Slow dynamics in a liquid crystal: 1H and 19F NMR relaxometry

Spin-lattice relaxation rates (R(1H) and R(1F)) of two nuclear species ((1)H and (19)F) are measured at different temperatures in the isotropic phase of a liquid crystal (4(')-butoxy-3(')-fluoro-4-isothiocyanatotolane-4OFTOL), over a wide range of Larmor frequency (10 kHz-50 MHz). Their dispersion profiles are found to be qualitatively very different, and the R(1F) in particular shows significant dispersion (varying over two orders of magnitude) in the entire isotropic range, unlike R(1H). The proton spin-lattice relaxation, as has been established, is mediated by time modulation of magnetic dipolar interactions with other protons (case of like spins), and the discernable dispersion in the mid-frequency range, observed as the isotropic to nematic transition is approached on cooling, is indicative of the critical slowing of the time fluctuations of the nematic order. Significant dispersion seen in the R(1F) extending to very low frequencies suggests a distinctly different relaxation path which is exclusively sensitive to the ultra slow modes apparently present in the system. We find that under the conditions of our experiment at low Zeeman fields, spin-rotation coupling of the fluorine with the molecular angular momentum is the dominant mechanism, and the observed dispersion is thus attributed to the presence of slow torques experienced by the molecules, arising clearly from collective modes. Following the arguments advanced to explain similar slow processes inferred from earlier detailed ESR measurements in liquid crystals, we propose that slowly relaxing local structures representing such dynamic processes could be the likely underlying mechanism providing the necessary slow molecular angular momentum correlations to manifest as the observed low frequency dispersion. We also find that the effects of the onset of cross-relaxation between the two nuclear species when their resonance lines start overlapping at very low Larmor frequencies (below ~400 kHz), provide an additional relaxation contribution.     

Mixed uncertainty sets for robust combinatorial optimization

In robust optimization, the uncertainty set is used to model all possible outcomes of uncertain parameters. In the classic setting, one assumes that this set is provided by the decision maker based on the data available to her. Only recently it has been recognized that the process of building useful uncertainty sets is in itself a challenging task that requires mathematical support. In this paper, we propose an approach to go beyond the classic setting, by assuming multiple uncertainty sets to be prepared, each with a weight showing the degree of belief that the set is a “true” model of uncertainty. We consider theoretical aspects of this approach and show that it is as easy to model as the classic setting. In an extensive computational study using a shortest path problem based on real-world data, we auto-tune uncertainty sets to the available data, and show that with regard to out-of-sample performance, the combination of multiple sets can give better results than each set on its own.

     

Fast separation for the three-index assignment problem

A critical step in a cutting plane algorithm is separation, i.e., establishing whether a given vector x violates an inequality belonging to a specific class. It is customary to express the time complexity of a separation algorithm in the number of variables n. Here, we argue that a separation algorithm may instead process the vector containing the positive components of x,  denoted as supp(x),  which offers a more compact representation, especially if x is sparse; we also propose to express the time complexity in terms of |supp(x)|. Although several well-known separation algorithms exploit the sparsity of x,  we revisit this idea in order to take sparsity explicitly into account in the time-complexity of separation and also design faster algorithms. We apply this approach to two classes of facet-defining inequalities for the three-index assignment problem, and obtain separation algorithms whose time complexity is linear in |supp(x)| instead of n. We indicate that this can be generalized to the axial k-index assignment problem and we show empirically how the separation algorithms exploiting sparsity improve on existing ones by running them on the largest instances reported in the literature.     

Perturbations in the A1Π, v = 0 state of 12C18O investigated via complementary spectroscopic techniques

The A¹Π(v = 0) level of ¹²C¹⁸O has been reinvestigated using three different high-resolution spectroscopic methods: (1) 2 + 1′ resonance-enhanced multiphoton ionisation of the A¹Π ? X¹Σ⁺(0, 0) band using narrowband lasers in a Doppler-free geometry; (2) Fourier-transform emission spectroscopy in the visible range probing the B¹Σ⁺ ? A¹Π(0, 0) band in a discharge; (3) Fourier-transform absorption spectroscopy in the vacuum-ultraviolet range measuring the A¹Π ? X¹Σ⁺(0, 0) and B¹Σ⁺ ? X¹Σ⁺(0, 0) bands at multiple temperatures ranging from 90 to 900 K. An effective-Hamiltonian analysis of A¹Π, v = 0 levels was performed up to J = 44 which quantitatively addresses perturbations by the e?³Σ^?(v = 1), d³Δ(v = 4), a′³Σ⁺(v = 9), D?¹Δ(v = 0), and I?¹Σ^?(v = 0, 1) levels.