Direct Neutrino Mass Experiments

With a mass at least six orders of magnitudes smaller than the mass of an electron -- but non-zero -- neutrinos are a clear misfit in the Standard Model of Particle Physics. On the one hand, its tiny mass makes the neutrino one of the most interesting particles, one that might hold the key to physics beyond the Standard Model. On the other hand this minute mass leads to great challenges in its experimental determination. Three approaches are currently pursued: An indirect neutrino mass determination via cosmological observables, the search for neutrinoless double $\beta$-decay, and a direct measurement based on the kinematics of single $\beta$-decay. In this paper the latter will be discussed in detail and the status and scientific reach of the current and near-future experiments will be presented.Comment: Talk presented at NuPhys2015 (London, 16-18 December 2015). 9 pages, LaTeX, 9 png figure

Computational Complexity for Physicists

These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.Comment: references updated, reprint available from http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm

A physicist's approach to number partitioning

The statistical physics approach to the number partioning problem, a classical NP-hard problem, is both simple and rewarding. Very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the ``easy-to-solve'' from the ``hard-to-solve'' phase of the NPP as well as for the probability distributions of the optimal and sub-optimal solutions. In addition, it can be shown that solving a number partioning problem of size $N$ to some extent corresponds to locating the minimum in an unsorted list of \bigo{2^N} numbers. Considering this correspondence it is not surprising that known heuristics for the partitioning problem are not significantly better than simple random search.Comment: 35 pages, to appear in J. Theor. Comp. Science, typo corrected in eq.1