Changeset cfc9c0 in git


Ignore:
Timestamp:
Apr 28, 2005, 6:32:06 PM (19 years ago)
Author:
Viktor Levandovskyy <levandov@…>
Branches:
(u'fieker-DuVal', '117eb8c30fc9e991c4decca4832b1d19036c4c65')(u'spielwiese', '4188d308699580d975efd0f6cca8dcb41c396f70')
Children:
fefe03c337c2d780101b98e7858eb2538ed41587
Parents:
fdebd3cb4759f0739bae5afd99a64ab39ae4efb4
Message:
*levandov: corrections, related to the documentation


git-svn-id: file:///usr/local/Singular/svn/trunk@7932 2c84dea3-7e68-4137-9b89-c4e89433aadc
Location:
Singular/LIB
Files:
3 edited

Legend:

Unmodified
Added
Removed
  • Singular/LIB/ncdecomp.lib

    rfdebd3 rcfc9c0  
    11///////////////////////////////////////////////////////////////////////////////
    2 version="$Id: ncdecomp.lib,v 1.7 2005-03-25 18:38:46 levandov Exp $";
     2version="$Id: ncdecomp.lib,v 1.8 2005-04-28 16:32:06 levandov Exp $";
    33category="Noncommutative";
    44info="
     
    105105///////////////////////////////////////////////////////////////////////////////
    106106proc CentralQuot(module I, ideal G)
    107 "USAGE:  CentralQuot(M, T), for a module M and an ideal T,
    108 RETURN:  module of the central quotient I:G,
     107"USAGE:  CentralQuot(M, G), for a module M and an ideal G,
     108PURPOSE: compute the central quotient I:G
     109RETURN:  module
    109110NOTE:    the output module is not necessarily a Groebner basis,
    110111SEE ALSO: CentralSaturation, CenCharDec
     
    139140proc CentralSaturation(module M, ideal T)
    140141"USAGE:  CentralSaturation(M, T), for a module M and an ideal T,
    141 RETURN:  module of the central saturation of M by T (also denoted by M:T^{\infty}),
     142PURPOSE: compute the central saturation of M by T (also denoted by M:T^{\infty})
     143RETURN:  module
    142144NOTE:    the output module is not necessarily a Groebner basis,
    143145SEE ALSO: CentralQuot, CenCharDec
     
    178180proc CenCharDec(module I, def #)
    179181"USAGE:  CenCharDec(I, C);  I a module, C an ideal/list of generators of the center;
    180 PURPOSE: compute a finite central character decomposition (or point out that there is no finite one),
     182PURPOSE: compute a finite central character decomposition (or determine that there is no finite one),
    181183RETURN:  a list L, where each entry consists of three records:
    182184@*       L[*][1] ('ideal' type), the central character as the maximal ideal in the center,
    183185@*       L[*][2] ('module' type), the Groebner basis of the weight module, corresponding to the character,
    184 @*       L[*][3] ('int' type) is the K-dimension of the weight module (-1 is returned for an infinite dimension);
    185 NOTE:     some modules have no finite decomposition (in such case one
    186           gets warning message)
     186@*       L[*][3] ('int' type) is the K-dimension of the weight module (-1 is returned in the case of infinite dimension);
     187NOTE:     actual decomposition is a sum of L[i][2];
     188@*        some modules have no finite decomposition (in such case one gets warning message)
    187189SEE ALSO: CentralQuot, CentralSaturation
    188190EXAMPLE: example CenCharDec; shows examples
     
    327329"USAGE:  IntersectWithSub(M,Z),  M an ideal, Z an ideal/list of pairwise commutative elements
    328330PURPOSE: computes an intersection of M with the subalgebra, generated by Z
    329 RETURN:  ideal (of two--sided generators, not a Groebner basis!)
     331RETURN:  ideal (of two-sided generators, not a Groebner basis!)
    330332NOTE:    usually one puts generators of the center into Z
    331333EXAMPLE: example IntersectWithSub; shows an example
  • Singular/LIB/nctools.lib

    rfdebd3 rcfc9c0  
    11///////////////////////////////////////////////////////////////////////////////
    2 version="$Id: nctools.lib,v 1.10 2005-02-23 18:10:46 levandov Exp $";
     2version="$Id: nctools.lib,v 1.11 2005-04-28 16:32:06 levandov Exp $";
    33category="Noncommutative";
    44info="
     
    1616RootOfUnity(n);         the minimal polynomial for the n-th primitive root of unity,
    1717Weyl([p]);              creates Weyl algebra structure in a basering (two different realizations),
    18 CreateWeyl(n, [p]);     returns n-th Weyl algebra in (x(i),D(i)) presentation; in char p, if an integer p is given,
    19 Heisenberg(N, [p,d]);   returns N-th  Heisenberg algebra in (x(i),y(i),h) realization,
     18CreateWeyl(n, [p]);     returns n-th Weyl algebra in (x(i),D(i)) presentation,
     19Heisenberg(N, [p,d]);   returns n-th  Heisenberg algebra in (x(i),y(i),h) realization,
    2020Exterior();             returns qring, the exterior algebra of a basering,
    21 Fin_dim_algebra(matrix M, list #); creates finite dimensional algebra structure from the basering and the multiplication matrix M,
     21fin_dim_algebra(matrix M, list #); creates finite dimensional algebra structure from the basering and the multiplication matrix M,
    2222
    2323AUXILIARY PROCEDURES:
    24 Newton(f);              Newton diagram of a polynomial f,
     24Newton(f);              Newton diagram of a polynomial,
    2525NCRelations(r);         recovers the non-commutative relations of a G-algebra,
    26 IsCentral(p,[v]);       check for the commutativity of polynomial p with the G-algebra,
     26IsCentral(p,[v]);       check for the commutativity of a polynomial in the G-algebra,
    2727Is_NC();                checks whether basering is noncommutative,
    2828UpOneMatrix(N);         returns NxN matrix with 1's in the whole upper triagle,
     
    3434///////////////////////////////////////////////////////////////////////////////
    3535
    36 // This procedure computes a weights vector for a G-Algebra
    37 // r must be a G-algebra
     36// This procedure computes a weights vector for a G-algebra r
    3837
    3938proc Gweights(def r)
    4039"USAGE:   Gweights(r); r a ring or a square matrix
    41 RETURN:   a weight vector for the G-algebra:
    42           r itself, if it is of the type ring,
     40PURPOSE: compute the weight vector for the G-algebra:
     41          either r itself, if it is of the type ring,
    4342          or for a G-algebra, defined by the square matrix r
     43RETURN:   intvec
    4444NOTE:    with Gweights you obtain a vector, which must be used to redefine the G-Algebra. If the input is a matrix and the output is the zero vector then there is not a G-algebra structure associated to these relations with respect to the given variables. Another possibility is to use wRing or weightedRing to obtain directly the G-Algebra.
    4545EXAMPLE: example Gweights; shows examples
     
    122122proc weightedRing(def r)
    123123"USAGE:   weightedRing(r); r a ring
    124 RETURN:  ring with the relations of r with order changed to comply with
    125 the ordering condition for G-algebras
    126 NOTE:    you have to activate this ring with the "setring" command
     124PURPOSE:  equip a ring with such weights,that the relations of a new ring
     125comply with the ordering condition for G-algebras
     126RETURN:  ring
     127NOTE:    you have to activate this ring with the \"setring\" command
    127128EXAMPLE: example weightedRing; shows examples
    128129SEE ALSO: wRing, Gweights
     
    337338proc Newton(poly f)
    338339"USAGE:   Newton(f); f a poly
    339 RETURN:  intmat, representing the Newton diagram of f
     340PURPOSE: compute the Newton diagram of f
     341RETURN:  intmat
    340342NOTE:    each row is the exponent of a monomial of f
    341343EXAMPLE: example Newton; shows examples
     
    373375proc NCRelations(def r)
    374376"USAGE:   NCRelations(r); r a ring
    375 RETURN:  a list with two elements, both elements are of type matrix and
    376          represent the matrices C,D defining the non-commutative relations
    377          of the G-algebra r
     377PURPOSE: recover the noncommutative relations via matrices C and D from
     378a noncommutative ring
     379RETURN:  list L with two elements, both elements are of type matrix:
     380         L[1] = matrix of coefficients C,
     381         L[2] = matrix of polynomials D
    378382EXAMPLE: example NCRelations; shows examples
    379383"{
     
    433437///////////////////////////////////////////////////////////////////////////////
    434438
    435 proc Fin_dim_algebra(matrix M, list #)
    436 "USAGE:   Fin_dim_algebra(M,[r]); M a matrix, r an optional ring.
    437 RETURN:  nothing. Creates finite dimensional algebra structure in a ring r
    438 (if it is given) or in a basering (by default) from the matrix M. Exports the ideal called Quot for further qring definition.
    439 NOTE: matrix M will be read according to relations x_j*x_i = M[i,j]; Quot is not given in its two-sided Groebner basis.
    440 EXAMPLE: Fin_dim_algebra; shows examples
     439proc fin_dim_algebra(matrix M, list #)
     440"USAGE:   fin_dim_algebra(M,[r]); M a matrix, r an optional ring.
     441PURPOSE: creates a finite dimensional algebra structure in a ring r
     442(if it is given) or in a basering (by default) from the matrix M.
     443Exports the ideal called Quot for further qring definition.
     444RETURN:  nothing
     445NOTE: matrix M will be read according to relations Xj*Xi = M[i,j]; Quot is not given in its two-sided Groebner basis.
     446EXAMPLE: example fin_dim_algebra; shows examples
    441447"
    442448{
     
    489495  matrix S[3][3];
    490496  S[2,3]=a*x(1); S[3,2]=-b*x(1);
    491   Fin_dim_algebra(S);
     497  fin_dim_algebra(S);
    492498  Quot = twostd(Quot);
    493499  qring Qr = Quot;
     
    498504
    499505proc IsCentral(poly p, list #)
    500 "USAGE:   IsCentral(p,[v]); p poly, v an integer (with v!=0 procedure will be verbose)
    501 RETURN:  integer (1 if p commutes with all variables, 0 otherwise)
     506"USAGE:   IsCentral(p,[v]); p poly, v an integer (with v!=0, procedure will be verbose)
     507PURPOSE: check whether p is central in a basering (that is, commutes with everything)
     508RETURN:  integer (1, if p commutes with all variables, 0 otherwise)
    502509EXAMPLE: example IsCentral; shows examples
    503510"{
     
    539546proc UpOneMatrix(int N)
    540547"USAGE:   UpOneMatrix(N); N an integer, the number of columns
    541 RETURN:  intmat, NxN matrix with 1's in the whole upper triagle
     548PURPOSE: compute an NxN matrix with 1's in the whole upper triagle
     549RETURN:  intmat
    542550NOTE: helpful for setting noncommutative algebras with complicated
    543551coefficient matrices
     
    567575///////////////////////////////////////////////////////////////////////////////
    568576proc ndc(list #)
    569 "USAGE:   ndc([v]); v an optional integer. If v!=0 procedure will be verbose
    570 RETURN:  ideal of non-degeneracy conditions of the basering
     577"USAGE:   ndc([v]); v an optional integer (if v!=0, procedure will be verbose)
     578PURPOSE: compute the non-degeneracy conditions of the basering,
     579RETURN:  ideal
    571580EXAMPLE: example ndc; shows examples
    572581"
     
    633642proc RootOfUnity(int n)
    634643"USAGE:   RootOfUnity(n); n an integer
    635 RETURN:  number, the n-th primitive root of unity (for use as minpoly)
     644PURPOSE: compute the minimal polynomial for the n-th primitive root of unity
     645RETURN:  number
    636646NOTE: works only in field extensions by one element
    637647EXAMPLE: example RootOfUnity; shows examples
     
    711721proc Weyl(list #)
    712722"USAGE:   Weyl([p]); p an optional integer.
    713 RETURN:  nothing. Creates Weyl algebra structure in a basering. By default
     723PURPOSE: create a Weyl algebra structure on a basering. By default
    714724mimics (x(1..N),d(1..N)) realization. If p is given and is not zero,
    715725uses (x(1),d(1),x(2),d(2),... ) realization.
     726RETURN:  nothing
     727SEE ALSO: CreateWeyl
    716728EXAMPLE: example Weyl; shows examples
    717729"
     
    769781proc Heisenberg(int N, list #)
    770782"USAGE:   Heisenberg(N, [p,d]); N an integer (setting 2*N+1 variables), p an optional integer (field characteristic), d an optional integer (power of h in the commutator)
    771 RETURN:  N-th  Heisenberg algebra in x(i),y(i),h realization
    772 NOTE: you have to activate this ring with the "setring" command
     783PURPOSE: create a N-th  Heisenberg algebra in x(i),y(i),h realization
     784RETURN: nothing
     785NOTE: you have to activate this ring with the \"setring\" command
    773786EXAMPLE: example Heisenberg; shows examples
    774787"
     
    805818  "EXAMPLE:";echo=2;
    806819  def a = Heisenberg(2);
    807   setring a;
    808   a;
     820  setring a;   a;
     821  def H3 = Heisenberg(3, 7, 2);
     822  setring H3;  H3;
    809823}
    810824
     
    813827proc Exterior(list #)
    814828"USAGE:   Exterior();
    815 RETURN:  qring, the exterior algebra of a basering
    816 NOTE: you have to activate this qring with the "setring" command
     829PURPOSE:  create the exterior algebra of a basering,
     830RETURN:  qring
     831NOTE: you have to activate this qring with the \"setring\" command
    817832EXAMPLE: example Exterior; shows examples
    818833"
     
    853868proc CreateWeyl(int n, list #)
    854869"USAGE:   CreateWeyl(n,[p]); n an integer, n>0; p an optional integer (field characteristic)
    855 RETURN:  a ring, describing n-th Weyl algebra
    856 NOTE:    You have to activate this ring with the "setring" command.
     870PURPOSE: create a n-th Weyl algebra
     871RETURN:  ring
     872NOTE:    you have to activate this ring with the \"setring\" command.
    857873         The presentation of n-th Weyl algebra is classical:
    858874         D(i)x(i)=x(i)D(i)+1
     
    896912proc Is_NC()
    897913"USAGE:   Is_NC();
    898 RETURN:   1, if basering is noncommutative; 0 otherwise.
     914PURPOSE: check whether a basering is commutative or not
     915RETURN:   int (1, if basering is noncommutative; 0 otherwise)
    899916EXAMPLE: example Is_NC; shows examples
    900917"{
  • Singular/LIB/qmatrix.lib

    rfdebd3 rcfc9c0  
    1 version="$Id: qmatrix.lib,v 1.8 2005-02-23 18:10:46 levandov Exp $";
     1version="$Id: qmatrix.lib,v 1.9 2005-04-28 16:32:06 levandov Exp $";
    22category="Noncommutative";
    33info="
     
    8080proc LengthSymElement(intvec v)
    8181"USAGE:   LengthSymElement(v); v an intvec representing an element of S(n)
    82 RETURN:  int, the length of v
     82PURPOSE: determine the length of v
     83RETURN:  int
    8384NOTE:    if v doesn't represent an element of S(n), the output may have no sense
    8485SEE ALSO: SymGroup, LengthSym
     
    108109proc LengthSym(intmat M)
    109110"USAGE:   LengthSym(M); M an intmat representing a subset of S(n) (each row must be an element of S(n))
    110 RETURN:  intvec, the i-th element is the length of the i-th row of M
     111PURPOSE: determine a vector, which i-th element is the length of the i-th row of M
     112RETURN:  intvec
    111113NOTE:    If M is not a subset of S(n), the output may not have meaning
    112114SEE ALSO: SymGroup, LengthSymElement
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