The Hydrogen trace of a complex
Motivated by the Hamiltonian of the Hydrogen atom, we can look at an anlogue operator for finite geometries and study the spectrum. There is an open conjecture about the trace of this operator.
Category: quantum calculus
A mass gap in the Barycentric limit
The Barycentric limit of the density of states of the connection Laplacian has a mass gap.
Functional integrals on finite geometries
We look at examples of functional integrals on finite geometries.
The finitist bunker
As Goedel has shown, mathematics can not tame the danger that some inconsistency develops within the system. One can build bunkers but never will be safe. But the danger is not as big as history has shown. Any crisis which developed has been very fruitful and led to new mathematics. (Zeno paradox->calculus, Epimenids paradox ->Goedel, irrationality crisis ->number fields etc.
The Helmholtz Hamiltonian System
As we have an internal energy for simplicial complexes and more generally for every element in the Grothendieck ring of CW complexes we can run a Hamiltonian system on each geometry. The Hamiltonian is the Helmholtz free energy of a quantum wave.
Energy theorem for Grothendieck ring
Energy theorem The energy theorem tells that given a finite abstract simplicial complex G, the connection Laplacian defined by L(x,y)=1 if x and y intersect and L(x,y)=0 else has an inverse g for which the total energy is equal to the Euler characteristic with . The determinant of is the Fermi characteristic . In the spring 2017 linear algebra Mathematica … ….
Helmholtz free energy for simplicial complexes
Over spring break, the Helmholtz paper [PDF] has finished. (Posted now on “On Helmholtz free energy for finite abstract simplicial complexes”.) As I will have little time during the rest of the semester, it got thrown out now. It is an interesting story, relating to one of the greatest scientist, Hermann von Helmholtz (1821-1894). It is probably one of the … ….
Energy, Entropy and Gibbs free Energy
Energy U and Entropy S are fundamental functionals on a simplicial complex equipped with a probability measure. Gibbs free energy U-S combines them and should lead to interesting minima.
Sphere Spectrum
This is a research in progress note while finding a proof of a conjecture formulated in the unimodularity theorem paper.
The unimodularity theorem proof
The proof of the unimodularity theorem is finished.