Quantum Calculus

Quantum Calculus

Calculus without limits
From Affinity over Vis Viva to Energy
From Affinity over Vis Viva to Energy
The history of the developent of energy and entropy are interesting. This page is a picture book featuring some of the people involved shaping the...
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Helmholtz free energy for simplicial complexes
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...
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Who is this famous person?
Who is this famous person?
A rather unfamiliar picture of a famous mathematician/physisist.
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Energy, Entropy and Gibbs free Energy
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...
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Shannon Entropy and Euler Characteristic
Shannon Entropy and Euler Characteristic
Entropy is the most important functional in probability theory, Euler characteristic is the most important functional in topology. Similarly as the twins Apollo and Artemis...
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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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About quantum calculus

What is it about?

Quantum calculus deals with other flavors of calculus and especially studies calculus on discrete sets. Examples of non-traditional calculus flavors are non-standard analysis, difference calculus or calculus on graphs. All these flavors of calculus can be seen as extensions of traditional calculus or calculus on Riemannian manifolds as the usual calculus can be seen as a special or limiting case.

What about traditional calculus?

The idea is to build notions which look exactly as the known knotions of calculus. An ideal generalization of traditional calculus to the discrete does not change language at all but just changes the meaning of the operations, possibly after an extension of language. This has already been realized in the form of internal set theory of Nelson. But this flavor of calculus is not easy to manage and teach and due to the reliance on more logic a bit more risky: its easy to make mistakes.One can imagine that under other historical circumstances, Quantum calculus could have been first developed and the calculus we know later been derived and have the status of generalized function theory or geometric measure theory. In the same way that we could in principle for practical purposes work with rational numbers alone, one could look exclusively at discrete geometries and look at continuum geometries as limiting cases. Any computation on a device is by nature finite as there is only a finite amount of memory available. In some sense, numerical analysis is already a quantized calculus. However, the theory as told in numerical analysis books is much less elegant than the actual theory so that this is not the right approach.

What is the main idea?

One of the useful ideas 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.

More generally, any geometric theory with a notion of exterior derivative on “forms” and integration of “forms” is a calculus flavor. A nice example is Riemannian geometry, which generalizes calculus in flat space. The frame work allows then to define notions like curvature or geodesics which are so central in modern theories of gravity. These notions can be carried over to discrete spaces.

But it is not only physics which motivates to look at calculus. A big “customer” of calculus ideas is computer science. Discrete versions of notions like gradient, curvature, surface etc allow a computer to “see” or to build new objects, never seen. All movies using some kind of CGI make heavy use of calculus ideas.

Why do we study calculus?

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.

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 probably the simplest quantum calculus with no limits: everything is finite and combinatorial. I prefer to work on graphs but one can work also with finite abstract simplicial complexes. It turns out that the category of complexes and especially graphs is quite powerful, despite its simplicity. Graphs are quite an adequate language because 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.

Open mind

Its not very helpful to look only for analogies to the continuum. Very general principles (numerical and computer science demonstrate constantly how finite machines mode things) show that the continuum can be emulated very well in the discrete. We also have to look out for new things. A surprise for example is that there is a Laplacian for discrete geometries which is always invertible: discovered in February 2016 and proven in the fall 2016, it leads to invariants and potential theory different from the usual Hodge Laplacian. Unlike the Hodge Laplacian which is the square of the Dirac Laplacian, this new Laplacian has a quantized and completely finite potential theory, no super symmetry. The total energy of the geometry is the Euler characteristic.

Project page

For more, see the project page on my personal web page. The entries here are more like drafts, sometimes expository, sometimes research logs which I leave as it is for me also interesting to see later, where and how I was stuck. So, it can happen that initially in an entry, the story has not yet been clear but at the end been understood.

About comments

At the moment, with very limited time at hand, comments are not turned on. This might change at some later if more material has been added.

From Affinity over Vis Viva to Energy
From Affinity over Vis Viva to Energy
The history of the developent of energy and entropy are interesting. This page is a picture book featuring some of the people involved shaping the...
Read More
Helmholtz free energy for simplicial complexes
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...
Read More
Who is this famous person?
Who is this famous person?
A rather unfamiliar picture of a famous mathematician/physisist.
Read More
Energy, Entropy and Gibbs free Energy
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...
Read More
Shannon Entropy and Euler Characteristic
Shannon Entropy and Euler Characteristic
Entropy is the most important functional in probability theory, Euler characteristic is the most important functional in topology. Similarly as the twins Apollo and Artemis...
Read More
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

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 … ….

Shannon Entropy and Euler Characteristic

Entropy is the most important functional in probability theory, Euler characteristic is the most important functional in topology. Similarly as the twins Apollo and Artemis displayed above they are closely related. Introduction This blog mentions some intriguing analogies between entropy and combinatorial notions. One can push the analogy in an other direction and compare random variables with simplicial complexes, Shannon … ….

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.