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adding dimensions for a reason - tommy1729 - 11/12/2012
In the imho very intresting post : http://math.eretrandre.org/tetrationforum/showthread.php?tid=499 I had the idea to " fix the problem ". And by that I mean to avoid the knots. How I intend to do this is by adding variables or dimensions if you like. Hence the title. Let be the cyclic group C8. Now construct the group ring by extending with the real . ( example of an element is see http://en.wikipedia.org/wiki/Integral_group_ring) Now we define a "complex absolute value" operator by . ... Lets call this group ring T8. Notice all functions on T8 are defined and computable apart from division by zerodivisors. Now let our starting value be z. Let f(z) map z to an element in T8 such that (f(z)) = z. There are many such f(z) and that is crucial here. Also C is far from a bijection , that is also crucial here. Let f*(z) be a suitable f(z) , where suitable is explained below. Let q be a positive real. Let r be a positive real. Let Q be the complex value of the q th iteration of the exp of z that is near a knot. Let Q2 be the complex value of the q+r th iteration of the exp of z that is near the same(!) knot , hence Q ~ Q2. Let Q' be a T8 value of the q th iteration of the exp of f*(z). Now Q' = Q = Q2 Let Q'' be a T8 value of the q+r th iteraton of f*(z). Now Q'' = Q = Q2 However suitable f*(z) means Q' =/= Q'' , which also mean that there is no knot (anymore in T8 space) ... and we have an invertible function ( continu!). this then solves the (original) problem for this particular knot. That is the basic idea. This is not the holy grail yet , but I think a promising idea. we thus need to work with sexp and slog on T8 space. and we need to find a way to work to find f*(z). And then there is ofcourse the fact of multiple knots rather than one. However statistically there exists a f*(z) that avoids all knots. statistically because if we consider the knots as collisions of a trajectory than adding dimensions means probability of hits goes down by the cardinality of the reals !! Plz do not confuse this with fixed points. There are fixpoints in both C and T8 but that is not directly related to this post. I wonder what you guys think. Regards Tommy1729 ps : sorry for my long absence. pps : where is bo ? |