During the summer and fall of 2016, Annie Rak did some URAF (a program formerly called HCRP) on partial differential equations on graphs. It led to a senior thesis in the applied mathematics department. Here is a project page and here [PDF] were some notes from the summer. The research of Annie mostly dealt with advection models on directed graphs … ….
Here are some slides about the paper. By the way, an appendix of the paper contains all the code for experimenting with the structures. To copy paste the code, one has to wait for the ArXiv version, where the LaTeX source is always included. Here is the unimodularity theorem again in a nutshell: Given a finite abstract simplicial complex G, … ….
The Christmas Theorem Because Pierre de Fermat announced his two square theorem to Marin Mersenne in a letter of December 25, 1640 (today exactly 376 years ago) the theorem A prime of the form 4k+1 is the sum of two squares. is also called the Christmas theorem. The converse, the fact that a prime p of the form 4k+3 is … ….
This is an informal overview over definitions of dimension, both in the continuum as well as in the discrete. It also contains suggestions for generalizations to general metric spaces.
The following picture illustrates the Euler and Fredholm theme in the special case of the prime graphs introduced in the Counting and Cohomology paper. The story there only dealt with the Euler characteristic, an additive valuation (in the sense of Klain and Rota). Since then, the work on the Fredholm characteristic has made more progress and is now understood. The … ….
Just uploaded a larger version of my 2013 Pecha-Kucha talk “If Archimedes knew functions…”. The Pecha-Kucha format of presenting 20 slides with 20 seconds time each is fantastic to keep talks concise and to the point. The video has been produced by Diane Andonica from the Bok Center for teaching and learning. The event had been organized by Curt Mc … ….
Zeta functions are ubiquitous in mathematics. One of the many zeta functions, the Bowen-Lanford Zeta function was introduced by my Phd dad Oscar Lanford and Rufus Bowen. I am in the process to wrap up a proof of a theorem which is so short that its statement can be done in 140 characters: A finite simple graph G=(V,E) defines H=(W,F) … ….
Paul Kustaanheimo (1924-1997) was a Finnish astronomer and mathematician. In celestial mechanics, his name is associated with the Kustaanheimo-Stiefel transform or shortly KS transform which allows to regularize the Kepler problem using Clifford algebras. In this elegant picture, the motion of the two bodies becomes a rotation in three dimensions rendering therefore the Kepler motion into a harmonic oscillator. The … ….
Source: Pride and Prejudice, 2005 Judy Dench plays the role of Lady Catherine de Bourgh. I recently posted a “Particles and Primes” as well as a “Counting and Cohomology” article on the ArXiv, because, as it is a truth universally acknowledged, that an article in possession of a good result, must be in want of a place to be read. … ….
There are various cohomologies for finite simplicial complexes. If the complex is the Whitney complex of a finite simple graph then many major results from Riemannian manifolds have discrete analogues. Simplicial cohomology has been constructed by Poincaré already for simplicial complexes. Since the Barycentric refinement of any abstract finite simplicial complex is always the Whitney complex of a finite simple graph, there is no loss of generality to study graphs instead of abstract simplicial complexes. This has many advantages, one of them is that graphs are intuitive, an other is that the data structure of graphs exists already in all higher order programming languages. A few lines of computer algebra system allow so to compute all cohomology groups. The matrices involved can however become large, so that alternative cohomologies are desired.