Packages

Lecture 12

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

From other sources

Oberwolfach References onMathematical 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

Oberwolfach References onMathematical SoftwareIdentifier Description01 Discrete Mathematics01.01 Convex and discrete geometry01.02 Graph theory01.03 Enumerative combinatorics01.04 Algebraic combinatorics01.05 Integer programming01.06 Ordered structures02 Algebra / Number Theory02.01 Mathematical logic and foundations02.02 Number theory02.03 Field theory02.04 Commutative rings and algebras02.04.01 Ideals, modules, homomorphisms02.04.02 Polynomial and power series rings02.04.03 Special rings02.04.04 Graded rings and Hilbert functions02.04.05 Integral dependence and normalization02.04.06 Dimension theory02.04.07 Factorization and primary decomposition02.04.08 Syzygies and resolutions02.04.09 Differential algebra02.04.10 Groebner bases02.05 Linear and multilinear algebra; matrix theory02.05.01 Linear equations02.05.02 Eigenvalues, singular values, and eigenvectors02.05.03 Canonical forms02.05.04 Matrix factorization02.05.05 Integral matrices02.05.06 Multilinear algebra02.05.07 Linear inequalities

02.06 Non-commutative and general rings and algebras02.06.01 Lie algebras02.07 Category theory; homological algebra02.08 Group theory and generalizations02.08.01 Permutation groups02.08.02 Matrix groups02.08.03 Finitely presented groups02.08.04 Polycyclicly presented groups02.08.05 Black box groups02.08.06 Group actions02.08.07 Subgroup lattices02.08.08 Group cohomology02.08.09 Semigroups02.09 Representation theory02.09.01 Ordinary Representations of Groups02.09.02 Ordinary Representations of Algebras02.09.03 Modular Representations of Groups02.09.04 Modular Representations of Algebras02.09.05 Character Theory02.09.06 Permutation Representations03 Geometry / Topology03.01 Algebraic geometry03.01.01 Local theory, singularities03.01.02 Cycles and subschemes03.01.03 Families, fibrations03.01.04 Birational theory03.01.05 Co(homology) theory03.01.06 Arithmetic problems03.01.07 Projective Geometry03.01.08 Algebraic groups and geometric invariant theory03.01.09 Special varieties03.01.10 Real algebraic geometry

Oberwolfach References onMathematical Software03.02 Topological groups, Lie groups03.03 Several complex variables and analytic spaces03.04 Geometry03.05 Differential geometry03.06 General topology03.07 Algebraic topology03.08 Manifolds and cell complexes03.09 Global analysis, analysis on manifolds03.10 Visualization04 Analysis04.01 Differentiation and Integration04.02 Sequences, series, limits04.03 Approximations and expansions04.04 Functional analysis and operator theory04.05 Special functions04.06 Calculus of variations04.07 Integral transforms, operational calculus05 Differential and Integral Equations05.34 Ordinary differential equations05.35 Partial differential equations05.37 Dynamical systems and ergodic theory05.39 Difference and functional equations05.45 Integral equations

06 Probability / Statistics06.01 Combinatorial probability06.02 Stochastic processes06.03 Stochastic analysis06.04 Stochastic geometry06.05 Distributions06.06 Statistics07 Applications of Mathematics07.70 Mechanics of particles and systems07.74 Mechanics of deformable solids07.76 Fluid mechanics07.78 Optics, electromagnetic theory07.80 Classical thermodynamics, heat transfer07.81 Quantum theory07.82 Statistical mechanics, structure of matter07.83 Relativity and gravitational theory07.85 Astronomy and astrophysics07.86 Geophysics07.87 Theoretical Chemistry07.88 Crystallography07.90 Operations research, mathematical programming07.91 Game theory, economics, social and behavioral sciences07.92 Biology and other natural sciences07.93 Systems theory; control07.94 Information and communication, circuits08 Teaching08.97 Mathematics education09 Logic/ Deduction/ Computational Logic

Comparison of computer algebra systemsFrom Wikipedia, the free encyclopedia

Comparison of computer algebra systemsFrom Wikipedia, the free encyclopedia

Google directory

Google directory

CAIN

CAIN (1998)

CAIN – Special purpose systems(Non)Commutative Algebra & Algebraic GeometryAlbertBergmanCALICASACLICALCLIFFORDCoCoAFELIXGANITHGBThe Grassman packageGRBGROEBNER (from RISC-Linz)GROEBNER (REDUCE package)IDEALSKANMacaulayMacaulay 2MASNCALGEBRASACLIBSingularWU

Differential Equation Solvers & ToolsA review of ODE SolversList of Symmetry Programs (PostScript, gzip

-compressed)CONTENT: Dynamical System SoftwareCRACKDELiADESIRDiffgrob2 (Manual: PostScript, gzip-formatted)DIMSYMFIDEODEtools (symmetry methods)LIE (A.K. Head, in MuMath)liesymmODESOLVEPDELIEPDEtools packagePhaser

: an Animator/Simulator for Dynamical Systems for IBM PC's

The Poincare packageSPDEStandardFormSYM_DESYMMGRP.MAX (Manual: PostScript, gzip

-formatted)

CAIN – Special purpose systemsFinite Element AnalysisPDEaseSENAC/FEM (PostScipt, gzip-formatted); see

also SENAC Group TheoryANU SoftwareCayleyCHEVIEGAPGRAPEGUAVALiELIE (REDUCE package)MagmaMeatAxeThe Magnus system for exploring infinite groupsSchurSisyphosSymmetricaWeyl Groups and Hecke Algebras

High Energy PhysicsSee also the FreeHEP Software database. FeynArtsFoamFORMThe Partials packageSchoonschipTracer Number TheoryGaloisKANTKASHLiDIAMALMNTLNumbersPARISIMATH

CAIN – Special purpose systems & experimentalTensor CalculusCARTANClassiEinsGRTensorMathTensorRedtenRicciSHEEPSTENSORTensors in Macsyma (R)Tools of Tensor Calculus

PC Shareware with Symbolic FeaturesAMPCalculus and Differential EquationsCC4CLAMathomaticPFSASymbMathX(PLORE)Various SystemsAUTOMATA, QUOTPIC & TESTISOMComputer Algebra KitFLACJACALMock-MmaNODES: Non linear Ordinary Differential Equations

SolverORMERepTilesAsirSimLab, Computer Tools for Analysis and Simulation

Other sourcesCAS

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/

Other sourcesProgram 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?group_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