We propose automated techniques for the verification and control of probabilistic real-time systems that are only partially observable. To formally model such systems, we define an extension of probabilistic timed automata in which local states are p
artially visible to an observer or controller. We give a probabilistic temporal logic that can express a range of quantitative properties of these models, relating to the probability of an events occurrence or the expected value of a reward measure. We then propose techniques to either verify that such a property holds or to synthesise a controller for the model which makes it true. Our approach is based on an integer discretisation of the models dense-time behaviour and a grid-based abstraction of the uncountable belief space induced by partial observability. The latter is necessarily approximate since the underlying problem is undecidable, however we show how both lower and upper bounds on numerical results can be generated. We illustrate the effectiveness of the approach by implementing it in the PRISM model checker and applying it to several case studies, from the domains of computer security and task scheduling.
We consider a two-band superconductor with relative phase $pi $ between the two order parameters as a model for the superconducting state in ferropnictides. Within this model we calculate the microwave response and the NMR relaxation rate. The influe
nce of intra- and interband impurity scattering beyond the Born and unitary limits is taken into account. We show that, depending on the scattering rate, various types of power law temperature dependencies of the magnetic field penetration depth and the NMR relaxation rate at low temperatures may take place.
The newly discovered iron pnictide superconductors apparently present an unusual case of interband-channel pairing superconductivity. Here we show that, in the limit where the pairing occurs within the interband channel, several surprising effects oc
cur quite naturally and generally: different density-of-states on the two bands lead to several unusual properties, including a gap ratio which behaves inversely to the ratio of density-of-states; the weak-coupling limit of the Eliashberg and the BCS theory, commonly taken as equivalent, in fact predict qualitatively different dependence of the $Delta_{1}/Delta_{2}$ and $Delta/T_{c}$ ratios on coupling constants. We show analytically that these effects follow directly from the interband character of superconductivity. Our results show that in the interband-only pairing model the maximal gap ratio is $sqrt{N_{2}/N_{1}}$ as strong-coupling effects act only to reduce this ratio. This suggests that if the large experimentally reported gap ratios (up to a factor 2) are correct, the pairing mechanism must include more intraband interaction than is usually assumed.
