## Fachbereich Mathematik

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- Locating stops along bus or railway lines - a bicriterial problem (2003)
- In this paper we consider the location of stops along the edges of an already existing public transportation network, as introduced in [SHLW02]. This can be the introduction of bus stops along some given bus routes, or of railway stations along the tracks in a railway network. The goal is to achieve a maximal covering of given demand points with a minimal number of stops. This bicriterial problem is in general NP-hard. We present a nite dominating set yielding an IP-formulation as a bicriterial set covering problem. We use this formulation to observe that along one single straight line the bicriterial stop location problem can be solved in polynomial time and present an e cient solution approach for this case. It can be used as the basis of an algorithm tackling real-world instances.

- Locational planning in the Mathematics Curriculum of High Schools (2002)
- Dealing with problems from locational planning in schools can enrich the mathematical education. In this report we describe planar locational problems which can be used in mathematical lessons. The problems production of a semiconductor plate, design of a fire brigade building and the warehouse problem are from real-world. The problems are worked out detailed so that the usage for school lessons is possible.

- Bills of material and linear algebra (2003)
- This publication tries to develop mathematical subjects for school from realistic problems. The center of this report are business planning and decision problems which occur in almost all companies. The main topics are: Calculation of raw material demand for given orders, consumption of existing stock and the lot sizing.

- Linear Optimization in School Mathematics (2003)
- Linear Optimization is an important area from applied mathematics. A lot of practical problems can be modelled and solved with this technique. This publication shall help to introduce this topic to pupils. The process of modelling, the reduction of problems to their significant attributes shall be described. The linear programms will be solved by using the simplex method. Many examples illustrate the topic.

- Wavelet Modelling of the Spherical Inverse Source Problem with Application to Geomagnetism (2004)
- The article is concerned with the modelling of ionospheric current systems from induced magnetic fields measured by satellites in a multiscale framework. Scaling functions and wavelets are used to realize a multiscale analysis of the function spaces under consideration and to establish a multiscale regularization procedure for the inversion of the considered vectorial operator equation. Based on the knowledge of the singular system a regularization technique in terms of certain product kernels and corresponding convolutions can be formed. In order to reconstruct ionospheric current systems from satellite magnetic field data, an inversion of the Biot-Savart's law in terms of multiscale regularization is derived. The corresponding operator is formulated and the singular values are calculated. The method is tested on real magnetic field data of the satellite CHAMP and the proposed satellite mission SWARM.

- Wavelet-Mie-Representations for Solenoidal Vector Fields with Applications to Ionospheric Geomagnetic Data (2004)
- A wavelet technique, the wavelet-Mie-representation, is introduced for the analysis and modelling of the Earth's magnetic field and corresponding electric current distributions from geomagnetic data obtained within the ionosphere. The considerations are essentially based on two well-known geomathematical keystones, (i) the Helmholtz-decomposition of spherical vector fields and (ii) the Mie-representation of solenoidal vector fields in terms of poloidal and toroidal parts. The wavelet-Mie-representation is shown to provide an adequate tool for geomagnetic modelling in the case of ionospheric magnetic contributions and currents which exhibit spatially localized features. An important example are ionospheric currents flowing radially onto or away from the Earth. To demonstrate the functionality of the approach, such radial currents are calculated from vectorial data of the MAGSAT and CHAMP satellite missions.

- Algebraic Systems Theory (2004)
- Control systems are usually described by differential equations, but their properties of interest are most naturally expressed in terms of the system trajectories, i.e., the set of all solutions to the equations. This is the central idea behind the so-called "behavioral approach" to systems and control theory. On the other hand, the manipulation of linear systems of differential equations can be formalized using algebra, more precisely, module theory and homological methods ("algebraic analysis"). The relationship between modules and systems is very rich, in fact, it is a categorical duality in many cases of practical interest. This leads to algebraic characterizations of structural systems properties such as autonomy, controllability, and observability. The aim of these lecture notes is to investigate this module-system correspondence. Particular emphasis is put on the application areas of one-dimensional rational systems (linear ODE with rational coefficients), and multi-dimensional constant systems (linear PDE with constant coefficients).

- Regularized Multiresolution Recovery of the Mass Density Distribution From Satellite Data of the Earth´s Gravitational Field (2004)
- The inverse problem of recovering the Earth's density distribution from data of the first or second derivative of the gravitational potential at satellite orbit height is discussed for a ball-shaped Earth. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbitheight is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG (satellite gravity gradiometry) satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust. Moreover, the noise sensitivity of the regularization technique is analyzed numerically.

- Frequency Filtering in Telecommunications (2004)
- Piezoelectric filters are used in telecommunication to filter electrical signals. This report deals with the problem of calculating passing and damped frequency intervals for a filter with given geometrical configurations and materials. Only periodic filters, which are widely used in practice, were considered. These filters consist of periodically arranged cells. For a small amount of cells a numerical procedure to visualise the wave propagation in the filter was developed. For a big number of cells another model of the filter was obtained. In this model it is assumed that the filter occupies an infinite domain. This leads to a differential equation, with periodic coefficients, that describes propagation of the wave with a given frequency in the filter. To analyse this equation the Spectral Theory for Periodic Operators had to be employed. Different ways -- analytical and numerical -- to apply the theory were proposed and analysed.

- A multiaxial stress-strain correction scheme (2005)
- A method to correct the elastic stress tensor at a fixed point of an elastoplastic body, which is subject to exterior loads, is presented and analysed. In contrast to uniaxial corrections (Neuber or ESED), our method takes multiaxial phenomena like ratchetting or cyclic hardening/softening into account by use of Jiang's model. Our numerical algorithm is designed for the case that the scalar load functions are piecewise linear and can be used in connection with critical plane/multiaxial rainflow methods in high cycle fatigue analysis. In addition, a local existence and uniqueness result of Jiang's equations is given.