07/04/2008, 09:01 PM
When mentioning continuos dimensions of space above, I do no mean fractal dimension
which should be used in this context, but topological dimension or some equivalent measure that is possible to extend to rational and real numbers. So it is more linked to:
Fractional Calculus
which is actually continuous real number iterates of differentiation/integration operations and involves gamma function ( factorial!).
Since integer differentation reduces dimension in exponent by 1, while integration increases, e.g.
d/dx (x^n) = n*x^(n-1)
it is reasonable to assume that any noninteger topological dimension (or what could be its equivalent) perhaps including negative and imaginary would be accesable via fractional differentiation/integration.
Ivars
which should be used in this context, but topological dimension or some equivalent measure that is possible to extend to rational and real numbers. So it is more linked to:
Fractional Calculus
which is actually continuous real number iterates of differentiation/integration operations and involves gamma function ( factorial!).
Since integer differentation reduces dimension in exponent by 1, while integration increases, e.g.
d/dx (x^n) = n*x^(n-1)
it is reasonable to assume that any noninteger topological dimension (or what could be its equivalent) perhaps including negative and imaginary would be accesable via fractional differentiation/integration.
Ivars