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Fundamental Principles of Tetration
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(03/09/2016, 10:33 AM)Daniel Wrote: Let and be holomorphic functions, then the Bell polynomials can be constructed using Faa Di Bruno's formula.
Thanks for your useful help!. I updated the thread.
Hopefully more can be deducted from it.

(03/09/2016, 10:33 AM)Daniel Wrote: Let me know if you have any questions.
How did you arrived to combinatorics and bell polynomials in relation to tetration?
It was across the Taylor series, or another way?

(03/09/2016, 10:33 AM)Daniel Wrote:
(03/08/2016, 06:58 PM)marraco Wrote: Also, many equations in use of different areas of mathematics already use iterated logarithms. But to consider those tetration to negative exponents, is necessary to consider tetration a multivalued function, with multiple branches.

Yes, if logarithms are infinitely multivalued, then tetration must account for it also. All my work is based on setting arbitrary fixed points to . That allows me to consider the dynamics of the infinite branches, because each of the branches has a fixed point for . Setting the entropy to being low for the exponential map can be achieved by setting close to unity in . Then the dynamics of neighboring fixed points can be computed from a fixed point.

Oh. I meant another thing. Let's take this expression:



It can be written , but only for some values of x and a, because for , is not defined for all values of x.

but it would be possible if is defined as a "function" with 3 branches (5 counting both asymptotes), because slog would be defined for any real value.

[Image: lynxfBI.jpg?1]

When I said iterated logarithms, I just meant the same, but for negative tetration exponents. Those iterated logarithms are the most commonly found on applications of tetration. For example, here, you find the use of



aside of that, there is a different issue: exponentiation and logarithms also produces multiple or infinite values. For example, has infinite results, so the plot of should be a surface or a fractal. What we do make of that?

I speculate that all those infinite numbers should be taken as a single number with r dimensions (as complex numbers are pairs of numbers, or bidimensional numbers). Maybe tetration introduce numbers whose dimension is a real value.
But that's pure speculation. I started this thread about it.
I didn't even managed to define a space with real dimension, and that's necessary to define operations, and tetration over those numbers.
I have the result, but I do not yet know how to get it.
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Fundamental Principles of Tetration - by Daniel - 03/08/2016, 03:58 AM

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