GAP (RCWA) - Chapter 8: The Algorithms Implemented in RCWA

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8 The Algorithms Implemented in RCWA

### 8 The Algorithms Implemented in RCWA

This chapter lists brief descriptions of the algorithms and methods implemented in this package. These descriptions are kept very informal and terse, 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 performing I/O operations, storing or recalling one or several values or doing very basic computations, 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 some of the algorithms and methods can be found in [Koh08].

`ActionOnRespectedPartition(G)`

  "Straightforward" after having computed a respected partition by `RespectedPartition`.

`AllElementsOfCTZWithGivenModulus(m)`

  This function first determines a list of all unordered partitions mathcalP of ℤ into m residue classes. Then for any such partition mathcalP it runs a loop over the elements of the symmetric group of degree m. For any σ ∈ S_m and any partition mathcalP it constructs the element of CT(Z) with modulus dividing m which maps the ordered partition {0(m), 1(m), dots, m-1(m)} to the ordered partition obtained from mathcalP by permuting the residue classes with σ. Finally it discards the elements whose modulus is a proper divisor of m, and returns the "rest".

`AllProducts(l,k)`

  "Straightforward".

`AllSmoothIntegers(maxp,maxn)`

  "Straightforward". -- The taken time is essentially proportional to the size of the resulting list.

`AssignGlobals(record)`

  "Trivial".

`Ball(G,g,r)`

  "Straightforward".

`Ball(G,p,r,act)`

  "Straightforward".

`ClassPairs(m)`

  Runs a loop over all 4-tuples of nonnegative integers less than m, and filters by congruence criteria and ordering of the entries.

`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 [Koh10].

`CycleRepresentativesAndLengths(g,S)`

  "Straightforward".

`CyclesOnFiniteOrbit(G,g,n)`

  "Straightforward".

`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 [Koh05].

`DifferencesList(l)`

  "Trivial".

`DirectProduct(G1,G2, ... )`

  Restricts the groups G1, G2, ... to disjoint residue classes. See `Restriction` and Corollary 2.3.3 in [Koh05].

`Display(f)`

  "Trivial".

`DistanceToNextSmallerPointInOrbit(G,n)`

  "Straightforward" -- computes balls of radius r about n for r = 1, 2, dots until a point smaller than n is found.

`Divisor(f)`

  Lcm of coefficients, as indicated.

`DrawGrid(U,range_y,range_x,filename)`

  "Straightforward".

`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.

`EpimorphismByGenerators(G,H)`

  "Trivial".

`EpimorphismFromFpGroup(G,r)`

  Computes orders of elements in the ball of radius r about 1 in G, and uses the corresponding relations if they affect the abelian invariants of G, G', G'', etc..

`EquivalenceClasses(list,relation)`, `EquivalenceClasses(list,classinvariant)`

  "Straightforward".

`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.

`ExponentOfPrime(n,p)`

  "Trivial".

`ExtRepOfObj(f)`

  "Trivial".

`FactorizationIntoCSCRCT(g)`, `Factorization(g)`

  The method used here is rather sophisticated, and 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 in a similar way 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".

`FixedResidueClasses(g,maxmod)`, `FixedResidueClasses(G,maxmod)`

  Runs a loop over all moduli m ≤ maxmod and all residues r modulo these moduli, and selects those residue classes r(m) which are mapped to itself by g, respectively, by all generators of G.

`FloatQuotientsList(l)`

  "Trivial".

`GeneratorsAndInverses(G)`

  "Trivial".

`GluckTaylorInvariant(a)`

  Evaluation of the given expression.

`GroupByResidueClasses(classes)`

  Finds all pairs of residue classes in the list classes which are disjoint, forms the corresponding class transpositions and returns the group generated by them.

`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 [Koh05].

`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 ≠ 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 [EN09], 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 run time 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 non-trivially. 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 [Koh05].

`IntegralizingConjugator(f)`, `IntegralizingConjugator(G)`

  Uses the algorithm described in the proof of Theorem 2.5.14 in [Koh05].

`Inverse(f)`

  Essentially inversion of affine mappings. See Lemma 1.3.1, Part (b) in [Koh05].

`IsBalanced(f)`

  Checks whether the sets of prime factors of the multiplier and the divisor of f are the same.

`IsBijective(f)`

  "Trivial", respectively, see `IsInjective` and `IsSurjective`.

`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), IsGeneralizedClassTransposition(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)`, `IsClassWiseTranslating(f)`

  "Trivial".

`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 [Koh05] to construct integral conjugates of f and g. Then essentially use the algorithm described in the proof of Theorem 2.6.7 in [Koh05] 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".

`IsNaturalCT(G)`, `IsNaturalRCWA(G)`

  Only checks a set flag.

`IsomorphismMatrixGroup(G)`

  Uses the algorithm described in the proof of Theorem 2.6.3 in [Koh05].

`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, cf. e.g. Section II.B. in [dlH00]). 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 rcwa permutations σ_r which satisfy 0(n_r)^σ_r = ℤ ∖ r(m).

  The method for free groups uses an adaptation of the construction given on page 27 in [dlH00] from PSL(2,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.

`IsOne(f)`

  "Trivial".

`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 ≥ 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 [Koh05], as well as of the criteria given in [Koh07a]. The criterion "surjective, but not injective means wild" (Theorem A.8 in [Koh05]) is the subject of [Koh07b]. 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 (cf. Theorem A.11 in [Koh05]).

`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 ↦ 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 ∪ 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`.

`IsTransitiveOnNonnegativeIntegersInSupport(G)`

  Computes balls about 1 with successively increasing radii, and checks whether the union of the sets where the elements of these balls are decreasing or shifting down equals the support of G. If a positive answer is found, transitivity on "small" points (nonnegative integers less than an explicit bound) is verified.

`IsZero(f)`

  "Trivial".

`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 [EN09] 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.

`LaTeXStringFactorsInt(n)`

  Integer factorization and straightforward string operations.

`LaTeXStringRcwaMapping(f)`, `LaTeXAndXDVI(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.

`ListOfPowers(g,n)`

  "Trivial" -- but uses only n-1 multiplications.

`LoadBitmapPicture(filename)`

  "Straightforward", interprets the format of bitmap picture files.

`LoadDatabaseOf...()`, `LoadRCWAExamples()`

  "Trivial". -- These functions do nothing more than reading in certain files.

`LocalizedRcwaMapping(f,p)`

  "Trivial".

`Log2HTML(logfilename)`

  Straightforward string operations.

`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 non-trivially with their images under f.

`MaximalShift(f)`

  "Trivial".

`MergerExtension(G,points,point)`

  As described in `MergerExtension` (3.1-4).

`Mirrored(g)`, `Mirrored(G)`

  Conjugates with n ↦ -n - 1, as indicated in the definition.

`mKnot(m)`

  "Straightforward", following the definition given in [Kel99].

`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.

`NrClassPairs(m)`

  Relatively straightforward. -- Practical for values of m ranging up into the hundreds and corresponding counts of $10^9$ and more.

`NrConjugacyClassesOfCTZOfOrder(ord)`,

  Evaluation of the expression `Length(Filtered(Combinations(DivisorsInt(ord)), l -> l <> [] and Lcm(l) = ord))`.

`NrConjugacyClassesOfRCWAZOfOrder(ord)`

  The class numbers are taken from Corollary 2.7.1 in [Koh05].

`ObjByExtRep(fam,l)`

  "Trivial".

`One(f)`, `One(G)`,

  "Trivial".

`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 mathcalP of G. Use mathcalP 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 mathcalP, but a subset of one of them. Put Ω := r(m). Unite the set Ω with its images under all the generators of G and their inverses. Repeat that until Ω does not change any more. Return Ω.

  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 [Koh05].

`PermutationOpNC(sigma,P,act)`

  Several different methods for different types of arguments, which either provide straightforward optimizations via computing with coefficients directly, or just delegate to `PermutationOp`.

`PreImage(f,S)`

  See `Image`.

`PreImagesRepresentative(phi,g)`, `PreImagesRepresentatives(phi,g)`

  As described 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 run time 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 [Koh05].

`ProjectionsToCoordinates(f)`

  Straightforward coefficient operations.

`ProjectionsToInvariantUnionsOfResidueClasses(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.

`QuotientsList(l)`

  "Trivial".

`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)`

  Performs basically the first part of the computations done by `KernelOfActionOnRespectedPartition`.

`Rcwa(R)`

  "Trivial". -- Attributes and properties set can be derived easily or hold by definition.

`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 [Koh05].

`RCWABuildManual()`

  Consists of a call to a function from the GAPDoc package.

`RcwaGroupByPermGroup(G)`

  Uses `RcwaMapping`, Part (d).

`RCWAInfo(n)`

  "Trivial".

`RcwaMapping`

  (a)-(c): "trivial", (d): `n^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 [Koh05], (j): uses a simple parser for the permitted expressions.

`RCWATestAll()`, `RCWATestInstall()`

  Just read in files running / containing the tests.

`RCWATestExamples()`

  Runs the example tester from the GAPDoc package.

`RepresentativeAction(G,src,dest,act)`, `RepresentativeActionPreImage`

  As described 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 every image of src or dest.

`RepresentativeAction(RCWA(Integers),P1,P2)`

  Arbitrary mapping: see Lemma 2.1.4 in [Koh05]. Tame mapping: see proof of Theorem 2.8.9 in [Koh05]. 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^x = g`.

`RespectedPartition(f)`, `RespectedPartition(G)`

  There are presently two sophisticated algorithms implemented for finding respected partitions. One of them has evolved from the algorithm described in the proof of Theorem 2.5.8 in [Koh05]. The other one starts with the coarsest partition of the base ring such that every generator of G is affine on every part. This partition is then refined successively until a respected partition is obtained. The refinement step is basically as follows: Take the images of the partition under all generators of G. This way one obtains as many further partitions of the base ring as there are generators of G. Then the "new" partition is the coarsest common refinement of all these partitions.

`RespectsPartition(G,P)`

  "Straightforward".

`RestrictedBall(G,g,r,modulusbound)`

  "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 ∏_i=2^l τ_r_1(m_1),r_i(m_i) of f a respected partition mathcalP into cycles ∏_i=1^l ∏_j=0^k-1 τ_r_1(km_1),r_i+jm_i(km_i) of the k-fold length on the refined partition which one gets from mathcalP by decomposing any r_i(m_i) ∈ mathcalP 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".

`RunDemonstration(filename)`

  "Trivial" -- only I/O operations.

`SaveAsBitmapPicture(picture,filename)`

  "Straightforward" -- only needs to know how to produce a bitmap picture file.

`SemilocalizedRcwaMapping(f,pi)`

  "Trivial".

`ShiftsDownOn(f)`, `ShiftsUpOn(f)`

  Straightforward coefficient- and residue class operations.

`ShortCycles(g,maxlng)`

  Looks for fixed points of affine partial mappings of powers of g.

`ShortCycles(g,S,maxlng)`, `ShortCycles(g,S,maxlng,maxn)`

  "Straightforward".

`ShortOrbits(G,S,maxlng)`, `ShortOrbits(G,S,maxlng,maxn)`

  "Straightforward".

`ShortResidueClassCycles(g,modulusbound,maxlng)`

  Different methods -- see source code in `pkg/rcwa/lib/rcwamap.gi`.

`ShortResidueClassOrbits(g,modulusbound,maxlng)`

  Different methods -- see source code in `pkg/rcwa/lib/rcwagrp.gi`.

`Sign(g)`

  Evaluation of the given expression. For the mathematical meaning (epimorphism!), see Theorem 2.12.8 in [Koh05].

`Sinks(f)`

  Computes the strongly connected components of the transition graph by the function `STRONGLY_CONNECTED_COMPONENTS_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.

`SmallGeneratingSet(G)`

  Eliminates generators g which can be found to be redundant easily, i.e. by checking whether the balls about 1 and g of some small radius r in the group generated by all generators of G except for g intersect nontrivially.

`Sources(f)`

  Computes the strongly connected components of the transition graph by the function `STRONGLY_CONNECTED_COMPONENTS_DIGRAPH`, and selects those which are proper supersets of their preimages and proper subsets of their images under f.

`SparseRep(f)`, `StandardRep(f)`

  Straightforward coefficient operations.

`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.

`String(obj)`

  "Trivial".

`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 τ_0(2),1(2) ⋅ τ_0(2),1(4). It is used that the images of 3(4) under powers of this mapping are pairwise disjoint residue classes.

`Zero(f)`

  "Trivial".

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