# Quantum Calculus

Calculus without limits ## The quantum plane riddle

This is a bit of an update on the problem to find the limiting law in the Barycentric central limit theorem. (See some older slides.) The distribution has first experimentally been found in the PeKeNePaPeTe paper in 2012. I proved universality in 2015 using a modification of the Lidski theorem …

## Three fantastic groups

About three interesting groups: Nichols-Rubik cube, Grigorchuk group, Gupta-Sidki group.

## Cayley, Zig-Zag and Rubik

We continue to look at natural groups.
There is some progress in seeing that the Rubik cube group is natural: semi direct products can be represented by zig-zag products of Cayley graphs.

## Group Extensions and Naturalness

A bit about group theory triggered by the observation that some of the non-natural groups are non-simple groups which do not split. The later problem is a central issue in the classification problem of finite groups. While the finite simple groups have been classified in a monster effort and finitalized …

## Graphs, Groups and Geometry

A bit more update on the project of natural spaces. Which groups are natural, which metric spaces are natural, which graphs are natural?

## Towards Ultra-Finite Taylor

A discrete ultra-finite Taylor theorem is a perfect expansion producing exact numbers (data fitting) with the property that evaluating the value of the sum at every point is a finite sum.

## Dihedral time

An update about natural spaces, something about time and the irreversibility riddle, a dihedral time postulate and something about natural graphs.

Is the dyadic group of integers natural in the sense that there is a metric space which forces the group structure on it?

## Natural Metric Groups

A natural metric space defines a unique group structure on the same set such that all group operations are isometries. The integers are not natural but the dihedral numbers are. The half integers are in this sense more natural than the integers.

## Ultra finite calculus

These days, I look a bit at an elementary problem. Deform the algebra of polynomials and derivatives so that all Taylor series are finite sums when evaluated at some point. This is a task which an ultra finitist is interested in. Ultra finitism is a branch of mathematics in which …

## Discrete Cauchy Integral Formula

The discrete Cauchy integral theorem can be formulated in the discrete so that it looks like in the continuum.

## More on Graph Arithmetic

A few more remarks [PDF] about graph arithmetic. Now on the ArXiv. (Previous documents are here (June 2017 ArXiv), here (August 2017 ArXiv) and here (May 2019 ArXiv). The talk below on youtube was used for me to get organized a bit. It does not look like much has changed, …

## Topology of Manifold Coloring

Last summer I have had some fun with codimension 2 manifolds M in a purely differential geometric setting: a positive curvature d-manifold which admits a circular action of isometries has a fixed point set K which consists of even codimension positive curvature manifold. The Grove-Searle situation https://arxiv.org/abs/2006.11973 is when K …

## Coloring Discrete Manifolds

It is now 7 years, since I started to think about coloring discrete manifolds, graphs for which unit spheres are (d-1)-spheres, where d-spheres are d-manifolds which after a removal of a vertex become contractible. The problem has not gained any attention which is a good thing because there are many …

Last week, I practiced a bit more enhanced talk presentation style in which, rather than with slides, the content is spoken and then enhanced in the video using additonal illustrations. The presentation deals with some things I have done in graph theory which I consider as part of quantum calculus …

## Homotopy Manifolds

The problem of discretization It is a question which probably was pondered first by philosophers like Democrit or Archimedes. What is the nature of space? It is made of discrete stuff or is it a true continuum? As Plato already noted, such questions border to being pointless as we live …

## Discrete Real Analysis

One dimension I have always also fun teaching single variable calculus. This year, the course site is here. One of the great pioneers in real analysis was Bernard Bolzano. He was not recognized enough during his life but he was one of the first to realize that theorems about continuous …

## Graph Complements of Cyclic Graphs

Graph complements of cylic graphs are homotopic to spheres or wedge sums of spheres. Their unit spheres are graph complements of path graphs and have Gauss-Bonnet curvature which converges to a limit.

## Calculating with Networks

Summary of update of current work Since a few weeks I work more on the arithmetic of graphs and especially the strong ring of simplicial complexes which can be extended to a unital commutative Banach algebra. This line of research had started late 2016, while preparing for a workshop on …

## More on Ringed Complexes

The results mentioned in the slides before are now written down. This document contains a proof of the energy relation . There are several reason for setting things up more generally and there is also some mentioning in the article: allowing general rings and not just division algebras extends the …

## Energy relation for Wu characteristic

The energy theorem for Euler characteristic X= sum h(x)was to express it as sum g(x,y)of Green function entries. We extend this to Wu characteristic w(G)= sum h(x) h(y) over intersecting sets. The new formula is w(G)=sum w(x) w(y) g(x,y)2, where w(x) =1 for even dimesnional x and w(x)=-1 for odd dimensional x.

## Mathematicians who did not die

As Hardy once pointed out in his apology, “languages die but mathematical ideas do not”. Hardy added that a mathematician “]has the best chance of becoming immortal whatever the word immortal might mean”. Let me illustrate this with three mathematicians, all teachers of mine, who have earned more than immortality. …

## Complex energized complexes

The energy theorem for simplicial complexes equipped with a complex energy comes with some surpises.

## A Sine Gordon Theme

Abstract for this post Analysis had not been my favorite topic at first. The subject appeared to me quite technical and tedious even when sweetened like in analytic number theory. The Sine-Gordon topic and other rather concrete topics in dynamical systems theory changed that a bit and I ended up …

## Geometries and Fields

Being in the process of wrapping up the latest summer calculus course, here are some thoughts about the frame work of calculus and more generally of classical field theories. It is an old theme for me which I think about often while teaching calculus. The question how to make calculus …

## Positive Curvature with Symmetry

A major open problem in Riemannian geometry is the classification of even dimensional positive curvature manifolds with symmetry. There is a reduction theory which produces a periodic system of elements. This picture has affinity with gauge bosons in physics.

## Physics on finite sets of sets?

Introduction The idea to base physics on the evolution finite set of sets is intriguing. It has been tried as an approach to quantum gravity. Examples are causal dynamical triangulation models or spin networks. It is necessary to bring in some time evolution as otherwise, a model has little chance …

## The Hopf Conjectures

The Hopf conjectures were first formulated by Hopf in print in 1931. The sign conjecture claims that positive curvature compact Riemannian 2d-manifolds have positive Euler characteristic and that negative curvature compact Riemannian 2d-manifolds have Euler characteristic with sign (-1)d . The product conjecture claims there is no positive curvature metric …

## Poincare-Hopf for Vector Fields on Graphs

The question In discrete Poincare-Hopf for graphs the question appeared how to generalize the result from gradient fields to directed graphs. The paper mentions already the problem what to do in the case of the triangle with circular orientation. The triangle has Euler characteristic 1. An integer index on vertices …

## Mickey Mouse Sphere Theorem

The Mickey mouse theorem assures that a connected positive curvature graph of positive dimension is a sphere.

## The joy of sets of sets

Finite sets of sets can be seen as a combintarorial oddity. There is a lot of mathematical structure available on such simple finitist constructs.

## Energized Simplicial Complexes

If a set of set is equipped with an energy function, one can define integer matrices for which the determinant, the eigenvalue signs are known. For constant energy the matrix is conjugated to its inverse and defines two isospectral multi-graphs.

## The counting matrix of a simplicial complex

The counting matrix of a simplicial complex has determinant 1 and is isospectral to its inverse. The sum of the matrix entries of the inverse is the number of elements in the complex.

## Poincare-Hopf and the Clique Problem

The parametrized poincare-hopf theorem allows to see the f-vector of a graph in terms of the f-vector s of parts of the unit spheres of the graph.

## Small Dehn-Sommerville Spaces

Dehn-Sommerville spaces are generalized spheres as they share many properties of spheres: Euler characteristic and more generally Dehn-Sommerville symmetries.

## On Numbers and Graphs

We have calculated with graphs from the very beginning: Humans computed with pebbles like in this scene of the Clan of the Cave Baer (1986)) or with line graphs when writing with tally sticks (see this lecture). In all of these cases, the addition of graphs is the disjoint union …

## Gauss-Bonnet for the f-function

The f-function of a graph minus 1 is the sum of the antiderivatives of the f-function anti-derivatives evaluated on the unit spheres.

## Dehn-Sommerville

Dehn-Sommerville relations are a symmetry for a class of geometries which are of Euclidean nature.

## Branko GrÃ¼nbaum

Branko GrÃ¼nbaum (1929-2018) passed away last September. Here is the obituary from the university of Washington. One of his master pieces is the book “Tilings and patterns”, written with G.C. Shephard. The well illustrated book is considered the bible on Tilings. Here is a page from that book: Links: Personal …

## Discrete Calculus etc

Some update about recent activities: a new calculus course, the Cartan magic formula and some programming about the coloring algorithm.

## Euler Game

We prove that any discrete surface has an Eulerian edge refinement. For a 2-disk, an Eulerian edge refinement is possible if and only if the boundary length is divisible by 3

## The Hamiltonian Manifold Theorem

We prove that connected combinatorial manifolds of positive dimension define finite simple graphs which are Hamiltonian.

## Connection Duality

A simplicial complex G defines the connection matrix L which is L(x,y)=1 if and only if x and y intersect. The dual matrix is K(x,y)=1 if and only if x and y do not intersect. It is the adjacency matrix of the dual connection graph.

## Interaction Cohomology (II)

This is an other blog entry about interaction cohomology [PDF], (now on the ArXiv), a draft which just got finished over spring break. The paper had been started more than 2 years ago and got delayed when the unimodularity of the connection Laplacian took over. There was an announcement [PDF] …

## 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.

## Cohomology in six lines

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 …

## Green Star Formula

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.

## A Perron-Frobenius Vector to Wu Characteristic

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.

## Is there physics for the connection Laplacian?

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 …

## Quest for a Green Function Formula

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, …

## More Green Function Values

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 …

## Isospectral Simplicial Complexes

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 …

## Wenjun Wu, 1919-2017

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 …

## Hearing the shape of a simplicial complex

A finite abstract simplicial complex has a natural connection Laplacian which is unimodular. The energy of the complex is the sum of the Green function entries. We see that the energy is also the number of positive eigenvalues minus the number of negative eigenvalues. One can therefore hear the Euler characteristic. Does the spectrum determine the complex?

## Symmetry via Ergodic Theory

One of the attempts to quantize space without losing too much symmetry is ergodic theory. Much of my thesis belongs to this program. It is a flavor of quantum calculus, as “no limits” are involved. The story is closely related to Jacob Feldman, one of my heroes of my graduate …

## What is geometry?

In the context of quantum calculus one is interested in discrete structures like graphs or finite abstract simplicial complexes studied primarily in combinatorics or combinatorial topology. Are they geometry? Are they calculus? What is geometry? In MathE320 I try to use the following definition: Geometry is the science of shape, …

## Jones Calculus

The mathematics of evolving fields with two complex components is known already in Jones calculus.

## A quaternion valued elliptic complex

This blog entry delivers an other example of an elliptic complex which can be used in discrete Atiyah-Singer or Atiyah-Bott type setups as examples. We had seen that when deforming an elliptic complex with an integrable Lax deformation, we get complex elliptic complexes. We had wondered in that blog entry …

## Discrete Atiyah-Singer and Atiyah-Bott

As a follow-up note to the strong ring note, I tried between summer and fall semester to formulate a discrete Atiyah-Singer and Atiyah-Bott result for simplicial complexes. The classical theorems from the sixties are heavy, as they involve virtually every field of mathematics. By searching for analogues in the discrete, …

## Strong Ring of Simplicial Complexes

The strong ring is a category of geometric objects G which are disjoint unions of products of
simplicial complexes. Each has a Dirac operator D and a connection operator L. Both are related in
various ways to topology.

## The Dirac operator of Products

Implementing the Dirac operator D for products of simplicial complexes without going to the Barycentric refined simplicial complex has numerical advantages. If G is a finite abstract simplicial complex with n elements and H is a finite abstract simplicial complex with m elements, then is a strong ring element with …

## Do Geometry and Calculus have to die?

In the book ‘This Idea Must Die: Scientific Theories That Are Blocking Progress’, there are two entries which caught my eye because they both belong to interests of mine: geometry and calculus. The two articles are provided below. [I believe it is “fair use” as a reprint of these two …

## The Two Operators

The strong ring The strong ring generated by simplicial complexes produces a category of geometric objects which carries a ring structure. Each element in the strong ring is a “geometric space” carrying cohomology (simplicial, and more general interaction cohomologies) and has nice spectral properties (like McKean Singer) and a “counting …

## Space and Particles

Elements in the strong ring within the Stanley-Reisner ring still can be seen as geometric objects for which mathematical theorems known in topology hold. But there is also arithemetic. We remark that the multiplicative primes in the ring are the simplicial complexes. The Sabidussi theorem imlies that additive primes (particles) have a unique prime factorization (into elementary particles).

## Graph limits with Mass Gap

The graph limit We can prove now that the graph limit of the connection graph of Ln x Ln which is the strong product of Ln‘ with itself has a mass gap in the limit n to infinity. The picture below shows this product graph for n=13, and to the …

## One ring to rule them all

Arithmetic with networks The paper “On the arithmetic of graphs” is posted. (An updated PDF). The paper is far from polished, the document already started to become more convoluted as more and more results were coming in. There had been some disappointment early June when realizing that the Zykov multiplication …

## Three Kepler Problems

Depending on scale, there are three different Kepler problems: the Hydrogen atom, the Newtonian Kepler problem as well as the binary Blackhole problem. The question whether there is a unifying model which covers all of them is part of the quest of finding a quantum theory of gravity.

## More about the ring of networks

The dual multiplication of the ring of networks is topological interesting as Kuenneth holds for this multiplication and Euler characteristic is a ring homomorphism from this dual ring to the ring of integers.

## Unique prime factorization for Zykov addition

We give two proofs that the additive Zykov monoid on the category of finite simple graphs has unique prime factorization. We can determine quickly whether a graph is prime and also produce its prime factorization.

## Hardy-Littlewood Prime Race

The Hardy-Littlewood race has been running now for more than a year on my machine. The Pari code is so short that it is even tweetable. Here are some slides which also mention Gaussian Goldbach: What do primes have to do with quantum calculus? First of all, analytic number theory …

## 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.

## The quantum plane

Update of May 27, 2017: I dug out some older unpublished slides authored in 2015 and early 2016. I added something about the quantum gap and something on the quantum plane at the very end. Here is the presentation, just spoken now. The quantum line In one dimension, there is …

## Tensor Products Everywhere

The tensor product is defined both for geometric objects as well as for morphisms between geometric objects. It appears naturally in connection calculus.

## 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 …

## From Affinity over Vis Viva to Energy

The history of the developent of energy and entropy is illustrated. This page is a picture book featuring some of the people involved shaping the concept of energy.

## 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, …

## 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.

## 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 …

## 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.

## 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 …

## 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 …

## 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 which the empty graph is the zero element and the one point graph is the one element. This ring contains the usual integers as a subring. In the form of positive and negative complete subgraphs.

## 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.

## Sphere Spectrum

This is a research in progress note while finding a proof of a conjecture formulated in the unimodularity theorem paper.

## 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 …

## 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 …

## 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 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 …

## 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 to general metric spaces.

## 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 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 …

## 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 be valid for more general structures, generalized cellular complexes. While for discrete CW complexes, the fredholm determinant is 1 or -1, in general it can now take more general values but the structures are also more strange: in the continuum much more general than CW complexes as the attached cells do not need to be bound by spheres but can be rather arbitrary.

## 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 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 …

## 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 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 …

## 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 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 …

## 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 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, …

## 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 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.

## 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 grand unified gauge group? Why are there three generations of particles? Why do neutrini oscillate? How is general relativity included? (See for example page 540 …

## 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 to be payed and this includes confusions when doing things in the discrete. Here are some examples: while in linear algebra we consider row and …

## Interaction cohomology

[Update, March 20, 2018: see the ArXiv text. See also an update blog entry with some Mathematica code. More mathematica code can be obtained from the TeX Source of the ArXiv article.]. Classical calculus we teach in single and multi variable calculus courses has an elegant analogue on finite simple …

## Wu Characteristic

Update: March 8, 2016: Handout for a mathtable talk on Wu characteristic. Gauss-Bonnet for multi-linear valuations deals with a number in discrete geometry. But since the number satisfies formulas which in the continuum need differential calculus, like curvature, the results can be seen in the light of quantum calculus. Here …

## Barycentric refinement

A finite graph has a natural Barycentric limiting space which can serve as the geometry on which to do quantum calculus or physics. The holographic picture has universal spectral properties.

## Why quantum calculus?

Quantum calculus is easy to learn, allows experimentation with small worlds and allows to use the same notation we are used to classically.

## Calculus on graphs

Calculus on graphs is a natural coordinate free frame work for discrete calculus.

## Exponential Function

We have seen that $f'(x)=Df(x) = (f(x+h)-f(x))/h$ satisfies $D[x]^n = n [x]^{n-1}$.
We will often leave the constant $h$ out of the notation and use terminology like $f'(x) = Df(x)$ for the “derivative”. It makes sense not to simplify $[x]^n$ to $x^n$ since the algebra structure is different.

Define the exponential function as $exp(x) = \sum_{k=0}^{\infty} [x]^k/k!$. It solves the equation $Df=f$. Because each of the approximating polynomials $exp_n(x) = \sum_{k=0}^{n} [x]^k/k!$ is monotone and positive also $exp(x)$ is monotone and positive for all $x$. The fixed point equation $Df=f$ reads $f(x+h) = f(x) + h f(x) = (1+h) f(x)$ so that for $h=1/n$ we have $f(x+1) = f(x+n h) = (1+h)^n f(x) = e_n f(x)$
where $e_n \to e$. Because $n \to e_n$ is monotone, we see that the exponential function $\exp(x)$ depends in a monotone manner on h and that for $h \to 0$ the graphs of $\exp(x)$ converge to the graph of $\exp(x)$ as $h \to 0$.

Since the just defined exponential function is monotone, it can be inverted on the positive real axes. Its inverse is called $\log(x)$. We can also define trigonometric functions by separating real and imaginary part of $\exp(i x) = \cos(x) + i \sin(x)$. Since $D\exp=\exp$, these functions satisfy $D\cos(x) = - \sin(x)$ and $D\sin(x) = \cos(x)$ and are so both solutions to $D^2 f = -f$.

## Fundamental Theorem

Let denote the discrete derivative of a continuous function f on the real line. In this post, I assume that all functions are continuous of have compact support. They are zero outside some large interval. No smoothness is required of course. Here is the simplest version of the fundamental theorem …

## Critical points

Assume f is a continuous function of one real variable. Lets call a point p a critical point of f if Df(p)=0 where Df(x) = f(x+1)-f(x) is the discrete derivative of f. As in classical calculus, a point p is called a local maximum of f, if there exists an …