## Monday, June 25, 2012

### Commutative decomposition of endofunctions

All endofunctions can be described as the composition of zap operations. If two different places are disjoint, then zap operations involving them commute. For example, the cube function only effects the magnitude of a number and - only effects its sign so these functions commute:
(= (compose (polynomial. [0 0 0 1]) (polynomial. [0 -1]))
(compose (polynomial. [0 -1]) (polynomial. [0 0 0 1])))

Doubling a number can be described as incrementing the multiplicity of the prime number factor 2 of the number, so multiplication commutes and $\mathbb{Q}+$ forms an abelian group. In general, two polynomials commute if they are either powers of x or chebyshev polynomials. First order homogeneous linear ordinary differential operators also commute under composition:
(= (compose (lodo. [0 1 1]) (lodo. [0 2 1]))
(compose (lodo. [0 2 1]) (lodo. [0 1 1])))

The behavior of the composition of linear ordinary differential operators corresponds to polynomial multiplication. These operators can be described as endomorphisms of vector spaces of differentiable functions.