02/11/2008, 10:15 PM
Infinite tetrates may be real or, indeed ...., complex.
My guess is that tetraton (y = b[4]x) is one multivalued real and/or complex "function" of x, depending on the value of base b > 0. Oscillations of y = b[4]x, for constant b, should be produced, at b < 1, by a complex y, multivalued "function" of a real variable x, which appear as such in a "real" projection on the yx plane. But, we must find such a "function", the smoothness of which would appear in this real projection (continuous and infinite-time derivable, continuity class Coo). Outside the yellow zone, the oscillations vanish for x -> oo. Inside the yellow zone, they remain persistent at x = +oo (sorry for my ... non standarization!).
Your question concerning the selfroot, if intended as the b-solution of y = b^y, is crucial. Either this solution is exclusively given by b = selfrt(y) = y^(1/y), and in this case we have to explain why the "yellow zone" appears, or the b-solution of y = b^y is not the selfroot alone, but it is accompanied by other "functions" or branches. In the second case the explicitation (extraction of b) in y = b^y would have (at least) two branches, the selfroot and the perimeter of the yellow zone. Its inverse, as I see it, must have (... at least) four branches, as it is shown by a "wild" graphical inversion.
The beginning of this analysis is the study of y = b^(b^y)), equivalent to y = b^y. But, I don't see how to proceed, in an exhaustive mathematical way. I found a candidate for the additional branches (yellow zone), but I cannot justify why it may be so. Nevertheless, it seems to work
!
The big problem is to show and correctly demonstrate all this. Unfortunately, at this date, I only have clear, but ... vague, guessings.
GFR
My guess is that tetraton (y = b[4]x) is one multivalued real and/or complex "function" of x, depending on the value of base b > 0. Oscillations of y = b[4]x, for constant b, should be produced, at b < 1, by a complex y, multivalued "function" of a real variable x, which appear as such in a "real" projection on the yx plane. But, we must find such a "function", the smoothness of which would appear in this real projection (continuous and infinite-time derivable, continuity class Coo). Outside the yellow zone, the oscillations vanish for x -> oo. Inside the yellow zone, they remain persistent at x = +oo (sorry for my ... non standarization!).
Your question concerning the selfroot, if intended as the b-solution of y = b^y, is crucial. Either this solution is exclusively given by b = selfrt(y) = y^(1/y), and in this case we have to explain why the "yellow zone" appears, or the b-solution of y = b^y is not the selfroot alone, but it is accompanied by other "functions" or branches. In the second case the explicitation (extraction of b) in y = b^y would have (at least) two branches, the selfroot and the perimeter of the yellow zone. Its inverse, as I see it, must have (... at least) four branches, as it is shown by a "wild" graphical inversion.
The beginning of this analysis is the study of y = b^(b^y)), equivalent to y = b^y. But, I don't see how to proceed, in an exhaustive mathematical way. I found a candidate for the additional branches (yellow zone), but I cannot justify why it may be so. Nevertheless, it seems to work
!The big problem is to show and correctly demonstrate all this. Unfortunately, at this date, I only have clear, but ... vague, guessings.
GFR
