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products/Sources/formale Sprachen/GAP/pkg/simpcomp/lib/ (GAP Algebra Version 4.15.1^©) Datei vom 18.2.2022 mit Größe 14 kB

Quellcode-Bibliothek pkghomalg.gi Sprache: Shell

################################################################################
##
## simpcomp / pkghomalg.gi
##
## Loaded when package `homalg' is available
##
## $Id$
##
################################################################################

# helper function that converts an integer modulus to an homalg ring according
# to the following table:
# modulus - homalg ring
# 0 - rationals
# 1 - integers
# n - integers mod n
SCIntFunc.ModulusToHomalgRing:=function(modulus)
 if(modulus<0) then
 return fail;
 elif(modulus=1) then
 return HomalgRingOfIntegers();
 elif(modulus=0) then
 return HomalgFieldOfRationals();
 else
 return HomalgRingOfIntegers(modulus);
 fi;
end;

SCIntFunc.bdOperator:=function(simplex)
 local b,cur,i,j,pos;
 b:=[];

 #all positions, i=kill entry
 for i in [1..Length(simplex)] do
 cur:=[];
 pos:=1;

 #extract one boundary face
 for j in [1..Length(simplex)] do
 if(i=j) then continue; fi;

 cur[pos]:=simplex[j];
 pos:=pos+1;
 od;

 #add to list of boundary faces
 Add(b,ShallowCopy(cur));
 od;

 return b;
end;

################################################################################
##<#GAPDoc Label="SCHomalgBoundaryMatrices">
## <ManSection>
## <Meth Name="SCHomalgBoundaryMatrices" Arg="complex,modulus"/>
## <Returns>a list of <Package>homalg</Package> objects upon success,
## <K>fail</K> otherwise.</Returns>
## <Description>
## This function computes the boundary operator matrices for the simplicial
## complex <Arg>complex</Arg> with a ring of coefficients as specified by
## <Arg>modulus</Arg>: a value of <C>0</C> yields <M>\mathbb{Q}</M>-matrices,
## a value of <C>1</C> yields <M>\mathbb{Z}</M>-matrices and a value of
## <C>q</C>, q a prime or a prime power, computes the
## <M>\mathbb{F}_q</M>-matrices.
## <Example><![CDATA[
## gap> SCLib.SearchByName("CP^2 (VT)");
## gap> c:=SCLib.Load(last[1][1]);;
## gap> SCHomalgBoundaryMatrices(c,0);
## ]]></Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
################################################################################
InstallMethod(SCHomalgBoundaryMatricesOp,
"for SCSimplicialComplex and Integer",
[SCIsSimplicialComplex, IsInt],
function(complex, modulus)
 local tmp,skel,x,matrices,one,minusone,k,p,i,j,b,ind,pos,res,d,ring;

 ring:=SCIntFunc.ModulusToHomalgRing(modulus);
 if(ring=fail) then
 return fail;
 fi;

 # for matrices
 matrices := [];

 # 1 and -1 in ring
 one:=One(ring);
 minusone:=MinusOne(ring);

 skel:=SCFaceLatticeEx(complex);
 d:=SCDim(complex);

 matrices[d+1]:=HomalgInitialMatrix(0,Length(skel[d+1]),ring);
 ResetFilterObj(matrices[d+1], IsInitialMatrix);
 for k in [2..d+1] do
 matrices[k-1] := HomalgInitialMatrix(Length(skel[k]),
 Length(skel[k-1]),ring);

 for i in [1..Length(skel[k])] do
 #get boundary of current simplex
 b:=SCIntFunc.bdOperator(skel[k][i]);

 for j in [1..Length(b)] do
 pos:=Position(skel[k-1],b[j]);
 if(pos<>fail) then
 if((j mod 2)=0) then
 AddToMatElm( matrices[k-1], i, pos, one );
 else
 AddToMatElm( matrices[k-1], i, pos, minusone );
 fi;
 else
 Info(InfoSimpcomp,1,"SCHomalgBoundaryOperatorMatrices: list of ",
 "boundary skel[k-1] is not complete k=",k,", b[j]=",b[j],"!");
 return fail;
 fi;
 od;
 od;
 ResetFilterObj(matrices[k-1],IsInitialMatrix);
 od;

 return matrices;
end);

################################################################################
##<#GAPDoc Label="SCHomalgCoboundaryMatrices">
## <ManSection>
## <Meth Name="SCHomalgCoboundaryMatrices" Arg="complex,modulus"/>
## <Returns>a list of <Package>homalg</Package> objects upon success,
## <K>fail</K> otherwise.</Returns>
## <Description>
## This function computes the coboundary operator matrices for the simplicial
## complex <Arg>complex</Arg> with a ring of coefficients as specified by
## <Arg>modulus</Arg>: a value of <C>0</C> yields <M>\mathbb{Q}</M>-matrices,
## a value of <C>1</C> yields <M>\mathbb{Z}</M>-matrices and a value of
## <C>q</C>, q a prime or a prime power, computes the
## <M>\mathbb{F}_q</M>-matrices.
## <Example><![CDATA[
## gap> SCLib.SearchByName("CP^2 (VT)");
## gap> c:=SCLib.Load(last[1][1]);;
## gap> SCHomalgCoboundaryMatrices(c,0);
## ]]></Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
################################################################################
InstallMethod(SCHomalgCoboundaryMatricesOp,
"for SCSimplicialComplex and Integer",
[SCIsSimplicialComplex, IsInt],
 function( complex, modulus )
 local tmp,skel,x,matrices,one,minusone,k,p,i,j,b,ind,pos,res,d,#############################################################################

 ring:=SCIntFunc.ModulusToHomalgRing(modulus);
 if(ring=fail) then
 return fail;
 fi;

 # for matrices
 matrices := [];

 # 1 and -1 in ring
 one:=One(ring);
 minusone:=MinusOne(ring);

 skel:=SCFaceLatticeEx(complex);
 d:=SCDim(complex);

 matrices[d+1]:=HomalgInitialMatrix(Length(skel[d+1]),0,ring);
 ResetFilterObj(matrices[d+1],IsInitialMatrix);
 for k in [2..d+1] do
 matrices[k-1]:=HomalgInitialMatrix(Length(skel[k-1]),Length(skel[k]),ring);

 for i in [1..Length(skel[k])] do
 #get boundary ofcurrent simplex
 b:=SCIntFunc.bdOperator(skel[k][i]);

 for#
 pos:=Position(skel[k-1],b[j]);
 if<> then## Id
 if( 2)=0 then
 (k-],osione)
 else
 AddToMatElm(matrices[k-1],pos,i,minusone#modulus - homalg ring
 fi;
 else
 Info(InfoSimpcomp,1,"SCHomalgCoboundaryOperatorMatrices: list of ",
 " 1 -
 returnfail
 fi;
 od;
 od;
 ResetFilterObj(matrices[k-1],IsInitialMatrix);
;
 returnmatrices;
;

#############################################
##<#GAPDoc Label="SCHomalgHomology">
## <ManSectionjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
# MethName=SCHomalgHomology" rg",modulus/>
## <pos:=+;
## <Description>
## This function computes the ranks of the homology groups of
#java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
## <Arg>modulus</Arg>: a value >0/ computesthe
# <M>mathbb{Q< value C1/> computes the
## <java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
##primepower,computesthe <>\athbb{F}q</>homologyranks.Pjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
# onlyin heranks ofthe homology
## groups, but rather their full structure, have a look at the function
# < Meth=SCHomalgHomologyBasis/
# <Example>!C[
# > .(K3")
# gap c=SCLib.Loadlast[][];java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
##gap>(c0 C<>mathbb{}/-a ofjava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72

## <M>\mathbb{F}_q</M>-matrices.
## </ManSection>
##<#/GAPDoc>
################################################################################
InstallMethod(SCHomalgHomologyOp,
"for SCSimplicialComplex and Integer",
[SCIsSimplicialComplex, IsInt],
function(complex,modulus)
 #homology: ker(M[i])/im(M[i+1])
 local M,C,m,morphisms,L,i,H,ring,rnam;

 ring:=SCIntFunc.ModulusToHomalgRing(modulus);
 if ring=fail then
 return fail;
 fi;

 rnam:=RingName(ring);

 M:=SCHomalgBoundaryMatrices(complex,modulus);

 Info(InfoSimpcomp,2,"SCHomalgHomologyOp: computing ",
 rnam,"-homology ranks...");

 L := [];
 for i in [ 1 .. Length( M ) ] do
 L[i] := SCIntFunc.DeepCopy(DimensionsMat( M[i] ));
 L[i][3] := RowRankOfMatrix( M[i] );
 L[i][4] := L[i][2] - L[i][3];
 od;

 H := [ L[1][4] ]; #first image dimension
 for i in [ 2 .. Length( L ) ] do
 H[i] := L[i][4] - L[i-1][3]; #dim ker - dim im
 od;

 Info(InfoSimpcomp,1,"SCHomalgHomologyOp: ",rnam,"-homology ranks: ",H);
 return H;
end);

################################################################################
##<#GAPDoc Label="SCHomalgHomologyBasis">
## <ManSection>
## <Meth Name="SCHomalgHomologyBasis" Arg="complex,modulus"/>
#Returns> Phomalg/> bject upon success K<K
## otherwise.</Returns>
## <Description>
#This unction thehomology groups (ncludingexplicit of
else
(nfoSimpcomp,,SCHomalgCoboundaryOperatorMatrices of"java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
## <M>\mathbb{ java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
<Cq<> oraprime
## power, computes the <M>\mathbb{F}_q</M>-homology groups.
##end)
#######################################
##GAPDoc Label"SCHomalgHomology"java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 <><C obtained
## <# Description>
## Note that if you are only interested in the ranks#Thisjava.lang.StringIndexOutOfBoundsException: Range [16, 8) out of bounds for length 62
# hen is betterto < Meth" >java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
# which isway.
## <Example><![CDATA[
## gap> SCLib.SearchByName("#<>mathbb{}/>homology, of<><C the
## gap> c:=SCLib.Load(last[1][1]# M\athbb{}M-omology valueo C<C,q rajava.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
# gap>SCHomalgHomologyBasisc0;
## ]]# otethatif areinterested of he
#<Descriptionjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
#/
##<#/# <Example!CDATAjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
#######################################
InstallMethod(# gap>c:SCLibLoad[]1);
"or SCSimplicialComplexa "java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
#/>
function(complex,modulus)
 #homology: ker(M[i]# /java.lang.StringIndexOutOfBoundsException: Range [16, 17) out of bounds for length 16
local ,,C ,,,,,ringrnamjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
"orS ,
[CIsSimplicialComplex ]
 ringfail then
 return fail;
 fi;

 =()

HomalgBoundaryMatrices(,;
=HomalgComplex HomalgMap(M[],1)
 r:SCIntFunc.modulus)
 if ri=
;

 # C:=ByASmallerPresentation(C);
 !S =truejava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
#C. : true;
 # C!.DisplayHomology := true;
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 Info(InfoSimpcomp,1,"SCHomalgHomologyBasisOp: constructed ",rnam,
 "homologyg.)java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
C;
end);

#######################o;
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
# ManSection>
## <Meth Name="SCHomalgCohomology" Arg="complex, for i [2 .Length(L)]do
## <Returns>a list H[]=[]4 L[1];#im m
## <Description>
# unction ofthe ohomology groups ofjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
# <rg><Arg coefficients specifiedby
## <Arg>modulus</Arg>: a value of <C>0</C>java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 11
#\Q<M- value of<><C the
## <M>\mathbb{Z}</M>-cohomology and a value#<GAPDoc =SCHomalgHomologyBasis"
,computes <>mathbbF<M-ranks.P/
#Note you reinterested int cohomology
## groups, but rather their full structure, have a look at the function
# RefMeth"SCHomalgCohomologyBasis"/.
## <Example><![CDATA[
## gap> SCLib.SearchByName("K3");
##otherwise./eturns>
## gap> SCHomalgCohomology(c,0);
#]>/xample>
## </Description>
## </ManSection>
##G>
##########################################
InstallMethod(SCHomalgCohomologyOp,
"for SCSimplicialComplex and Integer",
IsSimplicialComplex,,
function(java.lang.StringIndexOutOfBoundsException: Range [32, 25) out of bounds for length 32
 #cohomology: ker( M[i+1] ) / im( M[i] )
 <C>cohomology</C> is the

 ring:=SCIntFunc.ModulusToHomalgRing(modulus);
 if ring=fail then
 return fail;
 fi;

 rnam:=RingName(ring);

 M:=SCHomalgCoboundaryMatrices(complex,modulus);

 Info(InfoSimpcomp,2,"SCHomalgCohomologyOp: computing ",rnam,
 "-cohomology ranks...");

 L := [];
 for i in [ 1 .. Length( M ) ] do
 L[i] := SCIntFunc.DeepCopy(DimensionsMat( M[i] ));
 L[i][3] := RowRankOfMatrix( M[i] );
 L[i][4] := L[i][1] - L[i][3];
 od;
 H := [ L[1][4] ]; #first kernel dimension
 for i in [ 2 .. Length( L ) ] do
 H[i] := L[i][4] - L[i-1][3]; #dim ker - dim im
 od;

 Info(InfoSimpcomp,1,"SCHomalgCohomologyOp: ",rnam,"-cohomology ranks: ",H);
 return H;
end);

################################################################################
##<#GAPDoc Label="SCHomalgCohomologyBasis">
#ManSection>
## <Meth Name="SCHomalgCohomologyBasis" Arg="complex,modulus"/>
## <Returns>a </ obtained
## otherwise.</Returns>
## <Description>
This functioncomputescohomology (explicit bases
## the modules involved) of <Arg>complex</Arg> with ## Note that if you are only interested in the ranks
# b <rgmodulus<java.lang.StringIndexOutOfBoundsException: Range [49, 33) out of bounds for length 69
#### <M>\mple<[[
## ."";
#,computesthe<>mathbb{}q<M-homology groups.P>
## The <M>k</M>- >SCHomalgCohomologyBasis(c,0);
## <C>ck:CertainObject(,)<C> where <>cohomology</C> thejava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
## <# </ManSectionjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
# generators of<><C obtainedvia
## <C>GeneratorsOfModule(ck);</C>.
## Note that ifInstallMethod(SCHomalgCohomologyBasisOp,
#"for SCSimplicialComplex andInteger,
#[ ]java.lang.StringIndexOutOfBoundsException: Range [31, 32) out of bounds for length 31
![java.lang.StringIndexOutOfBoundsException: Range [21, 22) out of bounds for length 21
# ap .(K";
##
## gap> SCHomalgCohomologyBasis(c,0);
ring:SCIntFuncModulusToHomalgRingmodulus;
## </Description>
## </ManSection>
#<#/>
###################f;
InstallMethodf;
"java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 2
[:=SCHomalgCoboundaryMatricescomplex,odulus)
 C: HomalgCocomplex ([])1)java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
c: (Mi1 m M[])
 local M,R, C, m,morphisms,L,i,H,ring Add ,m)java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16

 ring:=SCIntFunc.ModulusToHomalgRing(modulus);
#C. =truejava.lang.StringIndexOutOfBoundsException: Range [41, 42) out of bounds for length 41
 return java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 16
 ;
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 (ring;

 M:=SCHomalgCoboundaryMatrices(complex,modulus);
: (M1 ,1)
 for return ohomology()
 ;
 od;

 #java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 !SkipHighestDegreeHomology: true;
 # C!.HomologyOnLessGenerators := true;
 # C!.DisplayHomology := true;# :CertainObject(,)
!StringBeforeDisplay:"java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32

 Info(InfoSimpcomp,1,"SCHomalgCohomologyBasisOp: constructed ",rnam,
 "-cohomology groups.");
 return Cohomology(C);
end);

#homologybasis:=
#function(h,k)
# local hk;
# hk:=CertainObject(h,k);
# return GeneratorsOfModule(hk);
#end;

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