9 papers
Calculus on Wave Fronts
Oliver Knill
We define a deformation of the exterior derivative that is a bounded operator and preserves the symmetries of the geometry. It satisfies a modified wave equation that honors the st…
Block Jacobi matrices, Barycentric limits and Manifolds
Oliver Knill
We deform block triangular Jacobi matrices appearing in geometry, look at multi-scale Barycentric limits of geometries and droplet boundary manifolds in Potts networks.
Remarks about Connection and Dirac matrices
Oliver Knill
The connection Laplacian L and the Dirac matrix D are both n x n matrices defined from a given finite simplicial complex G with n sets. In both cases, there is interlacing of the e…
Interacting Geodesics on Discrete Manifolds
Oliver Knill
We define an evolution of multiple particles on a discrete manifold . Each particle alone moves on geodesics and particles can interact if they are on the same facet. They move…
Geodesics for Discrete manifolds
Oliver Knill
The geodesic flow on a finite discrete q-manifold with or without boundary is defined as as a permutation of its ordered q-simplices. This allows to define geodesic sheets and a no…
Density of wave fronts
Emily Kang, Oliver Knill
We prove that wave fronts on a flat torus become dense. As a corollary, wave fronts become dense for a square billiard or for the geodesic flow on the flat Klein bottle or the cube…
Colorful Rings of Partition
Oliver Knill
We visualize the identity p(n) = sum s(k) p(n-k)/n for the integer partition function p(n) involving the divisor function s, add comments on the history of visualizations of number…
Gauss-Bonnet for Form Curvatures
Oliver Knill
We look at curvatures that are supported on k-dimensional parts of a simplicial complex G. These curvature all satisfy the Gauss-Bonnet theorem, provided that the k-dimensional sim…
Fusion inequality for quadratic cohomology
Oliver Knill
Classical simplicial cohomology on a simplicial complex G deals with functions on simplices x in G. Quadratic cohomology deals with functions on pairs of simplices (x,y) in G x G t…