Structure elucidation and X-ray crystallographic study of 3-(naphth-1-ylmethyl)indole

Synthesis and structure determination of 3-(naphth-1-ylmethyl)indole are reported. The molecular and crystal structures together with the molecular formula were determined by spectral and single crystal X-ray studies. X-ray crystallography revealed the presence of two conformers arising from the flipping of the naphthalene unit. The crystal of the compound belongs to the triclinic crystal system and space group \(P\bar 1\). Crystal data are as follows: a = 10.302(5) Å, b = 12.522(4) Å, c = 13.383(4) Å, α = 111.9(1)°, β = 116.86(6)°, γ = 71.65(5)°; V = 1726.429 ų; Z = 4. The final R and Rw are R = 0.0744 (on observed F′s); R = 0.0924 (all F data), R _w = 0.1757 (observed F ²) and R _w = 0.1834 (all F ² data).     

[Hydroxy(aryl)methylene]diphos­phonic acids,a class of drugs in bone pathology treatments,crystallize as head‐to‐head dimers

Two [hydroxy­(aryl)­methyl­ene]­di­phospho­nic acids have been crystallized as dimers. The first compound, [hydroxy­(phenyl)­methyl­ene]­di­phospho­nic acid monohydrate, C₇H₁₀O₇P₂·H₂O, crystallizes in the non‐centrosymmetric space group P2₁, with the two enantiomers related by a non‐crystallographic centre of inversion, while the second compound, [hydroxy(4‐nitro­phenyl)­methyl­ene]­di­phospho­nic acid tetra­hydro­furan disolvate, C₇H₉NO₉P₂·2C₄H₈O, crystallizes in the centrosymmetric space group P2₁/c and uses the centre of symmetry to form the same dimer.     

Galerkin's procedure,quasilinearization, and nonlinear boundary-value problems

Among the popular and successful techniques for solving boundary-value problems for nonlinear, ordinary differential equations (ODE) are quasilinearization and the Galerkin procedure. In this note, it is demonstrated that utilizing the Galerkin criterion followed by the Newton-Raphson scheme results in the same iteration process as that obtained by applying quasilinearization to the nonlinear ODE and then the Galerkin criterion to each linear ODE in the resulting sequence. This equivalence holds for only the Galerkin procedure in the broad class of weighted-residual methods.This work was supported in part by the National Science Foundation, Grant No. GJ-1075.     

Viscoelastic properties of isomeric alkylglucoside surfactants studied by surface light scattering

Surface light scattering (SLS) by capillary waves was used to investigate the adsorption behavior of non-ionic sugar surfactants at the air/liquid interface. SLS by the subphase (water) followed predictions from hydrodynamic theory. The viscoelastic properties (surface elasticity and surface viscosity) of monolayers formed by octyl beta-glucoside, octyl alpha-glucoside, and 2-ethylhexyl alpha-glucoside surfactants were quantified at submicellar concentrations. It is further concluded that a diffusional relaxation model describes the observed trends in high-frequency, nonintrusive laser light scattering experiments. The interfacial diffusion coefficients that resulted from fitting this diffusional relaxation model to surface elasticity values obtained with SLS reflect the molecular dynamics of the subphase near the interface. However, differences from the theoretical predictions indicate the existence of effects not accounted for such as thermal convection, molecular rearrangements, and other relaxation mechanisms within the monolayer. Our results demonstrate important differences in molecular packing at the air-water interface for the studied isomeric surfactants.     

Interactions of 2,2,2‐trifluoroethanol with melittin

Melittin dissolved in 42% trifluoroethanol‐water at pH 2 has been shown to be α‐helical between residues 6 and 12 and between residues 13 and 25, with the two helical regions separated by a bend at the Leu13 residue. The inter‐helix angle was found to be 154 ± 3° at 0 °C and 135 ± 3° at 25 °C. The dominant conformation of the peptide is thus similar to those observed by previous workers for the peptide in a variety of media. At 25 °C, intermolecular nuclear Overhauser effects arising from nuclear spin dipole‐dipole interactions between melittin hydrogens and fluorines of the solvent are essentially those expected for a system that is homogeneous as regards concentration and translational diffusion of the peptide and fluoroalcohol components. However, at 0 °C, peptide‐trifluoroethanol cross‐relaxation terms are negative, a result consistent with the conclusion that fluoroalcohol molecules associate with the peptide for times (～1 ns) that are long compared to the time of a typical peptide‐fluoroalcohol diffusive encounter (～0.2 ns). Such interactions may be responsible for the reduction of the translational diffusion coefficient of trifluoroethanol produced by dissolved peptides. Copyright © 2009 John Wiley & Sons, Ltd.     

The complexity of advice‐giving

Advice‐giving about personal problems is a common form of human interaction. However, an open question is whether there is an abstract and general logic that explains how advice‐giving works. In this study, we addressed this question from the perspective of dynamical systems. We measured the nonlinear dynamics of advice‐giving by using recurrence quantification analysis. Analyzing 600 texts of request for advice and the advice given, our results uncover a typical logic of advice‐giving, and suggest that advice‐giving may be understood as a dynamic manipulation of perspective‐taking. © 2009 Wiley Periodicals, Inc. Complexity 2009     

Radial Flow in a Bounded Randomly Heterogeneous Aquifer

Flow to wells in nonuniform geologic formations is of central interest to hydrogeologists and petroleum engineers. There are, however, very few mathematical analyses of such flow. We present analytical expressions for leading statistical moments of vertically averaged hydraulic head and flux under steady-state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer. Like in the widely used Thiem equation, we prescribe a constant pumping rate deterministically at the well and a constant head at a circular outer boundary of radius L. We model the natural logarithm Y = lnT of aquifer transmissivity T as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. Perturbation of these nonlocal equations leads to a system of local recursive moment equations that we solve analytically to second order in the standard deviation of Y. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. It yields expected values of head and flux, and the variance–covariance of these quantities, as functions of distance from the well. It also yields an apparent transmissivity, T _a, defined as the negative ratio between expected flux and head gradient at any radial distance. The solution is supported by numerical Monte Carlo simulations, which demonstrate that it is applicable to strongly heterogeneous aquifers, characterized by large values of log transmissivity variance. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness. Potential uses include the analysis of pumping tests and tracer test conducted in such wells, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells.     

Inequalities and Bounds for the Incomplete Gamma Function

Inequalities involving the incomplete gamma function are established. They are obtained using logarithmic convexity of some function associated with the function in question. Lower and upper bounds for the incomplete gamma function are also derived. Bounds for the error function erf are also established.