Michael Overesch Dissertation Format

This thesis was devoted to the study and optimization of solution approaches to the forward problem of electroencephalography (EEG). We investigated different numerical approaches regarding time consumption and particularly accuracy, two aspects that are crucial for solving the inverse problem of EEG, i.e., reconstructing source activity from measurement data. Several studies have shown that the accuracy of the solution highly depends on a realistic modeling of the human head. Besides others, two different numerical techniques are commonly used in combination with realistically shaped head models nowadays: Boundary element methods (BEM) and finite element methods (FEM). We evaluated a recently proposed BE approach, the symmetric BEM, and demonstrated that it leads to a considerably improved accuracy in multi-layer sphere models compared to former BE formulations. Furthermore, we enhanced accuracy and speed of different FE approaches by the application of leadfield interpolation, a technique originally proposed for the speed-up of BE calculations. Various investigations regarding the accuracy of the FE approaches in multi-layer sphere models were carried out before leadfield interpolation was applied to further optimize the results.  [pdf-file]
Johannes Vorwerk is now a PhD-Student at the Institute for Biomagnetism and Biosignalanalysis, WWU Münster.

  Optimal Consumption Using a Continuous Income Model
Daniel Raß, 2011 (Collaboration with Mark Trede, WWU Münster)

This thesis was concerned with the developement of a Hamilton-Jacobi-Bellman like equation arising in the optimal consumption problem using a continuous income model with stochastic uncertainty. Later on a finite differencing scheme was developed to determine the associated viscosity solution numerically. The same scheme was used to compute the lower border of an area with allowed income/wealth combinations under the additional condition no debts are allowed at the end of the life. With this border we were able to compute the maximal utilty gained by allowed income/wealth combinations.  [pdf-file]
Daniel Raß is now working for Saracus Consulting, Münster.


  Numerical methods for the determination of relative inverter positions in large PV-plants
Matthias Gröne, 2011 (Collaboration with Jens Klein, SMA Technologies, Kassel)

We provide an algorithm that is capable of determining relative inverter and string positions in PV-power plants. For this purpose, we harness clouds whose shadows cause declines in the power and current data allowing us to estimate the relative distances between the strings. The estimation of the positions from the relative distances leads to a problem of matrix factorization and is solved through an alternating least squares algorithm. This algorithm is analyzed theoretically and then applied to synthetic data. Among other procedures, we have to linearize cloud fronts to match the mathematical model in order to obtain reliable time differences from real data. We will show results from real data and also deal with problems that occur within them.
Matthias Gröne is now working for SMA Technologies, Kassel.


  Reconstruction of the Epicardial Potential from Body Surface Potential Maps
Patrick Verfürth, 2011 (Collaboration with Alexander Samol, UKM and Christian Vahlhaus, Klinikum Leer)

The standard 12-lead electrocardiogram only yields information about the electrical potential on the body surface. These potentials are generated by the electric currents along the heart surface and these currents determine the health status of the heart. Applying a larger number (120) of electrodes on the torso surface we record a Body Surface Potential Map (BSPM) and formulate the inverse problem of ECG, i.e. reconstruct the epicardial potential. This problem is severely ill-posed and regularization techniques are required. We applied different Tikhonov regularizations penalizing the derivations of the solutions. With the reconstructed electrical potential along the heart surface a better diagnosis might be possible in future.  [pdf-file]
Patrick Verfürth is now working for Mentz DV, Münster.


  Bildbasierte Lösung von Partiellen Differentialgleichungen mit Composite Finite Elements
Sebastian Westerheide, 2011 (Collaboration with Mario Ohlberger, WWU)

This thesis was concerned with the approximate solution of partial differential equations via discretization with composite finite elements. It deals with the application of this approach to problems where the computational domain is given through image data and thus possibly is of complex shape. We summarize the method's mathematical background, namely the Galerkin method and classical finite elements, and describe in detail the theory of composite finite elements as introduced by Hackbusch and Sauter. Based on simple elliptic model problems we further present an efficient strategy for calculating arbitrarily coarse approximations which nevertheless resolve the problem's computational domain and preserve characteristic properties of the exact solution. As part of the work, a program realizing the described method for Poisson-like problems in up to three dimensions was implemented in C++ using the DUNE framework. We use this program to practically verify the theoretical convergence properties of composite finite element discretizations and compare the potential of those discretizations with the potential of the classical finite element method.  [pdf-file]
Sebastian Westerheide is now a PhD Student at the Institute for Computational and Applied Mathematics, WWU Münster.


  Hierarchical Bayesian Approaches to the Inverse Problem of EEG/MEG Current Density Reconstruction
Felix Lucka, 2011 (Collaboration with Carsten Wolters, UKM)

This thesis was concerned with hierarchical Bayesian approaches to the inverse problem of EEG/MEG current density reconstruction. The estimation of the activity-related ion currents by measuring the induced electromagnetic fields outside the skull by means of current density reconstruction is both severely under-determined and ill-conditioned. A special class of statistical models, called hierarchical Bayesian models, were introduced. Different inference strategies based on these models were proposed and algorithmical aspects of the implementation and their practical use and properties within simulation studies were examined. The corresponding forward modeling was done with a realistic high resolution finite element (FE) model of a human head.  [pdf-file]
Felix Lucka is now a PhD Student at the Institute for Computational and Applied Mathematics and the Institute for Biomagnetism and Biosignal-Analysis, WWU Münster.


  Variationsmethoden in der Datenassimilation zur Wetter- und Preisvorhersage
Fabian Lenz, 2011 (Collaboration with Peter Markowich, Cambridge)

This thesis was concerned with an application of the 4DVAR model to weather prediction and price formation. In both situations we have an initial value problem where the initial condition has to be estimated using appropriate observations. The 4DVAR method reduces this task to a minimisation problem. This can be solved with the help of Gauss-Newton or regularisation methods. We have compared the numerical solutions obtained with this methods.  [pdf-file]


  Fokker-Planck Equations for Heterogeneous Trader Models
Kathrin Schulte, 2010

This thesis was concerned with the derivation, analysis, and numerical simulation of multi-dimensional Fokker-Planck equations arising from agent-based models of financial markets with heterogenous agents. The derivation of steady states and their existence and uniqueness is shown. The entropy approach yields convergence of the solution to the steady state. The positivity of the solution is shown for appropriate conditions. The derived systems in two and three dimensions are implemented with the Scharfetter-Gummel scheme and an adjusted Upwind scheme.  [pdf-file]


  Adjoint Methods for Hamilton-Jacobi Equations
Jan-Michael Schulte, 2010

This thesis was concerned with a novel approach for Hamilton-Jacobi-Bellman equations. These partial differential equations arise as a central aspect in optimal control theory. Furthermore, they play an important part in the recently introduced concept of mean field games, an approach for modelling the behavior of large numbers of interacting agents. We take advantage of this by deducing stable discretizations of Hamilton-Jacobi-Bellman equations and by applying the derived schemes to carry out mean field games simulations. As a last result, we show a new achievement in the theory of viscosity solutions and how these ideas may be generalized.  [pdf-file]
Jan Michael Schulte is now working for Oliver Wyman.


  Mean-Field and Kinetic Market Models
Veronika Penner, 2010

This thesis was concerned with the illustration and development of kinetic models, related to financial markets. Three models with particular attention to their long time behaviour were presented. The derivation of a novel model with an explicit analysis based on results from particle and statistical physics was the main purpose. In favour of a realistic illustration we considered homogeneous as well as heterogeneous conditions. In conclusion we yielded mean-field partial differential equations for the corresponding models.  [pdf-file]
Veronika Penner is now a PhD-Student at University of Kiel.


  Numerical Simulation of Water Transport into Porous Membranes
Thomas Grosser, 2010

In biophysics, one is often interested in the osmotic water flow through membranes, i.e., in investigating a membrane's osmotic permeability coefficient. As this coefficient cannot always be determined experimentally, a numerical simulation of this process is of interest. In this thesis, the process is modelled by partial differential equations which are then solved numerically. The solvers are implemented in C++ for the DOLFIN interface of FEniCS. .  [pdf-file]


  Rekonstruktionsbasierte Partialvolumenkorrektur in der Positronen-Emissions-Tomographie
Andre Gripshöfer, 2010 (Collaboration with: Klaus Schäfers, EIMI)

This thesis was concerned with the correction of the partial volume effect in positron emission tomography. For this purpose a new method was presented which corrects the partial volume effect directly in the reconstruction process. This method was tested on synthetically generated pet-images and the results were compared to those of standard correction methods which are applied to pet-images after reconstruction.  [pdf-file]


  NuBAkO - Entwicklung und Anwendung einer Toolbox zur Bewertung von Derivaten
Christoph Heuer, 2010 (Collaboration with: Nicole Branger, WWU)

This thesis was concerned with valuing European and American calls and puts. In order to calculate the prices of these options the Black-Scholes model, the Merton model, the Stochastic Volatility model, the SVJ model and the Duffie-Pan-Singleton model were derived. After the derivation of these models the resulting partial differential equations were discussed numerically and implemented in Matlab. The implementation also includes the imlied volatility with respect to the Black-Scholes model and the price with respect to moneyness.  [pdf-file]
Christof Heuer is now a PhD-Student at the Department of Mathematics, University of Sussex.


  Numerische Berechnung von Preis-Konsum Quotienten in verallgemeinerten Gleichgewichtsmodellen
Jann-Philipp Zocher, 2010 (Collaboration with: Nicole Branger, WWU)

Asset pricing models are an important part of today's world of finance. This thesis was concerned with the numerical approximation of the price- consumption ratio for a general equilibrium. The paper "Explaining pre- and post-1987 crash asset prices Within a unified general equilibrium framework" from Benzoni, Collin-Dufresne and Goldstein yields the basic model for this thesis. This financial market model contains the price-consumption ratio as a fundamental part of it. In particular the price-consumption ratio was analyzed with respect to the existence and uniqueness of an analytic solution as well as to the convergence of a numerical solution. The accuracy of the price-consumption ratio computation is improved by this thesis. Thereby the simulation results of the asset pricing model are more sophisticated.  [pdf-file]
Jann-Philipp Zocher is now working for Accenture.


  Fokker-Planck Equations for Agent-based Models of Financial Markets
Jörg Hagenberg, 2010

This thesis deals with an alternative method to analyze a financial market model. Instead of exploiting a time series, the Fokker-Planck equation is derived from the agent based model of Lux, Marchesi and Sheng. Some of its properties as the existence of a stationary solution, the large time behavior and the possibility of "fat tails'' are proven. Then this equation is compared with the Fokker-Planck equation obtained from the GARCH(1,1) model and a second Fokker-Planck equation that describes the dynamics of the price.  [pdf-file]


  Mathematische Modelle der Meinungsbildung
Lisa Richter, 2010

This thesis was concerned with the presentation and comparison of several models of opinion formation and some of their extensions. In particular the behaviour and characteristics of the popular models of Sznajd-Weron, Hegselmann-Krause and Deffuant-Weisbuch were considered. For some models the processes of opinion formation were described by partial differential equations. Especially for the Deffuant-Weisbuch-Model the dynamics for the time variations of opinions' density were derived and analyzed by computing stationary solutions. Finally the Deffuant-Weisbuch-Model was simulated numerically.  [pdf-file]
Lisa Richter is now working for Alte Leipziger Lebensversicherung.


  Mathematische Methoden zur QRS-Detektion in EKG-BSPM Signalen
Jan Bäumker, 2010 (Collaboration with: Frank Wübbeling, WWU, Alex Samol, UKM, Christian Vahlhaus, Klinikum Leer)

This thesis describes a variety of algorithms for automatical QRS-detection in ECG/BSPM-Signals. In it we give a overview of the heart and its structure, and how signalanalysis is used to work with ECG-signals. We explain what types of algorithms exist, how they work, and after that we give an in-depth explanation of two chosen algorithms. One of the algorithms, the Pan/Tompkins-algorithm, we code in a new version in MATLAB.  [pdf-file]


  Mass-preserving Registration of Medical Images
Lars Ruthotto, 2010 (Collaboration with: Carsten Wolters, WWU)

Image Registration is one of today's most challenging tasks in medical imaging. This thesis develops and investigates tailored registration approaches for two important problems in real life medicine demanding mass-preserving transformations. A variational approach to susceptibility correction of echo planar images (EPI) is proposed, validated on phantom data and applied to functional magnetic resonance images (fMRI) and diffusion tensor images (DTI). Secondly, gated positron emission tomography (PET) images are corrected for respiratory motion using a novel non linear mass preserving registration algorithm.  [pdf-file]
Lars Ruthotto is now a PhD-Student at MIC, University of Lübeck and the Institute for Computational and Applied Mathematics, WWU Münster.


  Opinion Dynamcis with Heterogeneous Agents
Ina Josek, 2009

This thesis investigated the modeling of dynamics of opinion formation between heterogeneous agents. The classical Deffuant-Weisbuch-Model was modified and extended to consider both the homogeneous and heterogeneous case. The development of opinions in time was characterized each with partial differential equations. In particular the heterogeneous model was analyzed with respect to the existence and uniqueness of solutions, the occurrence of stationary states and their stability. Furthermore numerical simulations of the model were carried out.  [pdf-file]
Ina Josek is now working for R+V Versicherung.


  SEGMEDIX: Development and Application of a Medical Image Segmentation Framework
Daniel Wirtz, 2009 (Collaboration with: Xiaoyi Jiang, WWU)

One recent aim of the Imaging Workgroup is the processing of medical MR/CT images in combination with ECG data to support noninvasive medical analysis and thus treatment of patients. As a first step this thesis is concerned with the development of a segmentation framework containing a collection of both standard and modern segmentation algorithms. Moreover, suitable pre- and post processing algorithms complete the MATLAB based program to a fully self-contained segmentation framework. In the second part the framework is applied to the MR patient data obtained through the current cooperation with several clinics and successful results are presented along with some experiments.  [pdf-file]
Daniel Wirtz is now a PhD-Student at University of Stuttgart.


  Spatial Structures in Geographical Economics: Mathematical Modelling, Simulation, and Inverse Problems
Ralf Engbers, 2009 (Collaboration with: Vincenzo Capasso, Milano)

This thesis was concerned with an extension of the Solow model which is used to describe economic growth. A spatial dimension was added to the model and a convex-concave production function was used instead of a strictly concave production function. The resulting direct problem was solved numerically and the parameter identification problem was solved via the adjoint method with exact and noisy data.  [pdf-file]
Ralf Engbers is now a PhD-Student at the Institute for Computational and Applied Mathematics, WWU Münster.


  Asset Allocation with Multiple State Variables
Tobias Neugebauer, 2009 (Collaboration with: Nicole Branger, WWU)

This thesis was concerned with the analysis and numerical solution of asset pricing problems with multiple underlyings, which are formulated as Hamilton-Jacobi-Bellman equations. The results are applied to an asset allocation problem with stochastic interest rate and income.  [pdf-file]


  Kollisionswahrscheinlichkeiten aus monokularen Bildfolgen
Jan Jatzkowski, 2009 (Collaboration with Bosch GmbH)

This thesis was concerned with the computation of a collision probability from motion scenes acquired with single cameras, for the potential use in a driving assistance system. The method presented combines the optical flow and the result of a segmentation to give the user an qualitative estimation of the risk, that an object will collide with the ego vehicle (i.e. the vehicle the user is driving). The method was tested by sequences of both contrived and real traffic scenes.


  An Adaptive Hybrid Method for 2D Crack Growth Simulation
Jan Hegemann, 2009 (Collaboration with: Joseph Teran, UCLA)

This thesis was concerned with the simulation of two dimensional quasi-static crack growth. The method presented combines the extended finite element method (XFEM) with a general cutting algorithm that provides the degrees of freedom for the crack to open as well as determines material connectivity. Also, a general and easy quadrature scheme was introduced, which uses an adaptively refined integration mesh embedded into a coarse simulation mesh. The approach is applied to several test settings, which highlight varying considerations, including accuracy, crack propagation, and the ability of the method to handle complicated crack geometries.  [pdf-file]
Jan Hegemann is now a PhD-Student at the Department of Mathemtics, UCLA,and the Institute for Computational and Applied Mathematics, WWU Münster.


  A Variational Approach for Sharpening High-Dimensional Images
Michael Möller, 2009 (Collaboration with: Andrea Bertozzi, Todd Whitman, UCLA)

This thesis was concerned with the pan-sharpening of high-dimensional and hyperspectral images by variational techniques. A novel variational model including wavelet based energies as well as an edge alignemnt term based on total variation is analyzed and solved numerically. The approach is applied to satellite images.  [pdf-file]
Michael Möller is now a PhD-Student at the Institute for Computational and Applied Mathematics, WWU Münster and UCLA.


  Influence of Volume Conduction on Beamformer Source Analysis in the Human Brain
Stephanie Sillekens, 2008 (Collaboration with: Carsten Wolters, Olaf Steinsträter, Markus Junghöfer, UKM)

This thesis was concerned with the investigation of beamforming in EEG/MEG source analysis. Beamforming is a variance-based technique for source localization, particular emphasis was laid on synthetic aperture magnetometry (SAM) here. The beamforming was applied together with different forward models at three different resolutions (sphere model, realistic geometry with isotropic conductivities, realistic geometry with anisotropic conductivities). In a series of tests the quality of beamforming reconstructions with the different models is tested in order to evaluate the potential need for realistic head models in certain source reconstruction tasks. [pdf-file]
Stefanie Sillekens is now working for ALDI-Sued.


  Long-Time Behaviour of Nonlinear Fokker-Planck Equations
Jan-Frederik Pietschmann, 2008 (Collaboration with: Peter Markowich, DAMTP, Cambridge University)

This thesis deals with the convergence of solution of a system of non-linear Fokker-Planck equations to a equilibrium or stationary state. To explore the behaviour of these equations, two techniques are used. The first one is the so-called entropy dissipation method. Here, an entropy functional is definied and then used to control the distance between a solution at time t and the steady state. The second aprroach is to define a manifold such that the Fokker-Planck equation can be understood as a gradient flow on this manifold. If it is furthermore possible to show that the Energy functional is convex, a contraction principle follows. The proper sense of convexity here is the so-called displacement convexity which reveals to connection to the Monge-Kantorovich problem of optimal transport.  [pdf-file]
Jan-Frederik Pietschmann is now a PhD-student at DAMTP, Cambridge University, and member of Trinity College.


  Parallel Total Variation Minimization
Jahn Müller, 2008 (Collaboration with: Sergej Gorlatch, Maraike Schellmann, Institut für Informatik, WWU Münster)

This thesis was developing a parallel algorithm for the solution of total variation denoising methods, to be used for high-resolution images. For this sake domain decomposition was applied to a primal-dual discretization of the variational problem. Optimal convergence behaviour was obtained by using a single primal-dual Newton step in each iteration of the domain decomposition. The scheme has been implemented using MPI and extensively tested on an HPC cluster. [pdf-file]
Jahn Müller is now a PhD-student at the Institute for Computational and Applied Mathematics and the European Institute for Molecular Imaging, WWU Münster.


  Valuation and Hedging of Carbon Derivatives
Matthias Tillmann, 2008 (Collaboration with: Magnus Wobben, , Institut für VWL, WWU Münster)

This thesis was concerned with options on carbon certificates, which receive growing interest due to climate preservation policies. The pricing is discussed based on jump-diffusion models with stochastic volatility. For several products explicit formulas can be derived, while American options are solved numerically using finite difference methods and appropriate iterative schemes. Calibration and application to real data is discussed, as well as dynamic hedging including transaction costs.  [pdf-file]
Matthias Tillmann is now working for Deutsche Bank AG.


  Early-Exercise Kontrakte in der Stromwirtschaft - Alternative Vermarktung von Erzeugungspotentialen
Arvind Sarin, 2008 (Collaboration with: Magnus Wobben, Institut für VWL, WWU Münster)

This thesis was concerned with the pricing of American options on power markets, hence extending the results of the thesis of Oleg Reichmann. The free boundary problem for the pricing PIDE was analyzed using approximation techniques and semigroup theory. After discretization by finite element methods, an iterative scheme based on a semismooth Newton method was implemented and analyzed also with respect to global convergence. In addition calibration and application to real data is discussed.  [pdf-file]
Arvind Sarin is now working for KPMG, Frankfurt.


  Fokker-Planck Equations for Agent-based Models of Financial Markets
Katharina Daniel, 2008

This thesis was concerned with the derivation, analysis, and numerical simulation of multi-dimensional Fokker-Planck equations arising from certain agent-based models of financial markets. The equations are derived by suitable asymptotics from the master equation and lead to a challenging structure with degenerate diffusivity and unbounded potential. Due to these issues standard results cannot be applied directly, and adaptations are carried out here in order to show existence, uniqueness and in order to analyze large-time behavior a relative entropy approach is carried out, and novel Hardy-Sobolev inequalities are derived. Moreover, a novel adjoint approach for computing autocorrelation functions in time is introduced, which allows to realize this task by solving a single Fokker-Planck equation with special initial value. For the robust numerical solution an implicit scheme based on operator splitting and Scharfetter-Gummel discretizations in time is introduced.  [pdf-file]


  Bewertung von Stromoptionen in Modellen mit Sprüngen (Valuation of Power Options in Models including Jumps)
Oleg Reichmann, 2008 (Collaboration with: Magnus Wobben, , Institut für VWL, WWU Münster)

This thesis was concerned with the modelling, numerical simulation, and calibration of European options on energy markets, using special characteristics of the latter. Option pricing is investigated using an underlying model based on Ohrnstein-Uhlenbeck and jump processes. The resulting PIDE is analyzed and solved numerically. Calibration to marked data is performed by solving a PIDE-constraint optimization problem. As an alternative option pricing based on FPNIG-processes is discussed, and the results of both approaches are compared.  [pdf-file]
Oleg Reichmann is now a PhD student at ETH Zürich.


  Identification of Cardiac Infarctions from ECG Measurements
Melanie Schröter 2008 (Collaboration with Bjorn Nielsen, SIMULA Research Labs, Oslo)

This thesis was concerned with the identification of infarcted region in the human heart from high-level ECG data. As a model for the electrical activity of the heart the bidomain equations have been used, where infarctions or ischemia can be modeled via unknown distributed parameters or subregions. The identification of these unknowns from ECG data was treated as a nonlinear inverse problem, which was analyzed in detail. An adjoint-based method for Tikhonov functionals was developed, which includes the first sensitivity analysis of the cardiac bidomain model. The Tikhonov regularízation has been adapted to a-priori knowledge about infarcted regions via a nonconvex term. Numerical implementations have been discussed as well as the asymptotic determination of small infarctions via topological derivatives.  [pdf-file]
Melanie Schröter is now working for LBS, Münster,


  A Nonlinear Variational Method for Improved Quantification of Myocardial Blood Flow using dynamic H215O PET
Martin Benning, 2008 (Collaboration with Klaus Schäfers, UKM & EIMI)

This thesis was concerned with the quantitative imaging of physiological parameters in the heart such as perfusion from PET data collected with radioactive water. The parameters are linked to the image intensity via nonlinear differential equations, hence the reconstruction from PET data becomes a nonlinear inverse problem, which is treated by nonlinear variational methods. The numerical solution is performed by a two-step method alternating EM iterations with the (approximate) solution of nonlinear parameter identification problems.  [pdf-file]
Martin Benning is now a PhD-student at the Institute for Computational and Applied Mathematics, WWU Münster.


  Numerical Methods for Equilibrium Problems in General Markets
Anna Weisweiler, 2008 (Collaboration with Nicole Branger, BWL)

This thesis was concerned with equilibria in general markets with heterogeneous agents or firms. The heterogeneity leads to coupled systems of partial differential equations of reaction-diffussion-convection type, usually with diffusion coefficients degenerating at the boundary. Such problems have been analyzed using fixed-point arguments and maximum principles. For the numerical solution robust iterative schemes have been derived and implemented. The approach is applied favourably to two particular market cases.  [pdf-file]
Anna Weisweiler is now working for GAD eg, Münster.


  Mathematical Models for Pedrestian Motion
Bärbel Schlake, 2008

This thesis was concerned with an investigation of models for the movement of pedestrians, in particular in evacuation situations, which have been proposed in the last years. A literature review has been given, writing the models into a unified mathematical form. Stability analyses for the most important models have been carried out as well as numerical simulations and some modifications. Moreover, continuum models have been derived in crowded situations, via an approach based on a BBGKY-hierarchy. Due to the crowding molecular chaos does not hold and suitable correlations between pedestrian positions needed to be introduced.  [pdf-file]
Bärbel Schlake is now a PhD-student at the Institute for Computational and Applied Mathematics, WWU Münster.


  Numerical simulation of oscillatory zoning
Tanja Mues, 2008 (Collaboration with Andreas Heuer, Institute for Physical Chemistry)

This thesis considered a model for oscillatory zoning, a particular case of crystal growth, based on partial diffferential equations and its extension to multiple spatial dimensions. Numerical algorithms for the simulation of the model have been developed and linear stability analyses have been carried out in various cases. Diffusion regularizations curing potential instabilities appearing in the original formulation of the model have been incorporated and investigated.  [pdf-file]
Tanja Mues is now a PhD-Student in the group of Andreas Heuer, Institute for Physical Chemistry, WWU Münster..


Когда вы отдали ей кольцо. Росио пожала плечами. - Сегодня днем. Примерно через час после того, как его получила.

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