06/04/2009, 01:16 AM

(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 , and , then, how to evaluate , assuming, that both and 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.