| Abstract:
The classical arithmetic mean AM(x,y) = (x+y)/2 is the mean for the
group law of addition in the sense that AM(x,y)+AM(x,y)=x+y and
similarly, the geometric mean is the mean corresponding to the group
law of multiplication in positive reals. It is natural to ask whether
there exists such a group operation (over positive real numbers) that
corresponds to the arithmetic-geometric mean (AGM) of Gauss. In other
words, the question is to construct a group operation x*y such that
AGM(x,y)*AGM(x,y)=x*y. In this talk, we recall the basic definition
and some of the elementary properties of the AGM and assuming that
such a group law exists, we will derive several formal equational
consequences valid for the binary algebra. Indeed, in 1999, Shinji
Tanimoto has succeeded in constructing such a (non-associative) loop
operation x*y using Jacobi's theta functions, but no such group law
is known to exist.
|
