Saturday, September 11, 2010

Elementary Mathematics from an Advanced Standpoint

When I first started at the STL research labs I came across a book entitled "Elementary Mathematics from an Advanced Standpoint" written for high-school teachers by the celebrated mathematician Felix Klein. This treated the stuff of sixth form mathematics (calculus, algebra, arithmetic, geometry) using the tools of advanced mathematics. There's a brief overview of the book here.

So here are some recent episodes of ordinary life treated from an 'advanced' standpoint.

On Thursday evening in the flat in Reading I re-experienced the joy of a tepid shower. Alex's boiler had broken again and the backup immersion heater had also failed, buried as it was under the encrustation of years of limescale. At 10 p.m. I discussed this matter with Alex who was looking forwards to a kettle and bowl experience the following morning. Was it worth getting the boiler and archaic plumbing system completely replaced and/or the immersion heater repaired? We briefly discussed the marginal benefit of such an investment over the period in question vs. the opportunity cost. Then we returned to our discussion of Java vs. scripting languages. I think he won't bother and will just fund an opportunistic fix (Tuesday apparently: - I look forwards to Monday night).

Our Roomba practically died trying to clean our new deep-pile carpets in Wells. In Alex's flat, its new home, it illustrates the ergodic theorem. Wherever it starts it eventually tunnels into the back of one of the recliners where it completely vanishes in the narrow space under the seat and gets completely stuck. So its trajectory phase space eventually gets arbitrarily (and sufficiently) close to every trap in the room. There are workarounds.

I was listening to music tracks on my mobile phone (loudspeaker mode - sounds crazy but the quality is not so bad and the volume OK in a quiet room: goes well with a whisky). I began to wonder about shuffling algorithms. Suppose there are ten tracks. You don't want to listen to them in the same order every time, but nor do you want to listen to the same track too often. So imagine a window of three tracks at the front of the queue. After you have heard a track it's put at the back of the ten-place queue. The first three tracks are randomised and the first one then gets played. It gets placed at the back and the process repeats.

Suppose you started with the tracks in order one through ten. Would this procedure end up eventually in randomising the whole sequence? Seems hand-wavily obvious but how would you show it? I figured out a cute answer as I was driving down to Wells yesterday evening but now I've momentarily forgotten it. Any vitamin B12 out there?