Classifying (almost)-Belyi maps with five exceptional points

We classify all rational functions $f:{\mathbb{P}}^1\to {\mathbb{P}}^1$ whose branching pattern above $0,1,\infty$ satisfy a certain regularity condition with precisely $d=5$ exceptions. This work is motivated by solving second order linear differential equations, with $d=5$ true singularities, in terms of hypergeometric functions. A similar problem was solved for $d=4$ in Vidunas and Filipuk (2013).     

Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilizationof the wave equation

We deal here with a second order elliptic mixed problem which is posed in a regular open bounded domain of . We study the regularity of its solution. We apply our results to the boundary stabilization of the wave equation.     

Modeling the performance of “up-flow anaerobic sludge blanket” reactor based wastewater treatment plant using linear and nonlinear approaches—A case study

The paper describes linear and nonlinear modeling of the wastewater data for the performance evaluation of an up-flow anaerobic sludge blanket (UASB) reactor based wastewater treatment plant (WWTP). Partial least squares regression (PLSR), multivariate polynomial regression (MPR) and artificial neural networks (ANNs) modeling methods were applied to predict the levels of biochemical oxygen demand (BOD) and chemical oxygen demand (COD) in the UASB reactor effluents using four input variables measured weekly in the influent wastewater during the peak (morning and evening) and non-peak (noon) hours over a period of 48 weeks. The performance of the models was assessed through the root mean squared error (RMSE), relative error of prediction in percentage (REP), the bias, the standard error of prediction (SEP), the coefficient of determination (R²), the Nash-Sutcliffe coefficient of efficiency (E_f), and the accuracy factor (A_f), computed from the measured and model predicted values of the dependent variables (BOD, COD) in the WWTP effluents. Goodness of the model fit to the data was also evaluated through the relationship between the residuals and the model predicted values of BOD and COD. Although, the model predicted values of BOD and COD by all the three modeling approaches (PLSR, MPR, ANN) were in good agreement with their respective measured values in the WWTP effluents, the nonlinear models (MPR, ANNs) performed relatively better than the linear ones. These models can be used as a tool for the performance evaluation of the WWTPs.     

Oxidative Prins and Prins/Friedel-Crafts cyclizations for the stereoselective synthesis of dioxabicycles and hexahydro-1H-benzo[f]isochromenes via the benzylic C-H activation

1-Benzyl ethers of (E)- and (Z)-hex-3-en-1,6-diols and hept-3-en-1,7-diols undergo a smooth oxidative cyclization with DDQ in the presence of In(OTf)(3) through a sequential C-H bond activation and an intramolecular Prins cyclization to afford the corresponding trans- and cis-fused hexahydro-2H-furo[3,2-c]pyrans and octahydropyrano[4,3-b]pyrans respectively in good yields with an excellent stereoselectivity. Aryl tethered homoallylbenzyl ethers such as benzyl ethers of (E)- and (Z)-6-arylhex-3-enyl alcohols undergo a tandem Prins/Friedel-Crafts cyclization in the presence of stoichiometric amounts of DDQ and SnCl(4)via the benzylic C-H bond activation to furnish the corresponding trans- and cis-fused hexahydro-1H-benzo[f]isochromenes in good yields with complete stereoselectivity.     

Direct C-allylation of aryl-alkyl/glycosyl carbinols: facile synthesis of C-linked carbo-beta2-/gamma2-/delta2-amino acids

A facile ZrCl4-catalyzed direct allylation of the p-methoxyphenyl-alkyl/glycosyl carbinols at room temperature, and the conversion of the derived aryl-glycosyl-alkenes into hitherto unknown C-linked carbo-beta2-/gamma2-/delta2-amino acids is reported.     

Formation of β-hairpins in l-Pro-Gly containing peptides facilitated by 3-amino benzoic acid

Designed β-hairpin peptides tolerate insertion of 3-amino benzoic acid (3-Aba) and also permit accommodation of both enantiomers of Pro-Gly turn motifs.     

The ring puckering vibrations in trimethylene oxide and trimethylene sulphide as studied by P.M.R. spectroscopy in a liquid crystalline medium

The P.M.R. spectra of trimethylene oxide and trimethylene sulphide dissolved in a nematic solvent have been investigated. The chemical shifts, the direct and the indirect spin-spin couplings were determined.

The interpretation of the direct couplings in terms of the molecular structure shows that the ring skeleton of these molecules is not rigidly planar due to ring puckering vibrations. Mean amplitudes for the vibrations are estimated assuming fixed values for the bond distances and for the HCH bond angles. The results obtained are in good agreement with the earlier microwave studies.     

Stereochemical control in the structures of linear δ,α‐hybrid tripeptides containing tetrahydrofuran amino acids

Structural and electronic properties of diastereomers of tetrahydrofuran amino acids (TAA) derived tripeptide, Boc‐TAA‐Leu‐Val‐OMe, are studied using density functional theory. Predicted secondary folding patterns with hydrogen bonded pseudocycles of different sizes in peptides containing (2R,5S)‐cis‐TAA and (2S,5R)‐cis‐TAA are confirmed by detailed NMR studies of both, and single crystal X‐ray analysis of the former. A novel unusual folding pattern emanating from three‐centered hydrogen bond is found in peptide with (2R,5S)‐cis relationship. Stereochemical control on the orientation of interacting sites is substantiated by structural analysis of the peptides. Using natural bonding orbital and atoms in molecules analyses, charge transfer interactions are analyzed. Copyright © 2010 John Wiley & Sons, Ltd.