Sunday, December 13, 2015

Fractions of Worms and Singular Limits

Do you know what's worse than taking a bite out of an apple and finding a worm? Taking a bite of an apple and finding half a worm. And do you know what's even worse than that? Taking a bite and finding a quarter of a worm? And even worse than that is taking a bite and finding an eighth of a worm.

I can continue like this, and finding progressively smaller fractions of a worm after the delicious bite is progressively worse --- until we get to the situation in which we take a bite of an apple and don't have any worm. That's the best situation, as we have a normal delicious apple without any worm in it.* In mathematics, we refer to situations like this as a singular limit: the situation of setting a value equal to zero is qualitatively different from the situation of considering a sequence of values that we let approach zero. This is a very important concept in what is called singular perturbation theory, and it shows up all over the place in mathematics and its applications (e.g., in fluid mechanics, quantum mechanics and semiclassical limits, and many other places).

*Unless, of course, one swallows an entire worm with the bite, in which case it's a regular limit and we lose.

(Note: I don't remember seeing this analogy for singular limits before, but if somebody used it and I blocked out where I should be giving credit, please let me know. I could imagine somebody like Steve Strogatz using this type of analogy, for example, but I don't remember seeing him or anyone else use it.)

Update: Dominic Vella reminds me that Michael Berry used this analogy in an article in Physics Today. I definitely read that article, so I assume I got the analogy from there and forgot about it. I did have a nagging feeling that I must have seen it somewhere before even though I wasn't able to place it. And, no, I am not surprised that Michael Berry would use such an analogy. He's very good with analogies (and many other things).

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