata . Matematiqki softver - nnoindent j e kvalitetna reprezentacij a i verifikacij a rezultata . n h f i l l Matematiqki softver − vrxe nj e simboliqkihski alati comma su name numeriqkihnj eni za iinovativno grafiqkih ,operacij interaktivno a comma i dinamiqko tako da po commad − u commaqava nj grafiqkee iz mo hyphen raznih oblasti matematike [ 2 ] . nnoindent ski alati su name $ nj $ eni za inovativno , interaktivno i dinamiqko po gu tsoft nosti raqunaraPostoj poma izhe niz u razliqitih da se matematika programskih vidi paketa comma name algebarskinj enih deo za softvera rad sa matematiq obez hyphen - $ d−u$ qava $nj$ eiz be dj uj e da sekim matematika sadr zhe radiaj ima comma , kao xto a koriste su Mathematica tsoft i izr to , Maple the power , Scientific of a-zhe WorkPlace aj nost programskog , Manipula j ezika ma hyphen Math with Java , Derive and Calculus T / L II , Algebra i drugi . Ve tsoft ina omogu nnoindent raznih oblasti matematike [ 2 ] . tematika stvaratsoft periodava period period .. quotedblleft hyphen 3 3 Dobri softverskivrxe paketinj e name simboliqkih nj eni uqe , numeriqkih nj u i nastavi i grafiqkih matematike operacij imaj a u , tako sle hyphen da , , grafiqke mo - n hspace ∗fn f i l l g Postoj i niz razliqitih programskih paketa name $ nj $ enih za rad sa matematiq − de tsoft e karakteristikegu tsoft :nosti raqunara poma zhe u da se matematika vidi , algebarski deo softvera bullet korisnik commaobez - pri upotrebi tehnologij e comma mo zhe e da ostvari potpunu kontrolu semicolon nnoindent kim sadr $ zhe $ aj ima , kao xto su Mathematica , Maple , Scientific WorkPlace , Manipula bullet podstiqube i motivixudj uj e da istr se matematika to the power radi of a-zhe , a koriste iva njtsoft e i ispitivai izra− njzhe eaj semicolon nost programskog j ezika bullet pru zhe ajma u - mogu tsoft nosti koj e bi bile nedostupne bez upotrebe tehnologij e semicolon nnoindent Math with Java , Derive and Calculus T / L II , Algebra i drugi . Ve $ tsoft $ bullet j asan i intuitivantematika stvara interfej . s . semicolon . \ - 3 3 ina omogu $ tsoft $ ava bullet j ednostavan unosDobri ulaznih softverski podataka paketi semicolon name nj eni uqe nj u i nastavi matematike imaj u sle - bullet momentalnade tsoft verifikacije karakteristike a izlaznih rezultata : open parenthesis umesto comma comma proveri svoj odgovor nnoindent vrxe $ nj $ e simboliqkih , numeriqkih i grafiqkih operacij a , tako da , , grafiqke mo − na kraj u k nj ige quotedblleft• korisnik , quotedblleft pri upotrebi sa tehnologij deta lj nim e , mo obrazlozhe e zhe da e ostvari nj em potpunusemicolon kontrolu ; bullet .. fleksibilna i intuitivna• manipulacijpodstiqu i motivixu a sa matematiqkimistra−zhe iva formamanj e i ispitiva commanj grafie ; hyphen nnoindent gu $ tsoft $ nosti raqunara poma $ zhe $ u da se matematika vidi , algebarski deo softvera obez − cima i tabelarnim• pru vrednostimazhe aj umogu semicolontsoft nosti koj e bi bile nedostupne bez upotrebe tehnologij e ; bullet 2 D ili 3 D prikaz ulaznih i izlaznih parametara• j asan i intuitivan semicolon interfej s ; nnoindent be $ dj $ uj e da se matematika radi , a koriste $ tsoft $ i $ izr ^f a−zhe g$ bullet .. omogu tsoft avaj u ponav lj• aj nj ednostavan e svakog unos koraka ulaznih transakcij podataka e za ; indivi d-u alno ve zhe ba nj e aj nost programskog j ezika ma − semicolon • momentalna verifikacij a izlaznih rezultata ( umesto , , proveri svoj odgovor bullet .. softver j e u potpunostina kraj u funkcij u k nj iige matematike \ \ sa deta endashlj nim korisnik obrazlo usvajzhe e anj matemaem ; hyphen nnoindent tematika stvara . nquad ` ` − 3 3 tiqke koncepte i vextine uz• povefleksibilna tsoft ano i intuitivna anga zhe ova manipulacij nj e comma a sa sa matematiqkim ve tsoft im stepenom formama razu , grafi hyphen - meva nj a period cima i tabelarnim vrednostima ; n hspace ∗fn f i l l g Dobri softverski paketi name $ nj $ eni uqe $ nj $ u i nastavi matematike imaj u sle − • 2 D ili 3 D prikaz ulaznih i izlaznih parametara ; • omogu tsoft avaj u ponav lj a nj e svakog koraka transakcij e za indivi d − u nnoindent de $ tsoft $ e karakteristike : alno ve zhe ba nj e ; • softver j e u potpunosti u funkcij i matematike { korisnik usvaj a matema - n centerline f $ n bullet $ korisnik , pri upotrebi tehnologij e , mo $ zhe $ e da ostvari potpunu kontrolu ; g tiqke koncepte i vextine uz pove tsoft ano anga zhe ova nj e , sa ve tsoft im stepenom razu - n centerline f $ n bullet $ podstiqu i motivixu $ istr ^f a−zhe g$ iva $ nj $ e i ispitiva meva nj a . $ nj $ e ; g n centerline f $ n bullet $ pru $ zhe $ aj u mogu $ tsoft $ nosti koj e bi bile nedostupne bez upotrebe tehnologij e ; g n centerline f $ n bullet $ j asan i intuitivan interfej s ; g n centerline f $ n bullet $ j ednostavan unos ulaznih podataka ; g n hspace ∗fn f i l l g $ n bullet $ momentalna verifikacij a izlaznih rezultata ( umesto , , proveri svoj odgovor n centerline fna kraj uk $ nj $ ige `` `` sa deta $ lj $ nim obrazlo $ zhe $ e $ nj $ em ; g n hspace ∗fn f i l l g $ n bullet $ nquad fleksibilna i intuitivna manipulacij a sa matematiqkim formama , grafi − n centerline fcima i tabelarnim vrednostima ; g n centerline f $ n bullet 2 $ D ili 3 D prikaz ulaznih i izlaznih parametara ; g n hspace ∗fn f i l l g $ n bullet $ nquad omogu $ tsoft $ avaj uponav $ lj $ a $ nj $ e svakog koraka transakcij e za indivi $ d−u$ alnove $zhe$ ba $nj$ e; n hspace ∗fn f i l l g $ n bullet $ nquad softver j e u potpunosti u funkcij i matematike −− korisnik usvaj a matema − n hspace ∗fn f i l l g tiqke koncepte i vextine uz pove $ tsoft $ ano anga $ zhe $ ova $ nj $ e , sa ve $ tsoft $ im stepenom razu − n centerline fmeva $ nj $ a . g Matematiqki s oftver ski alati tipa FOSS .. 25 n hspace1 period∗fn 1f period i l l g Matematiqki .. Matematiq k-i s apleti oftver ski alati tipa FOSS nquad 25 Matematiqki apleti su Java aplikacij e comma koj e se pokre tsoft u unutar Web hyphen stra hyphen n centerlinenice periodf ..1 Java . 1 apleti . nquad omoguMatematiq tsoft avaj u korisniku $ k−i da $ bez a p nekih l e t i posebnihg matematiqkih programa mogu operisati geometrij skim obj ektima i algebarskim kalkulacij ama period MatematiqkiJava apleti su apleti interaktivne su Java comma aplikacij intuitivne i j e ednostavne , koj e aplikacij se pokre e comma $ tsoft koj e mogu $ u biti unutar Web − s t r a − n i c e . nquad Java apleti omogu $ tsoft $ avaj u korisniku da bez nekih posebnih matematiqkih izvrsne za uqe nj e i po d-u qava nj e matematike period .. ZaMatematiqki pregled ovih s oftver stranica ski alati nu tipa zhe FOSS no j e 25 programa mogu operisati geometrij skim obj ektima i algebarskim kalkulacij ama . da korisnik na svom raqunaru ima instaliran1 . Java1 . programski Matematiq jk ezik− i commaapleti koj i j e u Java apleti su interaktivne , intuitivne i j ednostavne aplikacij e , koj e mogu biti xirokoj upotrebi naMatematiqki Internetu period apleti su Java aplikacij e , koj e se pokre tsoft u unutar Web - stra - izvrsne za uqe $ nj $ e i po $d−u $ qava $ nj $ e matematike . nquad Za pregled ovih stranica nu Jedna od riznicanice matematiqkih . Java apleti apleta omogu commatsoft nameavaj nj u korisniku ena osnovcima da bez i srednekih nj posebnih oxkol hyphen matematiqkih $ zhe $ no j e cima comma nalaziprograma se na mogu adresi operisati open square geometrij bracket skim 3 obj closing ektima square i algebarskim bracket period kalkulacij Na adresi ama . open Java square da korisnik na svom raqunaru ima instaliran Java programski j ezik , koj i j e u bracket 4 closing squareapleti bracket su interaktivne nalazi se , intuitivnesoftverski paketi j ednostavne Manipula aplikacij e , koj e mogu biti izvrsne za xirokoj upotrebi na Internetu . Math with Javauqe namenj nje i em po sredd − u njqava oxkolcimanj e matematike comma studentima .
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • Collaborative Use of Ketcindy and Free Cass for Making Materials 1
    Collaborative Use of KeTCindy and Free CASs for Making Materials Setsuo Takato1, Alasdair McAndrew2, Masataka Kaneko3 1 Toho University, Japan, [email protected] 2 Victoria University, Australia, [email protected] 3 Toho University, Japan, [email protected] 1 Introduction Many mathematics teachers at collegiate level use LATEX to write materials for dis- tribution to their classes. As is well known, LATEX can typeset complex mathe- matical formulas. On the other hand, it has poor ability to create figures. One possibility might be to use a third-party package, such as TiKZ. However, TiKZ coding is complicated and is not easy to read even for the following simple figure. y y = x y = sinx x O Figure 1 A simple example TiKZ can in fact produce figures of great complexity, but its power comes at the cost of a steep learning curve. In order to provide a system for easy creation of publication quality figures, the first author has developed KETpic, the first version of which was released in 2006. KETpic is a macro package of mathematical soft- ware such as Maple, Mathematica, Scilab and R. The recent version uses Scilab mainly and R secondarily. The flow of generating and inserting graphs with KETpic is as follows. 1. KETpic and Scilab commands are listed in a Scilab editor and executed by Scilab. 2. Scilab generates a LATEX file composed of codes for drawing figures. 3. That file can be inserted into a LATEX document with \input command. 4. The document can be compiled to produce a pdf file.
    [Show full text]
  • Computer Algebra Independent Integration Tests 0 Independent Test Suites/Hebisch Problems
    Computer algebra independent integration tests 0_Independent_test_suites/Hebisch_Problems Nasser M. Abbasi November 25, 2018 Compiled on November 25, 2018 at 10:22pm Contents 1 Introduction 2 2 detailed summary tables of results 9 3 Listing of integrals 11 4 Listing of Grading functions 25 1 2 1 Introduction This report gives the result of running the computer algebra independent integration problems.The listing of the problems are maintained by and can be downloaded from Albert Rich Rubi web site. 1.1 Listing of CAS systems tested The following systems were tested at this time. 1. Mathematica 11.3 (64 bit). 2. Rubi 4.15.2 in Mathematica 11.3. 3. Rubi in Sympy (Version 1.3) under Python 3.7.0 using Anaconda distribution. 4. Maple 2018.1 (64 bit). 5. Maxima 5.41 Using Lisp ECL 16.1.2. 6. Fricas 1.3.4. 7. Sympy 1.3 under Python 3.7.0 using Anaconda distribution. 8. Giac/Xcas 1.4.9. Maxima, Fricas and Giac/Xcas were called from inside SageMath version 8.3. This was done using SageMath integrate command by changing the name of the algorithm to use the different CAS systems. Sympy was called directly using Python. Rubi in Sympy was also called directly using sympy 1.3 in python. 1.2 Design of the test system The following diagram gives a high level view of the current test build system. POST PROCESSOR SCRIPT Mathematica script Result table Rubi script Result table Test files from Program that Albert Rich Rubi Maple script Result table generates the web site Latex report PDF using input Result table Latex report Python script to test sympy from the result tables Giac Result table SageMath/Python HTML Fricas Result table script to test SageMath Maxima, Fricas, Giac Maxima Result table One record (line) per one integral result.
    [Show full text]
  • Abstract Computer Algebra Systems Have to Deal with The
    Abstract Computer algebra systems have to deal with the confusion between “programming variables” and “mathematical symbols”. We claim that they should also deal with “unknowns”, i.e. elements whose values are unknown, but whose type is known. For example, xp =6 x if x is a symbol, but xp = x if x ∈ GF(p). We show how we have extended Axiom to deal with this concept. 1 The “Unknown” in Computer Algebra James Davenport∗ Christ`ele Faure† [email protected] [email protected] School of Mathematical Sciences University of Bath Bath BA2 7AY England 1 Introduction Computer Algebra and Numerical Computation are two means of computational mathematics which are different in the way they use and treat symbols. In numerical computation, symbols are considered as programming variables and are used to store intermediate results. Computer algebra systems take into account in polynomials etc. a second class of symbols: uninterpreted variables. But in mathematics, a third class of symbols is also involved in computation. One often begins mathematical texts with sentences like “let p be a polynomial” or “let n be an integer” . In those sentences, n and p are neither programming variables, nor uninterpreted variables, but are names of mathematical objects, with types but without known values, called unknowns — in future we will refer to such symbols as n as basic unknowns. One begins a mathematical text by “let p be a polynomial” to make the following text more precise. For example, the expression “n+m” can mean a lot of things: the sum of two matrices, the application of the concatenation operator to two lists etc., whereas this expression is clear if one declares that n and m are two “integers”.
    [Show full text]
  • Symbolic Tensor Calculus on Manifolds: a Sagemath Implementation Vol
    Journées Nationales de Calcul Formel Rencontre organisée par : Carole El Bacha, Luca De Feo, Pascal Giorgi, Marc Mezzarobba et Alban Quadrat 2018 Éric Gourgoulhon and Marco Mancini Symbolic tensor calculus on manifolds: a SageMath implementation Vol. 6, no 1 (2018), Course no I, p. 1-54. <http://ccirm.cedram.org/item?id=CCIRM_2018__6_1_A1_0> Centre international de rencontres mathématiques U.M.S. 822 C.N.R.S./S.M.F. Luminy (Marseille) France cedram Texte mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Les cours du C.I.R.M. Vol. 6 no 1 (2018) 1-54 Cours no I Symbolic tensor calculus on manifolds: a SageMath implementation Éric Gourgoulhon and Marco Mancini Preface These notes correspond to two lectures given by one of us (EG) at Journées Nationales de Calcul Formel 2018 (French Computer Algebra Days), which took place at Centre International de Rencontres Mathématiques (CIRM), in Marseille, France, on 22-26 January 2018. The slides, demo notebooks and videos of these lectures are available at https://sagemanifolds.obspm.fr/jncf2018/ EG warmly thanks Marc Mezzarobba and the organizers of JNCF 2018 for their invitation and the perfect organization of the lectures. He also acknowledges the great hospitality of CIRM and many fruitful exchanges with the conference participants. We are very grateful to Travis Scrimshaw for his help in the writing of these notes, especially for providing a customized LATEX environment to display Jupyter notebook cells. Course taught during the meeting “Journées Nationales de Calcul Formel” organized by Carole El Bacha, Luca De Feo, Pascal Giorgi, Marc Mezzarobba and Alban Quadrat.
    [Show full text]
  • An Approach to Teach Calculus/Mathematical Analysis (For
    Sandra Isabel Cardoso Gaspar Martins Mestre em Matemática Aplicada An approach to teach Calculus/Mathematical Analysis (for engineering students) using computers and active learning – its conception, development of materials and evaluation Dissertação para obtenção do Grau de Doutor em Ciências da Educação Orientador: Vitor Duarte Teodoro, Professor Auxiliar, Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa Co-orientador: Tiago Charters de Azevedo, Professor Adjunto, Instituto Superior de Engenharia de Lisboa do Instituto Politécnico de Lisboa Júri: Presidente: Prof. Doutor Fernando José Pires Santana Arguentes: Prof. Doutor Jorge António de Carvalho Sousa Valadares Prof. Doutor Jaime Maria Monteiro de Carvalho e Silva Vogais: Prof. Doutor Carlos Francisco Mafra Ceia Prof. Doutor Francisco José Brito Peixoto Prof. Doutor António Manuel Dias Domingos Prof. Doutor Vítor Manuel Neves Duarte Teodoro Prof. Doutor Tiago Gorjão Clara Charters de Azevedo Janeiro 2013 ii Título: An approach to teach Calculus/Mathematical Analysis (for engineering students) using computers and active learning – its conception, development of materials and evaluation 2012, Sandra Gaspar Martins (Autora), Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa e Universidade Nova de Lisboa A Faculdade de Ciências e Tecnologia e a Universidade Nova de Lisboa tem o direito, perpétuo e sem limites geográficos, de arquivar e publicar esta dissertação através de exemplares impressos reproduzidos em papel ou de forma digital, ou por qualquer outro meio conhecido ou que venha a ser inventado, e de a divulgar através de repositórios científicos e de admitir a sua cópia e distribuição com objectivos educacionais ou de investigação, não comerciais, desde que seja dado crédito ao autor e editor.
    [Show full text]
  • The Sage Project: Unifying Free Mathematical Software to Create a Viable Alternative to Magma, Maple, Mathematica and MATLAB
    The Sage Project: Unifying Free Mathematical Software to Create a Viable Alternative to Magma, Maple, Mathematica and MATLAB Bur¸cinEr¨ocal1 and William Stein2 1 Research Institute for Symbolic Computation Johannes Kepler University, Linz, Austria Supported by FWF grants P20347 and DK W1214. [email protected] 2 Department of Mathematics University of Washington [email protected] Supported by NSF grant DMS-0757627 and DMS-0555776. Abstract. Sage is a free, open source, self-contained distribution of mathematical software, including a large library that provides a unified interface to the components of this distribution. This library also builds on the components of Sage to implement novel algorithms covering a broad range of mathematical functionality from algebraic combinatorics to number theory and arithmetic geometry. Keywords: Python, Cython, Sage, Open Source, Interfaces 1 Introduction In order to use mathematical software for exploration, we often push the bound- aries of available computing resources and continuously try to improve our imple- mentations and algorithms. Most mathematical algorithms require basic build- ing blocks, such as multiprecision numbers, fast polynomial arithmetic, exact or numeric linear algebra, or more advanced algorithms such as Gr¨obnerbasis computation or integer factorization. Though implementing some of these basic foundations from scratch can be a good exercise, the resulting code may be slow and buggy. Instead, one can build on existing optimized implementations of these basic components, either by using a general computer algebra system, such as Magma, Maple, Mathematica or MATLAB, or by making use of the many high quality open source libraries that provide the desired functionality. These two approaches both have significant drawbacks.
    [Show full text]
  • A La Cryptologie
    Introduction `ala Cryptologie Fran¸coisDahmani Universit´eGrenoble Alpes, Institut Universitaire de France Master Math´ematiqueset Applications M1 Maths G´en´erales 2017 1 Table des mati`eres 1 S´eance1 : arithm´etiquemodulaire, Les anneaux Z=mZ 4 1.1 Bref rappels sur Z=mZ (en autonomie) . .4 1.2 Le Th´eor`emeChinois . .5 1.3 Le symbole de Legendre, et de Jacobi . .7 1.4 Exercices . 10 2 S´eance2 : Prise en main SageMath 11 3 S´eance3 : Complexit´e 17 3.1 D´efinition de la complexit´e . 17 3.2 Savez vous multiplier des nombres ? . 17 3.3 L'exponentielle modulaire . 19 3.4 L'algorithme d'Euclide . 20 4 S´eance4 : Codes correcteurs d'erreurs (lin´eaires) 22 4.1 D´efinitions. 22 4.2 Le r^oled'un code . 23 4.3 D´ecodage . 23 4.4 Exercices . 24 4.5 Parit´eg´en´eralis´ee; introduction `a F256 ................................. 25 4.6 Exercices/TP . 28 5 S´eance5 : Registres `ad´ecalages,et g´en´erateurs pseudo-al´eatoires 30 5.1 Registres `ad´ecalages. 30 5.2 Serie formelle caracteristique du processus . 31 5.3 Combinaisons, corr´elationset transform´eede Fourier . 33 5.4 Non lin´earit´e,un troisi`emecrit`ere. 36 5.5 TP..................................................... 37 6 S´eance6 : Techniques sym´etriques 38 6.1 Substitution : Vigenere, Vernam, one-time pad . 38 7 S´eance7 : Techniques asym´etriques 39 7.1 Fonctions `asens unique . 39 7.2 Une fonction `asens unique : l'exponentielle modulaire et le probl`emedu Log discret ? . 40 7.3 Protocoles li´es`al'exponentielle modulaire : Diffie-Hellman et El Gamal .
    [Show full text]
  • Open Source Computer Algebra Systems
    Open source computer algebra systems Riccardo Guida and Sylvain Ribault IPhT Saclay Tuesday 7 May 2019 Riccardo Guida and Sylvain Ribault (IPhT) OSCAS Tuesday 7 May 2019 1 / 9 Summary the trouble with Mathematica; a quick overview of the best Open Source Computer Algebra Systems (OSCAS) available today; a demonstration of SymPy. Riccardo Guida and Sylvain Ribault (IPhT) OSCAS Tuesday 7 May 2019 2 / 9 The trouble with Mathematica Mathematica is the most widespread computer algebra system in our field. However, it is expensive for non-academic researchers; it is not accessible by anyone anywhere (collaborators, readers of your articles, yourself after moving to another institute or ending your grant); availability of old versions is not guaranteed, and is subject to Wolfram's commercial policy. IPhT-specific problems: We currently pay 25ke/year for only 10 system-wide licenses. We used to pay the same price for 200 licenses. Then a new salesperson came. We bought \perpetual" licenses for 11 users on MacOS. But these 32-bits licences will be unusable on the next version of MacOS (64-bits only). Upgrade cost: ∼ 700 e per \perpetual" license. Riccardo Guida and Sylvain Ribault (IPhT) OSCAS Tuesday 7 May 2019 3 / 9 Open source software Open source software solves the problems of cost and license availability. Other advantages include: Verifiability of algorithms: no need to trust a black box. Influencing development of the software by reporting bugs, writing code yourself, or giving grant money for adding a feature you need. Anyone can check, reproduce, and improve your results, if you give them your code.
    [Show full text]
  • Introduction
    Introduction Christoph Koutschan Research Institute for Symbolic Computation Johannes Kepler Universit¨atLinz, Austria Computer Algebra Systems 4.10.2010 • Research Institute for Symbolic Computation (RISC) (Institut f¨urSymbolisches Rechnen) Self-Introduction • Dr. Christoph Koutschan [email protected] Office: Room 1.7-2, Schloss Hagenberg, A-4232 Hagenberg im M¨uhlkreis Self-Introduction • Dr. Christoph Koutschan [email protected] Office: Room 1.7-2, Schloss Hagenberg, A-4232 Hagenberg im M¨uhlkreis • Research Institute for Symbolic Computation (RISC) (Institut f¨urSymbolisches Rechnen) Computer Algebra System Definition: A computer algebra system (CAS) is a software for doing computer algebra, where computer algebra = symbolic computation: More concretely: a software for solving mathematical problems with symbolic methods. Symbolic Computation (1) p Problem: What is 72? Numeric Evaluation: 8:48528 Symbolic Solution: p 6 2 Symbolic Computation (2) Problem: What is the 100th harmonic number 100 X 1 ? n n=1 Numeric Evaluation: 5:18738 Symbolic Solution: 14466636279520351160221518043104131447711 2788815009188499086581352357412492142272 Symbolic Computation (3) Problem: 1 X 1 n4 n=1 Numeric Evaluation: 1:08232 Symbolic Solution: π4 90 Symbolic Computation (4) Problem: Find all solutions of the equation x3 + x2 + x − 1 = 0. Numeric Evaluation: x1 = 0:543689 x2 = − 0:771845 + 1:11514i x3 = − 0:771845 − 1:11514i Symbolic Solution: ! 1 2 q3 p x = −1 − + 17 + 3 33 1 p3 p 3 17 + 3 33 p 1 1 + i 3 1 p q3 p x = − + − 1 − i 3 17 + 3 33 2 p3 p 3 3 17 + 3 33 6 p 1 1 − i 3 1 p q3 p x = − + − 1 + i 3 17 + 3 33 3 p3 p 3 3 17 + 3 33 6 Symbolic Computation (5) Problem: Solve the differential equation x(x2 − 1)f 00(x) + (x2 + 1)f 0(x) + 4x3(x2 − 1)f(x) = 0 subject to the initial values f(1) = 1 and f 0(0) = 0.
    [Show full text]