adding dimensions for a reason tommy1729 Ultimate Fellow Posts: 1,395 Threads: 340 Joined: Feb 2009 11/12/2012, 05:29 PM (This post was last modified: 11/12/2012, 05:34 PM by tommy1729.) In the imho very intresting post : http://math.eretrandre.org/tetrationforu...hp?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 $1 , x , x_2 , x_3 , x_4 , x_5 , x_6 , x_7$ be the cyclic group C8. Now construct the group ring by extending with the real $R$. ( example of an element is $2,5 - 3x + 5x_2$ see http://en.wikipedia.org/wiki/Integral_group_ring) Now we define a "complex absolute value" operator by $C$. $C(a + b x + c x_2 + ... ) = a + b 1^{1/8} + c 1^{2/8} +$... 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 $C$(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 $C$Q' = Q = Q2 Let Q'' be a T8 value of the q+r th iteraton of f*(z). Now $C$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 ? « Next Oldest | Next Newest »