Inst. Mathematik

339 Items

Recent Submissions
Points, lines, and circles:

Scheucher, Manfred (2020)

In this dissertation we investigate some problems from the field of combinatorics and computational geometry which involve basic geometric entities (points, lines, and circles). In the first part we look at Erdös-Szekeres type problems: The classical theorem by Erdös and Szekeres from 1935 asserts that, for every natural number 𝑘, every sufficiently large point set in general position contai...

Error analysis and model adaptivity for flows in gas networks

Stolwijk, Jeroen Johannes ; Mehrmann, Volker (2018-11-22)

In the simulation and optimization of natural gas flow in a pipeline network, a hierarchy of models is used that employs different formulations of the Euler equations. While the optimization is performed on piecewise linear models, the flow simulation is based on the one to three dimensional Euler equations including the temperature distributions. To decide which model class in the hierarchy is...

Condition and homology in semialgebraic geometry

Tonelli Cueto, Josué (2019)

The computation of the homology groups of semialgebraic sets (given by Boolean formulas) remains one of the open challenges of computational semialgebraic geometry. Despite the search for an algorithm taking singly exponential time only on the number of variables, as of today, the existing algorithms are symbolic and doubly exponential. In this PhD thesis, we show how to obtain a numerical algo...

Derivation, asymptotic analysis and numerical solution of atomistically consistent phase-field models

Bergmann, Sibylle (2019)

In this thesis a systematic derivation and analysis of phase-field models with parameters based on molecular-dynamical simulations is developed. Applications of our models describe the anisotropic growth of solid-liquid interfaces during crystallization from a melt. We focus here on silicon, while other materials with different anisotropies and material parameters are also possible. We combine ...

Pesin's formula for translation invariant random dynamical systems

Senin, Vitalii (2019)

Pesin's formula asserts that metric entropy of a dynamical system is equal to the sum of its positive Lyapunov exponents, where metric entropy measures the chaoticity of the system, whereas Lyapunov exponents measure the asymptotic exponential rate of separation of nearby trajectories. It is well known, that this formula holds for dynamical systems on a compact Riemannian manifold with an invar...

Mathematical analysis of large-scale biological neural networks with delay

Mehri, Sima (2019)

It is well-known that the components of the solution to a system of N interacting stochastic differential equations with an averaged sum of interaction terms and with independent identically distributed (chaotic) initial values, as N tends to infinity, converge to the solutions of Vlasov-McKean equations, in which the averaged sum is replaced by the expectation. Since the solutions to the corre...

Methods for the temporal approximation of nonlinear, nonautonomous evolution equations

Eisenmann, Monika (2019)

Differential equations are an important building block for modeling processes in physics, biology, and social sciences. Usually, their exact solution is not known explicitly though. Therefore, numerical schemes to approximate the solution are of great importance. In this thesis, we consider the temporal approximation of nonlinear, nonautonomous evolution equations on a finite time horizon. We p...

Multicriteria linear optimisation with applications in sustainable manufacturing

Schenker, Sebastian (2019)

Multicriteria optimisation is concerned with optimising several objectives simultaneously. Assuming that all objectives are equally important but conflicting, we are interested in the set of nondominated points, i.e., the image set of solutions that cannot be improved in any objective without getting worse off in another one. This thesis comprises three parts. The first part considers the compu...

On the sign characteristics of Hermitian matrix polynomials

Mehrmann, Volker ; Noferini, Vanni ; Tisseur, Françoise ; Xu, Hongguo (2016-09-14)

The sign characteristics of Hermitian matrix polynomials are discussed, and in particular an appropriate definition of the sign characteristics associated with the eigenvalue infinity. The concept of sign characteristic arises in different forms in many scientific fields, and is essential for the stability analysis in Hamiltonian systems or the perturbation behavior of eigenvalues under structu...

Sparse Proteomics Analysis – a compressed sensing-based approach for feature selection and classification of high-dimensional proteomics mass spectrometry data

Conrad, Tim O. F. ; Genzel, Martin ; Cvetkovic, Nada ; Wulkow, Niklas ; Leichtle, Alexander ; Vybiral, Jan ; Kutyniok, Gitta ; Schütte, Christof (2017)

Background: High-throughput proteomics techniques, such as mass spectrometry (MS)-based approaches, produce very high-dimensional data-sets. In a clinical setting one is often interested in how mass spectra differ between patients of different classes, for example spectra from healthy patients vs. spectra from patients having a particular disease. Machine learning algorithms are needed to (a) i...

Properties of the Wright-Fisher diffusion with seed banks and multiple islands

Buzzoni, Eugenio (2019)

The main purpose of this thesis is the analysis under several viewpoints both of the Wright-Fisher diffusion with seed bank, introduced in [BGKWB16], and the two-island diffusion, investigated e.g. in [KZH08] and [NG93]. The former simulates a population in which some of the individuals can become inactive for long periods of time, like seeds or dormant bacteria, while the latter is used to inv...

Discrete Yamabe problem for polyhedral surfaces

Kourimska, Hana ; Springborn, Boris (2019-09-13)

We introduce a new discretization of the Gaussian curvature on piecewise at surfaces. As the prime new feature the curvature is scaled by the factor 1/r2 upon scaling the metric globally with the factor r. We develop a variational principle to tackle the corresponding discrete uniformisation theorem – we show that each piecewise at surface is discrete conformally equivalent to one with constant...

Convex geometry of numbers: covering, successive minima and Banach-Mazur distance

Xue, Fei (2019)

This thesis addresses several classical problems in convex geometry of numbers, including the lattice point covering problem, successive-minima-type inequalities and the Banach-Mazur distance of convex bodies. In the first chapter we will introduce basic concepts, definitions and results which provide the background for the problems in this thesis. Other concepts which are more specific or lim...

Efficient graph exploration

Hackfeld, Jan (2019)

The thesis “Efficient Graph Exploration” studies the following three closely related problems, where collaborating mobile agents move in a graph and have to jointly perform a certain task. Chapter 2 “Space Efficient Graph Exploration” considers the problem of deterministically exploring an undirected and initially unknown graph with n vertices either by a single agent equipped with a set of pe...

Towards single-valued polylogarithms in two variables for the seven-point remainder function in multi-Regge kinematics

Brödel, Johannes ; Sprenger, Martin ; Torres Orjuela, Alejandro (2016-12-27)

We investigate single-valued polylogarithms in two complex variables, which are relevant for the seven-point remainder function in super-Yang–Mills theory in the multi-Regge regime. After constructing these two-dimensional polylogarithms, we determine the leading logarithmic approximation of the seven-point remainder function up to and including five loops.

Discrete Yamabe problem for polyhedral surfaces

Kourimska, Hana ; Springborn, Boris (2019-09-13)

We introduce a new discretization of the Gaussian curvature on piecewise at surfaces. As the prime new feature the curvature is scaled by the factor 1/r2 upon scaling the metric globally with the factor r. We develop a variational principle to tackle the corresponding discrete uniformisation theorem – we show that each piecewise at surface is discrete conformally equivalent to one with constant...

Inexact methods for the solution of large scale Hermitian eigenvalue problems

Kandler, Ute (2019)

This thesis focuses on the solution of high dimensional Hermitian eigenproblems in situations where vector operations cannot be carried out exactly. To this end an inexact Arnoldi method with the aim to approximate extreme eigenvalues and eigenvectors is developed. This method is particularly wellsuited for large scale problems as it efficiently reduces the storage and computational requiremen...

Rough volatility models

Stemper, Benjamin Marco (2019)

So-called rough stochastic volatility models constitute the latest advancement in option price modeling. In contrast to popular bivariate diffusion models such as Heston, here the driving noise of volatility is modeled by a fractional Brownian motion (fBM) with scaling in the rough regime of Hurst parameter H < 1/2. A major appeal of such models lies in their ability to parsimoniously recover k...

Coupled system of Maxwell equations and circuit equations in electro-magnetism

Niroomand Rad, Helia (2019)

This thesis is devoted to modeling electro-magnetic coupling, the so-called crosstalk phenomenon, in circuit simulation, where a new modeling approach via bilateral coupling of the Maxwell equations with circuit equations is considered. Using the bilaterally coupled model allows full simulation of the crosstalk phenomenon in the evolution of time, when the disturbances generated by the excitati...

Information flow in stochastic optimal control and a stochastic representation theorem for Meyer-measurable processes

Beßlich, David (2019)

Stochastic control theory determines intervention policies optimizing the evolution of a system subject to randomness. Delicate issues arise when the considered system can jump due to both exogenous shocks and endogenous controls. Here one has to specify what the controller knows when about the exogenous shocks and how and when she can act on this information. Classical optimal control resolves...