- A talk about LDPC codes. This explains why I was making those perplexing pictures a few posts down.
- A talk about using a neural network to solve the 8 queens problem. These might not make a whole bunch of sense as the talk was for a course so requires background and when I gave the talk my network didn’t actually work. (Since then I managed to make it pretty much work so I’ve changed the ending of the talk).
- A movie of my neural network converging to a solution. It’s mp4 format which I can play in quicktime or VLC so hopefully you can play it in something*. The exciting swirly part is slightly before halfway.

*It’s unlikely you guys will be able to help me with this but do you know a good way to do video on a mac? I get a 430mb uncompressed avi file out of matlab and I used QTamateur then iSquint to compress it down to 600k. I’m sure there’s a better way to do this.

So, points for Dave for doing what the book did, Yuliya for getting approximately the right answer first and Leesa for having both the worst and possibly the best answer.

The answer of 0.3 is clearly terrible: if you are a frequentist that doesn’t believe in probability representing belief then your real answer should be that the 10 flips are not at all significant.

But the 0.3485 might be pretty good. What was your prior Leesa? I can certainly believe that a prior that is normal with mean 0.5 and some smallish standard deviation is perhaps more reasonable than a uniform prior and that should give us an answer above 1/3.

Guesses and answers with some calculations are both acceptable.

But just a few days ago I discovered* that there does exist a formula! (well, it can’t give you an answer of but apart from that it works). It is the rather mysterious statement . Wikipedia says that it’s related to half angle trigonometry but looking at the formula you should also be able to get it from a picture of a circle.

*reading something on Wikipedia is a form of discovery, right?

So here’s one way to come up with it: Draw the point in the complex plane and then draw the circle centred at the origin that goes through the point. Now consider the following diagram.

The angle at the boundary of the circle is half of that at the centre (that’s a theorem from circle geometry but it’s pretty easy to see from the picture in this case). The radius of the circle is so you have and this time you can invert with the standard function.

This seems quite weird to me: that this function works but the other doesn’t. It’s a stereographic projection I guess and is a better coordinate than …

This formula isn’t actually that useful but it was just what I wanted to find to do a question from my complex analysis homework 0, so it was helpful to someone somewhere.

http://www37.wolframalpha.com/input/?i=PolarPlot[(1+%2B+0.9+Cos[8+t])+(1+%2B+0.1+Cos[24+t])+(0.9+%2B+0.05+Cos[200+t])+(1+%2B+Sin[t]),+{t,+-Pi,+Pi}]

In his defense, I did spend one class just plotting wacky parametric equations.

Integral Lattices: This class was taught by a teaching robot – he was amazing, going for 75 minutes every Tuesday and Thursday without ever looking at his notes or stopping to breathe. He was very good at including all the details but not so good at explaining why we should be interested.
The topic of the course was lattices: basically abelian groups with a quadratic form that give you the length of a vector. Quadratic forms are used all over the place in number theory, geometry, algebra, physics, etc. but I still don’t really know what the point of some of the things we did were. I don’t really care how many even unimodular lattices of rank 24 there are. Some of the applications to sphere packing/coding theory were interesting although they are more useful to mathematicians than people in the real world. I gave a small talk about a coding theory result that implied a fact about lattices that you can look at here.

Channel Coding: This was an engineering course and required a fair bit of work but much of it was programming which is nice. I am now able to write mediocre C code, which will I’m sure be useful to me in my life.
In contrast to the high speed lectures of the two courses above this class went kind of slowly and there were lecture notes provided. It was occasionally frustrating – we covered the Berlekamp-Massey algorithm just by writing it down (no explanation of why it worked) and then going through amazingly complicated examples by hand. I certainly am pretty good at doing finite field calculations by now though.
But overall it was very nice to do something useful in the real world and see a lot of different topics.

As a postscript I realise that I may seem negative in my reviews of these courses but I do think they were taught pretty well. I would say that I’ve never taken an advanced maths course that I was entirely happy with. Still, lectures are the best way to learn things I think so I don’t really know how to improve them.

Final grades: 4 A’s, 7 B’s, 10 C’s, 3 D’s, 5 E’s.

So far I’ve had one request for extra credit to bump a D up to a C which I summarily denied.

Algebraic Geometry – writing an expository paper on Algebraic Geometry codes (see this book for example).

Integral Lattices – giving a talk on a paper of Koch titled “On Self-Dual, Doubly Even Codes of Length 32″.

Channel Coding – Writing computer programs to simulate CRC burst error detection and Berlekamp-Massey decoding of a Reed-Solomon code.