G. F. Romerio
Le Cannet  France ; Saluzzo  Italy
Registered: Sep 2004
Posts: 16 
Notes on Hyperoperations. Second Progress report.
Notes on Hyperoperations  Second Progress Reports
C. A. Rubtsov, G. F. Romerio  (NKS Forum III)
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In the attached notes (Notes on Hyperoperations  NKS Forum III), reference is made to previous documents prepared by the authors on hyperoperations and presented to WRI and to the NKS Forum, such as:
 a NKS Bibliography paper, Ackermann's Function and New Arithmetical operations; see also: http://www.rotarysaluzzo.it/filePDF...ioni%20(1).pdf ;
 the NKS Forum thread concerning comments on that paper, identified as NKS Forum I, News and Announcements, "Hyperoperations", see http://forum.wolframscience.com/sho...s=&threadid=579 ;
 the First Progress Report concerning some research activities on hyperoperations carried on by the authors (NKS Forum II), with the title: Hyperoperations, Tetration, Progress Report, see: http://forum.wolframscience.com/sho...s=&threadid=956 .
In the attached report, the authors present the results of their research in the hyperoperations' field, so far, particularly concerning the infinite tower, the square superroot, as well as tetration and pentation. An attempt of introducing infinite or noninteger hyperoperation ranks is also included for future analysis.
The problem of the full complete definition, as well of the domain of convergence, of the "infinite tower" function is analyzed. Heuristic "piecewise" attempts are described to find the formula of the attractor curve near the origin, which defines an area "avoided" by all the tetration curves of the type: y(x) = x # n (for integer n, even and odd). A similar "attractor" curve was apparently parametrically described by Euler (documentation missing). The results so obtained is an experimental curve described by x = k.[y^(y)1], where k = 0.148398483.. . Some possible plots are shown, by using the Mathematica software package. Comparison is made of the behaviour of the square superroot together with some tetrational curves. Indication is shown of their domains and ranges of existence.
Some recursive hyperoperations' formulas are recalled and an analysis of the values assumed by the tetrational and pentational functions, for negative integer operands, is attempted. A new mathematical entity is defined, called Number Theta, as the result of an infinite iteration of the log operator, applied to numbers 0 or 1. A new constant "sigma", satisfying expression: sigma = sln sigma = e # sigma and also found as the asymptotic value of function epentan, for n > oo, is defined. A new limit hyperoperation, called omegation, together with its inverses (omegaroot and omegalog) is defined. The graph of this "functions" still need to be carefully analyzed. Attempts are also made for defining hyperoperations with noninteger ranks, based on some observations made by Prof. H. Paul Williams, LSE, among which a possible "sesquation" operation, with rank s=3/2, seems to be justified by the Gauss "arithmeticgeometric mean", that can be defined by using the "complete elliptical integral of the first kind". For more information on the subject see:
http://mathworld.wolfram.com/Arithm...metricMean.html.
The authors are convinced that a lot of progress will be done in this area in the near future, with the successful achievements obtained in the framework of "Fractional Calculus" and of the analytic continuation of operators (see: http://mathworld.wolfram.com/FractionalCalculus.html). They also believe that hyperoperations will be an excellent tool for NKS developments and a very useful framework, in Science and Technology, particularly for designing new data formats for storing and handling extremely large numbers and practically avoiding computing overflow. It goes without saying that almost all hyperoperations commands can easily be included in Mathematica.
A Poster will jointly be presented by the authors at the International Conference of Mathematicians (ICM06), Madrid, 2030 August, 2006 (Hyperoperations as a Tool for Science and Engineering).
C. A. Rubtsov, G. F. Romerio
25th July 2006
Attachment: nks forum iii (final).pdf
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