Math Links

1. Properties of the function $x^x$ (Post-Doc Ergo Propter Hoc H/T Dave Richeson). Can you prove:
$$\int_{0}^{1} x^{x} = -\sum_{n=1}^{\infty} (-n)^{-n}$$
2. Steve Strogatz on Einstein's boyhood proofs (newyorker).

The style of his Pythagorean proof, elegant and seemingly effortless, also portends something of the later scientist. Einstein draws a single line in Step 1, after which the Pythagorean theorem falls out like a ripe avocado. The same spirit of minimalism characterizes all of Einstein’s adult work. Incredibly, in the part of his special-relativity paper where he revolutionized our notions of space and time, he used no math beyond high-school algebra and geometry.

3. Via Twitter, Steven Strogatz ‏@stevenstrogatz  Sep 17
A strange derivative: let $$f(x)=x^2 \sin(1/x^2)$$ for $x \ne 0$, and $f(0)=0$. This function has $f'(0)=0$ even though $$\lim_{x \rightarrow 0} f'(x),$$ does not exist.