09/20/2007, 11:14 AM
andydude Wrote:If I might suggest a name, I would say alternating series of hyper-powers, since without qualifier "series" could also mean "infinite series", and "hyper-powers" or more specifically hyper-4-powers (spow), is what you describe as "increasing tower-height".
Andrew Robbins
Well, I'm not sure, what the best naming scheme is. The number of variants for series grows.
If we have {b,x}^^h ,then we have
1) same base, increasing height (or level, GFR) of towers, top exponent x
{b,x}^^0 - {b,x}^^1 + {b,x}^^2 - ...
2) same base, same height, increasing top exponent
{b,0}^^h - {b,1}^^h + {b,2}^^h - ...
3) increasing base
{1,x}^^h - {2,x}^^h + {3,x}^^h - ...
Version 1) suggests the most naive idea, to possibly extend the characteristics of a geometric series. But I didn't find a remarkable relation yet.
Version 2) embeds the geometric series as a special case and generalizes the sum of geometric series of positive and negative exponents. After I found the relation, which I presented already here, I tend to use a name related to geometric series for this version
Version 3) would extend the harmonic series, or zeta series. In fact, I'm already fiddling with that series, but with no obviously interesting result so far.
Generally I feel, that the use of "hyper" should be reserved for attributing the *property* of being of higher order. So this is not an absolute qualifier, but always relative - relative to some other, which may be common in a context, but this context changes with the evolution of knowledge. So "tetra" may be the best choice. (The use of "hyper-geometric" was a similar bad choice, but may be understandable. At least, the related base, to what it is hyper, is preserved in the name here).
Before I'll stick to a final naming convention, I'd suggest to better see, in which context these operations / series are occuring in nature. We know, that the tetration-operation of height/level 2 occurs in the description of high-energy heat-equations and of ecological systems. If, for instance, version 1) would express some meaningful thing here, I'd bet, the most usable, significant and memorizable name comes from it.
Names like tetra-geometric series (for version 2) and tetra-harmonic-series (for version 3) would be my favourites, where for tetra-height of 1 the leading "tetra" could then simply be omitted.
For the version 1) I have thus no idea except the full description. "alternating series of tetra towers of increasing height" would express the precise meaning... "alternating series of tetra-growth towers" or "alternating tetra-growth-series"... but, well, the last was awful....
One may possibly allude to the picture of such a tower as a "stairway to heaven" ;-), finally...
Gottfried
Gottfried Helms, Kassel
