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Looking for google terms for some math stuff

September 27, 2013

If there are any mathematicians reading I have two questions.  Basically these are two mathematical “things” but I don’t know what the proper google term for these things are:

  • In standard algebraic topology, you have a shape and you use homologies to find the group structure of that shape.  But suppose the shape consists of links that are non-uniform.  For example, you have a road network, but you have freeways and dirt roads.  Or else, you are trying to graph the flow of say RMB between HK and Mainland China and you are looking at flows which are legal, quasi-legal, or illegal, and end legal and quasi-legal flows have capacity limits.  What’s the type of mathematics that lets you look at a non-uniform network, and find if there are any loops.
  • We have Black-Scholes which is based on geometric brownian motion.  Now the real world is not based on GBM, but you have a different stochastic process with fat tails.  What is the name for the process by which you can connect the microscopic stochastic process with the macroscopic PDE.  Also, Black-Scholes assumes that the random process is independent and identically distributed.  Suppose you have a process by which the random processes are not IID, how does this affect the PDE that gets created.   What’s weird is that we have things like GARCH models from econometric, and then we have stochastic calculus from physics, but there seems to be this missing piece between the two, that will let you move between a non-uniform random process and a PDE and vice-versa.

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