Arithmetic Laplacians

4 Connections to Laplacians

Laplacians of Covering Complexes

The Laplace operator on a simplicial complex encodes information about the adjacencies between simplices. A relationship between simplicial complexes does not always translate to a relationship between their Laplacians. In this paper we look at the case of covering complexes. A covering of a simplicial complex is built from many copies of simplices of the original complex, maintaining the adjac...

Laplacians for flow networks

We define a class of Laplacians for multicommodity, undirected flow networks, and bound their smallest nonzero eigenvalues with a generalization of the sparsest cut.

Laplacians on Shifted Multicomplexes

The Laplacian of an undirected graph is a square matrix, whose eigenvalues yield important information. We can regard graphs as one-dimensional simplicial complexes, and as whether there is a generalisation of the Laplacian operator to simplicial complexes. It turns out that there is, and that is useful for calculating real Betti numbers [8]. Duval and Reiner [5] have studied Laplacians of a sp...

Laplacians on lattices

We consider some lattices and look at discrete Laplacians on these lattices. In particular, we look at solutions of the equation △(1)φ = △(2)Z, where △(1) and △(2) denote two such Laplacians on the same lattice. We show that, in one dimension, when △(i) i = 1, 2 denote △(1)φ = φ(i+ 1)− φ(i− 1)− 2φ(i) and △(2)Z = Z(i+ 2)− Z(i− 2)− 2Z(i) this equation has a simple solution φ(i) = Z(i+ 1) + Z(i− 1...