On path lengths modulo three

We show that between any two nodes of a cubic, planar, three-connected graph there are three paths whose lengths are 0, 1, and 2 modulo 3, respectively. The proof is by a rather extensive case analysis. Counterexamples show that all three hypotheses (i.e., planarity, degree-three, and three-connectivity) are necessary.     

Vortex ring generation due to the coalescence of a water drop at a free surface

 Results are presented of an experimental investigation of vortex ring formation by a fluid drop contacting a free surface with negligible velocity. The pool fluid is mixed with fluorescein dye, and a laser sheet is used to illuminate a plane of the flow. A series of representative images is recorded by a CCD camera and speculation is made regarding specific sources of vorticity flux through the free surface. Two scaling analyses previously presented by other investigators are demonstrated to be equivalent under the assumptions of this experiment, and they provide the motivation for a series of test runs in which the duration of the coalescence process, τ_*, is related to variations in drop diameter L and fluid surface tension σ. Experimental results are in agreement with the analyses, showing τ_*∼σ^-1/2 and τ_*∼L ^3/2. Received: 22 December 1995 / Accepted: 15 October 1996     

Substitution of calcium by strontium within selected calcium phosphates

Sr incorporation in the molecules of amorphous calcium phosphate, apatitic tricalcium phosphate, hydroxyapatite, octacalcium phosphate and dicalcium phosphate dihydrate was investigated. The concentration of Sr ranged from 225 to 1010 μ g / g, i.e. it overlapped with the physiological range of Sr concentrations in human bone. The leading experimental technique was extended X-ray absorption fine structure (EXAFS) at the Sr K edge. Results of these studies demonstrated the following: (1) Sr incorporation in the calcium phosphates is compound-dependent, (2) the coordination of incorporated Sr atoms in the Ca-P molecules is similar to that of Ca atoms, but interatomic distances are ≈0.015 nm larger, (3) in apatitic tricalcium phosphate, hydroxyapatite and octacalcium phosphate lattices Sr atoms may occupy selected Ca sites, which was not the case for dicalcium phosphate dihydrate, (4) in the apatite lattice Sr atoms are coordinated by 6 PO₄ tetrahedrals and (5) EXAFS spectra at the K edge of the incorporated Sr may be used to distinguish the structures of amorphous calcium phosphate, dicalcium phosphate dihydrate as well as apatite and its derivatives (apatitic tricalcium phosphate, octacalcium phosphate).     

X-ray induced fluorescence spectrometry at grazing incidence for quantitative surface and layer analysis

Summary Single layers and layer systems on diverse substrates were measured by Total Reflection X-Ray Fluorescence (TXRF) spectrometry. The angular dependence of the fluorescence intensities at grazing incidence allows the elemental composition, density and thickness of the layers to be evaluated using model calculations.     

Unoccupied electronic states of epitaxial iron layers on Cu(100)

Inverse photoemission spectra were taken for thin epitaxial iron films on Cu(100). For a film thickness of eight monolayers the observed electronic states are characteristic for a fcc(100) surface. Thed-bands of iron show a ferromagnetic exchange splitting of 1.1 eV, considerably smaller than the bulk value of 1.8 eV, which we observe for film thicknesses above 18 monolayers.     

Investigation of the stability of metal species with respect to liquid chromatographic separations

Summary For accurate speciation analyses it is important to know the stability of the respective species, especially in the case of metal complexes. Factors affecting the chromatographic stability of such metal species are investigated. By using thermodynamic models for complex formation and chromatographic retention equilibria the influence of species concentration, stoichiometry and excess of ligand is calculated and compared with experimental results for iron complexes (lactate, gluconate and citrate species). Iron citrate is the only species, that is chromatographed as 1:2 complex (metal: ligand), while iron lactate and gluconate are transformed to 1:1 species. Problems resulting from the coelution of different species are discussed.Dedicated to Professor Dr. Wilhelm Fresenius on the occasion of his 80th birthday     

A globally convergent multi-grid algorithm for moving boundary problems of two-phase Stefan type

In this paper we deal with the numerical solution of movingboundary problems of two-phase Stefan type. Based on an implicitdiscretization in time and the use of continuous, piecewiselinear finite elements in the space variables with respect tothe weak formulation of the problem, a globally convergent multi-gridalgorithm is developed. That algorithm strongly relies on thevariational characterization of the fully discretized problemas the unconstrained minimization of a subdifferentiable convexobjective functional. Numerical results indicate a significantimprovement in efficiency compared with previous multi-gridapproaches.     

Adaptation of an echelle spectrograph to a large CCD detector

A broad range echelle spectrograph with a CCD matrix detector is developed for different applications in atomic spectroscopy. The compact optical design in combination with a dispersive entrance slit illumination offers a low stray light level and high throughput in the UV spectral region comparable only with spectrographs of higher focal length. Ray tracing calculations for typical wavelengths and the multi-line spectrum of an iron arc show the excellent overall spectrum quality. The subtraction of two partial spectra of different steel alloys demonstrates the spectra evaluation possibilities of the CCD spectrograph.     

TLC-screening of important pharmaceutical substances using fluorescence detection

Summary Observation of fluorescence reactions on TLC plates is a valuable additional tool within the scope of screening procedures for many toxicologically relevant substances commonly encountered in clinical- and forensic-toxicological analysis. The reactions are based on native fluorescence without any treatment and on reactions obtained with an approved derivatization procedure. Due to the enormous sensitivity of the fluorescence detection, the method is also applicable to very low concentrations and small amounts of biological materials. The procedures described in this article have proven their high pragmatic usefulness in many practical cases.Dedicated to Professor Dr. Wilhelm Fresenius on the occasion of his 80th birthday