On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives

Summary. The electronic Schrödinger equation describes the motion of electrons under Coulomb interaction forces in the field of clamped nuclei and forms the basis of quantum chemistry. The present article is devoted to the regularity properties of the corresponding wavefunctions that are compatible with the Pauli principle. It is shown that these wavefunctions possess certain square integrable mixed weak derivatives of order up to N+1 with N the number of electrons, across the singularities of the interaction potentials. The result is of particular importance for the analysis of approximation methods that are based on the idea of sparse grids or hyperbolic cross spaces. It indicates that such schemes could represent a promising alternative to current methods for the solution of the electronic Schrödinger equation and that it may even be possible to reduce the computational complexity of an N-electron problem to that of a one-electron problem.Mathematics Subject Classification (1991): 35J10, 35B65, 41A63, 65D99     

Organization of nanoscale objects via polymer demixing

We present an extremely versatile method for the lateral organization of nano-scale objects (NOs) based on the phenomenon of polymer demixing. NOs are suspended in a solution of two immiscible polymers, which is used to form a thin polymer film by spin coating. During spin coating the two polymers separate to give a microphase structure, whose length scale depends on the experimental conditions. The NOs spontaneously partition into one or other of the polymer phases resulting in their lateral organization. In this work, the organization of CdSe nanoparticles and fluorescent organic dyes was studied by fluorescence microscopy. The NOs were organized in the polymer film in stochastic patterns or in ordered designs on substrates pre-patterned by soft-lithography techniques. Single-particle measurements, using confocal microscopy, showed that at low concentrations there was little aggregation of the particles.     

A general approach to detect protein expression in vivo using fluorescent puromycin conjugates

Understanding the expression of known and unknown gene products represents one of the key challenges in the post-genomic world. Here, we have developed a new class of reagents to examine protein expression in vivo that does not require transfection, radiolabeling, or the prior choice of a candidate gene. To do this, we constructed a series of puromycin conjugates bearing various fluorescent and biotin moieties. These compounds are readily incorporated into expressed protein products in cell lysates in vitro and efficiently cross cell membranes to function in protein synthesis in vivo as indicated by flow cytometry, selective enrichment studies, and Western analysis. Overall, this work demonstrates that fluorescent-puromycin conjugates offer a general means to examine protein expression in vivo.     

Formation of dimers of inclusion cryptand/paraquat complexes driven by dipole-dipole and face-to-face pi-stacking interactions

Dimers of inclusion complexes were formed from a new cryptand and viologens (paraquats) driven by dipole-dipole and face-to-face pi-stacking interactions as shown by mass spectrometric characterization and X-ray analysis.     

At-sea measurements of sound penetration into sediments using a buried vertical synthetic array

Acoustic bottom penetration experiments were carried out in a medium-grain sandy bottom at a site in St. Andrews Bay, Florida. These investigations used a new buried, vertical, one-dimensional synthetic array system where a small hydrophone was water-jetted into the sediment to a depth of approximately 2 m. Once buried, this hydrophone was mounted to a vertical robotics stage that translated the hydrophone upward in 1-cm increments. A broadband (3 to 80 kHz) spherical source, positioned 50 cm above the sediment-water interface, was used to insonify the sediment. Measurements were made with insonification angles above and below the critical angle by changing the horizontal distance of the source relative to the insertion point. This new measurement system is detailed, and results are presented that include temporal, frequency, and wavenumber analysis for natural and roughened interfaces. The measured compressional sound speed and attenuation are shown to be self-consistent using the Kramers-Kronig relation. Furthermore, only a single fast compressional wave was observed. There was no observation of a second slower compressional wave as predicted by some applications of the Biot model to unconsolidated water-saturated porous media.     

Quantum mechanical predictions of the stereoselectivities of proline-catalyzed asymmetric intermolecular aldol reactions

Quantum mechanical calculations were employed to predict the ratio of four stereoisomeric products expected from two complex reactions involving the aldol reactions of cyclohexanone with benzaldehyde or with isobutyraldehyde catalyzed by (S)-proline. Experimental tests of these predictions provide an assessment of the state-of-the-art in quantum mechanical prediction of products of complex organic reactions in solution.     

Synthesis of 2,5-disubstituted tetrahydrofurans by stereospecific elimination-cyclization of 1-iodomethyl-1,5-bis-epoxides

[reaction: see text] Treatment of 1-iodomethyl-1,5-bis-epoxides with zinc in refluxing ethanol affords cis or trans 2-vinyl-5-(1-hydroxyethy)-substituted tetrahydrofurans stereospecifically in high yield.     

On Uniform Laws of Large Numbers for Ergodic Diffusions and Consistency of Estimators

Consider a regular diffusion process X with finite speed measure m. Denote the normalized speed measure by μ. We prove that the uniform law of large numbers holds if the class has an envelope function that is μ-integrable, or if is bounded in L _p(μ) for some p>1. In contrast with uniform laws of large numbers for i.i.d. random variables, we do not need conditions on the ‘size’ of the class in terms of bracketing or covering numbers. The result is a consequence of a number of asymptotic properties of diffusion local time that we derive. We apply our abstract results to improve consistency results for the local time estimator (LTE) and to prove consistency for a class of simple M-estimators. This revised version was published online in June 2006 with corrections to the Cover Date.