08/03/2009, 12:41 AM
(08/02/2009, 10:47 AM)bo198214 Wrote: However our widely used super-logarithm does not fit in this pattern,
thatswhy I followed Dmitriis suggestion: arcsuper-exponential
This is the inverse of the super-exponential and hence what we previously called super-logarithm. arcsuper-exponential means also Abel function of the exponential, so we *dont* introduce an extra arcsuperfunction.
I think "arcsuper" is a terrible idea. I personally dislike the "arctan" terminology, and prefer "inverse tangent" instead. What's wrong with "super-logarithm"? (aside from being a terrible Google search term)
(08/02/2009, 10:47 AM)bo198214 Wrote: @Andrew: Instead of with matrices I would like you to start with the equations, because the matrices are not really necessary to explain the approach. Something roughly like: From the Abel equation ... powerseries ... infinite equation system ... (represented with the Carleman matrix as ...) ambiguity of solutions .... intuitive solution ... etc.
Hmm... I have no idea how I would describe it without matrices... if you can do that, more power to you. For me it would be like trying to describe functions without sets. However, it would be good to generalize a little, because the definition of the Abel matrix doesn't apply to doing intuitive iteration of (x -> a x) because you have to use different truncations for this case. I remember we tried to use the Abel matrix for this (x -> a x) a while ago, and for the longest time, I've been wanting to follow through with this. It seems like even knowing the solution (log_a(x)) doesn't really make it any easier when discussing this example.