Although I agree with your conclusion, I would say it different. Instead of saying:
"towers and super-roots are disjoint" I would say:
"the ranges of tetrates and super-roots are pointwise disjoint over the domain (0, 1)."
Also, a slightly more accurate statement would be \( (\mathbb{N} = \mathbb{Z}^{+}) \):
"\( \text{srt}^{(2\mathbb{N})}(x) \ <\ x^{1/x} \ <\ \text{srt}^{(2\mathbb{N}+1)}(x) \ <\ x \ <\ {}^{(2\mathbb{N}+1)}x \ <\ {}^{\infty}x \ <\ {}^{(2\mathbb{N})}x \) for all \( 0 < x < 1 \)"
GFR Wrote:"\( \text{srt}^{({N})}(x) \ <\ x^{1/x} \)"
...
Do you agree? Sorry, I am still not familiar with TeX.
I agree with everything except the above part, this should read:
"\( x^{1/x} \ <\ \text{srt}^{(\mathbb{N})}(x) \)"
@GFR
By the way, I like your itemization of the "remarkable numbers" related to e. If you look at Robert Munafo's http://home.earthlink.net/~mrob/pub/math....html#la15 he has a bit about e^e, just to let you know.
What I meant is that physically infinite or even finite tetrations means process being INSIDE phase transition region. That is a fantastic place to be since no one has ever been able to stay there for long.
I just wish You success as the tools You are developing might be quite soon becoming a routinely needed in any science trying to deal with far from equlibrium processes.
With this I will probably depart, since I have found what I was looking for.
Best wishes and many many thanks to everyone who responded to my mathematically annoying attempts.
About GFR's "yellow zone" (YZ) or Ivars' "phase transition region" I was very interested to see the formula \( x = (y^{-y} -1)\frac{e^{-e}}{e^{1/e}-1} \). I think it is a very good approximation, but I think it is not accurate. In the YZ, the variables you gave should satisfy \( y = x^{x^y} \) for each of the two stems off the main graph. This equation has been discussed by both Knoebel ("Exponentials Reiterated"), and Galidakis ("... solving the p-th auxiliary equation..." or something), so we should probably see what they say about the subject. Until then I am including a graph which includes (black = GFR-approximation, blue = super-roots of height 10000/10001).
Thats odd... when I try and solve the equation \( y = x^{x^y} \), I get \( x = e^{1/(y \ln y)} \), but clearly, GFR, your approximation is better than this. I must be making a mistake somewhere... I have included a graph with (blue = 10000-10001 super-roots, green = GFR-approximation, red = my-approximation).
Anyways, Knoebel seems to focus on the limits of iteration of this equation and proving the disjoint-ness rather than parameterizing, while Galidakis focuses on the algebraic properties of his HW function, which is already a function defined as the inverse of something like \( c = yb^{b^y} \), it is no surprise that it can solve an equation of that form. I am interested though, because for some reason I thought one of them had solved it. I guess not. I suppose the "parameterization of the bifurcation of tetration" remains an open problem.
01/15/2008, 11:22 AM (This post was last modified: 01/15/2008, 11:23 AM by Ivars.)
I still want to be around
A quote from Corless: http://www.apmaths.uwo.ca/~rcorless/fram.../LambertW/
Robert M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, ``On the Lambert W Function", Advances in Computational Mathematics, volume 5, 1996, pp. 329--359.
Quote:
Euler observed in L. Euler De formulis exponentialibus replicatis'
that the equation g= z^z^g sometimes has real roots g that are
not roots of h=z^h. A complete analysis of such questions considering also the complex
roots involves the T function as shown by Hayes in ND Hayes "The roots of the equation x
=c (exp)^n (x) and the cycles of the substitution
x ce^x
Thank you for being around, IVARS! I shall study your citation.
@andydude: Let us examine carefully your approximation. The fact is that I cannot fully justify mine!!
The problem is that this "yellow zone" or "phase transition region" (the "off limits" area for ... integers) is of an extremely high importance for our field of research.
For our discussions, I accept the (Jaydfox' ?) idea to put "Eta = e^(1/e)" and, similarly, I propose to put "Beta = e^(-e)". In this case, the Euler's domain (and range) of the bases, for which the infinite towers converge, being defined as:
e^(-e) =< b < e^(1/e), i.e. Beta < b < Eta
could be called the "Eta-Beta" domain (and/or range). Which is amusing !!?!! (;-->)
More seriously, I should like to propose a (very probable) interpretation o the "off limits" area, the perimeter of which , when superexponent x -> infinity (see my simulations and also those provided by andydude for one case in 0 < b < Beta). In fact, a cross section of the y(b) diagram, for b < Beta shows three real numbers, one of which [plog(-ln b)/(-ln b), lower "branch"] possibly indicating the limit, for superexponent x -> +oo, of the "average" of y(b). The other two numbers given, for instance, by my approximation, would indicate the max/min extesions of the y oscillations. I can provide graphical evidence of that. In that domain the oscillation are persistent, for x -> +oo.
