Asymptotic Approximations for Prolate Spheroidal Wave Functions

Uniformly valid (with respect to the independent variable) asymptotic approximations to the radial, prolate spheroidal wave functions are constructed from Bessel-function and Coulomb-wave-function models for large values of the wave number c. The prolate angular functions also are considered, but more briefly. The emphasis is on qualitative accuracy (such as might be useful to the physicist), rather than on efficient algorithms for very accurate numerical computation, and the error factor for most of the approximations is 1 + O (1/c) as c↑∞.     

Resonant motion of a spherical pendulum

The weakly nonlinear, resonant response of a damped, spherical pendulum (length l, damping ratio δ, natural frequency ω₀) to the planar displacement εl cos ωt (ε ? 1) of its point of suspension is examined in a four-dimensional phase space in which the coordinates are slowly varying amplitudes of a sinusoidal motion. The loci of equilibrium points and the corresponding bifurcation points in this space are determined. The control parameters are $\text{α= 2δ/ε}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$ and . If α < 0.441 there is a finite interval of v within which no stable equilibrium points exist. As v decreases through the upper bound (a Hopf-bifurcation point) of this interval the motion in the phase space becomes periodic and then, following a period-doubling cascade, chaotic. There may be alternating sub-intervals of chaotic and periodic motion. The chaotic trajectories in the phase space appear to lie on fractal attractors.     

High-turnover supramolecular catalysis by a protected ruthenium(II) complex in aqueous solution

The design of a supramolecular catalyst capable of high-turnover catalysis is reported. A ruthenium(II) catalyst is incorporated into a water-soluble supramolecular assembly, imparting the ability to catalyze allyl alcohol isomerization. The catalyst is protected from decomposition by sequestration inside the host but retains its catalytic activity with scope governed by confinement within the host. This host-guest complex is a uniquely active supramolecular catalyst, capable of >1000 turnovers.     

Towards functional fluorous surfactants. Synthesis of hydrophilic fluorous 1,2,3-triazolylmethyl ethers and di(1,2,3-triazolylmethyl) ethers

Copper(I)-accelerated Huisgen–Meldal dipolar cycloaddition reactions between polyfluoroalkyl azides and propargyl ethers of n-octanol and of triethyleneglycol monomethyl ether exhibited variation in yield of 1,2,3-triazol-4-ylmethyl ethers. Microwave acceleration, and in situ generation of the azides, provided improvements in yield and efficiency. In contrast, very good yields of equivalent fluorous triazoles were obtained from a range of n-alkyl azides with propargyl ethers of perfluorohexylethanol and of perfluoroheptylmethanol through conventional copper(I)-promoted reactions. Together, the resulting substances with systematic variations in polyfluoroalkyl and alkyl substituent length and position of substitution, and degree of oxygen content, make up small libraries of hybrid fluorous 1,2,3-triazol-4-ylmethyl ethers as candidates for study as hydrophilic fluorous surfactants. In addition, a pilot sample of di(1,2,3-triazol-4-ylmethyl) ethers with 1′-octyl-1-polyfluoroalkyl-substituents and 1′-nonyl-1-perfluorooctylethyl substituents were synthesised for the first time in an effort to develop more functional, fluorous surfactants.     

Molecular mechanism of α-β phase transformation in poly-p-xylylene

The molecular mechanism of the α→β phase transformation of poly-p-xylylene was studied by transmission electron microscopy using solution-grown single crystals. The observations suggest that the phase transformation from the α to β-phase occurs by a coordinated lateral movement of the molecules from the lattice sites of the α-phase to the sites of the β-phase.     

Campedelli surfaces with fundamental group of order 8

Let S be a Campedelli surface (a minimal surface of general type with p _g  = 0, K ² = 2), and an etale cover of degree 8. We prove that the canonical model of Y is a complete intersection of four quadrics . As a consequence, Y is the universal cover of S, the covering group G = Gal(Y/S) is the topological fundamental group π ₁ S and G cannot be the dihedral group D ₄ of order 8. The first author is a member of the Centre for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Lisboa. The second is a member of G.N.S.A.G.A.–I.N.d.A.M.     

Expression of the DNA Repair Enzyme, Photolyase, in Developmental Tissues and Larvae, and in Response to Ambient UV-R in the Antarctic Sea Urchin Sterechinus neumayeri

Gene expression of the DNA repair enzyme, photolyase (E.C. 4.1.99.3) was examined in the gonads, eggs, embryos and larval stages of the Antarctic sea urchin, Sterechinus neumayeri . Partial sequencing of the gene revealed two highly conserved regions, including a 300 bp region representing the binding site for the cofactor flavin adenine dinucleotide. The second 1200 bp region, likely representing a second light-harvesting cofactor binding site, was identified in a second sea urchin species, Strongylocentrotus frascicanus . Probes for photolyase were developed from the shorter sequence, and expression in sea urchin developmental tissue and stages, and in response to in situ exposure to ultraviolet radiation was quantified using PCR and RT-qPCR, with concentrations of photolyase normalized to actin concentrations. Photolyase was expressed in all tissues and developmental stages examined. In controlled field-based experiments in McMurdo Sound, Antarctica, we found evidence of both constitutive expression of photolyase and induction in response to in situ exposure of embryos to UV-R. Induction of photolyase was observed in response to greater ambient UV-R (such as shallower water depths or sea ice-free regions).     

Total Synthesis of Rapamycin

Rapamycin (1) is a macrocyclic natural product, established as a potent immunosuppressant and currently of interest to the scientific community as the framework for a series of novel anticancer drugs. Extensive studies have culminated in a new convergent total synthesis of 1 , which features a number of group‐derived methodologies and an unusual catechol‐templating strategy for the construction of the challenging macrocyclic core.