Theory and Application of Simplicial Harmonic Spaces
Young Jin Kim · Seoul National University Open Repository (Seoul National University) · 2020
A harmonic cycle λ, also called a discrete harmonic form, is a solution of the Laplace's equation with the combinatorial Laplace operator obtained from the boundary operators of a simplicial chain complex.By combinatorial Hodge theory, harmonic spaces are isomorphic to homology groups with real coefficients.In particular, if a cell complex has a reduced homology with Betti number β i = 1 of a specific dimension i, it has a unique harmonic cycle up to scalar multiplication, which we call the standard harmonic cycle.We will present a formula for the standard harmonic cycle λ of a cell complex based on a high-dimensional generalization of cycletrees.Moreover, by using duality, we will define the standard harmonic cocycle λ * and show intriguing combinatorial properties of λ and λ * in relation to (dual) spanning trees, (dual) cycletrees, winding numbers w(•) and cutting numbers c(•) in high dimensions.Finally, we will also suggest two application methods; an analysis to detect oscillations by using winding number, and cutting number, and a network embedding method, called harmonic mirroring.