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Tetration algorithm
#1
Quote:Is your tetration algorithm mathematically equivalent to Kneser?  For example, Walker's infinitely differentable solution is only defined at the real axis, but is a different 1-cyclic solution than Kneser's.

[tex]\text{sexp}_w(z)=\text{sexp}_k(z+\theta(z));[\tex] where theta is a 1-cyclic periodic function - Sheldon

Background: I come from a different era in tetration research. In the late eighties and early nineties, even though I was networking with Stephen Wolfram and spending a lot of time at the Library of Congress, I was unable to find any relevant published research papers. Of course, that now seems strange. 

My first tetration related connection was with Stephen Wolfram, in 1986, across three meetings totaling five hours. My goal was to publish my work on tetration so that I would have the leverage to be able to become Wolfram's graduate assistant. I was bummed out when he exited academia to focus on Mathematica. By 1987 I had a general derivation for hyperbolic tetration, but Wolfram upped the ante.

I believe most folks on the Tetration Forum are interested in tetration's mathematical beauty, but I caught the bug from Wolfram for unifying maps and flows. This would be fundamental to mathematical physics. According to Wolfram and other sources, the two separate mathematical foundations for physics are partial differential equations and iterated functions. PDEs are nicely connected to the classical mathematics of the last few hundred years. But according to Wolfram, the weakness of PDEs is their inability to master chaotic systems while iterated functions are great for dealing with different types of chaos, but there is a significant mathematical impediment in expressing their connection with PDEs.   

Answer: I jumped from tetration extended to the complex numbers to the continuously iterated functions. With the exception of Yiannis Galidakis' research, my results have been more general than the work of others I had access to. Hyperbolic tetration in 1987, hyperbolic and parabolic iterated functions in 1994. Honestly, I don't get my kicks studying other's research when its goal is to give more limited results than my own research. Check out my work at Finding f such that f(f(x))=g(x) given g and note that Terry Tao weighs in. Did I or did I not answer a classical mathematical problem with a three-line Mathematica program? 

PS: The research I've shared with folks is fifteen years old or more. I now focus on things like the Ackerman function and the iterated functions of the General Linear group.
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#2
(07/14/2019, 05:05 AM)Daniel Wrote: I jumped from tetration extended to the complex numbers to the continuously iterated functions.... Hyperbolic tetration in 1987, hyperbolic and parabolic iterated functions in 1994...
PS: The research I've shared with folks is fifteen years old or more. I now focus on things like the Ackerman function and the iterated functions of the General Linear group.

Yeah, that makes sense; Kneser's paper is for base e, for the elliptic case where the fixed point conjugate pairs aren't real valued.
- Sheldon
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#3
In the referenced the mathoverflow question I see two answers of you. The first from 2014 (actually the one where your link is pointing to) and the second from 2017 (where you give the Mathematica code).

In the first one I don't get why you claim that one could also use non-integer numbers for in your iteration formula. appears in the summation limit, so how can one plug non-integer numbers there? Or don't you claim it at all, so the formula is for natural numbers only (and then why would you post it as answer to this question)?

In the second one you solve for , assuming and  are formal poweseries with constant term equal to 0. You give a short Mathematica expression there (unfortunately I dont possess Mathematica so I can not even try it). But the formal powerseries expansion is not everything. Typically one is interested in analytic solutions (or merely computation). Then you should have some ideas about the radius of convergence. Or at least a proof that this radius non-zero. Did you consider this?

Luckily the proof of convergence was done by others long before you in the hyperbolic case, and your formula is not applicable to the parabolic case because the denominator would be 0 in the second derivation. But even the formula for the parabolic case is typically not convergent at the fixed point.

You have several quite questionable statements in your posts:
Quote:Honestly, I don't get my kicks studying other's research when its goal is to give more limited results than my own research. 
Quote:my results have been more general than the work of others I had access to
Quote:Did I or did I not answer a classical mathematical problem with a three-line Mathematica program?
Just for this question I cant help replying:
You make it sound like you solved a (long standing) open problem, but for me it feels more like an exercise in a course about formal powerseries.

But back to the original question from Sheldon. He mentioned already that he was talking about the elliptic case, i.e. where there are no real fixed points of , i.e. . Do you have some material about that?

Also I would like to know which fixpoint you used in the case of hyperbolic tetration and why. Also what is the concrete meaning of tetration for you here? Is it the fractional iteration of , is it the super-exponential or super-logarithm that you calculate?
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