Inst. Mathematik

313 Items

Recent Submissions
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 ...

A stable, polynomial-time algorithm for the eigenpair problem

Armentano, Diego ; Beltrán, Carlos ; Bürgisser, Peter ; Cucker, Felipe ; Shub, Michael (2018)

We describe algorithms for computing eigenpairs (eigenvalue-eigenvector pairs) of a complex n×n matrix A. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomial-time). We do not believe they outperform in practice the algorithms currently used for this computational problem. The merit of our paper is to give a positive answer to a long-standing...

The Loewner equation and Lipschitz graphs

Rohde, Steffen ; Tran, Huy ; Zinsmeister, Michel (2018)

The proofs of continuity of Loewner traces in the stochastic and in the deterministic settings employ different techniques. In the former setting of the Schramm–Loewner evolution SLE, Hölder continuity of the conformal maps is shown by estimating the derivatives, whereas the latter setting uses the theory of quasiconformal maps. In this note, we adopt the former method to the deterministic sett...

The hyperbolic geometry of Markov’s theorem on Diophantine approximation and quadratic forms

Springborn, Boris (2017)

Markov’s theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov’s theorem using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra...

Moments of quantum Lévy areas using sticky shuffle Hopf algebras

Hudson, Robin ; Schauz, Uwe ; Wu, Yue (2018)

We study a family of quantum analogs of Lévy's stochastic area for planar Brownian motion depending on a variance parameter σ ≥ 1 which deform to the classical Lévy area as σ → ∞. They are defined as second rank iterated stochastic integrals against the components of planar Brownian motion, which are one-dimensional Brownian motions satisfying Heisenberg-type commutation relations. Such iterate...

A coupling approach to Doob’s theorem

Kulik, Alexei ; Scheutzow, Michael (2015)

We provide a coupling proof of Doob’s theorem which says that the transition probabilities of a regular Markov process which has an invariant probability measure μ converge to μ in the total variation distance. In addition we show that non-singularity (rather than equivalence) of the transition probabilities suffices to ensure convergence of the transition probabilities for μ-almost all initial...

Planar graphs and face areas: Area-Universality

Kleist, Linda (2018)

In this work, we study planar graphs with prescribed face areas. This field is inspired by cartograms. A 'cartogram' is a distorted map where the size of the regions are proportional to some statistical parameter such as the population, the total birth, the gross national product, or some other special property. As a mathematical abstraction we are interested in straight-line drawings of plane...

A bilevel shape optimization problem for the exterior Bernoulli free boundary value problem

Kasumba, Henry ; Kunisch, Karl ; Laurain, Antoine (2014)

A bilevel shape optimization problem with the exterior Bernoulli free boundary problem as lowerlevel problem and the control of the free boundary as the upper-level problem is considered. Using the shape of the inner boundary as the control, we aim at reaching a specific shape for the free boundary. A rigorous sensitivity analysis of the bilevel shape optimization in the infinite-dimensional se...

Anisotropic surface energy formulations and their effect on stability of a growing thin film

Korzec, Maciek D. ; Münch, Andreas ; Wagner, Barbara (2012)

In this paper we revisit models for the description of the evolution of crystalline films with anisotropic surface energies.We prove equivalences of symmetry properties of anisotropic surface energy models commonly used in the literature. Then we systematically develop a framework for the derivation of surface diffusion models for the self-assembly of quantum dots during Stranski-Krastanov grow...

Pinning of interfaces in random media

Dirr, Nicolas ; Dondl, Patrick W. ; Scheutzow, Michael (2011)

For a model for the propagation of a curvature sensitive interface in a time independent random medium, as well as for a linearized version which is commonly referred to as Quenched Edwards– Wilkinson equation, we prove existence of a stationary positive supersolution at non-vanishing applied load. This leads to the emergence of a hysteresis that does not vanish for slow loading, even though th...

Analysis and Numerical Approximation of an Integro-differential Equation Modeling Non-local Effects in Linear Elasticity

Emmrich, Etienne ; Weckner, Olaf (2007)

Long-range interactions for linearly elastic media resulting in nonlinear dispersion relations are modeled by an initial-value problem for an integro-differential equation (IDE) that incorporates non-local effects. Interpreting this IDE as an evolutionary equation of second order, well-posedness in L ∞(ℝ) as well as jump relations are proved. Moreover, the construction of the micromodulus funct...

Online railway delay management: Hardness, simulation and computation

Berger, André ; Hoffmann, Ralf ; Lorenz, Ulf ; Stiller, Sebastian (2011)

Delays in a railway network are a common problem that railway companies face in their daily operations. When a train is delayed, it may either be beneficial to let a connecting train wait so that passengers in the delayed train do not miss their connection, or it may be beneficial to let the connecting train depart on time to avoid further delays. These decisions naturally depend on the global ...

When do Projections Commute?

Rehder, W. (1980)

Necessary and sufficient conditions for commutativity of two projections in Hilbert space are given through properties of so-called conditional connectives which are derived from the conditional probability operator PQP. This approach unifies most of the known proofs, provides a few new criteria, and permits certain suggestive interpretations for compound properties of quantum-mechanical systems.

Lattice points in convex bodies

Berg, Sören Lennart (2018)

This thesis addresses classical lattice point problems in discrete and convex geometry. Integer points in convex bodies are the central objects of our studies. In the second chapter, we will prove bounds on the number of lattice points in centered convex bodies. The underlying problems are motivated by classical results in geometry of numbers. We will show that the assumption of centricity yie...

Continuum limits of variational systems

Vermeeren, Mats (2018)

In this thesis we examine how to recover continuous systems from discrete systems, i.e. differential equations from difference equations. In particular, we are interested in equations with a variational (Lagrangian) structure and the transferal of this structure from the discrete to the continuous. In the context of numerical integration, the differential equation corresponding to a given diff...

Finite-Size Effects on Traveling Wave Solutions to Neural Field Equations

Lang, Eva ; Stannat, Wilhelm (2017-07-06)

Neural field equations are used to describe the spatio-temporal evolution of the activity in a network of synaptically coupled populations of neurons in the continuum limit. Their heuristic derivation involves two approximation steps. Under the assumption that each population in the network is large, the activity is described in terms of a population average. The discrete network is then approx...