  The standard view is that PDEs are much more complex than ODEs, but, as will
be shown below, for finite derivatives this is not true. We consider the
$C^*$-algebras ${\mathscr H}_{N,M}$ consisting of $N$-dimensional finite
differential operators with $M\times M$-matrix-valued bounded periodic
coefficients. We show that any ${\mathscr H}_{N,M}$ is $*$-isomorphic to the
universal uniformly hyperfinite algebra (UHF algebra) $
  \bigotimes_{n=1}^{\infty}\mathbb{C}^{n\times n}. $ This is a complete
characterization of the differential algebras. In particular, for different
$N,M\in\mathbb{N}$ the algebras ${\mathscr H}_{N,M}$ are topologically and
algebraically isomorphic to each other. In this sense, there is no difference
between multidimensional matrix valued PDEs ${\mathscr H}_{N,M}$ and
one-dimensional scalar ODEs ${\mathscr H}_{1,1}$. Roughly speaking, the
multidimensional world can be emulated by the one-dimensional one.
