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<Chapter Label="ch:algorithms">
<Heading>The Algorithms Implemented in RCWA</Heading>
This chapter lists brief descriptions of many of the algorithms and methods
implemented in this package. These descriptions are kept very informal and
short, and some of them provide only rudimentary information. They are listed
in alphabetical order. The word <Q>trivial</Q> as a description means that
essentially nothing is done except of storing or recalling one or several
values, and <Q>straightforward</Q> means that no sophisticated algorithm
is used. Note that <Q>trivial</Q> and <Q>straightforward</Q> are to be read
as <E>mathematically</E> trivial respectively straightforward, and that the
code of a function or method attributed in this way can still be reasonably
long and complicated. Longer and better descriptions of many of the
algorithms and methods can be found in <Cite Key="Kohl07a"/>.
<List>
<Mark>
<C>ActionOnRespectedPartition(<A>G</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q> after having computed a respected partition
by <C>RespectedPartition</C>. One only needs to know how to compute
images of residue classes under affine mappings.
</Item>
<Mark>
<C>Ball(<A>G</A>,<A>g</A>,<A>r</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q>.
</Item>
<Mark>
<C>Ball(<A>G</A>,<A>p</A>,<A>r</A>,<A>act</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q>.
</Item>
<Mark>
<C>ClassPairs</C>
</Mark>
<Item>
Run over all 4-tuples, and filter by divisibility criteria,
size comparisons, ordering of the entries etc.
</Item>
<Mark>
<C>ClassReflection(<A>r</A>,<A>m</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>ClassRotation(<A>r</A>,<A>m</A>,<A>u</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>ClassShift(<A>r</A>,<A>m</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>ClassTransposition(<A>r1</A>,<A>m1</A>,<A>r2</A>,<A>m2</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>ClassWiseOrderPreservingOn(<A>f</A>)</C>, etc.
</Mark>
<Item>
Forms the union of the residue classes modulo the modulus
of <A>f</A> in whose corresponding coefficient triple
the first entry is positive, zero or negative, respectively.
</Item>
<Mark>
<C>Coefficients(<A>f</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>CommonRightInverse(<A>l</A>,<A>r</A>)</C>
</Mark>
<Item>
(See <C>RightInverse</C>.)
</Item>
<Mark>
<C>CT(<A>R</A>)</C>
</Mark>
<Item>
Attributes and properties are set according
to <Cite Key="Kohl06b"/>.
</Item>
<Mark>
<C>DecreasingOn(<A>f</A>)</C>
</Mark>
<Item>
Forms the union of the residue classes which are determined by the
coefficients as indicated.
</Item>
<Mark>
<C>DerivedSubgroup(<A>G</A>)</C>
</Mark>
<Item>
No genuine method -- &GAP; Library methods already work for tame groups.
</Item>
<Mark>
<C>Determinant(<A>g</A>)</C>
</Mark>
<Item>
Evaluation of the given expression.
For the mathematical meaning (epimorphism!), see Theorem 2.11.9
in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>DirectProduct(<A>G1</A>,<A>G2</A>, ... )</C>
</Mark>
<Item>
Restricts the groups <A>G1</A>, <A>G2</A>, ...
to disjoint residue classes. See <C>Restriction</C>
and Corollary 2.3.3 in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>Display(<A>f</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>Divisor(<A>f</A>)</C>
</Mark>
<Item>
Lcm of coefficients, as indicated.
</Item>
<Mark>
<C>DrawOrbitPicture</C>
</Mark>
<Item>
Compute spheres of radius <M>1, \dots, r</M> around the given point(s).
Choose the origin either in the lower left corner of the picture
(if all points lie in the first quadrant) or in the middle of the picture
(if they don't). Mark points of the ball with black pixels in case of
a monochrome picture. Choose colors from the given palette depending on
the distance from the starting points in case of a colored picture.
</Item>
<Mark>
<C>EpimorphismFromFpGroup(<A>G</A>,<A>r</A>)</C>
</Mark>
<Item>
If the package <Package>FR</Package> <Cite Key="FR"/> is loaded,
then use its function <C>FindGroupRelations</C> to find relations.
Otherwise proceed as follows: First compute the ball of
radius <A>r</A> around 1 in the free group whose rank is the
number of stored generators of <A>G</A>. Then compute the images
of the elements of that ball under the natural projection onto the
group <A>G</A>. Take pairs of elements of the ball whose images
coincide, and add their quotients to the set of known relations.
For images which have finite order, add the corresponding power
relations. Finally, regardless of whether <Package>FR</Package>
is present or not, simplify the finitely presented group with the
determined relations by the operation <C>IsomorphismSimplifiedFpGroup</C>
from the &GAP; Library, and return the natural epimorphism from it
to <A>G</A>.
</Item>
<Mark>
<C>Exponent(<A>G</A>)</C>
</Mark>
<Item>
Check whether <A>G</A> is finite. If it is, then use the &GAP; Library
method, applied to <C>Image(IsomorphismPermGroup(<A>G</A>))</C>.
Check whether <A>G</A> is tame. If yes, return <C>infinity</C>.
If not, run a loop over <A>G</A> until finding an element of infinite
order. Once one is found, return <C>infinity</C>. <P/>
The final loop to find a non-torsion element can be left away under
the assumption that any finitely generated wild rcwa group has a wild
element. It looks likely that this holds, but currently the author does
not know a proof.
</Item>
<Mark>
<C>FactorizationIntoCSCRCT(<A>g</A>)</C>
</Mark>
<Item>
This uses a rather sophisticated method which will likely
some time be published elsewhere. At the moment termination is not
guaranteed, but in case of termination the result is certain.
The strategy is roughly first to make the mapping class-wise
order-preserving and balanced, and then to remove all prime factors
from multiplier and divisor one after the other in decreasing order
by dividing by appropriate class transpositions.
The remaining integral mapping can be factored almost similarly
easily as a permutation of a finite set can be factored into
transpositions.
</Item>
<Mark>
<C>FactorizationOnConnectedComponents(<A>f</A>,<A>m</A>)</C>
</Mark>
<Item>
Calls <Package>GRAPE</Package> to get the connected components of the
transition graph, and then computes a partition of the suitably
<Q>blown up</Q> coefficient list corresponding to the connected
components.
</Item>
<Mark>
<C>FixedPointsOfAffinePartialMappings(<A>f</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q>.
</Item>
<Mark>
<C>GluckTaylorInvariant(<A>a</A>(</C>
</Mark>
<Item>
Evaluation of the given expression.
</Item>
<Mark>
<C>GuessedDivergence(<A>f</A>)</C>
</Mark>
<Item>
Numerical computation of the limit of some series, which seems to
converge <Q>often</Q>. Caution!!!
</Item>
<Mark>
<C>Image(<A>f</A>)</C>, <C>Image(<A>f</A>,<A>S</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q> if one can compute images of residue classes
under affine mappings and unite and intersect residue classes
(Chinese Remainder Theorem).
See Lemma 1.2.1 in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>ImageDensity(<A>f</A>)</C>
</Mark>
<Item>
Evaluation of the given expression.
</Item>
<Mark>
<C><A>g</A> in <A>G</A></C> (membership test for rcwa groups)
</Mark>
<Item>
Test whether the mapping <A>g</A> or its inverse is in the list of
generators of <A>G</A>. If it is, return <C>true</C>.
Test whether its prime set is a subset of the prime set of <A>G</A>.
If not, return <C>false</C>. Test whether the multiplier or the divisor
of <A>g</A> has a prime factor which does not divide the multiplier
of <A>G</A>. If yes, return <C>false</C>. Test if <A>G</A> is
class-wise order-preserving, and <A>g</A> is not. If so, return
<C>false</C>. Test if the sign of <A>g</A> is -1 and all generators
of <A>G</A> have sign 1. If yes, return <C>false</C>.
Test if <A>G</A> is class-wise order-preserving, all generators
of <A>G</A> have determinant 0 and <A>g</A> has
determinant <M>\neq 0</M>. If yes, return <C>false</C>.
Test whether the support of <A>g</A> is a subset of the support
of <A>G</A>. If not, return <C>false</C>.
Test whether <A>G</A> fixes the nonnegative integers setwise,
but <A>g</A> does not. If yes, return <C>false</C>. <P/>
If <A>G</A> is tame, proceed as follows:
Test whether the modulus of <A>g</A> divides the modulus of <A>G</A>.
If not, return <C>false</C>. Test whether <A>G</A> is finite and <A>g</A>
has infinite order. If so, return <C>false</C>. Test whether <A>g</A>
is tame. If not, return <C>false</C>.
Compute a respected partition <C>P</C> of <A>G</A> and the
finite permutation group <C>H</C> induced by <A>G</A> on it
(see <C>RespectedPartition</C>).
Check whether <A>g</A> permutes <C>P</C>. If not, return <C>false</C>.
Let <C>h</C> be the permutation induced by <A>g</A> on <C>P</C>.
Check whether <C>h</C> lies in <C>H</C>. If not, return <C>false</C>.
Compute an element <C>g1</C> of <A>G</A> which acts on <C>P</C>
like <A>g</A>. For this purpose, factor <A>h</A> into generators
of <C>H</C> using <C>PreImagesRepresentative</C>, and compute the
corresponding product of generators of <A>G</A>.
Let <C>k := g/g1</C>. The mapping <C>k</C> is always integral.
Compute the kernel <C>K</C> of the action of <A>G</A>
on <C>P</C> using <C>KernelOfActionOnRespectedPartition</C>. Check
whether <C>k</C> lies in <C>K</C>. This is done using the package
<Package>Polycyclic</Package> <Cite Key="Polycyclic"/>, and uses an
isomorphism from a supergroup of <C>K</C> which is isomorphic
to the <C>|P|</C>-fold direct product of the infinite dihedral group
and which always contains <C>k</C> to a polycyclically presented
group. If <C>k</C> lies in <C>K</C>, return <C>true</C>,
otherwise return <C>false</C>. <P/>
If <A>G</A> is not tame, proceed as follows:
Look for finite orbits of <A>G</A>. If some are found, test whether
<A>g</A> acts on them, and whether the induced permutations lie in
the permutation groups induced by <A>G</A>. If for one of the
examined orbits one of the latter two questions has a negative answer,
then return <C>false</C>.
Look for a positive integer <M>m</M> such that <A>g</A> does
not leave a partition of <M>\mathbb{Z}</M> into unions of residue
classes (mod <M>m</M>) invariant which is fixed by <A>G</A>.
If successful, return <C>false</C>. If not, try to factor <A>g</A> into
generators of <A>G</A> using <C>PreImagesRepresentative</C>.
If successful, return <C>true</C>. If <A>g</A> is in <A>G</A>, this
terminates after a finite number of steps. Both runtime and memory
requirements are exponential in the word length. If <A>g</A> is not
in <A>G</A> at this stage, the method runs into an infinite loop.
</Item>
<Mark>
<C><A>f</A> in <A>M</A></C> (membership test for rcwa monoids)
</Mark>
<Item>
Test whether the mapping <A>f</A> is in the list of generators
of <A>G</A>. If it is, return <C>true</C>.
Test whether the multiplier of <A>f</A> is zero, but all generators
of <A>M</A> have nonzero multiplier. If yes, return <C>false</C>.
Test if neither <A>f</A> nor any generator of <A>M</A>
has multiplier zero. If so, check whether the prime set of <A>f</A>
is a subset of the prime set of <A>M</A>, and whether the set of
prime factors of the multiplier of <A>f</A> is a subset of the union
of the sets of prime factors of the multipliers of the generators
of <A>M</A>. If one of these is not the case, return <C>false</C>.
Check whether the set of prime factors of the divisor
of <A>f</A> is a subset of the union of the sets of prime factors
of the divisors of the generators of <A>M</A>.
If not, return <C>false</C>.
If the underlying ring is <M>\mathbb{Z}</M> or a semilocalization
thereof, then check whether <A>f</A> is not class-wise order-preserving,
but <A>M</A> is. If so, return <C>false</C>. <P/>
If <A>f</A> is not injective, but all generators of <A>M</A> are,
then return <C>false</C>.
If <A>f</A> is not surjective, but all generators of <A>M</A> are,
then return <C>false</C>.
If the support of <A>f</A> is not a subset of the support
of <A>M</A>, then return <C>false</C>.
If <A>f</A> is not sign-preserving, but <A>M</A> is,
then return <C>false</C>.
Check whether <A>M</A> is tame. If so, then return <C>false</C> provided
that one of the following three conditions hold: 1. The modulus
of <A>f</A> does not divide the modulus of <A>M</A>.
2. <A>f</A> is not tame. 3. <A>M</A> is finite, and <A>f</A>
is bijective and has infinite order.
If membership has still not been decided, use <C>ShortOrbits</C> to look
for finite orbits of <A>M</A>, and check whether <A>f</A> fixes all
of them setwise. If a finite orbit is found which <A>f</A> does not map
to itself, then return <C>false</C>. <P/>
Finally compute balls of increasing radius around 1 until <A>f</A>
is found to lie in one of them. If that happens, return <C>true</C>.
If <A>f</A> is an element of <A>M</A>, this will eventually
terminate, but if at this stage <A>f</A> is not an element
of <A>M</A>, this will run into an infinite loop.
</Item>
<Mark>
<C><A>point</A> in <A>orbit</A></C> (membership test for orbits)
</Mark>
<Item>
Uses the equality test for orbits:
The orbit equality test computes balls of increasing radius around the
orbit representatives until they intersect nontrivially. Once they do so,
it returns <C>true</C>. If it finds that one or both of the orbits are
finite, it makes use of that information, and returns <C>false</C> if
appropriate. In between, i.e. after having computed balls to a certain
extent depending on the properties of the group, it chooses a suitable
modulus <M>m</M> and computes orbits (modulo <M>m</M>).
If the representatives of the orbits to be compared belong to different
orbits (mod <M>m</M>), it returns <C>false</C>. If this is not the
case although the orbits are different, the equality test runs into an
infinite loop.
</Item>
<Mark>
<C>IncreasingOn(<A>f</A>)</C>
</Mark>
<Item>
Forms the union of the residue classes which are determined by the
coefficients as indicated.
</Item>
<Mark>
<C>Index(<A>G</A>,<A>H</A>)</C>
</Mark>
<Item>
In general, i.e. if the underlying ring is not <M>\mathbb{Z}</M>,
proceed as follows: If both groups <A>G</A> and <A>H</A> are finite,
return the quotient of their orders. If <A>G</A> is infinite,
but <A>H</A> is finite, return <C>infinity</C>. Otherwise return the
number of right cosets of <A>H</A> in <A>G</A>, computed by the
&GAP; Library function <C>RightCosets</C>. <P/>
If the underlying ring is <M>\mathbb{Z}</M>, do additionally the
following before attempting to compute the list of right cosets:
If the group <A>G</A> is class-wise order-preserving, check whether one
of its generators has nonzero determinant, and whether all generators
of <A>H</A> have determinant zero.
If so, then return <C>infinity</C>.
Check whether <A>H</A> is tame, but <A>G</A> is not.
If so, then return <C>infinity</C>.
If <A>G</A> is tame, then check whether the rank of the largest free
abelian subgroup of the kernel of the action of <A>G</A> on a respected
partition is higher than the corresponding rank for <A>H</A>.
For this check, use <C>RankOfKernelOfActionOnRespectedPartition</C>.
If it is, then return <C>infinity</C>.
</Item>
<Mark>
<C>Induction(<A>g</A>,<A>f</A>)</C>
</Mark>
<Item>
Computes <C>f * g * RightInverse(<A>f</A>)</C>.
</Item>
<Mark>
<C>Induction(<A>G</A>,<A>f</A>)</C>
</Mark>
<Item>
Gets a set of generators by applying <C>Induction(<A>g</A>,<A>f</A>)</C>
to the generators <A>g</A> of <A>G</A>.
</Item>
<Mark>
<C>InjectiveAsMappingFrom(<A>f</A>)</C>
</Mark>
<Item>
The function starts with the entire source of <A>f</A> as <Q>preimage</Q>
<C>pre</C> and the empty set as <Q>image</Q> <C>im</C>.
It loops over the residue classes (mod <C>Mod(<A>f</A>)</C>).
For any such residue class <C>cl</C> the following is done: Firstly,
the image of <C>cl</C> under <A>f</A> is added to <C>im</C>.
Secondly, the intersection of the preimage of the intersection of the
image of <C>cl</C> under <A>f</A> and <C>im</C> under <A>f</A> and
<C>cl</C> is subtracted from <C>pre</C>.
</Item>
<Mark>
<C>IntegralConjugate(<A>f</A>)</C>,
<C>IntegralConjugate(<A>G</A>)</C>
</Mark>
<Item>
Uses the algorithm described in the proof of Theorem 2.5.14
in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>IntegralizingConjugator(<A>f</A>)</C>,
<C>IntegralizingConjugator(<A>G</A>)</C>
</Mark>
<Item>
Uses the algorithm described in the proof of Theorem 2.5.14
in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>Inverse(<A>f</A>)</C>
</Mark>
<Item>
Essentially inversion of affine mappings.
See Lemma 1.3.1, Part (b) in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>IsBalanced(<A>f</A>)</C>
</Mark>
<Item>
Checks whether the sets of prime factors of the multiplier and
the divisor of <A>f</A> are the same.
</Item>
<Mark>
<C>IsClassReflection(<A>g</A>)</C>
</Mark>
<Item>
Computes the support of <A>g</A>, and compares <A>g</A> with
the corresponding class reflection.
</Item>
<Mark>
<C>IsClassRotation(<A>g</A>)</C>
</Mark>
<Item>
Computes the support of <A>g</A>, extracts the possible rotation
factor from the coefficients and compares <A>g</A> with the
corresponding class rotation.
</Item>
<Mark>
<C>IsClassShift(<A>g</A>)</C>
</Mark>
<Item>
Computes the support of <A>g</A>, and compares <A>g</A> with
the corresponding class shift.
</Item>
<Mark>
<C>IsClassTransposition(<A>g</A>)</C>
</Mark>
<Item>
Computes the support of <A>g</A>, writes it as a disjoint union of
two residue classes and compares <A>g</A> with the class transposition
which interchanges them.
</Item>
<Mark>
<C>IsClassWiseOrderPreserving(<A>f</A>)</C>
</Mark>
<Item>
Tests whether the first entry of all coefficient triples is positive.
</Item>
<Mark>
<C>IsConjugate(RCWA(Integers),<A>f</A>,<A>g</A>)</C>
</Mark>
<Item>
Test whether <A>f</A> and <A>g</A> have the same order, and whether
either both or none of them is tame. If not, return <C>false</C>. <P/>
If the mappings are wild, use <C>ShortCycles</C> to search for finite
cycles not belonging to an infinite series, until their numbers for a
particular length differ. This may run into an infinite loop.
If it terminates, return <C>false</C>. <P/>
If the mappings are tame, use the method described in the proof of
Theorem 2.5.14 in <Cite Key="Kohl05"/> to construct
integral conjugates of <A>f</A> and <A>g</A>. Then essentially use
the algorithm described in the proof of Theorem 2.6.7
in <Cite Key="Kohl05"/> to compute <Q>standard representatives</Q>
of the conjugacy classes which the integral conjugates of <A>f</A>
and <A>g</A> belong to. Finally compare these standard representatives,
and return <C>true</C> if they are equal and <C>false</C> if not.
</Item>
<Mark>
<C>IsInjective(<A>f</A>)</C>
</Mark>
<Item>
See <C>Image</C>.
</Item>
<Mark>
<C>IsIntegral(<A>f</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>IsomorphismMatrixGroup(<A>G</A>)</C>
</Mark>
<Item>
Uses the algorithm described in the proof of Theorem 2.6.3
in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>IsomorphismPermGroup(<A>G</A>)</C>
</Mark>
<Item>
If the group <A>G</A> is finite and class-wise order-preserving,
use <C>ActionOnRespectedPartition</C>.
If <A>G</A> is finite, but not class-wise order-preserving, compute
the action on the respected partition which is obtained by splitting
any residue class <M>r(m)</M> in <C>RespectedPartition(<A>G</A></C>
into three residue classes <M>r(3m), r+m(3m), r+2m(3m)</M>.
If <A>G</A> is infinite, there is no isomorphism to a finite
permutation group, thus return <C>fail</C>.
</Item>
<Mark>
<C>IsomorphismRcwaGroup(<A>G</A>)</C>
</Mark>
<Item>
The method for finite groups uses <C>RcwaMapping</C>, Part (d). <P/>
The method for free products of finite groups uses the Table-Tennis
Lemma (which is also known as <E>Ping-Pong Lemma</E>,
cf. e.g. Section II.B. in <Cite Key="LaHarpe00"/>).
It uses regular permutation representations of the factors
<M>G_r</M> (<M>r = 0, \dots ,m-1</M>) of the free product on residue
classes modulo <M>n_r := |G_r|</M>.
The basic idea is that since point stabilizers in regular permutation
groups are trivial, all non-identity elements map any of the permuted
residue classes into their complements.
To get into a situation where the Table-Tennis Lemma is applicable,
the method computes conjugates of the images of the mentioned permutation
representations under bijective rcwa mappings <M>\sigma_r</M> which
satisfy <M>0(n_r)^{\sigma_r} = \mathbb{Z} \setminus r(m)</M>. <P/>
The method for free groups uses an adaptation of the construction
given on page 27 in <Cite Key="LaHarpe00"/> from
PSL(2,<M>\mathbb{C}</M>) to RCWA(<M>\mathbb{Z}</M>).
As an equivalent for the closed discs used there, the method takes
the residue classes modulo two times the rank of the free group.
</Item>
<Mark>
<C>IsPerfect(<A>G</A>)</C>
</Mark>
<Item>
If the group <A>G</A> is trivial, then return <C>true</C>.
Otherwise if it is abelian, then return <C>false</C>. <P/>
If the underlying ring is <M>\mathbb{Z}</M>, then do the following:
If one of the generators of <A>G</A> has sign -1, then
return <C>false</C>. If <A>G</A> is class-wise order-preserving
and one of the generators has nonzero determinant, then return
<C>false</C>. <P/>
If <A>G</A> is wild, and perfectness has not been decided so far,
then give up. If <A>G</A> is finite, then check the image of
<C>IsomorphismPermGroup(<A>G</A>)</C> for perfectness, and return
<C>true</C> or <C>false</C> accordingly. <P/>
If the group <A>G</A> is tame and if it acts transitively
on its stored respected partition, then return <C>true</C> or
<C>false</C> depending on whether the finite permutation group
<C>ActionOnRespectedPartition(<A>G</A>)</C> is perfect or not.
If <A>G</A> does not act transitively on its stored respected
partition, then give up.
</Item>
<Mark>
<C>IsPrimeSwitch(<A>g</A>)</C>
</Mark>
<Item>
Checks whether the multiplier of <A>g</A> is an odd prime,
and compares <A>g</A> with the corresponding prime switch.
</Item>
<Mark>
<C>IsSignPreserving(<A>f</A>)</C>
</Mark>
<Item>
If <A>f</A> is not class-wise order-preserving, then return <C>false</C>.
Otherwise let <M>c \geq 1</M> be greater than or equal to the maximum of
the absolute values of the coefficients <M>b_{r(m)}</M> of the affine
partial mappings of <A>f</A>, and check whether
the minimum of the image of <M>\{0, \dots, c\}</M> under <A>f</A>
is nonnegative and whether the maximum of the image of
<M>\{-c, \dots, -1\}</M> under <A>f</A> is negative.
If both is the case, then return <C>true</C>, otherwise
return <C>false</C>.
</Item>
<Mark>
<C>IsSolvable(<A>G</A>)</C>
</Mark>
<Item>
If <A>G</A> is abelian, then return <C>true</C>.
If <A>G</A> is tame, then return <C>true</C> or <C>false</C> depending on
whether <C>ActionOnRespectedPartition(<A>G</A>)</C> is solvable or not.
If <A>G</A> is wild, then give up.
</Item>
<Mark>
<C>IsSubset(<A>G</A>,<A>H</A>)</C> (checking for a subgroup relation)
</Mark>
<Item>
Check whether the set of stored generators of <A>H</A> is a subset
of the set of stored generators of <A>G</A>. If so, return
<C>true</C>. Check whether the prime set of <A>H</A> is a subset
of the prime set of <A>G</A>. If not, return <C>false</C>.
Check whether the support of <A>H</A> is a subset
of the support of <A>G</A>. If not, return <C>false</C>.
Check whether <A>G</A> is tame, but <A>H</A> is wild.
If so, return <C>false</C>. <P/>
If <A>G</A> and <A>H</A> are both tame, then proceed as follows:
If the multiplier of <A>H</A> does not divide the multiplier
of <A>G</A>, then return <C>false</C>.
If <A>H</A> does not respect the stored respected partition
of <A>G</A>, then return <C>false</C>.
Check whether the finite permutation group induced by <A>H</A> on
<C>RespectedPartition(<A>G</A>)</C> is a subgroup of
<C>ActionOnRespectedPartition(<A>G</A>)</C>. If yes, return <C>true</C>.
Check whether the order of <A>H</A> is greater than the order
of <A>G</A>. If so, return <C>false</C>. <P/>
Finally use the membership test to check whether all generators
of <A>H</A> lie in <A>G</A>, and return <C>true</C> or
<C>false</C> accordingly.
</Item>
<Mark>
<C>IsSurjective(<A>f</A>)</C>
</Mark>
<Item>
See <C>Image</C>.
</Item>
<Mark>
<C>IsTame(<A>G</A>)</C>
</Mark>
<Item>
Checks whether the modulus of the group is nonzero.
</Item>
<Mark>
<C>IsTame(<A>f</A>)</C>
</Mark>
<Item>
Application of the criteria given in Corollary 2.5.10
and 2.5.12 and Theorem A.8 and A.11
in <Cite Key="Kohl05"/>, as well as of the criteria given
in <Cite Key="Kohl07b"/>.
The criterion <Q>surjective, but not injective means wild</Q>
(Theorem A.8 in <Cite Key="Kohl05"/>) is the subject
of <Cite Key="Kohl06a"/>.
The package <Package>GRAPE</Package> is needed for the application
of the criterion which says that an rcwa permutation is wild if
a transition graph has a weakly-connected component which is not
strongly-connected (cf. Theorem A.11 in <Cite Key="Kohl05"/>).
</Item>
<Mark>
<C>IsTransitive(<A>G</A>,Integers)</C>
</Mark>
<Item>
Look for finite orbits, using <C>ShortOrbits</C> on a couple of
intervals. If a finite orbit is found, return <C>false</C>.
Test if <A>G</A> is finite. If yes, return <C>false</C>. <P/>
Search for an element <C>g</C> and a residue class <M>r(m)</M>
such that the restriction of <C>g</C> to <M>r(m)</M> is given
by <M>n \mapsto n + m</M>. Then the cyclic group generated by <C>g</C>
acts transitively on <M>r(m)</M>. The element <C>g</C> is searched
among the generators of <A>G</A>, its powers, its commutators,
powers of its commutators and products of few different generators.
The search for such an element may run into an infinite loop,
as there is no guarantee that the group has a suitable element. <P/>
If suitable <C>g</C> and <M>r(m)</M> are found, proceed as follows: <P/>
Put <M>S := r(m)</M>. Put <M>S := S \cup S^g</M> for all generators
<M>g</M> of <A>G</A>, and repeat this until <M>S</M> remains constant.
This may run into an infinite loop. <P/>
If it terminates: If <M>S = \mathbb{Z}</M>, return <C>true</C>,
otherwise return <C>false</C>.
</Item>
<Mark>
<C>KernelOfActionOnRespectedPartition(<A>G</A>)</C>
</Mark>
<Item>
First determine the abelian invariants of the kernel <C>K</C>.
For this, compute sufficiently many quotients of orders of
permutation groups induced by <A>G</A> on refinements of
the stored respected partition <C>P</C> by the order of the
permutation group induced by <A>G</A> on <C>P</C> itself.
Then use a random walk through the group <A>G</A>.
Compute powers of elements encountered along the way which
fix <C>P</C>.
Translate these kernel elements into elements of a polycyclically
presented group isomorphic to the <C>|P|</C>-fold direct product of
the infinite dihedral group (<C>K</C> certainly embeds into this group).
Use <Package>Polycyclic</Package> <Cite Key="Polycyclic"/>
to collect independent <Q>nice</Q> generators of <C>K</C>.
Proceed until the permutation groups induced by <C>K</C> on the
refined respected partitions all equal the initially stored quotients.
</Item>
<Mark>
<C>LargestSourcesOfAffineMappings(<A>f</A>)</C>
</Mark>
<Item>
Forms unions of residue classes modulo the modulus of the mapping,
whose corresponding coefficient triples are equal.
</Item>
<Mark>
<C>LaTeXObj(<A>f</A>)</C>
</Mark>
<Item>
Collects residue classes those corresponding coefficient triples
are equal.
</Item>
<Mark>
<C>LikelyContractionCentre(<A>f</A>,<A>maxn</A>,<A>bound</A>)</C>
</Mark>
<Item>
Computes trajectories with starting values from a given interval, until
a cycle is reached. Aborts if the trajectory exceeds the prescribed
bound. Form the union of the detected cycles.
</Item>
<Mark>
<C>LocalizedRcwaMapping(<A>f</A>,<A>p</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>Loops(<A>f</A>)</C>
</Mark>
<Item>
Runs over the residue classes modulo the modulus of <A>f</A>,
and selects those of them which <A>f</A> does not map to themselves,
but which intersect nontrivially with their images under <A>f</A>.
</Item>
<Mark>
<C>mKnot(<A>m</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q>, following the definition given in
<Cite Key="Keller99"/>.
</Item>
<Mark>
<C>Modulus(<A>G</A>)</C>
</Mark>
<Item>
Searches for a wild element in the group.
If unsuccessful, tries to construct a respected partition
(see <C>RespectedPartition</C>).
</Item>
<Mark>
<C>Modulus(<A>f</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>MovedPoints(<A>G</A>)</C>
</Mark>
<Item>
Needs only forming unions of residue classes and determining
fixed points of affine mappings.
</Item>
<Mark>
<C>Multiplier(<A>f</A>)</C>
</Mark>
<Item>
Lcm of coefficients, as indicated.
</Item>
<Mark>
<C>Multpk(<A>f</A>,<A>p</A>,<A>k</A>)</C>
</Mark>
<Item>
Forms the union of the residue classes modulo the modulus of the
mapping, which are determined by the given divisibility criteria
for the coefficients of the corresponding affine mapping.
</Item>
<Mark>
<C>NrConjugacyClassesOfRCWAZOfOrder(<A>ord</A>)</C>
</Mark>
<Item>
The class numbers are taken from Corollary 2.7.1
in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>Orbit(<A>G</A>,<A>pnt</A>,<A>gens</A>,<A>acts</A>,<A>act</A>)</C>
</Mark>
<Item>
Check if the orbit has length less than a certain bound.
If so, then return it as a list.
Otherwise test whether the group <A>G</A> is tame or wild. <P/>
If <A>G</A> is tame, then test whether <A>G</A> is finite.
If yes, then compute the orbit by the &GAP; Library method.
Otherwise proceed as follows:
Compute a respected partition <M>\mathcal{P}</M> of <A>G</A>.
Use <M>\mathcal{P}</M> to find a residue class <M>r(m)</M>
which is a subset of the orbit to be computed. In general, <M>r(m)</M>
will not be one of the residue classes in <M>\mathcal{P}</M>, but
a subset of one of them.
Put <M>\Omega := r(m)</M>. Unite the set <M>\Omega</M> with its
images under all the generators of <A>G</A> and their inverses.
Repeat that until <M>\Omega</M> does not change any more.
Return <M>\Omega</M>. <P/>
If <A>G</A> is wild, then return an orbit object which stores the
group <A>G</A>, the representative <A>rep</A> and the
action <A>act</A>.
</Item>
<Mark>
<C>OrbitsModulo(<A>f</A>,<A>m</A>)</C>
</Mark>
<Item>
Uses <Package>GRAPE</Package> to compute the connected
components of the transition graph.
</Item>
<Mark>
<C>OrbitsModulo(<A>G</A>,<A>m</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q>.
</Item>
<Mark>
<C>Order(<A>f</A>)</C>
</Mark>
<Item>
Test for <C>IsTame</C>.
If the mapping is not tame, then return <C>infinity</C>.
Otherwise use Corollary 2.5.10 in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>PreImage(<A>f</A>,<A>S</A>)</C>
</Mark>
<Item>
See <C>Image</C>.
</Item>
<Mark>
<C>PreImagesRepresentative(<A>phi</A>,<A>g</A>)</C>,
<C>PreImagesRepresentatives(<A>phi</A>,<A>g</A>)</C>
</Mark>
<Item>
As indicated in the documentation of these methods.
The underlying idea to successively compute two balls around 1
and <A>g</A> until they intersect nontrivially is standard
in computational group theory. For rcwa groups it would mean wasting
both memory and runtime to actually compute group elements.
Thus only images of tuples of points are computed and stored.
</Item>
<Mark>
<C>PrimeSet(<A>f</A>)</C>,
<C>PrimeSet(<A>G</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q>.
</Item>
<Mark>
<C>PrimeSwitch(<A>p</A>)</C>
</Mark>
<Item>
Multiplication of rcwa mappings as indicated.
</Item>
<Mark>
<C>Print(<A>f</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C><A>f</A>*<A>g</A></C>
</Mark>
<Item>
Essentially composition of affine mappings.
See Lemma 1.3.1, Part (a) in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>Projections(<A>G</A>,<A>m</A>)</C>
</Mark>
<Item>
Use <C>OrbitsModulo</C> to determine the supports of the images of the
epimorphisms to be determined, and use <C>RestrictedPerm</C> to compute
the images of the generators of <A>G</A> under these epimorphisms.
</Item>
<Mark>
<C>Random(RCWA(Integers))</C>
</Mark>
<Item>
Computes a product of <Q>randomly</Q> chosen class shifts,
class reflections and class transpositions. This seems to
be suitable for generating reasonably good examples.
</Item>
<Mark>
<C>RankOfKernelOfActionOnRespectedPartition(<A>G</A>)</C>
</Mark>
<Item>
This performs basically the first part of the computations done by
<C>KernelOfActionOnRespectedPartition</C>.
</Item>
<Mark>
<C>RCWA(<A>R</A>)</C>
</Mark>
<Item>
Attributes and properties are set according to Theorem 2.1.1,
Theorem 2.1.2, Corollary 2.1.6 and Theorem 2.12.8
in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>RcwaGroupByPermGroup(<A>G</A>)</C>
</Mark>
<Item>
Uses <C>RcwaMapping</C>, Part (d).
</Item>
<Mark>
<C>RcwaMapping</C>
</Mark>
<Item>
(a)-(c): <Q>trivial</Q>,
(d): <C>n&circum;perm - n</C> for determining the coefficients,
(e): <Q>affine mappings by values at two given points</Q>,
(f) and (g): <Q>trivial</Q>,
(h) and (i): correspond to Lemma 2.1.4 in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>RepresentativeAction(<A>G</A>,<A>src</A>,<A>dest</A>,<A>act</A>)</C>,
<C>RepresentativeActionPreImage</C>
</Mark>
<Item>
As indicated in the documentation of these methods.
The underlying idea to successively compute two balls around
<A>src</A> and <A>dest</A> until they intersect nontrivially
is standard in computational group theory. Words standing for
products of generators of <A>G</A> are stored for any image
of <A>src</A> or <A>dest</A>.
</Item>
<Mark>
<C>RepresentativeAction(RCWA(Integers),<A>P1</A>,<A>P2</A>)</C>
</Mark>
<Item>
Arbitrary mapping: see Lemma 2.1.4 in <Cite Key="Kohl05"/>.
Tame mapping: see proof of Theorem 2.8.9
in <Cite Key="Kohl05"/>. The former is almost trivial, while the
latter is a bit complicated and takes usually also much more time.
</Item>
<Mark>
<C>RepresentativeAction(RCWA(Integers),<A>f</A>,<A>g</A>)</C>
</Mark>
<Item>
The algorithm used by <C>IsConjugate</C> constructs actually also
an element <C>x</C> such that <C><A>f</A>&circum;x = <A>g</A></C>.
</Item>
<Mark>
<C>RespectedPartition(<A>f</A>)</C>,
<C>RespectedPartition(<A>G</A>)</C>
</Mark>
<Item>
Uses the algorithm described in the proof of Theorem 2.5.8
in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>RespectsPartition(<A>G</A>,<A>P</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q>.
</Item>
<Mark>
<C>RestrictedPerm(<A>g</A>,<A>S</A></C>
</Mark>
<Item>
<Q>Straightforward</Q>.
</Item>
<Mark>
<C>Restriction(<A>g</A>,<A>f</A>)</C>
</Mark>
<Item>
Computes the action of <C>RightInverse(<A>f</A>) * g * f</C> on the
image of <A>f</A>.
</Item>
<Mark>
<C>Restriction(<A>G</A>,<A>f</A>)</C>
</Mark>
<Item>
Gets a set of generators by applying
<C>Restriction(<A>g</A>,<A>f</A>)</C>
to the generators <A>g</A> of <A>G</A>.
</Item>
<Mark>
<C>RightInverse(<A>f</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q> if one knows how to compute images of residue
classes under affine mappings, and how to compute inverses of
affine mappings.
</Item>
<Mark>
<C>Root(<A>f</A>,<A>k</A>)</C>
</Mark>
<Item>
If <A>f</A> is bijective, class-wise order-preserving and has
finite order: <P/>
Find a conjugate of <A>f</A> which is a product of class transpositions.
Slice cycles <M>\prod_{i=2}^l \tau_{r_1(m_1),r_i(m_i)}</M>
of <A>f</A> a respected partition <M>\mathcal{P}</M> into cycles
<M>\prod_{i=1}^l \prod_{j=0}^{k-1} \tau_{r_1(km_1),r_i+jm_i(km_i)}</M>
of the <A>k</A>-fold
length on the refined partition which one gets from <M>\mathcal{P}</M>
by decomposing any <M>r_i(m_i) \in \mathcal{P}</M> into residue classes
(mod <M>km_i</M>). Finally conjugate the resulting permutation
back. <P/>
Other cases seem to be more difficult and are currently not covered.
</Item>
<Mark>
<C>RotationFactor(<A>g</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>SemilocalizedRcwaMapping(<A>f</A>,<A>pi</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>ShortCycles(<A>f</A>,<A>maxlng</A>)</C>
</Mark>
<Item>
Looks for fixed points of affine partial mappings of powers
of <A>f</A>.
</Item>
<Mark>
<C>ShortOrbits(<A>G</A>,<A>S</A>,<A>maxlng</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q>.
</Item>
<Mark>
<C>Sign(<A>g</A>)</C>
</Mark>
<Item>
Evaluation of the given expression.
For the mathematical meaning (epimorphism!), see Theorem 2.12.8
in <Cite Key="Kohl05"/>.
</Item>
<Mark>
<C>Sinks(<A>f</A>)</C>
</Mark>
<Item>
Computes the strongly connected components of the transition graph
by the function
<C>STRONGLY&uscore;CONNECTED&uscore;COMPONENTS&uscore;DIGRAPH</C>,
and selects those which are proper subsets of their preimages and
proper supersets of their images under <A>f</A>.
</Item>
<Mark>
<C>Size(<A>G</A>)</C> (order of an rcwa group)
</Mark>
<Item>
Test whether one of the generators of the group <A>G</A> has
infinite order. If so, return <C>infinity</C>.
Test whether the group <A>G</A> is tame. If not, return <C>infinity</C>.
Test whether <C>RankOfKernelOfActionOnRespectedPartition(<A>G</A>)</C>
is nonzero. If so, return <C>infinity</C>.
Otherwise if <A>G</A> is class-wise order-preserving, return the size
of the permutation group induced on the stored respected partition.
If <A>G</A> is not class-wise order-preserving, return the size
of the permutation group induced on the refinement of the stored
respected partition which is obtained by splitting each residue class
into three residue classes with equal moduli.
</Item>
<Mark>
<C>Size(<A>M</A>)</C> (order of an rcwa monoid)
</Mark>
<Item>
Check whether <A>M</A> is in fact an rcwa group. If so, use the method
for rcwa groups instead. Check whether one of the generators
of <A>M</A> is surjective, but not injective. If so, return
<C>infinity</C>. Check whether for all generators <M>f</M>
of <A>M</A>, the image of the union of the loops of <M>f</M>
under <M>f</M> is finite. If not, return <C>infinity</C>.
Check whether one of the generators of <A>M</A> is bijective and
has infinite order. If so, return <C>infinity</C>.
Check whether one of the generators of <A>M</A> is wild.
If so, return <C>infinity</C>.
Apply the above criteria to the elements of the ball of radius 2
around 1, and return <C>infinity</C> if appropriate.
Finally attempt to compute the list of elements of <A>M</A>.
If this is successful, return the length of the resulting list.
</Item>
<Mark>
<C>Sources(<A>f</A>)</C>
</Mark>
<Item>
Computes the strongly connected components of the transition graph
by the function
<C>STRONGLY&uscore;CONNECTED&uscore;COMPONENTS&uscore;DIGRAPH</C>,
and selects those which are proper supersets of their preimages and
proper subsets of their images under <A>f</A>.
</Item>
<Mark>
<C>SplittedClassTransposition(<A>ct</A>,<A>k</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q>.
</Item>
<Mark>
<C>StructureDescription(<A>G</A>)</C>
</Mark>
<Item>
This method uses a combination of techniques to obtain some basic
information on the structure of an rcwa group.
The returned description reflects the way the group has been built
(<C>DirectProduct</C>, <C>WreathProduct</C>, etc.).
</Item>
<Mark>
<C><A>f</A>+<A>g</A></C>
</Mark>
<Item>
Pointwise addition of affine mappings.
</Item>
<Mark>
<C>Support(<A>G</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q>.
</Item>
<Mark>
<C>Trajectory(<A>f</A>,<A>n</A>,...)</C>
</Mark>
<Item>
Iterated application of an rcwa mapping.
In the methods computing <Q>accumulated coefficients</Q>,
additionally composition of affine mappings.
</Item>
<Mark>
<C>TransitionGraph(<A>f</A>,<A>m</A>)</C>
</Mark>
<Item>
<Q>Straightforward</Q> -- just check a sufficiently long interval.
</Item>
<Mark>
<C>TransitionMatrix(<A>f</A>,<A>m</A>)</C>
</Mark>
<Item>
Evaluation of the given expression.
</Item>
<Mark>
<C>TransposedClasses(<A>g</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>View(<A>f</A>)</C>
</Mark>
<Item>
<Q>Trivial</Q>.
</Item>
<Mark>
<C>WreathProduct(<A>G</A>,<A>P</A>)</C>
</Mark>
<Item>
Uses <C>DirectProduct</C> to embed the <C>DegreeAction(<A>P</A>)</C>th
direct power of <A>G</A>, and <C>RcwaMapping</C>, Part (d)
to embed the finite permutation group <A>P</A>.
</Item>
<Mark>
<C>WreathProduct(<A>G</A>,<A>Z</A>)</C>
</Mark>
<Item>
Restricts <A>G</A> to the residue class 3(4), and encodes the
generator of <A>Z</A> as <M>\tau_{0(2),1(2)} \cdot
\tau_{0(2),1(4)}</M>. It is used that the images of 3(4) under
powers of this mapping are pairwise disjoint residue classes.
</Item>
</List>
</Chapter>
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