Hmm, Xorter, I'm not sure I do actually understand what you're after. But well, in case this is in line...
I've one time considered the relation between \( a + b \) and \( a * b \) as interpolatable. The key was that \( a * b = \log( \exp(a) + \exp(b) ) \) and the addition can be written as \( \log^0( \exp^0(a) + \exp^0(b) ) \) compared with the multiplication \( a * b = \log^1( \exp^1(a) + \exp^1(b) ) \) . So the idea was to define a continuum of fractional-order operators between "+" and "*" based on fractional iterates of \( \log() \) and \( \exp() \) .
Well, this has surely a lot of issues, for instance do we want to have that fractional-order operations associative, commutative and so on, remembering, that such properties are reduced when we extrapolate to higher/lower orders by higher/lower iterates of the \( \log() \) and \( \exp() \) -functions. Also I didn't find this really promising/interesting so I did no more engage much in this ansatz.
Gottfried
I've one time considered the relation between \( a + b \) and \( a * b \) as interpolatable. The key was that \( a * b = \log( \exp(a) + \exp(b) ) \) and the addition can be written as \( \log^0( \exp^0(a) + \exp^0(b) ) \) compared with the multiplication \( a * b = \log^1( \exp^1(a) + \exp^1(b) ) \) . So the idea was to define a continuum of fractional-order operators between "+" and "*" based on fractional iterates of \( \log() \) and \( \exp() \) .
Well, this has surely a lot of issues, for instance do we want to have that fractional-order operations associative, commutative and so on, remembering, that such properties are reduced when we extrapolate to higher/lower orders by higher/lower iterates of the \( \log() \) and \( \exp() \) -functions. Also I didn't find this really promising/interesting so I did no more engage much in this ansatz.
Gottfried
Gottfried Helms, Kassel