Tuesday, July 15, 2014

I've been working on a post explaining the Uncertainty Principle using induced matrix norms. It might surprise that such a simple thing (it's a very standard method of proof) to do might take time, but I want to define and illustrate every single theoretical term verbally, algebraically and geometrically, so it's a little time consuming. Uncertainty principles are a consequence of linearity. It's got me wondering if there are many more applications than are usually considered - "usually considered" meaning quantum mechanics and linear filters. I've seen real applications to game theory, which make me wonder about applications to wider economic models (in fact, just thinking about applications to things which are two player zero-sum games in disguise would make for a fun post). The invoked theorem is for a non-linear operator, which might make one hopeful. What about applications to, say Input-Output Models (since these are just LTI systems, all the theory should come over...)? GE models? How about other game theory models, such as signalling games? So much to think about!

Via Leisure of the Theory Class, a neat problem from an old novel - Typhoon by Joseph Conrad. A group of mostly Chinese and Indian laborers board a ship to return to the motherland. They store their life savings in chests aboard this ship. The chests are destroyed in a typhoon. How can you distribute the money in a way fair to each person? You know the amount of silver and can ask each person how much they stored. Here is one solution.

If I post and link much more about economics, people will think I'm an economist. Something closer to my heart. Some somewhat trivial, but instructive!, application of Morse Theory. With a pencil and paper, even one who is not intimate with the mysteries one can come to a really nice understanding of how Morse Theory implies a conserved quantity for figures in two dimensions, and getting Euler's Formula as a result is a nice plus! The important bit is understanding what he calls the "Morse Lemma", which shows that in a neighborhood around a critical point.

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