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(2023). . In: (eds) . SAGT 2023. Lecture Notes in Computer Science, vol 14238. Springer, Cham. 

Lead-208 is the heaviest known doubly magic nucleus and its structure is therefore of special interest. Despite this magicity, which acts to provide a strong restorative force toward sphericity, it is known to exhibit both strong octupole correlations and some of the strongest quadrupole collectivity observed in doubly magic systems. In this Letter, we employ state-of-the-art experimental equipment to conclusively demonstrate, through four Coulomb-excitation measurements, the presence of a large, negative, spectroscopic quadrupole moment for both the vibrational octupole $3_1^{-}$and quadrupole $2_1^{+}$state, indicative of a preference for prolate deformation of the states. The observed quadrupole moment is discussed in the context of the expected splitting of the $3^{-}\otimes 3^{-}$two-phonon states, due to the coupling of the quadrupole and octupole motion. These results are compared with theoretical values from three different methods, which are unable to reproduce both the sign and magnitude of this deformation. Thus, in spite of its well-studied nature, ${}^{208}\mathrm{Pb}$remains a puzzle for our understanding of nuclear structure. Published by the American Physical Society2025 

We present a methodology to robustly estimate the competitive equilibria (CE) of combinatorial markets under the assumption that buyers do not know their precise valuations for bundles of goods, but instead can only provide noisy estimates. We first show tight lower- and upper-bounds on the buyers' utility loss, and hence the set of CE, given a uniform approximation of one market by another. We then present two probably-approximately-correct algorithms for learning CE with finite-sample guarantees. The first is a baseline and the second leverages a connection between the first welfare theorem of economics and uniform approximations to adaptively prune value queries when it is determined that they are provably not part of a CE. Extensive experimentation shows that pruning achieves better estimates than the baseline with far fewer samples.