08/21/2007, 05:06 AM

Using gp, I computed Andrew's solution for base e, using a 50x50 matrix. (Side question: Andrew, how much faster are other libraries at solving these large matrices?)

Even with such a short truncation, only 50 terms, it's pretty clear that at least his first 6 odd derivatives (1, 3, 5, 7, 9, and 11) are convex. If this pattern continues, it would appear that maybe even all the odd derivatives will be convex as the limit of the number of terms goes to infinity.

While not strictly necessary to satisfy the basic constraints (iterated exponential property, infinitely differentiable (hopefully)), having all the odd derivatives be convex basically ensures uniqueness. That is, conceptually, I'm pretty sure that there can be only one solution that has all its odd derivatives convex. If we try to tweak the curve in the slightest (with a cyclic function, e.g., a Fourier series), somewhere, maybe in the 7th derivative, or the 25th, the disturbance will cause a loss of convexity, which shows up two derivatives later as a negative value for that derivative.

So, while my solution is the conversion from the unique solution of base eta (in the limiting case), it would seem that base conversion is only valid for integer increments of the superexponent. I suppose this is possible, since my formula can only be explicitly proven for integer increments, and I made the mistake of assuming (Occam's razor and all) that if it worked for all integer increments, it would work for fractional increments. Oops.

Even with such a short truncation, only 50 terms, it's pretty clear that at least his first 6 odd derivatives (1, 3, 5, 7, 9, and 11) are convex. If this pattern continues, it would appear that maybe even all the odd derivatives will be convex as the limit of the number of terms goes to infinity.

While not strictly necessary to satisfy the basic constraints (iterated exponential property, infinitely differentiable (hopefully)), having all the odd derivatives be convex basically ensures uniqueness. That is, conceptually, I'm pretty sure that there can be only one solution that has all its odd derivatives convex. If we try to tweak the curve in the slightest (with a cyclic function, e.g., a Fourier series), somewhere, maybe in the 7th derivative, or the 25th, the disturbance will cause a loss of convexity, which shows up two derivatives later as a negative value for that derivative.

So, while my solution is the conversion from the unique solution of base eta (in the limiting case), it would seem that base conversion is only valid for integer increments of the superexponent. I suppose this is possible, since my formula can only be explicitly proven for integer increments, and I made the mistake of assuming (Occam's razor and all) that if it worked for all integer increments, it would work for fractional increments. Oops.

~ Jay Daniel Fox