Sharp Gårding inequality on compact Lie groups

We establish the sharp Gårding inequality on compact Lie groups. The positivity condition is expressed in the non-commutative phase space in terms of the full matrix symbol, which is defined using the representations of the group. Applications are given to the L² and Sobolev boundedness of pseudo-differential operators.     

Effect of molecular size on the parity-non-conserving contributions to the nuclear magnetic resonance shielding constant

We present density-functional theory studies on the effects of molecular size on the parity-violating contribution to the nuclear magnetic shielding constant. We focus on models with different backbone and side chain lengths, as well as the details of geometry optimization for certain helical polysilylenes and investigate the parity-violating contribution to the shielding constant of the ²⁹Si nucleus of the backbone. Our calculations show that the molecular geometry has a large influence on the magnitude of the parity-violating shielding contribution, a result that is in line with the previous studies on much smaller molecules. In addition, we find convergence in the magnitude of the PV effect with respect to system size, when using geometries that preserve the helical Si backbone structure optimized for the largest of the present systems. This can be interpreted in terms of the non-size-extensive nature of the parity-violating operator influencing the leading-order effect on nuclear magnetic shielding, as opposed to the size-extensive interaction affecting the energy difference between enantiomers. Our molecules are truncated models of large polysilylene systems, for which a difference in the ²⁹Si chemical shift between enantiomers has been observed to be 0.06 ppm (Fujiki in Macromol Rapid Commun 22, 669–674, 2001). As expected based on earlier first principles studies of small molecules, we do not find support for the difference to be of the parity-violating origin. Instead, the predicted parity-violation-induced splitting of the ²⁹Si resonance is found to converge at values around 10^?8 ppm with increasingly large Si backbone.     

Phase equilibria on five binary systems containing 1-butanethiol and 3-methylthiophene in hydrocarbons

Isothermal vapor–liquid equilibrium (VLE) of the following systems was measured with a recirculation still: 1-butanethiol + methylcyclopentane at 343.15 K, 1-butanethiol + 2,2,4-trimethylpentane at 368.15 K, 3-methylthiophene + toluene at 383.15 K, 3-methylthiophene + o-xylene at 383.15 K, and 3-methylthiophene + 1,2,4-trimethylbenzene at 383.15 K. 1-Butanethiol + methylcyclopentane and 1-butanethiol + 2,2,4-trimethylpentane systems exhibit positive deviation from Raoult's law, whereas systems containing 3-methylthiophene in aromatic hydrocarbons exhibit only slight positive deviation from Raoult's law. A maximum pressure azeotrope was found in the system 1-butanethiol + 2,2,4-trimethylpentane (x₁ = 0.548, P = 100.65 kPa, T = 368.15 K). The experimental results were correlated with the Wilson model and compared with original UNIFAC and COSMO-SAC predictive models. Raoult's law can be used to describe the behavior of 3-methylthiophene in aromatic hydrocarbons at the experimental conditions in this work. Liquid and vapor-phase composition were determined with gas chromatography. All measured data sets passed the thermodynamic consistency tests applied. The activity coefficients at infinite dilution are also presented.     

Branched immersions and braids

Branch points of a real 2-surface Σ in a 4-manifold M generalize branch points of complex curves in complex surfaces: for example, they can occur as singularities of minimal surfaces. We investigate such a branch point p when Σ is topologically embedded. It defines a link L(p), the components of which are closed braids with the same axis up to orientation. If Σ is closed without boundary, the contribution of p to the degree of the normal bundle of Σ in M can be computed on the link L(p), in terms of the algebraic crossing numbers of its components and of their linking numbers with one another.      

Solubility of carbon dioxide in aqueous solutions of diisopropanolamine and methyldiethanolamine

The solubility of carbon dioxide in aqueous solutions of alkanolamines was measured by means of two experimental methods. The solubility of carbon dioxide was measured at 298 K with a static total pressure apparatus in solutions of water + diisopropanolamine (DIPA) having mass fraction of DIPA equal to 10.1%, 11.0% and 33.9%. The density of the water + DIPA solution was measured continuously during the experiments to investigate the changes in density introduced by the absorption of carbon dioxide. A correlation for the density of CO₂-loaded aqueous solutions of DIPA is presented.     

Conical upper density theorems and porosity of measures

We study how measures with finite lower density are distributed around (n−m)-planes in small balls in Rn. We also discuss relations between conical upper density theorems and porosity. Our results may be applied to a large collection of Hausdorff and packing type measures.     

Spectroscopic elucidation of pseudo-base formation from benzo[8,9]quinoliziono[4,5,6,7-fed]phenanthrindyliums

¹³C and ¹H NMR spectra and UV spectra are recorded and assigned for a variety of substituted and hetero derivatives of benzo[8,9]quinolizino[4,5,6,7-fed]phenanthrindyliums. Large specific effects of traces of water on these spectra are traced to pseudo-base formation.     

A 1,4-photochemical aryl shift

1-Aryl-2-(2′-benzimidazolyl)-4,6-diphenyl-pyridiniums are converted photochemically into 2-[2′-(1′-arylbenzimidazolyl)]-4,6-diphenylpyridines. X-Ray crystal analysis comfirms the structure for the $\text{p}$-chlorophenyl derivative.