GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
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##
#W cscrct.tst GAP4 Package `RCWA' Stefan Kohl
##
## This file contains automated tests related to series of rcwa permutations
## like class shifts, -reflections, -rotations and -transpositions.
##
#############################################################################
gap> START_TEST( "cscrct.tst" );
gap> RCWADoThingsToBeDoneBeforeTest();
gap> x := Indeterminate(GF(4),1);; SetName(x,"x");
gap> R1 := PolynomialRing(GF(4),1);
GF(2^2)[x]
gap> y := Indeterminate(GF(25),1);; SetName(y,"y");
gap> R2 := PolynomialRing(GF(25),1);
GF(5^2)[y]
gap> ClassShift(Integers,1,2);
ClassShift( 1(2) )
gap> ClassShift([Integers,1,2]);
ClassShift( 1(2) )
gap> last^2;
ClassShift( 1(2) )^2
gap> LaTeXString(last);
"{\\nu_{1(2)}}^2"
gap> last2^3;
ClassShift( 1(2) )^6
gap> LaTeXString(last);
"{\\nu_{1(2)}}^6"
gap> last2^-1;
ClassShift( 1(2) )^-6
gap> LaTeXString(last);
"{\\nu_{1(2)}}^{-6}"
gap> last2^-2;
ClassShift( 1(2) )^12
gap> LaTeXString(last);
"{\\nu_{1(2)}}^{12}"
gap> l := [ ClassShift(0,1), ClassReflection(0,1), ClassRotation(0,1,-1),
> ClassTransposition(0,2,1,2) ];;
gap> for g in l do SetName(g,"a"); od; l;
[ a, a, a, a ]
gap> ClassShift(Z_pi(2),0,4);
ClassShift( 0(4) )
gap> ClassShift(0,3);
ClassShift( 0(3) )
gap> ClassShift(R1,Zero(R1),x);
ClassShift( 0(x) )
gap> Source(last);
GF(2^2)[x]
gap> ClassShift(Zero(R1),x);
ClassShift( 0(x) )
gap> Display(last);
Rcwa permutation of GF(2)[x] with modulus x, of order 2
/
| P + x if P in 0(x)
P |-> < P if P in 1(x)
|
\
gap> ClassShift([Zero(R1),x]);
ClassShift( 0(x) )
gap> Source(last);
GF(2)[x]
gap> ClassShift(Integers,ResidueClass(2,3));
ClassShift( 2(3) )
gap> ClassShift(ResidueClass(2,3));
ClassShift( 2(3) )
gap> ClassShift(R1,ResidueClass(R1,x,Zero(x)));
ClassShift( 0(x) )
gap> Display(last);
Rcwa permutation of GF(2^2)[x] with modulus x, of order 2
/
| P + x if P in 0(x)
P |-> < P otherwise
|
\
gap> ClassShift(ResidueClass(R1,x,Zero(x)));
ClassShift( 0(x) )
gap> Source(last);
GF(2^2)[x]
gap> ClassShift(Integers);
ClassShift( Z )
gap> ClassShift(Z_pi([2,3]));
ClassShift( Z_( 2, 3 ) )
gap> ClassShift([Z_pi([2,3])]);
ClassShift( Z_( 2, 3 ) )
gap> ClassShift(R1);
ClassShift( GF(2^2)[x] )
gap> Display(last);
Rcwa permutation of GF(2^2)[x]: P -> P + Z(2)^0
gap> ClassReflection(Integers,1,2);
ClassReflection( 1(2) )
gap> ClassReflection([Integers,1,2]);
ClassReflection( 1(2) )
gap> ClassReflection(Z_pi(2),0,4);
ClassReflection( 0(4) )
gap> ClassReflection(0,3);
ClassReflection( 0(3) )
gap> ClassReflection(R1,Zero(R1),x);
IdentityMapping( GF(2^2)[x] )
gap> IsRcwaMapping(last);
true
gap> ClassReflection(R2,Zero(R2),y);
ClassReflection( 0(y) )
gap> last^2;
IdentityMapping( GF(5^2)[y] )
gap> ClassReflection(Zero(R2),y);
ClassReflection( 0(y) )
gap> ClassReflection([Zero(R2),y]);
ClassReflection( 0(y) )
gap> Source(last);
GF(5)[y]
gap> ClassReflection(Integers,ResidueClass(2,3));
ClassReflection( 2(3) )
gap> ClassReflection(ResidueClass(2,3));
ClassReflection( 2(3) )
gap> ClassReflection(R2,ResidueClass(R2,y,Zero(y)));
ClassReflection( 0(y) )
gap> Source(last);
GF(5^2)[y]
gap> ClassReflection(ResidueClass(R2,y,Zero(y)));
ClassReflection( 0(y) )
gap> Source(last);
GF(5^2)[y]
gap> ClassReflection(Integers);
ClassReflection( Z )
gap> ClassReflection(Z_pi([2,3]));
ClassReflection( Z_( 2, 3 ) )
gap> ClassReflection([Z_pi([2,3])]);
ClassReflection( Z_( 2, 3 ) )
gap> Display(last);
Rcwa permutation of Z_( 2, 3 ): n -> -n
gap> ClassReflection(R2);
ClassReflection( GF(5^2)[y] )
gap> Display(last);
Rcwa permutation of GF(5^2)[y]: P -> -P
gap> ClassRotation(Integers,-1);
ClassReflection( Z )
gap> ClassRotation(Integers,1);
IdentityMapping( Integers )
gap> ClassRotation(Z_pi(2),-1);
ClassReflection( Z_( 2 ) )
gap> ClassRotation(Z_pi(2),1);
IdentityMapping( Z_( 2 ) )
gap> ClassRotation(Z_pi(2),1/3);
ClassRotation( Z_( 2 ), 1/3 )
gap> ClassRotation(Z_pi(2),-1/3);
ClassRotation( Z_( 2 ), -1/3 )
gap> ClassRotation(Z_pi(2),1/3);
ClassRotation( Z_( 2 ), 1/3 )
gap> Display(last);
Tame rcwa permutation of Z_( 2 ): n -> 1/3 n
gap> ClassRotation(Z_pi(2),ResidueClass(Z_pi(2),2,1),3/5);
ClassRotation( 1(2), 3/5 )
gap> Display(last);
Tame rcwa permutation of Z_( 2 ) with modulus 2, of order infinity
/
| 3/5 n + 2/5 if n in 1(2)
n |-> < n if n in 0(2)
|
\
gap> ClassRotation(ResidueClass(Z_pi(2),2,1),3/5);
ClassRotation( 1(2), 3/5 )
gap> ClassRotation([ResidueClass(Z_pi(2),2,1),3/5]);
ClassRotation( 1(2), 3/5 )
gap> ClassRotation(R1,ResidueClass(R1,x,Zero(R1)),Z(4)*One(R1));
ClassRotation( 0(x), Z(2^2) )
gap> Display(last);
Rcwa permutation of GF(2^2)[x] with modulus x, of order 3
/
| Z(2^2)*P if P in 0(x)
P |-> < P otherwise
|
\
gap> last^-1;
ClassRotation( 0(x), Z(2^2) )^2
gap> ClassRotation(R1,Z(4)*One(R1));
ClassRotation( GF(2^2)[x], Z(2^2) )
gap> Display(last);
Rcwa permutation of GF(2^2)[x]: P -> Z(2^2)*P
gap> last^2;
ClassRotation( GF(2^2)[x], Z(2^2) )^2
gap> Display(last);
Rcwa permutation of GF(2^2)[x]: P -> Z(2^2)^2*P
gap> ClassRotation(R2,ResidueClass(R2,y^2,y+1),Z(25)*One(R2));
ClassRotation( y+1(y^2), Z(5^2) )
gap> last^2;
ClassRotation( y+1(y^2), Z(5^2) )^2
gap> last^5;
ClassRotation( y+1(y^2), Z(5^2) )^10
gap> last^12;
IdentityMapping( GF(5^2)[y] )
gap> ClassTransposition(0,2,1,2);
( 0(2), 1(2) )
gap> ClassTransposition(Integers,0,2,1,2);
( 0(2), 1(2) )
gap> ClassTransposition(Z_pi(2),0,2,1,2);
( 0(2), 1(2) )
gap> Display(last);
Rcwa permutation of Z_( 2 ) with modulus 2, of order 2
/
| n + 1 if n in 0(2)
n |-> < n - 1 if n in 1(2)
|
\
gap> LaTeXString(last);
"\\tau_{0(2),1(2)}"
gap> ClassTransposition(Z_pi(2),0,2,1,4);
( 0(2), 1(4) )
gap> Support(last);
Z_( 2 ) \ 3(4)
gap> ClassTransposition(ResidueClass(0,3),ResidueClass(1,3));
( 0(3), 1(3) )
gap> Support(last);
Z \ 2(3)
gap> ClassTransposition(Integers,ResidueClass(0,3),ResidueClass(1,3));
( 0(3), 1(3) )
gap> ClassTransposition(ResidueClass(Z_pi([2,3]),3,0),
> ResidueClass(Z_pi([2,3]),3,1));
( 0(3), 1(3) )
gap> Support(last);
Z_( 2, 3 ) \ 2(3)
gap> Display(last2);
Rcwa permutation of Z_( 2, 3 ) with modulus 3, of order 2
/
| n + 1 if n in 0(3)
n |-> < n - 1 if n in 1(3)
| n if n in 2(3)
\
gap> ClassTransposition(Z_pi([2,3]),
> ResidueClass(Z_pi([2,3]),3,0),
> ResidueClass(Z_pi([2,3]),3,1));
( 0(3), 1(3) )
gap> TransposedClasses(last);
[ 0(3), 1(3) ]
gap> IsClassTransposition(last2);
true
gap> ClassTransposition(R1,ResidueClass(R1,x,Zero(R1)),
> ResidueClass(R1,x^2,x+1));
( 0(x), x+1(x^2) )
gap> TransposedClasses(last);
[ 0(x), x+1(x^2) ]
gap> Support(last2);
0(x) U x+1(x^2)
gap> Source(last3);
GF(2^2)[x]
gap> ClassTransposition(ResidueClass(R1,x,Zero(R1)),
> ResidueClass(R1,x^2,x+1));
( 0(x), x+1(x^2) )
gap> Display(last);
Rcwa permutation of GF(2^2)[x] with modulus x^2, of order 2
/
| x*P + x+Z(2)^0 if P in 0(x)
P |-> < (P + x+Z(2)^0)/x if P in x+1(x^2)
| P otherwise
\
gap> last^2;
IdentityMapping( GF(2^2)[x] )
gap> IsRcwaMapping(last);
true
gap> ct := ClassTransposition(-100,2,141,20);
( [-100/2], [141/20] )
gap> IsGeneralizedClassTransposition(ct);
true
gap> IsClassTransposition(ct);
false
gap> Sign(ct);
1
gap> Factorization(ct);
[ ClassShift( 1(20) )^-57, ClassShift( 0(2) )^57, ( 0(2), 1(20) ) ]
gap> Product(last)/ct;
IdentityMapping( Integers )
gap> TransposedClasses(ct);
[ [-100/2], [141/20] ]
gap> ct = ClassTransposition(last);
true
gap> R := Integers^2;
( Integers^2 )
gap> L := [ [ 2, 1 ], [ -1, 2 ] ];;
gap> cls := AllResidueClassesModulo(R,L);
[ (0,0)+(1,3)Z+(0,5)Z, (0,1)+(1,3)Z+(0,5)Z, (0,2)+(1,3)Z+(0,5)Z,
(0,3)+(1,3)Z+(0,5)Z, (0,4)+(1,3)Z+(0,5)Z ]
gap> cls[2] := SplittedClass(cls[2],[2,3]);
[ (0,1)+(2,6)Z+(0,15)Z, (0,6)+(2,6)Z+(0,15)Z, (0,11)+(2,6)Z+(0,15)Z,
(1,4)+(2,6)Z+(0,15)Z, (1,9)+(2,6)Z+(0,15)Z, (1,14)+(2,6)Z+(0,15)Z ]
gap> cls := Flat(cls);
[ (0,0)+(1,3)Z+(0,5)Z, (0,1)+(2,6)Z+(0,15)Z, (0,6)+(2,6)Z+(0,15)Z,
(0,11)+(2,6)Z+(0,15)Z, (1,4)+(2,6)Z+(0,15)Z, (1,9)+(2,6)Z+(0,15)Z,
(1,14)+(2,6)Z+(0,15)Z, (0,2)+(1,3)Z+(0,5)Z, (0,3)+(1,3)Z+(0,5)Z,
(0,4)+(1,3)Z+(0,5)Z ]
gap> Union(cls);
( Integers^2 )
gap> Sum(List(cls,Density));
1
gap> ct := ClassTransposition(cls[1],cls[3]);
( (0,0)+(1,3)Z+(0,5)Z, (0,6)+(2,6)Z+(0,15)Z )
gap> ct = ClassTransposition(R,cls[1],cls[3]);
true
gap> ct = ClassTransposition([0,0],[[1,3],[0,5]],[0,6],[[2,6],[0,15]]);
true
gap> ct = ClassTransposition(R,[0,0],[[1,3],[0,5]],[0,6],[[2,6],[0,15]]);
true
gap> ct = ClassTransposition([R,[0,0],[[1,3],[0,5]],[0,6],[[2,6],[0,15]]]);
true
gap> ImageDensity(ct);
1
gap> ct*ct;
IdentityMapping( ( Integers^2 ) )
gap> Display(ct);
Rcwa permutation of Z^2 with modulus (2,6)Z+(0,15)Z, of order 2
/
| (2m,-3m+3n+6) if (m,n) in (0,0)+(1,3)Z+(0,5)Z
(m,n) |-> < (m/2,(3m+2n-12)/6) if (m,n) in (0,6)+(2,6)Z+(0,15)Z
| (m,n) otherwise
\
gap> Cycle(ct,[1,8]);
[ [ 1, 8 ], [ 2, 27 ] ]
gap> Support(ct);
(0,0)+(1,3)Z+(0,5)Z U (0,6)+(2,6)Z+(0,15)Z
gap> String(ct);
"ClassTransposition((Integers^2),[0,0],[[1,3],[0,5]],[0,6],[[2,6],[0,15]])"
gap> ViewString(ct);
"( (0,0)+(1,3)Z+(0,5)Z, (0,6)+(2,6)Z+(0,15)Z )"
gap> Print(ct,"\n");
ClassTransposition((Integers^2),[0,0],[[1,3],[0,5]],[0,6],[[2,6],[0,15]])
gap> cs1 := ClassShift(R,1);
ClassShift( Z^2, 1 )
gap> cs2 := ClassShift(R,2);
ClassShift( Z^2, 2 )
gap> Order(cs1);
infinity
gap> String(cs1);
"ClassShift((Integers^2),[0,0],[[1,0],[0,1]],1)"
gap> ViewString(cs1);
"ClassShift( Z^2, 1 )"
gap> Print(cs1,"\n");
ClassShift((Integers^2),[0,0],[[1,0],[0,1]],1)
gap> cs1 = ClassShift(R,1);
true
gap> cs1 = ClassShift(R,[0,0],[[1,0],[0,1]],1);
true
gap> cs1 = ClassShift([0,0],[[1,0],[0,1]],1);
true
gap> cs1 = ClassShift([[0,0],[[1,0],[0,1]],1]);
true
gap> cs1 = cs2;
false
gap> Display(cs1);
Tame rcwa permutation of Z^2: (m,n) -> (m+1,n)
gap> Display(cs2);
Tame rcwa permutation of Z^2: (m,n) -> (m,n+1)
gap> Display(cs1*cs2);
Rcwa permutation of Z^2: (m,n) -> (m+1,n+1)
gap> Comm(cs1,cs2);
IdentityMapping( ( Integers^2 ) )
gap> cs := ClassShift(cls[4],1);
ClassShift( (0,11)+(2,6)Z+(0,15)Z, 1 )
gap> cs = ClassShift([0,11],[[2,6],[0,15]],1);
true
gap> cs = ClassShift(R,[0,11],[[2,6],[0,15]],1);
true
gap> cs = ClassShift([R,[0,11],[[2,6],[0,15]],1]);
true
gap> Order(cs);
infinity
gap> cls[4]^cs = cls[4];
true
gap> Support(cs);
(0,11)+(2,6)Z+(0,15)Z
gap> Display(cs);
Tame rcwa permutation of Z^2 with modulus (2,6)Z+(0,15)Z, of order infinity
/
| (m+2,n+6) if (m,n) in (0,11)+(2,6)Z+(0,15)Z
(m,n) |-> < (m,n) otherwise
|
\
gap> String(cs);
"ClassShift((Integers^2),[0,11],[[2,6],[0,15]],1)"
gap> ViewString(cs);
"ClassShift( (0,11)+(2,6)Z+(0,15)Z, 1 )"
gap> Print(cs,"\n");
ClassShift((Integers^2),[0,11],[[2,6],[0,15]],1)
gap> cr := ClassReflection(R);
ClassReflection( Z^2 )
gap> Order(cr);
2
gap> Support(cr);
Z^2 \ [ [ 0, 0 ] ]
gap> Display(cr);
Rcwa permutation of Z^2: (m,n) -> (-m,-n)
gap> cr*cr;
IdentityMapping( ( Integers^2 ) )
gap> cr := ClassReflection(cls[1]);
ClassReflection( (0,0)+(1,3)Z+(0,5)Z )
gap> cr = ClassReflection(R,cls[1]);
true
gap> cr = ClassReflection(R,[0,0],[[1,3],[0,5]]);
true
gap> cr = ClassReflection([R,[0,0],[[1,3],[0,5]]]);
true
gap> Order(cr);
2
gap> Support(cr);
(0,0)+(1,3)Z+(0,5)Z \ [ [ 0, 0 ] ]
gap> Display(cr);
Rcwa permutation of Z^2 with modulus (1,3)Z+(0,5)Z, of order 2
/
| (-m,-n) if (m,n) in (0,0)+(1,3)Z+(0,5)Z
(m,n) |-> < (m,n) otherwise
|
\
gap> String(cr);
"ClassReflection((Integers^2),[0,0],[[1,3],[0,5]])"
gap> ViewString(cr);
"ClassReflection( (0,0)+(1,3)Z+(0,5)Z )"
gap> Print(cr,"\n");
ClassReflection((Integers^2),[0,0],[[1,3],[0,5]])
gap> cr := ClassRotation(cls[7],[[1,1],[0,1]]);
ClassRotation( (1,14)+(2,6)Z+(0,15)Z, [ [ 1, 1 ], [ 0, 1 ] ] )
gap> cr = ClassRotation(R,cls[7],[[1,1],[0,1]]);
true
gap> cr = ClassRotation([R,cls[7],[[1,1],[0,1]]]);
true
gap> cr = ClassRotation([R,[1,14],[[2,6],[0,15]],[[1,1],[0,1]]]);
true
gap> Order(cr);
infinity
gap> Support(cr);
(1,14)+(2,6)Z+(0,15)Z
gap> last^cr;
(1,14)+(2,6)Z+(0,15)Z
gap> Display(cr);
Tame rcwa permutation of Z^2 with modulus (2,6)Z+(0,15)Z, of order infinity
/
| (m,(15m+2n-15)/2) if (m,n) in (1,14)+(2,6)Z+(0,15)Z
(m,n) |-> < (m,n) otherwise
|
\
gap> String(cr);
"ClassRotation((Integers^2),[1,14],[[2,6],[0,15]],[[1,1],[0,1]])"
gap> ViewString(cr);
"ClassRotation( (1,14)+(2,6)Z+(0,15)Z, [ [ 1, 1 ], [ 0, 1 ] ] )"
gap> Print(cr,"\n");
ClassRotation((Integers^2),[1,14],[[2,6],[0,15]],[[1,1],[0,1]])
gap> RCWADoThingsToBeDoneAfterTest();
gap> STOP_TEST( "cscrct.tst", 1090000000 );
#############################################################################
##
#E cscrct.tst . . . . . . . . . . . . . . . . . . . . . . . . . . ends here