Differential form methods in the theory of variational systems and Lagrangian field theories

This paper will attempt to unify diverse material from physics and engineering in terms of differential forms on manifolds. A variational system will be defined by means of a scalar-valued differential form on a manifold and an ideal in the Grassmann algebra of differential forms on that manifold to serve as constraints. Two types of extremal submanifolds will be defined. The first-called the Euler-Lagrange extremals-will be defined by a method that is the generalization of the classical methods in the calculus of variations. The second—a generalization of a method used by Cartan in his treatise Leçons sur les invariants intégraux-will define extremals as integral submanifolds of an exterior differential system invariently attached to the variational system. As examples, the variational systems attached to string theories in Riemannian manifolds and Yang-Mills fields will be discussed from this differential form point of view. Finally, as application, the differential geometric properties and definition of energy will be presented from the differential form point of view.This work was supported by a grant from the Applied Mathematics program of the National Science Foundation.     

Asymptotic expansions for renewal measures in the plane

Summary Let P be a distribution in the plane and define the renewal measure R=P ^*nwhere * denotes convolution. The main results of this paper are three term asymptotic expansions for R far from the origin. As an application, expansions are obtained for distributions in linear boundary crossing problems.Research supported by NSF grants MCS-8102080 and DMS-8504708     

Diffusion coefficient measurement in a microfluidic analyzer using dual-beam microscale-refractive index gradient detection. Application to on-chip molecular size determination

We report a microchip-based detection scheme to determine the diffusion coefficient and molecular mass (to the extent correlated to molecular size) of analytes of interest. The device works by simultaneously measuring the refractive index gradient (RIG) between adjacent laminar flows at two different positions along a microchannel. The device, referred to as a microscale molecular mass sensor (micro-MMS), takes advantage of laminar flow conditions where the mixing of two streams occurs essentially by diffusion across the boundary between the two streams. Two flows merge on the microchip, one containing solvent only, referred to as the mobile phase stream and one which contains the analyte(s) of interest in the solvent, i.e. the sample stream. As these two streams merge and flow parallel to each other down the microchannel a RIG is created by the concentration gradient. The RIG is further influenced by analyte diffusion from the sample stream into the mobile phase stream. Measuring the RIG at a position close to the merging point (upstream signal) and simultaneously a selected distance further down the microchannel (downstream signal) provides real-time data related to the extent a given analyte has diffused, which can be readily correlated to analyte molecular mass by taking the ratio of the downstream-to-upstream signals. For the dual-beam RIG measurements, a diode laser output is coupled to a single mode fiber optic splitter with two output fibers. Light from each fiber passes through a graded refractive index (GRIN) lens forming a collimated beam that then passes through the microchannel and then on to a position sensitive detector (PSD). The RIG at both detection positions deflects the two collimated probe beams. The deflection angle of each beam is then measured on two separate PSDs. The micro-MMS was evaluated using polyethylene glycols (PEGs), sugars, and as a detector for size-exclusion chromatography (SEC). Peak purity can be readily identified using the micro-MMS with SEC. The limit of detection was 0.9 ppm (PEG at 11 840 g/mol) at the upstream detection position corresponding to a RI limit of detection (LOD) (3sigma) of 7-10(-8) RI. The pathlength for the RIG measurement was 200 microm and the angular LOD was 0.23 micro(rad) with a detection volume of 8 nl at both positions. The average molecular mass resolution was 9% (relative standard deviation) for a series of PEGs ranging in molecular mass from 106 to 22 800 g/mol. With this excellent mass resolution, small molecules such as monosaccharides, disaccharides, and so on, are readily distinguished. The sensor is demonstrated to readily determine unknown diffusion coefficients.     

Synthesis and structural characterization of 3-thienyl alkyl imides

Five new thiophene imides having the structures C₄H₃SC(O)N(H)C(O)R (where R=Me, Et, Pr, t-Bu, 1-4) and C₄H₃SCH₂C(O)N(H)C(O)CH₃5 have been prepared. Control over the orientation of the carbonyl groups was achieved by the introduction of a bulky group in the terminal position of the side chain, and molecular modeling calculations indicated that the energy difference between the ct and tt conformations is small. The acidity of 1-4 decreases from Me to t-Bu, and, although 1-4 will not undergo oxidative polymerization, the introduction of a single spacer CH₂ group allows 5 to polymerize.     

Extended applications and potential limitations of ring-fused 2,3-oxazolidinone thioglycosides in glycoconjugate synthesis

The utility of ring-fused 2,3-oxazolidinone derivatives of 2-amino-2-deoxy-thioglycosides in the synthesis of glycopeptide intermediates and building blocks for the synthesis of heparan sulfate oligosaccharides is reported. These unique ring-fused monosaccharides afford a novel and efficient route to alpha-O-linked 2-amino-2-deoxy amino acid derivatives and heparan sulfate oligosaccharide intermediates.     

Freezing temperatures and densities of solutions of some hydrocarbons and sodium oleate inN-methylacetamide. Evidence for a solvophobic interaction

The freezing temperatures and densities (at 31°C) of solutions of octane, nonane, decane, 3,3-diethylpentane, and sodium oleate inN-methylacetamide (NMA) have been measured. The molality of the freezing solution was calculated from the density. The solubilities of octane, nonane, and decane inN-methylacetamide are also reported. Apparent molal volumes calculated from the densities are close to the values in the pure hydrocarbons and are not strong functions of the concentration. This indicates the absence of any unusual packing effect. The calculated free energies of transfer of the hydrocarbons from pure hydrocarbon to NMA solution are much less negative than the corresponding values for water, showing that the bulk solvophobic interaction inN-methylacetamide is smaller than in water. This is consistent with the freezing temperatures of sodium oleate which show that micelles do not form below 0.1 mole-kg^–1. The osmotic coefficients of the hydrocarbons calculated from the freezing temperatures showed negative deviations from ideality that were larger for the hydrocarbons with the higher molecular weights. Two estimates of the pairwise solvophobic interaction inN-methylacetamide indicate that it is also smaller than in water. The solvophobic effect in this solvent does not include the large entropy and enthalpy effects found in aqueous solutions.     

The effect of covalent surface immobilization on the bactericidal efficacy of a quaternary ammonium compound

This study investigates the effect of surface immobilization on the bactericidal function of a quaternary ammonium compound. Quaternary ammonium silane (QAS) coated planar surfaces did not produce any measurable mortality of Staphylococcus aureus, while 1 µm QAS‐coated microparticles did produce S. aureus mortality. The experiments using QAS‐coated microparticles indicate that the ability of QAS molecules to disrupt the cell wall is not hindered by covalent immobilization of QAS to a surface. These results provide evidence that S. aureus cells on a QAS‐coated planar surface are not exposed to a sufficient number of QAS molecules to produce significant mortality. This result has important implications for the development of self‐decontaminating coatings. Covalent immobilization is used to prevent leaching of the bactericidal compound. However, covalent immobilization may result in a significant tradeoff in bactericidal performance. Published in 2007 by John Wiley & Sons, Ltd.