Heisenberg uniqueness pairs for the measure supported on the finite parallel lines
A Heisenberg uniqueness pair is a pair $\left(\Gamma, \Lambda\right)$, where $\Gamma$ is a smooth curve or a finite disjoint union of smooth curves and $\Lambda\subset\mathbb R^2$ is such that any finite Borel measure $\mu$ with support on $\Gamma$ and absolutely continuous with respect to the arc length of $\Gamma$ for which Fourier transform $\hat\mu\vert_\Lambda=0,$ implies $\mu=0.$ In this article, we find a characterization of the Heisenberg uniqueness pairs corresponding to finite number of parallel lines. In the latter case, we observe a phenomenon of interlacing of finite number of dispensable sets.
We describe preliminary results from an effort to quantify the uncertainties in parton distribution functions and the resulting uncertainties in predicted physical quantities. The production cross section of the $W$ boson is given as a first example. Constraints due to the full data sets of the CTEQ global analysis are used in this study. Two complementary approaches, based on the Hessian and the Lagrange multiplier method respectively, are outlined. We discuss issues on obtaining meaningful uncertainty estimates that include the effect of correlated experimental systematic uncertainties and illustrate them with detailed calculations using one set of precision DIS data.