symbolic comp.

Symbolic Computation
Higher Edu - Research dev card
Development from the higher education and research community
  • Creation or important update: 08/09/13
  • Minor correction: 08/09/13

IntegerVectorsModPermutationGroup : enumeration up to the action of a permutation group

This software was developed (or is under development) within the higher education and research community. Its stability can vary (see fields below) and its working state is not guaranteed.
  • Web site
  • System:
  • License(s): GPL
  • Status: stable release
  • Support: maintained, ongoing development
  • Designer(s): Nicolas Borie
  • Contact designer(s): nicolas.borie@univ-mlv.fr
  • Laboratory, service:

 

General software features

IntegerVectorsModPermutationGroup is an enumeration engine of integer vectors up to the action of a permutation group.

Let n a positif integer and G a permutation group, subgroup of the symmetric group of order n. This Sage module IntegerVectorsModPermutationGroup allows to enumerate tuples of length n modulo the action by position of G. This problem generalizes the enumeration of unlabelled graphs up to an isomorphism. One can also add some constraints like the sum of the entries or their maximum size.

This module is completly integrated in Sage since the version 4.7.

Exemple

Exemple for the cyclic group over 4 elements:

sage: G = PermutationGroup([[(1,2,3,4)]]); G
Permutation Group with generators [(1,2,3,4)]
sage: G.cardinality()
4
sage: S = IntegerVectorsModPermutationGroup(G); S
Integer vectors of length 4 enumerated up to the action of Permutation Group with generators [(1,2,3,4)]
sage: S.cardinality()
+Infinity
sage: it = iter(S)
sage: for i in range(25): v = it.next(); print v, " : ", S.orbit(v)
....:
[0, 0, 0, 0]  :  set([[0, 0, 0, 0]])
[1, 0, 0, 0]  :  set([[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]])
[2, 0, 0, 0]  :  set([[2, 0, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2], [0, 2, 0, 0]])
[1, 1, 0, 0]  :  set([[1, 0, 0, 1], [0, 0, 1, 1], [1, 1, 0, 0], [0, 1, 1, 0]])
[1, 0, 1, 0]  :  set([[0, 1, 0, 1], [1, 0, 1, 0]])
[3, 0, 0, 0]  :  set([[0, 0, 3, 0], [0, 3, 0, 0], [3, 0, 0, 0], [0, 0, 0, 3]])
[2, 1, 0, 0]  :  set([[0, 2, 1, 0], [0, 0, 2, 1], [1, 0, 0, 2], [2, 1, 0, 0]])
[2, 0, 1, 0]  :  set([[0, 1, 0, 2], [0, 2, 0, 1], [1, 0, 2, 0], [2, 0, 1, 0]])
[2, 0, 0, 1]  :  set([[2, 0, 0, 1], [0, 1, 2, 0], [1, 2, 0, 0], [0, 0, 1, 2]])
[1, 1, 1, 0]  :  set([[1, 1, 1, 0], [1, 1, 0, 1], [1, 0, 1, 1], [0, 1, 1, 1]])
[4, 0, 0, 0]  :  set([[4, 0, 0, 0], [0, 4, 0, 0], [0, 0, 4, 0], [0, 0, 0, 4]])
[3, 1, 0, 0]  :  set([[0, 0, 3, 1], [1, 0, 0, 3], [0, 3, 1, 0], [3, 1, 0, 0]])
[3, 0, 1, 0]  :  set([[0, 3, 0, 1], [0, 1, 0, 3], [3, 0, 1, 0], [1, 0, 3, 0]])
[3, 0, 0, 1]  :  set([[0, 0, 1, 3], [3, 0, 0, 1], [0, 1, 3, 0], [1, 3, 0, 0]])
[2, 2, 0, 0]  :  set([[0, 2, 2, 0], [2, 2, 0, 0], [2, 0, 0, 2], [0, 0, 2, 2]])
[2, 1, 1, 0]  :  set([[2, 1, 1, 0], [1, 0, 2, 1], [1, 1, 0, 2], [0, 2, 1, 1]])
[2, 1, 0, 1]  :  set([[0, 1, 2, 1], [1, 0, 1, 2], [2, 1, 0, 1], [1, 2, 1, 0]])
[2, 0, 2, 0]  :  set([[2, 0, 2, 0], [0, 2, 0, 2]])
[2, 0, 1, 1]  :  set([[1, 2, 0, 1], [2, 0, 1, 1], [0, 1, 1, 2], [1, 1, 2, 0]])
[1, 1, 1, 1]  :  set([[1, 1, 1, 1]])
[5, 0, 0, 0]  :  set([[0, 0, 0, 5], [5, 0, 0, 0], [0, 5, 0, 0], [0, 0, 5, 0]])
[4, 1, 0, 0]  :  set([[0, 0, 4, 1], [1, 0, 0, 4], [0, 4, 1, 0], [4, 1, 0, 0]])
[4, 0, 1, 0]  :  set([[0, 4, 0, 1], [1, 0, 4, 0], [0, 1, 0, 4], [4, 0, 1, 0]])
[4, 0, 0, 1]  :  set([[4, 0, 0, 1], [1, 4, 0, 0], [0, 0, 1, 4], [0, 1, 4, 0]])
[3, 2, 0, 0]  :  set([[3, 2, 0, 0], [0, 0, 3, 2], [2, 0, 0, 3], [0, 3, 2, 0]])

Context in which the software is used

The development of a such engine was necessary for the thesis work of the author. The thesis is about effective invariant theory. This module is also usefull in the following fields:

  • Effective invariant theory,
  • Effective Galois theory,
  • Structure Species theory.
Publications related to the software
Higher Edu - Research dev card
Development from the higher education and research community
  • Creation or important update: 07/09/12
  • Minor correction: 07/09/12

MuPAD-Combinat : algebraic combinatorics library for MuPAD

This software was developed (or is under development) within the higher education and research community. Its stability can vary (see fields below) and its working state is not guaranteed.
  • Web site
  • System:
  • Current version: dev-1.3.3 - june 2008
  • License(s): LGPL
  • Status: validated (according to PLUME), stable release
  • Support: not maintained, no ongoing development
  • Designer(s): Nicolas M. Thiéry et Florent Hivert
  • Contact designer(s): nthiery@users.sf.net
  • Laboratory, service: Departament de Llenguatges i Sistemes Informátics (Universitat Politècnica de Catalunya), Department of Mathematics (University of California Davis)

 

General software features
  • Provides an extensible toolbox for computer exploration in algebraic combinatorics
  • Fosters the software development in this area, with a mutualisation of the researchers' effort via free software
  • Supports the dissemination of research results
  • Supports combinatorics education

The developpement of MuPAD-Combinat was stopped in June 2008 and has evolved into Sage-Combinat.

Context in which the software is used

Tool for computer exploration in the Algebraic Combinatorics and Symbolic Computation research team, at the Gaspard-Monge computer software lab (LIGM).

Publications related to the software

Note : this last document gives a complet list of publications using or related to MuPAD-Combinat (until decembre 2008).

Higher Edu - Research dev card
Development from the higher education and research community
  • Creation or important update: 24/10/11
  • Minor correction: 25/10/11

Bianchi.gp : fundamental domain computation in hyperbolic space for the Bianchi groups

This software was developed (or is under development) within the higher education and research community. Its stability can vary (see fields below) and its working state is not guaranteed.
  • Web site
  • System:
  • Current version: 2-1-0
  • License(s): GPL
  • Status: stable release
  • Support: maintained, no ongoing development
  • Designer(s): Alexander D. Rahm
  • Contact designer(s): Alexander.Rahm@ujf-grenoble.fr
  • Laboratory, service:

 

General software features

This program carries out computations of the geometry of a certain class of arithmetic groups (the Bianchi groups), via a proper action on a contractible space. We access their group homology.

In more detail, consider an imaginary quadratic number field Q(√−m), where m is a square-free positive integer. Let A(m) be its ring of integers. The Bianchi groups are the groups SL_2(A(m)).

A wealth of information on the Bianchi groups can be found in monographs of Elstrodt/Grunewald/Mennicke, Benjamin Fine, Maclachlan/Reid, etc. These groups act in a natural way on hyperbolic three-space, which is isomorphic to their associated symmetric space.

The kernel of this action is the centre {±1} of the groups, which makes it useful to study the quotients PSL_2(A(m)) of the groups by their centre, which we again call Bianchi groups.

In 1892, Luigi Bianchi computed fundamental domains for this action for some of these groups. Such a fundamental domain has the shape of a hyperbolic polyhedron (up to some missing vertices), and we call it the Bianchi fundamental polyhedron. The computation of the the Bianchi fundamental polyhedron has been implemented in this program for all Bianchi groups.

The images under SL_2(A(m)) of the faces of this polyhedron provide hyperbolic space with a cellular structure. In order to view clearly the local geometry, we pass to the refined cell complex, which we obtain by subdiving this cell structure until the cell stabilisers in SL_2(A(m)) fix the cells pointwise. We can exploit this cell complex in different ways, in order to see different aspects of the geometry of these groups.

A database with cell complexes computed with Bianchi.gp is available at

http://www.wisdom.weizmann.ac.il/~rahm/cellComplex...

Context in which the software is used

This program is based on the computer algebra system Pari/GP, which is freely available.

The version 2.4.3 (or more recent) of Pari/GP needs to be installed in order to use Bianchi.gp.

Publications related to the software

The homological torsion of SL_2 of the imaginary quadratic integers, Alexander D. Rahm, Accepted for publication at the Transactions of the AMS.

Homologie et K-théorie des groupes de Bianchi, Alexander D. Rahm, Note bilingue anglais/francais, Comptes Rendus Mathématique of the Académie des Sciences - Paris, Volume 349, Issues 11-12, June 2011, Pages 615-619, doi:10.1016/j.crma.2011.05.014 .

The integral homology of PSL_2 of imaginary quadratic integers with nontrivial class group, Alexander D. Rahm and Mathias Fuchs, Journal of Pure and Applied Algebra, DOI: 10.1016/j.jpaa.2010.09.005 .

(Co)homologies et K-théorie de groupes de Bianchi par des modèles géométriques calculatoires, Alexander D. Rahm, Thèse de Doctorat, Université de Grenoble and Universität Göttingen, 2010.
Archived on the server TEL of the CNRS.

On level one cuspidal Bianchi modular forms, Alexander D. Rahm and Mehmet Haluk Sengun, preprint on HAL and arXiv.

Higher Edu - Research dev card
Development from the higher education and research community
  • Creation or important update: 15/02/10
  • Minor correction: 31/03/10

Giac/Xcas : the swiss knife for mathematics

This software was developed (or is under development) within the higher education and research community. Its stability can vary (see fields below) and its working state is not guaranteed.
  • Web site
  • System:
  • Current version: 0.8.5 (stable), 0.9.0 (developpement) - 2/2/2010
  • License(s): GPL - GPL 3
  • Status: validated (according to PLUME), stable release
  • Support: maintained, ongoing development
  • Designer(s): Bernard Parisse (code), Renée De Graeve (documentation)
  • Contact designer(s): bernard.parisse@ujf-grenoble.fr
  • Laboratory, service:

 

General software features
  • Giac is a C++ library for computer algebra. It is build on C and C++ libraries: PARI, NTL (arithmetic), CoCoA (Groebner basis), GSL (numerics), GMP (big integers), MPFR (bigfloats) and provides algorithms for basic polynomial operations (product, GCD) and symbolic computations (simplifications, limits/series, symbolic integration, sommation, ...). The library can be configured to accept Maple or TI syntax to ease the transition for users of these systems.
  • Xcas is a GUI application interfaced with Giac. It was first a GUI for symbolic computation, then several modules were added: 2-d and 3-d graphics, dynamic 2-d and 3-d geometry (exact or numeric), spreadsheet, programming environment
Context in which the software is used
  • The giac library is the computing kernel of Xcas online and of the CmathOOCas OpenOffice plugin.
  • Xcas is one of the software available for the French agrégation de maths Fiche Plume. Xcas is also used at the University of Grenoble (undergraduate level).
  • Xcas is one of the softwares used by math teachers in French secondary schools, especially for teaching algorithmic. It has been recently translated in Greek and might also be used in Greek secondary schools.
  • giac (commandline interface) is the computing kernel of some LaTeX tools for math teachers in secondary schools (professor, tablor, pgiac).
Higher Edu - Research dev card
Development from the higher education and research community
  • Creation or important update: 25/10/09
  • Minor correction: 29/08/12

Sage-Combinat : toolbox for computer exploration in (algebraic) combinatorics

This software was developed (or is under development) within the higher education and research community. Its stability can vary (see fields below) and its working state is not guaranteed.
  • Web site
  • System:
  • Current version: (integrated in Sage 4.7.2) - January 2012
  • License(s): GPL
  • Status: validated (according to PLUME), stable release, under development
  • Support: maintained, ongoing development
  • Designer(s): See the contributor's list
  • Contact designer(s): sage-combinat-devel _AT_ googlegroups.com
  • Laboratory, service: University of California at Davis, University of Washington at Seattle, York University, University of Pennsylvania, Stanford University, Kaist University

 

General software features

Sage-Combinat is a software project whose mission is: "to improve the open source mathematical system Sage as an extensible toolbox for computer exploration in algebraic combinatorics, and to foster code sharing between researchers in this area".

Sage-combinat was born as a partial port of MuPAD-Combinat, and is its natural successor.

Context in which the software is used

This software is used and developped by a community of researchers in combinatorics for their computer exploration needs. This page lists the projects and labs involved in Sage-Combinat.

Publications related to the software

Sage-Combinat and its predecessor MuPAD-Combinat played an essential role in more than 50 publications. See:

Note : this later document contains as appendix a complete list of publications using or about MuPAD-Combinat and Sage-Combinat (up to Decembre 2008).

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