Collatz-Intro - Notations |
Workshop |
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I found
some sources, which use an analoguous definition for the notation of the
collatz-transformation, but I think, that my proposal here is worth to be
favored. The original Collatz-problem seem to have been stated in the following form: For any natural number>0 define a Collatz-transformation from a to b CT_odd:= b = 3*a +1 if a is odd CT_even:= b = a/2 if a is even That means, after a CT_odd always a CT_even-transformation must follow, but it is possible, that many CT_even-transformations follow each other. The first enhancement in notation could then be to put all CT_even together in a parametrized transformation CT_even1:= b = a/2^A with A being the exponent, which makes b an odd natural. |
I put both transformation together in one formula and call that transformation T(a;A):= b = (3a+1)/2^A In this notation "A" normally is a parameter, which is not free but exactly determined by the value of "a" (this might confuse the appropriate understanding of the underlying problem). But for a general analysis it is useful, · if we deal with analytical manipulations · if we deal with a generalized view of the problem, where we allow intermediate rational values for a · if "a" is unknown beforehand and the collatz-process is analyzed regarding such exponents · if we deal with the inversion of the operation. The iteration of such transformations ("multi-step-transformation") may be noted as T(a;A,B):= T(T(a;A);B) to any extent, example b = T(a;1,4,2,1) means b = ((((a*3+1)/2)*3+1)/16)*3+1)/4)*3+1)/2 The inverse operation means C(b;A):= a = (b*2^A-1)/3 where A is now a free parameter. C(b;A) is the inverse of T(a;A) and T(C(a;A);A) = a Note that the inversion of a multi-step-transformation requires to write the exponents in reverse order: b = T(a;A,B,C) a = C(b;C,B,A) For the sake of not wasting too many letters in longer formulas, I sometimes write this a' = T(a;A,B,C,...) The quote is currently not meant as a derivation-symbol or a transposition in all these following text. |
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last update: 15.8.2004 |
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