Existence and uniqueness of rectilinear slit maps

We consider a generalization of the parallel slit uniformization in which the angle of inclination of each image slit is assigned independently. Koebe proved that for domains of finite connectivity there is, up to a normalization, a unique rectilinear slit map achieving any given angle assignment. Koebe's theorem is partially extended to domains of infinite connectivity. A uniqueness result is shown for domains of countable connectivity and arbitrary angle assignments, and an existence result is proved for arbitrary domains under the assumption that the angle assignment is continuous and has finite range. In order to prove the existence result a new extremal length tool, called the crossing-module, is introduced. The crossing-module allows greater freedom in the family of admissible arcs than the classical module. Several results known for the module are extended to the crossing-module. A generalization of Jenkins' module condition for the parallel slit problem is given for the rectilinear slit problem in terms of the crossing-module and it is shown that rectilinear slit maps satisfying this crossing-module condition exist.

     

Crystal Structure of Bis[hydrobis(3-phenylpyrazolyl)(5-phenylpyrazolyl)borato]-zinc(II)

Abstract  

The X-ray crystal structure of (C₅₄ H₄₂ B₂ N₁₂ Zn)*2(C H Cl₃) is reported. The title compound is a chloroform-solvated neutral complex crystallizing in the Triclinic space group, P − 1, with two half complexes (each located on a crystallographic center of inversion) and two full chloroform molecules appearing in the asymmetric unit. The capping ligands are related by symmetry and together form, essentially, an octahedral coordination sphere. Strain is apparent in the complex as indicated by asymmetric bond distances and angles. Final statistical data: R = 4.60% with I > 2σ(I), a = 11.2912(8) ?, b = 14.7912(11) ?, c = 17.0047(13) ?, α = 72.3560(10)°, β = 89.8950(10)°, γ = 88.1010(10)°, Z = 2.     

A study on the interactions of Aurein 2.5 with bacterial membranes

Aurein 2.5 (GLFDIVKKVVGAFGSL-NH₂) is an uncharacterised antimicrobial peptide. At an air/water interface, it exhibited strong surface activity (maximal surface pressure 25 mN m⁻¹) and molecular areas consistent with the adoption of α-helical structure orientated either perpendicular (1.72 nm² molecule⁻¹) or parallel (3.6 nm² molecule⁻¹) to the interface. Aurein 2.5 was strongly antibacterial, exhibiting a minimum inhibitory concentration (MIC) of 30 μM against Bacillus subtilis and Escherichia coli. The peptide induced maximal surface pressure changes of 9 mN m⁻¹ and 5 mN m⁻¹, respectively, in monolayers mimicking membranes of these organisms whilst compression isotherm analysis of these monolayers showed ΔG_Mix > 0, indicating destabilisation by Aurein 2.5. These combined data suggested that toxicity of the peptide to these organisms may involve membrane invasion via the use of oblique orientated α-helical structure. The peptide induced strong, comparable maximal surface changes in monolayers of DOPG (7.5 mN m⁻¹) and DOPE monolayers (6 mN m⁻¹) suggesting that the membrane interactions of Aurein 2.5 were driven by amphiphilicity rather than electrostatic interaction. Based on these data, it was suggested that the differing ability of Aurein 2.5 to insert into membranes of B. subtilis and E. coli was probably related to membrane-based factors such as differences in lipid packing characteristics. The peptide was active against both sessile E. coli and Staphylococcus aureus with an MIC of 125 μM. The broad-spectrum antibacterial activity and non-specific modes of membrane action used by Aurein 2.5 suggested use as an anti-biofilm agent such as in the decontamination of medical devices.     

Anionic polymerization with expansion in volume

New bicyclic and tricyclic compounds have been synthesized by the Lewis acid-catalyzed condensation of epoxides with cyclic acid anhydrides or cyclic bislactones. These materials take part in the anionic addition polymerization of amine-functional polyamides with diepoxides, and modify the shrinkage that is characteristic of the reaction. It is possible to control the extent of shrinkage by choosing an appropriate comonomer and by adjusting the concentration of this material in the reaction mixture.     

7,10A-Methanocycloocta[c,d]indazoles: Diisophoranes incorporating the pyrazole ring system

Compounds of the novel 7,10a-methanocycloocta[c,d]indazole ring-system 5, 7 are obtained by the action of hydrazines on 1-chloro(or hydroxy)diisophor-2,7-en-3-ones 1–3 . Their formulation agrees with their origin and their chemical and spectral characteristics. Analogous syntheses of two comparable ring-systems include the production of a substituted 7,10a-methanocycloocta[g,h][2,1]benzisoxazole 10 by a facile cyclodehydration of the oxime of diisophorone 9a , and that of the indole-analogue 13 by the action of aniline on diisophorone-1-carboxylic acid 11 .     

A Mixture Model of Truncated Zeta Distributions with Applications to Scientific Collaboration Networks

The degree distribution has attracted considerable attention from network scientists in the last few decades to have knowledge of the topological structure of networks. It is widely acknowledged that many real networks have power-law degree distributions. However, the deviation from such a behavior often appears when the range of degrees is small. Even worse, the conventional employment of the continuous power-law distribution usually causes an inaccurate inference as the degree should be discrete-valued. To remedy these obstacles, we propose a finite mixture model of truncated zeta distributions for a broad range of degrees that disobeys a power-law behavior in the range of small degrees while maintaining the scale-free behavior. The maximum likelihood algorithm alongside the model selection method is presented to estimate model parameters and the number of mixture components. The validity of the suggested algorithm is evidenced by Monte Carlo simulations. We apply our method to five disciplines of scientific collaboration networks with remarkable interpretations. The proposed model outperforms the other alternatives in terms of the goodness-of-fit.     

m-Chloroperbenzoic acid-oxchromium (VI)-mediated cleavage of 2,4,5-trisubstituted oxazoles

An array of 2-substituted-4,5-diphenyloxazoles were found to be cleaved to triacylamines and diacylamines (imides) using a reagent system composed of m-chloroperbenzoic acid (MCPBA) and 2,2′-bipyridinium chlorochromate (BPCC). The 2-alkyl-4,5-diphenyloxazoles give imides (38–60%) as the predominant cleavage product while the 2-aryl-4,5-diphenyloxazoles give triacylamines (44–71%). Two mechanisms involving intermediates such as cyclic endoperoxides or oxachromacycles were proposed. An application of the oxidative cleavage to the multi-step synthesis of (±)-phoracantholide I seco acid is detailed.