07/14/2019, 05:05 AM

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.