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Tetration FAQ Discussion
I just realized that the nth tetrate makes an incredibly unified terminology system if the other suffix used with tetration was used for iteration as well! I've been using the term "orbit" for the third way of viewing iteration but this is classically viewed as an indexed set, not a function per se. So this could be the new terminology:
  • iteration (binary), nth iterate , iterational (formerly orbit).
  • tetration (binary), nth tetrate (formerly hyperpower, tower), tetrational .

Andrew Robbins
Yes, I like the iterate/iterational and tetrate/tetrational terminology. I've been using "iterator" where you've been using orbit, though I like all three terms if I had to adopt one of them and stick with it.

And "multiple" at least rhymes with iterational, tetrational, and exponential (in the sense that the suffixes "-al" and "-le" sound the same, at least for a Californian like myself), making for a loosely consistent pattern.

One place it breaks down with other precedents is power/exponential. Calling x^3 the third exponentiate doesn't quite sound right (even if making an effort to pronounce it differently from the verb "to exponentiate"). But here, there is an ingrained terminology that may be causing a bias to what "sounds right".

Edit: Already the term "exponentiate" is starting to sound okay to me, used as a noun. Funny how the mind adapts.
~ Jay Daniel Fox
Just came across an article of Helmuth Kneser, which G.A.Edgar referenced in his reply:


Reelle analytische Lösungen der Gleichung ... und verwandter Funktionalgleichungen.
Autor: Kneser, Hellmuth
Journal für reine und angewandte Mathematik
Zeitschrift / Zeitschriften-Band 187 / Zeitschriften-Artikel

It covers many of our discussions here, but is in german lang.

Gottfried Helms, Kassel
What do you mean by "iteration formula"?

Andrew Robbins
The more I think about it, the more I think that the FAQ and the formula sheet should be combined.

Here is a combined outline for the possible merge of the two:
  • Preamble

  • Introduction to Hyper-operators
    • What is the Ackermann function?
    • Hyper-operator notation
    • Hyper-operator terminology
    • Hyper-operator identities

  • Introduction to Tetration
    • Tetration as the hyper-4-operator
    • Tetration notation and terminology
    • Sequence of n-th tetrates
    • Standard extensions of tetration
      • Extending tetration to zero base
      • Extending tetration to complex bases
      • Extending tetration to negative heights
      • Extending tetration to infinite heights

  • Introduction to Iteration
    • Dynamical systems translation equation
    • Abel's functional equation
    • Schroeder's functional equation
    • Boettcher's functional equation
    • Julia's functional equation
    • Tetration vs. Iterated Exp

  • Exponentiation and factorials
    • Table of exponentiation
    • Extension of exponentiation
    • Extension of factorials
    • Extension of super-factorials

  • Topological conjugacy
    • Lambert-W-like conjugates (srt2(x), W(x), x^^oo)
    • Exponential-like conjugates (b^x = (h^(1/h))^x, h^x - 1)
    • Topological conjugacy of the super-logarithm

  • Other extensions of Tetration
    • Ingolf Dahl's solution
    • Ioannis Galidakis' solution
    • Daniel Geisler's solution
    • Gottfried Helms' solution
    • Robert Munafo's solution
    • Cliff Nelson's solution
    • Peter Walker's solution
      • Andrew Robbins' solution
      • Jay D. Fox's solution

  • Other extensions of Hyper-operators
    • Bromer's mixed hyper-operators
    • Munafo's lower hyper-operators
    • Trappmann's binary-tree hyper-operators
    • Trappmann's associative/commutative hyper-operators
    • Trappmann's balanced hyper-operators

If you could provide the TeX source for your FAQ, I will combine it with the formula sheet...

Andrew Robbins
It has become large.

I think our intent is to make this a short introduction, not a full book on tetration, but at least the version I'm posting today has grown in length from the version Henryk started this thread with, but Henryk's content still needs to be merged in with this content. I have included Henryk's introduction, and in the file I have indicated where Henryk's sections would fit in with this outline.

In the version that I am posting, I have included many of my own results, results which are easy to verify, and results that do not need a great deal of mathematical background to verify. This is a list of my own work I have included:
  • Derivation of Munafo class using super-logarithms.
  • Extending hyper-N-operations to real number N.
  • Extending hyper-N-logarithms to real number N.
  • Extending Galidakis' Puiseux series to iterated exponentials. (trivial)
  • My Recurrence equation for Taylor series of tetration.
  • My Recurrence equation for Taylor series of iterated exponentials.
  • Topological conjugacy of exp., dec. exp., and scaled exp. (Exponential-like conjugates).
  • Topological conjugacy of prod. exp., self-power, and self-root (Lambert-like conjugates).

Of all of these, the theorem I was most reluctant to include is the Taylor series of tetration and iterated exponentials (pages 21-22). This recurrence equation took me several months of research to find, and as such, I want a little credit for it, but thats all. I have other results that could be much more valuable for publication, so I decided to include this recurrence equation after all.

Much of the content, like reviews of other people's work, I had actually written some time before, and was planning on writing a book about Tetration. Although this never happened, I thought these sections were appropriate for a FAQ, so I have included some of these sections verbatim. Since the topological conjugacy section involves commutative diagrams, I have included the package (diagrams.sty) in the folder I uploaded. Also included are the images used within the FAQ.

To everyone: please look over the material, see what you can improve, and what is missing, wrong, inappropriate, too advanced, misplaced, out of order, and so on. We may have to split this into two articles again, one for the FAQ and one for the collection of theorems, but I like the idea of having a single article to describe our progress.

Andrew Robbins

PS. If you don't want the sources, just download the PDF.

Attached Files
.zip (Size: 823.46 KB / Downloads: 619)
.pdf   FAQ_20080112.pdf (Size: 527.45 KB / Downloads: 8,171)
Hi Andy

I know I have to keep head cool, but the first fabulous graph on Your summary article is just the most general phase diagram possible in reals, and I just know it is true. It explains all existing phase diagrams and hints on some new. Next task is just to adapt parameters to fit experimental data-any. And the physics will pop out themselves from mathematical models You use.

Now imagine what it would look like when imaginary values are included and what would that mean.

It can be done as slowly as possible, of courseSmile

I appreciate the evolution of the FAQ draft.

May I submit to you all, and particularly to Henryk and Andrew, the attached notes that I prepared few months ago for my discussions with KAR ? They contain a bidimensional display trying to define the domains of existence of the "towers" and "superroots" and trying to show that they are disjoint. It needs to be carefully finalized.

I shall appreciate your kind comments on that. Thank you.


Attached Files
.pdf   Disjoint Tetrational Domains. Tetration Forum..pdf (Size: 78.64 KB / Downloads: 679)
Moderator Note: I split off the discussion about bifurcation, because this is mathematical discussion, not discussion about the FAQ.
andydude Wrote:It has become large.

Hey Andrew, you put really a lot of work in. Thanks for your contribution.
It seems however that we have to discuss what to put in the FAQ to keep it in a reasonable size.

Basically a FAQ shell prevent people starting to deal with this topic from reinventing the wheel, by clearly and elementary explaining the basics and give known answers to frequently asked questions or frequently occuring problems.

This should be our directive for designing the FAQ.
We should also be careful to not explain things based on our advanced background but instead being very explicit, elementary (and easy to read) and give a lot of examples.

As such I would seperate a (formula, theorem) reference from the FAQ.

I would ban the more exotic notations into a history chapter.

Also note that the box notation is not commonly used but rather our invention. The box tex-notation is also very difficult to use without predefined commands (for example between the tex-tags in this forum). I think we need a simple TeX-representation like a sub- or superscripted operator symbol (together with srt, slg, hrt, hlg for the inverse functions). Also we need for everything an ASCII equivalent which people can use when posting (for the non-tex-experienced people).

I append my own tex-source.

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