green function Archives - Quantum Calculus

Green Star Formula

By oliverknill January 28, 2018 February 5, 2018

We found a formula of the green function entries g(x,y). Where g is the inverse of the connection matrix of a finite abstract simplicial complex. The formula involves the Euler characteristic of the intersection of the stars of the simplices x and y, hence the name.

discrete geometry, Graph theory

Quest for a Green Function Formula

By oliverknill November 26, 2017 November 26, 2017

A simplicial complex G, a finite set of non-empty sets closed under the operation of taking finite non-empty subsets, has the Laplacian $L(x,y) = {\rm sign}(|x \cap y|)$. It is natural as it is always unimodular so that its inverse $g(x,y)$ is always integer valued. In a potential theoretical setup, …