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    Longitudinal dynamics of semiconductor lasers
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2001) Philip, Jan
    We investigate the longitudinal dynamics of semiconductor lasers using a model which couples a hyperbolic linear system of partial differential equations nonlinearly with ordinary differential equations. We prove the global existence and uniqueness of solutions using the theory of strongly continuous semigroups. Subsequently, we analyse the long-time behavior of the solutions in two steps. First, we find attracting invariant manifolds of low dimension benefitting from the fact that the system is singularly perturbed, i. e., the optical and the electronic variables operate on differente time-scales. The flow on these manifolds can be approximated by the so-called mode approximations. The dimension of these mode approximations depends on the number of critical eigenvalues of the linear hyperbolic operator. Next, we perform a detailed numerical and analytic bifurcation analysis for the two most common constellations. Starting from known results for the single-mode approximation, we investigate the two-mode approximation in the special case of a rapidly rotating phase difference between the two optical components. In this case, the first-order averaged model unveils the mechanisms for various phenomena observed in simulations of the complete system. Moreover, it predicts the existence of a more complex spatio-temporal behavior. In the scope of the averaged model, this is a bursting regime.
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    Classification and clustering: models, software and applications
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Mucha, Hans-Joachim; Ritter, Gunter
    We are pleased to present the report on the 30th Fall Meeting of the working group ``Data Analysis and Numerical Classification'' (AG-DANK) of the German Classification Society. The meeting took place at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, from Friday Nov. 14 till Saturday Nov. 15, 2008. Already 12 years ago, WIAS had hosted a traditional Fall Meeting with special focus on classification and multivariate graphics (Mucha and Bock, 1996). This time, the special topics were stability of clustering and classification, mixture decomposition, visualization, and statistical software.
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    Algebraische Modellierung von Tonsystemen : Musiktheorie mit mathematischen Mitteln
    (Mühltal : Verl. Allgemeine Wissenschaft, 2009) Winkler, Jan Thomas
    Das vorliegende Buch stellt Möglichkeiten einer algebraischen Modellierung von Tonsystemen vor. Dazu wird zunächst die von Rudolf Wille und Wilfried Neumaier formulierte extensionale Standardsprache der Musiktheorie aufbereitet und ein breites Spektrum musiktheoretischer Begriffe mathematisch gefasst. Diese Begriffe werden in ihrem Bezug zu Tonsystemen dargestellt und ausführlich analysiert. Zum Abschluss des Buches wird eine Interpretation von Tonsystemen in der Kategorientheorie formuliert und eine einfache Möglichkeit vorgestellt, strukturerhaltende Abbildungen zwischen Tonsystem-Kategorien zu konstruieren.
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    Contributions to continuum theories : anniversary volume for Krzysztof Wilma´nski
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2000) Albers, Bettina; Wilmanski, Krzysztof
    [no abstract available]
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    Reaktions-Diffusionsgleichungen in Heterostrukturen mit Anwendungen in der Halbleitertechnologie : Schlußbericht zu einem Vorhaben im BMBF-Förderprogramm Anwendungsorientierte Verbundvorhaben auf dem Gebiet der Mathematik
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 1997) Hünlich, Rolf; Glitzky, Annegret; Röpke, Wilfried
    Im Vorhaben wurden Beitraege zur Modellierung und Simulation relevanter Teilprozesse bei der Herstellung von Halbleiterbauelementen der Nanoelektronik geleistet. Behandelt wurden vorrangig Fragestellungen,die beim Verbundpartner, dem Institut fueur Halbleiterphysik Frankfurt (Oder), zur Entwicklung von SiGe--Heterojunction--Bipolartransistoren von Bedeutung waren. Schwerpunkte bildeten Fragen zur Diffusion von Fremdatomen in verspannten SiGe--Schichten sowie zu Feldeffekten bei der Diffusion elektrisch geladener Teilchen im Hochkonzentrationsfall. Gegenstand der analytischen und numerischen Untersuchungen waren verschiedene Klassen von Elektro-Reaktions-Diffusionsgleichungen in Heterostrukturen, die relevante Aufgaben aus der Halbleitertechnologie auf verschiedenen Niveaus modellieren. Hier wurden neue Aussagen zur globalen Existenz, Einzigkeit und zum asymptotischen Verhalten der Loesungen erhalten.Weiterhin wurden Diskretisierungsschemata, die Konvergenz von Naeherungsverfahren sowie die Reduktion der Modellgleichungen fuer singulaer gestoerte Faelle diskutiert.
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    Geometry of heteroclinic cascades in scalar parabolic differential equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 1998) Wolfrum, Matthias
    We investigate the geometrical properties of the attractor for semilinear scalar parabolic PDEs on a bounded interval with Neumann boundary conditions. Using the nodal properties of the stationary solutions which are determined by an ordinary boundary value problem, we obtain crucial information about the long-time behavior for the full PDE. Especially, we prove a criterion for the intersection of strong- stable and unstable manifolds in the finite dimensional Morse-Smale flow on the attractor.
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    Electro-reaction-diffusion systems in heterostructures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2000) Glitzky, Annegret; Hünlich, Rolf
    The paper is devoted to the mathematical investigation of a general class of electro-reaction-diffusion systems with nonsmooth data which arises in applications to semiconductor technology. Besides of a basic problem, a reduced problem is considered which is obtained if the kinetics of the free carriers is fast. For two dimensional domains we prove a global existence and uniqueness result. In addition, asymptotic properties of solutions are studied. Basic ideas are energy estimates, Moser iteration, regularization techniques and an existence result for electro-diffusion systems with weakly nonlinear volume and boundary source terms which is proved in the paper, too. The relationship between the property that the energy functional decays exponentially in time to its equilibrium value and the existence of global positive lower bounds for the densities of the species is investigated. We illustrate relations between the model and its reduced version in general and for concrete examples. Finally, we discuss the special features of heterostructures for simplified model problems.
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    Simulation of pulse propagation in nonlinear optical fibers
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2003) Bandelow, Uwe; Demircan, Ayhan; Kesting, Martin
    We solve numerically a generalized nonlinear Schroedinger equation by using a pseudospectral method. Integration is performed by using an eight-order Runge-Kutta scheme. The numerical method therefore differs from the commonly used split-step method. Effects such as the impact of group velocity dispersion (GVD) up to fourth-order dispersion, self phase modulation (SPM), self-steepening and intrapulse Raman scattering can be investigated with the code. Examples for the above effects are demonstrated, as well as their interplay in the context of soliton propagation and sub-picosecond pulses.
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    Transient numerical simulation of sublimation growth of SiC bulk single crystals : modeling, finite volume method, results
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2003) Philip, Peter
    This work treats transient numerical simulation of growth of silicon carbide (SiC) bulk single crystals by physical vapor transport (also called the modified Lely method). A transient mathematical model of the growth process is presented. Subsequently, the finite volume method for the discretization of evolution equations, which constitutes the basis for the numerical simulations presented in this work, is studied mathematically, proving the existence of discrete solutions. All material data used for numerical simulations in this work are collected in the appendix. Starting with a description of the physical growth procedure, problems arising during the growth process are discussed as well as techniques that are used for process control. It is explained why numerical simulation is an important tool for control, and the advantages of a transient approach are considered. Within the presented transient model, continuous mixture theory is used to obtain balance equations for energy, mass, and momentum inside the gas phase. In particular, reaction-diffusion equations are deduced. Heat conduction is treated inside solid materials. Heat transport by radiation is modeled via the net radiation method for diffuse-gray radiation to allow for radiative heat transfer between the surfaces of cavities. The model includes the semi-transparency of the single crystal via a band approximation. Induction heating is modeled by an axisymmetric complex-valued magnetic scalar potential that is determined as the solution of an elliptic problem. The resulting heat source distribution is calculated from the magnetic potential. The heat sources are updated continuously during the solution of the transient problem for the temperature evolution to allow for changes in the electrical conductivity depending on temperature and for changes due to a moving induction coil. The finite volume method is treated in a rigorous mathematical framework. It allows the discretization of parabolic, hyperbolic, and elliptic partial differential equations, as they arise from the mathematical model of the growth process, including nonlocal contributions due to radiative heat transfer. The general abstract setting consists of a system of nonlinear evolution equations in arbitrary finite space dimension, each evolution equation living on a different polytope domain. In general, each evolution equation has diffusive and convective contributions as well as source and sink terms. Each contribution is permitted to depend on the solution. Discontinuities of the solution are allowed at domain interfaces. Interface conditions in terms of the solution and its flux are considered. Moreover, nonlocal interface conditions are considered. Outer boundary conditions include Dirichlet conditions, flux conditions, emission conditions, and nonlocal conditions. Time discretization is performed by an implicit Euler scheme, where an explicit discretization is allowed in certain dependencies such that the temperature-dependent emissivities can be taken from the previous time step. As usual, the space discretization is performed by integrating the evolution equations over control volumes and then using quadrature formulas. As an axisymmetric setting and cylindrical coordinates are used in the simulations, a treatment of change of variables is included in the abstract considerations. For the case that the evolution equations constitute nonlinear heat equations, still allowing nonlinear diffusion, convection, and source and sink terms, as well as nonlocal interface and boundary conditions as they arise from modeling radiative heat transfer, discrete L∞ - L1 a priori estimates are established for the system resulting from the finite volume discretization. A fixed point argument is then used to prove the existence and uniqueness of discrete solutions. The presented numerical simulations are conducted in an axisymmetric setting. They constitute transient investigations of control parameters affecting the temperature evolution during the heating of the growth apparatus. A cylindrically symmetric finite volume scheme provides the discretization for both the transient nonlinear heat problem and the stationary magnetic potential problem. For different heating powers and different vertical coil positions, the temperature evolution is monitored at the surface of the crystal and at the surface of the source powder as well as at the top and at the bottom of the growth apparatus. It is studied how the temperature difference between source and seed, which is highly relevant to the growth process, is related to the measurable temperature difference between bottom and top. Results concerning the time lack between the heating of the surface of the source powder and the heating of its interior are considered. Finally, the global evolution of temperature and heat sources is investigated.
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    MAGNUS - mehrstufige Analyse grosser Netzwerke und Systeme
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 1994) Borchardt, Jürgen; Grund, Friedrich; Horn, Dietmar; Uhle, Manfred
    [no abstract available]