G. F. Romerio
Le Cannet  France ; Saluzzo  Italy
Registered: Sep 2004
Posts: 16 
Hyperoperations for Science and Technology
We are pleased to bring to your attention the collection of the research works jointly carried out by us during the last ten years in the field of the hyperoperation hierarchy, with the indispensable external collaboration of the WRI, particularly concerning the use and assistance in the field of the Mathematica tools.
The outputs of these research efforts have been put together by the authors (K. A. Rubtsov, G. F. Romerio), according to their historical developments, in a monograph with the title: Hyperoperations for Science and Technology, New algorithmic tools for computer science  LAP, Lambert Academic Publishing  2011  ISBN: 9783844315165. The book is available in the following Web sites:
https://www.morebooks.de/books/gb/p...ducts?page=155,
http://www.bod.com/index.php?id=3435&objk_id=508443 .
It can also be bought from Amazon at: http://www.amazon.com/s/ref=nb_sb_n...ions&x=12&y=19.
Part 5 of the monograph is an original presentation of the latest complementary information, as well as improvements, corrections, provisional conclusions and recommendations, particularly related to the performance of a very useful cubical approximation, based on an idea of Andrew Robbins (Maryland, USA), concerning the slog critical path, for the extension of tetration to the real numbers.
The principles that inspired that book were the object of four successful threads, posted by the authors to the NKS Forum since 2001, and have been presented at various international conferences, during the last 17 years, in particular by one of us (Rubtsov) in 1994 and, jointly, by both of us since 2006 (see bibliography, Part 5), i.e. at:
(a) Three International Conferences of Mathematicians (ICM), organized by the IMU (International Mathematical Union): ICM1994, Zurich, Switzerland; ICM2006, Madrid, Spain; ICM2010, Hyderabad, India.
(b) Two meetings of the International Scientific Conference of Mathematical Methods in Techniques and Technology: MMTT23, 2010, Saratov, Russia; MMTT24, 2011, Kiev, Ukraine.
In section 12 of Part 5 of the book, we described an 80bit tetrational machine number format, following the hyperformat rules proposed by the authors (RRH©) and conceived for the storage of extremely large and extremely small numbers. Based on this machine format, a hypercalculator and a number notation converter have been designed.
During three of the abovementioned meetings, the authors mentioned a practical application of tetration for the solution of differential equations, according to the Euler's finite differences method. Besides these practical examples, several other applications involving tetration and the superlog can be foreseen in future, e.g. in the probability and coding theories, in intensive computing, as well as in solving intrinsically difficult problems. It goes without saying that an analytically defined tetration would certainly be better than any excellent available approximation. We still hope that the application of a homomorphism with an appropriate mapping function could finally help in finding such analytical solution.
For the moment, we would like to mention the following objectives achieved so far and compare some numerical values obtained with those calculated by Prof. Ingolf Dahl, University of Goteborg, Sweden, by using his proposed 'fractional iteration' method:
 Tetration, y = b [4] x, can be defined as a function of x, for any constant base b > 0, and for the domain 2 < x < oo, always increasing, for b > 1, and oscillating, for 0 < b < 1.
 Superlog of x, base b, y = slog/b(x), can be defined as one of the inverse functions of tetration, for any b > 1; it is an always increasing function and it has a range depending on b.
 Superrot of x, of n order, can be easily exactly calculated for order n = 2, by using the Lambert function; the superroots for other orders are only calculable by using iterative processes.
 By using the cubical critical path (see the book), the approximate value of etetrapi is now:
y = e [4] pi = 3.66901 x 10^10 = 36' 690'101'018.425... , to be compared with:
y = e [4]pi = 37'150'463'906.547... (estimated by Ingolf Dahl);
 By using the cubical critical path, the approximate value of pitetrae is now:
y = pi [4] e = 1.94625 x 10^6 = 1'946'245.600..., to be compared with:
y = pi [4] e = 1'921'616.483... (estimated by Ingolf Dahl);
 By using the cubical critical path, the approximate value of etetrae is now:
y = e [4] e =2'119.349... .
We wish to apologize for (and correct) the values that we proposed using our provisional linear critical path approximation critical paths (see: http://en.wikipedia.org/wiki/Superlogarithm) and suggest the new abovementioned values, to be continuously verified as benchmarks for any future comparisons of the research results in this field. The numerical values obtained as results of some key calculations with different methods are now approaching and, certainly, more precise values will be obtained in the near future. Perhaps, we are just at the end of the... beginning and the 'Tetration Fortress'¯, with the help of the international mathematical community, will soon be taken, after its ... complete unconditional surrender.
It may be interesting to notice that y = e [4] e = e [5] 2, i.e.: epenta2, which suggests a first possible analysis of y = x [4] x = x [5] 2, the squaresupertower, or square supersuperpower of x.
It was also observed that expression i = e ^ (i*pi/2) = (e ^ pi/2) ^ i means that: (e ^ pi/2) # +oo = i (see page 165, i.e.: an infinite tower, with a real base and a unitary imaginary height, even more impressive than: e ^ i*pi = 1). The authors were so astonished by this fact that, when they reported it in the text, they didn't even realize the sudden appearance of an extended and very strange ... typing error (page 166).
Konstantin A. Rubtsov  Giovanni F. Romerio
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