Anyway, here is a proof of the irrationality of π. (Proof of the transcendence of is *π* left as an exercise for the reader. Hint: first prove *e* is transcendental, then use Euler’s formula!)

In other news, the last few years of grad school have led me to the conclusion that advanced math is essentially a black hole into which time and self-esteem are sucked, and from which nothing good ever escapes.

Assume π =

a/bwith positive integersaandb.Now, for some natural number

ndefine the functionsfandFas follows. Strictly speaking,fandFshould each havenas an index as they depend onnbut this would render things unreadable; remember thatnis always the same constant throughout this proof.Let

f(x)=x(^{n}abx)^{n}/n!and let

F(x)=f(x)+ … + (-1)^{j}f^{(2j)}(x)+ … + (-1)^{n}f^{(2n)}(x)where

f^{(2j)}denotes the 2j-th derivative off.Then

fandFhave the following properties:

fis a polynomial with coefficients that are integer, except for a factor of 1/n!

f(x)=f(π-x)0 <=

f(x)<= π/^{n}a^{n}n! for 0 <=x<= πFor 0 <=

j<n, thej-th derivative offis zero at 0 and π.For

n<=j, thej-th derivative offis integer at 0 and π

(inferred from (1.)).

F(0) andF(π)are integer (inferred from (4.) and (5.)).

F+F” =f(

F‘·sin –F·cos)’ =f·sin (inferred from (7.))Hence, the integral over

f·sin, taken from 0 to π, is integer.For sufficiently large

n, however, inequality (3.) tells us that this integral must be between 0 an 1. Hence, we could have chosennsuch that the assumption is ledad absurdum.

*Blatantly stolen from here.*

Like a lot of math/science geeks, I went through a phase of my undergrad where I was caught up in the beauty and elegance of “pure” math. Now, though, I find statistics much more interesting. Not that I read myself to sleep with stats textbooks, and the more esoteric it gets the less interesting I find it, but I do find myself more and more looking at the world using the tools of statistics (and its dark cousin, economics). Pure math, like programming, creates a perfect, orderly universe that can be mechanically understood, but statistics gives us tools to make sense of a messy, anarchic universe without taming it. But in order to use stats, you have to first pay attention to its world and try to understand it. And then stats will show you how wrong you are.

]]>also known as "NIPS", originally uploaded by Mister Wind-Up Bird.

Paper writing is pretty stressful. Code crashes, mistakes are found in equations, and you never have quite the experimental results you want. As deadlines approach, the pressure builds and sleep is abandoned. But I kind of welcome them. And not just for the rare flash of “hey, I’m actually doing *science*!” satisfaction. A good, hard deadline adds some much-needed structure and discipline to the grad student lifestyle. It forces you to stop tinkering with code and equations and write that shit down. And having just submitted a paper is a great excuse to drink heavily and spend a few days watching movies and playing video games before going back to work.

The paper I submitted yesterday was for the rather grand-sounding Advances in Neural Information Processing Systems. Or: “NIPS”. (For years now, I’ve been thinking about making various off-colour “I heart NIPS” T-shirts, and I swear, one day I’ll do it.) This is my fourth paper since February — the fifth if you count my PhD thesis proposal. I’m kind of looking forward to not writing another one for a while.

This screenshot is my work environment a little before the paper deadline. If you click on that there image, you can see the Flickr version, where I explain what all those windows are doing (more or less: it’s an anonymous-review conference, so I can’t say anything identifying about the research).

On a geeky note (well, even more geeky), I did all the coding, experiments and figures for this paper in Python using Pylab and SciPy, rather than MATLAB. After years of cursing MATLAB (generally by muttering “god, I fucking hate MATLAB” every time it eats up all the available memory and then crashes) and threatening to switch to something — *any*thing — else, I decided I had to either put up or shut up. And shutting up is not my way. Python is not a perfect replacement, but I’m quite happy to report that it worked very, very well and I expect to do most of my thesis work with Python.

Because he was so prolific and published so widely, as a tribute, his friends created the “Erdős number“, a kind of nerd version of the Kevin Bacon game. Erdős has a number of 0. People he co-authored papers with have a 1. People who co-authored papers with *them* have a 2, and so on. My Erdős Number, as near as I can determine, is 5.

That’s right, ladies: * five*.

**Paul Erdös**, F Harary and**Maria Klawe**. 1980.*Residually complete graphs.*Combinatorial mathematics, optimal designs and their applications, Proceedings of a Symposium on Combinatorial Mathematics and Optimal Design; Colorado State University, Fort Collins, Colorado, 1978, Annals of Discrete Mathematics 6, 1980:117-123- K Inkpen,
**Kellogg S Booth**, S D Gribble and**Maria Klawe**. 1995.*Give and take: Children collaborating on one computer.*CHI’95 Conference Companion, (Denver, Colorado). - A Csinger,
**Kellogg S Booth**and**David Poole**. 1994.*AI Meets Authoring: User models for untelligent multimedia.*Artificial Intelligence Review. Springer Netherlands. 8(5-6):447-468. - P Carbonetto, J Kisynski,
**Nando de Freitas**and**David Poole**. 2005.*Nonparametric Bayesian Logic*. Uncertainty in Artificial Intelligence 2005. **Eric Brochu**,**Nando de Freitas**and Kejie Bao. 2003.*The Sound of an Album Cover: Probabilistic Multimedia and AI*. AI-STATS 2003.

Actually, having a number of 5 isn’t particularly noteworthy, even for a student. What I think is interesting is the way that the connections spread not just through authors, but through fields. The first paper is a math paper. The second is about human-computer interaction (HCI): research on the way people use computers. The third is on user-modelling, which combines HCI and AI. The fourth paper is a stats-oriented AI paper, as is the fifth one (mine), though my focus is not theoretical, but applied. By this point we’re a very long way from the pure math of Paul Erdős. I kind of like the idea of different branches of research being so tightly networked together.

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