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Dissertation
Automatic computation of continued fraction representations as solutions of explicit differential equations
(2019)
The main focus of this thesis is to present a variation of an algorithm first presented by Maulat and Salvy, with which it is possible to algorithmically guess as well as prove continued fraction expansions of analytical expressions with the help of ordinary differential equations.
Dissertation
Aspekte der linearen Minimax-Schätzung
(FB 17, Mathematik/Informatik, Angewandte Mathematik, Analysis und Angewandte Mathematik, 2004-07-14)
Es werde das lineare Regressionsmodell y = X b + e mit den ueblichen Bedingungen betrachtet. Weiter werde angenommen, dass der Parametervektor aus einem Ellipsoid stammt. Ein optimaler Schaetzer fuer den Parametervektor ist durch den Minimax-Schaetzer gegeben. Nach der entscheidungstheoretischen Formulierung des Minimax-Schaetzproblems werden mit dem Bayesschen Ansatz, Spektralen Methoden und der Darstellung von Hoffmann und Laeuter Wege zur Bestimmung des Minimax- Schaetzers dargestellt und in Beziehung gebracht. ...
Dissertation
Development of a Preconditioning Scheme for Real Gases using Asymptotic Expansions
(2022)
Bei der Beschreibung von Strömungen wird klassischerweise zwischen inkompressiblen und kompressiblen Bereichen unterschieden. Während inkompressible Strömungen durch ein divergenzfreies Geschwindigkeitsfeld charakterisiert werden, sind kompressible Strömungsfelder durch Expansionsfächer, Kontaktunstetigkeiten und Stoßwellen gekennzeichnet. Die beiden Bereiche werden damit durch stark unterschiedliche Systeme partieller Differentialgleichungen beschrieben. Diese Unterscheidung zeigt sich auch in der numerischen ...
Dissertation
Optimal control of a rate-independent system constrained to parameterized balanced viscosity solutions
(2022)
In this dissertation, we analyze an optimal control problem governed by a rate-independent system in an abstract infinite-dimensional setting. The rate-independent system is characterized by a nonconvex stored energy functional, which depends on time via a time-dependent external loading, and by a convex dissipation potential, which is assumed to be bounded and positively homogeneous of degree one.
The optimal control problem uses the external load as control variable and is constrained to normalized ...
Dissertation
Beliefs von Lehramtsstudierenden zur doppelten Diskontinuität
(2021-09)
Das Ziel der Forschungsarbeit ist die Untersuchung von Überzeugungen von Lehramtsstudierenden zur sogenannten doppelten Diskontinuität. Genauer geht es um die Beforschung von Überzeugungen zur Kohärenz zwischen Schulmathematik und Hochschulmathematik und die Überzeugungen zur Relevanz der universitären Mathematik für die spätere Tätigkeit als Lehrkraft in der Schule. Dabei soll erstens der Frage nachgegangen werden, welche Überzeugungen Lehramtsstudierende zur doppelten Diskontinuität haben und zweitens eine Antwort ...
Dissertation
Modeling of human vitreous as viscoelastic fluid considering the orientation of collagen fibers
(2021-11)
For the most common treatment of retinal diseases worldwide by drug distribution in the human vitreous we developed the mathematical model of the vitreous. Compare to previous works we focus on the vitreous as a viscoelastic fluid including its heterogeneous property due to the orientation of collagen fibers. By using the incompressible viscoelastic Burgers-type model based on experimental data as the specific constitutive equation in the setting of continuum mechanics we considered its non-Newtonian nature. This ...
Dissertation
Algorithmic Reduction of Biochemical Reaction Networks
(2022-02-25)
The dynamics of species concentrations of chemical reaction networks are given by autonomous first-order ordinary differential equations. Singular perturbation methods allow the computation of approximate reduced systems that make explicit several time scales with corresponding invariant manifolds. This thesis presents:
1. An algorithmic approach for the computation of such reductions on solid analytical grounds. Required scalings are derived using tropical geometry. The existence of invariant manifolds is subject ...
Dissertation
Free Resolutions from Involutive Bases
(2016-11-02)
We show that the theory of involutive bases can be combined with discrete algebraic Morse Theory. For a graded k[x0 ...,xn]-module M, this yields a free resolution G, which in general is not minimal. We see that G is isomorphic to the resolution induced by an involutive basis. It is possible to identify involutive bases inside the resolution G. The shape of G is given by a concrete description. Regarding the differential dG, several rules are established for its computation, which are based on the fact that in the ...
Dissertation
Mathematical Modelling and Adaptive Finite Element Simulation of Viscoelastic Fluid-Structure Interaction Systems and Chemical Processes with Applications to Ophthalmology
(2023)
The aim of this thesis is the numerical analysis of nonlinear coupled partial differential equations and their application to ophthalmology. Firstly, we consider fluid-structure interaction problems where the fluid is either Newtonian or viscoelastic. The structure is modelled as a hyperelastic material. The application to ophthalmology lies in the interaction of the vitreous with its surrounding elastic structures like the sclera and the lens. The underlying flow in the vitreous is modelled by a viscoelastic Burgers ...