01/17/2008, 04:02 PM
Here is Andrews' observation from the thread tetration of h(0) :
I do not know how to copy Tex so I copy the link:
http://math.eretrandre.org/tetrationforu...825#pid825
When I was speaking in previous post about the missing dimension, I was thinking about rotating plane ( or rotation in plane) perpendicular the graphs of GFR and parallel to y axis.
Then I have not even a guess, but just a vision of these 3 definitions of h(x) forming a double helix (odd/even) with a thread (x^1/x) in the midlle in region e^-e<x<e^(1/e). The projection on GFR graph is just a single line, but projection on that perpendicular plane is 3 points at each x.
When bifurcation happens x< e^-e, we get 2 opening spirals rotating left (odd?) and right (even?) in that plane and I am not sure what happens with x^(1/x) central thread there-it may tend to become a rotating connection between these spirals of that circle.
At x=0 we may have a unit circle and x^(1/x) a rotating diameter or radius?
That perpendicular plane is necesserily complex plane.
Would that be possible to somehow then connect with Gotfrieds spider like imaginary zeros x>e^1/e of so that his complexplane is situated in this construction perpendicularly x but behind x> e^1/e? With all the branches.
Then we would be able to connect somehow that region above e^1/e on that complex plane with the region x(0, 1) in other complex plane I proposed via region in the middle-by moving this plane as a crossection of h(x) in 3 dimensions.
May be this would not work without involving superroots as well. Would that not allow to understand the phenomena of bifurcation as such better? And phase transitions?
Ivars
I do not know how to copy Tex so I copy the link:
http://math.eretrandre.org/tetrationforu...825#pid825
Quote: Each one of these definitions gives a different answer as x approaches 0. The strange thing is that all 3 definitions are equivalent for e^-e<x<e^1/e.
Andrew Robbins
When I was speaking in previous post about the missing dimension, I was thinking about rotating plane ( or rotation in plane) perpendicular the graphs of GFR and parallel to y axis.
Then I have not even a guess, but just a vision of these 3 definitions of h(x) forming a double helix (odd/even) with a thread (x^1/x) in the midlle in region e^-e<x<e^(1/e). The projection on GFR graph is just a single line, but projection on that perpendicular plane is 3 points at each x.
When bifurcation happens x< e^-e, we get 2 opening spirals rotating left (odd?) and right (even?) in that plane and I am not sure what happens with x^(1/x) central thread there-it may tend to become a rotating connection between these spirals of that circle.
At x=0 we may have a unit circle and x^(1/x) a rotating diameter or radius?
That perpendicular plane is necesserily complex plane.
Would that be possible to somehow then connect with Gotfrieds spider like imaginary zeros x>e^1/e of so that his complexplane is situated in this construction perpendicularly x but behind x> e^1/e? With all the branches.
Then we would be able to connect somehow that region above e^1/e on that complex plane with the region x(0, 1) in other complex plane I proposed via region in the middle-by moving this plane as a crossection of h(x) in 3 dimensions.
May be this would not work without involving superroots as well. Would that not allow to understand the phenomena of bifurcation as such better? And phase transitions?
Ivars