02/21/2009, 06:36 PM

sheldonison Wrote:no, I haven't read Jay's post, but I certainly will! that's exactly what I was looking for. Thank you very much for pointing me to it. It will take awhile for me to digest.Jay made really good contributions though he was a bit short in making things publically understandable. So I often had difficulties to follow him. But you are anyway closer to the topic as you made the same observation like he did. Perhaps we can help each other to get a better understanding of the topic.

Quote:For b=e^(1/e), the critical section is at x=infinity. But for any b>e^(1/e), the critical section is at x= some finite large number, and not at infinity. I realize I'm a long way from a formal mathematical definition, especially since it would involve all of the complexities of Jay's base conversion post (which I'm reading now), extended to dealing with arbitrarily large base conversion factors as b gets closer to e^(1/e).

...

When the tetration base is equal to "e", the linear section only extends from sexp(-1 to 0), and after that, the function is logarithmic on the left, and exponential on the right. But for smaller bases, the transition from logarithmic to super-exponential takes much longer, as the sexp function climbs gradually, slowly making the transition from close to linear, until super-exponential behavior takes over.

Ah, ok, now I get it. With base going towards the inflection point goes to infinity and the mainly linear interval gets longer (though indeed the question is how to get this mathematically caught, but I see what you mean). The starting point of this interval is going also to infinity (?).

The slope in this linear interval goes to 0 (?).

How do you make use of this increased linearity to define ( little bigger thean )?

Just in the usual way by and ?

Quote:Then, assuming that there is a valid base conversion forumula, I extrapolated from the base a little bit larger than e^(1/e) and used that to define tetration data for other bases, like base 2, 3, e and 10.

how?

Quote:The wobble turned out to be quite a bit larger than the error term for my estimates,How do you obtain the error term?

Quote:And the closer you get to e^(1/e), the more linear the tetration curve would be. The contributions made by all the higher derivatives go to zero, and goes to zero faster than the slope of the critical section goes to zero. A linear approximation over the critical section is always guaranteed to have a continuous 1st derivative, no matter what base is used. As the base approaches, but is larger than e^(1/e), the tetration curve becomes more and more defined, with a linear approximation for the critical section, and with fewer and fewer degrees of freedom for alternative definitions of the critical section for how to extend sexp to real numbers.

I see, the proper tetrational should become more and more linear when the base approaches from above, and is characterized by this demand.

Quote:defining tetration for a base approacing e^(1/e) defines the tetration sexp to real numbers for all bases.

Here I see some difficulties to put that mathematically. You want to define tetration only for one base, but "approaching" means that you have to define tetration at least for a sequence of bases.

But perhaps this is just a question of exactness. The more exact the tetration for arbitrary bases should be the closer to one have to choose the base of the initial tetration, something in that direction.

Quote:and sorry if my posts seem confusing,

Ya, mathematics is always somewhat confusing, especially if it is in the making. As long as in the end a mutual clear understanding can be achieved there is nothing bad about some fog on the way.