First of all, I agree with most of your notation and terminology decisions, but I do have some comments and suggestions regarding them.
To summarize your proposal, it seems you have focused on these systems:
Of all of these the ones I have a problem with are 2, 5, 8, and 9. I like the others. Points 2 and 5 are TeX-notation problems and point 8 is just a discussion. I don't recommend any changes to point 8. Point 9 you hardly talk about, but I would like to say a few things about it.
I agree with your ASCII symbols (+, *, ^, #), and your boxed notation, but the partial box notation for super-root and super-logarithm I do not agree with, partially because I don't understand how to use it in TeX. If you could perhaps post some example on how to use the partial-box notation in TeX, then I would love to use it myself.
I think that the Mnemonic representations of super-root (srt) and the super-logarithm (slog) are easier to use with TeX, but again if you can provide TeX macros for partial-boxed notation and your super-root notation, then I might be inclined to use it.
For your terminology for hyper-powers and hyper-exponentials, I agree with "tetrational" since I use this myself, but "tower" to me describes something more general, what you describe as "inhomogeneous towers". This is what I call nested exponentials. The term nested exponentials is analogous to nested radicals, nested logarithms, and nested integration. It is interesting that you chose that term, since I used to call them heterogeneous towers myself, so maybe we should go with that term instead. When I think of "tower" I think of "nested exponential", but we still need a term for the hyper-4-power function, and the sad thing is that J.F.MacDonnell has already popularized "hyper-power" for the hyper-4-power, which breaks the system (hyper- is supposed to be all ranks). This leaves us with the super-power function from which the mnemonic (spow) stems from. The problem with "super-power" is that it is a terrible search word, because of its use in politics. So after eliminating all possibilities we're left with no suitable terms for the hyper-4-power, except one, which is the generic term we've been using all along: hyper-4-power. But if you really want a shorter term, I would say that tower would be better than "hyper-power" or "super-power".
There are two people that have done some very interesting things with nested exponentials (heterogeneous towers, inhomogeneous towers), specifically Barrow [2], and Yukalov et.al. [3]. Barrow shows that it is possible to turn any function (such that f(0) = 1) into a tower with appropriate coefficients. He tried to find simple tower expansions of sine and cosine, but found that they were more complicated than the normal series expansion. Yukalov et.al. show that it is possible to construct a nested exponential that approximates any data-set, and that this approximation works very well for certain kinds of data like stock markets. While this may just be all hype, it looks very promising, and it may be that this tower interpolation could be a useful tool in approximation theory.
I gave a speech at my school a few months ago (quite an honor actually), and I made a handout for that speech called "Tetration in Context", and although it is a little out of context for this post, I figure it has a bit of what I've been talking about with the notations and stuff. So here it is: Tetration in Context.
Andrew Robbins
[1] J.F.MacDonnell, http://www.faculty.fairfield.edu/jmac/ther/tower.htm.
[2] D.F.Barrow, Infinite Exponentials, The Amer. Math. Monthly, Vol. 43, No. 3, (Mar 1936), pp. 150-160.
[3] V.I.Yukalov, S.Gluzman, Weighted Fixed Points in Self-Similar Analysis of Time Series, http://arxiv.org/abs/cond-mat/9907422.
To summarize your proposal, it seems you have focused on these systems:
- Box notation for hyper-operators.
- Partial-box notation for hyper-roots and hyper-logarithms.
- The hyper- prefix representing all ranks.
- The super- prefix representing rank 4.
- Symbolic representation of the super-root.
- Mnemonic representation of the super-logarithm (slog).
- Representation of hyper-operators through repetition (# = ^^ = ***).
- Terminology for hyper-n-powers, and hyper-n-exponentials.
- Terminology for nested exponentials (a^b^c^...^y^z).
Of all of these the ones I have a problem with are 2, 5, 8, and 9. I like the others. Points 2 and 5 are TeX-notation problems and point 8 is just a discussion. I don't recommend any changes to point 8. Point 9 you hardly talk about, but I would like to say a few things about it.
I agree with your ASCII symbols (+, *, ^, #), and your boxed notation, but the partial box notation for super-root and super-logarithm I do not agree with, partially because I don't understand how to use it in TeX. If you could perhaps post some example on how to use the partial-box notation in TeX, then I would love to use it myself.
I think that the Mnemonic representations of super-root (srt) and the super-logarithm (slog) are easier to use with TeX, but again if you can provide TeX macros for partial-boxed notation and your super-root notation, then I might be inclined to use it.
For your terminology for hyper-powers and hyper-exponentials, I agree with "tetrational" since I use this myself, but "tower" to me describes something more general, what you describe as "inhomogeneous towers". This is what I call nested exponentials. The term nested exponentials is analogous to nested radicals, nested logarithms, and nested integration. It is interesting that you chose that term, since I used to call them heterogeneous towers myself, so maybe we should go with that term instead. When I think of "tower" I think of "nested exponential", but we still need a term for the hyper-4-power function, and the sad thing is that J.F.MacDonnell has already popularized "hyper-power" for the hyper-4-power, which breaks the system (hyper- is supposed to be all ranks). This leaves us with the super-power function from which the mnemonic (spow) stems from. The problem with "super-power" is that it is a terrible search word, because of its use in politics. So after eliminating all possibilities we're left with no suitable terms for the hyper-4-power, except one, which is the generic term we've been using all along: hyper-4-power. But if you really want a shorter term, I would say that tower would be better than "hyper-power" or "super-power".
There are two people that have done some very interesting things with nested exponentials (heterogeneous towers, inhomogeneous towers), specifically Barrow [2], and Yukalov et.al. [3]. Barrow shows that it is possible to turn any function (such that f(0) = 1) into a tower with appropriate coefficients. He tried to find simple tower expansions of sine and cosine, but found that they were more complicated than the normal series expansion. Yukalov et.al. show that it is possible to construct a nested exponential that approximates any data-set, and that this approximation works very well for certain kinds of data like stock markets. While this may just be all hype, it looks very promising, and it may be that this tower interpolation could be a useful tool in approximation theory.
I gave a speech at my school a few months ago (quite an honor actually), and I made a handout for that speech called "Tetration in Context", and although it is a little out of context for this post, I figure it has a bit of what I've been talking about with the notations and stuff. So here it is: Tetration in Context.
Andrew Robbins
[1] J.F.MacDonnell, http://www.faculty.fairfield.edu/jmac/ther/tower.htm.
[2] D.F.Barrow, Infinite Exponentials, The Amer. Math. Monthly, Vol. 43, No. 3, (Mar 1936), pp. 150-160.
[3] V.I.Yukalov, S.Gluzman, Weighted Fixed Points in Self-Similar Analysis of Time Series, http://arxiv.org/abs/cond-mat/9907422.