One-particle and few-particle billiards

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles.     

Trapped modes due to narrow cracks in thin simply-supported elastic plates

We consider an elastic plate of infinite length and constant width supported simply along its two parallel edges and having a finite length crack along its centreline. In particular, we look for and find trapped modes (localised oscillations) in the presence of the crack. An explicit wide-spacing approximation based on the Wiener–Hopf technique applied to incident wave scattering by semi-infinite cracks is complemented by an exact formulation of the problem in the form of integro-differential equations. An application of a Galerkin method for the numerical calculation of results from the latter method leads to a novel explicit ‘small-spacing’ approximation. In combination with the wide-spacing results this is shown to provide accurate results for all lengths of crack.     

Dimensional reduction of a layered metal chalcogenide into a 1D near-IR direct band gap semiconductor

Reducing the dimensionality of inorganic lattices allows for the creation of new materials that have unique optoelectronic properties. We demonstrate that a layered metal chalcogenide lattice, TiS(2), can form a dimensionally reduced crystalline one-dimensional hybrid organic/inorganic TiS(2)(ethylenediamine) framework when synthesized from molecular precursors in solution. This solid has strong absorption above 1.70 eV and pronounced emission in the near-IR regime. The energy dependence of the absorption, the near-IR photoluminescence, and electronic band structure calculations confirm that TiS(2)(ethylenediamine) has a direct band gap.     

Social structure of Facebook networks

We study the social structure of Facebook “friendship” networks at one hundred American colleges and universities at a single point in time, and we examine the roles of user attributes–gender, class year, major, high school, and residence–at these institutions. We investigate the influence of common attributes at the dyad level in terms of assortativity coefficients and regression models. We then examine larger-scale groupings by detecting communities algorithmically and comparing them to network partitions based on user characteristics. We thereby examine the relative importance of different characteristics at different institutions, finding for example that common high school is more important to the social organization of large institutions and that the importance of common major varies significantly between institutions. Our calculations illustrate how microscopic and macroscopic perspectives give complementary insights on the social organization at universities and suggest future studies to investigate such phenomena further.     

The effects of electronic polarization on water adsorption in metal-organic frameworks: H(2)O in MIL-53(Cr)

The effects of electronic polarization on the adsorption of water in the MIL-53(Cr) metal-organic framework are investigated using molecular dynamics simulations. For this purpose a fully polarizable force field for MIL-53(Cr) was developed which is compatible with the ab initio-based TTM3-F water model. The analysis of the spatial distributions of the water molecules within the MIL-53(Cr) nanopores calculated as a function of loading indicates that polarization effects play an important role in the formation of hydrogen bonds between the water molecules and the hydroxyl groups of the framework. As a result, large qualitative differences are found between the radial distribution functions calculated with non-polarizable and polarizable force fields. The present analysis suggests that polarization effects can significantly impact molecular adsorption in metal-organic frameworks under hydrated conditions.     

Nonlinearity management in optics

We study nonlinearity management in optics by investigating the propagation of localized pulses and plane waves in a layered, cubically nonlinear (Kerr) medium that consists of alternating layers of glass and air. We show that such nonlinearity management delays the blow-up/collapse of pulses and leads to a band structure of modulationally unstable regions for plane waves. We find excellent agreement between experiments, numerical simulations, and theory. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Method of calibrating gel permeation chromatography with whole polymers

Column systems for gel permeation chromatography are usually calibrated by eluting a series of sharp polymer fractions of known molecular weights and by correlating their elution volumes with molecular size or weight. A different method for calibration is proposed in which only one polymer sample, with a broad, well-characterized molecular weight distribution, is used. The cumulative distribution and the integrated, normalized GPC chromatogram are successively superimposed, and molecular weights and corresponding elution volumes are correlated. It is found that calibration curves obtained in this manner show a definite curvature. A possible explanation and correction for this behavior is given, based on the concentration dependence of elution volumes.     

Carbon-13 nuclear magnetic resonance studies of coumarin and related compounds

Fourier-transform ¹³C NMR spectra of nine coumarinoid compounds of medicinal interest are reported. All of the carbon resonances are assigned with the aid of various spectral techniques and stable isotopic labeling. The substituent effects on the chemical shifts in several systems are also discussed.     

The nonlinear graviton: Superposition of plane waves

The nonlinear graviton construction of Penrose is given explicitly for the nonlinear superposition of a finite number of plane waves with arbitrary propagation directions, phase, and amplitude profiles. These are realized as generalizedH-spaces for which the asymptotic data are distributional and have singular support on a finite number of null generators ofI ⁺.