02/04/2008, 05:50 PM
Well, it's a long way... ! 
Let's start considering that, at rank s=4, tetration, we have that :
y = b^y = b[3]y <---> b = yth-rt(b) = selfrt(y) <---> y = b[4](+oo) = h <---> b = (+oo)th-srt(h).
We may then suppose that, very probably, we should also have that:
y = b§y = b[4]y <---> b = yth-srt(b) = selfsrt(y) <---> y = b[5](+oo) = g <---> b = (+oo)th-ssrt(g).
In other words, by calling "g" the infinite pentation (infinite super-tower .. !) to the base "b", we may discover that "g", at rank s=5 (like "h" for rank s=4) can also converge in some domains of "b". But this needs a strict mathematical demonstration, itself requiring the continuity (analyticity?) of both slog and sexp, at rank s=4. First, we must solve and consolidate this. Then, we shall see. The consequence of that would be that the infinite pentation "g(b)" would be depending on "b" and that g(b) should be the inverse of b(g), like h(b) is the inverse of b(h).
Nevertheless (...), even before any accurate demonstration, it can be easily seen that y = b[5]x = b§x = b-penta-x, at lest for bases near the value b = e, has an asymptotic behaviour, for x -> -oo (minus infinity). By synthetically examining the situation (please, ... try to remain cool, bo!), we must admit that, for x < 0, we indeed have that:
x = b#x <---> [ b]slog(x) = x <---> b#x = [ b]slog(x).
Now, both with the (GFR/KAR) "linear approximation" and with the Robbins (at present, more precise) "smoothings", we can easily obtain the value of "x", for which the before-mentioned relation is satisfied. We (KAR/GFR) checked it for base b = e and it worked nicely.
We also have the surprise to discover, for the "supertowers" the possibility of having "negative heights". How about the "push-down", Gottfried ?
As a (provisional) conclusion, I must say that the infinite supertowers (pentations) correspond to fixpoints of sexp(x) and slog(x), like the infinite towers (tetrations) were determined by the fixpoints of exp(x) and log(x). This "pattern" is probably repeated for all the hyperops hierarchy.
For base b = e, the sexp(x) and slog(x) admit one very clear fixpoint for x < 0, (we may call "sigma" the value of x satisfying that). Well, this "sigma" (if ... correctly calculated) is the ordinate of the horizontal asymptote of y = e[5]x.
In the other cases of b, the fixpoint are, maybe, two, three, ... oder ... ?
Here we are, for the moment.
GFR
Let's start considering that, at rank s=4, tetration, we have that :
y = b^y = b[3]y <---> b = yth-rt(b) = selfrt(y) <---> y = b[4](+oo) = h <---> b = (+oo)th-srt(h).
We may then suppose that, very probably, we should also have that:
y = b§y = b[4]y <---> b = yth-srt(b) = selfsrt(y) <---> y = b[5](+oo) = g <---> b = (+oo)th-ssrt(g).
In other words, by calling "g" the infinite pentation (infinite super-tower .. !) to the base "b", we may discover that "g", at rank s=5 (like "h" for rank s=4) can also converge in some domains of "b". But this needs a strict mathematical demonstration, itself requiring the continuity (analyticity?) of both slog and sexp, at rank s=4. First, we must solve and consolidate this. Then, we shall see. The consequence of that would be that the infinite pentation "g(b)" would be depending on "b" and that g(b) should be the inverse of b(g), like h(b) is the inverse of b(h).
Nevertheless (...), even before any accurate demonstration, it can be easily seen that y = b[5]x = b§x = b-penta-x, at lest for bases near the value b = e, has an asymptotic behaviour, for x -> -oo (minus infinity). By synthetically examining the situation (please, ... try to remain cool, bo!), we must admit that, for x < 0, we indeed have that:
x = b#x <---> [ b]slog(x) = x <---> b#x = [ b]slog(x).
Now, both with the (GFR/KAR) "linear approximation" and with the Robbins (at present, more precise) "smoothings", we can easily obtain the value of "x", for which the before-mentioned relation is satisfied. We (KAR/GFR) checked it for base b = e and it worked nicely.
We also have the surprise to discover, for the "supertowers" the possibility of having "negative heights". How about the "push-down", Gottfried ?
As a (provisional) conclusion, I must say that the infinite supertowers (pentations) correspond to fixpoints of sexp(x) and slog(x), like the infinite towers (tetrations) were determined by the fixpoints of exp(x) and log(x). This "pattern" is probably repeated for all the hyperops hierarchy.
For base b = e, the sexp(x) and slog(x) admit one very clear fixpoint for x < 0, (we may call "sigma" the value of x satisfying that). Well, this "sigma" (if ... correctly calculated) is the ordinate of the horizontal asymptote of y = e[5]x.
In the other cases of b, the fixpoint are, maybe, two, three, ... oder ... ?
Here we are, for the moment.
GFR