01/15/2008, 03:36 PM
Sorry again!
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.
GFR
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.
GFR