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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 ...
Preprint
Identifying critical demand scenarios for the robust capacitated network design problem using principal component analysis
(2021-11-30)
In this paper, we consider the single-commodity robust network design problem. Given an undirected graph with capacity installation costs on its edges and a set S of scenarios with associated flow balance vectors that represent different scenarios of node supplies and demands, the goal is to find integer edge capacities that minimize the total installation cost and permit a feasible single commodity flow for each scenario. This problem arises, for example, in the design of power networks, which are dimensioned to ...
Dissertation
Explicit Description Of Isogeny And Isomorphism Classes Of Drinfeld Modules Of Higher Rank Over Finite Fields
(2020)
When jumping from the number fields theory to the function fields theory, one cannot miss the deep analogy between rank 1 Drinfeld modules and the group of root of unity and the analogy between rank 2 Drinfeld modules and elliptic curves. But so far, there is no known structure in number fields theory that is analogous to the Drinfeld modules of higher rank r ≥ 3. In this thesis we investigate the classes of those Drinfeld modules of higher rank r ≥ 3 defined over a finite field L. We describe explicitly the Weil ...
Dissertation
Analysis of a Coupled Fluid-Elastic Interaction Problem
(2023-01)
In this thesis, a non-linear system of partial differential equations is studied, describing the motions of an elastic structure which is immersed into an incompressible viscous fluid. The displacement of the elastic structure is modelled by a Lamé system and the fluid velocity as well as the fluid pressure are described by the Navier-Stokes equations. The structure and the fluid are coupled via two boundary conditions at the interface which correspond to continuity of velocities and forces. As the elasticity is ...
Working paper
Nominal Science without Data
(2022-09-28)
Expanding upon literature on early digital computers, this paper shows the role mathematicians have undertaken in founding the academic fields of Game Theory and Operations Research, and details how they were supported by the mathematics departments of military agencies in branches of the US Armed Services. This paper claims that application is only decoration. Other than astronomy, physics and engineering, where experiments generate data analysed with the aid of models and appropriate software on computers, Game ...
Dissertation
Finite Element Simulations for the Design of Therapeutic Approaches for Retinal Diseases
(2022)
The retinal disease age-related macular degeneration is the most common cause of vision loss in industrialized countries. In this thesis, motivated by the drug (antibody) treatment of this disease, we designed long-term three dimensional Finite Element simulations of the drug distribution in the healthy human eye. The underlying model consists of a time-dependent convection-diffusion equation coupled to a stationary Darcy equation describing the flow of the aqueous humor through the vitreous medium. We replaced the ...
Habilitation
Computing Ground States for Fermi-Bose Mixtures through Efficient Numerical Methods
(2023-05)
In this work, we will first review the Quantum Mechanics theory to derive the main equations. Next, we will analyze these equations by Functional Analysis methods to find conditions for existence, uniqueness, multiplicity, and other properties as positivity. Next, we will review and develop some numerical methods for solving the nonlinear Schrödinger equation, its time version, generalizations with rotational terms, and systems of NLSE (NLSS). We notice that the main problem to run numerical methods is the memory ...
Dissertation
Generalized Involutive Bases and Their Induced Free Resolutions
(2022-05)
In this thesis, we generalize several types of involutive and marked bases for ideals in quotient rings of commutative polynomial rings. We apply these new types of bases to the analysis of infinite free resolutions and of Hilbert schemes defined over certain types of quotient rings. We are mostly concerned with Pommaret and Janet bases; the marked bases we consider are marked over monomial submodules that are quasi-stable, i.e., that possess finite Pommaret bases.
Involutive bases of the types we consider induce ...
Dissertation
Symmetrien von Differentialgleichungen via Vessiot-Theorie
(2021-04)
Die übliche Definition des Symmetriebegriffs einer Differentialgleichung lautet wie folgt: Symmetrien sind Transformationen, die Lösungen wieder in Lösungen überführen. Modelliert man Differentialgleichungen als Untermannigfaltigkeiten eines Jetbündels, so lassen sich zwei Arten von Symmetrien unterscheiden: innere und äußere. Der erste Fall entspricht einer Transformation, die ausschließlich auf der Differentialgleichung definiert ist. Im zweiten Fall ist die betrachtete Transformation auf dem gesamten umgebenden ...
Konferenzveröffentlichung
Propagation and branching strategies for job shop scheduling minimizing the weighted energy consumption
(2023)
We consider a job shop scheduling problem with time windows, flexible energy prices, and machines whose energy consumption depends on their operational state (offline, ramp-up, setup, processing, standby or ramp-down). The goal is to find a valid schedule that minimizes the overall energy cost. To solve this problem to optimality, we developed a branch-and-bound algorithm based on a time-indexed integer linear programming (ILP) formulation, which uses binary variables that describe blocks spanning multiple inactive ...