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<!-- ##  rcwamap.xml          RCWA documentation          Stefan Kohl  ## -->
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<Chapter Label="ch:RcwaMappings">
<Heading>Residue-Class-Wise Affine Mappings</Heading>

<Ignore Remark="settings for the example tester">
<Example>
<![CDATA[
gap> SizeScreen([75,24]);;
gap> SetAssertionLevel(0);
]]>
</Example>
</Ignore>

This chapter contains the basic definitions, and it describes how to
enter residue-class-wise affine mappings and how to compute with them. <P/>

How to compute with residue-class-wise affine groups is described in detail
in the next chapter. The reader is encouraged to look there already after
having read the first few pages of this chapter, and to look up definitions
as he needs to.

<!-- #################################################################### -->

<Section Label="sec:basicdefinitions">
<Heading>Basic definitions</Heading>

<Index Key="rcwa mapping" Subkey="definition">rcwa mapping</Index>
<Index Key="rcwa group" Subkey="definition">rcwa group</Index>

Residue-class-wise affine groups, or <E>rcwa</E> groups for short, are
permutation groups whose elements are bijective residue-class-wise affine
mappings. <P/>

A mapping <M>f: &ZZ; \rightarrow &ZZ;</M> is called
<E>residue-class-wise affine</E>, or for short an <E>rcwa</E> mapping,
if there is a positive integer&nbsp;<M>m</M> such that the restrictions
of&nbsp;<M>f</M> to the residue classes
<M>r(m) \in &ZZ;/m&ZZ;</M> are all affine,
i.e. given&nbsp;by
<Alt Only="LaTeX">
  <Display>
    f|_{r(m)}: \ r(m) \rightarrow \mathbb{Z}, \ \ \
    n \ \mapsto \ \frac{a_{r(m)} \cdot n + b_{r(m)}}{c_{r(m)}}
  </Display>
</Alt>
<Alt Only="HTML"><![CDATA[<center>
  <img src = "rcwamap.png" width = "366" height = "49"
       alt = "f|_r(m): n |-> (a_r(m) * n + b_r(m)) / c_r(m)" />
</center>]]></Alt>
<Alt Only="Text"><Verb>
                                        a_r(m) * n + b_r(m)
           f|_r(m):  r(m) -> Z,  n |->  -------------------
                                              c_r(m)
</Verb></Alt>
for certain coefficients <M>a_{r(m)}, b_{r(m)}, c_{r(m)} \in &ZZ;</M>
depending on&nbsp;<M>r(m)</M>.

<Index Key="modulus" Subkey="definition">modulus</Index>
<Index Key="rcwa mapping" Subkey="modulus">rcwa mapping</Index>
<Index Key="multiplier" Subkey="definition">multiplier</Index>
<Index Key="rcwa mapping" Subkey="multiplier">rcwa mapping</Index>
<Index Key="divisor" Subkey="definition">divisor</Index>
<Index Key="rcwa mapping" Subkey="divisor">rcwa mapping</Index>

The smallest possible <M>m</M> is called the <E>modulus</E>
of&nbsp;<M>f</M>. It is understood that all fractions are reduced,
i.e. that <M>\gcd( a_{r(m)}, b_{r(m)}, c_{r(m)} ) = 1</M>, and that
<M>c_{r(m)} &gt; 0</M>. The lcm of the coefficients <M>a_{r(m)}</M>
is called the <E>multiplier</E> of&nbsp;<M>f</M>, and the lcm of the
coefficients <M>c_{r(m)}</M> is called the <E>divisor</E>
of&nbsp;<M>f</M>. <P/>

It is easy to see that the residue-class-wise affine mappings of&nbsp;&ZZ;
form a monoid under composition, and that the residue-class-wise affine
permutations of&nbsp;&ZZ; form a countable subgroup of Sym(&ZZ;).
We denote the former by Rcwa(&ZZ;), and the latter by RCWA(&ZZ;). <P/>

<Index Key="tame" Subkey="rcwa mapping">tame</Index>
<Index Key="tame" Subkey="rcwa group">tame</Index>
<Index Key="wild" Subkey="rcwa mapping">wild</Index>
<Index Key="wild" Subkey="rcwa group">wild</Index>
<Index Key="rcwa mapping" Subkey="tame">rcwa mapping</Index>
<Index Key="rcwa group" Subkey="tame">rcwa group</Index>
<Index Key="rcwa mapping" Subkey="wild">rcwa mapping</Index>
<Index Key="rcwa group" Subkey="wild">rcwa group</Index>

An rcwa mapping is called <E>tame</E> if the set of moduli of its powers
is bounded, or equivalently if it permutes a partition
of&nbsp;&ZZ; into finitely many residue classes on all of which
it is affine. An rcwa group is called <E>tame</E> if there is a common such
partition for all of its elements, or equivalently if the set of moduli of
its elements is bounded. Rcwa mappings and -groups which are not tame
are called <E>wild</E>. Tame rcwa mappings and -groups are something which
one could call the <Q>trivial cases</Q> or <Q>basic building blocks</Q>,
while wild rcwa groups are the objects of primary interest. <P/>

The definitions of residue-class-wise affine mappings and -groups
can be generalized in the obvious way to suitable rings other than&nbsp;&ZZ;.
In fact, this package provides also some support for residue-class-wise
affine groups over <M>&ZZ;^2</M>, over semilocalizations of&nbsp;&ZZ;
and over univariate polynomial rings over finite fields.
The ring <M>&ZZ;^2</M> has been chosen as an example of a suitable ring
which is not a principal ideal domain, the semilocalizations of&nbsp;&ZZ;
have been chosen as examples of rings with only finitely many prime elements,
and the univariate polynomial rings over finite fields have been chosen
as examples of rings with nonzero characteristic.

</Section>

<!-- #################################################################### -->

<Section Label="sec:EnteringRcwaMappings">
<Heading>Entering residue-class-wise affine mappings</Heading>

Entering an rcwa mapping of&nbsp;&ZZ; requires giving the
modulus&nbsp;<M>m</M> and the coefficients <M>a_{r(m)}</M>,
<M>b_{r(m)}</M> and&nbsp;<M>c_{r(m)}</M> for <M>r(m)</M> running over the
residue classes (mod&nbsp;<M>m</M>). <P/>

This can be done easiest by <C>RcwaMapping( <A>coeffs</A> )</C>, where
<A>coeffs</A> is a list of <M>m</M> coefficient triples
<C>coeffs[</C><M>r+1</M><C>]&nbsp;= [</C><M>a_{r(m)}</M>, <M>b_{r(m)}</M>,
<M>c_{r(m)}</M><C>]</C>, with <M>r</M> running from 0 to&nbsp;<M>m-1</M>.
<P/>

If some coefficient <M>c_{r(m)}</M> is zero or if images of some integers
under the mapping to be defined would not be integers, an error message is
printed and a break loop is entered. For example, the coefficient triple
<C>[1,4,3]</C> is not allowed at the first position.
The reason for this is that not all integers congruent to
<M>1 \cdot 0 + 4 = 4</M>&nbsp;mod&nbsp;<M>m</M>
are divisible by&nbsp;3. <P/>

For the general constructor for rcwa mappings, see
<Ref Meth="RcwaMapping" Label="by ring, modulus and list of coefficients"/>.

<Example>
<![CDATA[
gap> T := RcwaMapping([[1,0,2],[3,1,2]]); # The Collatz mapping.
<rcwa mapping of Z with modulus 2>
gap> [ IsSurjective(T), IsInjective(T) ];
[ true, false ]
gap> Display(T);

Surjective rcwa mapping of Z with modulus 2

        /
        | n/2      if n in 0(2)
 n |-> <  (3n+1)/2 if n in 1(2)
        |
        \

gap> a := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]);
<rcwa mapping of Z with modulus 3>
gap> IsBijective(a);
true
gap> Display(a); # This is Collatz' permutation:

Rcwa permutation of Z with modulus 3

        /
        | 2n/3     if n in 0(3)
 n |-> <  (4n-1)/3 if n in 1(3)
        | (4n+1)/3 if n in 2(3)
        \

gap> Support(a);
Z \ [ -1, 0, 1 ]
gap> Cycle(a,44);
[ 44, 59, 79, 105, 70, 93, 62, 83, 111, 74, 99, 66 ]
]]>
</Example>

<Alt Only="LaTeX">\noindent</Alt>
There is computational evidence for the conjecture that any
residue-class-wise affine permutation of&nbsp;&ZZ; can be
factored into members of the following three series of permutations of
particularly simple structure
(cf. <Ref Attr="FactorizationIntoCSCRCT"
          Label="for an rcwa permutation of Z"/>):

<ManSection>
  <Func Name="ClassShift" Arg="r, m" Label="r, m"/>
  <Func Name="ClassShift" Arg="cl" Label="cl"/>
  <Returns> the class shift <M>\nu_{r(m)}</M>. </Returns>
  <Description>
    The <E>class shift</E> <M>\nu_{r(m)}</M> is the rcwa mapping
    of&nbsp;&ZZ; which maps <M>n \in r(m)</M> to <M>n + m</M>
    and which fixes <M>&ZZ; \setminus r(m)</M> pointwise. <P/>

    In the one-argument form, the argument&nbsp;<A>cl</A> stands for the
    residue class&nbsp;<M>r(m)</M>. Enclosing the argument list in list
    brackets is permitted.
<Example>
<![CDATA[
gap> Display(ClassShift(5,12));

Tame rcwa permutation of Z with modulus 12, of order infinity

        /
        | n+12 if n in 5(12)
 n |-> <  n    if n in Z \ 5(12)
        |
        \
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Func Name="ClassReflection" Arg="r, m" Label="r, m"/>
  <Func Name="ClassReflection" Arg="cl" Label="cl"/>
  <Returns> the class reflection <M>\varsigma_{r(m)}</M>. </Returns>
  <Description>
    The <E>class reflection</E> <M>\varsigma_{r(m)}</M> is the rcwa mapping
    of&nbsp;&ZZ; which maps <M>n \in r(m)</M> to <M>-n + 2r</M> and which
    fixes <M>&ZZ; \setminus r(m)</M> pointwise, where it is understood that
    <M>0 \leq r &lt; m</M>. <P/>

    In the one-argument form, the argument&nbsp;<A>cl</A> stands for the
    residue class&nbsp;<M>r(m)</M>. Enclosing the argument list in list
    brackets is permitted.
<Example>
<![CDATA[
gap> Display(ClassReflection(5,9));

Rcwa permutation of Z with modulus 9, of order 2

        /
        | -n+10 if n in 5(9)
 n |-> <  n     if n in Z \ 5(9)
        |
        \
]]>
</Example>
  </Description>
</ManSection>

<Alt Only="LaTeX">\pagebreak[4]</Alt>

<ManSection>
  <Func Name="ClassTransposition" Arg="r1, m1, r2, m2"
                                  Label="r1, m1, r2, m2"/>
  <Func Name="ClassTransposition" Arg="cl1, cl2"
                                  Label="cl1, cl2"/>
  <Returns>
    the class transposition <M>\tau_{r_1(m_1),r_2(m_2)}</M>.
  </Returns>
  <Description>
    Given two disjoint residue classes <M>r_1(m_1)</M> and <M>r_2(m_2)</M>
    of the integers, the <E>class transposition</E>
    <M>\tau_{r_1(m_1),r_2(m_2)}</M> <M>\in</M> RCWA(&ZZ;) is defined as the
    involution which interchanges <M>r_1+km_1</M> and <M>r_2+km_2</M> for
    any integer&nbsp;<M>k</M> and which fixes all other points.
    It is understood that <M>m_1</M> and&nbsp;<M>m_2</M> are positive,
    that <M>0 \leq r_1 &lt; m_1</M> and that <M>0 \leq r_2 &lt; m_2</M>.
    For a <E>generalized class transposition</E>, the latter assumptions
    are not made. <P/>

    The class transposition <M>\tau_{r_1(m_1),r_2(m_2)}</M> interchanges
    the residue classes <M>r_1(m_1)</M> and <M>r_2(m_2)</M> and fixes the
    complement of their union pointwise. <P/>

    <Index Key="TransposedClasses" Subkey="of a class transposition">
      <C>TransposedClasses</C>
    </Index>

    In the four-argument form, the arguments <A>r1</A>, <A>m1</A>, <A>r2</A>
    and&nbsp;<A>m2</A> stand for <M>r_1</M>, <M>m_1</M>, <M>r_2</M>
    and&nbsp;<M>m_2</M>, respectively.
    In the two-argument form, the arguments <A>cl1</A> and&nbsp;<A>cl2</A>
    stand for the residue classes <M>r_1(m_1)</M> and&nbsp;<M>r_2(m_2)</M>,
    respectively. Enclosing the argument list in list brackets is permitted.
    The residue classes <M>r_1(m_1)</M> and <M>r_2(m_2)</M> are stored
    as an attribute <C>TransposedClasses</C>. <P/>

    <Index Key="ClassPairs" Subkey="m"><C>ClassPairs</C></Index>

    A list of all class transpositions interchanging residue classes with
    moduli less than or equal to a given bound <A>m</A> can be obtained
    by <C>List(ClassPairs([<A>P</A>],<A>m</A>),ClassTransposition)</C>,
    where the function <C>ClassPairs</C> returns a list of all 4-tuples
    <M>(r_1,m_1,r_2,m_2)</M> of integers corresponding to the unordered
    pairs of disjoint residue classes <M>r_1(m_1)</M> and <M>r_2(m_2)</M>
    with <M>m_1</M> and <M>m_2</M> less than or equal to the specified
    bound. If a list of primes is given as optional argument <A>P</A>,
    then the returned list contains only those 4-tuples where all prime
    factors of <M>m_1</M> and <M>m_2</M> lie in <A>P</A>.
    If the option <C>divisors</C> is set, the returned list contains
    only the 4-tuples where <M>m_1</M> and <M>m_2</M> divide <A>m</A>. <P/>

    <Index Key="NrClassPairs" Subkey="m"><C>NrClassPairs</C></Index>
    <Index Key="ExtRepOfObj" Subkey="for a class transposition">
      <C>ExtRepOfObj</C>
    </Index>

    The function <C>NrClassPairs(<A>m</A>)</C> returns the length of the
    list <C>ClassPairs(<A>m</A>)</C>, where the result is computed much
    faster and without actually generating the list of tuples.
    Given a class transposition <A>ct</A>, the corresponding 4-tuple
    can be obtained by <C>ExtRepOfObj(<A>ct</A>)</C> <P/>

    <Index Key="SplittedClassTransposition"
           Subkey="for a class transposition and a number of factors">
      <C>SplittedClassTransposition</C>
    </Index>

    A class transposition can be written as a product of any given number
    <M>k</M> of class transpositions. Such a decomposition can be obtained
    by <C>SplittedClassTransposition(<A>ct</A>,<A>k</A>)</C>. <P/>
<Example>
<![CDATA[
gap> Display(ClassTransposition(1,2,8,10):CycleNotation:=false);

Rcwa permutation of Z with modulus 10, of order 2

        /
        | 5n+3    if n in 1(2)
 n |-> <  (n-3)/5 if n in 8(10)
        | n       if n in 0(2) \ 8(10)
        \

gap> List(ClassPairs(4),ClassTransposition);
[ ( 0(2), 1(2) ), ( 0(2), 1(4) ), ( 0(2), 3(4) ), ( 0(3), 1(3) ), 
  ( 0(3), 2(3) ), ( 0(4), 1(4) ), ( 0(4), 2(4) ), ( 0(4), 3(4) ), 
  ( 1(2), 0(4) ), ( 1(2), 2(4) ), ( 1(3), 2(3) ), ( 1(4), 2(4) ), 
  ( 1(4), 3(4) ), ( 2(4), 3(4) ) ]
gap> NrClassPairs(100);
3528138
gap> SplittedClassTransposition(ClassTransposition(0,2,1,4),3);
[ ( 0(6), 1(12) ), ( 2(6), 5(12) ), ( 4(6), 9(12) ) ]]]>
</Example>
  </Description>
</ManSection>

<Alt Only="LaTeX">\noindent</Alt>
The set of all class transpositions of the ring of integers generates the
simple group CT(&ZZ;) mentioned in Chapter&nbsp;<Ref Label="ch:AboutRCWA"/>.
This group has a representation as a &GAP; object -- see&nbsp;<Ref Func="CT"
Label="the group generated by all class transpositions of a ring"/>.
The set of all generalized class transpositions of&nbsp;&ZZ;
generates a simple group as well, cf.&nbsp;<Cite Key="Kohl09"/>. <P/>

Class shifts, class reflections and class transpositions of rings
<M>R</M> other than&nbsp;&ZZ; are defined in an entirely analogous
way -- all one needs to do is to replace &ZZ; by&nbsp;<M>R</M>
and to read <M>&lt;</M> and <M>\leq</M> in the sense of the ordering used
by&nbsp;&GAP;.
They can also be entered basically as described above -- just prepend
the desired ring&nbsp;<M>R</M> to the argument list. Often also a sensible
<Q>default ring</Q> (<M>\rightarrow</M> <C>DefaultRing</C> in the &GAP;
Reference Manual) is chosen if that optional first argument is omitted. <P/>

On rings which have more than two units, there is another basic series
of rcwa permutations which generalizes class reflections:

<ManSection>
  <Func Name="ClassRotation" Arg="r, m, u" Label="r, m, u"/>
  <Func Name="ClassRotation" Arg="cl, u" Label="cl, u"/>
  <Returns> the class rotation <M>\rho_{r(m),u}</M>. </Returns>
  <Description>
    Given a residue class <M>r(m)</M> and a unit&nbsp;<M>u</M> of a suitable
    ring&nbsp;<M>R</M>, the <E>class rotation</E> <M>\rho_{r(m),u}</M>
    is the rcwa mapping which maps <M>n \in r(m)</M> to <M>un + (1-u)r</M>
    and which fixes <M>R \setminus r(m)</M> pointwise.
    Class rotations generalize class reflections, as we have
    <M>\rho_{r(m),-1} = \varsigma_{r(m)}</M>. <P/>

    <Index Key="RotationFactor" Subkey="of a class rotation">
      <C>RotationFactor</C>
    </Index>

    In the two-argument form, the argument&nbsp;<A>cl</A> stands for the
    residue class&nbsp;<M>r(m)</M>. Enclosing the argument list in list
    brackets is permitted. The argument <A>u</A> is stored as an attribute
    <C>RotationFactor</C>.
<Example>
<![CDATA[
gap> Display(ClassRotation(ResidueClass(Z_pi(2),2,1),1/3));

Tame rcwa permutation of Z_( 2 ) with modulus 2, of order infinity

        /
        | 1/3 n + 2/3 if n in 1(2)
 n |-> <  n           if n in 0(2)
        |
        \

gap> x := Indeterminate(GF(8),1);; SetName(x,"x");
gap> R := PolynomialRing(GF(8),1);;
gap> cr := ClassRotation(1,x,Z(8)*One(R)); Support(cr);
ClassRotation( 1(x), Z(2^3) )
1(x) \ [ 1 ]
gap> Display(cr);

Rcwa permutation of GF(2^3)[x] with modulus x, of order 7

        /
        | Z(2^3)*P + Z(2^3)^3 if P in 1(x)
 P |-> <  P                   otherwise
        |
        \
]]>
</Example>
  </Description>
</ManSection>

<Index Key="IsClassShift" Subkey="for an rcwa mapping">
  <C>IsClassShift</C>
</Index>
<Index Key="IsClassReflection" Subkey="for an rcwa mapping">
  <C>IsClassReflection</C>
</Index>
<Index Key="IsClassRotation" Subkey="for an rcwa mapping">
  <C>IsClassRotation</C>
</Index>
<Index Key="IsClassTransposition" Subkey="for an rcwa mapping">
  <C>IsClassTransposition</C>
</Index>
<Index Key="IsGeneralizedClassTransposition" Subkey="for an rcwa mapping">
  <C>IsGeneralizedClassTransposition</C>
</Index>

<C>IsClassShift</C>, <C>IsClassReflection</C>,
<C>IsClassRotation</C>, <C>IsClassTransposition</C> and
<C>IsGeneralizedClassTransposition</C> are properties which indicate
whether a given rcwa mapping belongs to the corresponding series. <P/>

In the sequel we describe the general-purpose constructor for rcwa
mappings. The constructor may look a bit technical on a first glance,
but knowing all possible ways of entering an rcwa mapping is by no means
necessary for understanding this manual or for using this package.

<ManSection>
  <Heading> RcwaMapping (the general constructor) </Heading>
  <Meth Name="RcwaMapping" Arg="R, m, coeffs"
        Label="by ring, modulus and list of coefficients"/>
  <Meth Name="RcwaMapping" Arg="R, coeffs"
        Label="by ring and list of coefficients"/>
  <Meth Name="RcwaMapping" Arg="coeffs"
        Label="by list of coefficients"/>
  <Meth Name="RcwaMapping" Arg="perm, range"
        Label="by permutation and range"/>
  <Meth Name="RcwaMapping" Arg="m, values"
        Label="by modulus and list of values"/>
  <Meth Name="RcwaMapping" Arg="pi, coeffs"
        Label="by set of non-invertible primes and list of coefficients"/>
  <Meth Name="RcwaMapping" Arg="q, m, coeffs"
        Label="by finite field size, modulus and list of coefficients"/>
  <Meth Name="RcwaMapping" Arg="P1, P2"
        Label="by two partitions of a ring into residue classes"/>
  <Meth Name="RcwaMapping" Arg="cycles"
        Label="by residue class cycles"/>
  <Meth Name="RcwaMapping" Arg="expression"
        Label="by arithmetical expression"/>
  <Returns> an rcwa mapping. </Returns>
  <Description>
    In all cases the argument&nbsp;<A>R</A> is the underlying ring,
    <A>m</A> is the modulus and <A>coeffs</A> is the coefficient list.
    A <E>coefficient list</E> for an rcwa mapping with modulus <M>m</M>
    consists of <M>|R/mR|</M> coefficient triples
    <C>[</C><M>a_{r(m)}</M>, <M>b_{r(m)}</M>, <M>c_{r(m)}</M><C>]</C>.
    Their ordering is determined by the ordering of the representatives of
    the residue classes (mod&nbsp;<M>m</M>) in the sorted list returned by
    <C>AllResidues(<A>R</A>, <A>m</A>)</C>. In case <M>R = &ZZ;</M>
    this means that the coefficient triple for the residue class <M>0(m)</M>
    comes first and is followed by the one for <M>1(m)</M>, the one for
    <M>2(m)</M> and so on. <P/>

    If one or several of the arguments <A>R</A>, <A>m</A> and
    <A>coeffs</A> are omitted or replaced by other arguments, the former
    are either derived from the latter or default values are chosen.
    The meaning of the other arguments is defined in the detailed
    description of the particular methods given in the sequel.

    The above methods return the rcwa mapping
    <List>
      <Mark>(a)</Mark>
      <Item>
        of <A>R</A> with modulus <A>m</A> and coefficients <A>coeffs</A>,
      </Item>
      <Mark>(b)</Mark>
      <Item>
        of <A>R</A> = &ZZ; or <A>R</A> = <M>&ZZ;_{(\pi)}</M> with modulus
        <C>Length(<A>coeffs</A>)</C> and coefficients <A>coeffs</A>,
      </Item>
      <Mark>(c)</Mark>
      <Item>
        of <A>R</A> = &ZZ; with modulus <C>Length(<A>coeffs</A>)</C> and
        coefficients <A>coeffs</A>,
      </Item>
      <Mark>(d)</Mark>
      <Item>
        of <A>R</A> = &ZZ;, permuting any set
        <C><A>range</A>+k*Length(<A>range</A>)</C> like <A>perm</A>
        permutes <A>range</A>,
      </Item>
      <Mark>(e)</Mark>
      <Item>
        of <A>R</A> = &ZZ; with modulus <A>m</A> and values given by a list
        <A>val</A> of 2 pairs <C>[</C>preimage<C>, </C>image<C>]</C> per
        residue class (mod <A>m</A>),
      </Item>
      <Mark>(f)</Mark>
      <Item>
        of <A>R</A> = <M>&ZZ;_{(\pi)}</M> with modulus
        <C>Length(<A>coeffs</A>)</C> and coefficients <A>coeffs</A>
        (the set of primes <M>\pi</M> which denotes the underlying ring
        is passed as argument&nbsp;<A>pi</A>),
      </Item>
      <Mark>(g)</Mark>
      <Item>
        of <A>R</A> = GF(<A>q</A>)[<A>x</A>] with modulus&nbsp;<A>m</A> and
        coefficients <A>coeffs</A>,
      </Item>
      <Mark>(h)</Mark>
      <Item>
        an rcwa permutation which induces a bijection between
        the partitions <A>P1</A> and <A>P2</A> of&nbsp;<A>R</A> into residue
        classes and which is affine on the elements of <A>P1</A>,
      </Item>
      <Mark>(i)</Mark>
      <Item>
        an rcwa permutation with <Q>residue class cycles</Q> given
        by a list <A>cycles</A> of lists of pairwise disjoint residue classes,
        each of which it permutes cyclically, or
      </Item>
      <Mark>(j)</Mark>
      <Item>
        the rcwa permutation of&nbsp;&ZZ; given by the
        arithmetical expression <A>expression</A> -- a string consisting of
        class transpositions (e.g. <C>"(0(2),1(4))"</C>) or cycles permuting
        residue classes (e.g. <C>"(0(2),1(8),3(4),5(8))"</C>), class shifts
        (e.g. <C>"cs(4(6))"</C>, class reflections (e.g. <C>"cr(3(4))"</C>),
        arithmetical operators (<C>"*"</C>, <C>"/"</C> and <C>"&circum;"</C>)
        and brackets (<C>"("</C>, <C>")"</C>),
      </Item>
    </List>
    respectively.
    The methods for the operation <C>RcwaMapping</C> perform a number of
    argument checks, which can be skipped by using <C>RcwaMappingNC</C>
    instead.
<Example>
<![CDATA[
gap> R := PolynomialRing(GF(2),1);; x := X(GF(2),1);; SetName(x,"x");
gap> RcwaMapping(R,x+1,[[1,0,x+One(R)],[x+One(R),0,1]]*One(R));     # (a)
<rcwa mapping of GF(2)[x] with modulus x+1>
gap> RcwaMapping(Z_pi(2),[[1/3,0,1]]);                              # (b)
Rcwa mapping of Z_( 2 ): n -> 1/3 n
gap> a := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]);                  # (c)
<rcwa mapping of Z with modulus 3>
gap> RcwaMapping((1,2,3),[1..4]);                                   # (d)
( 1(4), 2(4), 3(4) )
gap> T = RcwaMapping(2,[[1,2],[2,1],[3,5],[4,2]]);                  # (e)
true
gap> RcwaMapping([2],[[1/3,0,1]]);                                  # (f)
Rcwa mapping of Z_( 2 ): n -> 1/3 n
gap> RcwaMapping(2,x+1,[[1,0,x+One(R)],[x+One(R),0,1]]*One(R));     # (g)
<rcwa mapping of GF(2)[x] with modulus x+1>
gap> a = RcwaMapping(List([[0,3],[1,3],[2,3]],ResidueClass),
>                    List([[0,2],[1,4],[3,4]],ResidueClass));       # (h)
true
gap> RcwaMapping([List([[0,2],[1,4],[3,8],[7,16]],ResidueClass)]);  # (i)
( 0(2), 1(4), 3(8), 7(16) )
gap> Cycle(last,ResidueClass(0,2));
[ 0(2), 1(4), 3(8), 7(16) ]
gap> g := RcwaMapping("((0(4),1(6))*cr(0(6)))^2/cs(2(8))");         # (j)
<rcwa permutation of Z with modulus 72>
gap> g = (ClassTransposition(0,4,1,6) * ClassReflection(0,6))^2/
>         ClassShift(2,8);
true
]]>
</Example>
  </Description>
</ManSection>

Rcwa mappings of &ZZ; can be <Q>translated</Q> to rcwa mappings
of some semilocalization <M>&ZZ;_{(\pi)}</M> of&nbsp;&ZZ;:

<ManSection>
  <Func Name="LocalizedRcwaMapping"
        Arg="f, p" Label="for an rcwa mapping of Z and a prime"/>
  <Func Name="SemilocalizedRcwaMapping"
        Arg="f, pi" Label="for an rcwa mapping of Z and a set of primes"/>
  <Returns>
    the rcwa mapping of <M>&ZZ;_{(p)}</M> respectively
    <M>&ZZ;_{(\pi)}</M> with the same coefficients as the rcwa mapping
    <A>f</A> of&nbsp;&ZZ;.
  </Returns>
  <Description>
    The argument <A>p</A> or <A>pi</A> must be a prime or a set of primes,
    respectively. The argument&nbsp;<A>f</A> must be an rcwa mapping
    of&nbsp;&ZZ; whose modulus is a power of&nbsp;<A>p</A>,
    or whose modulus has only prime divisors which lie in&nbsp;<A>pi</A>,
    respectively.
<Example>
<![CDATA[
gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping.
gap> Cycle(LocalizedRcwaMapping(T,2),131/13);
[ 131/13, 203/13, 311/13, 473/13, 716/13, 358/13, 179/13, 275/13, 
  419/13, 635/13, 959/13, 1445/13, 2174/13, 1087/13, 1637/13, 2462/13, 
  1231/13, 1853/13, 2786/13, 1393/13, 2096/13, 1048/13, 524/13, 262/13 ]
]]>
</Example>
  </Description>
</ManSection>

<Index Key="View" Subkey="for an rcwa mapping"><C>View</C></Index>
<Index Key="Display" Subkey="for an rcwa mapping"><C>Display</C></Index>
<Index Key="Print" Subkey="for an rcwa mapping"><C>Print</C></Index>
<Index Key="String" Subkey="for an rcwa mapping"><C>String</C></Index>

<Alt Only="LaTeX">\noindent</Alt>
Rcwa mappings can be <C>View</C>ed, <C>Display</C>ed, <C>Print</C>ed and
written to a <C>String</C>. The output of the <C>View</C> method is kept
reasonably short. In most cases it does not describe an rcwa mapping
completely. In these cases the output is enclosed in brackets.
There are options <C>CycleNotation</C>, <C>AsClassMapping</C>,
<C>PrintNotation</C> and <C>AbridgedNotation</C> to take influence on
how certain rcwa mappings are shown. These options can either be not set,
set to <C>true</C> or set to <C>false</C>.
If the option <C>CycleNotation</C> is set, it is tried harder to write
down an rcwa permutation of &ZZ; of finite order as a product of disjoint
residue class cycles, if this is possible.
If the option <C>AsClassMapping</C> is set, <C>Display</C> shows
which residue classes are mapped to which by the affine partial mappings,
and marks any loops.
The option <C>PrintNotation</C> influences the output in favour of
&GAP; - readability, and the option <C>AbridgedNotation</C> can be used to
abridge longer names like <C>ClassShift</C>, <C>ClassReflection</C> etc..
By default, the output of the methods for <C>Display</C> and <C>Print</C>
describes an rcwa mapping in full. The <C>Print</C>ed representation of
an rcwa mapping is &GAP; - readable if and only if the <C>Print</C>ed
representation of the elements of the underlying ring is so. <P/>

<Index Key="LaTeXStringRcwaMapping" Subkey="for an rcwa mapping">
  <C>LaTeXStringRcwaMapping</C>
</Index>

There is also an operation <C>LaTeXStringRcwaMapping</C>, which takes
as argument an rcwa mapping and returns a corresponding
&LaTeX; string. The output makes use of the &LaTeX; macro package
<Package>amsmath</Package>. If the option <A>Factorization</A> is set
and the argument is bijective, a factorization into class shifts,
class reflections, class transpositions and prime switches is printed
(cf. <Ref Attr="FactorizationIntoCSCRCT"
          Label="for an rcwa permutation of Z"/>).
For rcwa mappings with modulus greater than&nbsp;1, an indentation by
<A>Indentation</A> characters can be obtained by setting this option
value accordingly.

<Example>
<![CDATA[
gap> Print(LaTeXStringRcwaMapping(T));
n \ \mapsto \
\begin{cases}
  n/2      & \text{if} \ n \in 0(2), \\
  (3n+1)/2 & \text{if} \ n \in 1(2).
\end{cases}
]]>
</Example>

<Index Key="LaTeXAndXDVI" Subkey="for an rcwa mapping">
  <C>LaTeXAndXDVI</C>
</Index>

There is an operation <C>LaTeXAndXDVI</C> which displays an rcwa mapping
in an <Package>xdvi</Package> window.
This works as follows: The string returned by <C>LaTeXStringRcwaMapping</C>
is inserted into a &LaTeX; template file. This file is &LaTeX;'ed, and the
result is shown with <Package>xdvi</Package>.
Calling <C>Display</C> with option <A>xdvi</A> has the same effect.
The operation <C>LaTeXAndXDVI</C> is only available on UNIX systems,
and requires suitable installations of &LaTeX; and <Package>xdvi</Package>.

</Section>

<!-- #################################################################### -->

<Section Label="sec:BasicArithmeticForRcwaMappings">
<Heading>Basic arithmetic for residue-class-wise affine mappings</Heading>

<Index Key="rcwa mapping" Subkey="arithmetic operations">rcwa mapping</Index>
<Index Key="Order" Subkey="of an rcwa permutation"><C>Order</C></Index>
<Index Key="IsTame" Subkey="for an rcwa mapping"><C>IsTame</C></Index> <P/>

Testing rcwa mappings for equality requires only comparing their
coefficient lists, hence is cheap.
Rcwa mappings can be multiplied, thus there is a method for <C>*</C>.
Rcwa permutations can also be inverted, thus there is a method for
<C>Inverse</C>. The latter method is usually accessed by raising a mapping
to a power with negative exponent. Multiplying, inverting and computing
powers of tame rcwa mappings is cheap. Computing powers of wild mappings
is usually expensive -- run time and memory requirements normally grow
approximately exponentially with the exponent. How expensive multiplying
a couple of wild mappings is, varies very much. In any case, the amount of
memory required for storing an rcwa mapping is proportional to its modulus.
Whether a given mapping is tame or wild can be determined by the operation
<C>IsTame</C>. There is a method for <C>Order</C>, which can not only
compute a finite order, but which can also detect infinite order.

<Example>
<![CDATA[
gap> T := RcwaMapping([[1,0,2],[3,1,2]]);;          # The Collatz mapping.
gap> a := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]);; # Collatz' permutation.
gap> List([-4..4],k->Modulus(a^k));
[ 256, 64, 16, 4, 1, 3, 9, 27, 81 ]
gap> IsTame(T) or IsTame(a);
false
gap> IsTame(ClassShift(0,1)) and IsTame(ClassTransposition(0,2,1,2));
true
gap> T^2*a*T*a^-3;
<rcwa mapping of Z with modulus 768>
gap> (ClassShift(1,3)*ClassReflection(2,7))^1000000;
<rcwa permutation of Z with modulus 21>
]]>
</Example>

<Index Key="IsInjective" Subkey="for an rcwa mapping">
  <C>IsInjective</C>
</Index>
<Index Key="IsSurjective" Subkey="for an rcwa mapping">
  <C>IsSurjective</C>
</Index>
<Index Key="IsBijective" Subkey="for an rcwa mapping">
  <C>IsBijective</C>
</Index>
<Index Key="Image" Subkey="of an rcwa mapping">
  <C>Image</C>
</Index>

There are methods installed for <C>IsInjective</C>, <C>IsSurjective</C>,
<C>IsBijective</C> and <C>Image</C>.

<Example>
<![CDATA[
gap> [ IsInjective(T), IsSurjective(T), IsBijective(a) ];
[ false, true, true ]
gap> Image(RcwaMapping([[2,0,1]]));
0(2)
]]>
</Example>

<Index Key="rcwa mapping" Subkey="images under">rcwa mapping</Index>

Images of elements, of finite sets of elements and of unions of finitely
many residue classes of the source of an rcwa mapping can be computed with
<C>&circum;</C>, the same symbol as used for exponentiation and conjugation.
The same works for partitions of the source into a finite number of
residue classes.

<Example>
<![CDATA[
gap> 15^T;
23
gap> ResidueClass(1,2)^T;
2(3)
gap> List([[0,3],[1,3],[2,3]],ResidueClass)^a;
[ 0(2), 1(4), 3(4) ]
]]>
</Example>

<Index Key="PreImageElm" Subkey="of a ring element under an rcwa mapping">
  <C>PreImageElm</C>
</Index>
<Index Key="PreImagesElm" Subkey="of a ring element under an rcwa mapping">
  <C>PreImagesElm</C>
</Index>
<Index Key="PreImage"
       Subkey="of a set of ring elements under an rcwa mapping">
  <C>PreImage</C>
</Index>
<Index Key="PreImage"
       Subkey="of a residue class union under an rcwa mapping">
  <C>PreImage</C>
</Index>

For computing preimages of elements under rcwa mappings,
there are methods for <C>PreImageElm</C> and <C>PreImagesElm</C>.
The preimage of a finite set of ring elements or of a union of finitely
many residue classes under an rcwa mapping can be computed by
<C>PreImage</C>.

<Example>
<![CDATA[
gap> PreImagesElm(T,8);
[ 5, 16 ]
gap> PreImage(T,ResidueClass(Integers,3,2));
Z \ 0(6) U 2(6)
gap> M := [1];; l := [1];;
gap> while Length(M) < 5000 do M := PreImage(T,M); Add(l,Length(M)); od; l;
[ 1, 1, 2, 2, 4, 5, 8, 10, 14, 18, 26, 36, 50, 67, 89, 117, 157, 208, 
  277, 367, 488, 649, 869, 1154, 1534, 2039, 2721, 3629, 4843, 6458 ]
]]>
</Example>

<Index Key="Support" Subkey="of an rcwa mapping">
  <C>Support</C>
</Index>
<Index Key="MovedPoints" Subkey="of an rcwa mapping">
  <C>MovedPoints</C>
</Index>
<Index Key="RestrictedPerm"
       Subkey="for an rcwa permutation and a residue class union">
  <C>RestrictedPerm</C>
</Index>
<Index Key="DensityOfSupport"
       Subkey="of an rcwa mapping">
  <C>DensityOfSupport</C>
</Index>
<Index Key="DensityOfSetOfFixedPoints"
       Subkey="of an rcwa mapping">
  <C>DensityOfSetOfFixedPoints</C>
</Index>

There is a method for the operation <C>Support</C> for computing the
support of an rcwa mapping. A synonym for <C>Support</C> is
<C>MovedPoints</C>. The natural density of the support of an rcwa mapping
of &ZZ; can be computed efficiently with the operation <C>DensityOfSupport</C>.
Likewise, the natural density of the set of fixed points of an rcwa mapping
of &ZZ; can be computed efficiently with the operation
<C>DensityOfSetOfFixedPoints</C>.
There is also a method for <C>RestrictedPerm</C>
for computing the restriction of an rcwa permutation to a union
of residue classes which it fixes setwise.

<Example>
<![CDATA[
gap> List([a,a^2],Support);
[ Z \ [ -1, 0, 1 ], Z \ [ -3, -2, -1, 0, 1, 2, 3 ] ]
gap> RestrictedPerm(ClassShift(0,2)*ClassReflection(1,2),
>                   ResidueClass(0,2));
<rcwa mapping of Z with modulus 2>
gap> last = ClassShift(0,2);
true
]]>
</Example>

Rcwa mappings can be added and subtracted pointwise. However, please note
that the set of rcwa mappings of a ring does not form a ring under
<C>+</C> and <C>*</C>. <P/>

<Example>
<![CDATA[
gap> b := ClassShift(0,3) * a;;
gap> [ Image((a + b)), Image((a - b)) ];
[ 2(4), [ -2, 0 ] ]
]]>
</Example>

<Index Key="Modulus" Subkey="of an rcwa mapping"><C>Modulus</C></Index>
<Index Key="Mod" Subkey="for an rcwa mapping"><C>Mod</C></Index>
<Index Key="Coefficients" Subkey="of an rcwa mapping">
  <C>Coefficients</C>
</Index>
<Index Key="rcwa mapping" Subkey="coercion">rcwa mapping</Index>
<Index Key="rcwa group" Subkey="coercion">rcwa group</Index>

There are operations <C>Modulus</C> (abbreviated <C>Mod</C>) and
<C>Coefficients</C> for retrieving the modulus and the coefficient list
of an rcwa mapping. The meaning of the return values is as described
in Section&nbsp;<Ref Label="sec:EnteringRcwaMappings"/>. <P/>

General documentation for most operations mentioned in this section can
be found in the &GAP; reference manual. For rcwa mappings of rings other
than&nbsp;&ZZ;, not for all operations applicable methods are
available. <P/>

As in general a subring relation <M>R_1&lt;R_2</M> does <E>not</E> give
rise to a natural embedding of RCWA(<M>R_1</M>) into RCWA(<M>R_2</M>),
there is no coercion between rcwa mappings or rcwa groups over different
rings.

</Section>

<!-- #################################################################### -->

<Section Label="sec:AttributesAndPropertiesOfRcwaMappings">
<Heading>
  Attributes and properties of residue-class-wise affine mappings
</Heading>

<Index Key="class-wise translating" Subkey="definition">
  class-wise translating
</Index>
<Index Key="integral" Subkey="definition">integral</Index>
<Index Key="balanced" Subkey="definition">balanced</Index>
<Index Key="sign-preserving" Subkey="definition">sign-preserving</Index>
<Index Key="maximal shift" Subkey="definition">maximal shift</Index>

<Index Key="rcwa mapping" Subkey="prime set">rcwa mapping</Index>
<Index Key="rcwa mapping" Subkey="maximal shift">rcwa mapping</Index>
<Index Key="rcwa mapping" Subkey="class-wise translating">
  rcwa mapping
</Index>
<Index Key="rcwa mapping" Subkey="integral">rcwa mapping</Index>
<Index Key="rcwa mapping" Subkey="balanced">rcwa mapping</Index>
<Index Key="rcwa mapping"
       Subkey="class-wise order-preserving">
  rcwa mapping
</Index>
<Index Key="rcwa mapping" Subkey="sign-preserving">rcwa mapping</Index>

<Index Key="Multiplier" Subkey="of an rcwa mapping"><C>Multiplier</C></Index>
<Index Key="Mult" Subkey="for an rcwa mapping"><C>Mult</C></Index>
<Index Key="Divisor" Subkey="of an rcwa mapping"><C>Divisor</C></Index>
<Index Key="Div" Subkey="for an rcwa mapping"><C>Div</C></Index>
<Index Key="PrimeSet" Subkey="of an rcwa mapping"><C>PrimeSet</C></Index>
<Index Key="MaximalShift" Subkey="of an rcwa mapping of Z">
  <C>MaximalShift</C>
</Index>

<Index Key="IsClassWiseTranslating" Subkey="for an rcwa mapping">
  <C>IsClassWiseTranslating</C>
</Index>
<Index Key="IsIntegral" Subkey="for an rcwa mapping">
  <C>IsIntegral</C>
</Index>
<Index Key="IsBalanced" Subkey="for an rcwa mapping">
  <C>IsBalanced</C>
</Index>
<Index Key="IsClassWiseOrderPreserving" Subkey="for an rcwa mapping">
  <C>IsClassWiseOrderPreserving</C>
</Index>
<Index Key="IsSignPreserving" Subkey="for an rcwa mapping">
  <C>IsSignPreserving</C>
</Index>

A number of basic attributes and properties of an rcwa mapping are derived
immediately from the coefficients of its affine partial mappings. This holds
for example for the multiplier and the divisor. These two values are stored
as attributes <C>Multiplier</C> and <C>Divisor</C>, or for short <C>Mult</C>
and <C>Div</C>. The <E>prime set</E> of an rcwa mapping is the set of
prime divisors of the product of its modulus and its multiplier.
It is stored as an attribute <C>PrimeSet</C>.
The <E>maximal shift</E> of an rcwa mapping of &ZZ; is the maximum of
the absolute values of its coefficients <M>b_{r(m)}</M> in the notation
introduced in Section&nbsp;<Ref Label="sec:basicdefinitions"/>.
It is stored as an attribute <C>MaximalShift</C>.
An rcwa mapping is called <E>class-wise translating</E> if all of its affine
partial mappings are translations, it is called <E>integral</E> if its
divisor equals&nbsp;1, and it is called <E>balanced</E> if its multiplier
and its divisor have the same prime divisors. A class-wise translating
mapping has the property <C>IsClassWiseTranslating</C>, an integral mapping
has the property <C>IsIntegral</C> and a balanced mapping has the property
<C>IsBalanced</C>.
An rcwa mapping of the ring of integers or of one of its semilocalizations
is called <E>class-wise order-preserving</E> if and only if all coefficients
<M>a_{r(m)}</M> (cf.&nbsp;Section&nbsp;<Ref Label="sec:basicdefinitions"/>)
in the numerators of the affine partial mappings are positive.
The corresponding property is <C>IsClassWiseOrderPreserving</C>.
An rcwa mapping of&nbsp;&ZZ; is called <E>sign-preserving</E> if
it does not map nonnegative integers to negative integers or vice versa.
The corresponding property is <C>IsSignPreserving</C>. All elements of the
simple group CT(&ZZ;) generated by the set of all class transpositions are
sign-preserving.

<Example>
<![CDATA[
gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
gap> IsBijective(u);; Display(u);

Rcwa permutation of Z with modulus 5

        /
        | 3n/5     if n in 0(5)
        | (9n+1)/5 if n in 1(5)
 n |-> <  (3n-1)/5 if n in 2(5)
        | (9n-2)/5 if n in 3(5)
        | (9n+4)/5 if n in 4(5)
        \

gap> Multiplier(u);
9
gap> Divisor(u);
5
gap> PrimeSet(u);
[ 3, 5 ]
gap> IsIntegral(u) or IsBalanced(u);
false
gap> IsClassWiseOrderPreserving(u) and IsSignPreserving(u);
true
]]>
</Example>

There are a couple of further attributes and operations related to the
affine partial mappings of an rcwa mapping:

<ManSection>
  <Attr Name="LargestSourcesOfAffineMappings"
        Arg="f" Label="for an rcwa mapping"/>
  <Returns>
    the coarsest partition of <C>Source(<A>f</A>)</C> on whose
    elements the rcwa mapping&nbsp;<A>f</A> is affine.
  </Returns>
  <Description>
<Example>
<![CDATA[
gap> LargestSourcesOfAffineMappings(ClassShift(3,7));
[ Z \ 3(7), 3(7) ]
gap> LargestSourcesOfAffineMappings(ClassReflection(0,1));
[ Integers ]
gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
gap> List( [ u, u^-1 ], LargestSourcesOfAffineMappings );
[ [ 0(5), 1(5), 2(5), 3(5), 4(5) ], [ 0(3), 1(3), 2(9), 5(9), 8(9) ] ]
gap> kappa := ClassTransposition(2,4,3,4) * ClassTransposition(4,6,8,12)
>           * ClassTransposition(3,4,4,6);
<rcwa permutation of Z with modulus 12>
gap> LargestSourcesOfAffineMappings(kappa);
[ 2(4), 1(4) U 0(12), 3(12) U 7(12), 4(12), 8(12), 11(12) ]
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Attr Name="FixedPointsOfAffinePartialMappings"
        Arg="f" Label="for an rcwa mapping"/>
  <Returns>
    a list of the sets of fixed points of the affine partial mappings of
    the rcwa mapping&nbsp;<A>f</A> in the quotient field of its source.
  </Returns>
  <Description>
    The returned list contains entries for the restrictions
    of&nbsp;<A>f</A> to all residue classes modulo <C>Mod(<A>f</A>)</C>.
    A list entry can either be an empty set, the source of&nbsp;<A>f</A>
    or a set of cardinality&nbsp;1. The ordering of the entries corresponds
    to the ordering of the residues in
    <C>AllResidues(Source(<A>f</A>),<A>m</A>)</C>.
<Example>
<![CDATA[
gap> FixedPointsOfAffinePartialMappings(ClassShift(0,2));
[ [  ], Rationals ]
gap> List([1..3],k->FixedPointsOfAffinePartialMappings(T^k));
[ [ [ 0 ], [ -1 ] ], [ [ 0 ], [ 1 ], [ 2 ], [ -1 ] ], 
  [ [ 0 ], [ -7 ], [ 2/5 ], [ -5 ], [ 4/5 ], [ 1/5 ], [ -10 ], [ -1 ] ] ]
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Oper Name="Multpk" Arg="f, p, k"
        Label="for an rcwa mapping, a prime and an exponent"/>
  <Returns>
    the union of the residue classes <M>r(m)</M> such that
    <M>p^k||a_{r(m)}</M> if <M>k \geq 0</M>, and the union of the residue
    classes <M>r(m)</M> such that <M>p^k||c_{r(m)}</M> if <M>k \leq 0</M>.
    In this context, <M>m</M> denotes the modulus of&nbsp;<A>f</A>, and
    <M>a_{r(m)}</M> and <M>c_{r(m)}</M> denote the coefficients
    of&nbsp;<A>f</A> as introduced in
    Section&nbsp;<Ref Label="sec:basicdefinitions"/>.
  </Returns>
  <Description>
<Example>
<![CDATA[
gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping.
gap> [ Multpk(T,2,-1), Multpk(T,3,1) ];
[ Integers, 1(2) ]
gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
gap> [ Multpk(u,3,0), Multpk(u,3,1), Multpk(u,3,2), Multpk(u,5,-1) ];
[ [  ], 0(5) U 2(5), Z \ 0(5) U 2(5), Integers ]
]]>
</Example>
  </Description>
</ManSection>

<Index Key="ClassWiseOrderPreservingOn">
  <C>ClassWiseOrderPreservingOn</C>
</Index>
<Index Key="ClassWiseConstantOn">
  <C>ClassWiseConstantOn</C>
</Index>
<Index Key="ClassWiseOrderReversingOn">
  <C>ClassWiseOrderReversingOn</C>
</Index>

There are attributes <C>ClassWiseOrderPreservingOn</C>,
<C>ClassWiseConstantOn</C> and <C>ClassWiseOrderReversingOn</C> which
store the union of the residue classes (mod&nbsp;<C>Mod(<A>f</A>)</C>) on
which an rcwa mapping&nbsp;<A>f</A> of&nbsp;&ZZ; or of a semilocalization
thereof is class-wise order-preserving, class-wise constant or class-wise
order-reversing, respectively.

<Example>
<![CDATA[
gap> List([ClassTransposition(1,2,0,4),ClassShift(2,3),
>          ClassReflection(2,5)],ClassWiseOrderPreservingOn);
[ Integers, Integers, Z \ 2(5) ]
]]>
</Example>

<Index Key="ShiftsUpOn"><C>ShiftsUpOn</C></Index>
<Index Key="ShiftsDownOn"><C>ShiftsDownOn</C></Index>

Also there are attributes <C>ShiftsUpOn</C> and <C>ShiftsDownOn</C>
which store the union of the residue classes
(mod&nbsp;<C>Mod(<A>f</A>)</C>) on which an rcwa mapping&nbsp;<A>f</A>
of&nbsp;&ZZ; induces affine mappings <M>n \mapsto n + c</M> for
<M>c &gt; 0</M>, respectively, <M>c &lt; 0</M>. <P/>

Finally, there are epimorphisms from the subgroup of RCWA(&ZZ;)
formed by all class-wise order-preserving elements to (&ZZ;,+)
and from RCWA(&ZZ;) itself to the cyclic group of order&nbsp;2,
respectively:

<ManSection>
  <Meth Name="Determinant" Arg="f" Label="of an rcwa mapping of Z"/>
  <Returns>
    the determinant of the rcwa mapping&nbsp;<A>f</A> of&nbsp;&ZZ;.
  </Returns>
  <Description>
    The <E>determinant</E> of an affine mapping <M>n \mapsto (an+b)/c</M>
    whose source is a residue class <M>r(m)</M> is defined by <M>b/|a|m</M>.
    This definition is extended additively to determinants of rcwa mappings.
    <P/>

    Let <M>f</M> be an rcwa mapping of the integers, and let
    <M>m</M> denote its modulus. Using the notation
    <M>f|_{r(m)}: n \mapsto (a_{r(m)} \cdot n + b_{r(m)})/c_{r(m)}</M>
    for the affine partial mappings, the <E>determinant</E> det(<M>f</M>)
    of&nbsp;<M>f</M> is given by
    <Alt Only="LaTeX">
      <Display>
        \sum_{r(m) \in \mathbb{Z}/m\mathbb{Z}} b_{r(m)}/(|a_{r(m)}| \cdot m).
      </Display>
    </Alt>
    <Alt Only="HTML"><![CDATA[<center>
      <img src = "det.png" width = "177" height = "58"
           alt = "sum_{r(m) in Z/mZ}
                  b_{r(m)}/(|a_{r(m)}| \cdot m)."/>
    </center>]]></Alt>
    <Alt Only="Text"><Verb>
                          -----
                           \           b_r(m)
                            >     --------------
                           /       |a_{r(m)}| * m
                          -----
                      r(m) in Z/mZ                .
    </Verb></Alt> 
    The determinant mapping is an epimorphism from the group of all
    class-wise order-preserving rcwa permutations of&nbsp;&ZZ;
    to (&ZZ;,+), see&nbsp;<Cite Key="Kohl05"/>, Theorem&nbsp;2.11.9.
<Example>
<![CDATA[
gap> List([ClassTransposition(0,4,5,12),ClassShift(3,7)],Determinant);
[ 0, 1 ]
gap> Determinant(ClassTransposition(0,4,5,12)*ClassShift(3,7)^100);   
100
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Attr Name="Sign" Arg="g" Label="of an rcwa permutation of Z"/>
  <Returns>
    the sign of the rcwa permutation&nbsp;<A>g</A> of&nbsp;&ZZ;.
  </Returns>
  <Description>
    Let <M>\sigma</M> be an rcwa permutation of the integers, and let
    <M>m</M> denote its modulus. Using the notation
    <M>\sigma|_{r(m)}: n \mapsto (a_{r(m)} \cdot n + b_{r(m)})/c_{r(m)}</M>
    for the affine partial mappings, the <E>sign</E> of&nbsp;<M>\sigma</M>
    is defined by
    <Alt Only="LaTeX">
      <Display>
        (-1)^{\displaystyle{{\rm det}(\sigma)
            + \sum_{r(m): \ a_{r(m)} &lt; 0} \frac{m - 2r}{m}}}.
      </Display>
    </Alt>
    <Alt Only="HTML"><![CDATA[<center>
      <img src = "sgn.png" width = "284" height = "63"
           alt = "(-1)^(det(sigma)
                      + sum_{r(m): \ a_{r(m)} &lt; 0} (m - 2r)/m)."/>
    </center>]]></Alt>
    <Alt Only="Text"><Verb>
                                    -----
                                     \       m - 2r
                      det(sigma) +    >      -------
                                     /          m
                                    -----
                                 a_r(m) &lt; 0
              (-1)                                   .
    </Verb></Alt> 
    The sign mapping is an epimorphism from RCWA(&ZZ;) to the group
    <M>&ZZ;^\times</M> of units of&nbsp;&ZZ;, see&nbsp;<Cite Key="Kohl05"/>,
    Theorem&nbsp;2.12.8. Therefore the kernel of the sign mapping is a normal
    subgroup of RCWA(&ZZ;) of index&nbsp;2. The simple group CT(&ZZ;) is
    a subgroup of this kernel.
<Example>
<![CDATA[
gap> List([ClassTransposition(3,4,2,6),
>          ClassShift(0,3),ClassReflection(2,5)],Sign);
[ 1, -1, -1 ]
]]>
</Example>
  </Description>
</ManSection>

</Section>

<!-- #################################################################### -->

<Section Label="sec:FactoringRcwaMappings">
<Heading>Factoring residue-class-wise affine permutations</Heading>

Factoring group elements into the members of some <Q>nice</Q> set of
generators is often helpful. In this section we describe an operation which
attempts to solve this problem for the group RCWA(&ZZ;).
Elements of finitely generated rcwa groups can be factored into generators
<Q>as usual</Q>, see&nbsp;<Ref Meth="PreImagesRepresentative"
Label="for an epi. from a free group to an rcwa group"/>.

<ManSection>
  <Attr Name="FactorizationIntoCSCRCT"
        Arg="g" Label="for an rcwa permutation of Z"/>
  <Meth Name="Factorization" Arg="g" Label="for an rcwa permutation of Z"/>
  <Returns>
    a factorization of the rcwa permutation&nbsp;<A>g</A> of&nbsp;&ZZ;
    into class shifts, class reflections and class transpositions, provided
    that such a factorization exists and the method finds it.
  </Returns>
  <Description>
    The method may return <C>fail</C>, stop with an error message or run
    into an infinite loop. If it returns a result, this result is always
    correct. <P/>

    The problem of obtaining a factorization as described is algorithmically
    difficult, and this factorization routine is currently perhaps the most
    sophisticated part of the &RCWA; package. Information about the
    progress of the factorization process can be obtained by setting the
    info level of the Info class <Ref InfoClass="InfoRCWA"/> to&nbsp;2. <P/>

    By default, prime switches (<M>\rightarrow</M>
    <Ref Func="PrimeSwitch" Label="p"/>) are taken as one factor. 
    If the option <A>ExpandPrimeSwitches</A> is set, they are each
    decomposed into the 6 class transpositions given in the definition. <P/>

    By default, the factoring process begins with splitting off factors
    from the right. This can be changed by setting the option
    <A>Direction</A> to <C>"from the left"</C>. <P/>

    By default, a reasonably coarse respected partition of the integral
    mapping occurring in the final stage of the algorithm is computed.
    This can be suppressed by setting the option <A>ShortenPartition</A>
    equal to <C>false</C>. <P/>

    By default, at the end it is checked whether the product of the
    determined factors indeed equals <A>g</A>. This check can be
    suppressed by setting the option <A>NC</A>.
<Example>
<![CDATA[
gap> Factorization(Comm(ClassShift(0,3)*ClassReflection(1,2),
>                       ClassShift(0,2)));
[ ClassReflection( 2(3) ), ClassShift( 2(6) )^-1, ( 0(6), 2(6) ), 
  ( 0(6), 5(6) ) ]
]]>
</Example>
  </Description>
</ManSection>

For purposes of demonstrating the capabilities of the factorization routine,
in Section&nbsp;<Ref Label="sec:FactoringTheCollatzPermutation"/>
Collatz' permutation is factored. Lothar Collatz has investigated this
permutation in&nbsp;1932. Its cycle structure is unknown so far. <P/>

The permutations of the following kind play an important role in
factoring rcwa permutations of&nbsp;&ZZ; into class shifts,
class reflections and class transpositions:

<ManSection>
  <Meth Name="PrimeSwitch" Arg="p" Label="p"/>
  <Meth Name="PrimeSwitch" Arg="p, k" Label="p, k"/>
  <Meth Name="PrimeSwitch" Arg="p, r, m" Label="p, r, m"/>
  <Meth Name="PrimeSwitch" Arg="p, cl" Label="p, cl"/>
  <Returns>
    in the first form the <E>prime switch</E>
    <M>\sigma_p := \tau_{0(8),1(2p)} \cdot \tau_{4(8),-1(2p)} \cdot
    \tau_{0(4),1(2p)} \cdot \tau_{2(4),-1(2p)} \cdot \tau_{2(2p),1(4p)}
    \cdot \tau_{4(2p),2p+1(4p)}</M>, in the second form
    the restriction of <M>\sigma_p</M> by <M>n \mapsto kn</M>,
    and in the third and fourth form the <E>prime switch</E>
    <M>\sigma_{p,r(m)} := \tau_{r_1(m/2),r_2(m)} \cdot
    \tau_{r_2(m),r_1(pm/2)} \cdot \tau_{r(m/2),r_1(pm/2)}</M>.
    In the latter case, <A>cl</A> is the residue class
    <M>r(m)</M>, the residue <M>r_1</M> is <M>1-(r \mod 2)</M>, and
    <M>r_2</M> is defined by the equality <M>r(m) \cup r_2(m) = r(m/2)</M>.
  </Returns>
  <Description>
    For an odd prime <M>p</M>, the prime switch <M>\sigma_p</M> is
    an rcwa permutation of&nbsp;&ZZ; with modulus&nbsp;<M>4p</M>,
    multiplier&nbsp;<M>p</M> and divisor&nbsp;2.
    The prime switch <M>\sigma_{p,r(m)}</M> has
    multiplier&nbsp;<M>p</M> and divisor&nbsp;2, and the class where
    the multiplication by <M>p</M> occurs is just <M>r(m)</M>.
    The key mathematical property of a prime switch is that it is
    a product of class transpositions whose multiplier and divisor
    are coprime. <P/>

    Prime switches can be distinguished from other
    rcwa mappings by their &GAP; property <C>IsPrimeSwitch</C>.
    <Index Key="IsPrimeSwitch" Subkey="for an rcwa mapping">
       <C>IsPrimeSwitch</C>
    </Index>
<Example>
<![CDATA[
gap> Display(PrimeSwitch(3));

Wild rcwa permutation of Z with modulus 12

        /
        | (3n+4)/2 if n in 2(4)
        | n-1      if n in 5(6) U 8(12)
        | n+1      if n in 1(6)
 n |-> <  n/2      if n in 0(12)
        | n-3      if n in 4(12)
        | n        if n in 3(6)
        |
        \

gap> Display(PrimeSwitch(3):AsClassMapping);

Wild rcwa permutation of Z with modulus 12

  0(12) -> 0(6)  loop
   1(6) -> 2(6)
   2(4) -> 5(6)
   3(6) -> 3(6)  id
  4(12) -> 1(12)
   5(6) -> 4(6)
  8(12) -> 7(12)

gap> Factorization(PrimeSwitch(3));
[ ( 1(6), 0(8) ), ( 5(6), 4(8) ), ( 0(4), 1(6) ), ( 2(4), 5(6) ), 
  ( 2(6), 1(12) ), ( 4(6), 7(12) ) ]
gap> Display(PrimeSwitch(5,3,4));

Wild rcwa permutation of Z with modulus 20

        /
        | n+1     if n in 0(2)
        | 5n-5    if n in 3(4)
 n |-> <  (n-1)/2 if n in 1(4) \ 1(20)
        | n-1     if n in 1(20)
        |
        \

gap> Multpk(PrimeSwitch(5,3,4),5,1);
3(4)
gap> PrimeSwitch(5,3,4) = PrimeSwitch(5,ResidueClass(3,4));
true
gap> Factorization(PrimeSwitch(5,3,4));
[ ( 0(2), 1(4) ), ( 1(4), 0(10) ), ( 1(2), 0(10) ) ]
]]>
</Example>
  </Description>
</ManSection>

Obtaining a factorization of an rcwa permutation into class shifts,
class reflections and class transpositions is particularly difficult if
multiplier and divisor are coprime. A prototype of permutations which have
this property has been introduced in a different context
in&nbsp;<Cite Key="Keller99"/>: 

<ManSection>
  <Func Name="mKnot" Arg="m" Label="for an odd integer"/>
  <Returns>
    the permutation <M>g_m</M> as defined in&nbsp;<Cite Key="Keller99"/>.
  </Returns>
  <Description>
    The argument <A>m</A> must be an odd integer greater than&nbsp;1.
<Example>
<![CDATA[
gap> Display(mKnot(5));

Wild rcwa permutation of Z with modulus 5

        /
        | 6n/5     if n in 0(5)
        | (4n+1)/5 if n in 1(5)
 n |-> <  (6n-2)/5 if n in 2(5)
        | (4n+3)/5 if n in 3(5)
        | (6n-4)/5 if n in 4(5)
        \
]]>
</Example>
  </Description>
</ManSection>

<Alt Only="LaTeX">\noindent</Alt>
In his article, Timothy P. Keller shows that a permutation of this type
cannot have infinitely many cycles of any given finite length.

</Section>

<!-- #################################################################### -->

<Section Label="sec:ExtractingRoots">
<Heading>
  Extracting roots of residue-class-wise affine mappings
</Heading>

<ManSection>
  <Meth Name="Root"
        Arg="f, k" Label="k-th root of an rcwa mapping"/>
  <Returns>
    an rcwa mapping <C>g</C> such that <C>g&circum;<A>k</A>=<A>f</A></C>,
    provided that such a mapping exists and that there is a method available
    which can determine it.
  </Returns>
  <Description>
    Currently, extracting roots is implemented for rcwa permutations
    of finite order.
<Example>
<![CDATA[
gap> Root(ClassTransposition(0,2,1,2),100);
( 0(8), 2(8), 4(8), 6(8), 1(8), 3(8), 5(8), 7(8) )
gap> Display(last:CycleNotation:=false);

Tame rcwa permutation of Z with modulus 8

        /
        | n+2 if n in Z \ 6(8) U 7(8)
 n |-> <  n-5 if n in 6(8)
        | n-7 if n in 7(8)
        \

gap> last^100 = ClassTransposition(0,2,1,2);
true
]]>
</Example>
  </Description>
</ManSection>

</Section>

<!-- #################################################################### -->

<Section Label="sec:NonBijectiveMappings">
<Heading>
  Special functions for non-bijective mappings
</Heading>

<ManSection>
  <Attr Name="RightInverse" Arg="f" Label="of an injective rcwa mapping"/>
  <Returns>
    a right inverse of the injective rcwa mapping&nbsp;<A>f</A>,
    i.e. a mapping <M>g</M> such that <A>f</A><M>g</M>&nbsp;=&nbsp;1.
  </Returns>
  <Description>
<Example>
<![CDATA[
gap> twice := 2*IdentityRcwaMappingOfZ;
Rcwa mapping of Z: n -> 2n
gap> twice * RightInverse(twice);
IdentityMapping( Integers )
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Oper Name="CommonRightInverse" Arg="l, r"
        Label="of two injective rcwa mappings"/>
  <Returns>
    a mapping <M>d</M> such that <A>l</A><M>d</M> = <A>r</A><M>d</M> = 1.
  </Returns>
  <Description>
    The mappings <A>l</A> and <A>r</A> must be injective, and their images
    must form a partition of their source.
<Example>
<![CDATA[
gap> twice := 2*IdentityRcwaMappingOfZ; twiceplus1 := twice+1;
Rcwa mapping of Z: n -> 2n
Rcwa mapping of Z: n -> 2n + 1
gap> Display(CommonRightInverse(twice,twiceplus1));

Rcwa mapping of Z with modulus 2

        /
        | n/2     if n in 0(2)
 n |-> <  (n-1)/2 if n in 1(2)
        |
        \
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Attr Name="ImageDensity" Arg="f" Label="of an rcwa mapping"/>
  <Returns>
    the <E>image density</E> of the rcwa mapping&nbsp;<A>f</A>.
  </Returns>
  <Description>
    In the notation introduced in the definition of an rcwa mapping,
    the <E>image density</E> of an rcwa mapping&nbsp;<M>f</M> is defined by
    <Alt Not="LaTeX">1/m</Alt><Alt Only="LaTeX"><M>\frac{1}{m}</M></Alt>
    <M>\sum_{r(m) \in R/mR} |R/c_{r(m)}R|/|R/a_{r(m)}R|</M>.
    The image density of an injective rcwa mapping is <M>\leq 1</M>, and
    the image density of a surjective rcwa mapping is <M>\geq 1</M>
    (this can be seen easily). Thus in particular the image density of
    a bijective rcwa mapping is&nbsp;1.
<Example>
<![CDATA[
gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping.
gap> List( [ T, ClassShift(0,1), RcwaMapping([[2,0,1]]) ], ImageDensity );
[ 4/3, 1, 1/2 ]
]]>
</Example>
  </Description>
</ManSection>

<Index Key="InjectiveAsMappingFrom" Subkey="for an rcwa mapping">
  <C>InjectiveAsMappingFrom</C>
</Index>

Given an rcwa mapping <C>f</C>, the function <C>InjectiveAsMappingFrom</C>
returns a set <C>S</C> such that the restriction of <C>f</C> to&nbsp;<C>S</C>
is injective, and such that the image of <C>S</C> under&nbsp;<C>f</C> is the
entire image of&nbsp;<C>f</C>.

<Example>
<![CDATA[
gap> InjectiveAsMappingFrom(T);
0(2)
]]>
</Example>

</Section>

<!-- #################################################################### -->

<Section Label="sec:Trajectories">
<Heading>
  On trajectories and cycles of residue-class-wise affine mappings
</Heading>

&RCWA; provides various methods to compute trajectories of rcwa mappings:

<ManSection>
  <Heading> Trajectory (methods for rcwa mappings) </Heading>
  <Meth Name="Trajectory" Arg="f, n, length"
        Label="for rcwa mapping, starting point, length"/>
  <Meth Name="Trajectory" Arg="f, n, length, m"
        Label="for rcwa mapping, starting point, length, modulus"/>
  <Meth Name="Trajectory" Arg="f, n, terminal"
        Label="for rcwa mapping, starting point, set of end points"/>
  <Meth Name="Trajectory" Arg="f, n, terminal, m"
        Label="for rcwa mapping, starting point, set of end points, modulus"/>
  <Returns>
    the first <A>length</A> iterates in the trajectory of the
    rcwa mapping&nbsp;<A>f</A> starting at&nbsp;<A>n</A>, respectively the
    initial part of the trajectory of the rcwa mapping&nbsp;<A>f</A>
    starting at&nbsp;<A>n</A> which ends at the first occurrence of an
    iterate in the set <A>terminal</A>. If the argument <A>m</A> is given,
    the iterates are reduced (mod&nbsp;<A>m</A>).
  </Returns>
  <Description>
    To save memory when computing long trajectories containing huge
    iterates, the reduction (mod&nbsp;<A>m</A>) is done each time before
    storing an iterate.
    In place of the ring element&nbsp;<A>n</A>, the methods also accept
    a finite set of ring elements or a union of residue classes.
<Example>
<![CDATA[
gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping.
gap> Trajectory(T,27,15); Trajectory(T,27,20,5);
[ 27, 41, 62, 31, 47, 71, 107, 161, 242, 121, 182, 91, 137, 206, 103 ]
[ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 0, 3, 0, 0, 3 ]
gap> Trajectory(T,15,[1]); Trajectory(T,15,[1],2);
[ 15, 23, 35, 53, 80, 40, 20, 10, 5, 8, 4, 2, 1 ]
[ 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1 ]
gap> Trajectory(T,ResidueClass(Integers,3,0),Integers);
[ 0(3), 0(3) U 5(9), 0(3) U 5(9) U 7(9) U 8(27), 
  <union of 20 residue classes (mod 27) (6 classes)>, 
  <union of 73 residue classes (mod 81)>, Z \ 10(81) U 37(81), Integers ]
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Heading>
    Trajectory (methods for rcwa mappings -- <Q>accumulated coefficients</Q>)
  </Heading>
  <Meth Name="Trajectory" Arg="f, n, length, whichcoeffs"
        Label="for rcwa mapping, starting point, length, coeff.-spec."/>
  <Meth Name="Trajectory" Arg="f, n, terminal, whichcoeffs"
        Label="for rcwa mapping, starting point, set of end points, coeff.-spec."/>
  <Returns>
    either the list <C>c</C> of triples of coprime coefficients such that
    for any&nbsp;<C>k</C> it holds that
    <C><A>n</A>&circum;(<A>f</A>&circum;(k-1)) =
    (c[k][1]*<A>n</A> + c[k][2])/c[k][3]</C> or the last entry of that list,
    depending on whether <A>whichcoeffs</A> is <C>"AllCoeffs"</C> or
    <C>"LastCoeffs"</C>.
  </Returns>
  <Description>
    The meanings of the arguments <A>length</A> and <A>terminal</A> are
    the same as in the methods for the operation <C>Trajectory</C>
    described above. In general, computing only the last coefficient triple
    (<A>whichcoeffs</A> = <C>"LastCoeffs"</C>) needs considerably less
    memory than computing the entire list.
<Example>
<![CDATA[
gap> Trajectory(T,27,[1],"LastCoeffs");
[ 36472996377170786403, 195820718533800070543, 1180591620717411303424 ]
gap> (last[1]*27+last[2])/last[3];
1
]]>
</Example>
  </Description>
</ManSection>

When dealing with problems like the <M>3n+1</M>-Conjecture or when
determining the degree of transitivity of the natural action of an rcwa group
on its underlying ring, an important task is to determine the residue classes
whose elements get larger or smaller when applying a given rcwa mapping:

<ManSection>
  <Heading> IncreasingOn &amp; DecreasingOn (for an rcwa mapping) </Heading>
  <Attr Name="IncreasingOn" Arg="f" Label="for an rcwa mapping"/>
  <Attr Name="DecreasingOn" Arg="f" Label="for an rcwa mapping"/>
  <Returns>
    the union of all residue classes <M>r(m)</M> such that
    <M>|R/a_{r(m)}R| &gt; |R/c_{r(m)}R|</M> or
    <M>|R/a_{r(m)}R| &lt; |R/c_{r(m)}R|</M>, respectively, where <M>R</M>
    denotes the source, <M>m</M> denotes the modulus and <M>a_{r(m)}</M>,
    <M>b_{r(m)}</M> and <M>c_{r(m)}</M> denote the coefficients
    of&nbsp;<A>f</A> as introduced in
    Section&nbsp;<Ref Label="sec:basicdefinitions"/>.
  </Returns>
  <Description>
    If the argument is an rcwa mapping of &ZZ; in sparse representation,
    an option <C>classes</C> is interpreted; if set, the step of forming
    the union of the residue classes in question is omitted, and the
    list of residue classes is returned instead of their union.
    This may save time and memory if the modulus is large.
<Example>
<![CDATA[
gap> List([1..3],k->IncreasingOn(T^k));
[ 1(2), 3(4), 3(4) U 1(8) U 6(8) ]
gap> List([1..3],k->DecreasingOn(T^k));
[ 0(2), Z \ 3(4), 0(4) U 2(8) U 5(8) ]
gap> a := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]);; # Collatz' permutation
gap> List([-2..2],k->IncreasingOn(a^k));
[ Z \ 1(8) U 7(8), 0(2), [  ], Z \ 0(3), 1(9) U 4(9) U 5(9) U 8(9) ]
]]>
</Example>
  </Description>
</ManSection>

We assign certain directed graphs to rcwa mappings, which encode the order
in which trajectories may traverse the residue classes modulo some modulus:

<ManSection>
  <Oper Name="TransitionGraph"
        Arg="f, m" Label="for an rcwa mapping and a modulus"/>
  <Returns>
    the transition graph of the rcwa mapping&nbsp;<A>f</A>
    for modulus&nbsp;<A>m</A>.
  </Returns>
  <Description>
    <Index Key="rcwa mapping" Subkey="transition graph">rcwa mapping</Index>
    <Alt Only="LaTeX">\noindent</Alt>
    The <E>transition graph</E> <M>\Gamma_{f,m}</M> of&nbsp;<M>f</M> for
    modulus&nbsp;<M>m</M> is defined as follows:
    <Enum>
      <Item>
        The vertices are the residue classes (mod&nbsp;<M>m</M>).
      </Item>
      <Item>
        There is an edge from <M>r_1(m)</M> to <M>r_2(m)</M> if and only if
        there is some <M>n \in r_1(m)</M> such that
        <M>n^f \in r_2(m)</M>.
      </Item>
    </Enum>
    The assignment of the residue classes (mod&nbsp;<M>m</M>) to the
    vertices of the graph corresponds to the ordering of the residues in
    <C>AllResidues(Source(<A>f</A>),<A>m</A>)</C>.
    The result is returned in the format used by the package
    <Package>GRAPE</Package>&nbsp;<Cite Key="GRAPE"/>.
  </Description>
</ManSection>

There are a couple of operations and attributes which are based
on these graphs:

<ManSection>
  <Oper Name="OrbitsModulo"
        Arg="f, m" Label="for an rcwa mapping and a modulus"/>
  <Returns>
    the partition of <C>AllResidues(Source(<A>f</A>),<A>m</A>)</C>
    corresponding to the weakly connected components of the transition
    graph of the rcwa mapping&nbsp;<A>f</A> for modulus&nbsp;<A>m</A>.
  </Returns>
  <Description>
<Example>
<![CDATA[
gap> OrbitsModulo(ClassTransposition(0,2,1,4),8);
[ [ 0, 1, 4 ], [ 2, 5, 6 ], [ 3 ], [ 7 ] ]
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Oper Name ="FactorizationOnConnectedComponents"
        Arg = "f, m" Label="for an rcwa mapping and a modulus"/>
  <Returns>
    the set of restrictions of the rcwa mapping&nbsp;<A>f</A> to the
    weakly connected components of its transition graph <M>\Gamma_{f,m}</M>.
  </Returns>
  <Description>
    The product of the returned mappings is&nbsp;<A>f</A>.
    They have pairwise disjoint supports, hence any two of them commute.
<Example>
<![CDATA[
gap> sigma := ClassTransposition(1,4,2,4)  * ClassTransposition(1,4,3,4)
>           * ClassTransposition(3,9,6,18) * ClassTransposition(1,6,3,9);;
gap> List(FactorizationOnConnectedComponents(sigma,36),Support);
[ 33(36) U 34(36) U 35(36), 9(36) U 10(36) U 11(36), 
  <union of 23 residue classes (mod 36)> \ [ -6, 3 ] ]
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Oper Name="TransitionMatrix"
        Arg="f, m" Label="for an rcwa mapping and a modulus"/>
  <Returns>
    the transition matrix of the rcwa mapping&nbsp;<A>f</A> for
    modulus&nbsp;<A>m</A>.
  </Returns>
  <Description>
    Let <M>M</M> be this matrix. Then for any two residue classes <M>r_1(m),
    r_2(m) \in R/mR</M>, the entry <M>M_{r_1(m),r_2(m)}</M> is defined by
    <Alt Only="LaTeX">
      <Display>
        <![CDATA[M_{r_1(m),r_2(m)} \ := \
                 \displaystyle{\frac{|R/mR|}{|R/\hat{m}R|}} \cdot
                 \left|\left\{r(\hat{m}) \in R/\hat{m}R | \ r \in r_1(m)
                 \wedge r^f \in r_2(m)\right\}\right|,]]>
      </Display>
    </Alt>
    <Alt Only="HTML"><![CDATA[<center>
      <img src = "transmat.png" width = "599" height = "47"
           alt = "(see the PDF version of the manual),"/>
    </center>]]></Alt>
    <Alt Only="Text">
      (see the PDF- or HTML version of the manual),
    </Alt>
    where <M>\hat{m}</M> is the product of <A>m</A> and the square of the
    modulus of&nbsp;<A>f</A>.
    The assignment of the residue classes (mod&nbsp;<A>m</A>) to the rows and
    columns of the matrix corresponds to the ordering of the residues in
    <C>AllResidues(Source(<A>f</A>),<A>m</A>)</C>. <P/>

    The transition matrix is a weighted adjacency matrix of the corresponding
    transition graph <C>TransitionGraph(<A>f</A>,<A>m</A>)</C>.
    The sums of the rows of a transition matrix are always equal to&nbsp;1.
<Example>
<![CDATA[
gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping.
gap> Display(TransitionMatrix(T^3,3));
[ [  1/8,  1/4,  5/8 ],
  [    0,  1/4,  3/4 ],
  [    0,  3/8,  5/8 ] ]
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Heading> Sources &amp; Sinks (of an rcwa mapping) </Heading>
  <Attr Name="Sources" Arg="f" Label="of an rcwa mapping"/>
  <Attr Name="Sinks" Arg="f" Label="of an rcwa mapping"/>
  <Returns>
    a list of unions of residue classes modulo the modulus&nbsp;<M>m</M>
    of the rcwa mapping&nbsp;<A>f</A>, as described below.
  </Returns>
  <Description>
    The returned list contains an entry for any strongly
    connected component of the transition graph of&nbsp;<A>f</A> for
    modulus&nbsp;<C>Mod(<A>f</A>)</C> which has only outgoing edges
    (<Q>source</Q>) or which has only ingoing edges (<Q>sink</Q>),
    respectively. The list entry corresponding to such a component
    is the union of the vertices belonging to it.
<Example>
<![CDATA[
gap> g := ClassTransposition(0,2,1,2)*ClassTransposition(0,2,1,4);;
gap> Sources(g); Sinks(g);
[ 0(4) ]
[ 1(4) ]
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Attr Name="Loops" Arg="f" Label="of an rcwa mapping"/>
  <Returns>
    if <A>f</A> is bijective, the list of non-isolated vertices of the
    transition graph of&nbsp;<A>f</A> for modulus <C>Mod(<A>f</A>)</C>
    which carry a loop. In general, the list of vertices of that transition
    graph which carry a loop, but which <A>f</A> does not fix setwise.
  </Returns>
  <Description>
    The returned list may also include supersets of the named
    residue classes instead if <A>f</A> is affine even on these.
<Example>
<![CDATA[
gap> Loops(ClassTransposition(0,2,1,2)*ClassTransposition(0,2,1,4));
[ 0(4), 1(4) ]
]]>
</Example>
  </Description>
</ManSection>

There is a nice invariant of trajectories of the Collatz mapping:

<ManSection>
  <Func Name="GluckTaylorInvariant" Arg="a" Label="of a trajectory"/>
  <Returns>
    the invariant defined in&nbsp;<Cite Key="GluckTaylor02"/>. This is
    <M>(\sum_{i=1}^l a_i \cdot a_{i \mod l + 1})/(\sum_{i=1}^l a_i^2)</M>,
    where <M>l</M> denotes the length of&nbsp;<A>a</A>.
  </Returns>
  <Description>
    The argument <A>a</A> must be a list of integers.
    In&nbsp;<Cite Key="GluckTaylor02"/> it is shown that if <A>a</A>
    is a trajectory of the `original' Collatz mapping <M>n</M>
    <M>\mapsto</M> (<M>n/2</M> if <M>n</M> even, <M>3n+1</M> if
    <M>n</M> odd) starting at an odd integer <M>\geq 3</M> and ending
    at&nbsp;1, then the invariant lies in the interval <M>]9/13,5/7[</M>.
<Example>
<![CDATA[
gap> C := RcwaMapping([[1,0,2],[3,1,1]]);;
gap> List([3,5..49],n->Float(GluckTaylorInvariant(Trajectory(C,n,[1]))));
[ 0.701053, 0.696721, 0.708528, 0.707684, 0.706635, 0.695636, 0.711769,
  0.699714, 0.707409, 0.693833, 0.710432, 0.706294, 0.714242, 0.699935,
  0.714242, 0.705383, 0.706591, 0.698198, 0.712222, 0.714242, 0.709048,
  0.69612, 0.714241, 0.701076 ]
]]>
</Example>
  </Description>
</ManSection>

Quite often one can make certain <Q>educated guesses</Q> on the overall
behaviour of the trajectories of a given rcwa mapping. For example it is
reasonably straightforward to make the conjecture that all trajectories of
the Collatz mapping eventually enter the finite set <M>\{-136, -91, -82, -68,
-61, -55, -41, -37, -34, -25, -17, -10, -7, -5, -1, 0, 1, 2 \}</M>, or that
<Q>on average</Q> the next number in a trajectory of the Collatz mapping is
smaller than the preceding one by a factor of <M>\sqrt{3}/2</M>. However it
is clear that such guesses can be wrong, and that they therefore cannot be
used to prove anything. Nevertheless they can sometimes be useful:

<ManSection>
  <Oper Name="LikelyContractionCentre"
        Arg="f, maxn, bound" Label="of an rcwa mapping"/>
  <Returns> a list of ring elements (see below). </Returns>
  <Description>
    This operation tries to compute the <E>contraction centre</E> of the
    rcwa mapping <A>f</A>. Assuming its existence this is the unique finite
    subset <M>S_0</M> of the source of&nbsp;<A>f</A> on which <A>f</A>
    induces a permutation and which intersects non-trivially with any
    trajectory of&nbsp;<A>f</A>. The mapping&nbsp;<A>f</A> is assumed
    to be <E>contracting</E>, i.e. to have such a contraction centre.
    As in general contraction centres are likely not computable, the methods
    for this operation are probabilistic and may return wrong results.
    The argument <A>maxn</A> is a bound on the starting
    value and <A>bound</A> is a bound on the elements of the trajectories
    to be searched.
    If the limit <A>bound</A> is exceeded, an Info message on Info
    level&nbsp;3 of <C>InfoRCWA</C> is given.
<Example>
<![CDATA[
gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping.
gap> S0 := LikelyContractionCentre(T,100,1000);
#I  Warning: `LikelyContractionCentre' is highly probabilistic.
The returned result can only be regarded as a rough guess.
See ?LikelyContractionCentre for more information.
[ -136, -91, -82, -68, -61, -55, -41, -37, -34, -25, -17, -10, -7, -5, 
  -1, 0, 1, 2 ]
]]>
</Example>
  </Description>
</ManSection>

<ManSection>
  <Oper Name="GuessedDivergence" Arg="f" Label="of an rcwa mapping"/>
  <Returns>
    a floating point value which is intended to be a rough guess on how fast
    the trajectories of the rcwa mapping&nbsp;<A>f</A> diverge (return value
    greater than&nbsp;1) or converge (return value smaller than&nbsp;1).
  </Returns>
  <Description>
    Nothing particular is guaranteed.
<Example>
<![CDATA[
gap> GuessedDivergence(T);
#I  Warning: GuessedDivergence: no particular return value is guaranteed.
0.866025
]]>
</Example>
  </Description>
</ManSection>

</Section>

<!-- #################################################################### -->

<Section Label="sec:SparseRepresentationOfRcwaMappings">
<Heading>
  Saving memory -- the sparse representation of rcwa mappings
</Heading>

<Index Key="rcwa mapping" Subkey="sparse representation">
  rcwa mapping
</Index>
<Index Key="RcwaMapping"
       Subkey="by list of coefficients, sparse representation">
  <C>RcwaMapping</C>
</Index>

It is quite common that an rcwa mapping with large modulus
has only few distinct affine partial mappings. In this case the
<Q>standard</Q> representation which stores a coefficient triple for each
residue class modulo the modulus is unsuitable. For this reason there is
a second representation of rcwa mappings, the <Q>sparse</Q> representation.
Depending on the rcwa mappings involved, using this representation may
speed up computations and reduce memory requirements by orders of
magnitude. For rcwa mappings with almost as many distinct affine partial
mappings as there are residue classes modulo the modulus, using sparse
representation makes computations slower and more memory-consuming.
Presently, the sparse representation is only available for rcwa mappings
of &ZZ;. <P/>

The sparse representation of an rcwa mapping consists of the modulus and
a list of 5-tuples <M>(r,m,a_{r(m)},b_{r(m)},c_{r(m)})</M> of integers.
Any such 5-tuple specifies the coefficients of the restriction
<M>n \mapsto (a_{r(m)} \cdot n + b_{r(m)})/c_{r(m)}</M> of the mapping
to a residue class <M>r(m)</M>. The <M>r(m)</M> are chosen to form the
coarsest possible partition of &ZZ; into residue classes such that the
restriction of the mapping to any of them is affine. Also the list of
coefficient tuples is sorted, all <M>c_{r(m)}</M> are positive and
<M>\gcd(c_{r(m)},\gcd(a_{r(m)},b_{r(m)})) = 1</M>. This way the coefficient
list of an rcwa mapping of &ZZ; is unique. <P/>

Changing the representation of rcwa mappings does not change their
behaviour with respect to <Q><C>=</C></Q> and <Q><C>&lt;</C></Q> 
The product of two rcwa mappings in sparse representation is in sparse
representation again, just like the product of two rcwa mappings in
standard representation is in standard representation. Also, inverses
are in the same representation. The product of two rcwa mappings in
different representation may be in any of the representations of the
factors.

<ManSection>
  <Oper Name="SparseRepresentation" Arg="f" Label="of an rcwa mapping"/>
  <Oper Name="SparseRep" Arg="f" Label="of an rcwa mapping"/>
  <Oper Name="StandardRepresentation" Arg="f" Label="of an rcwa mapping"/>
  <Oper Name="StandardRep" Arg="f" Label="of an rcwa mapping"/>
  <Returns>
    the rcwa mapping <A>f</A> in sparse, respectively, standard
    representation.
  </Returns>
  <Description>
    Appropriate attribute values and properties are copied over
    to the rcwa mapping in the <Q>new</Q> representation. 
<Example>
<![CDATA[
gap> a := ClassTransposition(1,2,4,6);
( 1(2), 4(6) )
gap> b := ClassTransposition(1,3,2,6);
( 1(3), 2(6) )
gap> c := ClassTransposition(2,3,4,6);
( 2(3), 4(6) )
gap> g := (b*a*c)^2*a;
<rcwa permutation of Z with modulus 288>
gap> h := SparseRep(g);
<rcwa permutation of Z with modulus 288 and 21 affine parts>
gap> g = h;
true
gap> Coefficients(h);
[ [ 0, 6, 1, 0, 1 ], [ 1, 3, 16, -1, 3 ], [ 2, 96, 9, 14, 16 ], 
  [ 3, 24, 9, 5, 4 ], [ 5, 24, 3, 1, 4 ], [ 8, 36, 2, -7, 9 ], 
  [ 9, 48, 27, 29, 8 ], [ 11, 24, 9, 5, 4 ], [ 14, 48, 27, 38, 8 ], 
  [ 15, 24, 27, 19, 4 ], [ 17, 48, 9, 7, 8 ], [ 20, 72, 3, 4, 4 ], 
  [ 21, 24, 1, -3, 6 ], [ 23, 24, 27, 19, 4 ], [ 26, 48, 3, 2, 8 ], 
  [ 32, 36, 4, -11, 9 ], [ 33, 48, 9, 7, 8 ], [ 38, 48, 9, 10, 8 ], 
  [ 41, 48, 27, 29, 8 ], [ 50, 96, 27, 58, 16 ], [ 56, 72, 1, 0, 4 ] ]
gap> h^2;
<rcwa permutation of Z with modulus 13824 and 71 affine parts>
gap> h^3;
<rcwa permutation of Z with modulus 663552 and 201 affine parts>
]]>
</Example>
    Memory consumption may differ a lot between sparse- and standard
    representation:
<Log>
gap> MemoryUsage(h^3);               # on a 64-bit machine
18254
gap> MemoryUsage(StandardRep(h^3));  # on a 64-bit machine
42467894
</Log>
  </Description>
</ManSection>

</Section>

<!-- #################################################################### -->

<Section Label="sec:CategoriesOfRcwaMappings">
<Heading>The categories and families of rcwa mappings</Heading>

<ManSection>
  <Filt Name="IsRcwaMapping" Arg="f"/>
  <Filt Name="IsRcwaMappingOfZ" Arg="f"/>
  <Filt Name="IsRcwaMappingOfZ_pi" Arg="f"/>
  <Filt Name="IsRcwaMappingOfGFqx" Arg="f"/>
  <Returns>
    <C>true</C> if <A>f</A> is an rcwa mapping,
    an rcwa mapping of the ring of integers,
    an rcwa mapping of a semilocalization of the ring of integers or
    an rcwa mapping of a polynomial ring in one variable over a finite field,
    respectively, and <C>false</C> otherwise.
  </Returns>
  <Description>
    Often the same methods can be used for rcwa mappings of the ring of
    integers and of its semilocalizations. For this reason there is
    a category <C>IsRcwaMappingOfZOrZ&uscore;pi</C> which is the union of
    <C>IsRcwaMappingOfZ</C> and <C>IsRcwaMappingOfZ&uscore;pi</C>.
    <Index Key="IsRcwaMappingOfZOrZ_pi">
      <C>IsRcwaMappingOfZOrZ&uscore;pi</C>
    </Index>
    The internal representation of rcwa mappings is called
    <C>IsRcwaMappingStandardRep</C>.
    <Index Key="IsRcwaMappingStandardRep">
      <C>IsRcwaMappingStandardRep</C>
    </Index>
    There are methods available for <C>ExtRepOfObj</C> and
    <C>ObjByExtRep</C>.
    <Index Key="ExtRepOfObj"><C>ExtRepOfObj</C></Index>
    <Index Key="ObjByExtRep"><C>ObjByExtRep</C></Index>
  </Description>
</ManSection>

<ManSection>
  <Func Name="RcwaMappingsFamily" Arg="R" Label="of a ring"/>
  <Returns>
    the family of rcwa mappings of the ring&nbsp;<A>R</A>.
  </Returns>
</ManSection>

</Section>

<!-- #################################################################### -->

</Chapter>

<!-- #################################################################### -->
