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Question about tetration methods
#11
(06/18/2022, 07:22 AM)Gottfried Wrote:
(06/18/2022, 04:29 AM)Daniel Wrote: My work is experimentally consistent with Bell matrices. Are Bell matrices considered along with Carlemann matrices in people's research?

The Carleman-matrix is simply a factorially similarity scaling of the Bell matrix, and transposed. The idea is to work on the coefficients of the taylor-series, and so the work is practically identical between Carleman- and Bell-notation.
But note: it provides its concise arithmetic/algebra on *taylor-series* only, not, for instance, laurent series - and for analysis and bulding a "toolbox" for many prominent problems, you need
the inclusion of $c_{-1} \cdot x^{-1}$ terms.

The only work with such an extension of the Carleman-ansatz I have ever seen is an article of Eri Jabotinsky, and when I read it it was over my head, and I had only spurious time for this matter then, so I did not step in. However I think it is a serious AND required extension of the range of applicability, it makes the Carleman-ansatz more useful and then surely wider known.

Gottfried

An exposition "from the ground" is in ContinuousfunctionalIteration (at my webspace for tetration) ; I think you'll find in it many things known to yourself (for instance the description in terms of derivatives) but I wrote this for the beginner as well as for a self-reflection of my knowledge.

I'd like to add, that Carleman matrices are guaranteed to exist. So, Kneser doesn't prove that they exist. Kneser proves there is a certain Riemann mapping which makes a solution to tetration. But, in and of itself, this means we can use Carleman matrices to find a solution. So, Carleman matrix method will work; there is some sequence of matrices that does work. But if you try to construct Kneser from scratch using Carleman; then you have a very different problem.

This is where I shy away from Carleman... It's like trying to start a fire with sticks and stones, when rocketfuel is right next door.
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#12
(06/18/2022, 09:41 AM)JmsNxn Wrote:
(06/18/2022, 07:22 AM)Gottfried Wrote:
(06/18/2022, 04:29 AM)Daniel Wrote: My work is experimentally consistent with Bell matrices. Are Bell matrices considered along with Carlemann matrices in people's research?

The Carleman-matrix is simply a factorially similarity scaling of the Bell matrix, and transposed. The idea is to work on the coefficients of the taylor-series, and so the work is practically identical between Carleman- and Bell-notation.
But note: it provides its concise arithmetic/algebra on *taylor-series* only, not, for instance, laurent series - and for analysis and bulding a "toolbox" for many prominent problems, you need
the inclusion of $c_{-1} \cdot x^{-1}$ terms.

The only work with such an extension of the Carleman-ansatz I have ever seen is an article of Eri Jabotinsky, and when I read it it was over my head, and I had only spurious time for this matter then, so I did not step in. However I think it is a serious AND required extension of the range of applicability, it makes the Carleman-ansatz more useful and then surely wider known.

Gottfried

An exposition "from the ground" is in ContinuousfunctionalIteration (at my webspace for tetration) ; I think you'll find in it many things known to yourself (for instance the description in terms of derivatives) but I wrote this for the beginner as well as for a self-reflection of my knowledge.

I'd like to add, that Carleman matrices are guaranteed to exist. So, Kneser doesn't prove that they exist. Kneser proves there is a certain Riemann mapping which makes a solution to tetration. But, in and of itself, this means we can use Carleman matrices to find a solution. So, Carleman matrix method will work; there is some sequence of matrices that does work. But if you try to construct Kneser from scratch using Carleman; then you have a very different problem.

This is where I shy away from Carleman... It's like trying to start a fire with sticks and stones, when rocketfuel is right next door.

never start a fire with rocketfuel
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#13
(06/18/2022, 11:15 PM)tommy1729 Wrote: never start a fire with rocketfuel

LET IT BURN!
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#14
(06/18/2022, 03:25 AM)JmsNxn Wrote: ...
As Gottfried spoke, he meant that he can think of less than 5 solutions to tetration which are intrinsically unique. This is true. There is Kneser. There is Sheldon. There is Carlemann (programmed by Gottfried). There is Kouznetsov. There is Paulsen and Cowgill.

...




Thanks, James, for your insightful msg!        



I just get the feeling, the ("Carleman-") matrix-approach is sometimes taken with a slight mystique expectations around it. So I'll reiterate some explanation (about which you might well be aware of, but likely not all other readers around).                             



Basically the matrix-approach is nothing else than a notational framework for the manipulation of power series (in terms of their coefficients). For instance, the Schröder-procedere for iteration of analytic functions having powerseries \(f(x)= \sum_{k=1}^\infty a_k x^k\) can be coded by appropriate matrix-operations, involving such "Carleman"-matrices, or we may say, that what we have to do to write down the manipulations of power series to implement the Schröder-functionality can be concisely be denoted by the matrix-formulae.    

When I started my engagement with tetration I didn't know anything about Schröder-/Abelfunction, Kneser, and all this stuff, and as well I did not know the name "Bell", nor "Carleman" matrix; I just fiddled with the idea of inventing a method to operate on the coefficients of powerseries, their powers, and their iterations.           



My primary object has been the Pascal-matrix, where I observed, that that matrix mapped the function \(f_1(x)=x\) to \(g_1(x)=x+1\) and in the same one sweep as well \(f_2(x)=x^2\) to \(g_2(x)=(x+1)^2\) and so on, when just set in the most simple matrix-multiplication scheme. Of course, powers of the pascalmatrix P made then iterations of that maps (discussion-article), and fiddling the same thing with other functions I came then to \(f(x)=\exp(x)\) where then soon the contact with the tetration-forum happened.   Still I did not know about any of the earlier research in this directions, only after Andrew Robbins coined the name "Bell"-matrix (and/or "Carleman"matrix) it was that I got aware that this naive playing around with the coefficients of powerseries had been done before (and has nothing of magic, for instance the fractional iteration of the \(\exp(x)-1\) by manipulating its powerseries has been discussed in the book of Comtet down to the operations with the coefficients, only that Comtet did not introduce a full-fledged matrix-notation for that algebra which is involved). The rediscovery of the Schröder-function in the context of this matrix-algebraic formulae was then only to understand that the completely common folklore of matrix-eigensystem-decomposition simply gives the coefficients that Schröder(?) found the other way around for his function.                       



So, for instance, it seems to me that there is a misconception of this all when I read that "the existence of Carlemanmatrices is/must be proven" or the like... It even might be, that the Riemann-map can be formulated in that matrix-algebraic notation, I only can't say this: since I do not understand enough of this mapping at all; if it can be expressed in terms of building powerseries at all then it should be possible, and it is not needed that the occuring matrices are "Carleman", they might be "Vandermonde", or whatelse ever.



Hmm - a long comment, perhaps not needed. On the other hand, if this comment is useful for understanding anyway, I could perhaps inserrt something in the hyperop-wiki (or in a separate thread)...      



Btw. it seems there is a good journey underway here, this days - may the summer be fruitful and glorious :-) 



Gottfried
Gottfried Helms, Kassel
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#15
No Gottfried, great comment.
It helped me sorting out some details about that. Also I find your explanation papers amazing.
The point that I was needed to be reminded of is that, everything about the matrix description is mainly about the ring of formal power-series. In other words... it is an algebraic framework where then you consider convergence radii in a second moment.

I believe that matrix descriptions of things are very powerful. I was always matrix-phobic tbh, but after a course on linear algebra and linear geometry I started to appreciate them.

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
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#16
(06/20/2022, 05:42 PM)Gottfried Wrote:
(06/18/2022, 03:25 AM)JmsNxn Wrote: ...
As Gottfried spoke, he meant that he can think of less than 5 solutions to tetration which are intrinsically unique. This is true. There is Kneser. There is Sheldon. There is Carlemann (programmed by Gottfried). There is Kouznetsov. There is Paulsen and Cowgill.

...



So, for instance, it seems to me that there is a misconception of this all when I read that "the existence of Carlemanmatrices is/must be proven" or the like... It even might be, that the Riemann-map can be formulated in that matrix-algebraic notation, I only can't say this: since I do not understand enough of this mapping at all; if it can be expressed in terms of building powerseries at all then it should be possible, and it is not needed that the occuring matrices are "Carleman", they might be "Vandermonde", or whatelse ever.

Gottfried


Sorry, when I meant Carlemann, I meant using Carlemann to produce Kneser. Carlemann for Schroder maps is actually very natural. My head gets fuzzy thinking about it for Kneser though, it just seems unnatural to me. Though, there definitely exists Carlemann matrices which solve the equation--can't imagine the construction method though.
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#17
(06/21/2022, 10:08 PM)JmsNxn Wrote: My head gets fuzzy thinking about it for Kneser though, it just seems unnatural to me.

Yes, perhaps it is too fuzzy, and "eats up" the joy of discovery...                

One point when it became too much for me was the procedere of Ecalle, when it went to involve the logarithm of the Schroeder-function (that's simple enough), but then to use the reciprocal (which might still be doable/expressible though) and then including the integral term. It seemed to me that to express this with Carleman-matrices needed the extension to negative indexes, so to introduce the ability to work with Laurent series or even fully 2-way-infinite series \( \sum_{k=-\infty}^{+\infty} a_k x^k \). I've seen, as I mentioned earlier, that Eri Jabotinsky worked with such an extension, but I gave up for my part... (feeling dried out, natural process, simply).

So also for me, one might say, the "carlemanization" of the analysis of the tetration-function became unnatural somehow/somewhere, and since the Ecalle-result/-process seems to be an important and a good one I didn't try to go further, ... and don't make bold statements since :-) ...

(06/21/2022, 10:08 PM)JmsNxn Wrote: Though, there definitely exists Carlemann matrices which solve the equation--can't imagine the construction method though.

I agree, with the words above. Only that there remains still a tickling of curiosity, how ... perhaps ...
Gottfried Helms, Kassel
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#18
The existance proof of kneser is no issue for me. It is just doing the mapping itself that is hard.

Numerical methods for it exist by polygon approximations to map to a circle.
But that does not " satisfy me ".

I think I can prove that it has no singularities in the upper plane, not sure if that is still an issue.

That that property is unique is another matter.
But Im thinking about it.

the fun this is if I can prove it for other functions super , it also holds for exp superfunctions.

regards

tommy1729
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