Discrete Calculus etc
Some update about recent activities: a new calculus course, the Cartan magic formula and some programming about the coloring algorithm.
Category: discrete geometry
Euler Game
We prove that any discrete surface has an Eulerian edge refinement. For a 2-disk, an Eulerian edge refinement is possible if and only if the boundary length is divisible by 3
The Hamiltonian Manifold Theorem
We prove that connected combinatorial manifolds of positive dimension define finite simple graphs which are Hamiltonian.
The beginnings of Shellability
About the origin of the definitino of shellability.
Combinatorial Alexander Duality
The beautiful Alexander duality theorem for finite abstract simplicial complexes.
Interaction cohomology Example
We compute the quadratic interaction cohomology in the simplest case.
Interaction Cohomology (II)
This is an other blog entry about interaction cohomology [PDF], (now on the ArXiv), a draft which just got finished over spring break. The paper had been started more than 2 years ago and got delayed when the unimodularity of the connection Laplacian took over. There was an announcement [PDF] which is now included as an appendix. [Not to appear … ….
The Hydrogen Relation
For a one-dimensional simplicial complex, the sign less Hodge operator can be written as L-g, where g is the inverse of L. This leads to a Laplace equation shows solutions are given by a two-sided random walk.
Cohomology in six lines
Here is the code to compute a basis of the cohomology groups of an arbitrary simplicial complex. It takes 6 lines in mathematica without any outside libraries. The input is a simplicial complex, the out put is the basis for $H^0,H^1,H^2 etc$. The length of the code compares in complexity with computations in basic planimetric computations in a triangle (Example … ….
Green Star Formula
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.