# Tetration Forum

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Happy year 2015! I have several news.

1. I could not recover my previous institute site at tori.ils.uec.ac.jp
But now I have clone at http://mizugadro.mydns.jp/t
Is it seen from other countries?

2. I have constructed the natural pentation (superfunction of tetration):
http://article.sciencepublishinggroup.co...306.14.pdf
http://www.ils.uec.ac.jp/~dima/PAPERS/2014acker.pdf
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

$u+{\rm i} v={\rm pen}(x+{\rm i} y)$

${\rm tet}({\rm pen}(z))={\rm pen}(z+1)$
${\rm pen}(0)=1$

3. I have Book about Superfunctions in Russian:
https://www.morebooks.de/store/ru/book/С...59-56202-0
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
I am working on the English version and I hope to finish it this year.
If Henryk submits, en fin, our review article "Bunch", I shall be glad to cite it.

4. I need someone to criticise my activity. So, the feedback should be greatly appreciated.
Thanks and happy new year to you.

1- From italy the site is available. Anyways congratulations for the richness of the website, is really great. Even if, given my poor knowledge of the sophisticated analisys methods used there, I can't understand half of the contenents seems really really interesting, and plots look amazing!

2-About Pentation: l have to notice that long ago, in 2006, Rubtsov and Romerio [1] were able to compute a first approximation of the fixed point $L_{e,4,0}=-1.850354529...$. Their first approximation where denoted by $\sigma=-1.84140566...$ and $pent_e (-\infty)=sln(\sigma)={{}^\sigma}e=\sigma...$ (where pent is computed using their approximation).

4 - Sorry but I cant help you, I'm just an amateur mathematician

PS: The way you plot function is really nice, is the same way one draws geographical map using contour lines! Is a common method to plot complex functions?

[1] Rubtsov, Romerio - Notes on Hyper-Operations, Progress Report -NKS forum III, Final review 3, 2006.
http://math.eretrandre.org/tetrationforu...hp?aid=222
(01/11/2015, 02:32 PM)MphLee Wrote: [ -> ]1- From italy the site is available.
Thanks for the confirmation.

Quote:..I can't understand half of the contenents..
for questions. But if with formulas, here seems to be easier.
You may use also http://math.eretrandre.org/hyperops_wiki...Kouznetsov

Quote: About Pentation: l have to notice that long ago, in 2006, Rubtsov and Romerio [1] were able to compute a first approximation of the fixed point $L_{e,4,0}=-1.850354529...$. Their first approximation where denoted by $\sigma=-1.84140566...$ and $pent_e (-\infty)=sln(\sigma)={{}^\sigma}e=\sigma...$ (where pent is computed using their approximation). ..
[1] Rubtsov, Romerio - Notes on Hyper-Operations, Progress Report -NKS forum III, Final review 3, 2006.
http://math.eretrandre.org/tetrationforu...hp?aid=222
Do you understand, how do they calculate the tetration?
I think, my algorithms are more efficient, because Rubtsov and Romerio do not present any complex map of tetration, nor pentation.

Quote: The way you plot function is really nice, is the same way one draws geographical map using contour lines! Is a common method to plot complex functions?
Some complex maps are presented in the handbook of mathematical functions, see, for example, figure 7.3 at
http://people.math.sfu.ca/~cbm/aands/abr...stegun.pdf
This refers to the past century. In this sense, already common.
(01/11/2015, 06:05 AM)Kouznetsov Wrote: [ -> ]Happy year 2015! I have several news.
Same to you.
Nice to have you back here.

Quote:1. I could not recover my previous institute site at tori.ils.uec.ac.jp
But now I have clone at http://mizugadro.mydns.jp/t
Is it seen from other countries?

But not Always ?
A restart (F5) or restart the computer helps.

Maybe its my safety settings or so ...

Quote:2. I have constructed the natural pentation (superfunction of tetration):
http://article.sciencepublishinggroup.co...306.14.pdf
http://www.ils.uec.ac.jp/~dima/PAPERS/2014acker.pdf
D.Kouznetsov. Evaluation of holomorphic ackermanns. Applied and Computational Mathematics. Vol. 3, No. 6, 2014, pp. 307-314.

$u+{\rm i} v={\rm pen}(x+{\rm i} y)$

${\rm tet}({\rm pen}(z))={\rm pen}(z+1)$
${\rm pen}(0)=1$

From an academic / career viewpoint these may be good papers.

But I feel the lack of complete explaination and motivation , at least in what I read -at first sight -.

For instance many methods of computation depend on a good " guess " ( I wonder if things can start from the " fake functions " but thats going offtopic ) and then I see things such as (say)

f(z) = 1.23515 log(d).

So natural questions are : what is 1.23515 ?

Does it has a closed form ? How was it computed ? Do you mean 1.23515... ? What is the property of this number ; does the iteration only work if 1.23515 log(d) < f(z) < 1.23516 log(d) ?

How did you find that solution ?
Does the iterate then work for all initial values of z , or do we have a fractal zone where it works ?

Are you talking about bases smaller then eta only in your formula ?
Cant we just rewrite 1.23515 by 1.2 and get the same result ?

***

I prefer color wheel graphs ...

***

Quote:3. I have Book about Superfunctions in Russian:
https://www.morebooks.de/store/ru/book/С...59-56202-0
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
I am working on the English version and I hope to finish it this year.
If Henryk submits, en fin, our review article "Bunch", I shall be glad to cite it.

4. I need someone to criticise my activity. So, the feedback should be greatly appreciated.

I hope my criticism does not upset you.

At first sight it seems you have 2 topics :

pentation

" Tania functions "

For pentation see the comment above + Why the focus on L and exp(kz) ?

I do not know much about Tania functions or the related physics , but I like the idea of naming functions after girls ??

Tania seems like a sister of Lambert.
Apart from the physics and the identity in the table , does Tania serve another purpose ?

MAYBE you said this before but why did you choose - assuming choice !? - pentation to be periodic ?

Welcome back

regards

tommy1729
(01/11/2015, 06:05 AM)Kouznetsov Wrote: [ -> ]Happy year 2015! I have several news.

1. I could not recover my previous institute site at tori.ils.uec.ac.jp
But now I have clone at http://mizugadro.mydns.jp/t
Is it seen from other countries?

2. I have constructed the natural pentation (superfunction of tetration)...

Hi Dimitrii,

Thanks for getting your tetration site back online. The graphs, as usual, are really nice. I thought I would limit my comments to your Pentation wiki; although I now see that the pdf has the same material. There is pari-gp pentation program posted here on eretrandre; I haven't played with it for awhile. The results match your results, for the period, and the fixed points, and the slope at the fixed point for pentation.

Where are the singularities of pentation? You wrote: "Pentation is holomorphic at least in the part of the complex plane, while the real part of the argument does not exceed some constant. For b=e, this constant is about −2.5"; I calculated the first singularity is -2.31527062760141 +/- 1.68383807835630i, so that would also match. One thing that I have puzzled over (but haven't posted) is that for pentation in the complex plane, is if pent(z) is a negative integer less than -1, then pent(z+1) is a singularity. This seems to imply that as real(z) increases, there are an infinite number of singularities arbitrarily close to the real axis, as pent(z) grows arbitrarily large.

You mention that tetration works for b>1. See the post by Nuinho, also see my post#28, on what he calls the super-euler number, b=1.6353244967, 1.6353244967 ^^^ oo ~= 3.0885549441, since sexp(z) has an upper real valued parabolic fixed point for this base=1.635324496, where pent(infinity) goes to a constant ...

Then for this base, and bases smaller, there are multiple real valued fixed points for Tet(z)=z. In my pari-gp code, sexpupfixed will generate this peculiar pentation base.
Perhaps, I should load the most important articles to
http://math.eretrandre.org/hyperops_wiki
It seems to be more stable.

Quote:But I feel the lack of complete explaination and motivation , at least in what I read -at first sight -.
.. Does it has a closed form? How was it computed? Do you mean 1.23515... ? What is the property of this number ; does the iteration only work if 1.23515 log(d) < f(z) < 1.23516 log(d) ?
How did you find that solution ?

I am not sure if I understand you well.. Perhaps, you refer to the Sheldon base..
Sheldon Levenstein asked me to make tetration namely for that base. I describe, how do I find the solution.
First, I implemented function, that evaluates the appropriate fixed points of the exponent and logarithm to the given base, I call it "filog", http://mizugadro.mydns.jp/t/index.php/Filog
Then, for base $b$, I POSTULATE, that the $\mathrm{tet}_b(z)$ should approach
$\mathrm{filog}(b)$ at $z\rightarrow \mathrm i \infty$ and
$\mathrm{filog}(b^*)^*$ at $z\rightarrow - \mathrm i \infty$
Then, with complex double arithmetics, the Cauchi integral gives of order of 14 correct decimal digits of $\mathrm{tet}_b(z)$; I invite you to do it for any base you like.
I did not check that the algorithm converges for every complex base,
but the claim by Sheldon Levenstein, that it fails for the "Sheldon base", is refuted.

Quote:Does the iterate then work for all initial values of z , or do we have a fractal zone where it works ?
Henryk Trappmann asked me to extend the algorithm beyond the cutline, to see the behaviour of another branch.. There is some fractal-like behaviour there. It is not difficult, you also can plot the beautiful pictures.
The algorithm for $n$th iterates of a transfer function $T$ is based on the superfunction $F$ and abelfuntion $G$:
$T^n(z)=F(n+G(z))$

Usually, abelfunction $G$ has branchpoints and cutlines.
Superfunction $G$ also may have some. So, in some cases, for some regions of argument $z$, the non-integer iterates may look scratched.

Quote: Are you talking about bases smaller then eta only in your formula ?
I talk about complex bases. For complex base, in general, the tetration can be evaluated through the Cauchi integral.
Also, it works for real base, greater than eta.
For real base equal or smaller than eta, the algorithm above cannot be applied. For these cases, two other algorithms are suggested, regular iteration and exotic iteration.
I think, all these algorithms evaluate the same tetration, I mean holomorphism of $\mathrm{tet})b(z)$ with respect to base $b$ at fixed $z$.

Quote: Cant we just rewrite 1.23515 by 1.2 and get the same result ?
What does it mean, "the same"?
If the base changes, the explicit plots and the complex maps change. But the result is the same in the sense, that the algorithm allow the evaluation and this evaluation passes series of various tests. We have Henrik Trappmann, who can write the long complicated formulas as a proof, that my simple algorithm provide the unique solution: http://www.springerlink.com/content/u7327836m2850246/ H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)

Quote:I prefer color wheel graphs ...
You can plot the graphs in any way you like.
If you meet any difficulties with the running of my algorithms,

Quote:I hope my criticism does not upset you.

Quote: For pentation see the comment above + Why the focus on L and exp(kz) ?
Because it provides the efficient way of evaluation of superfunction;
it is not specific for pentation: many other superfunctions can be constructed in this way. For example, superfactorial.

Quote:Tania seems like a sister of Lambert.
Apart from the physics and the identity in the table , does Tania serve another purpose?
Yes. But the Lambert, as it is implemented in Mathematica and other software, has ugly behaviour in the complex plane, while Tania has only two branchpoints and only two cutlines. In order to avoid the confusions, I need the specific name, different from Lambert.
This Tania has many uses.

Quote: I do not know much about Tania functions or the related physics , but I like the idea of naming functions after girls ??

Past century, Tania Kuznetsova asked me to solve numerically some equations, similar to $f'(x)=f(x)/(1+f(x))$
Yes, it comes from physics. We have published the serious of articles.. May be, one day, I'll describe how did that happen, it may be as a funny detective story.

Quote:MAYBE you said this before but why did you choose - assuming choice !? - pentation to be periodic ?
I think, I am first to provide the efficient algorithm for evaluation of superfunction of tetration, and to provide the detailed complex map of this superfunction, and to publish this. I use my prerogative to choose name for this function. I call it <b>pentation</b> and denote with symbol pen. By construction, this function is periodic. The period is determined by the derivative of tetration at its "zeroth" fixed point.

You may chose the specific base, $b=\tau\approx 1.63532$, such that graphic $y=\mathrm{tet}_{\tau}(x)$ touches the graphic of the identity function, $y=x$, and for this base construct the non-periodic supertetration.
The method is described in
http://www.ils.uec.ac.jp/~dima/PAPERS/2011e1e.pdf
http://mizugadro.mydns.jp/PAPERS/2011e1e.pdf H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of computation, 2012 February 8. ISSN 1088-6842(e) ISSN 0025-5718(p)

I suggest, that you call this supertetration with some specific name, for example, SuTet (Super Tetration), in order to keep name pen (pentation) for the periodic superfunction of tetration, id est, the superfunction, shown in the map above.

I think, you can implement the algorithm and plot the pictures and describe them by yourself, and submit them to some "Mathematics of Computation". If finish it within few months, I shall be glad to cite your publication in the English version of my Book. If you need any kind of help about this, let me know.
(01/12/2015, 05:11 AM)Kouznetsov Wrote: [ -> ].... I think, I am first to provide the efficient algorithm for evaluation of superfunction of tetration, and to provide the detailed complex map of this superfunction

Was there anything of interest to you in Andy's 2009 pentation thread? My first post to the thread includes pari-gp code, post#10, Oct30th 2010 and gives the fixed points and Taylor series for Pentation to higher precision than your results
(01/13/2015, 02:56 PM)sheldonison Wrote: [ -> ]Was there anything of interest to you in [url=http://math.eretrandre.org/tetrationforum/showthread.php?
No. Sorry. My Mathematica fails to run the code suggested,
<< Tetration
NaturalIterate[Series[Tetrate[E, x], {x, 0, 3}], z]

Quote:mode=linear&tid=372]Andy's 2009 pentation thread[/url]? My first post to the thread includes pari-gp code, post#10, Oct30th 2010 and gives the fixed points and Taylor series for Pentation to higher precision than your results
? How can I see that the precision is better?
Have you calculated the residual at the substitution into the transfer equation?
Or comparison with the explicitly–holomorphic Taylor expansion?
(01/13/2015, 06:52 PM)Kouznetsov Wrote: [ -> ]
(01/13/2015, 02:56 PM)sheldonison Wrote: [ -> ]Was there anything of interest to you in [url=http://math.eretrandre.org/tetrationforum/showthread.php?
No. Sorry. My Mathematica fails to run the code suggested,
<< Tetration
NaturalIterate[Series[Tetrate[E, x], {x, 0, 3}], z]

Quote:mode=linear&tid=372]Andy's 2009 pentation thread[/url]? My first post to the thread includes pari-gp code, post#10, Oct30th 2010 and gives the fixed points and Taylor series for Pentation to higher precision than your results
? How can I see that the precision is better?
Have you calculated the residual at the substitution into the transfer equation?
Or comparison with the explicitly–holomorphic Taylor expansion?

There is a Taylor series in that post#10, for Pent(x-1); you could compare it to results you have. When I rerun in higher precision today, the Taylor series coefficients I first posted are accurate to approximately 21 decimal digits; that Pentation result used a 32 decimal digits Tetration implementation for its base. The pentation pari-gp program hasn't been updated in over three years ... I would probably clean it up if I were writing it today, and also write more math equations (instead of just pari-gp code). The most recent version I have (roughly the same as post#13) works fine, and can be run in arbitrary precision and gives pent(-0.5)=0.4910543386356481974128179471452718984517, which gives an error term vs that very first post of pent(-0.5)=-7E-22.
(01/13/2015, 09:02 PM)sheldonison Wrote: [ -> ]
(01/13/2015, 06:52 PM)Kouznetsov Wrote: [ -> ]? How can I see that the precision is better?
Have you calculated the residual at the substitution into the transfer equation?
Or comparison with the explicitly–holomorphic Taylor expansion?
There is a Taylor series in that post#10, for Pent(x-1); you could compare it to results you have.
Ah, now I understand. The truncated Taylor series, by construction, is holomorphic; so, we can substitute it into the transfer equation and see the agreement. Let the truncated Taylor series be called F; let the transfer function be called T. Could you plot the map of agreement
$
A(z)= - \lg \left( \frac
{|T(F(z-1))-F(z)|}
{|T(F(z-1))|+|F(z)|}
\right)
$

? This agreement indicates, with how many decimal digits the transfer equation is satisfied.

Quote:When I rerun in higher precision today, the Taylor series coefficients I first posted are accurate to approximately 21 decimal digits; that Pentation result used a 32 decimal digits Tetration implementation for its base. The pentation pari-gp program hasn't been updated in over three years ... I would probably clean it up if I were writing it today, and also write more math equations (instead of just pari-gp code). The most recent version I have (roughly the same as post#13) works fine, and can be run in arbitrary precision and gives pent(-0.5)=0.4910543386356481974128179471452718984517, which gives an error term vs that very first post of pent(-0.5)=-7E-22.
How wide is area of agreement with 21 decimal figures?
Is it possible to implement your algorithm in C++ and/or in Mathematica?
Could you submit your result to some mathematical journal?
Then, it may be easier to cite it.
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