## 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.

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.

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.

The Wu characteristic of a simplicial complex is the eigenvalue of an

eigenvector to a matrix L J, where L is the connection Laplacian and J

a checkerboard matrix. The eigenvector has components whicih are

Wu intersection numbers.

The classical potential $V(x,y) = 1/|x-y|$ has infinite range which violently clashes with relativity. Solving this problem had required a completely new theory: GR. It remains also a fundamental problem still in general relativity: a Gedanken experiment in which the particles in the sun suddenly transition to particles without mass shows this. [This is forbidden by energy conservation but energy … ….

A simplicial complex G, a finite set of non-empty sets closed under the operation of taking finite non-empty subsets, has the Laplacian $L(x,y) = {\rm sign}(|x \cap y|)$. It is natural as it is always unimodular so that its inverse $g(x,y)$ is always integer valued. In a potential theoretical setup, the Green function values $g(x,y)$ measure a potential energy between … ….

We have seen that for a finite abstract simplicial complex $G$, the connection Laplacian L has an inverse g with integer entries and that $g(x,x) = 1-X(S(x))$, where $S(x)$ is the unit sphere of $x$ in the graph $G_1=(V,E)$, where $V=G$ and where (a,b) in E if and only if $a \subset b$ or $b \subset a$. We have also … ….

One can not hear a complex! After some hope that some kind of algebraic miracle allows to recover the complex from the spectrum (for example by looking for the minimal polynomial which an eigenvalue has and expecting that the factorization reflects some order structure in the abstract simplicial complex), I wondered whether there is an argument proving that that there … ….

According to Wikipedia, the mathematician Wen-Tsun Wu passed away earlier this year. I encountered some mathematics developed by Wu when working on Wu characteristic. See the Slides and the paper on multi-linear valuations. There is an other paper on this in preparation, especially dealing with the cohomology belonging to Wu characteristics. Just as a reminder, the Wu characteristic of a … ….