The Algorithms Implemented in RCWA

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 trivial as a description means that essentially nothing is done except of storing or recalling one or several values, and straightforward means that no sophisticated algorithm is used. Note that trivial and straightforward are to be read as mathematically 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 .

ActionOnRespectedPartition(G)
Straightforward
after having computed a respected partition by RespectedPartition. One only needs to know how to compute images of residue classes under affine mappings.
Ball(G,g,r)
Straightforward
.
Ball(G,p,r,act)
Straightforward
.
ClassPairs
Run over all 4-tuples, and filter by divisibility criteria, size comparisons, ordering of the entries etc.
ClassReflection(r,m)
Trivial
.
ClassRotation(r,m,u)
Trivial
.
ClassShift(r,m)
Trivial
.
ClassTransposition(r1,m1,r2,m2)
Trivial
.
ClassWiseOrderPreservingOn(f)
, etc.
Forms the union of the residue classes modulo the modulus of f in whose corresponding coefficient triple the first entry is positive, zero or negative, respectively.
Coefficients(f)
Trivial
.
CommonRightInverse(l,r)
(See RightInverse.)
CT(R)
Attributes and properties are set according to .
DecreasingOn(f)
Forms the union of the residue classes which are determined by the coefficients as indicated.
DerivedSubgroup(G)
No genuine method -- &GAP; Library methods already work for tame groups.
Determinant(g)
Evaluation of the given expression. For the mathematical meaning (epimorphism!), see Theorem 2.11.9 in .
DirectProduct(G1,G2, ... )
Restricts the groups G1, G2, ... to disjoint residue classes. See Restriction and Corollary 2.3.3 in .
Display(f)
Trivial
.
Divisor(f)
Lcm of coefficients, as indicated.
DrawOrbitPicture
Compute spheres of radius 1, \dots, r 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.
EpimorphismFromFpGroup(G,r)
If the package `FR` is loaded, then use its function FindGroupRelations to find relations. Otherwise proceed as follows: First compute the ball of radius r around 1 in the free group whose rank is the number of stored generators of G. Then compute the images of the elements of that ball under the natural projection onto the group G. 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 `FR` is present or not, simplify the finitely presented group with the determined relations by the operation IsomorphismSimplifiedFpGroup from the &GAP; Library, and return the natural epimorphism from it to G.
Exponent(G)
Check whether G is finite. If it is, then use the &GAP; Library method, applied to Image(IsomorphismPermGroup(G)). Check whether G is tame. If yes, return infinity. If not, run a loop over G until finding an element of infinite order. Once one is found, return infinity.

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.
FactorizationIntoCSCRCT(g)
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.
FactorizationOnConnectedComponents(f,m)
Calls `GRAPE` to get the connected components of the transition graph, and then computes a partition of the suitably blown up coefficient list corresponding to the connected components.
FixedPointsOfAffinePartialMappings(f)
Straightforward
.
GluckTaylorInvariant(a(
Evaluation of the given expression.
GuessedDivergence(f)
Numerical computation of the limit of some series, which seems to converge often. Caution!!!
Image(f)
, Image(f,S)
Straightforward
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 .
ImageDensity(f)
Evaluation of the given expression.
g in G
(membership test for rcwa groups)
Test whether the mapping g or its inverse is in the list of generators of G. If it is, return true. Test whether its prime set is a subset of the prime set of G. If not, return false. Test whether the multiplier or the divisor of g has a prime factor which does not divide the multiplier of G. If yes, return false. Test if G is class-wise order-preserving, and g is not. If so, return false. Test if the sign of g is -1 and all generators of G have sign 1. If yes, return false. Test if G is class-wise order-preserving, all generators of G have determinant 0 and g has determinant \neq 0. If yes, return false. Test whether the support of g is a subset of the support of G. If not, return false. Test whether G fixes the nonnegative integers setwise, but g does not. If yes, return false.

If G is tame, proceed as follows: Test whether the modulus of g divides the modulus of G. If not, return false. Test whether G is finite and g has infinite order. If so, return false. Test whether g is tame. If not, return false. Compute a respected partition P of G and the finite permutation group H induced by G on it (see RespectedPartition). Check whether g permutes P. If not, return false. Let h be the permutation induced by g on P. Check whether h lies in H. If not, return false. Compute an element g1 of G which acts on P like g. For this purpose, factor h into generators of H using PreImagesRepresentative, and compute the corresponding product of generators of G. Let k := g/g1. The mapping k is always integral. Compute the kernel K of the action of G on P using KernelOfActionOnRespectedPartition. Check whether k lies in K. This is done using the package `Polycyclic` , and uses an isomorphism from a supergroup of K which is isomorphic to the |P|-fold direct product of the infinite dihedral group and which always contains k to a polycyclically presented group. If k lies in K, return true, otherwise return false.

If G is not tame, proceed as follows: Look for finite orbits of G. If some are found, test whether g acts on them, and whether the induced permutations lie in the permutation groups induced by G. If for one of the examined orbits one of the latter two questions has a negative answer, then return false. Look for a positive integer m such that g does not leave a partition of \Z into unions of residue classes (mod m) invariant which is fixed by G. If successful, return false. If not, try to factor g into generators of G using PreImagesRepresentative. If successful, return true. If g is in G, this terminates after a finite number of steps. Both runtime and memory requirements are exponential in the word length. If g is not in G at this stage, the method runs into an infinite loop.
f in M
(membership test for rcwa monoids)
Test whether the mapping f is in the list of generators of G. If it is, return true. Test whether the multiplier of f is zero, but all generators of M have nonzero multiplier. If yes, return false. Test if neither f nor any generator of M has multiplier zero. If so, check whether the prime set of f is a subset of the prime set of M, and whether the set of prime factors of the multiplier of f is a subset of the union of the sets of prime factors of the multipliers of the generators of M. If one of these is not the case, return false. Check whether the set of prime factors of the divisor of f is a subset of the union of the sets of prime factors of the divisors of the generators of M. If not, return false. If the underlying ring is \Z or a semilocalization thereof, then check whether f is not class-wise order-preserving, but M is. If so, return false.

If f is not injective, but all generators of M are, then return false. If f is not surjective, but all generators of M are, then return false. If the support of f is not a subset of the support of M, then return false. If f is not sign-preserving, but M is, then return false. Check whether M is tame. If so, then return false provided that one of the following three conditions hold: 1. The modulus of f does not divide the modulus of M. 2. f is not tame. 3. M is finite, and f is bijective and has infinite order. If membership has still not been decided, use ShortOrbits to look for finite orbits of M, and check whether f fixes all of them setwise. If a finite orbit is found which f does not map to itself, then return false.

Finally compute balls of increasing radius around 1 until f is found to lie in one of them. If that happens, return true. If f is an element of M, this will eventually terminate, but if at this stage f is not an element of M, this will run into an infinite loop.
point in orbit
(membership test for orbits)
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 true. If it finds that one or both of the orbits are finite, it makes use of that information, and returns false 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 and computes orbits (modulo m). If the representatives of the orbits to be compared belong to different orbits (mod m), it returns false. If this is not the case although the orbits are different, the equality test runs into an infinite loop.
IncreasingOn(f)
Forms the union of the residue classes which are determined by the coefficients as indicated.
Index(G,H)
In general, i.e. if the underlying ring is not \Z, proceed as follows: If both groups G and H are finite, return the quotient of their orders. If G is infinite, but H is finite, return infinity. Otherwise return the number of right cosets of H in G, computed by the &GAP; Library function RightCosets.

If the underlying ring is \Z, do additionally the following before attempting to compute the list of right cosets: If the group G is class-wise order-preserving, check whether one of its generators has nonzero determinant, and whether all generators of H have determinant zero. If so, then return infinity. Check whether H is tame, but G is not. If so, then return infinity. If G is tame, then check whether the rank of the largest free abelian subgroup of the kernel of the action of G on a respected partition is higher than the corresponding rank for H. For this check, use RankOfKernelOfActionOnRespectedPartition. If it is, then return infinity.
Induction(g,f)
Computes f * g * RightInverse(f).
Induction(G,f)
Gets a set of generators by applying Induction(g,f) to the generators g of G.
InjectiveAsMappingFrom(f)
The function starts with the entire source of f as preimage pre and the empty set as image im. It loops over the residue classes (mod Mod(f)). For any such residue class cl the following is done: Firstly, the image of cl under f is added to im. Secondly, the intersection of the preimage of the intersection of the image of cl under f and im under f and cl is subtracted from pre.
IntegralConjugate(f)
, IntegralConjugate(G)
Uses the algorithm described in the proof of Theorem 2.5.14 in .
IntegralizingConjugator(f)
, IntegralizingConjugator(G)
Uses the algorithm described in the proof of Theorem 2.5.14 in .
Inverse(f)
Essentially inversion of affine mappings. See Lemma 1.3.1, Part (b) in .
IsBalanced(f)
Checks whether the sets of prime factors of the multiplier and the divisor of f are the same.
IsClassReflection(g)
Computes the support of g, and compares g with the corresponding class reflection.
IsClassRotation(g)
Computes the support of g, extracts the possible rotation factor from the coefficients and compares g with the corresponding class rotation.
IsClassShift(g)
Computes the support of g, and compares g with the corresponding class shift.
IsClassTransposition(g)
Computes the support of g, writes it as a disjoint union of two residue classes and compares g with the class transposition which interchanges them.
IsClassWiseOrderPreserving(f)
Tests whether the first entry of all coefficient triples is positive.
IsConjugate(RCWA(Integers),f,g)
Test whether f and g have the same order, and whether either both or none of them is tame. If not, return false.

If the mappings are wild, use ShortCycles 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 false.

If the mappings are tame, use the method described in the proof of Theorem 2.5.14 in to construct integral conjugates of f and g. Then essentially use the algorithm described in the proof of Theorem 2.6.7 in to compute standard representatives of the conjugacy classes which the integral conjugates of f and g belong to. Finally compare these standard representatives, and return true if they are equal and false if not.
IsInjective(f)
See Image.
IsIntegral(f)
Trivial
.
IsomorphismMatrixGroup(G)
Uses the algorithm described in the proof of Theorem 2.6.3 in .
IsomorphismPermGroup(G)
If the group G is finite and class-wise order-preserving, use ActionOnRespectedPartition. If G is finite, but not class-wise order-preserving, compute the action on the respected partition which is obtained by splitting any residue class r(m) in RespectedPartition(G into three residue classes r(3m), r+m(3m), r+2m(3m). If G is infinite, there is no isomorphism to a finite permutation group, thus return fail.
IsomorphismRcwaGroup(G)
The method for finite groups uses RcwaMapping, Part (d).

The method for free products of finite groups uses the Table-Tennis Lemma (which is also known as Ping-Pong Lemma, cp. e.g. Section II.B. in ). It uses regular permutation representations of the factors G_r (r = 0, \dots ,m-1) of the free product on residue classes modulo n_r := |G_r|. 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 \sigma_r which satisfy 0(n_r)^{\sigma_r} = \Z \setminus r(m).

The method for free groups uses an adaptation of the construction given on page 27 in from PSL(2,\mathbb{C}) to RCWA(\Z). As an equivalent for the closed discs used there, the method takes the residue classes modulo two times the rank of the free group.
IsPerfect(G)
If the group G is trivial, then return true. Otherwise if it is abelian, then return false.

If the underlying ring is \Z, then do the following: If one of the generators of G has sign -1, then return false. If G is class-wise order-preserving and one of the generators has nonzero determinant, then return false.

If G is wild, and perfectness has not been decided so far, then give up. If G is finite, then check the image of IsomorphismPermGroup(G) for perfectness, and return true or false accordingly.

If the group G is tame and if it acts transitively on its stored respected partition, then return true or false depending on whether the finite permutation group ActionOnRespectedPartition(G) is perfect or not. If G does not act transitively on its stored respected partition, then give up.
IsPrimeSwitch(g)
Checks whether the multiplier of g is an odd prime, and compares g with the corresponding prime switch.
IsSignPreserving(f)
If f is not class-wise order-preserving, then return false. Otherwise let c \geq 1 be greater than or equal to the maximum of the absolute values of the coefficients b_{r(m)} of the affine partial mappings of f, and check whether the minimum of the image of \{0, \dots, c\} under f is nonnegative and whether the maximum of the image of \{-c, \dots, -1\} under f is negative. If both is the case, then return true, otherwise return false.
IsSolvable(G)
If G is abelian, then return true. If G is tame, then return true or false depending on whether ActionOnRespectedPartition(G) is solvable or not. If G is wild, then give up.
IsSubset(G,H)
(checking for a subgroup relation)
Check whether the set of stored generators of H is a subset of the set of stored generators of G. If so, return true. Check whether the prime set of H is a subset of the prime set of G. If not, return false. Check whether the support of H is a subset of the support of G. If not, return false. Check whether G is tame, but H is wild. If so, return false.

If G and H are both tame, then proceed as follows: If the multiplier of H does not divide the multiplier of G, then return false. If H does not respect the stored respected partition of G, then return false. Check whether the finite permutation group induced by H on RespectedPartition(G) is a subgroup of ActionOnRespectedPartition(G). If yes, return true. Check whether the order of H is greater than the order of G. If so, return false.

Finally use the membership test to check whether all generators of H lie in G, and return true or false accordingly.
IsSurjective(f)
See Image.
IsTame(G)
Checks whether the modulus of the group is nonzero.
IsTame(f)
Application of the criteria given in Corollary 2.5.10 and 2.5.12 and Theorem A.8 and A.11 in , as well as of the criteria given in .

The criterion surjective, but not injective means wild (Theorem A.8 in ) is the subject of .

The package `GRAPE` 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 (cp. Theorem A.11 in ).
IsTransitive(G,Integers)
Look for finite orbits, using ShortOrbits on a couple of intervals. If a finite orbit is found, return false. Test if G is finite. If yes, return false.

Search for an element g and a residue class r(m) such that the restriction of g to r(m) is given by n \mapsto n + m. Then the cyclic group generated by g acts transitively on r(m). The element g is searched among the generators of G, 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.

If suitable g and r(m) are found, proceed as follows:

Put S := r(m). Put S := S \cup S^g for all generators g of G, and repeat this until S remains constant. This may run into an infinite loop.

If it terminates: If S = \Z, return true, otherwise return false.
KernelOfActionOnRespectedPartition(G)
First determine the abelian invariants of the kernel K. For this, compute sufficiently many quotients of orders of permutation groups induced by G on refinements of the stored respected partition P by the order of the permutation group induced by G on P itself. Then use a random walk through the group G. Compute powers of elements encountered along the way which fix P. Translate these kernel elements into elements of a polycyclically presented group isomorphic to the |P|-fold direct product of the infinite dihedral group (K certainly embeds into this group). Use `Polycyclic` to collect independent nice generators of K. Proceed until the permutation groups induced by K on the refined respected partitions all equal the initially stored quotients.
LargestSourcesOfAffineMappings(f)
Forms unions of residue classes modulo the modulus of the mapping, whose corresponding coefficient triples are equal.
LaTeXObj(f)
Collects residue classes those corresponding coefficient triples are equal.
LikelyContractionCentre(f,maxn,bound)
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.
LocalizedRcwaMapping(f,p)
Trivial
.
Loops(f)
Runs over the residue classes modulo the modulus of f, and selects those of them which f does not map to themselves, but which intersect nontrivially with their images under f.
mKnot(m)
Straightforward
, following the definition given in .
Modulus(G)
Searches for a wild element in the group. If unsuccessful, tries to construct a respected partition (see RespectedPartition).
Modulus(f)
Trivial
.
MovedPoints(G)
Needs only forming unions of residue classes and determining fixed points of affine mappings.
Multiplier(f)
Lcm of coefficients, as indicated.
Multpk(f,p,k)
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.
NrConjugacyClassesOfRCWAZOfOrder(ord)
The class numbers are taken from Corollary 2.7.1 in .
Orbit(G,pnt,gens,acts,act)
Check if the orbit has length less than a certain bound. If so, then return it as a list. Otherwise test whether the group G is tame or wild.

If G is tame, then test whether G is finite. If yes, then compute the orbit by the &GAP; Library method. Otherwise proceed as follows: Compute a respected partition \mathcal{P} of G. Use \mathcal{P} to find a residue class r(m) which is a subset of the orbit to be computed. In general, r(m) will not be one of the residue classes in \mathcal{P}, but a subset of one of them. Put \Omega := r(m). Unite the set \Omega with its images under all the generators of G and their inverses. Repeat that until \Omega does not change any more. Return \Omega.

If G is wild, then return an orbit object which stores the group G, the representative rep and the action act.
OrbitsModulo(f,m)
Uses `GRAPE` to compute the connected components of the transition graph.
OrbitsModulo(G,m)
Straightforward
.
Order(f)
Test for IsTame. If the mapping is not tame, then return infinity. Otherwise use Corollary 2.5.10 in .
PreImage(f,S)
See Image.
PreImagesRepresentative(phi,g)
, PreImagesRepresentatives(phi,g)
As indicated in the documentation of these methods. The underlying idea to successively compute two balls around 1 and g until they intersect non-trivially 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.
PrimeSet(f)
, PrimeSet(G)
Straightforward
.
PrimeSwitch(p)
Multiplication of rcwa mappings as indicated.
Print(f)
Trivial
.
f*g
Essentially composition of affine mappings. See Lemma 1.3.1, Part (a) in .
Projections(G,m)
Use OrbitsModulo to determine the supports of the images of the epimorphisms to be determined, and use RestrictedPerm to compute the images of the generators of G under these epimorphisms.
Random(RCWA(Integers))
Computes a product of randomly chosen class shifts, class reflections and class transpositions. This seems to be suitable for generating reasonably good examples.
RankOfKernelOfActionOnRespectedPartition(G)
This performs basically the first part of the computations done by KernelOfActionOnRespectedPartition.
RCWA(R)
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 .
RcwaGroupByPermGroup(G)
Uses RcwaMapping, Part (d).
RcwaMapping
(a)-(c): trivial, (d): n&circum;perm - n for determining the coefficients, (e): affine mappings by values at two given points, (f) and (g): trivial, (h) and (i): correspond to Lemma 2.1.4 in .
RepresentativeAction(G,src,dest,act)
, RepresentativeActionPreImage
As indicated in the documentation of these methods. The underlying idea to successively compute two balls around src and dest until they intersect non-trivially is standard in computational group theory. Words standing for products of generators of G are stored for any image of src or dest.
RepresentativeAction(RCWA(Integers),P1,P2)
Arbitrary mapping: see Lemma 2.1.4 in . Tame mapping: see proof of Theorem 2.8.9 in . The former is almost trivial, while the latter is a bit complicated and takes usually also much more time.
RepresentativeAction(RCWA(Integers),f,g)
The algorithm used by IsConjugate constructs actually also an element x such that f&circum;x = g.
RespectedPartition(f)
, RespectedPartition(G)
Uses the algorithm described in the proof of Theorem 2.5.8 in .
RespectsPartition(G,P)
Straightforward
.
RestrictedPerm(g,S
Straightforward
.
Restriction(g,f)
Computes the action of RightInverse(f) * g * f on the image of f.
Restriction(G,f)
Gets a set of generators by applying Restriction(g,f) to the generators g of G.
RightInverse(f)
Straightforward
if one knows how to compute images of residue classes under affine mappings, and how to compute inverses of affine mappings.
Root(f,k)
If f is bijective, class-wise order-preserving and has finite order:

Find a conjugate of f which is a product of class transpositions. Slice cycles \prod_{i=2}^l \tau_{r_1(m_1),r_i(m_i)} of f a respected partition \mathcal{P} into cycles \prod_{i=1}^l \prod_{j=0}^{k-1} \tau_{r_1(km_1),r_i+jm_i(km_i)} of the k-fold length on the refined partition which one gets from \mathcal{P} by decomposing any r_i(m_i) \in \mathcal{P} into residue classes (mod km_i). Finally conjugate the resulting permutation back.

Other cases seem to be more difficult and are currently not covered.
RotationFactor(g)
Trivial
.
SemilocalizedRcwaMapping(f,pi)
Trivial
.
ShortCycles(f,maxlng)
Looks for fixed points of affine partial mappings of powers of f.
ShortOrbits(G,S,maxlng)
Straightforward
.
Sign(g)
Evaluation of the given expression. For the mathematical meaning (epimorphism!), see Theorem 2.12.8 in .
Sinks(f)
Computes the strongly connected components of the transition graph by the function STRONGLY&uscore;CONNECTED&uscore;COMPONENTS&uscore;DIGRAPH, and selects those which are proper subsets of their preimages and proper supersets of their images under f.
Size(G)
(order of an rcwa group)
Test whether one of the generators of the group G has infinite order. If so, return infinity. Test whether the group G is tame. If not, return infinity. Test whether RankOfKernelOfActionOnRespectedPartition(G) is nonzero. If so, return infinity. Otherwise if G is class-wise order-preserving, return the size of the permutation group induced on the stored respected partition. If G 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.
Size(M)
(order of an rcwa monoid)
Check whether M is in fact an rcwa group. If so, use the method for rcwa groups instead. Check whether one of the generators of M is surjective, but not injective. If so, return infinity. Check whether for all generators f of M, the image of the union of the loops of f under f is finite. If not, return infinity. Check whether one of the generators of M is bijective and has infinite order. If so, return infinity. Check whether one of the generators of M is wild. If so, return infinity. Apply the above criteria to the elements of the ball of radius 2 around 1, and return infinity if appropriate. Finally attempt to compute the list of elements of M. If this is successful, return the length of the resulting list.
Sources(f)
Computes the strongly connected components of the transition graph by the function STRONGLY&uscore;CONNECTED&uscore;COMPONENTS&uscore;DIGRAPH, and selects those which are proper supersets of their preimages and proper subsets of their images under f.
SplittedClassTransposition(ct,k)
Straightforward
.
StructureDescription(G)
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 (DirectProduct, WreathProduct, etc.).
f+g
Pointwise addition of affine mappings.
Support(G)
Straightforward
.
Trajectory(f,n,...)
Iterated application of an rcwa mapping. In the methods computing accumulated coefficients, additionally composition of affine mappings.
TransitionGraph(f,m)
Straightforward
-- just check a sufficiently long interval.
TransitionMatrix(f,m)
Evaluation of the given expression.
TransposedClasses(g)
Trivial
.
View(f)
Trivial
.
WreathProduct(G,P)
Uses DirectProduct to embed the DegreeAction(P)th direct power of G, and RcwaMapping, Part (d) to embed the finite permutation group P.
WreathProduct(G,Z)
Restricts G to the residue class 3(4), and encodes the generator of Z as \tau_{0(2),1(2)} \cdot \tau_{0(2),1(4)}. It is used that the images of 3(4) under powers of this mapping are pairwise disjoint residue classes.