## Discrete Calculus etc

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

## 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.

## Connection Duality

A simplicial complex G defines the connection matrix L which is L(x,y)=1 if and only if x and y intersect. The dual matrix is K(x,y)=1 if and only if x and y do not intersect. It is the adjacency matrix of the dual connection graph.

## 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.

## The dunce hat and Lusternik-Schnirelmann

The interaction cohomology of the dunce hat is computed. We then comment on the discrete Lusternik-Schnirelmann theorem.

## 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.

## 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.

## A Dyadic Riemann hypothesis

When replacing the circle group with the dyadic group of integers, the Riemann zeta function becomes an explicit entire function for which all roots are on the imaginary axes. This is the Dyadic Riemann Hypothesis.