You might say that I could have spent my time better but after a few hours of work I was able to make this rather inscrutable picture. Any guesses as to what it’s doing?

1/3 is ‘correct’ by exactly Dave’s solution. I was going to provide a more detailed and motivated exposition but I don’t think I can be bothered. My knowledge of Bayesian inference comes exclusively from reading the first half of section two of McKay’s textbook Information Theory, Inference and Learning Algorithms so you can read that and know as much as I do.

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.

You toss a (not necessarily fair) coin 10 times and get three heads. What is the probability of getting a head on the next toss?

Guesses and answers with some calculations are both acceptable.

The argument of a complex number is defined to be a number which represents the angle of the vector corresponding to the number measured from the positive real axis. If you draw a picture of this setup your first thought is that . Unfortunately this can't work because the range of is . It is true that but ever since I learnt about complex numbers 8 years ago or so I just knew that you had to draw the picture and think about it to solve this equation for .

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.

A link one of my students sent me

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.

Algebraic Geometry: This was a pretty scary class and the professor on the last day said “You probably haven’t learnt anything in the last 6 months.” He may have been right in my case. The other great quote was when he said “Hartshorne was a wimp” (he later clarified that he only meant he was a wimp in how he deals with arithmetic).
We talked about schemes, sheaves, Cech cohomology and various topics like line bundles and the Riemann-Roch theorem. The nice thing about the approach taken was that it was very geometric: the horrible commutative algebra that underlies everything was kind of ignored. We didn’t prove a whole lot but I feel that I now have some idea of this vast edifice that is algebraic geometry.
There ended up being no assessment at all in this class but I did spend many hours trying to figure out what the notes I took were going on about.

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.

My class’s average on the final was 67% and the course average was 65%.

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.

As usual, every class I have requires an end of semester project. I’m kind of obsessed with coding theory at the moment so I’m doing all my projects on topics related to that.

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.

A talk I gave today in the applied math graduate seminar. You might be sick of looking at my talks but this one has pictures and stuff – it’s meant for applied mathematicians.

or: my descent into crackpottery

So I am quite unhappy with the foundations of calculus as it’s currently taught, in particular with how complicated the proof of the product rule is. Let’s recap one proof:

.

It’s pretty awful. Maybe it isn’t that bad a proof as analysis goes but really, it’s unnecessary to subject normal people to such things: that can wait until a real analysis class.

The concept of limit is really difficult for a lot of people. So the solution (or one possible solution) is differentials. This is just replacing one vague concept (limits) with another that is even harder to formalise (differentials are really elements of the cotangent space to something, which is a bit scarier than epsilons and deltas).

But it’s historically accurate and as long as you are willing to accept a little bit of magic works quite well.

If y depends on x (basically is a function locally) then an ‘infinitesimal’ (really small but positive) change of x, called dx, leads to an infinitesimal change in y, called dy.

For example if then

Notice . You can axiomatise this sort of behaviour (this is really or maybe this computation is in ) but this is expected behaviour for infinitesimals: they’re small enough that only linear terms matter.

The above calculation is ‘using the definition’. After you’ve done derivative rules you would just take d of both sides to go from

to

One property of the derivative is now that . This is (arguably) the real meaning of the derivative: it gives the best linear approximation of a function and infinitesimally this approximation is correct.

Now the proof of the product rule is much nicer:
if

This is a little more difficult to extend to integrals but who ever needs to use the definition of Riemann integral anyway? You can use whatever kind of approximations you want to find areas under curves: the only time you use the Riemann integral is when you have an antiderivative so can use the fundamental theorem of calculus.

The other big problem I have with calculus is that we pretend that functions are important when in science and geometry what you really care about are relations between things. We can do calculus on circles, and who knows whether distance is a function of velocity or the other way around….

Thoughts of a few other people on the matter:
Putting Differentials back. A very interesting paper on the issue with good references.

Calculus without Limits – Almost. A rather crazy textbook.

Other ways to teach calculus:
- These notes from Karl Heinz Dovermann define differentiability by a Lipschitz condition with exponent 2.
- Non standard analysis as seen in Keisler’s book
- Apparently there is a book that does everything with differential forms but I can’t find anything that is truly a single variable calculus text like this.