Inst. Mathematik

328 Items

Recent Submissions
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...

Error analysis and adaptive control for gas flow in networks

Stolwijk, Jeroen Johannes (2019)

In this thesis, uniform estimators are derived for the different errors that a numerical solution could contain, namely modeling, discretization, iteration, data uncertainty, and rounding errors. Subsequently, the errors are adaptively controlled on a network in order to bring the total error below a prescribed tolerance while keeping the computational cost low. As example problems, the simulat...

Homogen-universelle metrische Räume

Rothacker, Edgar (1976)

Homogeneous universal metric spaces are constructed in the Jónsson class of metric spaces starting from finite metric spaces over the rationals and then completing the construction over the reals; the main theorem deals with the finite homgeneity property established by Urysohn for his famous universal metric space: we show that this can be extended to totally bounded metric spaces, thereby ext...

Homogen-universelle verallgemeinerte metrische Räume unter besonderer Berücksichtigung nicht-archimedischer Metriken

Rothacker, Edgar (1978)

Homogeneous universal metric spaces are constructed over generalized distance sets such as totally ordered abelian groups or linear orderings without any further structure; the latter yields the basis for the study of hom.-univ. ultrametric spaces in the third part of this work; finally the fourth part studies the Cauchy completeness in connection with the construction of the appropriately ge...

The Mismatch Principle and L1-analysis compressed sensing

Genzel, Martin (2019)

This thesis contributes to several mathematical aspects and problems at the interface of statistical learning theory and signal processing. Although based on a common theoretical foundation, the main results of this thesis can be divided into two major topics of independent interest. The first one deals with the estimation capacity of the generalized Lasso, i.e., least squares minimization comb...

Message routeing and percolation in interference limited multihop networks

Tóbiás, András József (2019)

This thesis consists of two main parts. In the first part, we investigate a probabilistic model for routeing of messages in relay-augmented multihop ad-hoc networks, where each transmitter sends one message to the origin. Given the (random) transmitter locations, we weight the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectori...

Convergence of gradient methods on hierarchical tensor varieties

Kutschan, Benjamin (2019)

Subject of the attached dissertation are the sets of tensors of bounded hierarchical rank. They are algebraic varieties. The central result is a parametrization of the tangent cones of these varieties. Using this result a Riemannian gradient method is constructed. The global convergence of this gradient method is proven using a Lojasiewicz inequality.

Subspace concentration of geometric measures

Pollehn, Hannes (2019)

In this work we study geometric measures in two different extensions of the Brunn-Minkowski theory. The first part of this thesis is concerned with problems in Lp Brunn-Minkowski theory, that is based on the concept of p-addition of convex bodies, which was first introduced by Firey for p = 1 and later considered for all real p by Lutwak et al. The interplay of the volume and other functionals ...

Connectedness of random set attractors

Scheutzow, Michael ; Vorkastner, Isabell (2018-12-28)

We examine the question whether random set attractors for continuous-time random dynamical systems on a connected state space are connected. In the deterministic case, these attractors are known to be connected. In the probabilistic setup, however, connectedness has only been shown under stronger connectedness assumptions on the state space. Under a weak continuity condition on the random dynam...

Low rank tensor decompositions for high dimensional data approximation, recovery and prediction

Wolf, Alexander Sebastian Johannes Wolf (2019)

In this thesis, we examine different approaches for efficient high dimensional data acquisition and reconstruction using low rank tensor decomposition techniques. High dimensional here refers to the order of the ambient tensor space in which the data is contained. Examples of such data include tomographic videos, solutions to parametric differential equations and quantum states of many particle...

Stability analysis in the inverse Robin transmission problem

Meftahi, Houcine (2016)

In this paper, we consider the conductivity problem with piecewise‐constant conductivity and Robin‐type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non‐monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to...

Spectral properties of the random conductance model

Flegel, Franziska (2019)

Charge and exciton transport in disordered media plays an essential role in modern technologies. Classical examples are amorphous and organic semiconductors where the disorder can give rise to localized electron states. These localized electrons effectively behave like discrete particles hopping between discrete sites in an inhomogeneous environment. A popular model for such a hopping process ...