# Tetration Forum

Full Version: overview paper co-author invitation
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(06/01/2009, 07:59 PM)Ansus Wrote: [ -> ]Why not to include the Lagrange and Newton's methods as well along with the definition through indefinite sum?

Ok, I actually started a survery article about all the different methods for real-analytic tetration we encountered so far on the forum.
Have a look at http://bitbucket.org/bo198214/bunch/raw/...5/main.pdf whether you would like to participate. (If so you should become familiar with mercurial version control system, as we would concurrently work on http://bitbucket.org/bo198214/bunch/. Just give me your public ssh-keys that I can add you to the project.)

In the best case we can compare the different methods in the complex plane, where they should reveal their difference more obviously than just on the real axis.
This should become a professional article with propositions and proofs and concise introductions into the needed tools (e.g regular iteration, matrix powers, etc).
I hope however to make it not too long.
Actually this is a second try after a hard disk crash that destroyed a lot of already written material.
I hope we can use our combined alertness to get rid of any sneaking errors, and also for everyone who likes to participate there should be interestings tasks ahead like proving equalities, implementing algorithms, performing numerical tests, drawing plots and pictures.
bo198214 Wrote:I actually started a survery article about all the different methods for real-analytic tetration we encountered so far on the forum.
Have a look at http://bitbucket.org/bo198214/bunch/raw/...5/main.pdf whether you would like to participate. (If so you should become familiar with mercurial version control system, as we would concurrently work on http://bitbucket.org/bo198214/bunch/. Just give me your public ssh-keys that I can add you to the project.)
Henryk, I looked at your project. After to send the paper about fractal behavior of tetration, I would like to participate.

About Kneser's expansion: it would be good to check, that we evaluate the same function expanding it at fixed point L and doing that at L^*.
I tried to do this, and I failed. Numerically, I could get only few correct digits in vicinity of the real axis...
Moderator's note: I moved the last two posts into the thread Kneser's super logarithm.
(06/02/2009, 01:06 AM)bo198214 Wrote: [ -> ]
(06/01/2009, 07:59 PM)Ansus Wrote: [ -> ]Why not to include the Lagrange and Newton's methods as well along with the definition through indefinite sum?

Ok, I actually started a survery article about all the different methods for real-analytic tetration we encountered so far on the forum.
Have a look at http://bitbucket.org/bo198214/bunch/raw/...5/main.pdf whether you would like to participate. (If so you should become familiar with mercurial version control system, as we would concurrently work on http://bitbucket.org/bo198214/bunch/. Just give me your public ssh-keys that I can add you to the project.)

In the best case we can compare the different methods in the complex plane, where they should reveal their difference more obviously than just on the real axis.
This should become a professional article with propositions and proofs and concise introductions into the needed tools (e.g regular iteration, matrix powers, etc).
I hope however to make it not too long.
Actually this is a second try after a hard disk crash that destroyed a lot of already written material.
I hope we can use our combined alertness to get rid of any sneaking errors, and also for everyone who likes to participate there should be interestings tasks ahead like proving equalities, implementing algorithms, performing numerical tests, drawing plots and pictures.

i think we all deserve some credits anyways.

its clear many ideas came from many people.

considering the sincerity and intelligence of the regular posters and the forum members in general , all members with more than 5 posts should be mentioned IMHO.

regards

tommy1729
(06/03/2009, 10:06 PM)tommy1729 Wrote: [ -> ]i think we all deserve some credits anyways.
its clear many ideas came from many people.
considering the sincerity and intelligence of the regular posters and the forum members in general , all members with more than 5 posts should be mentioned IMHO.

I suggest that participants summarize their most important ideas and results about tetration (for example, from the 5 their most important posts), and let tommy1729 writes the section about the historical development of the evaluation of tetrational. Then we save Henryk time and he will better write the mathematical deduction.

The article Henryk suggests, is the only part of the Big Job.
I believe, we are making the new branch of the Functional Analysis.
I would mention the following topics:

1. Comparison of evaluations. – that Henryk is doing and in that he asks for helpers.

There is a lot of other work which also should be acknowledged.

2. Various implementations of each of the algorithms. Each method of evaluation needs the numerical implementation for the verification. It would be good to have the portable codes in various languages. Such codes would allow everyone to confirm our results (or indicate an error, if any). I suggest that you post your codes also in some other places, easy for downloading: Your homepage, citizendium, some wiki, etc., with mutual links to the Forum. I usually supply the free copyleft license to each my code; but if you hope to make some money of your code, add the sentence "if non-commertial" to the permission before the posting.

I suggest that tommy1729 or any other volunteer takes the burden of coordination of this work. (Unification of style, supply the short description, make each algorithm easy to find, check that it compiles and runs at various platforms, avoid conflicts in the names of the functions, etc.)

3. Construction of algorithms for super-function of various exponentials. As a good wish, such an algorithm should treat the base of tetrational as an additional parameter.

4. Constructions of algorithms for super-function of any holomophic function. Test this algorithm, evaluating, for example, the super-functions of factorial and tetrational (pentational, sextational, etc.)

5. Generalization of various Ackermann functions. The interpretation in terms of super-functions. Until now, only one of them (fourth) is plotted in my preprint; as for others, even the analysis of the fixed points is not yet reported.

6. Construction of super-functions (in particular, super-logarithms), that are singular only in the fixed points and regular in the most of the complex plane.

7. Relations between various super-functions. For example, there is some number so huge, that it cannot be stored with "floating-point" representation. The only we can count, how many times should we take a logarithm of it, in order to make the result of order of unity. For example, one researchers estimates, that we need to take the binary logarithm M times; another one has estimates, that we should take the natural logarithm N times. Both researchers bring their results to the Forum. How to check, do their estimates agree or not?

8. Implementation of operations with numbers stored in the tetrational form. For example, if $P=\mathrm{tet}(p)$, and $Q=\mathrm{tet}(q)$, then, how to evaluate $r=\mathrm{tet}^{-1}(PQ)$, assuming, that both $P$ and $Q$ are huge and cannot be evaluated in your computer?

It should be some analogy of expression exp(p) exp(q) = exp(p+q) .

I hope, tommy1729 can collect the results related to some of these topics in some "review" treads. Then, I shall ask Tommy to convert such a "reviews" into Latex and submit them (or some of them) to some mathematical journal(s). Let tommy1729 invites volunteers to help with this job. From my side, I shall be glad to be invited to participate in a paper, describing any of topics, mentioned above. However, you may suggest other interesting and important topics too.
Sounds good, I like it so far!

I would love to help in any way that I can. I have much more free time recently, so I will definitely be around more. I can send you my sshpubkey by email if you'd like.

Also, I was wondering if we could use any of the discussions from my other papers:

I personally think the applications would also help in an overview paper.

Andrew Robbins
how about a tetration meeting ?

we live far apart i assume , but still.

regular posters are welcome at my place ...

with all respect for this forum and all written papers and posted ideas , such communications might be faster and more efficient than a forum or a pdf 'alone' ...

regards

tommy1729
well i can't attend the meeting. (what if one of you guys are internet predators or something)

you guys can instead create a pdf file summarizing the discussion that was held during the meeting, and i can comment on it on this forum.

this tetration thing isn't going too well, obviously. i;m pretty sure they weren't stuck in a quagmire like this when they developed real exponentiation, or iterated multiplication... multiplication, which exponentiation is based on, is both commutative and associative, leading to all of the nice properties, like distributivity of powers over multiplication and a^(x+y) = a^x*a^y.... I saw one day, that the problem is that is addition is commutative and associative, but exponentiation is not, so no function can consistently map addition of numbers to exponentiatiation of numbers. so ${}^{x+y} a \ne ({}^x a)^{({}^y a)}$, and ${}^x (a^b) \ne ({}^x a)^{({}^x b)}$
So guys,

Thats Dmitrii, Ansus and Andrew.
Everyone of you should have an account on http://bitbucket.org/.
There is an e-mail system built in, which we will use for communication.
There is even a bug-tracking system which we may use for the organization of changes.
To test the system, please e-mail me your public ssh-keys via e-mail on bitbucket.
(if the bit-bucket e-mail system is too primitive we may switch to normal e-mail communication.)

A hg clone of the sources should have been successful for all participants.
Did everyone manage to do a clone?

I think the tasks are as follows:
• Ansus: is responsible for the Newton and Lagrange-method.
• Andrew: is responsible for the intuitive Abel function and watches over the use of the English language. (Andrew, sorry for the name change, but I think its better to change the name than keeping confusing names, the old name "natural ...tion" also did not yet appear in a journal article, so I think we can still change).
• Dmitrii: is responsible for the Cauchy-integral method.
• Henryk: coordinates and does all the things nobody else wants to to do
• all: reviewing (and understanding) what the others write

There are a lot of methods we will present. So the focus is on a rigorous, short but appealing introduction into each method, its particular application to exponentiation as base function (with proofs of convergence/well-definition where possible), and proofs of the equality of methods.
Where we dont know about the equality we want at least show difference/equality numerically in the complex plane.
Thats really ALOT. To keep the paper publishable (i.e. short! I aim at under 30 pages) we really need to squeeze as much as possible, but keep it readable and appealing. Otherwise we may drop rather exotic methods.

@tommy:
A meeting would not be a bad idea. However I am personally not able to travel anywhere due to a catastrophic financial situation.
The purpose of the article is not to summarize everything is written on the forum.
It has a clearly cut topic: methods for real-analytic tetration (though for some methods we only conjecture that they are real-analytic).
The article will feature only methods that are either theoretically safe, or that have a working numerical implementation.
As far as it is known of course the first creator of a method will be mentioned.
(06/14/2009, 08:55 PM)bo198214 Wrote: [ -> ]Thats Dmitrii, Ansus and Andrew.
Everyone of you should have an account on http://bitbucket.org/.
There is an e-mail system built in, which we will use for communication.
There is even a bug-tracking system which we may use for the organization of changes.
To test the system, please e-mail me your public ssh-keys via e-mail on bitbucket.
Henryk, I made the accont http://bitbucket.org/kouznetsov/
(for simplicity, the same name as here)

I am not successful with email:

I am not successful with the public key. I copypast my intent:
D:~/BITBU>ls /.ssh/id_rsa.pu
ls: /.ssh/id_rsa.pu: No such file or directory
D:~/BITBU>ls ~/.ssh/id_dsa.pub
ls: /Users/D/.ssh/id_dsa.pub: No such file or directory
D:~/BITBU>ssh -v
OpenSSH_3.8.1p1, OpenSSL 0.9.7i 14 Oct 2005
usage: ssh [-1246AaCfghkNnqsTtVvXxY] [-b bind_address] [-c cipher_spec]
[-D port] [-e escape_char] [-F configfile] [-i identity_file]
[-L port:host:hostport] [-l login_name] [-m mac_spec] [-o option]
[-p port] [-R port:host:hostport] [user@]hostname [command]
D:~/BITBU>ssh-keygen
You must specify a key type (-t).
Usage: ssh-keygen [options]
Options:
-b bits Number of bits in the key to create.
-c Change comment in private and public key files.
-e Convert OpenSSH to IETF SECSH key file.
-f filename Filename of the key file.
-g Use generic DNS resource record format.
-i Convert IETF SECSH to OpenSSH key file.
-l Show fingerprint of key file.
-p Change passphrase of private key file.
-q Quiet.
-y Read private key file and print public key.
-t type Specify type of key to create.
-B Show bubblebabble digest of key file.
-C comment Provide new comment.
-N phrase Provide new passphrase.
-P phrase Provide old passphrase.
-r hostname Print DNS resource record.
-G file Generate candidates for DH-GEX moduli
-T file Screen candidates for DH-GEX moduli
D:~/BITBU>ssh-keygen -t
ssh-keygen: option requires an argument -- t
Usage: ssh-keygen [options]
Options:
-b bits Number of bits in the key to create.
-c Change comment in private and public key files.
-e Convert OpenSSH to IETF SECSH key file.
-f filename Filename of the key file.
-g Use generic DNS resource record format.
-i Convert IETF SECSH to OpenSSH key file.
-l Show fingerprint of key file.
-p Change passphrase of private key file.
-q Quiet.
-y Read private key file and print public key.
-t type Specify type of key to create.
-B Show bubblebabble digest of key file.
-C comment Provide new comment.
-N phrase Provide new passphrase.
-P phrase Provide old passphrase.
-r hostname Print DNS resource record.
-G file Generate candidates for DH-GEX moduli
-T file Screen candidates for DH-GEX moduli
D:~/BITBU>ssh-keygen -t public
unknown key type public
--------------------------
Why it does not want to geberate the public key?
What other type of key can be generated instead?
------------------------
I have created the folder
http://bitbucket.org/kouznetsov/cauchi-integral/
Can you access this folder?
Should I upload there the latex files, C++ codes, pics?
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