The logarithm function log(x.y) = log(x) + log(y) so it is a group homomorphism R+->R. The exponential function e^(x+y) = e^x . e^y so it too is a homomorphism R->R+ Another note: Formalizing intuition (of existence of natural isomorphism between vector space V and its 'dual dual') is a motivation for the development of category theory.

Functional Analyses

Somehow, using a single parameter can fit any dataset (arbitrarily close, depending on Tau). The function is fα(x) = sin^2(2^xτ * arcsin √α)