Finite PDEs and finite ODEs are isomorphic

TL;DR

This paper demonstrates that finite-dimensional PDEs and ODEs are algebraically isomorphic, showing that multidimensional differential operators can be represented by one-dimensional scalar operators, thus challenging the traditional complexity hierarchy.

Contribution

It provides a complete algebraic characterization showing finite PDEs are isomorphic to finite ODEs via $C^*$-algebras, unifying their structure.

Findings

Finite PDEs are $*$-isomorphic to UHF algebras.

Algebras ${ m extbf{H}}_{N,M}$ are topologically and algebraically isomorphic for different N, M.

Multidimensional matrix-valued PDEs can be emulated by one-dimensional scalar ODEs.

Abstract

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.