An approximation method using approximate approximations

The aim of this paper is to 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 Laplace equation in two dimensions using approximate approximations. 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.

Stability of preconditioned finite volume schemes at low Mach numbers

In [4], Guillard and Viozat propose a finite volume method for the simulation of inviscid steady as well as unsteady flows at low Mach numbers, based on a preconditioning technique. The scheme satisfies the results of a single scale asymptotic analysis in a discrete sense and comprises the advantage that this can be derived by a slight modification of the dissipation term within the numerical flux function. Unfortunately, it can be observed by numerical experiments that the preconditioned approach combined with an explicit time integration scheme turns out to be unstable if the time step Dt does not satisfy the requirement to be O(M2) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Dt=O(M), M to 0, which results from the well-known CFL-condition. We present a comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M-2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. Thereby, we present statements for both the standard preconditioner used by Guillard and Viozat [4] and the more general one due to Turkel [21]. The theoretical results are after wards confirmed by numerical experiments.

A computer-algebraic approach to the study of the symmetry properties of molecules and clusters

This thesis work is dedicated to use the computer-algebraic approach for dealing with the group symmetries and studying the symmetry properties of molecules and clusters. The Maple package Bethe, created to extract and manipulate the group-theoretical data and to simplify some of the symmetry applications, is introduced. First of all the advantages of using Bethe to generate the group theoretical data are demonstrated. In the current version, the data of 72 frequently applied point groups can be used, together with the data for all of the corresponding double groups. The emphasize of this work is placed to the applications of this package in physics of molecules and clusters. Apart from the analysis of the spectral activity of molecules with point-group symmetry, it is demonstrated how Bethe can be used to understand the field splitting in crystals or to construct the corresponding wave functions. Several examples are worked out to display (some of) the present features of the Bethe program. While we cannot show all the details explicitly, these examples certainly demonstrate the great potential in applying computer algebraic techniques to study the symmetry properties of molecules and clusters. A special attention is placed in this thesis work on the flexibility of the Bethe package, which makes it possible to implement another applications of symmetry. This implementation is very reasonable, because some of the most complicated steps of the possible future applications are already realized within the Bethe. For instance, the vibrational coordinates in terms of the internal displacement vectors for the Wilson's method and the same coordinates in terms of cartesian displacement vectors as well as the Clebsch-Gordan coefficients for the Jahn-Teller problem are generated in the present version of the program. For the Jahn-Teller problem, moreover, use of the computer-algebraic tool seems to be even inevitable, because this problem demands an analytical access to the adiabatic potential and, therefore, can not be realized by the numerical algorithm. However, the ability of the Bethe package is not exhausted by applications, mentioned in this thesis work. There are various directions in which the Bethe program could be developed in the future. Apart from (i) studying of the magnetic properties of materials and (ii) optical transitions, interest can be pointed out for (iii) the vibronic spectroscopy, and many others. Implementation of these applications into the package can make Bethe a much more powerful tool.

The Surface Plasmon Resonance of Supported Noble Metal Nanoparticles: Characterization, Laser Tailoring, and SERS Application

This work deals with the optical properties of supported noble metal nanoparticles, which are dominated by the so-called Mie resonance and are strongly dependent on the particles’ morphology. For this reason, characterization and control of the dimension of these systems are desired in order to optimize their applications. Gold and silver nanoparticles have been produced on dielectric supports like quartz glass, sapphire and rutile, by the technique of vapor deposition under ultra-high vacuum conditions. During the preparation, coalescence is observed as an important mechanism of cluster growth. The particles have been studied in situ by optical transmission spectroscopy and ex situ by atomic force microscopy. It is shown that the morphology of the aggregates can be regarded as oblate spheroids. A theoretical treatment of their optical properties, based on the quasistatic approximation, and its combination with results obtained by atomic force microscopy give a detailed characterization of the nanoparticles. This method has been compared with transmission electron microscopy and the results are in excellent agreement. Tailoring of the clusters’ dimensions by irradiation with nanosecond-pulsed laser light has been investigated. Selected particles are heated within the ensemble by excitation of the Mie resonance under irradiation with a tunable laser source. Laser-induced coalescence prevents strongly tailoring of the particle size. Nevertheless, control of the particle shape is possible. Laser-tailored ensembles have been tested as substrates for surface-enhanced Raman spectroscopy (SERS), leading to an improvement of the results. Moreover, they constitute reproducible, robust and tunable SERS-substrates with a high potential for specific applications, in the present case focused on environmental protection. Thereby, these SERS-substrates are ideally suited for routine measurements.

On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.

Theoretical study of the structural dependence of nuclear quadrupole frequencies in high-T_c superconductors

The remarkable difference between the nuclear quadrupole frequencies v_Q of Cu(1) and Cu(2) in YBa_2Cu_3O_6 and YBa_2Cu_3O_7 is analyzed. We calculate the ionic contribution to the electric field gradients and estimate, by using experimental results for Cu_2O and La_2CuO_4, the contribution of the d valence electrons. Thus, we determine v_Q1, v_Q2, and the asymmetry parameter η for YBa_2Cu_3O_6 and YBa_2Cu_3O_7. The number of holes in dthe Cu-O planes and chains is found to be important for the different behavior of v_Q1 and v_Q2.

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.

First results of experimental and theoretical investigations on sextet states of doubly-excited five-electron ions

In continuation of our previous work on doubly-excited ions with three and four electrons we present the first results on optical transitions in the term system of doubly-excited ions with five electrons. Transitions between such sextet states were identified in beam-foil spectra of the ions nitrogen, oxygen and fluorine. Assignments were first established by comparison with Multi-Configuration Dirac-Fock calculations. Later assignments were aided by Multi-Configuration Hartree-Fock calculations (see the contribution by G. Miecznik et al. in this issue). Decay curves were recorded for all six candidate lines. The lifetime results are compared to theoretical values which confirm most of the assignments qualitatively.

Theoretical study of the physicochemical properties of the light transactinides

A theoretical study of the physicochemical properties of elements 104, 105, and 106 and their compounds in the gas phase and aqueous solutions has been undertaken using relativistic atomic and molecular codes. Trends in properties such as bonding, ionization potentials, electron affinities, energies of electronic transitions, stabilities of oxidation states etc. have been defined within the corresponding chemical groups and within the transactinides. These trends are shown to be determined by increasing relativistic effects within the groups. The behaviour of some gas phase compounds and complexes in solutions is predicted for the gas chromatography and solvent extraction experiments. Redox potentials in aqueous solutions of these elements are estimated.

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.

Applications and modelling in mathematics teaching

The aim of this paper is a comprehensive presentation of some important basic and general aspects of the topic applications and modelling, with emphasis on the secondary school level. Owing to the review character of this paper, some overlap with the survey paper Blum and Niss (1989) for ICME-6 in Budapest is inevitable. The paper will consist of three parts. In part 1, I shall try to clarify some basic concepts and remind the reader of a few application and modelling examples suitable for teaching. In part 2, I shall formulate some general aims for mathematics instruction and, on that basis, summarise the most important arguments for and against applications and modelling in mathematics teaching. Finally, in part 3, I shall discuss some relevant instructional aspects resulting from the considerations in part 2.