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Attempting to compute the kslog numerically (i.e., Kneser's construction)
#4
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 Tongue ). 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
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Messages In This Thread
RE: Attempting to compute the kslog numerically (i.e., Kneser's construction) - by jaydfox - 09/24/2009, 11:39 PM

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