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 Fundamental Principles of Tetration marraco Fellow Posts: 93 Threads: 11 Joined: Apr 2011 03/13/2016, 04:13 AM (This post was last modified: 03/13/2016, 04:38 AM by marraco.) (03/09/2016, 10:33 AM)Daniel Wrote: Let $f(z)$ and $g(z)$ 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 $z=0$. That allows me to consider the dynamics of the infinite branches, because each of the branches has a fixed point for $^{-\infty}a$. Setting the entropy to being low for the exponential map can be achieved by setting $a$ close to unity in $^za$. Then the dynamics of neighboring fixed points can be computed from a fixed point. Oh. I meant another thing. Let's take this expression: ${a_1}^{{a_2}^{{{\cdot^\cdot}^\cdot}^{a_n}^x}}$ It can be written $\vspace{25}{a_1}^{{a_2}^{{{\cdot^\cdot}^\cdot}^{a_n}^x}} \,=\,\, ^{n+slog_a(x)}a$, but only for some values of x and a, because for $\vspace{15} 0 \leq a \leq e^{e^{-1}}$, $\vspace{15}slog_a(x)$ is not defined for all values of x. but it would be possible if $\vspace{10} ^{x}a$ is defined as a "function" with 3 branches (5 counting both asymptotes), because slog would be defined for any real value. 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 $\vspace{20}ln(ln(ln(n))) \,=\, {\,}^{-3+slog_e(n)}e$ aside of that, there is a different issue: exponentiation and logarithms also produces multiple or infinite values. For example, $\vspace{15}a^{\pi}$ has infinite results, so the plot of $\vspace{10}^{x}a$ 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. « Next Oldest | Next Newest »

 Messages In This Thread Fundamental Principles of Tetration - by Daniel - 03/08/2016, 03:58 AM RE: Fundamental Principles of Tetration - by sheldonison - 03/08/2016, 10:18 AM RE: Fundamental Principles of Tetration - by Daniel - 03/10/2016, 02:14 AM RE: Fundamental Principles of Tetration - by sheldonison - 03/10/2016, 03:06 AM RE: Fundamental Principles of Tetration - by sheldonison - 03/10/2016, 04:16 AM RE: Fundamental Principles of Tetration - by Daniel - 03/10/2016, 04:55 AM RE: Fundamental Principles of Tetration - by sheldonison - 03/11/2016, 08:50 AM RE: Fundamental Principles of Tetration - by marraco - 03/08/2016, 06:58 PM RE: Fundamental Principles of Tetration - by Daniel - 03/09/2016, 10:33 AM RE: Fundamental Principles of Tetration - by Gottfried - 03/11/2016, 08:52 AM RE: Fundamental Principles of Tetration - by tommy1729 - 03/12/2016, 01:27 PM RE: Fundamental Principles of Tetration - by tommy1729 - 03/15/2016, 12:11 AM

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