Complex coordinates and the Stark effect

The theories of the dilatation, r → r e^iθ, and translation, x → x + iq, transformations as related to the Stark problem are reviewed, and new results obtained. Results for the hydrogen atom n = 1 and n = 2 levels and the ¹P⁰, 2s2p H^? shape resonance in dc fields are presented, and the extension to the ac Stark effect made. Spectral estimates are made using the technique of the numerical range and via discussion of several model problems, using both coordinate rotation and coordinate translation.     

Fiber optic multi-channel protein detector for use in preparative continuous annular chromatography

On-line coupling continuous-flow liquid membrane extraction (CFLME) with HPLC, a novel automatic system was developed for the determination of sulfonylurea herbicides in water. After an automatic trace-enrichment process by CFLME, which is the combination of continuous flow liquid-liquid extraction and support liquid membrane (SLM) extraction, the target analytes were concentrated in 50 microl of 0.2 M Na2CO3-NaHCO3 (pH 10.0) buffer. The concentrated sample solutions were injected directly onto a C18 analytical column with a valve, and detected at 240 nm with a diode array detector. Metsulfuron methyl (MSM), and DPX-A 7881 were baseline separated with a mobile phase consisting of methanol and 67 mM KH2PO4-Na2HPO4 (pH 5.91) buffer (45+55, v+v) at a flow-rate of 1.0 ml min(-1). With an enrichment time of 10 min and enrichment sample volume of 20 ml, the enrichment factors and detection limits are 100 and 0.05 microg l(-1) for MSM, and 96 and 0.1 microg l(-1) for DPX-A 7881, respectively. The linear range and precision (RSD) are 0.1-50 microg l(-1) and 7.0% for MSM, and 0.2-50 microg l(-1) and 9.2% for DPX-A 7881, respectively. This proposed method was applied to determine MSM and DPX-A 7881 in seawater, tap water, and bottled mineral water with spiked recoveries in the range of 83-95% for MSM and 88-100% for DPX-A 7881, respectively.     

Synthesis of new two-photon absorbing fluorene derivatives via Cu-mediated Ullmann condensations

The Ullmann amination reaction was utilized to provide access to a number of fluorene analogues from common intermediates, via facile functionalization at positions 2, 7, and 9 of the fluorene ring. Through variation of amine or iodofluorene derivative, analogues bearing substitutents with varying electron-donating and electron-withdrawing ability, e.g., diphenylamino, bis-(4-methoxyphenyl)amine, nitro, and benzothiazole, were synthesized in good yield. The novel fluorene derivatives were fully characterized, including absorption and emission spectra. Didecylation at the 9-position afforded remarkably soluble derivatives. Target compounds 4, 5, and 9 are potentially useful as fluorophores in two-photon fluorescence microscopy. Their UV-vis spectra display desirable absorption in the range of interest suitable for two-photon excitation by near-IR femtosecond lasers. Preliminary measurements of two-photon absorption indicate the derivatives exhibit high two-photon absorptivity, affirming their potential as two-photon fluorophores. For example, using a 1,210 nm femtosecond pump beam, diphenylaminobenzothiazolylfluorene 4 exhibited nondegenerate two-photon absorption, with two-photon absorptivity (delta) of ca. 820 x 10(-50) cm(4) s photon(-1) molecule(-1) at the femtosecond white light continuum probe wavelength of 615 nm.     

Very long-lived doubly vibrationally excited states above dissociation threshold: non-RRKM behavior in model triatomic systems

For couplings and mass ratios appropriate to molecules such as H₂O, D₂O, and CH₂ large volumes of the classical vibrational phase space are found to be non-dissociating even well above threshold. Fully quantum calculations yield families of exceptionally long-lived quantum states corresponding to these trapped volumes.     

Uniform semiclassical self-consistent-field calculations of vibrational resonance energies and widths

Using the Breit—Wigner definition of a resonance, we calculate both resonance positions and widths for a model non-separable hamiltonian, using a uniform semiclassical self-consistent-field (SCF) method. Excellent agreement with recent complex-coordinate quantum SCF calculations is obtained.     

Computing (Un)stable Manifolds with Validated Error Bounds: Non-resonant and Resonant Spectra

We develop techniques for computing the (un)stable manifold at a hyperbolic equilibrium of an analytic vector field. Our approach is based on the so-called parametrization method for invariant manifolds. A feature of this approach is that it leads to a posteriori analysis of truncation errors which, when combined with careful management of round off errors, yields a mathematically rigorous enclosure of the manifold. The main novelty of the present work is that, by conjugating the dynamics on the manifold to a polynomial rather than a linear vector field, the computer-assisted analysis is successful even in the case when the eigenvalues fail to satisfy non-resonance conditions. This generically occurs in parametrized families of vector fields. As an example, we use the method as a crucial ingredient in a computational existence proof of a connecting orbit in an amplitude equation related to a pattern formation model that features eigenvalue resonances.     

On Planar Toeplitz Graphs

We describe several classes of finite, planar Toeplitz graphs and present results on their chromatic number. We then turn to counting maximal independent sets in these graphs and determine recurrence equations and generating functions for some special cases.