On the Inverse Scattering Method for Integrable PDEs on a Star Graph

We present a framework to solve the open problem of formulating the inverse scattering method (ISM) for an integrable PDE on a star-graph. The idea is to map the problem on the graph to a matrix initial-boundary value (IBV) problem and then to extend the unified method of Fokas to such a matrix IBV problem. The nonlinear Schrödinger equation is chosen to illustrate the method. The framework unifies all previously known examples which are recovered as particular cases. The case of general Robin conditions at the vertex is discussed: the notion of linearizable initial-boundary conditions is introduced. For such conditions, the method is shown to be as efficient as the ISM on the full-line.     

Hypersurfaces in hyperbolic space with support function

Based on [19], we develop a global correspondence between immersed hypersurfaces ?:Mⁿ→Hⁿ⁺¹$?:{\mathrm{M}}^n\to {\mathbb{H}}^{n+1}$ satisfying an exterior horosphere condition, also called here horospherically concave hypersurfaces, and complete conformal metrics e^2ρ_gSn$e^{2\rho }{g}_{{\mathbb{S}}^n}$ on domains Ω in the boundary Sⁿ${\mathbb{S}}^n$ at infinity of Hⁿ⁺¹${\mathbb{H}}^{n+1}$, where ρ   is the horospherical support function, _∂∞?(Mⁿ)=∂Ω${\partial }_{\infty }?({\mathrm{M}}^n)=\partial \mathrm{\Omega }$, and Ω is the image of the Gauss map G:Mⁿ→Sⁿ$G:{\mathrm{M}}^n\to {\mathbb{S}}^n$. To do so we first establish results on when the Gauss map G:Mⁿ→Sⁿ$G:{\mathrm{M}}^n\to {\mathbb{S}}^n$ is injective. We also discuss when an immersed horospherically concave hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles for elliptic problems of both immersed hypersurfaces in Hⁿ⁺¹${\mathbb{H}}^{n+1}$ and conformal metrics on domains in Sⁿ${\mathbb{S}}^n$. Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically concave hypersurface in Hⁿ⁺¹${\mathbb{H}}^{n+1}$ of constant mean curvature.     

H2 evolution and molecular electrocatalysts: determination of overpotentials and effect of homoconjugation

In an effort to standardize the determination of overpotential values for H(2)-evolving catalysts in non-aqueous solvents and allow a reliable comparison of catalysts prepared and assayed by different groups, we propose to adopt the half-wave potential as reference potential. We provide a simple method for measuring it from usual stationary cyclic voltammograms, and we derive the formulas to which the measured potential should be compared, taking into account the effect of homoconjugation. We also revisit tabulated values of the standard reduction potential of protons in nonaqueous solvents, E(H+/H(2))°.     

Effects of surface properties of different substrates on fine structure of plasma-polymerized SiOCH films prepared from hexamethyldisiloxane (HMDSO)

In this study, Doppler broadening energy spectroscopy (DBES) combined with slow positron beam was used to discuss the effect of substrate types on the fine structure of a plasma-polymerized SiOCH layer as a function of depth. From the SEM pictures, the SiOCH films formed on different substrates showed hemispherical macrostructures, and the deposition rate was dependent on the mean pore size. It appears that the morphology of the plasma-polymerized SiOCH films was associated with the porosity-related characteristics of the substrate such as the size/shape of pores. As deposited on the MCE-022 substrate (mixed cellulose esters membrane with a mean pore size of 0.22 μm) with a nodular structure, the SiOCH films had pillar-like structures and high gas permeabilities. DBES results showed that the SiOCH films deposited on different substrates were composed of three layers: the SiOCH bulk layer, the transition layer, and the substrate. It was observed that the microstructure of the SiOCH films was affected layer by layer; a higher surface pore size in the substrates induced thicker transition layers with higher microporosities and led to thinner bulk layers having higher S parameter values during the plasma polymerization. It was also observed that the change in O(2)/N(2) selectivity was consistent with the DBES analysis results. The gas separation performance and DBES analysis results agreed with each other.     

The origin of the conformational preference of N,N'-diaryl-N,N'-dimethyl ureas

Poly aromatic ureas and poly aromatic amides are important classes of foldamers-oligomers with well defined conformations. We have explored the origins of the conformational preference of some N,N'-diaryl-N,N'-dimethyl ureas by a combination of NMR spectroscopy and electronic structure calculations using both a recently developed density functional (M06-2X) and a DFT approach (DFT-D) having empirical corrections for dispersive interactions. We have validated the DFT-D approach for structures of this type using high level wavefunction calculations, (CCSD(T)), of the unsubstituted N,N'-diphenyl-N,N'-dimethyl urea. For the N,N'-diaryl-N,N'-dimethyl ureas we have identified a number of 'endo' conformers (i.e. having an E,E geometrical conformation about the two urea C-N bonds), both π- and tert-butyl-stacked, as well as 'exo' structures (having a Z geometrical conformation about at least one of the C-N bonds), and have computed the relative energies of these conformers as well as the barriers for their interconversion. We find that the relative energies of the 'endo' structures closely follow the relative values of the dispersive interactions. The calculations have allowed us to associate different conformers with the various peaks in the NMR spectra, which point to relatively small differences in energy between the conformers. Somewhat larger energy differences are predicted by the two computational approaches, with the M06-2X functional performing the better of the two. It is suggested that the continuum model employed may not be sufficiently accurate to reflect the solvation of the various conformers.     

Fractionation and characterization of gold nanoparticles in aqueous solution: asymmetric-flow field flow fractionation with MALS, DLS, and UV–Vis detection

Asymmetrical-flow field flow fractionation (AFFF) separates constituents based on hydrodynamic size and is emerging as a powerful tool for obtaining high-resolution information on the size, molecular weight, composition, and stability of nanoscale particles in liquid media. We employ a customized AFFF system combining on-line detectors for multi-angle light scattering, dynamic light scattering, and UV–Vis absorption. Our objective is to develop optimized measurement protocols for the characterization of gold nanoparticles (GNPs), which are widely utilized in biomedical research and other nanotechnology applications. Experimental conditions have been optimized by controlling key parameters, including injection volume and solids concentration, mobile phase composition, membrane type and pore size, and ratio of channel-to-cross-flow rates. Individual citrate-stabilized GNP components (nominally 10, 20, 30, 40, and 60 nm) and GNPs functionalized with polyethylene glycol were separated from multicomponent GNP mixtures by AFFF and subsequently characterized. We discuss the effects due to variations in measurement parameters and GNP surface modification on observed retention, recovery, and peak resolution.     

Characterization and surface-reactivity of nanocrystalline anatase in aqueous solutions

The chemical and electrostatic interactions at mineral-water interfaces are of fundamental importance in many geochemical, materials science, and technological processes; however, the effects of particle size at the nanoscale on these interactions are poorly known. Therefore, comprehensive experimental and characterization studies were completed, to begin to assess the effects of particle size on the surface reactivity and charging of metal-oxide nanoparticles in aqueous solutions. Commercially available crystalline anatase (TiO2) particles were characterized using neutron and X-ray small-angle scattering, electron microscopy, and laser diffraction techniques. The 4 nm primary nanoparticles were found to exist almost exclusively in a hierarchy of agglomerated structures. Potentiometric and electrophoretic mobility titrations were completed in NaCl media at ionic strengths from (0.005 to 0.3) mol/kg, and 25 degrees C, with these two experimental techniques matched as closely as the different procedures permitted. The pH of zero net proton charge (pHznpc, from potentiometric titration) and isoelectric point pH value (pHiep, from electrophoretic mobility titrations) were both in near perfect agreement (6.85 +/- 0.02). At high ionic strengths the apparent pHznpc value was offset slightly toward lower pH values, which suggests some specific adsorption of the Na+ electrolyte ions. Proton-induced surface charge curves of nanocrystalline anatase were very similar to those of larger rutile crystallites when expressed relative to their respective pHznpc values, indicating that the development of positive and negative surface charge away from the pHznpc for nanocrystalline anatase is similar to that of larger TiO2 crystallites.