I've been (not surprisingly) drinking quite a bit of coffee lately, and I've noticed that many corregated coffee cup holders include a bit of loose glue. At first, I thought this was a mistake, an oversight in the perfection of the coffee cup holder design.

I'm sometimes bored while flying, and I like looking out the window (though if I can, I usually pick aisle seats so I can exit more quickly).

Let $f(n)$ be the number of Google hits for the integer $n$. Then $f(578)$ is about 100 million, and $f(1156)$, that is, the number of hits for a number twice as big, is about 40 million, a bit less than half as big. Doubling the input continues to halve the output: $f(2312)$ is about 20 million (half again!), and $f(4624)$ is about 8 million, and $f(9248)$ is about 4 million.

I searched for each number between 1 and 500 on Google, and recorded the (estimated) number of hits. I'm not aware of anyone having done this before; in any case, I made a chart:

In seminar today, Okun pointed out the following interesting observation; for any finitely generated group $G$, you can define its growth series $G(t) = \sum_{g \in G} t^{\ell(g)}$, where $\ell(g)$ is the length of the shortest word for $g$. The first observation is that $G(t)$ is often a rational function, in which case $G(1)$ makes sense. The second observation is that $G(1)$ is "often" equal to $\chi(G)$. This is an example of weighted $L^2$ cohomology.

Cartan proved that every finite-dimensional real Lie algebra $\germ g$ comes from a connected, simply-connected Lie group $G$. I hadn't known the proof of this result (and apparently it is rather uglier than one might hope), but

What can be said about the history of static electricity? Did Greek science know about it? Any medieval experiments with static electricity?