"Do not attempt to disprove the four-colour theorem on your flag!"

Did I ever tell you about that website I found, where this guy gave grades to all the flags of the world? Well, I've finally found it again, the guy's name is Josh Parsons, and the site is hilarious. Go read it.

Narasimhan makes a side note

Complex analysis today. Narasimhan has just proved Rouché's Theorem, and he decides to give an application. Let φ:Ω→C holomorphic, {|z|≤r}⊆Ω, |z|=r ⇒ |φ(z)|<r. Then φ has a unique fixpoint in {|z|≤r}. Proof: Let f(z)=φ(z)-z, g(z)=-z. Then on {|z|=r}, |f(z)-g(z)|=|φ(z)|<r=|g(z)|, so, by Rouché, on {|z|≤r}, f has the same number of zeroes of g, i.e. φ has exactly one fixpoint.

"I just came up with this last night, I'm sure it must be well known," he said. I think also said something about, in the case r=1, anyway, it also coming from φ being a contraction mapping under the hyperbolic distance? Hm.

-Harry