## Discrete Calculus etc

Some update about recent activities: a new calculus course, the Cartan magic formula and some programming about the coloring algorithm.

Some update about recent activities: a new calculus course, the Cartan magic formula and some programming about the coloring algorithm.

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

We prove that connected combinatorial manifolds of positive dimension define finite simple graphs which are Hamiltonian.

About the origin of the definitino of shellability.

The beautiful Alexander duality theorem for finite abstract simplicial complexes.

We compute the quadratic interaction cohomology in the simplest case.

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 … ….

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.

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 … ….

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.