  
  [1m[4m[31m1. Preface[0m
  
  
  [1m[4m[31m1.1 Motivation[0m
  
  The  development of this package has originally been motivated by the famous
  3n+1  -  Conjecture,  which asserts that iterated application of the Collatz
  mapping
  
  
                                         /
                                        | n/2 if n even,
                 T:  Z -> Z,   n  |->  <
                                        | (3n+1)/2 if n odd
                                         \
  
  
  to any given positive integer eventually yields 1.
  
  This  has  been  conjectured by Lothar Collatz in the 1930s, and is still an
  unsolved  problem  today.  Jeffrey  C.  Lagarias has written and maintains a
  commented  bibliography [L06], which currently lists about 200 references to
  publications  related to Collatz' conjecture. None of the articles mentioned
  there  tries to attack the problem by means of group theory, or investigates
  the  structure  of groups generated by bijective mappings which are "similar
  to  the  Collatz  mapping", i.e. [22m[36mresidue class-wise affine[0m. In fact, [22m[36mresidue
  class-wise  affine  groups[0m  apparently have not been treated anywhere in the
  literature before.
  
  After  having  investigated  these objects for a couple of years, the author
  feels that this is a gap which is worth to be filled.
  
  
  [1m[4m[31m1.2 Purpose of this package[0m
  
  So far, compared to classes of groups like for example matrix groups, finite
  permutation  groups  or polycyclic groups, only relatively basic facts about
  residue  class-wise  affine  groups  are  known. This package is intended to
  serve  as  a  tool  for  obtaining  a better understanding of their rich and
  interesting group theoretical and combinatorial structure.
  
  This  manual is pure software documentation, and as such it does not contain
  any  theorems or proofs. In a few places, where this is absolutely necessary
  for understanding what some function is good for, corresponding mathematical
  assertions  are  made.  Proofs  of  all  of  them  as  well  as  a  detailed
  introduction into the subject can be found in the author's PhD thesis [K05].
  A  copy  of  this  thesis and an english translation thereof are distributed
  with this package (see [1mthesis/thesis.pdf[0m resp. [1mthesis/thesis_e.pdf[0m).
  
  
  [1m[4m[31m1.3 Scope of this package[0m
  
  This  package  being a research tool which can be applied in various ways to
  various  different  problems,  it  is simply not possible to say what can be
  found  out  with  it  about which mappings or groups. The best way to get an
  idea  about this is likely to experiment with the examples discussed in this
  manual and included in the file [1mpkg/rcwa/examples/examples.g[0m. Another source
  of  examples  is  the  [1m[34mRandom[0m ([1m3.1-2[0m) - function. If you have LaTeX and [1mxdvi[0m
  installed,  you  can  nicely  display  examples of residue class-wise affine
  mappings by repeatedly issueing [22m[32mDisplay(Random(RCWA(Integers)):xdvi);[0m.
  
  Often  the  package  does not provide an out-of-the-box solution for a given
  problem.  At  the  beginning you will perhaps notice extremely long runtimes
  for  seemingly  trivial  things.  But with some experience you will learn to
  estimate in advance how long something will take and to see why raising some
  harmlessly-looking mapping to the 20th power would take terabytes of memory,
  while one can easily find out nontrivial things about some group which looks
  much  more  complicate.  Quite  often it is possible to find an answer for a
  given question by using an interactive trial-and-error approach.
  
  Among  many  other results, with substancial help of this package the author
  has  found  a  non-trivial  normal  subgroup  of  the  group  of all residue
  class-wise  affine  permutations  of the integers. Interactive sessions with
  this  package  have  also  lead to the development of a method for factoring
  residue  class-wise  affine  permutations  into  involutions  which  have  a
  particularly simple structure (see [1m[34mFactorizationIntoCSCRCT[0m ([1m2.4-1[0m)).
  
  
  [1m[4m[31m1.4 Acknowledgements[0m
  
  I  would like to thank Bettina Eick for her kind help in trying to make this
  package and in particular its documentation more useful and more interesting
  for  a  larger  number  of people. Furthermore I would like to thank the two
  anonymous referees for their constructive criticism and helpful suggestions.
  
  If  you use [1mRCWA[0m in some work leading to a publication, I ask you to cite it
  just  as  you  would cite a journal article. I would be grateful for any bug
  reports,  comments or suggestions and of course for reports of results found
  with the help of this package.
  
