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In german pupils competition ("Jugend forscht") Markus Müller presented 1996(?) an article, called "Reihenalgebra", where he discussed higher operators like tetration, pentation and so on. Two aspects have been of special interest to me:
1) He started the index of operators differently:

1 - do nothing
2 - shift by 1
4 - multiplication
5 - exponentiation
6 - tetration
and so on.

Then he indexed the inverses like

1 - do nothing
1/2 - shift by -1
1/3 - subtraction
1/4 - division

and so on (considering also the different inverses of higher index)

2) Based on this he even tried to define operations of fractional order

The original article is a nice treatise, recalling that it was done by a pupil; unfortunately it is in windows write 3.11 format and not directly transferable to current word-/pdf-formats.

Recently I found an update of this concept
http://www.math.tu-berlin.de/~mueller/reihenalgebra.pdf
where Markus Müller seems to be affiliated to the TU of Berlin.

Possibly the administrator would like to invite Markus Müller to this forum (provided he is still interested in the subject). I would like to know, what he has to say about the current research.

Gottfried
Thats indeed interesting.
I just sent him an e-mail (haha I also studied at the TU-Berlin!).

Btw. I also invited Constantin Rubtsov (Zeration) to this forum but the e-mail "rubcov@russia.crosswinds.net" bounced. Does anyone know his actual, or at least another e-mail address?
Markus is currently quite involved in preparation of his phd disputation (perhaps he joins us later).
That reminds me of not neglecting my own preparation too!
I've long been interested in math and physics, and this exercise has made me want to pursue a Master's degree in math or physics. I currently have a bachelor's degree in Computer science with an "option" in math/science, roughly equivalent to a little more than a minor in math, a little less than a minor in physics.
jaydfox Wrote:I currently have a bachelor's degree in Computer science with an "option" in math/science, roughly equivalent to a little more than a minor in math, a little less than a minor in physics.

I think you have talent, you merely need to learn how to express your ideas in the mathematical framework. Wish you the best!
I think one of Markus Mueller's most interesting discussions is that of his arrows in and of themselves, not exactly how he defines them, though. He defines them with implicit exponentiation, whereas Knuth's arrows use implicit multiplication, a major problem between the two systems. But I think using Mueller's arrow, so much about iteration is much easier to express:

$x ({\uparrow}A) y = x A (x A \cdots A (x A x))$
$x ({\downarrow}A) y = ((x A x) A \cdots A x) A x$

Using $x{\uparrow}y = x({\uparrow}\cdot)y = x^y$ (Knuth) instead of $x{\uparrow}y = x({\uparrow}\^)y = {}^{y}x$ (Mueller) we can still use the notation for both higher and lower hyper-operation hierarchies, as well as iteration itself. The iteration of a function has been written in various ways in this forum making $f^{[n]}(x) = f^{\circ n}(x) = (f {\uparrow} \circ n) \circ x$ yet another notation we could use. The only advantage of this would consistency, which we could get from using any notation if we used it a lot.

Andrew Robbins