## Network Theory and Discrete Calculus – Introduction

I’ve enjoyed applying discrete calculus to various problems since Urs Schreiber and I wrote our paper together back in 2004

**Discrete differential geometry on causal graphs**

Shortly after that, I wrote an informal paper applying the theory to finance in

**Financial modeling using discrete stochastic calculus**

From there I wrote up some private notes laying the foundations for applying a higher-dimensional version of discrete calculus to interest rate models. However, life intervened, I went to work on Wall Street followed by various career twists leading me to Hong Kong where I am today. The research has laid fairly dormant since then.

I started picking this up again recently when my friend, John Baez, effectively changed careers and started the Azimuth Project. In particular, I’ve recently developed a discrete Burgers equation with corresponding discrete Cole-Hopf transformation, which is summarized – including numerical simulation results – on the Azimuth Forum here:

**Discrete Burgers equation revisited**

Motivated by these results, I started looking at a reformulation of the Navier-Stokes equation in

**Towards Navier-Stokes from noncommutative geometry**

This is still a work-in-progress, but sorting this out is a necessary step to writing down the discrete Navier-Stokes equation.

Even more recently, John began a series of very interesting Azimuth Blog posts on network theory. I knew that network theory and discrete calculus should link up together naturally, but it took a while to see the connection. It finally clicked one night as I laid in bed half asleep in one of those rare “Eureka!” moments. I wrote up the details in

There is much more to be said about the connection between network theory and discrete calculus. I intend to write a series of subsequent posts in parallel to John’s highlighting how his work with Brendan Fong can be presented in terms of discrete calculus.

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Network Theory and Discrete Calculus – Notation Revisited « Phorgy PhynanceNovember 19, 2011 at 11:27 pm

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Network Theory and Discrete Calculus – Noether’s Theorem « Phorgy PhynanceDecember 25, 2011 at 9:09 am