Geometric, variational discretization of continuum theories

This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincaré systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes.     

Self-Reciprocal Irreducible Polynomials Over Finite Fields

We classify monic self-reciprocal irreducible polynomials over finite fields in terms of their orders. We also study the weights of these polynomials.     

Analytical representations of the Mössbauer recoilless fraction,Debye-Waller factors,lattice energies,and heat capacities

Simple analytical expansions are given for the recoilless fraction in Mössbauer spectroscopy, the Debye-Waller factor in X-ray scattering, and the lattice energy and heat capacity of solids. While this problem has been discussed in an earlier paper [1], computer technology has now advanced to the point that direct evaluations of the simple expansions of these quantities are useful for quick curve fitting to experimental data at any desired temperature, and these expansions are easier to evaluate than using graphs to estimate recoilless fractions and Debye temperatures. We compare this approach with a polynomial expansion in terms of Bernoulli numbers, which has only a limited domain of convergence. We explicitly evaluate the convergence of these Debye integral expansions as a function of the number of terms used and the time required.This work was prepared with the support of the U.S. Department of Energy, Grant No. DE-FG02-85 ER 45199.     

Check Character Systems Using Quasigroups: II

In this part of the article we continue to research check character systems with one check character over quasigroups under check equations which have one permutation. These systems always detect all single errors (i.e. errors in only one component of a code word) and can detect some other errors arising during transmission of data. We study check character systems over T-quasigroups. These quasigroups are isotopic to abelian groups and generalize the well-known class of medial quasigroups. We establish some properties of a T-quasigroup so that the check character systems over it are able to detect transpositions, jump transpositions, twin errors and jump twin errors. We also give some models of T-quasigroups, which satisfy all of the required properties for detection of errors of each of the considered types. Communicated by: P. Wild     

Determination of arsenic in high-purity copper by flameless atomic-absorption spectrophotometry

Arsenic is separated from an ammoniacal solution containing 1 g or less of copper(II) by co-precipitation with ferric hydroxide. The precipitate is dissolved in nitric acid, the solution almost neutralized with ammonia, made up to volume and the As determined by flameless atomic-absorption spectrophotometry. Less than 0.5 ppm can be determined.     

Interactions of poly(amidoamine) dendrimers with Survanta lung surfactant: the importance of lipid domains

The interaction of generation 5 (G5) and 7 (G7) poly(amidoamine) (PAMAM) dendrimers with mica-supported Survanta bilayers is studied with atomic force microscopy (AFM). In these experiments, Survanta forms distinct gel and fluid domains with differing lipid composition. Nanoscale defects are induced by the PAMAM dendrimers. The positively charged dendrimers remove lipid from the fluid domains at a significantly greater rate than for the gel domains. Dendrimer accumulation on lipid edges and terraces preceding lipid removal has been directly imaged. Immediately following lipid removal, the mica surface is clean, indicating that lipid defects are not induced by dendrimers binding to the mica substrate and displacing the lipid.     

Toward multistation rotaxanes using metalloporphyrin coordination templating

This paper describes some approaches toward the templated synthesis of rotaxanes incorporating strapped metalloporphyrin moieties as the shuttle unit, with the thread component containing both a neutral diimide "station" and a functionalized pyridine moiety, the latter acting not only as a template but also as a second binding motif. In the first instance, the use of appropriately 3,5-difunctionalized pyridine esters and naphthoquinol-strapped rhodium(III) chloride porphyrins in a stoppering approach to rotaxanes produced only unlinked components: the flexibility of the strap allowed sufficient room for the potential thread unit to bind on the same face of the porphyrin as the strap, while not being interlocked through it. An alternative strategy involving a 1,3-dipolar cycloaddition reaction (a "click" reaction) between azides and alkynes, producing triazole linkers in the thread component of rotaxanes, was more successful. Both porphyrinic (zinc, free base, and rhodium(III) derivatives) and crown ether rotaxanes were successfully produced, with multifunctional (triazole and naphthodiimide) thread units. The potential for molecular motion through the use of stimuli such as acid, solvent, and competing ligands was investigated, with limited success. The same cycloaddition methodology was extended to pyridine-templated analogues of the thread components in the Rh(III)-strapped porphyrins, but again, only unlinked thread and porphyrin shuttle units were produced.     

Near-complete external difference families

We introduce and explore near-complete external difference families, a partitioning of the nonidentity elements of a group so that each nonidentity element is expressible as a difference of elements from distinct subsets a fixed number of times. We show that the existence of such an object implies the existence of a near-resolvable design. We provide examples and general constructions of these objects, some of which lead to new parameter families of near-resolvable designs on a non-prime-power number of points. Our constructions employ cyclotomy, partial difference sets, and Galois rings.