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 Fundamental Principles of Tetration tommy1729 Ultimate Fellow Posts: 1,424 Threads: 346 Joined: Feb 2009 03/12/2016, 01:27 PM I have thought about a method that is fixpoint independant. Cant find it now , but it is here. I apparantly have the 2sinh method wich is a coo solution , probably nowhere analytic. And with sheldon i developped fake function theory. Regards Tommy1729 marraco Fellow Posts: 100 Threads: 12 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. tommy1729 Ultimate Fellow Posts: 1,424 Threads: 346 Joined: Feb 2009 03/15/2016, 12:11 AM I have been quite active lately. This is another C^oo solution for tetration ( super in base e ). - again the question " analytic ? " - http://math.eretrandre.org/tetrationforu...p?tid=1028 Also a related open question on MSE copied from here by my friend mick. http://math.stackexchange.com/questions/...l-equation --- The continuum sum / product got alot of attention here too. In particular from mike3. I wrote about parabolic fixpoints and its limit formula's. In general 99% of limit questions are solved ( excluding matrices , analytic ? ). See http://math.eretrandre.org/tetrationforu...hp?tid=951 Another thing that got popular is the " J - function " This is not in main , but mentioned occasionally so , therefore http://math.eretrandre.org/tetrationforu...hp?tid=911 Basically , you can easily find the popular ideas By looking at 1) open problems section - incomplete for clarity !! - 2) sorting by votes , replies and views ... But take with a grain of salt , Some are a bit overrated and others underrated ... Its just a statistically logic idea to search like that. Title choices can bias and there are not many votes. 3) keywords in other threads. You can search the tag " tetration " on MSE to find Some ideas from here. My student mick , sheldon and gottfried are on MSE too and i dare to say they are a reference in combination with the tag tetration. An open problem that i discussed with mick is http://math.stackexchange.com/questions/...-a-n-a-n-1 Myself , mick and sheldon imho tend to have as main keywords Complex analysis , periodic and limit. So i guess that is the style of us 3. I already mentioned fake function theory and my 2sinh in a reply here in this thread. I have a - very unfinished - google page. In fact it needs attention. https://sites.google.com/site/tommy1729/tetration Im actually a number theorist. https://sites.google.com/site/tommy1729/...conjecture And i do not like set theory in Some ways ( ZFC ! ). See https://sites.google.com/site/tommy1729/...urable-set And here http://math.eretrandre.org/tetrationforu...p?tid=1042 And http://math.eretrandre.org/tetrationforu...hp?tid=695 Hope this helps. My apologies for not posting Tex and plots or code alot. I lack time Probably gives you An idea of the way i do things and think about them. Regards Tommy1729 Daniel Fellow Posts: 79 Threads: 29 Joined: Aug 2007 03/16/2016, 09:40 PM (03/13/2016, 04:13 AM)marraco Wrote: (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? See combinatorics for the story of how I discovered the combinatorial nature of tetration. « Next Oldest | Next Newest »

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