Oscillatory Critical Amplitudes in Hierarchical Models and the Harris Function of Branching Processes

Oscillatory critical amplitudes have been repeatedly observed in hierarchical models and, in the cases that have been taken into consideration, these oscillations are so small to be hardly detectable. Hierarchical models are tightly related to iteration of maps and, in fact, very similar phenomena have been repeatedly reported in many fields of mathematics, like combinatorial evaluations and discrete branching processes. It is precisely in the context of branching processes with bounded off-spring that T. Harris, in 1948, first set forth the possibility that the logarithm of the moment generating function of the rescaled population size, in the super-critical regime, does not grow near infinity as a power, but it has an oscillatory prefactor (the Harris function). These oscillations have been observed numerically only much later and, while the origin is clearly tied to the discrete character of the iteration, the amplitude size is not so well understood. The purpose of this note is to reconsider the issue for hierarchical models and in what is arguably the most elementary setting—the pinning model—that actually just boils down to iteration of polynomial maps (and, notably, quadratic maps). In this note we show that the oscillatory critical amplitude for pinning models and the Harris function coincide. Moreover we make explicit the link between these oscillatory functions and the geometry of the Julia set of the map, making thus rigorous and quantitative some ideas set forth in Derrida et al. (Commun. Math. Phys. 94:115–132, 1984).     

"Cooking the sample": radiofrequency induced heating during solid-state NMR experiments

Dissipation of radiofrequency (RF) energy as heat during continuous wave decoupling in solid-state NMR experiment was examined outside the conventional realm of such phenomena. A significant temperature increase could occur while performing dynamic NMR measurements provided the sample contains polar molecules and the sequence calls for relatively long applications of RF power. It was shown that the methyl flip motion in dimethylsulfone (DMS) is activated by the decoupling RF energy conversion to heat during a CODEX pulse sequence. This introduced a significant bias in the correlation time–temperature dependency measurement used to obtain the activation energy of the motion. By investigating the dependency of the temperature increase in hydrated lead nitrate on experimental parameters during high-power decoupling one-pulse experiments, the mechanisms for the RF energy deposition was identified. The samples were heated due to dissipation of the energy absorbed by dielectric losses, a phenomenon commonly known as “microwave” heating. It was thus established that during solid-state NMR experiments at moderate B₀ fields, RF heating could lead to the heating of samples containing polar molecules such as hydrated polymers and inorganic solids. In particular, this could result in systematic errors for slow dynamics measurements by solid-state NMR.     

Posetic quantitative superstructure/activity relationships (QSSARs) for chlorobenzenes

As a result of the widespread industrial use of polychlorinated hydrocarbons, they have accumulated in nearly all types of environmental compartments, especially in aquatic systems. Particularly, chloroaromatics are among the most undesirable industrial effluents because of their persistence and toxicity. To predict chlorobenzene (CB) toxicities, we make use of a novel scheme that looks beyond simple molecular structure to the manner in which such a structure embeds in an overall reaction network. Thence, a resultant modeling gives a quantitative superstructure/activity relationship (QSSAR) with the (chloro-substitution) reaction network viewed mathematically as a partially ordered set (or poset). Different numerical fittings to the overall poset lead to different QSSAR models, of which we investigate three: average poset, cluster expansion, and splinoid poset QSSAR models for the CBs' toxicities against various species (Poecilia reticulata,Pimephales promelas, Daphnia magna, Rana japonica, etc). Excellent results are obtained for all QSSAR toxicity models. On the basis of the poset reaction diagram, all three of these QSSAR models reflect, in distinct ways, the topology of the network that describes the interconversion of chemical species. Although in the majority of investigated datasets all poset QSSAR models give very good predictions, in some cases, they complement each other. These differences show that more reliable predictions can be obtained by using a consensus prediction that combines data from the three posetic models.     

Probing structure in the polymorphic domain of the L-enantiomer of N-benzoyl-phenylalanine by means of 2D solid-state NMR spectroscopy and DFT calculations

A study of polymorphism using a range of solid-state NMR techniques is presented. We demonstrate the existence of at least six polymorphs in a sample of N-benzoyl-L-phenylalanine. We also present methodology for the characterization of the protonation state, hydrogen bonding, and molecular conformation for the polymorphs, together with results of such a characterization for one of the polymorphs present in our sample. DFT modeling is used to investigate the separate effects hydrogen bonding and molecular conformation have on the chemical shift tensor.     

Determination of membrane protein structure and dynamics by magic-angle-spinning solid-state NMR spectroscopy

It is shown that molecular structure and dynamics of a uniformly labeled membrane protein can be studied under magic-angle-spinning conditions. For this purpose, dipolar recoupling experiments are combined with novel through-bond correlation schemes that probe mobile protein segments. These NMR schemes are demonstrated on a uniformly [13C,15N] variant of the 52-residue polypeptide phospholamban. When reconstituted in lipid bilayers, the NMR data are consistent with an alpha-helical trans-membrane segment and a cytoplasmic domain that exhibits a high degree of structural disorder.     

Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials

This paper studies the scattering matrix S(E;?) of the problem

−?²ψ^″(x)+V(x)ψ(x)=Eψ(x)     

Slow dynamics in glassy methyl alpha-l-rhamnopyranoside studied by 1D NMR exchange experiments

It is widely known that the ability of sugar glasses to preserve anhydrobiotic systems in nature is important but the process is not yet fully understood. Molecular motions in the glassy state are likely to be important in the process but until now have remained largely uncharacterized. Here we describe the use of 1D 13C NMR exchange experiments using CODEX (centreband only detection of exchange) methods to study the dynamics of the well characterised model glassy monosaccharide, methyl alpha-l-rhamnopyranoside. The glass was prepared by fast cooling of a melt inside an NMR rotor. Molecular motions in the range of seconds to milliseconds were observed in the glass, whereas identical experiments using the crystalline material displayed no observable motions in the time-scales covered by the experiment. At 13 to 14 K above Tg the nature of the motion in the glass changed probably due to the onset of larger scale reorientation. A bimodal distribution of jump angles combined with a broad distribution of correlation times was found to best represent the observed motions.     

Intermediate motions as studied by solid-state separated local field NMR experiments

In this report, the application of a class of separated local field NMR experiments named dipolar chemical shift correlation (DIPSHIFT) for probing motions in the intermediate regime is discussed. Simple analytical procedures based on the Anderson-Weiss (AW) approximation are presented. In order to establish limits of validity of the AW based formulas, a comparison with spin dynamics simulations based on the solution of the stochastic Liouville-von-Neumann equation is presented. It is shown that at short evolution times (less than 30% of the rotor period), the AW based formulas are suitable for fitting the DIPSHIFT curves and extracting kinetic parameters even in the case of jumplike motions. However, full spin dynamics simulations provide a more reliable treatment and extend the frequency range of the molecular motions accessible by DIPSHIFT experiments. As an experimental test, molecular jumps of imidazol methyl sulfonate and trimethylsulfoxonium iodide, as well as the side-chain motions in the photoluminescent polymer poly[2-methoxy-5-(2'-ethylhexyloxy)-1,4-phenylenevinylene], were characterized. Possible extensions are also discussed.