\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter }Preface}}{7}{chapter.1}
\contentsline {section}{\numberline {1.1}\leavevmode {\color {Chapter }Motivation}}{7}{section.1.1}
\contentsline {section}{\numberline {1.2}\leavevmode {\color {Chapter }Purpose of this package}}{7}{section.1.2}
\contentsline {section}{\numberline {1.3}\leavevmode {\color {Chapter }Scope of this package}}{8}{section.1.3}
\contentsline {section}{\numberline {1.4}\leavevmode {\color {Chapter }Acknowledgements}}{8}{section.1.4}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter }Residue Class-Wise Affine Mappings}}{9}{chapter.2}
\contentsline {section}{\numberline {2.1}\leavevmode {\color {Chapter }Basic definitions}}{9}{section.2.1}
\contentsline {section}{\numberline {2.2}\leavevmode {\color {Chapter }Entering residue class-wise affine mappings}}{10}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}\leavevmode {\color {Chapter }ClassShift (r, m)}}{11}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}\leavevmode {\color {Chapter }ClassReflection (r, m)}}{11}{subsection.2.2.2}
\contentsline {subsection}{\numberline {2.2.3}\leavevmode {\color {Chapter }ClassTransposition (r1, m1, r2, m2)}}{12}{subsection.2.2.3}
\contentsline {subsection}{\numberline {2.2.4}\leavevmode {\color {Chapter }PrimeSwitch (p)}}{12}{subsection.2.2.4}
\contentsline {subsection}{\numberline {2.2.5}\leavevmode {\color {Chapter }RcwaMapping (R, m, coeffs)}}{13}{subsection.2.2.5}
\contentsline {subsection}{\numberline {2.2.6}\leavevmode {\color {Chapter }LaTeXObj (f)}}{15}{subsection.2.2.6}
\contentsline {section}{\numberline {2.3}\leavevmode {\color {Chapter }Basic functionality for rcwa mappings}}{15}{section.2.3}
\contentsline {section}{\numberline {2.4}\leavevmode {\color {Chapter }Factoring rcwa mappings}}{17}{section.2.4}
\contentsline {subsection}{\numberline {2.4.1}\leavevmode {\color {Chapter }FactorizationIntoCSCRCT (g)}}{17}{subsection.2.4.1}
\contentsline {subsection}{\numberline {2.4.2}\leavevmode {\color {Chapter }mKnot (m)}}{18}{subsection.2.4.2}
\contentsline {section}{\numberline {2.5}\leavevmode {\color {Chapter }Determinant and sign}}{19}{section.2.5}
\contentsline {subsection}{\numberline {2.5.1}\leavevmode {\color {Chapter }Determinant (sigma)}}{19}{subsection.2.5.1}
\contentsline {subsection}{\numberline {2.5.2}\leavevmode {\color {Chapter }Sign (sigma)}}{19}{subsection.2.5.2}
\contentsline {section}{\numberline {2.6}\leavevmode {\color {Chapter }Attributes and properties derived from the coefficients}}{20}{section.2.6}
\contentsline {subsection}{\numberline {2.6.1}\leavevmode {\color {Chapter }Multiplier (f)}}{20}{subsection.2.6.1}
\contentsline {subsection}{\numberline {2.6.2}\leavevmode {\color {Chapter }Divisor (f)}}{20}{subsection.2.6.2}
\contentsline {subsection}{\numberline {2.6.3}\leavevmode {\color {Chapter }PrimeSet (f)}}{20}{subsection.2.6.3}
\contentsline {subsection}{\numberline {2.6.4}\leavevmode {\color {Chapter }IsIntegral (f)}}{21}{subsection.2.6.4}
\contentsline {subsection}{\numberline {2.6.5}\leavevmode {\color {Chapter }IsClassWiseOrderPreserving (f)}}{21}{subsection.2.6.5}
\contentsline {section}{\numberline {2.7}\leavevmode {\color {Chapter }Functionality related to the affine partial mappings}}{21}{section.2.7}
\contentsline {subsection}{\numberline {2.7.1}\leavevmode {\color {Chapter }LargestSourcesOfAffineMappings (f)}}{21}{subsection.2.7.1}
\contentsline {subsection}{\numberline {2.7.2}\leavevmode {\color {Chapter }Multpk (f, p, k)}}{22}{subsection.2.7.2}
\contentsline {subsection}{\numberline {2.7.3}\leavevmode {\color {Chapter }SetOnWhichMappingIsClassWiseOrderPreserving (f)}}{22}{subsection.2.7.3}
\contentsline {subsection}{\numberline {2.7.4}\leavevmode {\color {Chapter }FixedPointsOfAffinePartialMappings (f)}}{22}{subsection.2.7.4}
\contentsline {section}{\numberline {2.8}\leavevmode {\color {Chapter }Transition graphs and transition matrices}}{23}{section.2.8}
\contentsline {subsection}{\numberline {2.8.1}\leavevmode {\color {Chapter }TransitionGraph (f, m)}}{23}{subsection.2.8.1}
\contentsline {subsection}{\numberline {2.8.2}\leavevmode {\color {Chapter }OrbitsModulo (f, m)}}{23}{subsection.2.8.2}
\contentsline {subsection}{\numberline {2.8.3}\leavevmode {\color {Chapter }FactorizationOnConnectedComponents (f, m)}}{23}{subsection.2.8.3}
\contentsline {subsection}{\numberline {2.8.4}\leavevmode {\color {Chapter }TransitionMatrix (f, m)}}{24}{subsection.2.8.4}
\contentsline {subsection}{\numberline {2.8.5}\leavevmode {\color {Chapter }Sources (f)}}{25}{subsection.2.8.5}
\contentsline {subsection}{\numberline {2.8.6}\leavevmode {\color {Chapter }Sinks (f)}}{25}{subsection.2.8.6}
\contentsline {subsection}{\numberline {2.8.7}\leavevmode {\color {Chapter }Loops (f)}}{25}{subsection.2.8.7}
\contentsline {section}{\numberline {2.9}\leavevmode {\color {Chapter }Trajectories}}{26}{section.2.9}
\contentsline {subsection}{\numberline {2.9.1}\leavevmode {\color {Chapter }Trajectory (f, n, length)}}{26}{subsection.2.9.1}
\contentsline {subsection}{\numberline {2.9.2}\leavevmode {\color {Chapter }Trajectory (f, n, length, whichcoeffs)}}{26}{subsection.2.9.2}
\contentsline {subsection}{\numberline {2.9.3}\leavevmode {\color {Chapter }IncreasingOn (f)}}{27}{subsection.2.9.3}
\contentsline {section}{\numberline {2.10}\leavevmode {\color {Chapter } Localizations of rcwa mappings of the integers }}{27}{section.2.10}
\contentsline {subsection}{\numberline {2.10.1}\leavevmode {\color {Chapter }LocalizedRcwaMapping (f, p)}}{27}{subsection.2.10.1}
\contentsline {section}{\numberline {2.11}\leavevmode {\color {Chapter } Extracting roots of rcwa mappings }}{27}{section.2.11}
\contentsline {subsection}{\numberline {2.11.1}\leavevmode {\color {Chapter }Root (f, k)}}{27}{subsection.2.11.1}
\contentsline {section}{\numberline {2.12}\leavevmode {\color {Chapter } Special functions for non-bijective mappings }}{28}{section.2.12}
\contentsline {subsection}{\numberline {2.12.1}\leavevmode {\color {Chapter }RightInverse (f)}}{28}{subsection.2.12.1}
\contentsline {subsection}{\numberline {2.12.2}\leavevmode {\color {Chapter }CommonRightInverse (l, r)}}{28}{subsection.2.12.2}
\contentsline {subsection}{\numberline {2.12.3}\leavevmode {\color {Chapter }ImageDensity (f)}}{28}{subsection.2.12.3}
\contentsline {section}{\numberline {2.13}\leavevmode {\color {Chapter } Probabilistic guesses on the behaviour of trajectories }}{29}{section.2.13}
\contentsline {subsection}{\numberline {2.13.1}\leavevmode {\color {Chapter }LikelyContractionCentre (f, maxn, bound)}}{29}{subsection.2.13.1}
\contentsline {subsection}{\numberline {2.13.2}\leavevmode {\color {Chapter }GuessedDivergence (f)}}{29}{subsection.2.13.2}
\contentsline {section}{\numberline {2.14}\leavevmode {\color {Chapter }The categories and families of rcwa mappings}}{30}{section.2.14}
\contentsline {subsection}{\numberline {2.14.1}\leavevmode {\color {Chapter }IsRcwaMapping (f)}}{30}{subsection.2.14.1}
\contentsline {subsection}{\numberline {2.14.2}\leavevmode {\color {Chapter }RcwaMappingsFamily (R)}}{30}{subsection.2.14.2}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter }Residue Class-Wise Affine Groups}}{31}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter }Constructing residue class-wise affine groups}}{31}{section.3.1}
\contentsline {subsection}{\numberline {3.1.1}\leavevmode {\color {Chapter }RCWA (R)}}{31}{subsection.3.1.1}
\contentsline {subsection}{\numberline {3.1.2}\leavevmode {\color {Chapter }Random (RCWA( Integers ))}}{32}{subsection.3.1.2}
\contentsline {subsection}{\numberline {3.1.3}\leavevmode {\color {Chapter }IsomorphismRcwaGroupOverZ (G)}}{34}{subsection.3.1.3}
\contentsline {section}{\numberline {3.2}\leavevmode {\color {Chapter }Attributes and properties of rcwa groups}}{35}{section.3.2}
\contentsline {subsection}{\numberline {3.2.1}\leavevmode {\color {Chapter }Modulus (G)}}{35}{subsection.3.2.1}
\contentsline {subsection}{\numberline {3.2.2}\leavevmode {\color {Chapter }IsTame (G)}}{35}{subsection.3.2.2}
\contentsline {subsection}{\numberline {3.2.3}\leavevmode {\color {Chapter }PrimeSet (G)}}{36}{subsection.3.2.3}
\contentsline {section}{\numberline {3.3}\leavevmode {\color {Chapter } Membership testing, order computation, permutation- / matrix representations }}{36}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }in (g, G)}}{36}{subsection.3.3.1}
\contentsline {subsection}{\numberline {3.3.2}\leavevmode {\color {Chapter }Size (G)}}{36}{subsection.3.3.2}
\contentsline {subsection}{\numberline {3.3.3}\leavevmode {\color {Chapter }IsomorphismPermGroup (G)}}{37}{subsection.3.3.3}
\contentsline {subsection}{\numberline {3.3.4}\leavevmode {\color {Chapter }IsomorphismMatrixGroup (G)}}{37}{subsection.3.3.4}
\contentsline {section}{\numberline {3.4}\leavevmode {\color {Chapter } Factoring elements into generators }}{38}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}\leavevmode {\color {Chapter }PreImagesRepresentative (phi, g)}}{38}{subsection.3.4.1}
\contentsline {subsection}{\numberline {3.4.2}\leavevmode {\color {Chapter }PreImagesRepresentatives (phi, g)}}{39}{subsection.3.4.2}
\contentsline {section}{\numberline {3.5}\leavevmode {\color {Chapter }The action of an rcwa group on the underlying ring R}}{40}{section.3.5}
\contentsline {subsection}{\numberline {3.5.1}\leavevmode {\color {Chapter }IsTransitive (G, Integers)}}{40}{subsection.3.5.1}
\contentsline {subsection}{\numberline {3.5.2}\leavevmode {\color {Chapter }RepresentativeAction (G, src, dest, act)}}{41}{subsection.3.5.2}
\contentsline {subsection}{\numberline {3.5.3}\leavevmode {\color {Chapter }RepresentativeActionPreImage (G, src, dest, act, F)}}{42}{subsection.3.5.3}
\contentsline {subsection}{\numberline {3.5.4}\leavevmode {\color {Chapter }RepresentativeAction (RCWA( Integers ), P1, P2)}}{42}{subsection.3.5.4}
\contentsline {subsection}{\numberline {3.5.5}\leavevmode {\color {Chapter }ShortOrbits (G, S, maxlng)}}{43}{subsection.3.5.5}
\contentsline {subsection}{\numberline {3.5.6}\leavevmode {\color {Chapter }OrbitsModulo (G, m)}}{44}{subsection.3.5.6}
\contentsline {subsection}{\numberline {3.5.7}\leavevmode {\color {Chapter }Ball (G, p, d, act)}}{44}{subsection.3.5.7}
\contentsline {section}{\numberline {3.6}\leavevmode {\color {Chapter }Conjugacy in RCWA(R)}}{45}{section.3.6}
\contentsline {subsection}{\numberline {3.6.1}\leavevmode {\color {Chapter }IsConjugate (RCWA( Integers ), f, g)}}{45}{subsection.3.6.1}
\contentsline {subsection}{\numberline {3.6.2}\leavevmode {\color {Chapter }RepresentativeAction (RCWA( Integers ), f, g)}}{45}{subsection.3.6.2}
\contentsline {subsection}{\numberline {3.6.3}\leavevmode {\color {Chapter }ShortCycles (f, maxlng)}}{46}{subsection.3.6.3}
\contentsline {subsection}{\numberline {3.6.4}\leavevmode {\color {Chapter }NrConjugacyClassesOfRCWAZOfOrder (ord)}}{46}{subsection.3.6.4}
\contentsline {section}{\numberline {3.7}\leavevmode {\color {Chapter }Restriction and induction}}{47}{section.3.7}
\contentsline {subsection}{\numberline {3.7.1}\leavevmode {\color {Chapter }Restriction (g, f)}}{47}{subsection.3.7.1}
\contentsline {subsection}{\numberline {3.7.2}\leavevmode {\color {Chapter }Restriction (G, f)}}{47}{subsection.3.7.2}
\contentsline {subsection}{\numberline {3.7.3}\leavevmode {\color {Chapter }Induction (g, f)}}{47}{subsection.3.7.3}
\contentsline {subsection}{\numberline {3.7.4}\leavevmode {\color {Chapter }DirectProduct (G1, G2, ...)}}{48}{subsection.3.7.4}
\contentsline {subsection}{\numberline {3.7.5}\leavevmode {\color {Chapter }WreathProduct (G, P)}}{48}{subsection.3.7.5}
\contentsline {section}{\numberline {3.8}\leavevmode {\color {Chapter }Special attributes for tame rcwa groups}}{49}{section.3.8}
\contentsline {subsection}{\numberline {3.8.1}\leavevmode {\color {Chapter }RespectedPartition (G)}}{49}{subsection.3.8.1}
\contentsline {subsection}{\numberline {3.8.2}\leavevmode {\color {Chapter }ActionOnRespectedPartition (G)}}{50}{subsection.3.8.2}
\contentsline {subsection}{\numberline {3.8.3}\leavevmode {\color {Chapter }IntegralConjugate (G)}}{50}{subsection.3.8.3}
\contentsline {subsection}{\numberline {3.8.4}\leavevmode {\color {Chapter }IntegralizingConjugator (G)}}{51}{subsection.3.8.4}
\contentsline {section}{\numberline {3.9}\leavevmode {\color {Chapter }The categories of rcwa groups}}{51}{section.3.9}
\contentsline {subsection}{\numberline {3.9.1}\leavevmode {\color {Chapter }IsRcwaGroup (G)}}{51}{subsection.3.9.1}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }Examples}}{52}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter } Factoring Collatz' permutation of the integers }}{52}{section.4.1}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter } An rcwa mapping which seems to be contracting, but very slow }}{59}{section.4.2}
\contentsline {section}{\numberline {4.3}\leavevmode {\color {Chapter }Checking a result by P. Andaloro}}{61}{section.4.3}
\contentsline {section}{\numberline {4.4}\leavevmode {\color {Chapter }Two examples by Matthews and Leigh}}{62}{section.4.4}
\contentsline {section}{\numberline {4.5}\leavevmode {\color {Chapter }Exploring the structure of a wild rcwa group}}{64}{section.4.5}
\contentsline {section}{\numberline {4.6}\leavevmode {\color {Chapter }A wild rcwa mapping which has only finite cycles}}{66}{section.4.6}
\contentsline {section}{\numberline {4.7}\leavevmode {\color {Chapter }An abelian rcwa group over a polynomial ring}}{70}{section.4.7}
\contentsline {section}{\numberline {4.8}\leavevmode {\color {Chapter }An rcwa representation of a small group}}{71}{section.4.8}
\contentsline {section}{\numberline {4.9}\leavevmode {\color {Chapter } An rcwa representation of the symmetric group on 10 points }}{72}{section.4.9}
\contentsline {section}{\numberline {4.10}\leavevmode {\color {Chapter }Checking for solvability}}{75}{section.4.10}
\contentsline {section}{\numberline {4.11}\leavevmode {\color {Chapter }Some examples over (semi)localizations of the integers}}{76}{section.4.11}
\contentsline {section}{\numberline {4.12}\leavevmode {\color {Chapter } Twisting 257-cycles into an rcwa mapping with modulus 32 }}{79}{section.4.12}
\contentsline {section}{\numberline {4.13}\leavevmode {\color {Chapter } The behaviour of the moduli of powers }}{80}{section.4.13}
\contentsline {section}{\numberline {4.14}\leavevmode {\color {Chapter } Images and preimages under the Collatz mapping }}{81}{section.4.14}
\contentsline {section}{\numberline {4.15}\leavevmode {\color {Chapter } A group which acts 4-transitively on the positive integers }}{83}{section.4.15}
\contentsline {section}{\numberline {4.16}\leavevmode {\color {Chapter } A group which acts 3-transitively, but not 4-transitively on Z }}{92}{section.4.16}
\contentsline {section}{\numberline {4.17}\leavevmode {\color {Chapter } Grigorchuk groups }}{95}{section.4.17}
\contentsline {section}{\numberline {4.18}\leavevmode {\color {Chapter } Forward orbits of a monoid with 2 generators }}{97}{section.4.18}
\contentsline {section}{\numberline {4.19}\leavevmode {\color {Chapter } Representations of the free group of rank 2 }}{98}{section.4.19}
\contentsline {section}{\numberline {4.20}\leavevmode {\color {Chapter } Representations of the modular group PSL(2,Z) }}{99}{section.4.20}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter }The Algorithms Implemented in RCWA}}{101}{chapter.5}
\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter }Installation and auxiliary functions}}{108}{chapter.6}
\contentsline {section}{\numberline {6.1}\leavevmode {\color {Chapter }Requirements}}{108}{section.6.1}
\contentsline {section}{\numberline {6.2}\leavevmode {\color {Chapter }Installation}}{108}{section.6.2}
\contentsline {section}{\numberline {6.3}\leavevmode {\color {Chapter }The Info class of the package}}{108}{section.6.3}
\contentsline {subsection}{\numberline {6.3.1}\leavevmode {\color {Chapter }InfoRCWA}}{108}{subsection.6.3.1}
\contentsline {section}{\numberline {6.4}\leavevmode {\color {Chapter }The testing routine}}{108}{section.6.4}
\contentsline {subsection}{\numberline {6.4.1}\leavevmode {\color {Chapter }RCWATest}}{108}{subsection.6.4.1}
\contentsline {section}{\numberline {6.5}\leavevmode {\color {Chapter }Building the manual}}{109}{section.6.5}
\contentsline {subsection}{\numberline {6.5.1}\leavevmode {\color {Chapter }RCWABuildManual}}{109}{subsection.6.5.1}
