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Neutron-deficient $^{177-185}$Hg isotopes were studied using in-source laser resonance-ionization spectroscopy at the CERN-ISOLDE radioactive ion-beam facility, in an experiment combining different detection methods tailored to the studied isotopes. These include either alpha-decay tagging or Multi-reflection Time-of-Flight gating to identify the isotopes of interest. The endpoint of the odd-even nuclear shape staggering in mercury was observed directly by measuring for the first time the isotope shifts and hyperfine structures of $^{177-180}$Hg. Changes in the mean-square charge radii for all mentioned isotopes, magnetic dipole and electric quadrupole moments of the odd-A isotopes and arguments in favor of $I = 7/2$ spin assignment for $^{177,179}$Hg were deduced. Experimental results are compared with Density Functional Theory (DFT) and Monte-Carlo Shell Model (MCSM) calculations. DFT calculations with several Skyrme parameterizations predict a large jump in the charge radius around the neutron $N = 104$ mid shell, with an odd-even staggering pattern related to the coexistence of nearly-degenerate oblate and prolate minima. This near-degeneracy is highly sensitive to many aspects of the effective interaction, a fact that renders perfect agreement with experiment out of reach for current functionals. Despite this inherent diffculty, the SLy5s1 and a modified UNEDF1^{SO} parameterization predict a qualitatively correct staggering that is off by two neutron numbers. MCSM calculations of states with the experimental spins and parities show good agreement for both electromagnetic moments and the observed charge radii. A clear mechanism for the origin of shape staggering within this context is identified: a substantial change in occupancy of the proton $\pi h_{9/2}$ and neutron $\nu i_{13/2}$ orbitals., Comment: 19 pages, 18 figures

We investigate the locality number, a recently introduced structural parameter for strings (with applications in pattern matching with variables), and its connection to two important graph-parameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NP-hard, but fixed-parameter tractable, if parameterised by the locality number or by the alphabet size, which has been formulated as open problems in the literature. Moreover, the locality number can be approximated with ratio O(sqrt(log(opt)) log(n)). An important aspect of our work -- that is relevant in its own right and of independent interest -- is that we identify connections between the string parameter of the locality number on the one hand, and the famous graph parameters of cutwidth and pathwidth, on the other hand. These two parameters have been jointly investigated in the literature and are arguably among the most central graph parameters that are based on "linearisations" of graphs. In this way, we also identify a direct approximation preserving reduction from cutwidth to pathwidth, which shows that any polynomial f(opt,|V|)-approximation algorithm for pathwidth yields a polynomial 2f(2 opt,h)-approximation algorithm for cutwidth on multigraphs (where h is the number of edges). In particular, this translates known approximation ratios for pathwidth into new approximation ratios for cutwidth, namely O(sqrt(log(opt)) log(h)) and O(sqrt(log(opt)) opt) for (multi) graphs with h edges.

For $k\geq 3$, a k-rollercoaster is a sequence of numbers whose every maximal contiguous subsequence, that is increasing or decreasing, has length at least $k$; $3$-rollercoasters are called simply rollercoasters. Given a sequence of distinct numbers, we are interested in computing its maximum-length (not necessarily contiguous) subsequence that is a $k$-rollercoaster. Biedl et al. [ICALP 2018] have shown that each sequence of $n$ distinct real numbers contains a rollercoaster of length at least $\lceil n/2\rceil$ for $n>7$, and that a longest rollercoaster contained in such a sequence can be computed in $O(n\log n)$-time. They have also shown that every sequence of $n\geq (k-1)^2+1$ distinct real numbers contains a $k$-rollercoaster of length at least $\frac{n}{2(k-1)}-\frac{3k}{2}$, and gave an $O(nk\log n)$-time algorithm computing a longest $k$-rollercoaster in a sequence of length $n$. In this paper, we give an $O(nk^2)$-time algorithm computing the length of a longest $k$-rollercoaster contained in a sequence of $n$ distinct real numbers; hence, for constant $k$, our algorithm computes the length of a longest $k$-rollercoaster in optimal linear time. The algorithm can be easily adapted to output the respective $k$-rollercoaster. In particular, this improves the results of Biedl et al. [ICALP 2018], by showing that a longest rollercoaster can be computed in optimal linear time. We also present an algorithm computing the length of a longest $k$-rollercoaster in $O(n \log^2 n)$-time, that is, subquadratic even for large values of $k\leq n$. Again, the rollercoaster can be easily retrieved. Finally, we show an $\Omega(n \log k)$ lower bound for the number of comparisons in any comparison-based algorithm computing the length of a longest $k$-rollercoaster.

The neutron-rich isotopes $^{58-63}$Cr were produced for the first time at the ISOLDE facility and their masses were measured with the ISOLTRAP spectrometer. The new values are up to 300 times more precise than those in the literature and indicate significantly different nuclear structure from the new mass-surface trend. A gradual onset of deformation is found in this proton and neutron mid-shell region, which is a gateway to the second island of inversion around \emph{N}=40. In addition to comparisons with density-functional theory and large-scale shell-model calculations, we present predictions from the valence-space formulation of the \emph{ab initio} in-medium similarity renormalization group, the first such results for open-shell chromium isotopes., Comment: 6 pages, 3 figures, 1 table