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<a id="X7FD73FCB8510050E" name="X7FD73FCB8510050E"></a>
<div class="ChapSects"><a href="chap2_mj.html#X7FD73FCB8510050E">2 Residue-Class-Wise Affine Mappings</a>
<div class="ContSect"> <a href="chap2_mj.html#X78ED07E37FC2BD46">2.1 Basic definitions</a>

</div>
<div class="ContSect"> <a href="chap2_mj.html#X86BC55648302D643">2.2 Entering residue-class-wise affine mappings</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X86B611BD7EED62A1">2.2-1 ClassShift</a>
 <a href="chap2_mj.html#X7896C5417E3692B4">2.2-2 ClassReflection</a>
 <a href="chap2_mj.html#X8716A75F7DD1C46B">2.2-3 ClassTransposition</a>
 <a href="chap2_mj.html#X87EB8C1C87F78A17">2.2-4 ClassRotation</a>
 <a href="chap2_mj.html#X8799551B83644B37">2.2-5 RcwaMapping (the general constructor) </a>

 <a href="chap2_mj.html#X7F1A559387D0226E">2.2-6 LocalizedRcwaMapping</a>
</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X78E796B8824C4FC8">2.3 Basic arithmetic for residue-class-wise affine mappings</a>

</div>
<div class="ContSect"> <a href="chap2_mj.html#X7C16D22C7BD40FDC">2.4 
 Attributes and properties of residue-class-wise affine mappings
</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X7C21406085B69C30">2.4-1 LargestSourcesOfAffineMappings</a>
 <a href="chap2_mj.html#X7D6D0F2783AD02F4">2.4-2 FixedPointsOfAffinePartialMappings</a>
 <a href="chap2_mj.html#X7A2E308C860B46E3">2.4-3 Multpk</a>
 <a href="chap2_mj.html#X7B1E53127D9AE52F">2.4-4 Determinant</a>
 <a href="chap2_mj.html#X8365EEEB82C946FD">2.4-5 Sign</a>
</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X8475F844869DD060">2.5 Factoring residue-class-wise affine permutations</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X829BA0537F2372FF">2.5-1 CTCSCRSplit</a>
 <a href="chap2_mj.html#X853885A182EC5104">2.5-2 FactorizationIntoCSCRCT</a>
 <a href="chap2_mj.html#X861C74E97AE5DA3B">2.5-3 PrimeSwitch</a>
 <a href="chap2_mj.html#X789CB69C7D97B0C4">2.5-4 mKnot</a>
</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X8141065381B0942B">2.6 
 Extracting roots of residue-class-wise affine mappings
</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X873692CE78433859">2.6-1 Root</a>
</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X8322C6848305EC4C">2.7 
 Special functions for non-bijective mappings
</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X7AEFF16E86533633">2.7-1 RightInverse</a>
 <a href="chap2_mj.html#X87C5B9CA7E319233">2.7-2 CommonRightInverse</a>
 <a href="chap2_mj.html#X808D9EDF7BA27467">2.7-3 ImageDensity</a>
</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X7A34724386A2E9F3">2.8 
 On trajectories and cycles of residue-class-wise affine mappings
</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X7C72174D7CCB6348">2.8-1 Trajectory (methods for rcwa mappings) </a>

 <a href="chap2_mj.html#X7FFD09837E934853">2.8-2 
 Trajectory (methods for rcwa mappings -- <q>accumulated coefficients</q>)
 </a>

 <a href="chap2_mj.html#X7E0244A386744185">2.8-3 IncreasingOn & DecreasingOn (for an rcwa mapping) </a>

 <a href="chap2_mj.html#X780841E07CAE7543">2.8-4 TransitionGraph</a>
 <a href="chap2_mj.html#X7F03CC4179424AA9">2.8-5 OrbitsModulo</a>
 <a href="chap2_mj.html#X7F11051E866C197F">2.8-6 FactorizationOnConnectedComponents</a>
 <a href="chap2_mj.html#X7B6833D67D916EF9">2.8-7 TransitionMatrix</a>
 <a href="chap2_mj.html#X81DBA2D58526BE7E">2.8-8 Sources & Sinks (of an rcwa mapping) </a>

 <a href="chap2_mj.html#X80221A4D81AF7453">2.8-9 Loops</a>
 <a href="chap2_mj.html#X8773152E81A30123">2.8-10 GluckTaylorInvariant</a>
 <a href="chap2_mj.html#X84F6A29280E2F925">2.8-11 LikelyContractionCentre</a>
 <a href="chap2_mj.html#X81E0D8E3817B3D16">2.8-12 GuessedDivergence</a>
</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X86F0E0D17E6A9663">2.9 
 Saving memory -- the sparse representation of rcwa mappings
</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X879451B17AD78B07">2.9-1 SparseRepresentation</a>
</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X83FA71DD842377F0">2.10 The categories and families of rcwa mappings</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X7927C13782729CE9">2.10-1 IsRcwaMapping</a>
 <a href="chap2_mj.html#X825DD365822934AF">2.10-2 RcwaMappingsFamily</a>
</div></div>
</div>

<h3>2 Residue-Class-Wise Affine Mappings</h3>

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

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.

<a id="X78ED07E37FC2BD46" name="X78ED07E37FC2BD46"></a>

<h4>2.1 Basic definitions</h4>

Residue-class-wise affine groups, or rcwa groups for short, are permutation groups whose elements are bijective residue-class-wise affine mappings.

A mapping \(f: ℤ \rightarrow ℤ\) is called residue-class-wise affine, or for short an rcwa mapping, if there is a positive integer \(m\) such that the restrictions of \(f\) to the residue classes \(r(m) \in ℤ/mℤ\) are all affine, i.e. given by

<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>

for certain coefficients \(a_{r(m)}, b_{r(m)}, c_{r(m)} \in ℤ\) depending on \(r(m)\). The smallest possible \(m\) is called the modulus of \(f\). It is understood that all fractions are reduced, i.e. that \(\gcd( a_{r(m)}, b_{r(m)}, c_{r(m)} ) = 1\), and that \(c_{r(m)} > 0\). The lcm of the coefficients \(a_{r(m)}\) is called the multiplier of \(f\), and the lcm of the coefficients \(c_{r(m)}\) is called the divisor of \(f\).

It is easy to see that the residue-class-wise affine mappings of ℤ form a monoid under composition, and that the residue-class-wise affine permutations of ℤ form a countable subgroup of Sym(ℤ). We denote the former by Rcwa(ℤ), and the latter by RCWA(ℤ).

An rcwa mapping is called tame if the set of moduli of its powers is bounded, or equivalently if it permutes a partition of ℤ into finitely many residue classes on all of which it is affine. An rcwa group is called tame 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 wild. 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.

The definitions of residue-class-wise affine mappings and -groups can be generalized in the obvious way to suitable rings other than ℤ. In fact, this package provides also some support for residue-class-wise affine groups over \(ℤ^2\), over semilocalizations of ℤ and over univariate polynomial rings over finite fields. The ring \(ℤ^2\) has been chosen as an example of a suitable ring which is not a principal ideal domain, the semilocalizations of ℤ 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.

<a id="X86BC55648302D643" name="X86BC55648302D643"></a>

<h4>2.2 Entering residue-class-wise affine mappings</h4>

Entering an rcwa mapping of ℤ requires giving the modulus \(m\) and the coefficients \(a_{r(m)}\), \(b_{r(m)}\) and \(c_{r(m)}\) for \(r(m)\) running over the residue classes (mod \(m\)).

This can be done easiest by <code class="code">RcwaMapping( <var class="Arg">coeffs</var> )</code>, where <var class="Arg">coeffs</var> is a list of \(m\) coefficient triples <code class="code">coeffs[</code>\(r+1\)<code class="code">] = [</code>\(a_{r(m)}\), \(b_{r(m)}\), \(c_{r(m)}\)<code class="code">]</code>, with \(r\) running from 0 to \(m-1\).

If some coefficient \(c_{r(m)}\) 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 <code class="code">[1,4,3]</code> is not allowed at the first position. The reason for this is that not all integers congruent to \(1 \cdot 0 + 4 = 4\) mod \(m\) are divisible by 3.

For the general constructor for rcwa mappings, see <code class="func">RcwaMapping</code> (<a href="chap2_mj.html#X8799551B83644B37">2.2-5</a>).

<div class="example"><pre>

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 ]

</pre></div>

There is computational evidence for the conjecture that any residue-class-wise affine permutation of ℤ can be factored into members of the following three series of permutations of particularly simple structure (cf. <code class="func">FactorizationIntoCSCRCT</code> (<a href="chap2_mj.html#X853885A182EC5104">2.5-2</a>)):

<a id="X86B611BD7EED62A1" name="X86B611BD7EED62A1"></a>

<h5>2.2-1 ClassShift</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassShift</code>( <var class="Arg">r</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassShift</code>( <var class="Arg">cl</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the class shift \(\nu_{r(m)}\).

The class shift \(\nu_{r(m)}\) is the rcwa mapping of ℤ which maps \(n \in r(m)\) to \(n + m\) and which fixes \(ℤ \setminus r(m)\) pointwise.

In the one-argument form, the argument <var class="Arg">cl</var> stands for the residue class \(r(m)\). Enclosing the argument list in list brackets is permitted.

<div class="example"><pre>

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)
 |
 \

</pre></div>

<a id="X7896C5417E3692B4" name="X7896C5417E3692B4"></a>

<h5>2.2-2 ClassReflection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassReflection</code>( <var class="Arg">r</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassReflection</code>( <var class="Arg">cl</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the class reflection \(\varsigma_{r(m)}\).

The class reflection \(\varsigma_{r(m)}\) is the rcwa mapping of ℤ which maps \(n \in r(m)\) to \(-n + 2r\) and which fixes \(ℤ \setminus r(m)\) pointwise, where it is understood that \(0 \leq r < m\).

In the one-argument form, the argument <var class="Arg">cl</var> stands for the residue class \(r(m)\). Enclosing the argument list in list brackets is permitted.

<div class="example"><pre>

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)
 |
 \

</pre></div>

<a id="X8716A75F7DD1C46B" name="X8716A75F7DD1C46B"></a>

<h5>2.2-3 ClassTransposition</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassTransposition</code>( <var class="Arg">r1</var>, <var class="Arg">m1</var>, <var class="Arg">r2</var>, <var class="Arg">m2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassTransposition</code>( <var class="Arg">cl1</var>, <var class="Arg">cl2</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the class transposition \(\tau_{r_1(m_1),r_2(m_2)}\).

Given two disjoint residue classes \(r_1(m_1)\) and \(r_2(m_2)\) of the integers, the class transposition \(\tau_{r_1(m_1),r_2(m_2)}\) \(\in\) RCWA(ℤ) is defined as the involution which interchanges \(r_1+km_1\) and \(r_2+km_2\) for any integer \(k\) and which fixes all other points. It is understood that \(m_1\) and \(m_2\) are positive, that \(0 \leq r_1 < m_1\) and that \(0 \leq r_2 < m_2\). For a generalized class transposition, the latter assumptions are not made.

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

In the four-argument form, the arguments <var class="Arg">r1</var>, <var class="Arg">m1</var>, <var class="Arg">r2</var> and <var class="Arg">m2</var> stand for \(r_1\), \(m_1\), \(r_2\) and \(m_2\), respectively. In the two-argument form, the arguments <var class="Arg">cl1</var> and <var class="Arg">cl2</var> stand for the residue classes \(r_1(m_1)\) and \(r_2(m_2)\), respectively. Enclosing the argument list in list brackets is permitted. The residue classes \(r_1(m_1)\) and \(r_2(m_2)\) are stored as an attribute <code class="code">TransposedClasses</code>.

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

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

A class transposition can be written as a product of any given number \(k\) of class transpositions. Such a decomposition can be obtained by <code class="code">SplittedClassTransposition(<var class="Arg">ct</var>,<var class="Arg">k</var>)</code>.

<div class="example"><pre>

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) ) ]
</pre></div>

The set of all class transpositions of the ring of integers generates the simple group CT(ℤ) mentioned in Chapter <a href="chap1_mj.html#X83A8C2927FAE2C23">1</a>. This group has a representation as a GAP object -- see <code class="func">CT</code> (<a href="chap3_mj.html#X7BD42D8481300E25">3.1-9</a>). The set of all generalized class transpositions of ℤ generates a simple group as well, cf. <a href="chapBib_mj.html#biBKohl09">[Koh10]</a>.

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

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

<a id="X87EB8C1C87F78A17" name="X87EB8C1C87F78A17"></a>

<h5>2.2-4 ClassRotation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassRotation</code>( <var class="Arg">r</var>, <var class="Arg">m</var>, <var class="Arg">u</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassRotation</code>( <var class="Arg">cl</var>, <var class="Arg">u</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the class rotation \(\rho_{r(m),u}\).

Given a residue class \(r(m)\) and a unit \(u\) of a suitable ring \(R\), the class rotation \(\rho_{r(m),u}\) is the rcwa mapping which maps \(n \in r(m)\) to \(un + (1-u)r\) and which fixes \(R \setminus r(m)\) pointwise. Class rotations generalize class reflections, as we have \(\rho_{r(m),-1} = \varsigma_{r(m)}\).

In the two-argument form, the argument <var class="Arg">cl</var> stands for the residue class \(r(m)\). Enclosing the argument list in list brackets is permitted. The argument <var class="Arg">u</var> is stored as an attribute <code class="code">RotationFactor</code>.

<div class="example"><pre>

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 |-> </div>

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

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.

<a id="X8799551B83644B37" name="X8799551B83644B37"></a>

<h5>2.2-5 RcwaMapping (the general constructor) </h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">R</var>, <var class="Arg">m</var>, <var class="Arg">coeffs</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">R</var>, <var class="Arg">coeffs</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">coeffs</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">perm</var>, <var class="Arg">range</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">m</var>, <var class="Arg">values</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">pi</var>, <var class="Arg">coeffs</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">q</var>, <var class="Arg">m</var>, <var class="Arg">coeffs</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">P1</var>, <var class="Arg">P2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">cycles</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">expression</var> )</td><td class="tdright">( method )</td></tr></table></div>
Returns: an rcwa mapping.

In all cases the argument <var class="Arg">R</var> is the underlying ring, <var class="Arg">m</var> is the modulus and <var class="Arg">coeffs</var> is the coefficient list. A coefficient list for an rcwa mapping with modulus \(m\) consists of \(|R/mR|\) coefficient triples <code class="code">[</code>\(a_{r(m)}\), \(b_{r(m)}\), \(c_{r(m)}\)<code class="code">]</code>. Their ordering is determined by the ordering of the representatives of the residue classes (mod \(m\)) in the sorted list returned by <code class="code">AllResidues(<var class="Arg">R</var>, <var class="Arg">m</var>)</code>. In case \(R = ℤ\) this means that the coefficient triple for the residue class \(0(m)\) comes first and is followed by the one for \(1(m)\), the one for \(2(m)\) and so on.

If one or several of the arguments <var class="Arg">R</var>, <var class="Arg">m</var> and <var class="Arg">coeffs</var> 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

<dl>
<dt>(a)</dt>
<dd>of <var class="Arg">R</var> with modulus <var class="Arg">m</var> and coefficients <var class="Arg">coeffs</var>,

</dd>
<dt>(b)</dt>
<dd>of <var class="Arg">R</var> = ℤ or <var class="Arg">R</var> = \(ℤ_{(\pi)}\) with modulus <code class="code">Length(<var class="Arg">coeffs</var>)</code> and coefficients <var class="Arg">coeffs</var>,

</dd>
<dt>(c)</dt>
<dd>of <var class="Arg">R</var> = ℤ with modulus <code class="code">Length(<var class="Arg">coeffs</var>)</code> and coefficients <var class="Arg">coeffs</var>,

</dd>
<dt>(d)</dt>
<dd>of <var class="Arg">R</var> = ℤ, permuting any set <code class="code"><var class="Arg">range</var>+k*Length(<var class="Arg">range</var>)</code> like <var class="Arg">perm</var> permutes <var class="Arg">range</var>,

</dd>
<dt>(e)</dt>
<dd>of <var class="Arg">R</var> = ℤ with modulus <var class="Arg">m</var> and values given by a list <var class="Arg">val</var> of 2 pairs <code class="code">[</code>preimage<code class="code">, </code>image<code class="code">]</code> per residue class (mod <var class="Arg">m</var>),

</dd>
<dt>(f)</dt>
<dd>of <var class="Arg">R</var> = \(ℤ_{(\pi)}\) with modulus <code class="code">Length(<var class="Arg">coeffs</var>)</code> and coefficients <var class="Arg">coeffs</var> (the set of primes \(\pi\) which denotes the underlying ring is passed as argument <var class="Arg">pi</var>),

</dd>
<dt>(g)</dt>
<dd>of <var class="Arg">R</var> = GF(<var class="Arg">q</var>)[<var class="Arg">x</var>] with modulus <var class="Arg">m</var> and coefficients <var class="Arg">coeffs</var>,

</dd>
<dt>(h)</dt>
<dd>an rcwa permutation which induces a bijection between the partitions <var class="Arg">P1</var> and <var class="Arg">P2</var> of <var class="Arg">R</var> into residue classes and which is affine on the elements of <var class="Arg">P1</var>,

</dd>
<dt>(i)</dt>
<dd>an rcwa permutation with <q>residue class cycles</q> given by a list <var class="Arg">cycles</var> of lists of pairwise disjoint residue classes, each of which it permutes cyclically, or

</dd>
<dt>(j)</dt>
<dd>the rcwa permutation of ℤ given by the arithmetical expression <var class="Arg">expression</var> -- a string consisting of class transpositions (e.g. <code class="code">"(0(2),1(4))"</code>) or cycles permuting residue classes (e.g. <code class="code">"(0(2),1(8),3(4),5(8))"</code>), class shifts (e.g. <code class="code">"cs(4(6))"</code>, class reflections (e.g. <code class="code">"cr(3(4))"</code>), arithmetical operators (<code class="code">"*"</code>, <code class="code">"/"</code> and <code class="code">"^"</code>) and brackets (<code class="code">"("</code>, <code class="code">")"</code>),

</dd>
</dl>
respectively. The methods for the operation <code class="code">RcwaMapping</code> perform a number of argument checks, which can be skipped by using <code class="code">RcwaMappingNC</code> instead.

<div class="example"><pre>

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

</pre></div>

Rcwa mappings of ℤ can be <q>translated</q> to rcwa mappings of some semilocalization \(ℤ_{(\pi)}\) of ℤ:

<a id="X7F1A559387D0226E" name="X7F1A559387D0226E"></a>

<h5>2.2-6 LocalizedRcwaMapping</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalizedRcwaMapping</code>( <var class="Arg">f</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemilocalizedRcwaMapping</code>( <var class="Arg">f</var>, <var class="Arg">pi</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the rcwa mapping of \(ℤ_{(p)}\) respectively \(ℤ_{(\pi)}\) with the same coefficients as the rcwa mapping <var class="Arg">f</var> of ℤ.

The argument <var class="Arg">p</var> or <var class="Arg">pi</var> must be a prime or a set of primes, respectively. The argument <var class="Arg">f</var> must be an rcwa mapping of ℤ whose modulus is a power of <var class="Arg">p</var>, or whose modulus has only prime divisors which lie in <var class="Arg">pi</var>, respectively.

<div class="example"><pre>

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 ]

</pre></div>

Rcwa mappings can be <code class="code">View</code>ed, <code class="code">Display</code>ed, <code class="code">Print</code>ed and written to a <code class="code">String</code>. The output of the <code class="code">View</code> 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 <code class="code">CycleNotation</code>, <code class="code">AsClassMapping</code>, <code class="code">PrintNotation</code> and <code class="code">AbridgedNotation</code> to take influence on how certain rcwa mappings are shown. These options can either be not set, set to <code class="code">true</code> or set to <code class="code">false</code>. If the option <code class="code">CycleNotation</code> is set, it is tried harder to write down an rcwa permutation of ℤ of finite order as a product of disjoint residue class cycles, if this is possible. If the option <code class="code">AsClassMapping</code> is set, <code class="code">Display</code> shows which residue classes are mapped to which by the affine partial mappings, and marks any loops. The option <code class="code">PrintNotation</code> influences the output in favour of GAP - readability, and the option <code class="code">AbridgedNotation</code> can be used to abridge longer names like <code class="code">ClassShift</code>, <code class="code">ClassReflection</code> etc.. By default, the output of the methods for <code class="code">Display</code> and <code class="code">Print</code> describes an rcwa mapping in full. The <code class="code">Print</code>ed representation of an rcwa mapping is GAP - readable if and only if the <code class="code">Print</code>ed representation of the elements of the underlying ring is so.

There is also an operation <code class="code">LaTeXStringRcwaMapping</code>, which takes as argument an rcwa mapping and returns a corresponding LaTeX string. The output makes use of the LaTeX macro package amsmath. If the option <var class="Arg">Factorization</var> is set and the argument is bijective, a factorization into class shifts, class reflections, class transpositions and prime switches is printed (cf. <code class="func">FactorizationIntoCSCRCT</code> (<a href="chap2_mj.html#X853885A182EC5104">2.5-2</a>)). For rcwa mappings with modulus greater than 1, an indentation by <var class="Arg">Indentation</var> characters can be obtained by setting this option value accordingly.

<div class="example"><pre>

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}

</pre></div>

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

<a id="X78E796B8824C4FC8" name="X78E796B8824C4FC8"></a>

<h4>2.3 Basic arithmetic for residue-class-wise affine mappings</h4>

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 <code class="code">*</code>. Rcwa permutations can also be inverted, thus there is a method for <code class="code">Inverse</code>. 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 <code class="code">IsTame</code>. There is a method for <code class="code">Order</code>, which can not only compute a finite order, but which can also detect infinite order.

<div class="example"><pre>

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>

</pre></div>

There are methods installed for <code class="code">IsInjective</code>, <code class="code">IsSurjective</code>, <code class="code">IsBijective</code> and <code class="code">Image</code>.

<div class="example"><pre>

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

</pre></div>

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 <code class="code">^</code>, the same symbol as used for exponentiation and conjugation. The same works for partitions of the source into a finite number of residue classes.

<div class="example"><pre>

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) ]

</pre></div>

For computing preimages of elements under rcwa mappings, there are methods for <code class="code">PreImageElm</code> and <code class="code">PreImagesElm</code>. 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 <code class="code">PreImage</code>.

<div class="example"><pre>

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 ]

</pre></div>

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

<div class="example"><pre>

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

</pre></div>

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 <code class="code">+</code> and <code class="code">*</code>.

<div class="example"><pre>

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

</pre></div>

There are operations <code class="code">Modulus</code> (abbreviated <code class="code">Mod</code>) and <code class="code">Coefficients</code> for retrieving the modulus and the coefficient list of an rcwa mapping. The meaning of the return values is as described in Section <a href="chap2_mj.html#X86BC55648302D643">2.2</a>.

General documentation for most operations mentioned in this section can be found in the GAP reference manual. For rcwa mappings of rings other than ℤ, not for all operations applicable methods are available.

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

<a id="X7C16D22C7BD40FDC" name="X7C16D22C7BD40FDC"></a>

<h4>2.4 
 Attributes and properties of residue-class-wise affine mappings
</h4>

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 <code class="code">Multiplier</code> and <code class="code">Divisor</code>, or for short <code class="code">Mult</code> and <code class="code">Div</code>. The prime set 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 <code class="code">PrimeSet</code>. The maximal shift of an rcwa mapping of ℤ is the maximum of the absolute values of its coefficients \(b_{r(m)}\) in the notation introduced in Section <a href="chap2_mj.html#X78ED07E37FC2BD46">2.1</a>. It is stored as an attribute <code class="code">MaximalShift</code>. An rcwa mapping is called class-wise translating if all of its affine partial mappings are translations, it is called integral if its divisor equals 1, and it is called balanced if its multiplier and its divisor have the same prime divisors. A class-wise translating mapping has the property <code class="code">IsClassWiseTranslating</code>, an integral mapping has the property <code class="code">IsIntegral</code> and a balanced mapping has the property <code class="code">IsBalanced</code>. An rcwa mapping of the ring of integers or of one of its semilocalizations is called class-wise order-preserving if and only if all coefficients \(a_{r(m)}\) (cf. Section <a href="chap2_mj.html#X78ED07E37FC2BD46">2.1</a>) in the numerators of the affine partial mappings are positive. The corresponding property is <code class="code">IsClassWiseOrderPreserving</code>. An rcwa mapping of ℤ is called sign-preserving if it does not map nonnegative integers to negative integers or vice versa. The corresponding property is <code class="code">IsSignPreserving</code>. All elements of the simple group CT(ℤ) generated by the set of all class transpositions are sign-preserving.

<div class="example"><pre>

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

</pre></div>

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

<a id="X7C21406085B69C30" name="X7C21406085B69C30"></a>

<h5>2.4-1 LargestSourcesOfAffineMappings</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LargestSourcesOfAffineMappings</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: the coarsest partition of <code class="code">Source(<var class="Arg">f</var>)</code> on whose elements the rcwa mapping <var class="Arg">f</var> is affine.

<div class="example"><pre>

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) ]

</pre></div>

<a id="X7D6D0F2783AD02F4" name="X7D6D0F2783AD02F4"></a>

<h5>2.4-2 FixedPointsOfAffinePartialMappings</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FixedPointsOfAffinePartialMappings</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: a list of the sets of fixed points of the affine partial mappings of the rcwa mapping <var class="Arg">f</var> in the quotient field of its source.

The returned list contains entries for the restrictions of <var class="Arg">f</var> to all residue classes modulo <code class="code">Mod(<var class="Arg">f</var>)</code>. A list entry can either be an empty set, the source of <var class="Arg">f</var> or a set of cardinality 1. The ordering of the entries corresponds to the ordering of the residues in <code class="code">AllResidues(Source(<var class="Arg">f</var>),<var class="Arg">m</var>)</code>.

<div class="example"><pre>

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 ] ] ]

</pre></div>

<a id="X7A2E308C860B46E3" name="X7A2E308C860B46E3"></a>

<h5>2.4-3 Multpk</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Multpk</code>( <var class="Arg">f</var>, <var class="Arg">p</var>, <var class="Arg">k</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: the union of the residue classes \(r(m)\) such that \(p^k||a_{r(m)}\) if \(k \geq 0\), and the union of the residue classes \(r(m)\) such that \(p^k||c_{r(m)}\) if \(k \leq 0\). In this context, \(m\) denotes the modulus of <var class="Arg">f</var>, and \(a_{r(m)}\) and \(c_{r(m)}\) denote the coefficients of <var class="Arg">f</var> as introduced in Section <a href="chap2_mj.html#X78ED07E37FC2BD46">2.1</a>.

<div class="example"><pre>

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 ]

</pre></div>

There are attributes <code class="code">ClassWiseOrderPreservingOn</code>, <code class="code">ClassWiseConstantOn</code> and <code class="code">ClassWiseOrderReversingOn</code> which store the union of the residue classes (mod <code class="code">Mod(<var class="Arg">f</var>)</code>) on which an rcwa mapping <var class="Arg">f</var> of ℤ or of a semilocalization thereof is class-wise order-preserving, class-wise constant or class-wise order-reversing, respectively.

<div class="example"><pre>

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

</pre></div>

Also there are attributes <code class="code">ShiftsUpOn</code> and <code class="code">ShiftsDownOn</code> which store the union of the residue classes (mod <code class="code">Mod(<var class="Arg">f</var>)</code>) on which an rcwa mapping <var class="Arg">f</var> of ℤ induces affine mappings \(n \mapsto n + c\) for \(c > 0\), respectively, \(c < 0\).

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

<a id="X7B1E53127D9AE52F" name="X7B1E53127D9AE52F"></a>

<h5>2.4-4 Determinant</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Determinant</code>( <var class="Arg">f</var> )</td><td class="tdright">( method )</td></tr></table></div>
Returns: the determinant of the rcwa mapping <var class="Arg">f</var> of ℤ.

The determinant of an affine mapping \(n \mapsto (an+b)/c\) whose source is a residue class \(r(m)\) is defined by \(b/|a|m\). This definition is extended additively to determinants of rcwa mappings.

Let \(f\) be an rcwa mapping of the integers, and let \(m\) denote its modulus. Using the notation \(f|_{r(m)}: n \mapsto (a_{r(m)} \cdot n + b_{r(m)})/c_{r(m)}\) for the affine partial mappings, the determinant det(\(f\)) of \(f\) is given by

<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>

The determinant mapping is an epimorphism from the group of all class-wise order-preserving rcwa permutations of ℤ to (ℤ,+), see <a href="chapBib_mj.html#biBKohl05">[Koh05]</a>, Theorem 2.11.9.

<div class="example"><pre>

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

</pre></div>

<a id="X8365EEEB82C946FD" name="X8365EEEB82C946FD"></a>

<h5>2.4-5 Sign</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Sign</code>( <var class="Arg">g</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: the sign of the rcwa permutation <var class="Arg">g</var> of ℤ.

Let \(\sigma\) be an rcwa permutation of the integers, and let \(m\) denote its modulus. Using the notation \(\sigma|_{r(m)}: n \mapsto (a_{r(m)} \cdot n + b_{r(m)})/c_{r(m)}\) for the affine partial mappings, the sign of \(\sigma\) is defined by

<center> <img src = "sgn.png" width = "284" height = "63" alt = "(-1)^(det(sigma) + sum_{r(m): \ a_{r(m)} < 0} (m - 2r)/m)."/> </center>

The sign mapping is an epimorphism from RCWA(ℤ) to the group \(ℤ^\times\) of units of ℤ, see <a href="chapBib_mj.html#biBKohl05">[Koh05]</a>, Theorem 2.12.8. Therefore the kernel of the sign mapping is a normal subgroup of RCWA(ℤ) of index 2. The simple group CT(ℤ) is a subgroup of this kernel.

<div class="example"><pre>

gap> List([ClassTransposition(3,4,2,6),
> ClassShift(0,3),ClassReflection(2,5)],Sign);
[ 1, -1, -1 ]

</pre></div>

<a id="X8475F844869DD060" name="X8475F844869DD060"></a>

<h4>2.5 Factoring residue-class-wise affine permutations</h4>

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