Dear Henryk!
I went out of the main field, a little bit rooming around and, therefore, you said:
Nevertheless, both the supersquare root (for 1/Eta < x < 1) and the "infinite-tower" h (for 1 < b < Eta) seem to appear as two-alued "functions", for the analysis of which the existing mathematical tools are not completely available yet (don't be nervous, please, and try to accept my ... critical descriptive innocent language). The theory of analytic functions is very respectable and extremely serious. It is what it can be made available now, but this is only the present status of the affairs. Tomorrow, your little little children will perhaps have also other more efficient (and complicated) instruments. Unfortunately, our eyes will not be around to see it.
You seemed to accept my way of presenting the inversion of function y = x^x (square tetration?), i.e. of x = ssqrt(y), by the use of the logical union of two branches of the Lambert Function (orders -1 and 0). I tried, then, to propose using the same methodology for representing the two h branches, which (together) represent the "inverse" of b = h^(1/h) = selfroot(h).
For the moment, these are only practical "tricks" and I dared to imagine a future theory justifying them. I see and respect the role of analytic functions, as well as the exigencies put forward by the necessary serial developments.
A good theory should be the most appropriate to "justifying the past and foreseeing the future". My wild ideas didn't completely cover the first part of the enlightened exigencies.
Sorry about that.
GFR
I went out of the main field, a little bit rooming around and, therefore, you said:
bo198214 Wrote:Hey Gianfranco,Actually "Thom's Catastrophe Theory" is not (... yet) a consistent mathematical theory, but only a set of ideas, to be further developed. On the contrary, the expression "multivalued functions" covers (little bit) more official mathematical concepts.
open your eyes! Catastrophe theory will not solve your problems with multivalued functions, nor will the use of the term "multivalued function".
I think I clarified in detail what terms are there and how to use them:
Nevertheless, both the supersquare root (for 1/Eta < x < 1) and the "infinite-tower" h (for 1 < b < Eta) seem to appear as two-alued "functions", for the analysis of which the existing mathematical tools are not completely available yet (don't be nervous, please, and try to accept my ... critical descriptive innocent language). The theory of analytic functions is very respectable and extremely serious. It is what it can be made available now, but this is only the present status of the affairs. Tomorrow, your little little children will perhaps have also other more efficient (and complicated) instruments. Unfortunately, our eyes will not be around to see it.
You seemed to accept my way of presenting the inversion of function y = x^x (square tetration?), i.e. of x = ssqrt(y), by the use of the logical union of two branches of the Lambert Function (orders -1 and 0). I tried, then, to propose using the same methodology for representing the two h branches, which (together) represent the "inverse" of b = h^(1/h) = selfroot(h).
For the moment, these are only practical "tricks" and I dared to imagine a future theory justifying them. I see and respect the role of analytic functions, as well as the exigencies put forward by the necessary serial developments.
A good theory should be the most appropriate to "justifying the past and foreseeing the future". My wild ideas didn't completely cover the first part of the enlightened exigencies.
Sorry about that.
GFR