Yay popcorn!

[sniffnoy]

You know the famous "continuous precisely at irrationals" function? Wikipedia has a name for it: the popcorn function.

Also, yay for having finished my... uh... well, I suppose the best name for it is the writeup. Of last quarter's math final, that is... I think I mentioned that, right? That he wants us to do the whole thing over the break? Well, now I have.

Comments

[fensef.livejournal.com]

Say, could I somehow get hold of a copy of that? I'd be interested in seeing some of the problems.

[sniffnoy.livejournal.com]

They're mostly not actually hard problems, the test is hard basically because of time pressure. I'm sure you've seen a bunch of these before. If you want, though, they're here.

[fensef.livejournal.com]

Actually, a lot of these I'm lost on because I don't know a thing about p-adics! But I'll go learn about them at some point. Wikipedia aside, can you recommend a good reference?

12 is groovy because I like isometries...
By the way, the Killing-Hopf theorem of which I was telling you has no article on wikipedia and in fact shows up very rarely on google (44 matches). Perhaps it is known under a different name, or maybe it is really just not very well known. It's pretty damn useful in classifying Riemannian surfaces of constant curvature though, so it's worth looking into. A good ref is Geometry of Surfaces by John Stillwell (1992).

-Avi

[sniffnoy.livejournal.com]

Oh, right, you weren't there last year. Well, the p-adics are the completion of Q under the p-adic metric; while this hardly makes their basic properties obvious, if you already have an idea of what they look like (like you might get from Wikipedia) it's pretty easy to show their basic properties... for more advanced p-adic stuff, I have no idea. I can't really recommend a good reference for anything. :P

By the way, if you want the easier version of problem 8, it was originally just in metric spaces, not necessarily complete.

Haven't you seen the isometries of Rⁿ before?

[fensef.livejournal.com]

I'm familiar with isometries of R^n from Stillwell (Geom. Surf.) and Hall (Applied Group Theory).

I'll look into more p-adics. They're elegantly constructed, but why are they so important?

Re: :A use of the popcorn function

[sniffnoy.livejournal.com]

Well really it's pretty obvious that the irrationals are G-delta without looking at the popcorn function, seeing as how the rationals are countable... I've seen the proof that the rationals are not G-delta, but only *assuming* R is a Baire space... I'm told all complete metric spaces are Baire, but I've not actually proved it before or seen the proof, so I may as well try that. :D

[mathnerdguy.livejournal.com]

I've heard it called the Stars Over Babylon function. The SOB function, in other words.