Continuing the Discussion

There's an old saw in mathematics that a mathematician is a machine for turning coffee into theorems. There's a newer saw that then says a comathematician is a machine for turning cotheorems into ffee, but that's a bit of a category theory joke. So, I've posted a picture of my new coffee cup for reference into how much of a functor I am lately. Anyway, I thought Pops raised some interesting points in the comments to my second to last (is "penultimate" appropriate here?) post, and for lengthy discussion, the blog post format seems more apt. I had been using this cup to hold change, by the way, but I decided it would be more convenient just to drink two of these instead of five normal cups, and it amuses me.

First, abelian groups. Or maybe Abelian groups. I'm not sure if that should be capitalized, as it's a common noun if we compare it to similar terms, such as simple group or finite group, but it's named after Niels Henrik Abel, who was quite proper. Regardless, I'll catch you up on what that means. A group is a set with a well-behaved "multiplication," that is a set G, and a function m: G x G -> G, that is associative, and the set is equipped with an identity and inverses. Generally, we drop the m notation and just write it like multiplication, so that all we require is that for all a,b,c in G, we have

a(bc) = (ab)c

an element e in G such that

ae = ea = a

and an element a^(-1) such that

aa^(1) = a^(-1)a =e.

You'll recognize this as being pretty normal, since all the sets of numbers you've ever worked with should meet these conditions for addition, and if you remove 0, they'll work for multiplication, too. Square matrices of a given size provide another convenient example for both operations, as long as you restrict to invertible matrices in the multiplicative case. If you don't know what that means, it just means matrices that meet that last condition. Certain sets of functions also provide common examples, but I won't go into too much detail.

The matrix example is nice because if you've ever done multiplication with them, you'll notice that AB = BA is generally not true, so we have an example of a non-abelian group. Abelian groups are groups where the multiplication does commute, so they are especially nice. Real numbers, rationals, integers, and anything you can come up with built around those will probably be abelian.

A group is called finite if, unsurprisingly, the underlying set G is finite. A nice example would be integers modulo an integer, where the "multiplication" is given by addition. To make that concrete, take all the integers (positive and negative) and divide by a fixed integer, say 4, but stop at the remainder step. You'll notice that you can only get 0,1,2, or 3. Now, choose two integers and add them, and then take their remainder when divided by 4. You'll notice that you get the same result whether you took the remainders first and then added or if you added first and then took the remainder, as long as you interpret adding remainders 2 + 3 = 5 to mean the remainder of 5, which is 1, etc. So it makes sense to add remainders, and you get a nice operation that maybe you never thought of before. You get weird looking relations like 2 + 2 = 0 and 3 + 3 = 2, but that's group theory for you.

The point I was making about finite abelian groups is that even though there are an infinite number of them (in fact the class of finite abelian groups is a proper class, not a set, unless you mod out by isomorphism), they are easily classified. Every finite abelian group it turns out is just a direct sum of this type of group, where the remainder is taken after division by a prime power. A direct sum just means an n-tuple in this case. It seems to me that chemical reactions should behave in a similar way, that more complicated molecules could be thought of as sums of simpler molecules, so that we should always be able to do arithmetic in higher molecule spaces using arithmetic in lower molecule spaces, if that makes any sense. Maybe not, though.

Now, on to the story that amuses me. As I mentioned in the comments, a lot of my students seem incapable of thinking, or at least content to go on avoiding it for as long as possible, so they just shotgun a bunch of terminology every time they come to a question they don't immediately understand. I like to use this to amuse myself and see if I can get them to write unnecessary things that they otherwise would never include in an answer. So, for example, one time we had a lab that focused on a non-parametric test, a permutation test. This type of test basically works on data that is in two groups, and we want to see if permuting which group each datum is in affects anything. If not, then we can attribute any differences we see in the groups to the randomness of the sampling. So, it requires a computer to generate random permutations of data, which is something computers are notoriously bad at doing. In fact, they are incapable of doing anything random by the deterministic nature of their operation, so programmers get around this by making the computer take strange inputs like the number of milliseconds we're at, and running them through bizarre functions that should produce seemingly random outputs. Of course, it's not random, so numbers generated in this fashion are called pseudo-random. None of this knowledge is necessary to explain how the test works because you could do it manually by generating random permutations yourself, but unless you have a few years to kill, I wouldn't suggest it. So, I decided to talk all about pseudo-random numbers, and lo and behold, students' lab reports contained all sorts of usage of the terms "pseudo-random" and "deterministic," which made me chuckle. Keep up the good work, bio department!