01/15/2008, 09:30 AM
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.
Andrew Robbins
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.
Andrew Robbins