On 17/05/08, Dan Asimov <dasimov@earthlink.net> wrote:
Here's a question that probably has an obvious answer, but I have to ask it:
Does the Fibonacci multiplication (which Knuth denotes with just a circle: x o y) have a continuous extension to the positive reals that remains commutative and associative ?
Firstly, if we try to interpolate more values in between those of x o y --- which, along with rescaling the arguments, is what happens in defining x @ y --- the associativity goes to pot. Secondly, the identity x o y = 3 x y - x [(y+1)/phi^2] - y [(x+1)/phi^2] induces immediately a canonical extension of x o y from natural numbers to reals --- but is evidently discontinuous at multiples of phi^2 == tau^2. So one way and another, I now reckon the answer to this question is --- almost certainly not. Fred Lunnon PS I notice that in my earlier theorem statements "tau" has become accidentally transmogrified into "phi" --- they both denote the golden section number: tau = phi = (1 + sqrt5)/2. WFL