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.
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.