Wednesday, January 29, 2014

magic of the Gaussian distribution

the width $\sigma$ of a Gaussian distribution
\begin{equation}
p(x)\mathrm{d}x =
 \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{x^2}{2\sigma^2}\right)\mathrm{d}x
\end{equation}
can be obtained by integration of the second moment, $\langle x^2\rangle = \int\mathrm{d}x\:p(x)x^2 = \sigma^2$ or by differentiation and solving $\mathrm{d}^2p/\mathrm{d}x^2=0$. can you show that this property is unique for the Gaussian distribution?

Wednesday, January 22, 2014

relativistic natural system of units

can you define fundamental scales for length, time and mass from the speed of light $c$, the gravitational coupling constant $G$ and the cosmological constant $\Lambda$? please compare size, age and density of the Universe to these fundamental scales.

Wednesday, January 15, 2014

subway to New Zealand

imagine there's a tunnel drilled across the centre of the Earth where a capsule moves under gravity serving as transport between Europe and the Antipodes. can you estimate the travel time? what would be the largest velocity? is there something funny about the result if you assume homogeneity?

Wednesday, January 8, 2014

nucleosynthesis with unstable nuclei

what qualitative changes would there be in the abundance of light elements if (1) neutrons had a shorter life-time or if (2) deuterium would be unstable?