03/03/2009, 07:27 PM
(This post was last modified: 03/04/2009, 02:42 PM by sheldonison.)
bo198214 Wrote:Are you sure that it is 1-periodic?You are correct, but it turns out not to matter that much. First, the two bases I'm comparing are the same. One is the ideal \( \text{sexp}_b \), for b a little bigger than \( \eta \), and the other is \( \text{sexp}_b \) converted from base e to base b. So slog(sexp(x))-x would have to be 1-cyclic. Since the base is approaching \( \eta \), and since I'm comparing sexp in the critical section, which is very linear, so I can get away with a short cut of just subtracting the two sexp functions, since I don't have an slog function in my spreadsheet. But for an sexp with a linear approximation over the critical section, its not that big a deal. One of the two waves is a line segment, and the other is a line segment plus a sinusoid with an amplitude of 0.0004 times the slope of the line segment. The difference is too small to be seen in the rough graphs I made.
I mean it is well known that
\( f^{-1}(g(x))-x=\theta(x) \) must be 1-periodic
for two superexponentials f and g.
This implies that
\( g(x)=f(\theta(x)+x) \)
But
\( f(\theta(x)+x)-f(x) \) does not look 1-periodic?
\( f(\theta(x+1)+x+1)-f(x+1)=\exp(f(\theta(x)+x)))-\exp(f(x))\neq f(\theta(x)+x)-f(x) \) mostly
Mostly, I'm looking for a curve fit to a sine wave for the 1-periodic transfer function, as opposed to a 1-cyclic wave with higher order terms. If it is a sine wave, then there is hope for calculating it theoretically, as opposed to empirically. Once we have the delta sinusoid, in theory the sine wave can be applied to the critical section for a base approaching \( \eta \), and then used to generate sexp for another base with all positive odd derivatives. I realize that this is pretty hypothetical, and that the only thing I'm basing this on is how good a curve fit there is between the graph, and an ideal sine wave, but an ideal sine wave makes the math a lot cleaner.
Maybe Dimitrii could verify my results, by graphing
\( \text{slog}_b(\text{sexp}_e(x))-x \) for b approaching \( \eta \).
If I'm right, the result will approach a perfect sine wave, as b approaches \( \eta \). The amplitude of the sine wave will converge on about 0.0004 as the conversion base approaches approaches \( \eta \).
new prediction: its even possible that all base conversions for Dimitrii's extension of sexp to real numbers will converge on 1-periodic sine waves, but surely someone would've noticed this already. Its all possible that it will be a sine wave in the sexp domain, (at the critical section, as the limit approaches e^(1/e)), but only an approximate sine wave in the slog domain.
