A Caste System, Green's Functions, and The Beta Distribution

Today I Learned:
1) There's a species of ant with a really strange caste system. They have queens and workers, just like most ant species, but they are maintained in separate genetic lineages. Queens can produce either queen or worker offspring, but only if they mate with a male of the appropriate line. If the queen only mates with worker-line males, then all of her children are workers or queen-line males. If she only mates with queen-line males, then all of her offspring are queens (and the colony usually dies). How this system came to be is of some debate right now.

2) A Green's function is the impulse response of a bounded linear dynamical system with specified boundary conditions.

Let me break that down a bit. First, an "impulse", in this particular mathematical context, is a function that is zero everywhere except one value, at which it is one. If you think about a digital time signal, an impulse is a flat zero signal with a single spike at one time. There are a couple of cool things about impulses. Firstly, they're simple. They're, like, as simple as a function can get without being uniform. Secondly, they can often by produced experimentally, at least to an approximation, relatively easily. This is especially true in signal processing, where impulses are used all the time. Finally, you can define any function in terms of some number of impulses with various locations and strengths.

A dynamical system is a set of differential equations -- that means it's a system where you can define how quickly each variable changes in terms of the values of each variable at that time. Being bounded and linear and having specified boundary conditions means the system is "nice" in some way.

So, a Green's function is simply what a function does when you spike in an impulse. For instance, you could imagine taking a pool of perfectly still water and instantaneously pulling the surface up by a foot at one spot. The resulting pattern of waves and ripples and splashes would be the Green function for that pool. The nice thing about a Green function is that if you know the Green function for a system, you can add together those functions just like you can add together impulses -- the Green function for two impulses is just the sum of the two impulses' Green function. So once you know the Green function for a system, you know the behavior of the system for all possible starting conditions (with whatever *boundary* conditions the Green function is defined for).

3) The beta distribution, with properly chosen parameters, is a good stand-in for a Gaussian when you only have probabilities defined on the range [0,1].