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Categories of Tetration and Iteration
#1
This category being empty is a perfect opportunity to think about the proper way to categorize this subject. If you go to Math Atlas you may find bits and pieces about iteration and tetration in Functional Equations, in Dynamical Systems, or Foundations or higher-level Statistics even. These are not the subjects of iteration/tetration. I believe that there are 4 sub-subjects within the subject that most people are interested in (whatever its called - you know - the field tetration researchers are in) and I think that these 4 sub-subjects are a natural separation of tetration-related research:

* Generalized Hyper-operators
* Generalized Iteration
* Iterated Exponentials
* Nested Exponentials

All four naturally have a lot to do with iteration, and as such there are 4 commonly encountered function types which people tend to perform iteration upon (Bennet actually classifies around 15 function types in his "Analytic Iteration"):

* All Functions
* Analytic Functions
* Analytic Functions with a Hyperbolic Fixed-point (ex: f(0) = 0 and f'(0) < 1)
* Analytic Functions with a Parabolic Fixed-point (ex: f(0) = 0 and f'(0) = 1)

Any post talking about iteration should explicitly specify which kind of functions they apply to, as it would facilitate searching, and cataloging the content posted. This can be done with keywords, like "parabolic" or "hyperbolic iteration" or something to that effect.

I have also noticed some commonalities between various methods of approaching generalized iteration of analytic functions:

* Functional Equations (Abel, Schroeder, Boettcher, Julia, ref: Szekeres)
* Double-Binomial Expansions (Woon's, Trappmann's)
* Iterate-Derivative Matrices (ref: Bell's "Iterated Exponential Integers")
* Power-Derivative Matrices (Bell matrix, Carleman matrix, refs:Koch,Jabotinsky)

I hesitate to call these separate methods, because they all seem to produce identical results for (parabolic FP) analytic functions, and some may apply to (hyperbolic FP) analytic functions as well. The only method I have encountered so far that doesn't produce results consistent with the 4 types of methods above is Iga's method which is not an exact "analytic iteration" method but a set-of-all-possibilities "continuous iteration" method, and as such, one member of Iga's set MAY be "analytic iteration", but he gives no clues as to how to find it.

Also, in http://en.wikipedia.org/wiki/Dynamical_s...inition%29 a wonderful notation for iteration is given, in three parts:

* As a 1-variable function Phi^t(x), (sometimes called an _Iterate_ or the "t-fold iterate of f")
* As a 1-variable function Phi_x(t), (sometimes called an _Orbit_ or the "orbit of f from x")
* As a 2-variable function Phi(t, x), (sometimes called a _Flow_)

But I am less interested in the notation, and more in the terminology. As far as I have seen, the three terms (iterate, orbit, flow) are not very common, but they are VERY useful in talking about iteration. Three analogous terms have been used for exponentiation for quite some time: powers, exponentials, and exponentiation respectively. I also think that the analogous terms for tetration are also emerging, and if we are going to talk at length about tetration we should begin to use them accordingly (notice how "power-tower" is nowhere in the list, I personally discourage the term "power-tower"):

* Hyper-power functions (hpr_n(x) == x^^n) where n is constant
* Tetrational functions (tet_x(y) == x^^y) where x is constant
* Tetration (hyper4(x, y) == x^^y)

After extensive meditation, I realized that hyper-operators are based on orbits, not iterates, so naturally, a better definition of tetration would be: "tetration is the orbit of an exponential from 1".

Lastly, although Galidakis gives a nearly exhaustive list of references, I think there should be place set aside for links and references that other people can add to if possible. For example, for many of the integer sequences I have been researching, I have found a variant of them on OEIS, and most of the time I find they were posted by Vladeta Jovovic. Although I have yet to contact Jovovic, if i didn't know better, I would guess that Jovovic is researching tetration as well! Perhaps if this forum takes off, people like Jovovic can post here directly, and not have to distill their research into an integer sequence suitable for OEIS.

Andrew Robbins
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#2
Hey Andrew,

For the actual division of the forum I decided to keep it simple in the beginning. It can then be extended/restructured if there will be too many threads in one category.

Hopefully all the methods of analytic iteration with fixed points you mention will be sometimes covered/presented on this forum.

Regarding terminlogy however for me it looks as if the people from dynamical systems are a different folk. Probably the terms orbit and flow will not establish in the tetration community. But that is of course up to you all.

Your distinction of hyperpower function, tetrational function and tetration (as I guess chosen similar to the names: power function, exponential function and exponentiation operation) is a good one.
For me personally however the prefix "hyper" (or "super") is somewhat unspecific as it means anything above. You dont know which one above while it is clear that "tetration" is the fourth operation.
For example if you take pentation x ^^^ n with fixed exponent n you would also call it hyperpower function.

My counter proposal would be "tetration exponential", "tetration power" and "tetration operation". Or perhaps even shorter "4-exponential", "4-power" and "4-operation", or "4th exponential", "4th power" and "4th operation". Though this is quite uncommon and will probably not establish, the use of "tetration base" and "tetration exponent" to denote and respectively in should be quite useful.

Your post is besides a quite sophisticated overview of the topic.
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#3
How about a terminology based on the hyper-n terminologies:
  • hyper-n-power (1-variable) (or tetra-power, penta-power, ...)
  • hyper-n-ational (1-variable) (or tetrational, pentational, ...)
  • hyper-n (2-variables) (or tetration, pentation, ...)

and a similar terminology for the reverse operations:
  • hyper-n-root (inverse of hyper-powers) (or tetra-root, penta-root, ...)
  • hyper-n-log (inverse of hyper-ationals) (i still don't like tetra-log)

and provisions should be made for the lesser and mixed hyper-operators, like those based on left-associative or right-associative iteration or left-right and right-left associative iteration. Since these only appear after hyper-3, I would imagine a logical way to name these would be to assign hyper-R = hyper-4 hyper-RR = hyper-5, hyper-RRR = hyper-6 since they all depend on right-associative iteration, and perhaps hyper-L, hyper-LL, and hyper-LLL for the "lower" or "hypo"-operator sequence. Using this terminology, we can refer to all of the above, for example, as hyper-LR-power, hyper-LR-ational, hyper-LR-root, hyper-LR-log, and so on. This works well for the hyper-operators based on simple iteration, but not for your Binary-Tree hyper-operators. How would that work for the Binary-Tree hyper-operators?

Andrew Robbins
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#4
Hm, I dont know. There is also the thing that one usually dont need for example a hyper-25-root. The most used number is 4, everything above probably then works in the same scheme. So perhaps we best stay with the established hyper power, hyper exponential, hyper root and hyper log (or similar super prefix).

andydude Wrote:and provisions should be made for the lesser and mixed hyper-operators, like those based on left-associative or right-associative iteration or left-right and right-left associative iteration. Since these only appear after hyper-3, I would imagine a logical way to name these would be to assign hyper-R = hyper-4 hyper-RR = hyper-5, hyper-RRR = hyper-6 since they all depend on right-associative iteration, and perhaps hyper-L, hyper-LL, and hyper-LLL for the "lower" or "hypo"-operator sequence. Using this terminology, we can refer to all of the above, for example, as hyper-LR-power, hyper-LR-ational, hyper-LR-root, hyper-LR-log, and so on.
Not to forget the balanced bracketing, for example , perhaps denote with a B(alanced) or M(iddle). There are however several different types of balanced bracketing.

Quote:This works well for the hyper-operators based on simple iteration, but not for your Binary-Tree hyper-operators. How would that work for the Binary-Tree hyper-operators?
Oh they are designed for containing all bracketings. Every binary tree encodes a bracketing. So one dont need to care about left bracketing or right bracketing, the bracketing is simply performed as it is encoded in the first operand, which is a binary tree. Thatswhy there is a natural sequence of hyper operations, not prefering a certain bracketing over others.
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#5
Dear Participants in tha Forum,

Taking into consideration the problems of dealing with new subjects, such as Hyperoperations and, particularly, Tetration, I take the liberty of proposing (attached) the terminology and symbols that, both KAR and myself, we used in our common research, as it also appears in the NKS Forum, News and Announcements. By this, I am trying to follow, whenever it's possible the ideas put forward by various researchers in this new mathematical field.

I hope it will be useful.

GFR


Attached Files
.pdf   Hyperoperations (terminology).pdf (Size: 60.49 KB / Downloads: 603)
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#6
Thanks for the proposal, Gianfranco.
I think that most of it makes sense.
What regards the symbols I would suggest:
and for the hyper operations of rank n. So that for example and and and .

But anyway the 4th operation is the bottle neck. If we solved this satisfactory then all the other higher operations probably follow at once. So nobody perhaps will speak particularely about but rather about all hyper logarithms. Thatswhy it is important to have the separate writings and the and .

Also the hyperroots are not yet discussed here. Appropriately I have no alternate suggestion Wink.
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#7
Hi Henryk!

Coirrect! However, also limiting ourselves to rank 4 (tetration), appropriate writings for the superexp or tower (tetra, sexp or tow, with base b) and its right-inverse, superlog (slog, to the base b) are not sufficient.
We should not forget its left-inverse, superroot (srt, n degree). Think, for instance, of the super-squareroot or square-superroot, solution of equation y = x ^ x, that we can write y = x # 2, with x = ssrt(y). Superlog and superroot are the two inverse operations of the superexp, tetra(tion), or tower, operation. They all belong to the tetration rank (s=4). Other general hyperroots mightl indeed come later.

GFR
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#8
GFR Wrote:Dear Participants in tha Forum,

Taking into consideration the problems of dealing with new subjects, such as Hyperoperations and, particularly, Tetration, I take the liberty of proposing (attached) the terminology and symbols that, both KAR and myself, we used in our common research, as it also appears in the NKS Forum, News and Announcements. By this, I am trying to follow, whenever it's possible the ideas put forward by various researchers in this new mathematical field.

I hope it will be useful.

GFR

Dear Gianfranco & Forum -

I took a rest in the last days, and used it to fiddle a bit with operator-definitions like at the kitchen-table. The basic ideas can surely be found elsewhere, but I liked it to express a scheme by myself.
The advantage is, that the notation is one-to-one translatable to a matrix-operator-approach for each common operation.

Maybe it is helpful too; and I'm thinking it's worth to be considered in this or that way -

Gottfried
Operators
Gottfried Helms, Kassel
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#9
Dear Gottfried,

I am just back from a trip and I saw your interesting "Operators" scheme. Certainly useful. Particularly concerning the automatic translation to matrix operators. Let me think about that during the next ... few hours. I shall come back to that. Probably, it's too early for establishing standard approaches. However, let me think about ... .

GFR
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#10
First of all, I agree with most of your notation and terminology decisions, but I do have some comments and suggestions regarding them.

To summarize your proposal, it seems you have focused on these systems:
  1. Box notation for hyper-operators.
  2. Partial-box notation for hyper-roots and hyper-logarithms.
  3. The hyper- prefix representing all ranks.
  4. The super- prefix representing rank 4.
  5. Symbolic representation of the super-root.
  6. Mnemonic representation of the super-logarithm (slog).
  7. Representation of hyper-operators through repetition (# = ^^ = ***).
  8. Terminology for hyper-n-powers, and hyper-n-exponentials.
  9. Terminology for nested exponentials (a^b^c^...^y^z).

Of all of these the ones I have a problem with are 2, 5, 8, and 9. I like the others. Points 2 and 5 are TeX-notation problems and point 8 is just a discussion. I don't recommend any changes to point 8. Point 9 you hardly talk about, but I would like to say a few things about it.

I agree with your ASCII symbols (+, *, ^, #), and your boxed notation, but the partial box notation for super-root and super-logarithm I do not agree with, partially because I don't understand how to use it in TeX. If you could perhaps post some example on how to use the partial-box notation in TeX, then I would love to use it myself.

I think that the Mnemonic representations of super-root (srt) and the super-logarithm (slog) are easier to use with TeX, but again if you can provide TeX macros for partial-boxed notation and your super-root notation, then I might be inclined to use it.

For your terminology for hyper-powers and hyper-exponentials, I agree with "tetrational" since I use this myself, but "tower" to me describes something more general, what you describe as "inhomogeneous towers". This is what I call nested exponentials. The term nested exponentials is analogous to nested radicals, nested logarithms, and nested integration. It is interesting that you chose that term, since I used to call them heterogeneous towers myself, so maybe we should go with that term instead. When I think of "tower" I think of "nested exponential", but we still need a term for the hyper-4-power function, and the sad thing is that J.F.MacDonnell has already popularized "hyper-power" for the hyper-4-power, which breaks the system (hyper- is supposed to be all ranks). This leaves us with the super-power function from which the mnemonic (spow) stems from. The problem with "super-power" is that it is a terrible search word, because of its use in politics. So after eliminating all possibilities we're left with no suitable terms for the hyper-4-power, except one, which is the generic term we've been using all along: hyper-4-power. But if you really want a shorter term, I would say that tower would be better than "hyper-power" or "super-power".

There are two people that have done some very interesting things with nested exponentials (heterogeneous towers, inhomogeneous towers), specifically Barrow [2], and Yukalov et.al. [3]. Barrow shows that it is possible to turn any function (such that f(0) = 1) into a tower with appropriate coefficients. He tried to find simple tower expansions of sine and cosine, but found that they were more complicated than the normal series expansion. Yukalov et.al. show that it is possible to construct a nested exponential that approximates any data-set, and that this approximation works very well for certain kinds of data like stock markets. While this may just be all hype, it looks very promising, and it may be that this tower interpolation could be a useful tool in approximation theory.

I gave a speech at my school a few months ago (quite an honor actually), and I made a handout for that speech called "Tetration in Context", and although it is a little out of context for this post, I figure it has a bit of what I've been talking about with the notations and stuff. So here it is: Tetration in Context.

Andrew Robbins

[1] J.F.MacDonnell, http://www.faculty.fairfield.edu/jmac/ther/tower.htm.
[2] D.F.Barrow, Infinite Exponentials, The Amer. Math. Monthly, Vol. 43, No. 3, (Mar 1936), pp. 150-160.
[3] V.I.Yukalov, S.Gluzman, Weighted Fixed Points in Self-Similar Analysis of Time Series, http://arxiv.org/abs/cond-mat/9907422.
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