Spooky Action from the FUTURE!

I have 2 machines that produce entangled pairs of photons. Call them thing 1 and thing 2.

Thing 1 sends a photon to Dick. Thing 2 sends a photon to Sally. They both send the other photons to the cat.

Dick and Sally measure their photons in whatever basis they choose.

Later, the cat decides whether or not to entangle the two photons it got.

If the cat did nothing, Dick and Sally's measurements have nothing to do with each other. There is no way they could talk to each other.

But if the cat did entangle them, the two measurements are correlated. That's weird, the cat acted in the future and influenced what Dick and Sally saw before.

An actual experiment

http://arxiv.org/ftp/arxiv/papers/1203/1203.4834.pdf

It reminds me of those time travelling neutrinos that came back and loosened that cable so they wouldn't be found out. (Credit Kate)

Some cute integrals

Normally I would hate evaluating integrals with trig functions with the white hot intensity of a thousand suns, but these are so cute they can get away with it.



Everybody get  . Good!
Next one



Everybody get  . Good!
Next one



Keep doing these all the way until



Big surprise!  What did you get?

Before doing the next one, guess the next one.

If you guessed   , you would be wrong. It is actually about 2.31e-11 lower than that. Still a rational multiple of   though. If you managed to guess that I tip my hat to you.

You can't possibly be satisfied with this. Something has to explain why the jump suddenly happened so far into the sequence.

Well, to do these better double the integral by going over the entire real line. Now use the property that integrating in real space and in momentum space gives the same results. The products of sincs becomes convolutions of box functions. In fact you might as well do this with a general sequence like so:

 instead of the reciprocals of the odd integers like we did before.

Look at the running sums of the alphas we had before. You see that



The step where things mess up is when the sum jumps above 2. This is generally true as well. To show why this is true you need to look at the widths of the convolutions and how that grows.

Next time someone tells you to fill in the pattern



you can confidently tell them you have no fucking clue.

Plant a seed and grow a cluster

As the title says start with a seed.

It is a n tuple of variables.

Also give an n by n skew symmetric matrix that is called the exchange matrix.

For each i between 1 and n, there is a mutation.

The action on the exchange matrix is given as follows.

View the exchange matrix as the adjacency matrix for a quiver. Now flip the arrows that touch vertex i. Also create shortcuts whenever you flip two arrows that are lined up. Cancel any loops you might have created and then give back the adjacency matrix.

The action on the seed is as follows.

Leave everything except the ith in the tuple alone. Call that one y and replace it with w. The other elements in the tuple are t's.

Now we have an n valence tree where each vertex has a seed and an exchange matrix and traversing an edge tells you to mutate in that direction.

A cluster algebra is called finite type if you end up only getting a finite number of seeds. It has been shown that the classification of finite type cluster algebras is the same as the classification of semisimple Lie algebras and finite root systems.

Baker, baker give me recurrence as fast as you can

The Baker's map is a simple dynamical system that has deterministic chaos. The map is defined on a square. Cut the dough in half vertically then stack them on top of each other after kneading each piece. It is illustrated below.

The exponential growth between two different points is especially easy to see for two points that are just separated in the horizontal direction and not in the vertical direction at all. The distance between them doubles every time, at least until they end up on different sides of the vertical cut, then its not so easy to say what happens.

It doubles the horizontal direction and halves the vertical direction, so it will preserve area for every rectangle, which is a basis. So extending this preservation to the sigma algebra they generate, you see that this map preserves Lebesgue measure.

A picture of the chaotic nature.

By composition, we get an action of this map on functions on the square. It is unitary on the Hilbert space of square integrable ones. Once we figure out this operator's spectrum, we will be able to exactly say what this map does to arbitrary square integrable functions. Express the function in the diagonalized basis then use the eigenvalues to determine the action on each summand.

Each eigenvalue will have some power N that takes it as close to 1 as you want. If you want an entire square integrable function to come back to look like itself, take the least common multiple of each N of the summands that contribute a significant amount. Some analysis will give you exactly which of the components you'll need to consider. Let's call that applying the baker map M times.

The punchline to the above is you can get a function that is arbitrary close after M iterations. This demonstrates Poincare recurrence for this dynamical system. So it starts out looking more and more disordered, but then it comes back to the original state or at least close enough.

If you deleted a bit when it was in it's chaotic state, then you would be screwed. Think like if you forgot the states in the Black Hole and didn't remember to keep track of all the Hawking radiation.