02/09/2008, 11:50 AM
Dear Gottfried, concerning:
I read your interesting report and I have to confess that I didn't assimilate all its contents. I measured, in doing so, the achievement of my ... perfect incompetence level, according to the Peter's Principle.
I shall read it, calmly, again !
In particular, I read with a lot of interest the list of important questions that still need to be answered, among which: "Why tetration oscillates for b < e^(-e)?". As a matter of fact, I think that it also oscillates when e^(-e) < b < 1, before reaching a constant real asymptotic value, depending on b, when x -> oo.
Why that ? Perhaps, because y = b[4]x is complex, for b < 1 ? But, we need a demonstration for saying that. Maybe your matrix approach will help. Please find attached some considerations on serial developments, based on information found in the Web. It may be useful.
But, if y = b[4]x oscillates at b < 1, what will be the meaning of the superlog, in that domain? Think of a simple "wild" graphical inversion or of a more mathematically correct non-inversiblity of it. Have we to consider that, perhaps, the slog "cannot be defined" for b < 1? This would have immediate consequences in the definition of "pentation" in that base area.
GFR
NB: This is my second posting, slightly modified. The last one was lost. Error at my side? Obscured, because badly formulated ?
By the way, "latte*makkiato", in general, is not equal to "kaffé*makkiato", but there is a fixpoint. Something like "kapputschino". Amazing !
Gottfried Wrote:ContinuousIteration, ... may of interest.
.......
merely more or less speculation, and only motivated to find any usble entry-point to access the problem, which I focus in this thread.
I read your interesting report and I have to confess that I didn't assimilate all its contents. I measured, in doing so, the achievement of my ... perfect incompetence level, according to the Peter's Principle.
I shall read it, calmly, again !In particular, I read with a lot of interest the list of important questions that still need to be answered, among which: "Why tetration oscillates for b < e^(-e)?". As a matter of fact, I think that it also oscillates when e^(-e) < b < 1, before reaching a constant real asymptotic value, depending on b, when x -> oo.
Why that ? Perhaps, because y = b[4]x is complex, for b < 1 ? But, we need a demonstration for saying that. Maybe your matrix approach will help. Please find attached some considerations on serial developments, based on information found in the Web. It may be useful.
But, if y = b[4]x oscillates at b < 1, what will be the meaning of the superlog, in that domain? Think of a simple "wild" graphical inversion or of a more mathematically correct non-inversiblity of it. Have we to consider that, perhaps, the slog "cannot be defined" for b < 1? This would have immediate consequences in the definition of "pentation" in that base area.
GFR
NB: This is my second posting, slightly modified. The last one was lost. Error at my side? Obscured, because badly formulated ?
By the way, "latte*makkiato", in general, is not equal to "kaffé*makkiato", but there is a fixpoint. Something like "kapputschino". Amazing !