1. Packages Lecture 12

2. ANU Polycyclic Quotient Programs The study of groups defined by finite presentations is one of the classical areas in group theory. Despite the fact that it has been shown that many general questions about finitely presented groups are algorithmically unsolvable, there exist algorithms and their computer implementations for studying these groups. For example, one family of algorithms investigates groups given by finite presentations by computing presentations for polycyclic factor groups. These includes the ANU quotient algorithms designed to compute presentations for quotients of a finitely presented group that have prim-power order, are nilpotent or are finite soluble. A polycyclic group has a descending series of subgroups, such that each is normal in the previous one and the quotient of two successive subgroups is cyclic. In addition to the p-quotient algorithm, the ANU p-Quotient Program50 offers access to: p-group generation algorithm, an algorithm to decide isomorphism of p-groups, an algorithm to compute the automorphism group of a p-group. In addition to the nilpotent factor groups, the ANU Nilpotent Quotient Program provides facilities for the computation of nilpotent groups. The program is implemented in C and is available as stand-alone or as part of the systems Gap, Magma, and Quotpic.

3. Arep The special purpose library (Abstract Represenation) provides data structures and algorithms for calculating symbolically with structures matrix representations of finite groups. It is possible to work with inductions or tensor products of representations without constructing large matrices explicitly. The main distinction of Arep from other libraries for representation theory is that representations are manipulated up to equality and not just up to equivalence. Arep is a Gap share package and is distributed together with Gap. Arep consists of four major building blocks: structured matrices (recursive data type; basic building blocks are permutation matrices, monomial matrices); structured matrix representations of finite groups; functions for decomposing monomial representations into irreducibles; functions for combinatorial search of certain types of symmetries in matrices. The original motivation to develop Arep was to construct fast algorithms for give discrete linear signal transforms, like the fast Fourier transform, automatically.

4. CALI Is a Reduce package that contains algorithms for computations in commutative algebra closely related to the Groebner algorithm for ideals and modules. Its hearts is an improved Reduce implementation of the Groebner algorithm. It allows also for the computations of syzygies and is applicable to submodules of free modules with generators represented as rows of a matrix. As main topics Cali contains facilities for: defining rings, ideals, modules; computing Groebner bases and local standard bases; computing syzygies, resolutions and Betti numbers; computing Hilbert series, multiplicities, independent sets, and dimensions; computing normal forms and representations; sums, products, intersections, quotients, elimination ideals; primality tests, computation of radicals, unmixed radicals, equidimensional parts, primary decompositions of ideals and modules; advanced application of Groebner bases; application of linear algebra techniques to zero-dimensional ideals; splitting polynomial systems of equations mixing Groebner algorithm with factorization, triangular systems; etc.

5. CLN C++ library for doing multiple-precision computations with high efficiency. It supports algebraic syntax and provides for automatic memory management. It is a free software available via Internet. CLN’s data types are integers, rational numbers, floating-point numbers, complex numbers, modular integers, and univariate polynomials, all of them with unlimited precision. CLN implements elementary, logical and transcendental functions. Its efficiency comes from: the Katatsuba and Schoenhage-Strassen multiplication algorithms it implements; from consequent use of the binary splitting technique for transcendental function evaluation; from the GMP kernel which it integrates. CLN is currently used in the domains of number theory, cryptography, complex analysis, and physics.

6. Crack, LiePDE, ApplySym and ConLaw To investigate non-linear differential equations or partial differential equations, for which no general solution techniques are known, one either relies on numerics, or, if one is interested in exact results, one investigates special properties, like symmetries or conservation laws. Any such result may give a better understanding of the problem, provide the basis for using more adequate numerical algorithms or even solve the problem analytically. Crack an related application programs are applicable whenever smooth analytic properties of differential equations or geometric objects including manifolds are investigated. To solve overdeterminated PDE systems, the package Crack contains a dozen modules for the integration of different types of PDEs, direct and indirect separation of PDEs, computation of a pseudo differential Groebner basis, substitution of equations into each other, length reduction of equations, solution of undetermined linear ODE/PDEs, reduction of redundant arbitrary constants and functions in solutions of differential equations and performing point transformations. The programs LiePDE, ApplySym and ConLaw use the package Crack to solve the PDEsystems they generate.

7. Crack, LiePDE, ApplySym and ConLaw ConLaw is a package of four programs implementing four different approaches for the computation of the first integrals of single or systems of ordinary differential equations or conservation laws for single or systems of PDEs. LiePDE is a program for the determination of ininitesimal point-, contact- and generalized higher order symmetries of single/systems of differential equations. With the program ApplySym point symmetries, computed with LiePDE, can be integrated to yield symmetry-, and similarity variables, i.e. a symmetry reduction. Crack has been used in general relativity either to investigate a special method to find exact solutions of field equations or to characterize space times by determining their symmetries. LiePDE has been used for the classification of PDE-systems. Examples of computations that become possible with ConLaw including new conservation laws. The required system is Reduce. Source files are freely available.

8. Dimsym A Reduce package primarily for the determination of symmetries of differential equations. It also can be used to compute symmetries distributions of vector fields or differential forms on finite dimensional manifolds, symmetries of geometric objects (e.g. isometries), and also to solve linear partial differential equations. 1. The user specifies a system of ordinary and/or partial differential equations and the type of symmetry to be found; 2. Dimsym produces the corresponding determining equations and it proceeds to solve these equations, reporting any special conditions required to produce a solution; 3. finally it gives the generators of the symmetry group. The programs allows to computer Lie brackets, directional derivatives and it has an interface with the Reduce package ExCalc so that all the machinery of calculus on manifolds can be utilized from within the program. Dimsym is a freely distributed package which includes the program (Rlisp) source code, a manual and a set of examples.

9. EinS EinS is a package for Mathematica intended for calculations with indexed objects (may be in particular tensors). It handles automatically dummy indices and Einstein’s summation notation, enables one to define new indexed objects and to assign symmetries to that objects. It is an efficient simplification algorithm based on pattern matching technique which takes full account for the symmetries of the objects and the possibility to rename dummy indices. EinS runs under a version of Mathematica. The package is available via Internet. A typical application field of EinS is various calculations in the post- Newtonian approximation scheme of metric gravity theories. An example of problem that was treated the help of EinS is the problem of constructing of a local reference system for a massive extended body in the framework of the parametrized post-Newtonian formalism.

10. FeynArts, FormCalc and FeynCalc Feynman diagrams are a powerful and intuitive technique of field theory for evaluating perturbative expansions of Green functions and observables, usually the S-matrix. The accuracy of a calculation is linked to the number of loops in the Feynman diagrams, and already a one-loop calculation can easily involve several hundreds of diagrams, particularly so in models with many particles. FeynArts is a system for the generation and visualization of Feynman diagrams and amplitudes based on Mathematica. It is an open-source package and can be obtained via Internet. FormCalc is a Mathematica-based program that simplifies one-loop Feynman diagrams. It reads amplitudes generated by FeynArts and returns the results in a way well suited for further numerical or analytical evaluation. The associated package LoopTools implements one-loop integrals; it is accessible in Fortran, C++ and Mathematica.

11. FeynArts, FormCalc and FeynCalc Internally, FormCalc delegates the work to Form. Thus FormCalc is merely a driver that threads the FeynArts amplitudes through Form in an appropriate way. FormCalc prepares the symbolic expressions of the diagrams in an input file for Form, runs Form, and retrieves the results. FeynCalc is a Mathematica package for algebraic calculations in elementary particle physics. It is partially based on earlier FeynCalc version. It provides: Lorentz index contraction; color factor calculation and simplification; automatic Feynman rule derivation; automatic 1- loop diagram simplification, general noncommutative algebra and special noncommutative operator algebra, tables for Feynman parameter integrals and Mellin transforms, convolutions and Feynman rules, special translation to and from Form, optimized Fortran generation.

12. GRAPE Grape is a Gap share package for computing finite graphs endowed with groups of automorphisms. It is designed primarily for constructing and analyzing graphs related to groups, designs and finite geometries. Special emphasis is placed on the determination of regularity properties and subgraph structure. Applications are including the discovery and analysis of ceratin distance-regular graphs, the analysis of vertex-transitive graphs for low-rank representations of sporadic groups, and the discovery, analysis and classification of designs and finite geometries of various types. Grape includes functions to construct graphs, to determine connected components, diameter and girth, to compute induced subgraphs and geodesics in graphs, to determine regularity parameters of graphs, to determine complete subgraphs of given wight-sum in a vertex-weighted graph, to calculate automorphism groups and test for graph isomorphism, to classify distance-regular graphs with a given vertex-transitive group of automorphism, and to classify partial linear spaces with given point graph and parameters. The automorphism group and isomorphism testing functions are using nauty package. Grape can be downloaded freely by Internet.

13. Molgen CA package for the generation of structural formulae of chemical molecules, i.e. of molecular graphs or connectivity isomers. It generates all the mathematically existing molecular graph that correspond to given data from spectroscopy. Typical cases are given chemical formula, an interval for the possible ring sizes, and prescribed as well as forbidden substructures of the molecule in question. Molgen is applied to molecular structure elucidation as well as in combinatorial chemistry, where a library of molecules has to be generated from a central part and further building blocks and reactions according to which the building blocks react with the central molecule. Its mathematical tools are taken both from combinatorics and algebra, in particular orderly generation, group actions, double coset methods, and the homomorphism principle are used. Molgen is a collection of routines written in C++. Several extensions are devoted to special purposes, for spectroscopy, structure elucidation using mass spectroscopy, combinatorial chemistry. It has a graphical user interface and presents its results with automatically generated 2D and 3D-drawings according to the rules of chemists.

14. Orme For the purpose of making clear and nice proofs of their completeness, completion procedures are described by transition rules. The philosophy of Orme is to use for implementation the same paradigm that was shown so useful for proofs. This way one gets readable programs and a good view on high level optimizations. Orme provides a tool box for easily build prototypes in equational reasoning. An implementation of an associative and commutative completion was fulfilled. Orme contains also tools for proving termination based on polynomial and elementary functions.

15. Ratappr The Ratappr is a special package for numerical minimax approximations of functions by rational expressions and balanced rational splines. It has been made for Maple.

16. TTC Tools of Tensor Calculus is a Mathematica package for doing tensor ad exterior calculus on differentiable manifolds. Due to the generic character of TTC their applications are the ones differential geometry has. Some of the typical fields of application are: general relativity, electromagnetism, continuum mechanics.

17. From other sources

18. Oberwolfach References on Mathematical Software The classification consists of a hierarchical class identifier together with a class description: xx (top level) e.g. 02 - Algebra / Number Theory xx.yy (subclass level) e.g. 02.04 - Commutative rings and algebras xx.yy.zz (subsubclass level) e.g. 02.04.04 - Graded rings and Hilbert functions The development of the classification is still in progress

19. Oberwolfach References on Mathematical Software Identifier Description 01 Discrete Mathematics 01.01 Convex and discrete geometry 01.02 Graph theory 01.03 Enumerative combinatorics 01.04 Algebraic combinatorics 01.05 Integer programming 01.06 Ordered structures 02 Algebra / Number Theory 02.01 Mathematical logic and foundations 02.02 Number theory 02.03 Field theory 02.04 Commutative rings and algebras 02.04.01 Ideals, modules, homomorphisms 02.04.02 Polynomial and power series rings 02.04.03 Special rings 02.04.04 Graded rings and Hilbert functions 02.04.05 Integral dependence and normalization 02.04.06 Dimension theory 02.04.07 Factorization and primary decomposition 02.04.08 Syzygies and resolutions 02.04.09 Differential algebra 02.04.10 Groebner bases 02.05 Linear and multilinear algebra; matrix theory 02.05.01 Linear equations 02.05.02 Eigenvalues, singular values, and eigenvectors 02.05.03 Canonical forms 02.05.04 Matrix factorization 02.05.05 Integral matrices 02.05.06 Multilinear algebra 02.05.07 Linear inequalities 02.06 Non-commutative and general rings and algebras 02.06.01 Lie algebras 02.07 Category theory; homological algebra 02.08 Group theory and generalizations 02.08.01 Permutation groups 02.08.02 Matrix groups 02.08.03 Finitely presented groups 02.08.04 Polycyclicly presented groups 02.08.05 Black box groups 02.08.06 Group actions 02.08.07 Subgroup lattices 02.08.08 Group cohomology 02.08.09 Semigroups 02.09 Representation theory 02.09.01 Ordinary Representations of Groups 02.09.02 Ordinary Representations of Algebras 02.09.03 Modular Representations of Groups 02.09.04 Modular Representations of Algebras 02.09.05 Character Theory 02.09.06 Permutation Representations 03 Geometry / Topology 03.01 Algebraic geometry 03.01.01 Local theory, singularities 03.01.02 Cycles and subschemes 03.01.03 Families, fibrations 03.01.04 Birational theory 03.01.05 Co(homology) theory 03.01.06 Arithmetic problems 03.01.07 Projective Geometry 03.01.08 Algebraic groups and geometric invariant theory 03.01.09 Special varieties 03.01.10 Real algebraic geometry

20. Oberwolfach References on Mathematical Software 03.02 Topological groups, Lie groups 03.03 Several complex variables and analytic spaces 03.04 Geometry 03.05 Differential geometry 03.06 General topology 03.07 Algebraic topology 03.08 Manifolds and cell complexes 03.09 Global analysis, analysis on manifolds 03.10 Visualization 04 Analysis 04.01 Differentiation and Integration 04.02 Sequences, series, limits 04.03 Approximations and expansions 04.04 Functional analysis and operator theory 04.05 Special functions 04.06 Calculus of variations 04.07 Integral transforms, operational calculus 05 Differential and Integral Equations 05.34 Ordinary differential equations 05.35 Partial differential equations 05.37 Dynamical systems and ergodic theory 05.39 Difference and functional equations 05.45 Integral equations 06 Probability / Statistics 06.01 Combinatorial probability 06.02 Stochastic processes 06.03 Stochastic analysis 06.04 Stochastic geometry 06.05 Distributions 06.06 Statistics 07 Applications of Mathematics 07.70 Mechanics of particles and systems 07.74 Mechanics of deformable solids 07.76 Fluid mechanics 07.78 Optics, electromagnetic theory 07.80 Classical thermodynamics, heat transfer 07.81 Quantum theory 07.82 Statistical mechanics, structure of matter 07.83 Relativity and gravitational theory 07.85 Astronomy and astrophysics 07.86 Geophysics 07.87 Theoretical Chemistry 07.88 Crystallography 07.90 Operations research, mathematical programming 07.91 Game theory, economics, social and behavioral sciences 07.92 Biology and other natural sciences 07.93 Systems theory; control 07.94 Information and communication, circuits 08 Teaching 08.97 Mathematics education 09 Logic/ Deduction/ Computational Logic

21. Comparison of computer algebra systems From Wikipedia, the free encyclopedia

23. Google directory

25. CAIN

26. CAIN (1998)

27. CAIN – Special purpose systems (Non)Commutative Algebra & Algebraic Geometry Albert Bergman CALI CASA CLICAL CLIFFORD CoCoA FELIX GANITH GB The Grassman package GRB GROEBNER (from RISC-Linz) GROEBNER (REDUCE package) IDEALS KAN Macaulay Macaulay 2 MAS NCALGEBRA SACLIB Singular WU Differential Equation Solvers & Tools A review of ODE Solvers List of Symmetry Programs (PostScript, gzip- compressed) CONTENT: Dynamical System Software CRACK DELiA DESIR Diffgrob2 (Manual: PostScript, gzip-formatted) DIMSYM FIDE ODEtools (symmetry methods) LIE (A.K. Head, in MuMath) liesymm ODESOLVE PDELIE PDEtools package Phaser: an Animator/Simulator for Dynamical Systems for IBM PC's The Poincare package SPDE StandardForm SYM_DE SYMMGRP.MAX (Manual: PostScript, gzip- formatted)

28. CAIN – Special purpose systems Finite Element Analysis PDEase SENAC/FEM (PostScipt, gzip-formatted)SENAC/FEM (PostScipt, gzip-formatted); see also SENACSENAC Group Theory ANU Software Cayley CHEVIE GAP GRAPE GUAVA LiE LIE (REDUCE package) Magma MeatAxe The Magnus system for exploring infinite groups Schur Sisyphos Symmetrica Weyl Groups and Hecke Algebras High Energy Physics See also the FreeHEP Software database.FreeHEP Software database FeynArts Foam FORM The Partials package Schoonschip Tracer Number Theory Galois KANT KASH LiDIA MALM NTL Numbers PARI SIMATH

29. CAIN – Special purpose systems & experimental Tensor Calculus CARTAN Classi Eins GRTensor MathTensor Redten Ricci SHEEP STENSOR Tensors in Macsyma (R) Tools of Tensor Calculus PC Shareware with Symbolic Features AMP Calculus and Differential Equations CC4 CLA Mathomatic PFSA SymbMath X(PLORE) Various Systems AUTOMATA, QUOTPIC & TESTISOM Computer Algebra Kit FLAC JACAL Mock-Mma NODES: Non linear Ordinary Differential Equations Solver ORME RepTiles Asir SimLab, Computer Tools for Analysis and Simulation

30. Other sources CAS GAP, http://www-gap.mcs.st-and.ac.uk/ MuPAD (mupad-2.5.2-54.1.i586.rpm), de la cab047 Kant/Kash, http://www.math.tu-berlin.de/~kant/kash.html Axiom, http://wiki.axiom-developer.org/AboutAxiom Maxima, http://maxima.sourceforge.net/ CARAT, http://wwwb.math.rwth-aachen.de/carat/ CoCoA, http://cocoa.dima.unige.it/ Discreta, http://www.mathe2.uni-bayreuth.de/discreta/ Felix, http://felix.hgb-leipzig.de/ Fermat, http://www.bway.net/~lewis/ (pe CD s-ar putea sa fie free, dar nu ultima versiune) Kan, http://www.math.kobe-u.ac.jp/KAN/ Macaulay, http://www.math.uiuc.edu/Macaulay2/ MAS, http://alice.fmi.uni-passau.de/mas.html, http://krum.rz.uni- mannheim.de/mas.html PARI-GP, http://pari.math.u-bordeaux.fr/ ReDuX, ftp://ftp.informatik.uni-tuebingen.de/pub/SR/ReDuX Simath, http://tnt.math.metro-u.ac.jp/simath/ Singular, http://www.singular.uni-kl.de/ Form, http://www.nikhef.nl/~form/ JACAL, http://swissnet.ai.mit.edu/~jaffer/JACAL.html Simath, http://tnt.math.metro-u.ac.jp/simath/ YACAS, http://www.xs4all.nl/~apinkus/yacas.html GIAC, http://www-fourier.ujf-grenoble.fr/~parisse/english.html Gtybalt, http://wwwthep.physik.uni-mainz.de/~stefanw/gtybalt/ Magnus, http://sourceforge.net/projects/magnus Asir, http://www.asir.org DoCon, http://www.haskell.org/docon/ DCAS, http://sourceforge.net/projects/dcas/ Mathomatic, http://mathomatic.orgserve.de/math/ Sage, http://sage.math.washington.edu/sage/ Isabelle, http://isabelle.in.tum.de/index.html Languages Aldor, Aldor & OpenMath, Aldor & MathML, http://www.aldor.org/ GiNaC pt. C++, http://www.ginac.de/ OpenXM, http://www.openxm.org Libraries: CLN, http://www.ginac.de/CLN/ NTL, http://www.shoup.net/ntl/ Apfloat, http://www.apfloat.org/ SacLib, http://www.cis.udel.edu/~saclib/ LiDIA, http://www.cdc.informatik.tu-darmstadt.de/TI/LiDIA/ GMP, http://www.swox.com/gmp/ Bergman in LISP, http://servus.math.su.se/bergman/ LiE, http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/ AREP for GAP, http://www.ece.cmu.edu/~smart/arep/arep.html GRAPE for GAP, http://www.maths.qmul.ac.uk/~leonard/grape/ muEC for MuPAD, http://igm.univ- mlv.fr/LabInfo/equipe/combinatoire/MUEC/ GB for MuPAD, http://fgbrs.lip6.fr/jcf/Software/Gb/index.html GiANT for Kash, http://giantsystem.sourceforge.net/ Synaps, http://www-sop.inria.fr/galaad/software/synaps/main.html Piologie, http://www.zetagrid.net/zeta/sourcecode.html JAS, http://krum.rz.uni-mannheim.de/jas/ JScience, http://www.jscience.org/ Commons-Math, http://jakarta.apache.org/commons/math/ Meditor, http://jscl-meditor.sourceforge.net/ Math.Net, http://www.cdrnet.net/projects/nmath/ Perisic, http://ring.perisic.com/ Galois, http://www.partow.net/projects/galois/ LinBox, http://linalg.org/ CGAL, http://www.cgal.org/

31. Other sources Program collections: Albert, http://www.cs.clemson.edu/~dpj/albertstuff/albert.html Meataxe, http://www.math.rwth-aachen.de/~MTX/ FoxBox, DagWood, DSC, WiLiSS, LinBox, AppFac, http://www4.ncsu.edu/~kaltofen/ link software Quotpic, http://www.maths.warwick.ac.uk/~dfh/ link isom_quotpic GB, ftp://ftp.risc.uni-linz.ac.at/pub/GB Symmetrica, http://www.mathe2.uni- bayreuth.de/axel/symneu_engl.html ANU, http://www.mathematik.tu- darmstadt.de/~nickel/anuqs.html HartMath, http://sourceforge.net/project/showfiles.php?gro up_id=5083 Web Symbmath, http://www.symbmath.com/ OGB, http://grobner.nuigalway.ie/ngb/basis.html MathEclipse, http://www.matheclipse.org/me/ JavaView, http://www.javaview.de/ MTAC, http://mtac.sourceforge.net/ CAS lists: http://www.math.fsu.edu/Virtual/index.php?f=21 http://www.symbolicnet.org/ http://krum.rz.uni- mannheim.de/cabench/cawww.html http://www.g4g4.com/software1.htm http://www.computeralgebra.nl/ http://www.mat.niu.edu/~rusin/known-math/98/CAS http://www.cs.ru.nl/~freek/digimath/xindex.html http://dmoz.org/Science/Math/Software/ http://www.mat.univie.ac.at/~slc/divers/software.html http://orms.mfo.de/classif-browser.php Comparisons: http://en.wikipedia.org/wiki/List_of_computer_algebra _systems http://www.fordham.edu/lewis/cacomp.html http://krum.rz.uni-mannheim.de/cafgbench.html