Generalized frieze pattern determinants and higher angulations of polygons

Frieze patterns (in the sense of Conway and Coxeter) are in close connection to triangulations of polygons. Broline, Crowe and Isaacs have assigned a symmetric matrix to each polygon triangulation and computed the determinant. In this paper we consider d-angulations of polygons and generalize the combinatorial algorithm for computing the entries in the associated symmetric matrices; we compute their determinants and the Smith normal forms. It turns out that both are independent of the particular d  -angulation, the determinant is a power of d−1$d-1$, and the elementary divisors only take values d−1$d-1$ and 1. We also show that in the generalized frieze patterns obtained in our setting every adjacent 2×2$2\times 2$-determinant is 0 or 1, and we give a combinatorial criterion for when they are 1, which in the case d=3$d=3$ gives back the Conway–Coxeter condition on frieze patterns.     

An atomistic study of a poly(styrene sulfonate)/poly(diallyldimethylammonium) bilayer: the role of surface properties and charge reversal

In this work, we present the first molecular dynamics simulation on the formation of a polyelectrolyte bilayer resolved in atomistic detail, extending a previous study of the adsorption of poly(styrene sulfonate) [B. Qiao, J. J. Cerdà and C. Holm, Macromolecules, 2011, 44, 1707-1718.] to the formation of a poly(styrene sulfonate)/poly(diallyldimethylammonium) bilayer. Extensive molecular dynamics simulations of the adsorption process on different substrates (hydrophilic/hydrophobic, charged/neutral) were performed. Our results seem to indicate that a high enough surface charge density (0.164 C m(-2) here) may be required to achieve a multilayer linear growth in the framework of the electrostatic driven mechanism for PEM growth. We furthermore demonstrate that the use of stiff hydroxyl groups for creating a hydrophilic surface from a hydrophilic one can lead to severe simulation artifacts, and we discuss a simple remedy for this problem.     

Methylated β-cyclodextrins: influence of degree and pattern of substitution on the thermodynamics of complexation with tauro- and glyco-conjugated bile salts

The complexation of 6 bile salts with various methylated β-cyclodextrins was studied to elucidate how the degree and pattern of substitution affects the binding. The structures of the CDs were determined by mass spectrometry and NMR techniques, and the structures of the inclusion complexes were characterized from the complexation-induced shifts of (13)C nuclei as well as by 2D ROESY NMR. Thermodynamic data were generated using isothermal titration calorimetry. The structure-properties analysis showed that methylation at O3 hinders complexation by partially blocking the cavity entrance, while methyl groups at O2 promote complexation by extending the hydrophobic cavity. Like in the case of 2-hydroxypropylated cyclodextrins, the methyl substituents cause an increased release of ordered water from the hydration shell of the bile salts, resulting in a strong increase in both the enthalpy and the entropy of complexation with increased number of methyl substituents. Due to enthalpy-entropy compensation the effect on the stability constant is relatively limited. However, when all hydroxyl groups are methylated, the rigid structure of the free cyclodextrin is lost and the complexes are severely destabilized due to very unfavorable entropies.     

Combining Enzymatic Esterification with Conventional Alkaline Transesterification in an Integrated Biodiesel Process

An integrated biodiesel process that combines enzymatic esterification and alkaline transesterification is suggested. With focus on the enzymatic step, the paper provides proof of concept and suggestions for further process development. Hence, palm fatty acid distillate (PFAD) has been enzymatically converted to fatty acid methyl esters in a two-step process using the immobilized lipase Novozym 435 in packed-bed columns. With only a small excess of methanol, the first reaction stage could reduce the free fatty acid (FFA) content from 85% to 5%. After removal of water by simple phase separation, it was possible to lower the FFA content to 2.5% in a second reaction stage. Both reaction stages are relatively fast with suggested reaction times of 15 min in column 1 (productivity 10 kg/kg/h) and 30 min in column 2 (productivity 5 kg/kg/h), resulting in 15% FFA after column 1 and 5% FFA after column 2. A lifetime study indicated that approximately 3,500 kg PFAD/kg Novozym 435 can be treated in the first reaction stage before the enzyme has become fully inactivated. With further optimization, the enzymatic process could be a real alternative to today’s sulfuric acid catalyzed process.     

Wavelength stabilization of extended-cavity tapered lasers with volume Bragg gratings

The wavelength stabilization of high-brightness extended-cavity lasers at 810-nm by the use of volume Bragg gratings is described. Narrow linewidth (<?20 pm), high power (2.5 W) and good beam quality (M²<4) operation has been obtained in a robust and simple design. The impact of the beam focusing into the Bragg grating on the external cavity performance has been theoretically and experimentally investigated. Finally, second-harmonic generation of the infrared beam has been obtained in a ppKTP crystal, demonstrating a non-linear conversion efficiency of 0.8%/W and up to 8 mW at 405 nm.