_Chapter_Examples_and_Tests.xml

<?xml version="1.0" encoding="UTF-8"?>

<!-- This is an automatically generated file. -->
<Chapter Label="Chapter_Examples_and_Tests">
<Heading>Examples and Tests</Heading>

<Section Label="Chapter_Examples_and_Tests_Section_Basic_Commands">
<Heading>Basic Commands</Heading>

<Example><![CDATA[
gap> Q := HomalgFieldOfRationals();;
gap> A := VectorSpaceObject( 4, Q );;
gap> B := VectorSpaceObject( 3, Q );;
gap> C := VectorSpaceObject( 2, Q );;
gap> alpha := VectorSpaceMorphism( A, 
> HomalgMatrix( [ [ 1, 1, 1 ], [ 0, 1, 1 ], 
> [ 1, 0, 1 ], [ 1, 1, 0 ] ], 4, 3, Q ), B );;
gap> gamma := VectorSpaceMorphism( C, 
> HomalgMatrix( [ [ -1, 1, -1 ], [ 1, 0, -1 ] ], 2, 3, Q ), B );;
gap> p := ProjectionInFactorOfFiberProduct( [ alpha, gamma ], 1 );;
gap> q := ProjectionInFactorOfFiberProduct( [ alpha, gamma ], 2 );;
gap> PreCompose( AsGeneralizedMorphism( alpha ), GeneralizedInverse( gamma ) );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> gen1 := PreCompose( AsGeneralizedMorphism( alpha ), 
>                        GeneralizedInverse( gamma ) );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> gen2 := PreCompose( GeneralizedInverse( p ), AsGeneralizedMorphism( q ) );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> IsCongruentForMorphisms( gen1, gen2 );
true
]]></Example>


</Section>


<Section Label="Chapter_Examples_and_Tests_Section_Intersection_of_Nodal_Curve_and_Cusp">
<Heading>Intersection of Nodal Curve and Cusp</Heading>

We are going to intersect the nodal curve
<Math>f = y^2 - x^2(x+1)</Math>
and the cusp <Math>g = (x+y)^2 - (y-x)^3</Math>.
The two curves are arranged in a way such that they intersect
at <Math>(0,0)</Math> with intersection number as high as possible.
We are going to compute this intersection number
using the definition of the intersection number as the
length of the module <Math>R/(f,g)</Math> localized at <Math>(0,0)</Math>.
In order to model modules over the localization of <Math>Q[x,y]</Math> at
<Math>(0,0)</Math>, we use a suitable Serre quotient category.
1
2
1
1
true
We are going to intersect the nodal curve
<Math>f = y^2 - x^2(x+1)</Math>
and the cusp <Math>g = (x+y)^2 - (y-x)^3</Math>.
The two curves are arranged in a way such that they intersect
at <Math>(0,0)</Math> with intersection number as high as possible.
We are going to compute this intersection number
using the definition of the intersection number as the
length of the module <Math>R/(f,g)</Math> localized at <Math>(0,0)</Math>.
In order to model modules over the localization of <Math>Q[x,y]</Math> at
<Math>(0,0)</Math>, we use a suitable Serre quotient category.
1
2
1
1
true
We are going to intersect the nodal curve
<Math>f = y^2 - x^2(x+1)</Math>
and the cusp <Math>g = (x+y)^2 - (y-x)^3</Math>.
The two curves are arranged in a way such that they intersect
at <Math>(0,0)</Math> with intersection number as high as possible.
We are going to compute this intersection number
using the definition of the intersection number as the
length of the module <Math>R/(f,g)</Math> localized at <Math>(0,0)</Math>.
In order to model modules over the localization of <Math>Q[x,y]</Math> at
<Math>(0,0)</Math>, we use a suitable Serre quotient category.
1
2
1
1
true
</Section>


<Section Label="Chapter_Examples_and_Tests_Section_Sweep">
<Heading>Sweep</Heading>

<Math>\href{https://terrytao.wordpress.com/2015/10/07/sweeping-a-matrix-rotates-its-graph/}{\textrm{Geometric interpretation of sweeping a matrix by Terence Tao.}}</Math>
<Example><![CDATA[
gap> Q := HomalgFieldOfRationals();;
gap> V := VectorSpaceObject( 3, Q );;
gap> mat := HomalgMatrix( [ [ 9, 8, 7 ], [ 6, 5, 4 ], [ 3, 2, 1 ] ], 3, 3, Q );;
gap> alpha := VectorSpaceMorphism( V, mat, V );;
gap> graph := FiberProductEmbeddingInDirectSum( 
>             [ alpha, IdentityMorphism( V ) ] );;
gap> Display( graph );
[ [     1,    -2,     1,     0,     0,     0 ],
  [  -4/3,   7/3,     0,     2,     1,     0 ],
  [   5/3,  -8/3,     0,    -1,     0,     1 ] ]

A split monomorphism in Category of matrices over Q
gap> D := DirectSum( V, V );;
gap> rotmat := HomalgMatrix( [ [ 0, 0, 0, -1, 0, 0 ],
>                              [ 0, 1, 0, 0, 0, 0 ],
>                              [ 0, 0, 1, 0, 0, 0 ],
>                              [ 1, 0, 0, 0, 0, 0 ],
>                              [ 0, 0, 0, 0, 1, 0 ],
>                              [ 0, 0, 0, 0, 0, 1 ] ],
>                              6, 6, Q );;
gap> rot := VectorSpaceMorphism( D, rotmat, D );;
gap> p := PreCompose( graph, rot );;
gap> Display( p );
[ [     0,    -2,     1,    -1,     0,     0 ],
  [     2,   7/3,     0,   4/3,     1,     0 ],
  [    -1,  -8/3,     0,  -5/3,     0,     1 ] ]

A morphism in Category of matrices over Q
gap> pi1 := ProjectionInFactorOfDirectSum( [ V, V ], 1 );;
gap> pi2 := ProjectionInFactorOfDirectSum( [ V, V ], 2 );;
gap> reversed_arrow := PreCompose( p, pi1 );;
gap> arrow := PreCompose( p, pi2 );;
gap> g := GeneralizedMorphismBySpan( reversed_arrow, arrow );;
gap> IsHonest( g );
true
gap> sweep_1_alpha := HonestRepresentative( g );;
gap> Display( sweep_1_alpha );
[ [  -1/9,   8/9,   7/9 ],
  [   2/3,  -1/3,  -2/3 ],
  [   1/3,  -2/3,  -4/3 ] ]

A morphism in Category of matrices over Q
gap> Display( alpha );
[ [  9,  8,  7 ],
  [  6,  5,  4 ],
  [  3,  2,  1 ] ]

A morphism in Category of matrices over Q
]]></Example>


</Section>


</Chapter>