The Loewner Equation: Maps and Shapes

An approach called Schramm–Loewner evolution (SLE) provides a new method for dealing with a wide variety of scale-invariant problems in two dimensions. This approach is based upon an older method called Loewner Evolution (LE), which connects analytic and geometrical constructions in the complex plane. In this paper, the bases of LE and SLE are described and some simple applications are discussed in relatively non-technical form. A bibliography of the subject is presented.     

On numerical ranges and roots

Existence of the fractional powers is established in Banach algebra setting, in terms of the numerical ranges of elements involved. The behavior of the spectra and (for Hermitian ∗-algebras satisfying some additional hypotheses) the ∗-numerical range under taking these powers also is investigated.     

Myoglobin-CO substate structures and dynamics: multidimensional vibrational echoes and molecular dynamics simulations

Spectrally resolved infrared stimulated vibrational echo data were obtained for sperm whale carbonmonoxymyoglobin (MbCO) at 300 K. The measured dephasing dynamics of the CO ligand are in agreement with dephasing dynamics calculated with molecular dynamics (MD) simulations for MbCO with the residue histidine-64 (His64) having its imidazole epsilon nitrogen protonated (N(epsilon)-H). The two conformational substate structures B(epsilon) and R(epsilon) observed in the MD simulations are assigned to the spectroscopic A(1) and A(3) conformational substates of MbCO, respectively, based on the agreement between the experimentally measured and calculated dephasing dynamics for these substates. In the A(1) substate, the N(epsilon)-H proton and N(delta) of His64 are approximately equidistant from the CO ligand, while in the A(3) substate, the N(epsilon)-H of His64 is oriented toward the CO, and the N(delta) is on the surface of the protein. The MD simulations show that dynamics of His64 represent the major source of vibrational dephasing of the CO ligand in the A(3) state on both femtosecond and picosecond time scales. Dephasing in the A(1) state is controlled by His64 on femtosecond time scales, and by the rest of the protein and the water solvent on longer time scales.     

Comparability of analytical results obtained in proficiency testing based on a metrological approach

A “yes–no” type of criterion is proposed for the assessment of comparability of proficiency testing (PT) results when the PT scheme is based on a metrological approach, i.e. on the use of a reference material as the test sample, etc. The criterion tests a null hypothesis concerning the insignificance of a bias of the mean of the results from a traceable value certified in the reference material used for the PT. Reliability of such assessment is determined by the probabilities of not rejecting the null hypothesis when it is true, and rejecting it when it is false (the alternative hypothesis is true). It is shown that a number of chemical, metrological and statistical reasons should be taken into account for careful formulation of the hypotheses, enabling the avoidance of an erroneous assessment of the comparability. The criterion can be helpful for PT providers and laboratory accreditation bodies in analysis of PT results.     

Banach algebras of singular integral operators with piecewise continuous coefficients. General contour and weight

The Banach algebra generated by one-dimensional linear singular integral operators with matrix valued piecewise continuous coefficients in the spaceL _p (,) with an arbitrary weight is studied. The contour consists of a finite number of closed curves and open arcs with satisfy the Carleson condition. The contour may have a finite number of points of selfintersection. The symbol calculus in this algebra is the main result of the paper.     

Reflections on spontaneous asymmetric synthesis by amplifying autocatalysis

Spontaneous generation of chirality was observed in the course of studying the mechanism of asymmetric autocatalysis by NMR in ZnR2 alkylation of pyrimidin-5-aldehydes. A systematic study was carried out in order to discover its origins. Even in clean fresh non-glass reaction vessels spontaneous ee was clearly observed, and was not dependent on any single reaction parameter. For comparison it was demonstrated that enantiomerically pure Zn alkoxide catalyst could control the configuration of the reaction product even when present at below micromolar concentrations. The high propensity of the Soai reaction system to produce an enantiomerically enriched product without initial bias is suggested to result from stochastic effects. These are especially important in autocatalysis because all the final products can be derived by breeding from a small number of initial events. The statistical excess of one enantiomer in that set is sufficient to generate a measurable ee in the product. The process is aided by the requirement for dimerisation before the product is an active catalyst. An enumeration that rationalises these observations is provided.     

Stepwise delivery of two methoxy groups of arylaldehyde acetals across the phenyl ring. Vacant site-controlled palladium catalysis

A new catalytic cyclization of o-alkynylbenzaldehyde acetals 1 to the functionalized indenes 2 was found to be strictly controlled by the number of triphenylphosphine ligands on the Pd catalyst. Only complexes with three available coordination sites on Pd catalyze this reaction. Mechanistic study suggests that pi-coordination of Pd to the benzene ring is a key step controlled by the number of vacant coordination sites.     

High-resolution electron spin resonance spectroscopy of XeF* in solid argon. The hyperfine structure constants as a probe of relativistic effects in the chemical bonding properties of a heavy noble gas atom

Xenon fluoride radicals were generated by solid-state chemical reactions of mobile fluorine atoms with xenon atoms trapped in Ar matrix. Highly resolved electron spin resonance spectra of XeF* were obtained in the temperature range of 5-25 K and the anisotropic hyperfine parameters were determined for magnetic nuclei 19F, 129Xe, and 131Xe using naturally occurring and isotopically enriched xenon. Signs of parallel and perpendicular hyperfine components were established from analysis of temperature changes in the spectra and from numerical solutions of the spin Hamiltonian for two nonequivalent magnetic nuclei. Thus, the complete set of components of hyperfine- and g-factor tensors of XeF* were obtained: 19F (Aiso=435, Adip=1249 MHz) and 129Xe (Aiso=-1340, Adip=-485 MHz); g(parallel)=1.9822 and g(perpendicular)=2.0570. Comparison of the measured hyperfine parameters with those predicted by density-functional theory (DFT) calculations indicates, that relativistic DFT gives true electron spin distribution in the 2Sigma+ ground-state, whereas nonrelativistic theory underestimates dramatically the electron-nuclear contact Fermi interaction (Aiso) on the Xe atom. Analysis of the obtained magnetic-dipole interaction constants (Adip) shows that fluorine 2p and xenon 5p atomic orbitals make a major contribution to the spin density distribution in XeF*. Both relativistic and nonrelativistic calculations give close magnetic-dipole interaction constants, which are in agreement with the measured values. The other relativistic feature is considerable anisotropy of g-tensor, which results from spin-orbit interaction. The orbital contribution appears due to mixing of the ionic 2Pi states with the 2Sigma+ ground state, and the spin-orbit interaction plays a significant role in the chemical bonding of XeF*.     

Synthesis of new tropinone derivatives by palladium-catalyzed couplings of 8-azabicyclo[3.2.1]oct-2-enyl nonaflates

Whereas tropinone derived nonaflate 3 was no suitable precursor for Heck-reactions, the related carbamate 7 was an excellent substrate for palladium-catalyzed processes. Nonaflate 7 was either isolated in excellent yield by LDA treatment of ketone 5 followed by trapping with NfF (nonafluorobutanesulfonyl fluoride) or generated in situ by fluoride-catalyzed reaction of silyl enol ether 6 with NfF. The desired 1,3-diene 8 was prepared by conventional Heck-reaction of nonaflate 7 with methyl acrylate in almost quantitative yield. Alternatively, the one-pot nonaflation-Heck protocol starting from silyl enol ether 6 provided 8 in good yield. The couplings of acrylonitrile or phenyl vinyl sulfone were also performed with in situ generated 7 and they afforded the expected 1,3-dienes 9 and 10 in good yields. The Sonogashira-reaction with phenylacetylene also started from silyl enol ether 6 and provided enyne 11 via 7 in good yield. A Diels-Alder reaction of 1,3-diene 8 with N-phenyl maleimide at 100 °C furnished tetracyclic adduct 12 in good yield, with excellent diastereofacial selectivity, but with low endo-exo-selectivity. Nonaflate 14 was easily obtained from the corresponding unsaturated bicyclic ketone 13. It behaved differently in an attempted Heck-reaction and mainly led to fragmentation products 15 and 16, whereas the expected 1,3-diene 17 was formed only as minor component. However, 14 could successfully be used in a Sonogashira-reaction with phenylacetylene to afford compound 18. These transformations demonstrate the great potential of tropinone derived alkenyl nonaflates for diversity oriented syntheses of interesting compounds containing an 8-azabicyclo[3.2.1]octane scaffold.