We exploit the mirror and complementary symmetries of elementary cellular automata (ECAs) to rewrite their rules in terms logical operators. The operator representation based on these fundamental enables us construct a periodic table ECAs that maps all unique clusters similar asymptotic behavior. also expand automaton (ECA) dynamics by introducing parameter scales pace with which operators iter...

We prove the following statements about bounded linear operators on a separable, complex Hilbert space: (1) Every normal operator N that is similar to a Hilbert-Schmidt perturbation of a diagonal operator D is unitarily equivalent to a Hilbert-Schmidt perturbation of D; (2) For every normal operator A', diagonal operator D and bounded operator X, the Hilbert-Schmidt norms (finite or infinite) o...