misc.tex

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\Chapter{Some functions for everyday use}

This chapter contains a number of functions that did not fit in
anywhere else. Some of them might be useful for other people, too, so
they were included here.

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\Section{Groups and actions}

\>OnSubgroups( <subgroup>, <aut> ) F

For a group $G$ and an automorphism <aut> of $G$, 
`OnSubgroups(<subgroup>,<aut>)' is the image of <subgroup> under <aut>


\beginexample
gap> G:=Group((1,2,3),(2,3));
Group([ (1,2,3), (2,3) ])
gap> alpha:=InnerAutomorphism(G,(1,2,3));
^(1,2,3)
gap> OnSubgroups(Subgroup(G,[(2,3)]),alpha);
Group([ (1,3) ])
\endexample

\>RepsCClassesGivenOrder( <group>, <order> ) O

`RepsCClassesGivenOrder( <group>, <order> )' returns all elements of 
order <order> up to conjugacy. Note that the representatives are *not*
always the smallest elements of each conjugacy class.


\beginexample
gap> RepsCClassesGivenOrder(SymmetricGroup(5),2);
[ (4,5), (2,3)(4,5) ]
\endexample

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\Section{Iterators}

\>CartesianIterator( <tuplelist> ) O

Returns an iterator for `Cartesian(<tuplelist>)'



\>ConcatenationOfIterators( <iterlist> ) F

`ConcatenationOfIterators(<iterlist>)' returns an iterator which runs 
through all iterators in <iterlist>. Note that the returned iterator loops 
over the iterators in <iterlist> *sequentially* beginning with the first 
one.




\beginexample
gap> it:=Iterator([1,2,3]);;
gap> it2:=CartesianIterator([[9,10],[11]]);;
gap> cit:=ConcatenationOfIterators([it,it2]);;
gap> repeat
> Print(NextIterator(cit),",\c");
> until IsDoneIterator(cit);
1,2,3,[ 9, 11 ],[ 10, 11 ],
\endexample

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\Section{Lists and Matrices}

\>IsListOfIntegers( <list> ) P

`IsListOfIntegers( <list> )' returns `IsSubset(Integers, <list> )' if <list>
is a dense list and `false' otherwise.



\>List2Tuples( <list>, <int> ) O

If `Size( <list> )' is divisible by <int>, `List2Tuples( <list>,<int>)'
returns a list <list2> of size <int> such that 
`Concatenation( <list2> )= <list>' and every element of <list2> has the 
same size.


\beginexample
gap> List2Tuples([1..6],2);
[ [ 1, 2, 3 ], [ 4, 5, 6 ] ]
\endexample

\>MatTimesTransMat( <mat> ) O

does the same as `<mat>*TransposedMat( <mat> )' but uses slightly less 
space and time for large matrices.



\>PartitionByFunctionNF( <list>, <f> ) O

`PartitionByFunctionNF( <list>, <f> )' partitions the list <list> 
according to the values of the function <f> defined on <list>. 
If <f> returns `fail' for some element of <list>, 
`PartitionByFunctionNF( <list>, <f> )' enters a break loop. 
Leaving the break loop with 'return;' is safe because 
`PartitionByFunctionNF' treats `fail' as all other results of <f>.



\>PartitionByFunction( <list>, <f> ) O

`PartitionByFunction( <list>, <f> )' partitions the list <list> 
according to the values of the function <f> defined on <list>. 
All elements, for which <f> returns `fail' are omitted, so 
`PartitionByFunction' does not necessarily return a partition.
If `InfoLevel(InfoRDS)'\index{InfoRDS@{\tt InfoRDS}} is at least 2, the number of 
elements for which <f> returns `fail' is shown 
(if `fail' is returned at all).


\begintt

gap> PartitionByFunctionNF([-1..5],x->x^2);
[ [ 0 ], [ -1, 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ] ]
gap> test:=function(x) 
> if x>0 then return Sqrt(x);
>  else return fail;
> fi;
> end;
function( x ) ... end
gap> PartitionByFunction([-1..5],test);  
[ [ 1 ], [ 4 ], [ 5 ], [ 2 ], [ 3 ] ]
gap> SetInfoLevel(InfoRDS,2);
gap> PartitionByFunction([-1..5],test);  
#I  -2-
[ [ 1 ], [ 4 ], [ 5 ], [ 2 ], [ 3 ] ]
gap> PartitionByFunctionNF([-1..5],test);
Error, function returned <fail> called from
...
brk> return;
[ [ 1 ], [ 4 ], [ 5 ], [ 2 ], [ 3 ], [ -1, 0 ] ]
\endtt

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\Section{Cyclotomic numbers}

\>IsRootOfUnity( <cyc> ) P

`IsRootOfUnity' tests if a given cyclotomic is actually a root of unity.



\>CoeffList2CyclotomicList( <list>, <root> ) O

`CoeffList2CyclogomicList( <list>, <root> )' takes a list of integers 
<list> and a root of unity <root> and returns a list <list2>, where 
<list2[i]=list[i]\* root^(i-1)>.



\>AbssquareInCyclotomics( <list>, <root> ) O

For a list of integers and a root of unity, 
`AbssquareInCyclotomics( <list>, <root> )' returns 
the modulus of `Sum(CoeffList2CyclotomicList( <list>, <root> ))'.



\>CycsGivenCoeffSum( <sum>, <root> ) O

`CycsGivenCoeffSum( <sum>, <root> )' returns all elements of $\Z[ root ]$
such that the coefficient sum is <sum> and all coefficients are 
non-negative.
The returned list has the following form:
The cyclotomic numbers are represented by coefficients. 
"CoeffList2CyclotomicList" can be used to get the 
algebraic number represented by <list>.
The list is partitioned into equivalence classes of elements having the 
same modulus.
For each class the modulus is returned.
This means that `CycsGivenCoeffSum' returns a list of pairs where the first
entry of each pair is the square of the modulus of an element of the 
second entry. And the second entry is a list of coefficient lists of 
cyclotomics in $\Z[ <root> ]$ having the coefficient sum <sum>. 


\beginexample
gap> CycsGivenCoeffSum(3,E(3));
[ [ 0, [ [ 1, 1, 1 ] ] ], 
  [ 3, [ [ 0, 1, 2 ], [ 0, 2, 1 ], [ 1, 0, 2 ], [ 1, 2, 0 ], [ 2, 0, 1 ], 
          [ 2, 1, 0 ] ] ], [ 9, [ [ 0, 0, 3 ], [ 0, 3, 0 ], [ 3, 0, 0 ] ] ] ]
gap> CycsGivenCoeffSum(2,E(2));
[ [ 0, [ [ 1, 1 ] ] ], [ 4, [ [ 0, 2 ], [ 2, 0 ] ] ] ]
\endexample

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\Section{Filters and Categories}

The following was originally posted at the \GAP\ forum by Thomas
Breuer \cite{BreuersAnswer}. 

Each filter in \GAP\ is either a simple filter or a meet of filters.
For example, `IsInt' and `IsPosRat' are simple filters,
and `IsPosInt' is defined as their meet `IsInt and IsPosRat'.

Each *simple filter* is of one of the following kinds.


1. property:
   Such a filter is an operation, the filter value can be computed.
   The (unary) methods of this operation must return `true' or `false',
   and the return value is stored in the argument,
   except if the argument is of a basic data type such as cyclotomic
   (including rationals and integers), finite field element, permutation,
   or internally represented list --the latter with a few exceptions.
   Examples of properties are `IsFinite', `IsAbelian', `IsSSortedList'.

2. attribute tester:
   Such a filter is associated to an operation that has been created
   via `DeclareAttribute',
   in the sense that the value is `true' if and only if a return value
   for (a unary method of) this operation is stored in the argument.
   Currently, attribute values are stored in objects in the filter
   `IsAttributeStoringRep'.
   Examples of attribute testers are `HasSize', `HasCentre',
   `HasDerivedSubgroup'.

2.' property tester:
   Such a filter is similar to an attribute tester,
   but the associated operation is a property.
   So property testers can return `true' also if the argument is not in
   the filter `IsAttributeStoringRep'.
   Examples of property testers are `HasIsFinite', `HasIsAbelian',
   `HasIsSSortedList'.

3. category or representation:
   These filters are not associated to operations, their values cannot
   be computed but are set upon creation of an object and should not be
   changed later, such that for a filter of this kind, one can rely on
   the fact that if the value is `true' then it was `true' already when
   the object in question was created.

   The distinction between representation and category is intended to
   express dependency on or independence of the way how the object is
   stored internally.
   For example, `IsPositionalObjectRep', `IsComponentObjectRep', and
   `IsInternalRep' are filters of the representation kind;
   the idea is that such filters are used in low level methods,
   and that higher level methods can be implemented without referring
   to these filters.

   Examples of categories are `IsInt', `IsRat', `IsPerm', `IsFFE',
   and filters expressing algebraic structures,
   such as `IsMagma', `IsMagmaWithOne', `IsAdditiveMagma'.
   When one calls such a filter, one can be sure that no computation is
   triggered.
   For example, whenever a quotient of two integers is formed, the result
   is clearly in the filter `IsRat', but the system also stores the value
   of `IsInt', i.e., \GAP\ does not support ``unevaluated rationals'' for
   which the `IsInt' value is computed on demand and then stored.

4. other filters:
   Some filters do not belong to the above kinds,
   they are not associated to operations but they are intended to be
   set (or even reset) by the user or by functions also after the creation
   of objects.
   Examples are `IsQuickPositionList', `CanEasilyTestMembership',
   `IsHandledByNiceBasis'.

Each *meet of filters* can involve computable simple filters (properties,
attribute and property testers) and not computable simple filters
(categories, representations, other filters).
When one calls a meet of two filters then the two filters from which
the meet was formed are evaluated (if necessary).
So a meet of filters is computable only if at least one computable simple
filter is involved.

\>IsComputableFilter( <filter> ) F




'IsComputableFilter(<filter>)' returns <true> if a the filter <filter>
is computable. Filters for which 'IsComputableFilter' returns <false>
may be used in 'DeclareOperation'.

\beginexample
    gap> IsComputableFilter( IsFinite );
    true
    gap> IsComputableFilter( HasSize );
    true
    gap> IsComputableFilter( HasIsFinite );
    true
    gap> IsComputableFilter( IsPositionalObjectRep );
    false
    gap> IsComputableFilter( IsInt );
    false
    gap> IsComputableFilter( IsQuickPositionList );
    false
    gap> IsComputableFilter( IsInt and IsPosRat );
    false
    gap> IsComputableFilter( IsMagma );
    false
\endexample

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