Lagrangian approximations and weak solutions of the Navier-Stokes equations

The motion of a viscous incompressible fluid flow in bounded domains with a smooth boundary can be described by the nonlinear Navier-Stokes equations. This description corresponds to the so-called Eulerian approach. We develop a new approximation method for the Navier-Stokes equations in both the stationary and the non-stationary case by a suitable coupling of the Eulerian and the Lagrangian representation of the flow, where the latter is defined by the trajectories of the particles of the fluid. The method leads to a sequence of uniquely determined approximate solutions with a high degree of regularity containing a convergent subsequence with limit function v such that v is a weak solution of the Navier-Stokes equations.

Redox reactions of group 5 elements, including element 105, in aqueous solutions

Standard redox potentials E^0(M^z+x/M^z+) in acidic solutions for group 5 elements including element 105 (Ha) and the actinide, Pa, have been estimated on the basis of the ionization potentials calculated via the multiconfiguration Dirac-Fock method. Stability of the pentavalent state was shown to increase along the group from V to Ha, while that of the tetra- and trivalent states decreases in this direction. Our estimates have shown no extra stability of the trivalent state of hahnium. Element 105 should form mixed-valence complexes by analogy with Nb due to the similar values of their potentials E^0(M^3+/M^2+). The stability of the maximumoxidation state of the elements decreases in the direction 103 > 104 > 105.

The electronic structure of anionic halide complexes of element 105 in aqueous solutions and their extraction by aliphatic amines

To study the complex formation of group 5 elements (Nb, Ta, Ha, and pseudoanalog Pa) in aqueous HCI solutions of medium and high concentrations the electronic structures of anionic complexes of these elements [MCl_6]^-, [MOCl_4]^-, [M(OH)-2 Cl_4]^-, and [MOCl_5]^2- have been calculated using the relativistic Dirac-Slater Discrete-Variational Method. The charge density distribution analysis has shown that tantalum occupies a specific position in the group and has the highest tendency to form the pure halide complex, [TaCl_6-. This fact along with a high covalency of this complex explains its good extractability into aliphatic amines. Niobium has equal trends to form pure halide [NbCl_6]^- and oxyhalide [NbOCl_5]^2- species at medium and high acid concentrations. Protactinium has a slight preference for the [PaOCl_5]^2- form or for the pure halide complexes with coordination number higher than 6 under these conditions. Element 105 at high HCl concentrations will have a preference to form oxyhalide anionic complex [HaOCl_5]^2- rather than [HaCl_6]^-. For the same sort of anionic oxychloride complexes an estimate has been done of their partition between the organic and aqueous phases in the extraction by aliphatic amines, which shows the following succession of the partition coefficients: P_Nb < P_Ha < P_Pa.

Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes' equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes' equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.

On Solutions of Holonomic Divided-Difference Equations on Non-Uniform Lattices

The main aim of this paper is the development of suitable bases (replacing the power basis x^n (n\in\IN_\le 0) which enable the direct series representation of orthogonal polynomial systems on non-uniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows to write solutions of arbitrary divided-difference equations in terms of series representations extending results given in [16] for the q-case. Furthermore it enables the representation of the Stieltjes function which can be used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in [5], see also [6].

Controlling the transport of an ion: classical and quantum mechanical solutions

The accurate transport of an ion over macroscopic distances represents a challenging control problem due to the different length and time scales that enter and the experimental limitations on the controls that need to be accounted for. Here, we investigate the performance of different control techniques for ion transport in state-of-the-art segmented miniaturized ion traps. We employ numerical optimization of classical trajectories and quantum wavepacket propagation as well as analytical solutions derived from invariant based inverse engineering and geometric optimal control. The applicability of each of the control methods depends on the length and time scales of the transport. Our comprehensive set of tools allows us make a number of observations. We find that accurate shuttling can be performed with operation times below the trap oscillation period. The maximum speed is limited by the maximum acceleration that can be exerted on the ion. When using controls obtained from classical dynamics for wavepacket propagation, wavepacket squeezing is the only quantum effect that comes into play for a large range of trapping parameters. We show that this can be corrected by a compensating force derived from invariant based inverse engineering, without a significant increase in the operation time.

„Alles hat seine Stunde. Für jedes Geschehen unter dem Himmel gibt es eine bestimmte Zeit“ (Koh 3,1)

Armut/Reichtum, Arbeit, Glück, Tod, Gott und Mensch sowie Zeit – Dies alles sind Themen, die als existentiell bezeichnet werden können. Mit ihnen beschäftigt sich das alttestamentliche Buch Kohelet, das ca. zwischen 250 und 190 v. Chr. entstanden ist. Trotz des Alters dieses Buches behandelt Kohelet Fragen, die hoch aktuell sind: Gibt es einen Gott? Greift dieser ein? Ist Reichtum etwas Erstrebenswertes? Warum mühen wir uns so ab, wenn wir doch sowieso sterben und all das Erarbeitete hinterlassen müssen? Und haben wir überhaupt Entscheidungs- und Handlungsfreiheit? Ist nicht vielleicht sogar alles von Gott bestimmt? Die Fragen hängen eng mit dem eigenen Selbstbild zusammen. Besonders im Jugendalter befindet sich der Mensch auf der Suche nach einer Identität, nach Erkenntnissen und einer eigenen Weltanschauung. Kohelets Fragen gewinnen in dieser Zeit an Relevanz. Das alttestamentliche Buch kann insbesondere in dieser Lebensphase fruchtbar gemacht werden und anregend sein. Es kann Heranwachsende dazu auffordern, das bisherige Selbstbild und die eigene Weltanschauung zu hinterfragen. Aus diesen Überlegungen heraus ergibt sich das Buch Kohelet als wertvoller Gesprächspartner Jugendlicher. Daneben erscheint es auch naheliegend, Jugendliche mit Kohelet ins Gespräch zu bringen, weil der Adressat des Buches nach Schmitz der junge Erwachsene sei (vgl. Schmitz, 2004, S. 325). Obwohl eine Relevanz des Buches für Jugendliche auszumachen ist, kann das Buch Kohelet bisher als „bibeldidaktisches Stiefkind“ (G. Langenhorst, 2004, S. 317) angesehen werden. Vereinzelt finden sich Umsetzungsvorschläge des Buches im Unterricht. In Lehrplänen wird es jedoch kaum aufgegriffen. Hinzu kommt, dass sich kaum Ausführungen dazu finden, wie Jugendliche die Gedanken Kohelets überhaupt verstehen. Aus diesem Grund hat die vorliegende Arbeit das Ziel, dies in Erfahrung zu bringen. Um die Ausgangsfrage zu beantworten, wurde eine empirische Untersuchung der Rezeption durchgeführt. Dabei wurde das Zeitgedicht des Koheletbuches fokussiert, das das menschliche Leben als Ganzes thematisiert und implizit ein Menschenbild entfaltet. Über die Frage der Rezeption durch OberstufenschülerInnen hinaus, möchte die Arbeit auch Rückschlüsse auf eine mögliche Beschäftigung mit Kohelet im Religionsunterricht schließen. Zunächst wird das hier grundgelegte Verständnis von Textrezeption erläutert. Daraufhin wird der Blick auf die Texteinheit gerichtet. Diese wird eingehend exegetisch untersucht, wobei auch die verschiedenen Lesarten thematisiert werden. In einem nächsten Kapitel rücken die SchülerInnen als Adoleszente ins Zentrum, woraufhin die Untersuchung des Dialogs zwischen Text und SchülerInnen und somit der Rezeption vollzogen wird. Den Schluss bildet ein Fazit mit einem Ausblick.