May 26th, 2006

beartato phd

(no subject)

I did my speaking skills talk, and it went okay. I got one faculty pass, and (≥) one student pass, and the other faculty member that was supposed to show up didn't, so I have to wait until he watches the recording to find out whether I actually passed, since he is the deciding vote. I think I have a decent chance.

Neighborhood of Infinity is a cool math blog. From it I learned about Stephen Schanuel (a frequent collaborator of category theory superstar Bill Lawvere) and his extremely cute construction of the real numbers directly from the integers. The intuition is to represent a real number r as a function ZZ whose slope is r.

It goes like this:

A quasihomomorphism on the integers Z is a map f : ZZ with the property that the quantity |f(x+y)-f(x)-(y)| is bounded by a constant as x,y vary over Z. That is, it's almost a homomorphism from the additive group of integers to itself: the amount by which it fails to be is just a constant.

(in big-O notation: f is a qh if f(x+y) = f(x) + f(y) + O(1))

Two quasihomomorphisms f, g are considered equivalent if their difference |f(x)-g(x)| is bounded as x varies over Z.

(in big-O notation, f ~ g if f(x) = g(x) + O(1))

The set of real numbers is just the quasihomomorphisms quotiented out by this equivalence. That's it!

Well, except for the arithmetic stuff on top; two add two of these real numbers, add them pointwise as functions, and to multiply them, just compose them. Say a quasihomomorphism is ≥ 0 if it's bounded below on inputs ≥ 0, and that lets you define ordering.

Here's a paper by Ross Street, one by Arthan, and A'Campo's independent discovery of all this.
beartato phd


Wait, no, alert and hyper is exactly how I'm not feeling. Had some delicious breakfast-for-dinner over at E'n'P, read some books about the history of animation. I didn't know Ren and Stimpy creator John Kricfalusi was totally the protégé of Ralph Bakshi, who was famous for crazy stuff like Fritz the Cat, Fire & Ice, and Cool World, none of which I've actually seen (except for fragments of "Cool World") but now I am tempted to track some of them down. But first, it is definitely time for me to go to sleep at like before 10pm on a friday night because I am awesome.