Quelle lists.xml Sprache: XML

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

<Chapter Label="chap-lists">
<Heading>Lists, Sets and Strings</Heading>

<Section Label="sec-lists">
<Heading>Functions for lists</Heading>

<ManSection>
   <Func Name="DifferencesList"
         Arg="L" />
<Description>
This function has been transferred from package &ResClasses;.
<P/>
It takes a list <M>L</M> of length <M>n</M> and outputs
the list of length <M>n-1</M> containing all the differences
<M>L[i]-L[i-1]</M>.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> List( [1..12], n->n^3 );
[ 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 ]
gap> DifferencesList( last );
[ 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397 ]
gap> DifferencesList( last );
[ 12, 18, 24, 30, 36, 42, 48, 54, 60, 66 ]
gap> DifferencesList( last );
[ 6, 6, 6, 6, 6, 6, 6, 6, 6 ]
]]>
</Example>

<ManSection>
   <Func Name="QuotientsList"
         Arg="L" />
   <Func Name="FloatQuotientsList"
         Arg="L" />
<Description>
These functions have been transferred from package &ResClasses;.
<P/>
They take a list <M>L</M> of length <M>n</M> and output the quotients
<M>L[i]/L[i-1]</M> of consecutive entries in <M>L</M>.
An error is returned if an entry is zero.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> List( [0..10], n -> Factorial(n) );
[ 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 ]
gap> QuotientsList( last );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
gap> L := [ 1, 3, 5, -1, -3, -5 ];;
gap> QuotientsList( L );
[ 3, 5/3, -1/5, 3, 5/3 ]
gap> FloatQuotientsList( L );
[ 3., 1.66667, -0.2, 3., 1.66667 ]
gap> QuotientsList( [ 2, 1, 0, -1, -2 ] );
[ 1/2, 0, fail, 2 ]
gap> FloatQuotientsList( [1..10] );
[ 2., 1.5, 1.33333, 1.25, 1.2, 1.16667, 1.14286, 1.125, 1.11111 ]
gap> Product( last );
10.
]]>
</Example>

<ManSection>
   <Oper Name="SearchCycle"
         Arg="L" />
<Description>
This function has been transferred from package &RCWA;.
<P/>
<C>SearchCycle</C> is a tool to find likely cycles in lists.
What, precisely, a <E>cycle</E> is, is deliberately fuzzy here,
and may possibly even change.
The idea is that the beginning of the list may be anything,
following that the same pattern needs to be repeated several times
in order to be recognized as a cycle.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> L := [1..20];;  L[1]:=13;;
gap> for i in [1..19] do
>        if IsOddInt(L[i]) then L[i+1]:=3*L[i]+1; else L[i+1]:=L[i]/2; fi;
>    od;
gap> L;
[ 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4 ]
gap> SearchCycle( L );
[ 1, 4, 2 ]
gap> n := 1;;  L := [n];;
gap> for i in [1..100] do  n:=(n^2+1) mod 1093;  Add(L,n);  od;
gap> L;
[ 1, 2, 5, 26, 677, 363, 610, 481, 739, 715, 795, 272, 754, 157, 604, 848,
  1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271,
  211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521,
  378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272,
  754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604,
  848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004,
  271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004 ]
gap> C := SearchCycle( L );
[ 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754 ]
gap> P := Positions( L, 157 );
[ 14, 26, 38, 50, 62, 74, 86, 98 ]
gap> Length( C );  DifferencesList( P );
12
[ 12, 12, 12, 12, 12, 12, 12 ]
]]>
</Example>

<ManSection>
   <Oper Name="RandomCombination"
         Arg="S k" />
<Description>
This function has been transferred from package &ResClasses;.
<P/>
It returns a random unordered <M>k</M>-tuple of distinct elements
of a set <M>S</M>.
<P/>
</Description>
</ManSection>
<Log>
<![CDATA[
gap> ## "6 aus 49" is a common lottery in Germany
gap> RandomCombination( [1..49], 6 );
[ 2, 16, 24, 26, 37, 47 ]
]]>
</Log>

</Section>

<Section><Heading>Distinct and Common Representatives</Heading>
<Index>distinct and common representatives</Index>

<ManSection>
   <Oper Name="DistinctRepresentatives"
         Arg="list" />
   <Oper Name="CommonRepresentatives"
         Arg="list" />
   <Oper Name="CommonTransversal"
         Arg="grp subgrp" />
   <Oper Name="IsCommonTransversal"
         Arg="grp subgrp list" />
<Description>
These operations have been transferred from package &XMod;.
<P/>
They deal with lists of subsets of <M>[1 \ldots n]</M>
and construct systems of distinct and common representatives using
simple, non-recursive, combinatorial algorithms.
<P/>
When <M>L</M> is a set of <M>n</M> subsets of <M>[1 \ldots n]</M>
and the Hall condition is satisfied
(the union of any <M>k</M> subsets has at least <M>k</M> elements),
a set of <C>DistinctRepresentatives</C> exists.
<P/>
When <M>J,K</M> are both lists of <M>n</M> sets,
the operation <C>CommonRepresentatives</C> returns two lists:
the set of representatives,
and a permutation of the subsets of the second list.
<P/>
The operation <C>CommonTransversal</C> may be used to provide a
common transversal for the sets of left and right cosets
of a subgroup <M>H</M> of a group <M>G</M>,
although a greedy algorithm is usually quicker.
</Description>
</ManSection>
<P/>
<Example>
<![CDATA[
gap> J := [ [1,2,3], [3,4], [3,4], [1,2,4] ];;
gap> DistinctRepresentatives( J );
[ 1, 3, 4, 2 ]
gap> K := [ [3,4], [1,2], [2,3], [2,3,4] ];;
gap> CommonRepresentatives( J, K );
[ [ 3, 3, 3, 1 ], [ 1, 3, 4, 2 ] ]
gap> d16 := DihedralGroup( IsPermGroup, 16 );
Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ])
gap> SetName( d16, "d16" );
gap> c4 := Subgroup( d16, [ d16.1^2 ] );
Group([ (1,3,5,7)(2,4,6,8) ])
gap> SetName( c4, "c4" );
gap> RightCosets( d16, c4 );
[ RightCoset(c4,()), RightCoset(c4,(2,8)(3,7)(4,6)), RightCoset(c4,(1,8,7,6,5,
   4,3,2)), RightCoset(c4,(1,8)(2,7)(3,6)(4,5)) ]
gap> trans := CommonTransversal( d16, c4 );
[ (), (2,8)(3,7)(4,6), (1,2,3,4,5,6,7,8), (1,2)(3,8)(4,7)(5,6) ]
gap> IsCommonTransversal( d16, c4, trans );
true
]]>
</Example>

</Section>

<Section Label="sec-strings">
<Heading>Functions for strings</Heading>

<ManSection>
   <Func Name="BlankFreeString"
         Arg="obj" />
<Description>
This function has been transferred from package &ResClasses;.
<P/>
The result of <C>BlankFreeString( obj );</C>
is a composite of the functions <C>String( obj )</C> and
<C>RemoveCharacters( obj, " " );</C>.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> gens := GeneratorsOfGroup( DihedralGroup(12) );
[ f1, f2, f3 ]
gap> String( gens );
"[ f1, f2, f3 ]"
gap> BlankFreeString( gens );
"[f1,f2,f3]"
]]>
</Example>

</Section>

</Chapter>

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