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 Tetration FAQ Discussion andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 11/13/2007, 09:09 AM @Gottfried Oh, about your top-down towers as opposed to bottom up towers, I've put some thought into those as well, and I've come up with a notation for those based on Barrow/Shell's tower notation. ${\overset{k=1}{\underset{n}{\text{\huge \rm T}}}}\, a_k := {\underset{k=1}{\overset{n}{\text{\huge \rm T}}}}\, {a}_{n-k+1} = {a}_{n}^{{a}_{n-1}^{\cdot^{\cdot^{a_2^{a_1}}}}}$ This way of looking at it makes it not so much a matter of notation (as yours is) but a matter of indexing. The reversed indexing allows you to take the limit in the other direction as I think you were talking about. Also using this notation, your downward towers can be written: $\lim_{n \rightarrow \infty} \, \overset{k=1}{\underset{n}{\text{\huge \rm T}}}\, (a_k; z) = {}_{\cdot^\cdot}{a_2^{a_1^z}}$ My only worry with this kind of tower is that the limits will be found to always evaluate to ${}^{\infty}\left(\lim_{n\rightarrow\infty} a_n\right)$ and will not provide a very "rich" convergence landscape. Andrew Robbins PS. I also find it interesting that the "k=1" at the top of the big-T fits on the line of the T which is bigger and the "n" fits at the bottom of the T which is smaller, so the top-down notation fits better with the T notation, just as the top-down view of nested exponentials is how they are actually evaluated, bringing us closer to reality. « Next Oldest | Next Newest »

 Messages In This Thread Tetration FAQ Discussion - by bo198214 - 08/09/2007, 12:07 AM Literature - by Gottfried - 11/23/2007, 04:30 PM RE: Literature - by andydude - 12/17/2007, 07:31 PM RE: Tetration FAQ - by andydude - 01/13/2008, 08:43 AM RE: Tetration FAQ - by Ivars - 01/13/2008, 10:17 AM RE: Tetration FAQ - by GFR - 01/13/2008, 04:02 PM RE: Tetration FAQ - by andydude - 01/16/2008, 04:13 AM RE: Tetration FAQ Discussion - by Gottfried - 07/17/2008, 10:34 AM FAQ-discuss: What is a fixpoint? - by Gottfried - 07/17/2008, 10:45 AM FAQ-discuss: What does f o g mean? - by Gottfried - 07/17/2008, 10:47 AM FAQ-discuss: What does half-/ fractional-/continuous-/complex iteration mean? - by Gottfried - 07/17/2008, 10:48 AM RE: FAQ-discuss: What does half-/ fractional-/continuous-/complex iteration mean? - by andydude - 07/17/2008, 02:52 PM Intro and Formula sheet - by andydude - 11/13/2007, 01:21 AM RE: Intro and Formula sheet - by jaydfox - 11/13/2007, 02:39 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 03:15 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 03:25 AM RE: Intro and Formula sheet - by Gottfried - 11/13/2007, 06:08 AM RE: Intro and Formula sheet - by Gottfried - 11/13/2007, 06:12 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 08:37 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 09:09 AM RE: Intro and Formula sheet - by bo198214 - 11/13/2007, 10:42 AM RE: Intro and Formula sheet - by andydude - 11/15/2007, 03:16 AM RE: Intro and Formula sheet - by jaydfox - 11/15/2007, 03:38 AM RE: Intro and Formula sheet - by andydude - 11/30/2007, 05:32 PM [split] Tetration FAQ - by Ivars - 07/11/2008, 06:28 AM

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