ChapterZero

if someone would solve this PDE analytically: on the unit square with on the right hand and bottom sides, and on the right hand side, on the top side. Any takers?

We’re supposed to solve this numerically (at least, I believe this is the correct PDE– it was a poisson problem with mixed BCs in a nonstandard geometry, this is the result of a change of coordinates). The instructor gave a reference solution for a choice of the parameters, but my results don’t match; I’m going crazy debugging.

Did I ever mention that I love the Tech? Only here have I seen such an irrelevant newspaper– the first edition I saw ran a review of Debbie does Dallas. The April 13th edition is one large insult to MIT. I especially love this opinion piece on LOTR. Here’s one of the jokes:

Heisenberg was driving his new Mercedes when he was pulled over by a police officer. “Do you know how fast you were going?” asked the officer. “No, but I know exactly where I am, ” replied Heisenberg. “You little smartass, ” said the officer, beating the snot out of Heisenberg with the butt of his Luger. “How long will this beating last?” asked Heisenberg. “I don’t know, but I know exactly how much energy I’m expending, ” said the officer.

I got my Sansa e260 mp3 player this weekend, and I’m stoked. Unfortunately, I only have Rufus Wainwright’s Poses, Chris Cornell Unplugged in Sweden, and four seasons of soundtracks from Xena on it (yeah, I’m a freak about that show). Of all these, the only ones I bought were the Xena soundtracks (because they were cheaper than air). I have a Sarah McLachlan cd, and two more Rufus Wainwrights, but I need to figure out how to convert them to an mp3 format the player will read.

I’d like to — *gasp* — buy more music. That’s unheard of– me spending money on something you can download for free with just a little effort. But the fact is, I’d like to stop ripping artists off. Seems like that’s going to be hard: Rhapsody and Napster, the only two subscription music services, are intimately tied to windows, for some reason having to do with the DRM (digital rights management) implementation they use. If I had a more adventurous spirit, I would exploit this obviously ripe opportunity for entrepenuership.

I have a short list of artists I’d like to buy albums from. It includes people like Damien Rice, Sondre Lerche, Sarah McLachlan, Jeff Buckley, The Decemberists, The Shins, Coldplay, and whoever last.fm recommends a la them. Any suggestions? Either for similar artists or linux friendly music outlets? Any offers for free music? I might end up buying cds from offline stores. Suppose that’s good– supporting neighborhood music outlets and such. But it requires effort.

As an aside, last.fm is really good– it’s a free service that tracks the music you listen to, either on your computer or on an iPod, and suggests similar artists and lets you see people with similar interests. I’ve used it to look up artists similar to Rufus Wainwright a couple of days ago, and I still have the player open on my school account so I can keep listening to the tracks it suggests. They’ve been on spot so far. There are a couple of Linux apps that let you integrate fast.fm with media players like Xine.

Also, I have to congratulate the Linux mp3 player community. They’ve done a good job with tools like gnomad and amarok, which try to make it easy to use devices under Linux that are for the most part targeted towards Windows users. It’s just my bad luck that neither of them seem to want to work for me.

here’s a link to google’s newest service. Be sure to click on the “more about” link.

On the first stochastic controls homework we have the following bonus problem (which is apparently now superbonus, since the instructor sent out an email saying we aren’t expected to be able to do this rigorously ),

Suppose that we have a sequence which are i.i.d. Gaussian random variables with mean zero and unit variance under , and such that are i.i.d. Gaussian random variables with mean zero and unit variance under . Give a necessary and sufficient condition on the non-random sequence such that . In the case that , give the corresponding Radon-Nikodym derivative. If , find an event so that but . in theory, how would you solve the hypothesis testing problem when ? How about when ?

(Here I’m assuming that where , and similarly for .)

I’m stumped on it. I have the intution that the sequence of means should be bounded, and I can construct an example where the fact that the aren’t bounded is crucial in establishing that . I can’t establish either the necessity or sufficiency, however. Aaargh!

For the necessity, given an unbounded sequence of means, I’ve been trying to construct a sequence of sets such that , but , using the fact that the ‘mass’ of the are moving away from the origin. The easiest way of doing this that occured to me is to take the sets such that under the Gaussian measure , shift them, and define so , but these sets don’t move away from the origin fast enough to get . Stumped there.

For the sufficiency, given a bounded set of means, I’ve been attempting to use the facts

and likewise for , where the inf is taken over the in the algebra of finite Borel cylinders. Then if I could show that goes to zero fast enough when does, just for the finite Borel cylinders, then I’d be done. But that fast enough is eluding me.

Any hints?

Update I got the necessity; it was actually trivial. My complaint above can be avoided by just taking an (absolutely) increasing subsequence of means which move away from the origin fast enough, doing the construction on it, and ignoring the other means by setting the other … Now on to sufficiency.

Yesterday I copied a couple of chapters from two of my prof’s large deviations books (Shwartz and Weiss, Dembo and Zeitouni), and today I started to read them. I’m stuck on the following exercise:

Show that in the case of fair coin flips, if is the number of heads obtained in flips and is an integer,

and does not grow with ! Hint:

I think the way to do this is:

• Estimate with some large deviation principle
• Use the fact that

Being able to approximate the quantity in the hint doesn’t help too much. A generalization for would maybe help, but …

The hint is wrong (and so is the generalization). Sure, you have and the first term on the right hand side goes to , but the second term grows too fast for you to say . The error actually grows with .

Aargh! I just spent 30 minutes writing a Mathematica code to see if that expectation does indeed converge to , only to realize when I finished that it’s virtually useless. Can you tell me why? I wonder if I can use my shiny new skill-set from last term to make a MCMC algorithm for testing the convergence …

I saw an alcohol vaporizer on Law and Order– never heard of such a thing– so I looked it up, and discovered AWOL, the Alcohol without liquid device. The user puts in a cartridge with their choice of beverage, and the device percolates oxygen bubbles through the liquor; the user then inhales the fumes. These are supposedly the shit in Asia and the UK.

The comments at Engadget are hilarious:

If only they could figure out way to combine cigarette smoke with the alcohol — that would be so much more efficient. Oxygen? Who needs that?!

Sitting at the bar and casually sipping your martini, or sitting at the bar and sucking down fumes from a large plastic dick?

In a freak coincedence, while checking my gmail I saw a link to a more interesting engadget blurb on a new Darpa project proposal, asking for shape-shifting robots. I’d like to read the white papers that are submitted.

Correct me if I’m wrong, but it seems the third part of the following problem serves as one of those ‘out of context’ problems that I mentioned earlier:

Let be a probability space on which is defined a sequence of i.i.d. Gaussian random variables with zero mean and unit variance. Consider the recursion where . This is a crude model for some nonnegative quantity that grows or shrinks randomly in every time step; for example, we could model the price of a stock this way, albeit in discrete time.

1. Under which conditions on and do we have in ?
2. Show that if in for some , then a.s.
3. Show that if there is no such that in , then in any sense
4. If we interpret as the price of stock, then is the amount of dollars our stock is worth by time if we invest one dollar in the stock at time 0. If a.s., this means we eventually use our investment with unit probability. However, it is possible for and to be such that a.s., but nonetheless our expected winnings ! Find such . Would you consider investing in such a stock?

To prove that assertion, it’s enough to show that the sequence doesn’t converge in distribution. In class, in distribution iff for any bounded continuous . I spent forever trying to work with this definition, but I couldn’t get anything from it (can you?). But then I broke down and decided to use another definition: in distribution iff at every point of continuity of ( and are the cdfs of and ). In this case, the problem reduces to calculus.

Has someone disproven the Riemann Hypothesis? No really, I’m asking: I barely understand the statement of it, much less the tools used in the tentative proof.