\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter }About the RCWA Package}}{6}{chapter.1}
\contentsline {section}{\numberline {1.1}\leavevmode {\color {Chapter }Motivation}}{6}{section.1.1}
\contentsline {section}{\numberline {1.2}\leavevmode {\color {Chapter }Purpose of this package}}{6}{section.1.2}
\contentsline {section}{\numberline {1.3}\leavevmode {\color {Chapter }Groups which this package can deal with}}{7}{section.1.3}
\contentsline {section}{\numberline {1.4}\leavevmode {\color {Chapter }Scope of this package}}{7}{section.1.4}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter }Residue-Class-Wise Affine Mappings}}{8}{chapter.2}
\contentsline {section}{\numberline {2.1}\leavevmode {\color {Chapter }Basic definitions}}{8}{section.2.1}
\contentsline {section}{\numberline {2.2}\leavevmode {\color {Chapter }Entering residue-class-wise affine mappings}}{9}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}\leavevmode {\color {Chapter }ClassShift (r, m)}}{10}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}\leavevmode {\color {Chapter }ClassReflection (r, m)}}{10}{subsection.2.2.2}
\contentsline {subsection}{\numberline {2.2.3}\leavevmode {\color {Chapter }ClassTransposition (r1, m1, r2, m2)}}{11}{subsection.2.2.3}
\contentsline {subsection}{\numberline {2.2.4}\leavevmode {\color {Chapter }ClassRotation (r, m, u)}}{12}{subsection.2.2.4}
\contentsline {subsection}{\numberline {2.2.5}\leavevmode {\color {Chapter } RcwaMapping (the general constructor) }}{13}{subsection.2.2.5}
\contentsline {subsection}{\numberline {2.2.6}\leavevmode {\color {Chapter }LocalizedRcwaMapping (for an rcwa mapping of Z and a prime)}}{14}{subsection.2.2.6}
\contentsline {section}{\numberline {2.3}\leavevmode {\color {Chapter }Basic arithmetic for residue-class-wise affine mappings}}{15}{section.2.3}
\contentsline {section}{\numberline {2.4}\leavevmode {\color {Chapter } Attributes and properties of residue-class-wise affine mappings }}{17}{section.2.4}
\contentsline {subsection}{\numberline {2.4.1}\leavevmode {\color {Chapter }LargestSourcesOfAffineMappings (for an rcwa mapping)}}{18}{subsection.2.4.1}
\contentsline {subsection}{\numberline {2.4.2}\leavevmode {\color {Chapter }FixedPointsOfAffinePartialMappings (for an rcwa mapping)}}{18}{subsection.2.4.2}
\contentsline {subsection}{\numberline {2.4.3}\leavevmode {\color {Chapter }Multpk (for an rcwa mapping, a prime and an exponent)}}{19}{subsection.2.4.3}
\contentsline {subsection}{\numberline {2.4.4}\leavevmode {\color {Chapter }Determinant (of an rcwa mapping of Z)}}{19}{subsection.2.4.4}
\contentsline {subsection}{\numberline {2.4.5}\leavevmode {\color {Chapter }Sign (of an rcwa permutation of Z)}}{20}{subsection.2.4.5}
\contentsline {section}{\numberline {2.5}\leavevmode {\color {Chapter }Factoring residue-class-wise affine permutations}}{20}{section.2.5}
\contentsline {subsection}{\numberline {2.5.1}\leavevmode {\color {Chapter }FactorizationIntoCSCRCT (for an rcwa permutation of Z)}}{20}{subsection.2.5.1}
\contentsline {subsection}{\numberline {2.5.2}\leavevmode {\color {Chapter }PrimeSwitch (p)}}{21}{subsection.2.5.2}
\contentsline {subsection}{\numberline {2.5.3}\leavevmode {\color {Chapter }mKnot (for an odd integer)}}{22}{subsection.2.5.3}
\contentsline {section}{\numberline {2.6}\leavevmode {\color {Chapter } Extracting roots of residue-class-wise affine mappings }}{22}{section.2.6}
\contentsline {subsection}{\numberline {2.6.1}\leavevmode {\color {Chapter }Root (k-th root of an rcwa mapping)}}{22}{subsection.2.6.1}
\contentsline {section}{\numberline {2.7}\leavevmode {\color {Chapter } Special functions for non-bijective mappings }}{23}{section.2.7}
\contentsline {subsection}{\numberline {2.7.1}\leavevmode {\color {Chapter }RightInverse (of an injective rcwa mapping)}}{23}{subsection.2.7.1}
\contentsline {subsection}{\numberline {2.7.2}\leavevmode {\color {Chapter }CommonRightInverse (of two injective rcwa mappings)}}{23}{subsection.2.7.2}
\contentsline {subsection}{\numberline {2.7.3}\leavevmode {\color {Chapter }ImageDensity (of an rcwa mapping)}}{23}{subsection.2.7.3}
\contentsline {section}{\numberline {2.8}\leavevmode {\color {Chapter } On trajectories and cycles of residue-class-wise affine mappings }}{24}{section.2.8}
\contentsline {subsection}{\numberline {2.8.1}\leavevmode {\color {Chapter } Trajectory (methods for rcwa mappings) }}{24}{subsection.2.8.1}
\contentsline {subsection}{\numberline {2.8.2}\leavevmode {\color {Chapter } Trajectory (methods for rcwa mappings -- ``accumulated coefficients'') }}{24}{subsection.2.8.2}
\contentsline {subsection}{\numberline {2.8.3}\leavevmode {\color {Chapter } IncreasingOn \& DecreasingOn (for an rcwa mapping) }}{25}{subsection.2.8.3}
\contentsline {subsection}{\numberline {2.8.4}\leavevmode {\color {Chapter }TransitionGraph (for an rcwa mapping and a modulus)}}{25}{subsection.2.8.4}
\contentsline {subsection}{\numberline {2.8.5}\leavevmode {\color {Chapter }OrbitsModulo (for an rcwa mapping and a modulus)}}{26}{subsection.2.8.5}
\contentsline {subsection}{\numberline {2.8.6}\leavevmode {\color {Chapter }FactorizationOnConnectedComponents (for an rcwa mapping and a modulus)}}{26}{subsection.2.8.6}
\contentsline {subsection}{\numberline {2.8.7}\leavevmode {\color {Chapter }TransitionMatrix (for an rcwa mapping and a modulus)}}{26}{subsection.2.8.7}
\contentsline {subsection}{\numberline {2.8.8}\leavevmode {\color {Chapter } Sources \& Sinks (of an rcwa mapping) }}{27}{subsection.2.8.8}
\contentsline {subsection}{\numberline {2.8.9}\leavevmode {\color {Chapter }Loops (of an rcwa mapping)}}{27}{subsection.2.8.9}
\contentsline {subsection}{\numberline {2.8.10}\leavevmode {\color {Chapter }GluckTaylorInvariant (of a trajectory)}}{27}{subsection.2.8.10}
\contentsline {subsection}{\numberline {2.8.11}\leavevmode {\color {Chapter }LikelyContractionCentre (of an rcwa mapping)}}{28}{subsection.2.8.11}
\contentsline {subsection}{\numberline {2.8.12}\leavevmode {\color {Chapter }GuessedDivergence (of an rcwa mapping)}}{28}{subsection.2.8.12}
\contentsline {section}{\numberline {2.9}\leavevmode {\color {Chapter }The categories and families of rcwa mappings}}{29}{section.2.9}
\contentsline {subsection}{\numberline {2.9.1}\leavevmode {\color {Chapter }IsRcwaMapping}}{29}{subsection.2.9.1}
\contentsline {subsection}{\numberline {2.9.2}\leavevmode {\color {Chapter }RcwaMappingsFamily (of a ring)}}{29}{subsection.2.9.2}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter }Residue-Class-Wise Affine Groups}}{30}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter }Constructing residue-class-wise affine groups}}{30}{section.3.1}
\contentsline {subsection}{\numberline {3.1.1}\leavevmode {\color {Chapter }RCWA (the group of all rcwa permutations of a ring)}}{30}{subsection.3.1.1}
\contentsline {subsection}{\numberline {3.1.2}\leavevmode {\color {Chapter }CT (the group generated by all class transpositions of a ring)}}{31}{subsection.3.1.2}
\contentsline {subsection}{\numberline {3.1.3}\leavevmode {\color {Chapter }IsomorphismRcwaGroup (for a group, over a given ring)}}{31}{subsection.3.1.3}
\contentsline {subsection}{\numberline {3.1.4}\leavevmode {\color {Chapter }DirectProduct (for rcwa groups over Z)}}{32}{subsection.3.1.4}
\contentsline {subsection}{\numberline {3.1.5}\leavevmode {\color {Chapter } WreathProduct (for an rcwa group over Z, with a permutation group or ($\@mathbb {Z}$,+)) }}{32}{subsection.3.1.5}
\contentsline {subsection}{\numberline {3.1.6}\leavevmode {\color {Chapter } Restriction (of an rcwa mapping or -group, by an injective rcwa mapping) }}{33}{subsection.3.1.6}
\contentsline {subsection}{\numberline {3.1.7}\leavevmode {\color {Chapter } Induction (of an rcwa mapping or -group, by an injective rcwa mapping) }}{33}{subsection.3.1.7}
\contentsline {section}{\numberline {3.2}\leavevmode {\color {Chapter } Basic routines for investigating residue-class-wise affine groups }}{34}{section.3.2}
\contentsline {subsection}{\numberline {3.2.1}\leavevmode {\color {Chapter }StructureDescription (for an rcwa group)}}{34}{subsection.3.2.1}
\contentsline {subsection}{\numberline {3.2.2}\leavevmode {\color {Chapter }EpimorphismFromFpGroup (for an rcwa group and a search radius)}}{37}{subsection.3.2.2}
\contentsline {subsection}{\numberline {3.2.3}\leavevmode {\color {Chapter }PreImagesRepresentative (for an epi. from a free group to an rcwa group)}}{38}{subsection.3.2.3}
\contentsline {section}{\numberline {3.3}\leavevmode {\color {Chapter } The natural action of an rcwa group on the underlying ring }}{39}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}\leavevmode {\color {Chapter } Orbit (for an rcwa group and either a point or a set) }}{39}{subsection.3.3.1}
\contentsline {subsection}{\numberline {3.3.2}\leavevmode {\color {Chapter } ShortOrbits (for rcwa groups) \& ShortCycles (for rcwa permutations) }}{40}{subsection.3.3.2}
\contentsline {subsection}{\numberline {3.3.3}\leavevmode {\color {Chapter } Ball (for group, element and radius or group, point, radius and action) }}{40}{subsection.3.3.3}
\contentsline {subsection}{\numberline {3.3.4}\leavevmode {\color {Chapter }RepresentativeAction (G, source, destination, action)}}{41}{subsection.3.3.4}
\contentsline {subsection}{\numberline {3.3.5}\leavevmode {\color {Chapter }Projections (for an rcwa group and a modulus)}}{42}{subsection.3.3.5}
\contentsline {subsection}{\numberline {3.3.6}\leavevmode {\color {Chapter }RepresentativeAction (for RCWA(R) and 2 partitions of R into residue classes)}}{43}{subsection.3.3.6}
\contentsline {section}{\numberline {3.4}\leavevmode {\color {Chapter } Special attributes of tame residue-class-wise affine groups }}{43}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}\leavevmode {\color {Chapter } RespectedPartition (of a tame rcwa group or -permutation) }}{44}{subsection.3.4.1}
\contentsline {subsection}{\numberline {3.4.2}\leavevmode {\color {Chapter } ActionOnRespectedPartition \& KernelOfActionOnRespectedPartition }}{44}{subsection.3.4.2}
\contentsline {section}{\numberline {3.5}\leavevmode {\color {Chapter }Generating pseudo-random elements of RCWA(R) and CT(R)}}{45}{section.3.5}
\contentsline {section}{\numberline {3.6}\leavevmode {\color {Chapter }Drawing pictures of orbits on $\@mathbb {Z}^2$ }}{46}{section.3.6}
\contentsline {subsection}{\numberline {3.6.1}\leavevmode {\color {Chapter }DrawOrbitPicture (G, p0, r, h, w, colored, palette, filename)}}{46}{subsection.3.6.1}
\contentsline {subsection}{\numberline {3.6.2}\leavevmode {\color {Chapter }SaveAsBitmapPicture (picture, filename)}}{47}{subsection.3.6.2}
\contentsline {section}{\numberline {3.7}\leavevmode {\color {Chapter }Some general utility functions}}{47}{section.3.7}
\contentsline {section}{\numberline {3.8}\leavevmode {\color {Chapter }The categories of residue-class-wise affine groups}}{48}{section.3.8}
\contentsline {subsection}{\numberline {3.8.1}\leavevmode {\color {Chapter }IsRcwaGroup}}{48}{subsection.3.8.1}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }Residue-Class-Wise Affine Monoids}}{49}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }Constructing residue-class-wise affine monoids}}{49}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}\leavevmode {\color {Chapter }Rcwa (the monoid of all rcwa mappings of a ring)}}{50}{subsection.4.1.1}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter }Computing with residue-class-wise affine monoids}}{50}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}\leavevmode {\color {Chapter }ShortOrbits (for rcwa monoid, set of points and bound on length)}}{51}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}\leavevmode {\color {Chapter } Ball (for monoid, element and radius or monoid, point, radius and action) }}{52}{subsection.4.2.2}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter }Examples}}{53}{chapter.5}
\contentsline {section}{\numberline {5.1}\leavevmode {\color {Chapter } Factoring Collatz' permutation of the integers }}{53}{section.5.1}
\contentsline {section}{\numberline {5.2}\leavevmode {\color {Chapter } An rcwa mapping which seems to be contracting, but very slow }}{55}{section.5.2}
\contentsline {section}{\numberline {5.3}\leavevmode {\color {Chapter }Checking a result by P. Andaloro}}{57}{section.5.3}
\contentsline {section}{\numberline {5.4}\leavevmode {\color {Chapter }Two examples by Matthews and Leigh}}{58}{section.5.4}
\contentsline {section}{\numberline {5.5}\leavevmode {\color {Chapter }Exploring the structure of a wild rcwa group}}{60}{section.5.5}
\contentsline {section}{\numberline {5.6}\leavevmode {\color {Chapter }A wild rcwa mapping which has only finite cycles}}{62}{section.5.6}
\contentsline {section}{\numberline {5.7}\leavevmode {\color {Chapter }An abelian rcwa group over a polynomial ring}}{66}{section.5.7}
\contentsline {section}{\numberline {5.8}\leavevmode {\color {Chapter } A tame group generated by commutators of wild permutations }}{67}{section.5.8}
\contentsline {section}{\numberline {5.9}\leavevmode {\color {Chapter }Checking for solvability}}{70}{section.5.9}
\contentsline {section}{\numberline {5.10}\leavevmode {\color {Chapter }Some examples over (semi)localizations of the integers}}{71}{section.5.10}
\contentsline {section}{\numberline {5.11}\leavevmode {\color {Chapter } Twisting 257-cycles into an rcwa mapping with modulus 32 }}{74}{section.5.11}
\contentsline {section}{\numberline {5.12}\leavevmode {\color {Chapter } The behaviour of the moduli of powers }}{75}{section.5.12}
\contentsline {section}{\numberline {5.13}\leavevmode {\color {Chapter } Images and preimages under the Collatz mapping }}{76}{section.5.13}
\contentsline {section}{\numberline {5.14}\leavevmode {\color {Chapter } A group which acts 4-transitively on the positive integers }}{78}{section.5.14}
\contentsline {section}{\numberline {5.15}\leavevmode {\color {Chapter } A group which acts 3-transitively, but not 4-transitively on Z }}{87}{section.5.15}
\contentsline {section}{\numberline {5.16}\leavevmode {\color {Chapter } Grigorchuk groups }}{90}{section.5.16}
\contentsline {section}{\numberline {5.17}\leavevmode {\color {Chapter } Forward orbits of a monoid with 2 generators }}{92}{section.5.17}
\contentsline {section}{\numberline {5.18}\leavevmode {\color {Chapter } Representations of the free group of rank 2 }}{93}{section.5.18}
\contentsline {section}{\numberline {5.19}\leavevmode {\color {Chapter } Representations of the modular group PSL(2,Z) }}{94}{section.5.19}
\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter }The Algorithms Implemented in RCWA}}{96}{chapter.6}
\contentsline {chapter}{\numberline {7}\leavevmode {\color {Chapter }Installation and auxiliary functions}}{107}{chapter.7}
\contentsline {section}{\numberline {7.1}\leavevmode {\color {Chapter }Requirements}}{107}{section.7.1}
\contentsline {section}{\numberline {7.2}\leavevmode {\color {Chapter }Installation}}{107}{section.7.2}
\contentsline {section}{\numberline {7.3}\leavevmode {\color {Chapter }The Info class of the package}}{107}{section.7.3}
\contentsline {subsection}{\numberline {7.3.1}\leavevmode {\color {Chapter }InfoRCWA}}{107}{subsection.7.3.1}
\contentsline {section}{\numberline {7.4}\leavevmode {\color {Chapter }The testing routine}}{107}{section.7.4}
\contentsline {subsection}{\numberline {7.4.1}\leavevmode {\color {Chapter }RCWATest}}{107}{subsection.7.4.1}
\contentsline {section}{\numberline {7.5}\leavevmode {\color {Chapter }Building the manual}}{108}{section.7.5}
\contentsline {subsection}{\numberline {7.5.1}\leavevmode {\color {Chapter }RCWABuildManual}}{108}{subsection.7.5.1}
