02/21/2009, 02:59 PM
(This post was last modified: 02/21/2009, 05:25 PM by sheldonison.)

bo198214 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.

And otherwise of course you surely read already Jay's post about change of base, which attributes the same effect you describe here that converges/should converge to a constant for if . Moreover he gives a formula to derive from any given such that exactly this condition is satisfied.

Quote:I have only a rough idea about what you suggest, so please let me ask some more questionsFor 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).

Let , when we consider we realize that , and . So the critical region is at infinity, where there .

Vice versa and .

I still dont see how you will use this for defining .

Quote:What do you mean by "there are fewer and fewer degrees of freedom for how to extend the sexp function to the real numbers"?

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.

Because of my earlier interest in the inflection point, I took that as a good region of the curve to use to estimate the tetration curve, using bases a little bit larger than e^(1/e). This is the most linear part of the curve, and for bases a little larger than e^(1/e), the linear section lasts a long time.

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.

The data I got seemed a little wobbly, or 1-cyclic as you described it, for the equation slog(sexp(x)), where I'm using Kouznetsov's taylor series versus the data I generated converting from a smaller base to base e. The wobble turned out to be quite a bit larger than the error term for my estimates, so I conjectered that there is no exact base conversion formula, and that the result will be approximately some number, converging to 1-cyclic.

Then you suggested that base conversion could be a uniqueness criteria, and --- a lightbulb went off!

I suddenly realized that one could use any base, as long as its larger than e^(1/e) to do the conversion. 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.

If there is a valid base conversion formula, as opposed to a 1-cyclic base conversion formula, then defining tetration for a base approacing e^(1/e) defines the tetration sexp to real numbers for all bases.

Kind regards, and sorry if my posts seem confusing,

- Sheldon Levenstein