# Tetration Forum

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(08/10/2009, 06:15 PM)Tetratophile Wrote: [ -> ]so make iter[] an operator like i said. how are we gonna pronounce it anyway.

That does not solve the problem of wording.
And btw. there is no iter operator because there are different methods to define extended iteration (thats the board about! )

Extended iteration is not an operator but a property, something is an iterational of something else, if it satisfies blabla. There can be several iterationals for one something else.

Only integer iteration is an operator.
(08/10/2009, 06:15 PM)Tetratophile Wrote: [ -> ]so make iter[] an operator like i said. how are we gonna pronounce it anyway.

Tetratophile Wrote:just make iter[] an operator on functions. then we can square it like any other operators. iter^2 means doing iter twice.
iter^-1[] is the inverse of iter[], it extracts the function you built the iter-function out of. so iter^-1[tetra-exponential]= exponential. this is more generalizable.
also, abel[f] := [iter(f)]^-1, iter[f] considered as a function.
examples:
we may be able to do away with all the hyper-n-operator s*it.
As Henryk says - "iter"/"abel" are not (good) prefixes. But anyway nice words: they makes the others prefixes: (taking your proposal
multiplication-abel = ...
iter-abel = <...oh good grief...>

G
So guys,

to solve our current problems, I come up with the following quite memorizable and hopefully non-confusing suggestion:

Roughly:
An $f$-exponential $\eta_z(w)=v$ is the $w$ times application of $f$ to $z$.
An $f$-logarithm $\gamma_z(v)=w$ returns the number $w$ of iterations of $f$ necessary to be applied on $z$ to obtain $v$.
An $f$-power $\varrho^w(z)=v$ is the $w$ times application of $f$ to $z$ (same as $f$-exponential but function in $z$, not in $w$.)

Implicitely there is contained the concept of the Abel function: $\gamma_z$ is an Abel function for each $z$ and that of a superfunction/iterational $\eta_z$ is a superfunction for each $z$.

Also ability to specify an initial value $z$ on which the iterations are applied, was often asked for here on the forum.

Another thing is that imho it lets one think more in the direction of "a" $f$-exponential, while the prefix "super" rather suggests "the" superfunction, super-exponential.
"$g$ is an $f$-exponential". (How do the native speakers think about it?)

So we can leave the old super-notation in there previous meaning (though I think it is an unnecessary notation as everything can be specified with tetra-, penta-, etc. prefixes.)
(08/10/2009, 06:12 PM)bo198214 Wrote: [ -> ]That was not the main reason, as I said "additionally".
The main reason is, that it is just not a proper prefix. Proper prefixes are rather super, hyper, ultra, tetra, etc.

Actually, it is a prefix: see here. Perhaps you think it is not because it is used so rarely. Rare prefixes are a better choice than "super" because there is less chance for ambiguity.

Although I think its a terrible idea, we could steal some other rare prefixes and re-use them for something different entirely: like these. But I don't know which would be which.

Andrew Robbins
andydude Wrote:Actually, it is a prefix
no, -iter- is not a prefix, it's a word-stem of latin that means repeating.
but i have no problem adopting it as a prefix. it's clearer than "super".

(08/11/2009, 12:06 AM)bo198214 Wrote: [ -> ]So guys,

to solve our current problems, I come up with the following quite memorizable and hopefully non-confusing suggestion:

Roughly:
An $f$-exponential $\eta_z(w)=v$ is the $w$ times application of $f$ to $z$.
An $f$-logarithm $\gamma_z(v)=w$ returns the number $w$ of iterations of $f$ necessary to be applied on $z$ to obtain $v$.
An $f$-power $\varrho^w(z)=v$ is the $w$ times application of $f$ to $z$ (same as $f$-exponential but function in $z$, not in $w$.)

Implicitely there is contained the concept of the Abel function: $\gamma_z$ is an Abel function for each $z$ and that of a superfunction/iterational $\eta_z$ is a superfunction for each $z$.

Also ability to specify an initial value $z$ on which the iterations are applied, was often asked for here on the forum.

Another thing is that imho it lets one think more in the direction of "a" $f$-exponential, while the prefix "super" rather suggests "the" superfunction, super-exponential.
"$g$ is an $f$-exponential". (How do the native speakers think about it?)

So we can leave the old super-notation in there previous meaning (though I think it is an unnecessary notation as everything can be specified with tetra-, penta-, etc. prefixes.)

it will be confusing w/ my greek prefix terminology (tetra-, etc.). is a tetra-exponential an iterate of tetration b[4]x, as your terminology would suggest, or is it an iteration of b^x?

but you don't need to worry about english, i thought you were a native speaker the first time i was here, i never knew you were henrik trappmann, a guy in germany
Also, about the "pus" thing, because Abel as already been mis-capitalized (abelian groups), we should really take a brake from him, and use the prefix "fato" for Fatou. This way "iter" would sound bad in German, and "fato" would sound bad in English. Everybody will laugh, which will help people enjoy using the terminology.

Another option would using "meta-F" for the superfunction and "para-F" for the Abel function (as I indicated above).

Another option would be to take advantage of the UXP terminology, and use "ultra-F" for superfunction and "infra-F" for the Abel function. This would allow "UXP" to continue being used and also allow "superlogarithm" to continue being used as well. I have also heard of the "infra-logarithm" which would be the only conflict with this approach.
(08/11/2009, 02:17 AM)andydude Wrote: [ -> ]Also, about the "pus" thing, because Abel as already been mis-capitalized (abelian groups), we should really take a brake from him, and use the prefix "fato" for Fatou. This way "iter" would sound bad in German, and "fato" would sound bad in English. Everybody will laugh, which will help people enjoy using the terminology.
rotflolmao. but sorry, it will only work for english dialects which are non-rhotic, ie. don't say the r in "fart" (eg. boston, british)

but enough joking around. let's take a poll or somethin.

ps. then the tetra-logarithm would be infra-exponential, not infra-logarithm.
(08/11/2009, 02:17 AM)andydude Wrote: [ -> ]I have also heard of the "infra-logarithm" which would be the only conflict with this approach.
(08/11/2009, 02:37 AM)Tetratophile Wrote: [ -> ]ps. then the tetra-logarithm would be infra-exponential, not infra-logarithm.

Ultra and Infra are made from observations of growth.
Ultraexponential grows faster than the exponential
Infralogarithm grows slower than the logarithm.
Thatswhy Infra-F is not the Abel function but the Abel function of the inverse of F. (which is theoretically not really useful)

In the same way "super" is a pretty memorable concept (bigger, grows faster) if it was not already used for super-logarithm. Which is a waste of terminology: super-logarithm grows not faster than the logarithm; the only bigger thing is the rank in the hierarchy, which can be more directly expressed with tetra-.

But we know the drawbacks of "super": it conflicts with our historical use of "super-logarithm" (though I think this was not yet used in published papers) and many dont want to have an arcsuper.

(08/11/2009, 02:14 AM)Tetratophile Wrote: [ -> ]it will be confusing w/ my greek prefix terminology (tetra-, etc.). is a tetra-exponential an iterate of tetration b[4]x, as your terminology would suggest, or is it an iteration of b^x?

"tetrational exponential" for a pentational.
And "tetraexponential" or "rank4 exponential" for a tetrational (inside the hyperoperation sequence).
"exponential exponential" or "exp-exponential" for "tetrational" (outside the hyperoperation sequence).

Though one have to adapt to the $f$-power, $f$-exponential, $f$-logarithm terminology, it is quite memorable as one can intuitively give meaning to how to apply the concepts power, exponential and logarithm to the iteration/application of the function $f$ to $z$.
They also can be used as shortcut for more classic sounding phrases:
"$f$-power" is short for "functional power of $f$",
"$f$-logarithm" is short for "functional logarithm of $f$"
etc.
e.g. "tetrational is just another word for functional exponential of the exponential."

Here however we have to be cautious and can not use the term "iterational" instead of "functional" as the phrase "iterative logarithm" is already reserved for the Julia function.

Ya well on the other hand nothing is as succinct as a prefix.
I think I had something like that before...
see my "hyper-iteration" post.
Tetratophile Wrote:Along the same lines as "functional root" $\sqrt[n]{f}(x),$ (a function which, iterated n times, gives f(x)), the "functional logarithm" can be defined so that $\operatorname{flog}_f (f^n(x)) = n$ for all n.
any differences betweeen bo's terminology and mine? It FEELS different...
(08/11/2009, 01:45 PM)Tetratophile Wrote: [ -> ]any differences betweeen bo's terminology and mine? It FEELS different...
imho its the same, though I wasnt remembering that notation of yours.
Just be aware that there are different possible flogs, etc.
At least this supports that it is a quite intuitively understandable naming.
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