Pairs of oblique duals in spaces of periodic functions

We construct non-tight frames in finite-dimensional spaces consisting of periodic functions. In order for these frames to be useful in practice one needs to calculate a dual frame; while the canonical dual frame might be cumbersome to work with, the setup presented here enables us to obtain explicit constructions of some particularly convenient oblique duals. We also provide explicit oblique duals belonging to prescribed spaces different from the space where we obtain the expansion. In particular this leads to oblique duals that are trigonometric polynomials.     

High-throughput approaches for the discovery and optimization of new olefin polymerization catalysts

The discovery of new olefin polymerization catalysts is currently a time-intensive trial-and-error process with no guarantee of success. A fully integrated high-throughput screening workflow for the discovery of new catalysts for polyolefin production has been implemented at Symyx Technologies. The workflow includes the design of the metal-ligand libraries using custom-made computer software, automated delivery of metal precursors and ligands into the reactors using a liquid-handling robot, and a rapid primary screen that serves to assess the potential of each metalligand-activator combination as an olefin polymerization catalyst. "Hits" from the primary screen are subjected to secondary screens using a 48-cell parallel polymerization reactor. Individual polymerization reactions are monitored in real time under conditions that provide meaningful information about the performance capabilities of each catalyst. Rapid polymer characterization techniques support the primary and secondary screens. We have discovered many new and interesting catalyst classes using this technology.     

Analysis of High-Resolution FTIR Spectrum of the nu6 Band of H13COOH

The FTIR spectrum of the nu6 band of H13COOH has been collected at a resolution of 0.004 cm-1 in the frequency range of 1030-1160 cm-1. The nu6 band was analyzed to be an A-B hybrid band and perturbed by the nearby nu8 band through a-, and b-Coriolis coupling terms. Using a Watson's A-reduced Hamiltonian in the Ir representation, with the inclusion of a- and b-Coriolis resonances, a total of 3004 IR transitions of nu6 have been analyzed to provide rovibrational constants for the v6 = 1 state with a standard deviation of 0.00047 cm-1. The nu6 band was found to be centered at 1095.40365 +/- 0.00003 cm-1. Copyright 1999 Academic Press.     

Dimers from α-alkoxybenzyl radicals. An NMR study

meso and dl Dimers (ArCHOR)₂ where R is Me, Et, ⁱPr, ^tBu, cyclohexyl and 1-adamantyl may readily be differentiated by their NMR spectra; the benzylic protons of the meso isomer always absorb at a slightly higher field than those of the dl isomer in each of the solvents used. Differences in chemical shift are discussed in terms of preferences in conformer distribution. The formation of equal amounts of both dimers from the corresponding radical ArCHOR shows that steric and polar factors are not important in influencing the dimerization. Magnetic non-equivalence due to the presence of asymmetric centres was found in some of the compounds discussed above.     

Dispersion and current-voltage characteristics of helical polyacetylene single fibers

To study the transport properties of individual helical polyacetylene (PA) fibers, we developed a method to extract a single fiber from tightly entangled ropes of helical PA bulk film. After a few minutes of sonication of a piece of helical PA bulk film in an organic solution containing surfactant, a droplet of solution is deposited on the pre-pattened electrode under argon atmosphere. AFM images show that extracted helical PA fibers are typically 10 mum in length and 100-200 nm in diameter. We found that the helicity of bulk materials is conserved. We present the temperature dependencies of current-voltage characteristics of individual helical PA fibers doped with iodine.     

Ultrastructure and assembly of segmental long spacing collagen studied by atomic force microscopy

The in vitro formation of segmental long spacing (SLS) collagen as induced by the addition of ATP to acidified Type I collagen solutions has been examined with the atomic force microscope (AFM). AFM images obtained suggest that the assembly proceeds in a stepwise manner, through an intermediate stage of oligomers, which then associate laterally to form the so-called "SLS crystallites". Attempts to induce SLS formation by the addition of other polyanionic species to monomeric collagen solutions met with mixed success; ATP-gamma-S and GTP produced SLS crystallites, whereas inorganic phosphate and other polyanionic dyes did not. This indicates that the formation of SLS cannot simply be attributed to the negation of positive charges believed to be located on the end of the collagen monomer, but rather it is a complex function of the structure and charge of both the collagen monomer and polyanion.     

Computer-generated holograms with a commercial microfilm

We have successfully used a commercial microfilm to produce good quality two-dimensional and three-dimensional computer-generated binary holograms.     

Greatest Descent Algorithms in Unconstrained Optimization

We show that, for an unconstrained optimization problem, the long-term optimal trajectory consists of a sequence of greatest descent directions and a Newton method in the final iteration. The greatest descent direction can be computed approximately by using a Levenberg-Marquardt like formula. This implies the view that the Newton method approximates a Levenberg-Marquardt like formula at a finite distance from the minimum point, instead of the standard view that the Levenberg-Marquadt formula is a way to approximate the Newton method. With the insight gained from this analysis, we develop a two-dimensional version of a Levenberg-Marquardt like formula. We make use of the two numerically largest components of the gradient vector to define here new search directions. In this way, we avoid the need of inverting a high-dimensional matrix. This reduces also the storage requirements for the full Hessian matrix in problems with a large number of variables. The author thanks Mark Wu, Professors Sanyang Liu, Junmin Li, Shuisheng Zhou and Feng Ye for support and help in this research as well as the referees for helpful comments.