  
  [1m[4m[31m1. Preface[0m
  
  
  [1m[4m[31m1.1 Motivation[0m
  
  The  development  of  this  package  has been motivated by the famous 3n+1 -
  Conjecture, which states that iterated application of the Collatz mapping
  
  T: Z -> Z, n |-> n/2 if n even, and 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.  Replacing  the Collatz mapping T by its conjugate
  under some permutation sigma of Z fixing 1 and setwisely fixing the positive
  integers   turns  the  3n+1  -  Conjecture  into  the  following  equivalent
  conjecture:  forall  n  in  N exists k in N_0: n^(T^sigma)^k = 1. The 3n+1 -
  Conjecture  is  true  if  and only if there is such a permutation sigma such
  that  T^sigma  maps all integers n > 1 to smaller ones. Hence the problem is
  to find a suitable normal form for the Collatz mapping T.
  
  Investigating  arbitrary  permutations  of finite sets is both theoretically
  and  computationally  a  hard problem. It is straightforward to restrict the
  considerations  to  permutations  which are "similar" to the Collatz mapping
  itself.  Probably  the  most  natural class of such mappings is the class of
  residue class-wise affine permutations.
  
  Jeffrey  C.  Lagarias  has  written  and  maintains a commented bibliography
  [L04],  which  currently lists 193 references to publications related to the
  3n+1  - Conjecture. None of the articles which are referenced there tries to
  attack  the  problem by means of group theory, or investigates the structure
  of  groups  generated  by  bijective  residue class-wise affine mappings. In
  fact,  the  subject [22m[36mresidue class-wise affine groups[0m apparently has not been
  treated anywhere in the literature before.
  
  The  author  feels  that  this  is a gap which is worth to be filled, and in
  particular that the group of all residue class-wise affine bijections of the
  integers is an object of natural interest in its own.
  
  
  [1m[4m[31m1.2 Purpose of this package[0m
  
  The  purpose  of  this  package  is  to  serve  as  a tool for computational
  investigations of residue class-wise affine mappings and -groups mainly over
  the ring of the integers.
  
  Perhaps  the only sensible reason for using this package is being fascinated
  by the mathematical beauty of the objects it helps to investigate.
  
  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  for  all  of  them  as  well  as  a detailed
  development  of  the  theoretical background for the subject can be found in
  the author's PhD thesis [K05].
  
  
  [1m[4m[31m1.3 Scope of this package[0m
  
  There  are  relatively elaborate methods for dealing with residue class-wise
  affine  groups  whose  elements  have  a  bounded number of different affine
  partial  mappings. These groups have a relatively easy structure. Hence they
  are  called  [22m[36mtame[0m.  Understanding tame residue class-wise affine groups is a
  necessary  prerequisite  for  being  able to investigate those which are not
  tame.
  
  The  groups  which  are  not  tame are much more difficult to examine. Hence
  these  groups  are  called [22m[36mwild[0m. Nevertheless, some computations can be done
  also  in  them  --  there  is  e.g. an algorithm for factoring elements into
  generators, which usually works reasonably well if the resulting word is not
  "too long". Often it is also possible to get useful information about a wild
  group  by considering its action on finite orbits -- of course provided that
  such orbits exist.
  
  It cannot be said in a few sentences what can be found out with this package
  about which mappings or groups. The best way to get an idea about this is to
  experiment  yourself with the examples discussed in this manual and included
  in  the file [1mpkg/rcwa/examples/examples.g[0m. Another useful source of examples
  is  the  [1m[34mRandom[0m  ([1m3.1-2[0m)  -  function. Often the package does not provide an
  out-of-the-box  solution.  At  the  beginning  you  will perhaps be bored by
  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
  non-trivial  things  about  some  group  which  looks  much more complicate.
  Provided  that  you  have  LaTeX  and [1mxdvi[0m installed, you can nicely display
  examples      of      rcwa      mappings      by     repeatedly     issueing
  [22m[32mDisplay(Random(RCWA(Integers)):xdvi);[0m. Quite often it is possible to find an
  answer  for  a  given  question  by  using  an  interactive  trial-and-error
  approach.
  
  The  author  for  example  has found with substancial help of this package a
  non-trivial  normal  subgroup  of the group of all residue class-wise affine
  bijections of the integers.
  
  
  [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.
  
