## Fachbereich Mathematik

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- 2003 (39) (remove)

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#### Keywords

- Lineare Algebra (4)
- Mathematikunterricht (4)
- Modellierung (4)
- linear algebra (4)
- modelling (4)
- praxisorientiert (4)
- Mehrskalenanalyse (3)
- Wavelet (3)
- mathematical education (3)
- Gravitationsfeld (2)

- Optimizacíon lineal en las clases de matemáticas (2003)
- Las matemáticas son atribuidas en general a algo no claro y sólo para matemáticos. La imagen de las matemáticas para los escolares, es la de una ciencia, la cual se sirve sólo de si misma. Es importante hacer frente al prejuicio de que las matemáticas distan lejos de toda utilidad práctica. La matemática es una ciencia al servicio de todas las dem´as ciencias, de cuya ayuda se necesita en casi todos los campos de la vida. La matemática de la escuela debería despertar en cualquier ámbito de la vida de los escolares el interés sobre ...

- 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.

- Lista de productos y álgebra lineal (2003)
- A mediados del año 1997 la publicación de los denominados TIMMS-Estudios (Third International Mathematics and Science Study) causó un importante impacto en el público alemán. El motivo de esto fue el rendimiento escolar conseguido en la rama de matemáticas y ciencias naturales del octavo curso, el cual estaba situado en un campo internacional, donde particularmente en el ámbito matemático el conjunto de los estados del norte-, oeste-, y del este de Europa que forman parte del TIMSS - sin mencionar a la mayoría de los paises asiáticos - habían conseguido claramente mejores rendimiento. En definitiva mostraban un peor rendimiento los escolares alemanes con respecto a los paises vecinos y con los ....

- 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.

- Multiscale Modeling of CHAMP-Data (2003)
- The following three papers present recent developments in multiscale gravitational field modeling by the use of CHAMP or CHAMP-related data. Part A - The Model SWITCH-03: Observed orbit perturbations of the near-Earth orbiting satellite CHAMP are analyzed to recover the long-wavelength features of the Earth's gravitational potential. More precisely, by tracking the low-flying satellite CHAMP by the high-flying satellites of the Global Positioning System (GPS) a kinematic orbit of CHAMP is obtainable from GPS tracking observations, i.e. the ephemeris in cartesian coordinates in an Earth-fixed coordinate frame (WGS84) becomes available. In this study we are concerned with two tasks: First we present new methods for preprocessing, modelling and analyzing the emerging tracking data. Then, in a first step we demonstrate the strength of our approach by applying it to simulated CHAMP orbit data. In a second step we present results obtained by operating on a data set derived from real CHAMP data. The modelling is mainly based on a connection between non-bandlimited spherical splines and least square adjustment techniques to take into account the non-sphericity of the trajectory. Furthermore, harmonic regularization wavelets for solving the underlying Satellite-to-Satellite Tracking (SST) problem are used within the framework of multiscale recovery of the Earth's gravitational potential leading to SWITCH-03 (Spline and Wavelet Inverse Tikhonov regularized CHamp data). Further it is shown how regularization parameters can be adapted adequately to a specific region improving a globally resolved model. Finally we give a comparison of the developed model to the EGM96 model, the model UCPH2002_02_0.5 from the University of Copenhagen and the GFZ models EIGEN-1s and EIGEN-2. Part B - Multiscale Solutions from CHAMP: CHAMP orbits and accelerometer data are used to recover the long- to medium- wavelength features of the Earth's gravitational potential. In this study we are concerned with analyzing preprocessed data in a framework of multiscale recovery of the Earth's gravitational potential, allowing both global and regional solutions. The energy conservation approach has been used to convert orbits and accelerometer data into in-situ potential. Our modelling is spacewise, based on (1) non-bandlimited least square adjustment splines to take into account the true (non-spherical) shape of the trajectory (2) harmonic regularization wavelets for solving the underlying inverse problem of downward continuation. Furthermore we can show that by adapting regularization parameters to specific regions local solutions can improve considerably on global ones. We apply this concept to kinematic CHAMP orbits, and, for test purposes, to dynamic orbits. Finally we compare our recovered model to the EGM96 model, and the GFZ models EIGEN-2 and EIGEN-GRACE01s. Part C - Multiscale Modeling from EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, EGM96: Spherical wavelets have been developed by the Geomathematics Group Kaiserslautern for several years and have been successfully applied to georelevant problems. Wavelets can be considered as consecutive band-pass filters and allow local approximations. The wavelet transform can also be applied to spherical harmonic models of the Earth's gravitational field like the most up-to-date EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, and the well-known EGM96. Thereby, wavelet coefficients arise and these shall be made available to other interested groups. These wavelet coefficients allow the reconstruction of the wavelet approximations. Different types of wavelets are considered: bandlimited wavelets (here: Shannon and Cubic Polynomial (CP)) as well as non-bandlimited ones (in our case: Abel-Poisson). For these types wavelet coefficients are computed and wavelet variances are given. The data format of the wavelet coefficients is also included.

- Numerical solution of coupled flow in plain and porous media (2003)
- The present thesis deals with coupled steady state laminar flows of isothermal incompressible viscous Newtonian fluids in plain and in porous media. The flow in the pure fluid region is usually described by the (Navier-)Stokes system of equations. The most popular models for the flow in the porous media are those suggested by Darcy and by Brinkman. Interface conditions, proposed in the mathematical literature for coupling Darcy and Navier-Stokes equations, are shortly reviewed in the thesis. The coupling of Navier-Stokes and Brinkman equations in the literature is based on the so called continuous stress tensor interface conditions. One of the main tasks of this thesis is to investigate another type of interface conditions, namely, the recently suggested stress tensor jump interface conditions. The mathematical models based on these interface conditions were not carefully investigated from the mathematical point of view, and also their validity was a subject of discussions. The considerations within this thesis are a step toward better understanding of these interface conditions. Several aspects of the numerical simulations of such coupled flows are considered: -the choice of proper interface conditions between the plain and porous media -analysis of the well-posedness of the arising systems of partial differential equations; -developing numerical algorithm for the stress tensor jump interface conditions, coupling Navier-Stokes equations in the pure liquid media with the Navier-Stokes-Brinkman equations in the porous media; -validation of the macroscale mathematical models on the base of a comparison with the results from a direct numerical simulation of model representative problems, allowing for grid resolution of the pore level geometry; -developing software and performing numerical simulation of 3-D industrial flows, namely of oil flows through car filters.

- 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.

- Semiparametric estimation of conditional quantiles for time series, with applications in finance (2003)
- The estimation of conditional quantiles has become an increasingly important issue in insurance and financial risk management. The stylized facts of financial time series data has rendered direct applications of extreme value theory methodologies, in the estimation of extreme conditional quantiles, inappropriate. On the other hand, quantile regression based procedures work well in nonextreme parts of a given data but breaks down in extreme probability levels. In order to solve this problem, we combine nonparametric regressions for time series and extreme value theory approaches in the estimation of extreme conditional quantiles for financial time series. To do so, a class of time series models that is similar to nonparametric AR-(G)ARCH models but which does not depend on distributional and moments assumptions, is introduced. We discuss estimation procedures for the nonextreme levels using the models and consider the estimates obtained by inverting conditional distribution estimators and by direct estimation using Koenker-Basset (1978) version for kernels. Under some regularity conditions, the asymptotic normality and uniform convergence, with rates, of the conditional quantile estimator for strong mixing time series, are established. We study the estimation of scale function in the introduced models using similar procedures and show that under some regularity conditions, the scale estimate is weakly consistent and asymptotically normal. The application of introduced models in the estimation of extreme conditional quantiles is achieved by augmenting them with methods in extreme value theory. It is shown that the overal extreme conditional quantiles estimator is consistent. A Monte Carlo study is carried out to illustrate the good performance of the estimates and real data are used to demonstrate the estimation of Value-at-Risk and conditional expected shortfall in financial risk management and their multiperiod predictions discussed.

- Nichtparametrische Trend- und Volatilitätsschätzer für Finanzzeitreihen (2003)
- Zuerst einmal werden die Grundlagen der nichtparametrischen Regression sowie die der Kleinste-Quadrate-Schätzer behandelt und unser verwendetes Modell hergeleitet. Kapitel 3 führt dann in die Theorie der gewichteten Kernschätzer ein, wobei auch das asymptotische Verhalten genauer untersucht wird. Des Weiteren wird ein numerischer Algorithmus zur Berechnung der Kernschätzer angegeben. Die Simulationsstudie der gewichteten Kernschätzer anhand von Regressionsdaten und Zeitreihendaten sowie die praktische Beurteilung erfolgen in Kapitel 4 und 5. Reale Zeitreihendaten bilden danach im sechsten Kapitel die Grundlage für die praktische Betrachtung der neuen Schätzer. Im letzten Kapitel folgt dann ein Resümee und ein kleiner Ausblick auf die gewichteten Kernschätzer für allgemeinere Modelle.