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 Fundamental Principles of Tetration Daniel Fellow Posts: 69 Threads: 29 Joined: Aug 2007 03/09/2016, 10:33 AM @marraco, you brought up a couple of issues of interest to me. (03/08/2016, 06:58 PM)marraco Wrote: There is a direct connection to partition numbers (number theory), in the Taylor series.See Combinatorics. There are several types of set partitions. Let $f(z)$ and $g(z)$ be holomorphic functions, then the Bell polynomials can be constructed using Faa Di Bruno's formula. $D^nf(g(z))=\sum_{\pi(n)} \frac{n!}{k_1! \cdots k_n!} (D^kf)(g(z))\left(\frac{Dg(z)}{1!}\right)^{k_1} \cdots \left(\frac{D^ng(z)}{n!}\right)^{k_n}$ A partition of $n$ is $\pi(n)$, usually denoted by $1^{k_1}2^{k_2}\cdots n^{k_n}$ with $k_1+2k_2+ \cdots nk_n=k$; where $k_i$ is the number of parts of size $i$. The partition function $p(n)$ is a decategorized version of $\pi(n)$, the function $\pi(n)$ enumerates the integer partitions of $n$, while $p(n)$ is the cardinality of the enumeration of $\pi(n)$. Setting $g(z) = f^{t-1}(z)$ results in $D^n f^t(z) = \sum_{\pi(n)} \frac{n!}{k_1! \cdots k_n!} (D^k f)(f^{t-1}(z))\left(\frac{Df^{t-1}(z)}{1!}\right)^{k_1} \cdots \left(\frac{D^n f^{t-1}(z)}{n!}\right)^{k_n}$ The Taylors series of $f^t(z)$ is derived by evaluating the derivatives of the iterated function at a fixed point $f^t(0)$ by setting $z=0$ and separating out the $k_n$ term of the summation that is dependent on $D^n f^{t-1}(0)$. $D^n f^t(0) = \sum \frac{n!(D^k f)(0)}{k_1! \cdots k_{n-1}!} \left(\frac{Df^{t-1}(0)}{1!}\right)^{k_1} \cdots \left(\frac{D^n f^{t-1}(0)}{(n-1)!}\right)^{k_{n-1}} + (D f)(0) D^n f^{t-1}(0)$ The remaining $p(n)-1$ terms of the summation are only dependent on $D^k f^{t-1}(0)$, where $0. Let me know if you have any questions. (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. « 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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