Quasi-Interpolation von Funktionen und Anwendungen auf Potentialprobleme

Ausgangspunkt der Dissertation ist ein von V. Maz'ya entwickeltes Verfahren, eine gegebene Funktion f : Rn ! R durch eine Linearkombination fh radialer glatter exponentiell fallender Basisfunktionen zu approximieren, die im Gegensatz zu den Splines lediglich eine näherungsweise Zerlegung der Eins bilden und somit ein für h ! 0 nicht konvergentes Verfahren definieren. Dieses Verfahren wurde unter dem Namen Approximate Approximations bekannt. Es zeigt sich jedoch, dass diese fehlende Konvergenz für die Praxis nicht relevant ist, da der Fehler zwischen f und der Approximation fh über gewisse Parameter unterhalb der Maschinengenauigkeit heutiger Rechner eingestellt werden kann. Darüber hinaus besitzt das Verfahren große Vorteile bei der numerischen Lösung von Cauchy-Problemen der Form Lu = f mit einem geeigneten linearen partiellen Differentialoperator L im Rn. Approximiert man die rechte Seite f durch fh, so lassen sich in vielen Fällen explizite Formeln für die entsprechenden approximativen Volumenpotentiale uh angeben, die nur noch eine eindimensionale Integration (z.B. die Errorfunktion) enthalten. Zur numerischen Lösung von Randwertproblemen ist das von Maz'ya entwickelte Verfahren bisher noch nicht genutzt worden, mit Ausnahme heuristischer bzw. experimenteller Betrachtungen zur sogenannten Randpunktmethode. Hier setzt die Dissertation ein. Auf der Grundlage radialer Basisfunktionen wird ein neues Approximationsverfahren entwickelt, welches die Vorzüge der von Maz'ya für Cauchy-Probleme entwickelten Methode auf die numerische Lösung von Randwertproblemen überträgt. Dabei werden stellvertretend das innere Dirichlet-Problem für die Laplace-Gleichung und für die Stokes-Gleichungen im R2 behandelt, wobei für jeden der einzelnen Approximationsschritte Konvergenzuntersuchungen durchgeführt und Fehlerabschätzungen angegeben werden.

X-Ray spectra of superheavy and quasi-superheavy elements

We discuss the possibility of identifying superheavy elements from the observation of their M-shell x-ray spectra, which might occur during the collision of a superheavy element with a heavy target. The same question is discussed for the possible observation of the x-rays from the quasimolecule (quasi-superheavy element) which is formed during such a heavy-ion collision. It is shown that it is very difficult, if not impossible, to determine any information about the interesting quantum electrodynamical effects from the M-shell x-ray spectra of these quasimolecules.

Evidence for an additional coupling of the innermost shells in very heavy quasi-molecular ion-atom collisions

Due to the tremendous spin-orbit splitting of quasi-molecular levels in superheavy collision systems (Z = Z_1 + Z_2 {\ge\approx} 137) bombarding energy 0.5-6 MeV N{^-1}, unusual couplings may occur around Z \simeq 165. Experimental evidence for such a theoretically predicted coupling is discussed.

Theoretical evidence for quasi-molecular structure at small internuclear distances in elastic ion-atom scattering

The potential energy curve of the system Ne-Ne is calculated for small internuclear distances from 0.005 to 3.0 au using a newly developed relativistic molecular Dirac-Fock-Slater code. A significant structure in the potential energy curve is found which leads to a nearly complete agreement with experimental differential elastic scattering cross sections. This demonstrates the presence of quasi-molecular effects in elastic ion-atom collisions at keV energies.

Biological Photonic Crystals: Diatoms

The aim of this work was to produce a variety of fluorescent diatom cell wall material as a basis for spectroscopic investigations of the influence of the photonic structure on the emission of an incorporated laser dye. This goal was achieved by the method of in vivo-fluorochromation, in which the fluorescence dyes are incorporated by the diatom cells during cell wall formation. Several fluorescent dyes (mostly rhodamines) known as strong laser dyes, were tested for a possible application within this method. The results of this work show that half of the tested rhodamines can be applied for an in vivo-fluorochromation of diatom cells. For a successful incorporation into the diatom cell wall, a relatively low toxicity to diatom cells is necessary. Replacement of the carbon acid function at the carboxyphenyl ring of the rhodamine by a methyl or ethylester function showed to convert a rhodamine of relatively low toxicity to a rhodamine leading to severe lethal effects within the cells. In contrast to their carbon acid forms, which posses a net neutral charge of the molecule, rhodamine esters exhibit a net positive charge. The enhanced toxicological effects seem to be due to an increased accumulation of positive charged rhodamines within the mitochondria, an increased hydrophobicity due to the attachment of an alkyl substituent, an increased retention time of the dyes within the mitochondria and a therefore stronger negative effect on the mitochondrial membrane bound energy processes of the diatom cell. Therefore rhodamines with a positive net charge deriving from a methyl or ethylester function at the carboxy phenyl ring instead of a carbon acid substituent showed not to be suitable for long-term investigations/ biomineralization studies of diatoms. Investigations performed on diatom species of different orders showed that rhodamine 19, rhodamine B, and rhodamine 101 can presumably be successfully applied for in vivo-fluorochromation to all diatom species. The results obtained here can help to find further laser dyes for an in vivo-fluorochromation of diatom cells and therefore for the production of fluorescent nanostructural elements for a detailed optical investigation of the diatom cell wall. First optical measurements performed on in vivo-fluorochromated cell walls did not give any hints concerning the photonic structure of the diatom cell. Cell wall parts with different nanostructural elements were investigated and by comparison of the obtained fluorescence emission spectra, no special features that might derive from photonic structural effects could be observed. Results concerning the concentration dependent shifts within the emission spectra, as well as the decrease of fluorescence intensity of the stained cell wall structures with increasing dye concentration, depict that several effects occurring by interaction of the molecules within the cell wall can have an impact on the technical application of fluorescent cell walls. It can be assumed that the investigation of the photonic crystal behaviour and the possibility to achieve laser action within the diatom cell wall can be hampered by molecular interactions. The results give hints to prevent such obstacles. Comparison of the recent findings and state of the art of in vivo-fluorochromation of diatom cell wall material, make clear that the here presented results are of importance and can offer a considerable contribution to the development and establishment of new biosilification markers, for diatoms as well as for other biosilifying organisms.