# Quantum Calculus

Calculus without limits
##### 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...
##### 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...
##### 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...
##### Aspects of Discrete Geometry
The area of discrete geometry is a maze. There are various flavors.
##### 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...
##### 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...
##### 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...
##### 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...
##### 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...
##### The Dirac operator of Products
Implementing the Dirac operator D for products $latex G \times H$ of simplicial complexes $latex G,H$ without going to the Barycentric refined simplicial complex has...
##### 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...
##### 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...
##### 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...
##### 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...

## 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 of calculus in a discrete setting: Theorem: . Proof. Choose … ….

## 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 open neighborhood U of a, such that for all x … ….

Taylor series