Quantum Calculus

Quantum Calculus

Calculus without limits
A Gauss-Bonnet connection
A Gauss-Bonnet connection
An experimental observation : the sum over all Green function values is the Euler characteristic. There seems to be a Gauss-Bonnet connection.
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Sphere spectrum paper
Sphere spectrum paper
The sphere spectrum paper is submitted to the ArXiv. A local copy. It is an addition to the unimodularity theorem and solves part of the...
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Spectra of Sums of Networks
Spectra of Sums of Networks
What happens with the spectrum of the Laplacian $latex L$ if we add some graphs or simplicial complexes? (I owe this question to An Huang)....
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A ring of networks
A ring of networks
Assuming the join operation to be the addition, we found a multiplication which produces a ring of oriented networks. We have a commutative ring in...
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Arithmetic with networks
Arithmetic with networks
The join operation on graphs produces a monoid on which one can ask whether there exists an analogue of the fundamental theorem of arithmetic. The...
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Sphere Spectrum
Sphere Spectrum
This is a research in progress note while finding a proof of a conjecture formulated in the unimodularity theorem paper.
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Partial differential equations on graphs
Partial differential equations on graphs
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...
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Unimodularity theorem slides
Unimodularity theorem slides
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...
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From the Christmas Theorem to Particle Physics
From the Christmas Theorem to Particle Physics
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...
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The unimodularity theorem proof
The unimodularity theorem proof
The proof of the unimodularity theorem is finished.
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Dimension: from discrete to general metric spaces
Dimension: from discrete to general metric spaces
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...
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Euler and Fredholm
Euler and Fredholm
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...
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The Unimodularity Theorem for CW Complexes
The Unimodularity Theorem for CW Complexes
The unimodularity theorem equates a fredholm determinant with a product of indices. It originally was formulated for graphs or simplicial complexes. It turns out to...
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Quantum calculus talk of 2013
Quantum calculus talk of 2013
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...
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On Bowen-Lanford Zeta Functions
On Bowen-Lanford Zeta Functions
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...
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The Kustaanheimo prime
The Kustaanheimo prime
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...
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Particles and Primes, Counting and Cohomology
Particles and Primes, Counting and Cohomology
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...
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Counting and Cohomology
Counting and Cohomology
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...
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Quaternions and Particles
Quaternions and Particles
The standard model of particle physics is not so pretty, but it is successful. Many lose ends and major big questions remain: is there a...
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Bosonic and Fermionic Calculus
Bosonic and Fermionic Calculus
Traditional calculus often mixes up different spaces, mostly due to pedagogical reasons. Its a bit like function overload in programming but there is a prize...
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About quantum calculus

Why?
This page exists because I wanted to experiment with the WordPress content management system. The subject deals with calculus flavors related to quantum calculus or `calculus without limits'. Examples of non-traditional calculus flavors are non-standard analysis (i.e. internal set theory) or calculus on graphs (which is currently my focus). All these flavors of calculus are extensions of traditional calculus. The usual school calculus can be seen as a special or limiting case.
How come?
The idea of calculus is to look at ``rates of changes" in order to ``predict the future". This idea is everywhere: look at the sequence 4, 15, 40, 85, 156, 259, 400, 585 for example. How does it continue? In order to figure that out, we take derivatives 11, 25, 45, 71, 103, 141, 185, then again derivatives 14, 20, 26, 32, 38, 44 and again derivatives 6,6,6,6,6,6. We see now a pattern and can integrate the three times starting with 6,6,6,6,6,6,6 always adjusting the constant. This gives us the next term 820. We can predict the future by analyzing the past. Thats what calculus is about in the simplest case.
What for?
Even ergodic theory or operator theory are related to calculus. For example, as differentiation and integration are operators, one can not separate operator theory from calculus. Because a measure preserving transformation allows to define a derivative df = f(T)-f, there are calculus versions in ergodic theory for example (a lot of my thesis is about that) but the cohomologies are difficult to understand even in the simplest cases like the cohomology of the set of all measurable sets on the circle modulo all coboundaries Z +T(Z), where T is an irrational rotation. These cohomologies matter for example when studying chaos and in particular the Lyapunov exponents, which quantifies chaos.
Why now?
The power, richness and applicability of calculus are all reasons why we teach it. Calculus is a wonderful and classical construct, rich of historical connections and related with many other fields. We hardly have to mention all the applications of calculus (the last 2 minutes of this 15 minute review for single variable calculus give a few). We have barely scratched the surface of what is possible when extending calculus and how it can be applied in the future. At the moment (2016), I work on a generalized interaction calculus.
Traditional calculus
The traditional exterior derivatives (like div,curl grad) or curvature notions based on differential forms define a traditional calculus. Integration and derivatives lead to theorems like Stokes, Gauss-Bonnet, Poincaré-Hopf or Brouwer-Lefschetz. In classical calculus, the basic building blocks of space are simplices. Differential forms are functions of these simplices. In the continuum, one can not see these infinitesimal simplices. To remedy this, sheaf theoretical constructs like tensor calculus have been developed, notably by Cartan. In the discrete, when looking at graphs, the structure is simple and transparent. The theorems become easy. Here, here and here are some write ups.
Calculus on graphs
Calculus on graphs is a quantum calculus with no limits: everything is finite and combinatorial. I prefer to work on graphs, despite the fact graphs are considered one-dimensional simplicial complexes for most of the mathematical community. It turns out that the category of graphs is much more powerful. This especially happens when a graph is equipped with the Whitney complex. Graphs can describe so an abstract finite simplicial complex: the barycentric refinement of an abstract finite simplicial complex is always the Whitney complex of a finite simple graph. Having ``trivialized calculus", one can look at more complex constructs.
Interaction calculus
In interaction calculus, functions on pairs (or more generally k-tuples) of interacting (=intersecting) simplices in the graph are at the center of attention. Exterior derivatives again lead to cohomology. While in classical calculus, the derivative is df(x)=f(dx) (which is Stokes theorem as dx is the boundary chain), in second level interaction calculus, the derivative is df(x,y)=f(dx,y)+(-1)dim(x) f(x,dy). The chain complex is bigger. The cohomology groups more interesting.
Interaction cohomology
The analogue of Euler characteristic is the more general Wu characteristic. There are generalized versions of the just mentioned theorems. At the moment (2016), only a glimpse of the power of interaction calculus is visible. There are indications that it could be powerful: it allows to distinguish topological spaces, which traditional calculus does not: here is a case study.
Project page
For more, see the project page on my personal web page.
A Gauss-Bonnet connection
A Gauss-Bonnet connection
An experimental observation : the sum over all Green function values is the Euler characteristic. There seems to be a Gauss-Bonnet connection.
Read More
Sphere spectrum paper
Sphere spectrum paper
The sphere spectrum paper is submitted to the ArXiv. A local copy. It is an addition to the unimodularity theorem and solves part of the...
Read More
Spectra of Sums of Networks
Spectra of Sums of Networks
What happens with the spectrum of the Laplacian $latex L$ if we add some graphs or simplicial complexes? (I owe this question to An Huang)....
Read More
A ring of networks
A ring of networks
Assuming the join operation to be the addition, we found a multiplication which produces a ring of oriented networks. We have a commutative ring in...
Read More
Arithmetic with networks
Arithmetic with networks
The join operation on graphs produces a monoid on which one can ask whether there exists an analogue of the fundamental theorem of arithmetic. The...
Read More
Sphere Spectrum
Sphere Spectrum
This is a research in progress note while finding a proof of a conjecture formulated in the unimodularity theorem paper.
Read More
Partial differential equations on graphs
Partial differential equations on graphs
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...
Read More
Unimodularity theorem slides
Unimodularity theorem slides
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...
Read More

Sphere spectrum paper

The sphere spectrum paper is submitted to the ArXiv. A local copy. It is an addition to the unimodularity theorem and solves part of the riddle about the Green function values, the diagonal elements of the inverse of the matrix 1+A’ where A’ is the adjacency matrix of the connection graph of the simplicial complex. The paper contains two main … ….

Spectra of Sums of Networks

What happens with the spectrum of the Laplacian if we add some graphs or simplicial complexes? (I owe this question to An Huang). Here is an example, where we sum a circular graph G=C4 and a star graph H=S4. The Laplace spectrum of G is {4,2,2,0}, the spectrum of H is {5,1,1,1,0}. The spectrum of G+H is {9, 9, 9, … ….

Arithmetic with networks

The join operation on graphs produces a monoid on which one can ask whether there exists an analogue of the fundamental theorem of arithmetic. The join operation mirrors the corresponding join operation in the continuum. It leaves spheres invariant. We prove the existence of infinitely many primes in each dimension and also establish Euclid’s lemma, the existence of prime factorizations. An important open question is whether there is a fundamental theorem of arithmetic for graphs.

Partial differential equations on graphs

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 … ….

Unimodularity theorem slides

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, … ….