Nonlinear n-port decomposition via the Laplace operator

We give a general solution to a previously open problem in the decomposition of nonlinear n-ports. Any resistive (or capacitive or inductive) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port can be decomposed into a particular interconnection of two simpler <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports. The first is reciprocal, and the second can be further decomposed into <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">( n^2 - n)/2</tex> reciprocal n-ports and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n^2 - n)/2</tex> linear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n-</tex> ports. The technique, which we believe is completely new to network theory, is based on certain algebraic properties of the Laplace operator. It is related to the Hodge theorem from differential geometry, applied to l-forms on Euclidean space.