Positive Liouville theorems and asymptotic behavior for p-Laplacian type elliptic equations with a Fuchsian potential

We study positive Liouville theorems and the asymptotic behavior of positive solutions of p-Laplacian type elliptic equations of the form Q'(u):= - pLaplace(u) + V |u|^{p-2} u = 0 in X, where X is a domain in R^d, d > 1, and 1<p<infty. We assume that the potential V has a Fuchsian type singularity at a point zeta, where either zeta=infty and X is a truncated C^2-cone, or zeta=0 and zeta is either an isolated point of a boundary of X or belongs to a C^2-portion of the boundary of X.Comment: 39 pages. Stronger results in the radial case, other results and conclusions are unchanged, considerable restructuring of the paper, introduction is modified, typos corrected, references adde

Bayesian quantum frequency estimation in presence of collective dephasing

We advocate a Bayesian approach to optimal quantum frequency estimation - an important issue for future quantum enhanced atomic clock operation. The approach provides a clear insight into the interplay between decoherence and the extent of the prior knowledge in determining the optimal interrogation times and optimal estimation strategies. We propose a general framework capable of describing local oscillator noise as well as additional collective atomic dephasing effects. For a Gaussian noise the average Bayesian cost can be expressed using the quantum Fisher information and thus we establish a direct link between the two, often competing, approaches to quantum estimation theoryComment: 15 pages, 3 figure