G. F. Romerio
Le Cannet - France ; Saluzzo - Italy
Registered: Sep 2004
Hyper-operations. Progress Report. Zeration.
C. A. Rubtsov – G. F. Romerio
By grouping the answers to the questions received about hyperoperations, as well as concerning some issues raised during Internet debates, we attach an introduction to the “zeration” operation, hyper-operation of rank s=0, with reference to a paper cited in the Stephen Wolfram’s book “A New Kind of Science”, under the title “Ackermann’s Function and New Arithmetical Operation”.
Zeration is a new arithmetical operation that belongs to the Grzegorczyk hierarchy and coincides, in some particular cases, with the well known “successor” unary operation. It is not associative, but it is commutative and, therefore, it has a unique inverse operation, called “deltation”. It follows its own algebraic rules and generates, with its inverse, a new set of numbers, called the “delta numbers” that can be put in bijection with the logarithms of negative numbers. Functions built with the help of zeration are discontinuous and singular. For these reasons, zeration can be used for defining Boolean operators and fundamental discontinuous functions, such as the Dirac function and the Heaviside function, with possible practical applications in science and engineering, in the framework of digital signals analysis. A poster concerning this issue has been presented at the International Congress of Mathematicians (ICM), Madrid, 22-30 August, 2006.
(See also: http://icm2006.org/AbsDef/Posters/abs_0480.pdf ).
From a philosophical and theoretical point of view, zeration opens a new way for approaching singular mathematical objects, such as “actual” infinite, with a methodology similar to the procedures used in non-standard analysis. It also allows to show an alternative possible way for representing multi-valued complex numbers, which will probably contribute to the solution of a paradox, subject of a famous dispute between Euler and d’Alembert, in the 18th century.
The authors encourage the readers of the attachment to freely use the terminology and symbols that they have proposed. They will welcome any suggestion and comment and, in case of partial or total text citations, they would appreciate the reference to: “C. (or K.) A. Rubtsov – G. F. Romerio; Hyper-operations. Progress Report. Zeration. The Wolfram Research Institute NKS Forum, Jan 6th, 2006”.
KAR / GFR
Attachment: nks forum iv - final.pdf
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Last edited by G. F. Romerio on 10-23-2012 at 04:43 PM
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