Finite Element Methods for Geometric Problems

In the herewith presented work we numerically treat geometric partial differential equations using finite element methods. Problems of this type appear in many applications from physics, biology and engineering use. We may partition the work in two blocks. The first one, including the chapters two to five, is about the approximation of stationary points of conformally invariant, nonlinear, elliptic energy functionals. Main interest is a compactness result for accumulation points of their discrete counterparts. The corresponding Euler-Lagrange equations are nonlinear, elliptic and of second order. They contain critical nonlinearities that are quadratic in the first derivatives. Thus, accumulation points of solutions to the discrete problem are not solutions of the continuous problem in general. We deduce a weak formulation in a mixed form and chose appropriate spaces for the discretization. First we show existence of discrete solutions and then, by the use of compensated compactness and standard finite element arguments, we establish convergence. Finally we introduce an iterative algorithm for the numerical realization and run different simulations. Hereby we confirm theoretical predictions derived in the stability analysis. The second part is about the derivation of gradient flows for shape functionals and their discretization with parametric finite elements. First, we consider the Willmore energy of a twodimensional surface in the threedimensional ambient space and deduce its first variation. Afterwards we phrase the corresponding gradient flow in a weak form and discuss possible discretizations. During the further progress of the work we modell cell membranes and the effects of surface active agents on the shape of these cells. Numerical simulations with closed surface give promising results and a reason to intensify the research in this field.Finite Elemente Methoden für Geometrische Probleme In der vorliegenden Dissertationsschrift geht es um die numerische Behandlung geometrischer partieller Differentialgleichungen unter Verwendung von Finite Elemente Methoden. Probleme dieser Art treten in einer Vielzahl von physikalischen, technischen und biologischen Anwendungen auf. Thematisch lässt sich die Arbeit in zwei Blöcke aufteilen. In den Kapiteln zwei bis fünf geht es um die Approximation stationärer Punkte konform invarianter, nichtlinearer, elliptischer Energiefunktionale. Das Hauptaugenmerk liegt dabei auf einem Kompaktheitsresultat für Häufungspunkte der diskretisierten Energiefunktionale. Die Euler Lagrange Gleichungen sind elliptisch und von zweiter Ordnung. Sie beinhalten kritische Nichtlinearitäten welche quadratisch von den ersten Ableitungen abhängen. Dies f¨hrt dazu, dass Häufungspunkte von Lösungen der diskretisierten Gleichung nicht zwangsläufig Lösungen der ursprünglichen Gleichung sind. Wir leiten eine schwache Formulierung der Gleichung in gemischter Form her und wählen stabile Finite Elemente Paare für die Diskretisierung. Zunächst zeigen wir, dass Lösungen der diskreten gemischten Formulierung Sattelpunkte eines erweiterten diskreten Energiefunktionals sind und schließen daraus auf die Existenz diskreter Löosungen. Um zu beweisen, dass Häufungspunkte der diskreten Sattelpunkte tatsächlich Lösungen der schwachen Formulierung sind bedienen wir uns einigen Resultaten der kompensierten Kompaktheit sowie bekannten Techniken aus dem Bereich der Finiten Elemente. Schließlich stellen wir einen iterativen Algorithmus für die numerische Realisierung auf und föhren mehrere Simulationen durch. Theoretische Stabilitätsergebnisse für den Algorithmus werden dabei numerisch bestätigt. Im zweiten Teil stehen die Herleitung von Gradientenflüssen von Flächenfunktionalen (shape functional) sowie deren Diskretisierung unter Verwendung von Parametrischen Finite Elemente Methoden im Mittelpunkt. Wir betrachten zunächst die sogenannte Willmore Energie einer zweidimensionalen Fläche im dreidimensionalen Raum und bestimmen deren erste Variation. Anschließend formulieren wir den zugehörigen Gradientenfluss in schwacher Form und diskutieren eine Diskretisierung mittels parametrischer Finite Elemente. Im weiteren Verlauf diskutieren wir die Modellierung von Zellmembranen und die Wirkung von oberflächenaktiven Substanzen (surfactants) auf die Form von Zellen. Numerische Simulationen mit geschlossenen Flächen liefern viel versprechende Resultate und geben Anlass zu weiteren Forschungsarbeiten in diesem Bereich

Cigarette smoking cessation increases plasma levels of several antioxidant micronutrients and improves resistance towards oxidative challenge

Cigarette smoking is associated epidemiologically with increased risk of cardiovascular diseases, but the pathophysiological mechanisms are still not fully understood. There is evidence that smoking is related to increased free radical production and antioxidant depletion, but the effects of smoking cessation on plasma concentrations of antioxidants and susceptibility to oxidative stress are largely unknown. Plasma levels of vitamins A, C, E, uric acid, total thiols, carotenoids (including lutein, zeaxanthin, β-cryptoxanthin, lycopene, α- and β-carotene) and malondialdehyde (MDA, a biomarker of lipid peroxidation) were measured in fifteen healthy, normolipidaemic subjects (seven males, eight females, 35·2 (sd2·3) years) before and 4 weeks after smoking cessation. To determine plasma resistance towards oxidative challenge, plasma was incubated for up to 5h with the peroxyl radical-generator 2,2′-azobis(2-amidinopropane) (AAPH); MDA and ascorbate levels were measured at various time points. The concentrations of all plasma antioxidants were lower before smoking cessation than afterwards; MDA levels were higher before than after termination of smoking. Upon AAPH exposure, the consumption of plasma ascorbate and the production of MDA occurred at a significantly faster rate before smoking cessation as compared with afterwards. Cigarette smoking cessation is followed by a marked increase in plasma antioxidant concentrations and substantially improves plasma resistance towards oxidative challenge. Given the importance of cigarette smoking as a risk factor for cardiovascular diseases and the pathophysiological role played by oxidative stress in these illnesses, quitting smoking represents an irreplaceable preventive strategy against tobacco-induced oxidative stress and vascular damage

Die Wortstellung im Russischen

Die Aufgabe der vorliegenden Untersuchung ist es, mit Hilfe von Material aus den einzelnen slavischen Sprachen die Ursachen für die Häufigkeit bzw. Seltenheit der einzelnen Wortstellungen, überhaupt die Gründe für ihr Auftreten zu untersuchen und darzustellen

The Centrosomal Protein Pericentrin Identified at the Basal Body Complex of the Connecting Cilium in Mouse Photoreceptors

BACKGROUND: Pericentrin (Pcnt), a conserved protein of the pericentriolar material, serves as a multifunctional scaffold for numerous proteins and plays an important role in microtubule organization. Recent studies indicate that Pcnt mutations are associated with a range of diseases including primordial dwarfism and ciliopathies. To date, three Pcnt splice variants from orthologous genes in mice and humans are known. PRINCIPAL FINDINGS: We generated a specific Pcnt antiserum detecting all known Pcnt splice variants and examined the cellular and subcellular distribution of Pcnt in ciliated tissues of the mouse, the olfactory epithelium and the retina. For the first time, we identified Pcnt and its centrosomal interaction partners at the basal body complex of mouse retinal photoreceptors. Photoreceptors are morphologically and functionally subdivided into the light sensitive outer segment and the inner segment comprising the metabolic function of the cell. The two compartments are linked via a modified, specialized, non-motile cilium, the connecting cilium. Here, Pcnt colocalized with the whole protein machinery responsible for transport processes between the two compartments. Surprisingly, photoreceptors expressed a small Pcnt splice transcript - most likely a modified variant of Pcnt S - which was not present in receptor neurons of the olfactory epithelium. CONCLUSIONS: Our findings suggest distinct functional roles of several Pcnt variants in different ciliated tissues and sensory neurons, like the olfactory epithelium and the retina of the mouse. The individual patchwork of different Pcnt splice transcripts seems to reflect the complexity of Pcnt function, an assumption corroborated by the heterogeneous clinical manifestations associated with mutations in the Pcnt gene