# Research Polymath

## August 2, 2009

### Application

Filed under: Uncategorized — ramanujantao @ 2:17 am
Tags: ,

Hardy used to say that number theory is the purest subject. He was proud of the fact that it had no applications in that time. Now however, it has many applications to cryptography, code breaking etc.. In the same way, quantum entanglement does not seem to have any real practical applications. However this following problem and thought process helped someone solve a problem he was having trouble on involving statistical entanglement. So it goes to show you, many subjects, no matter how pure they are, can have real world applications.

Question. Consider three cities: Philadelphia, New York City, and Trenton. These can be represented by subsystems $A$, $B$ and $C$. Now if we measure $A$, $B$ and $C$ separately, it is impossible to obtain information about the entire system. The information is encoded in the nonlocal correlations between the subsystems.

This problem helped someone in a real world setting.

One thing I notice about Rudin in general is that he tends to “pull rabbits out of hats.” For instance, on pg.2, he does the following: Let $A = \{p: p^2<2, \ p \in \mathbb{Q}, \ p>0 \}$ and $B = \{p: p^2>2, \ p \in \mathbb{Q}, \ p>0 \}$. He wants to show that $A$ contains no largest element and $B$ contains no smallest. Then all of a sudden he does the following: For every $p>0$ associate $q = p-\frac{p^2-2}{p+2} = \frac{2p+2}{p+2}$. Then consider $q^2-2$ to show that $q \in A$ or $q \in B$.