Sally Clark, 1965-2007

Sally Clark, the solictor wrongly imprisoned between 1999 and 2003 for the murder of both her young sons, died this morning, according to BBC News.

Her conviction, and subsequent acquittal, became very famous in the UK, particular because of some of the statistics offered by an expert witness, Sir Roy Meadow. Meadow’s evidence was, basically, that the chance of both of Sally Clark’s sons dying of natural causes was so slim, that one had to assume that they were murdered. The structure of his argument was as follows:

The chance of a randomly chosen child dying or Sids (Sudden Infant Death Syndome, popularly known as cot death) is about 1 in 3000.

Amongst non-smoking, older parents, with at least one wage, this rises to about 1 in 8500.

The chance of both Sally Clark’s sons dying of Sids is about 1 in 8500×8500, or 1 in 73 million.

The chance that the two boys were not murdered is 1 in 73 million.

QOTW: Erdős on Ramsey Theory

Aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack.

Paul Erdős

AOTW: Fractional Calculus

This week’s Article of the Week features Alan Beardon on Fractional Calculus (I, II, III). Find out how do integrate 2-and-a-half times!

The articles are from the website NRICH. It’s aimed at schoolkids, so goes gently enough. So if, for example, you already know what the Gamma function is, you can skip the first part. But I learned something, nonetheless.

Survey Articles on An Ergodic Walk

An Ergodic Walk is starting to put together a list of survey articles for communication theorists. Not having started my PhD yet, I can’t really add to the list. But he’s still my hero.

Unfortunately, I rather seemed to have copied his blog design, which is terribly confusing…

The Optional Stopping Theorem

Someone once asked my what my favourite theorem was. This seemed an odd question. I sort of blustered and said “bluh, uh, optional stopping?”, although later wondered if I should have gone for max-flow min-cut.

Sometime later, I decided to make a blog. “Optional Stopping” seemed to be a terribly good title for two reasons: 1) because it is Officially My Favourite Theorem, and 2) because it’s hi-lar-ious pun on the fact that I only write blog posts when I’m bored with doing proper work. Optional stopping. I choose to stop work. Geddit? Oh, never mind.

So, in case you don’t know, It thought this would be a good opportunity to explain what the optional stopping theorem is.

In this article, we look at stopping times, martingales and the OST itself, before giving a couple of (I think) nifty applications to random walks.

George Dantzig and the Simplex Algorithm

Ars Mathematica informs me of the latest issue of Notices of the AMS, which include two features – here (pdf) and here (pdf) – on George Dantzig, who died in 2005.

An anecdote Dantzig often told was this:

During my first year at Berkeley I arrived late one day to one of Neyman’s classes. On the blackboard were two problems which I assumed had been assigned for homework. I copied them down. A few days later I apologized to Neyman for taking so long to do the homework – the problems seemed to be a little harder to do than usual. I asked him if he still wanted the work. He told me to throw it on his desk. I did so reluctantly because his desk was covered with such a heap of papers that I feared my homework would be lost there forever.

About six weeks later, one Sunday morning about eight o’clock, Anne and I were awakened by someone banging on our front door. It was Neyman. He rushed in with papers in hand, all excited: “I’ve just written an introduction to one of your papers. Read it so I can send it out right away for publication.” For a minute I had no idea what he was talking about. To make a long story short, the problems on the blackboard which I had solved thinking they were homework were in fact two famous unsolved problems in statistics. That was the first inkling I had that there was anything special about them.

Dantzig is most famous for his simplex algorithm. The simplex algorithm is a method of solving linear programming problems, that is problems that look like

Maximize

subject to

You can find out more about this remarkable algorithm – apparently one of the top 10 algorithms of all time – at the second article above (pdf), in more detail by Spyros Reveliotis, or less detail on Wikipedia.

AOTW: Does Mathematics Nees a Philosophy?

In Article of the Week, I intend to post something that I’ve enjoyed reading in the last the last seven days, but don’t really have anything to say about. So it will usually be outside my areas of study.

To kick us off, Fields medalist Tim Gowers asks Cuspomms (Cambridge University society for the philosophy of maths and the mathematical sciences) Does mathematics need a philosophy?

I would like to advance a rather cheeky thesis: that modern mathematicians are formalists, even if they profess otherwise, and that it is good that they are.

This is the sort of evidence I have in mind. When mathematicians discuss unsolved problems, what they are doing is not so much trying to uncover the truth as trying to find proofs. Suppose somebody suggests an approach to an unsolved problem that involves proving an intermediate lemma. It is common to hear assessments such as, “Well, your lemma certainly looks true, but it is very similar to the following unsolved problem that is known to be hard,” or, “What makes you think that the lemma isn’t more or less equivalent to the whole problem?” The probable truth and apparent relevance of the lemma are basic minimal requirements, but what matters more is whether it forms part of a realistic-looking research strategy, and what that means is that one should be able to imagine, however dimly, an argument that involves it.

Another next week!

QOTW: Bonds of Hell

Quote of the Week beings you a thoughtful or humorous maths quote every week. -ish.

The good Christian should beware of mathematicians [...] The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.

St Augustine of Hippo

Testing LaTeX

In their infinite wisdom, WordPress.com have wisely decided to include LaTeX on their blogs. Hurrah. Now I have even less excuse for not righting stuff.

Me messing about after the fold…