09/24/2009, 11:39 PM
Okay, now for some details.
I've basically followed the steps outlined by Henryk in his summary of Kneser's paper:
Kneser's Super Logarithm
Before I start, I'm going to put up a legend of sorts. Kneser's paper makes use of a strange font for labelling the various images of curves and regions, so I thought I'd use his convention, to give a frame of reference to his paper. In the legend I show the scripted letter and the same letter in a plain font, for reference, as well as the relationship between the various images.
So, I start with the same regions Henryk showed in his post. I don't actually use the blue lines in my calculations; they are there merely to show graphically the relationship between the various graphs (important when we get to the region that I map to the unit disk).
I iteratively perform the natural logarithm and rescale to get the chi function. Note that my graph is rotated 180 degrees relative to Henryk's, but mine matches Kneser's, and I've triple-checked my results:
First note that I've added C-1, D''-1, D'2, and C2. This is to allow me to close off all three regions, K-1, K0, and K1. Note that I've continued the D curves further than Kneser (who was merely sketching, and didn't have access to SAGE or a similar product
). I did this for two reasons. One, the curves have a fractal nature that is fascinating in its own right, quite apart from its relevance here. See my thread:
The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!]
The second reason is that it helps us see that the D curves are not simple, and thus we cannot hope for a simple description of them. This extends likewise to the F curves (the image of the logarithm of the D curves). Kneser makes a detailed argument that each region L0, L1, etc., is simply connected to its immediate neighbors and is disjoint with all other regions. This seems obvious at first, but is quite a bit less obvious when you see the fractal curves of F0, F1, etc., and the way that they seem to overlap (they don't).
I've basically followed the steps outlined by Henryk in his summary of Kneser's paper:
Kneser's Super Logarithm
Before I start, I'm going to put up a legend of sorts. Kneser's paper makes use of a strange font for labelling the various images of curves and regions, so I thought I'd use his convention, to give a frame of reference to his paper. In the legend I show the scripted letter and the same letter in a plain font, for reference, as well as the relationship between the various images.
So, I start with the same regions Henryk showed in his post. I don't actually use the blue lines in my calculations; they are there merely to show graphically the relationship between the various graphs (important when we get to the region that I map to the unit disk).
I iteratively perform the natural logarithm and rescale to get the chi function. Note that my graph is rotated 180 degrees relative to Henryk's, but mine matches Kneser's, and I've triple-checked my results:
First note that I've added C-1, D''-1, D'2, and C2. This is to allow me to close off all three regions, K-1, K0, and K1. Note that I've continued the D curves further than Kneser (who was merely sketching, and didn't have access to SAGE or a similar product
). I did this for two reasons. One, the curves have a fractal nature that is fascinating in its own right, quite apart from its relevance here. See my thread:The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!]
The second reason is that it helps us see that the D curves are not simple, and thus we cannot hope for a simple description of them. This extends likewise to the F curves (the image of the logarithm of the D curves). Kneser makes a detailed argument that each region L0, L1, etc., is simply connected to its immediate neighbors and is disjoint with all other regions. This seems obvious at first, but is quite a bit less obvious when you see the fractal curves of F0, F1, etc., and the way that they seem to overlap (they don't).
~ Jay Daniel Fox
