Institut für Mathematik

234 Items

Recent Submissions
Sharp-interface formation during lithium intercalation into silicon

Meca, Esteban ; Münch, Andreas ; Wagner, Barbara (2017)

In this study, we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as amorphous silicon (a-Si). The governing equations couple a viscous Cahn–Hilliard-Reaction model with elasticity in the framework of the Cahn–Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the forma...

Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation

Dondl, Patrick W. ; Scheutzow, Michael ; Throm, Sebastian (2015)

For a model of a driven interface in an elastic medium with random obstacles we prove the existence of a stationary positive supersolution at non-vanishing driving force. This shows the emergence of a rate-independent hysteresis through the interaction of the interface with the obstacles despite a linear (force = velocity) microscopic kinetic relation. We also prove a percolation result, namely...

Topological and symbolic dynamics for hyperbolic systems with holes

Bundfuss, Stefan ; Krüger, Tyll ; Troubetzkoy, Serge (2010)

We consider an axiom A diffeomorphism or a Markov map of an interval and the invariant set Ω* of orbits which never falls into a fixed hole. We study various aspects of the symbolic representation of Ω* and of its non-wandering set Ωnw. Our results are on the cardinality of the set of topologically transitive components of Ωnw and their structure. We also prove that Ω* is generically a subshif...

Motion and structure of atmospheric mesoscale baroclinic vortices: dry air and weak environmental shear

Päschke, Eileen ; Marschalik, Patrik ; Owinoh, Antony Z. ; Klein, Rupert (2012)

A strongly tilted, nearly axisymmetric vortex in dry air with asymmetric diabatic heating is analysed here by matched asymptotic expansions. The vortex is in gradient wind balance, with vortex Rossby numbers of order unity, and embedded in a quasigeostrophic (QG) background wind with weak vertical shear. With wind speeds of 60–120 km h-1, such vortices correspond to tropical storms or nascent h...

On the random dynamics of Volterra quadratic operators

Jamilov, U. U. ; Scheutzow, M. ; Wilke-Berenguer, M. (2015)

We consider random dynamical systems generated by a special class of Volterra quadratic stochastic operators on the simplex Sm-1. We prove that in contrast to the deterministic set-up the trajectories of the random dynamical system almost surely converge to one of the vertices of the simplex Sm-1, implying the survival of only one species. We also show that the minimal random point attractor o...

Singular analysis and coupled cluster theory

Flad, Heinz-Jürgen ; Harutyunyan, Gohar ; Schulze, Bert-Wolfgang (2015)

The primary motivation for systematic bases in first principles electronic structure simulations is to derive physical and chemical properties of molecules and solids with predetermined accuracy. This requires a detailed understanding of the asymptotic behaviour of many-particle Coulomb systems near coalescence points of particles. Singular analysis provides a convenient framework to study the ...

Backward stochastic differential equations with jumps are stable

Saplaouras, Alexandros (2017)

A backward stochastic differential equation is a stochastic differential equation whose terminal value is known, in contrast to a (forward) stochastic differential equation whose initial value is known, and whose solution has to be adapted to a given filtration. The main aim of this thesis is to provide the suitable framework for the stability of stochastic differential equations with jumps, he...

Numerical and statistical aspects of tensor decompositions

Breiding, Paul (2017)

In this work we study numerical and statistical properties of tensor decompositions, namely the canonical-polyadic decomposition— commonly known as tensor-rank decomposition—and the computation of tensor eigenpairs. After a preliminary section, in which we consider tensors and their properties, explain the use of condition numbers in numerical analysis and give a short introduction to random te...

On the Lagrangian structure of integrable hierarchies

Suris, Yuri B. ; Vermeeren, Mats (2016)

We develop the concept of pluri-Lagrangian structures for integrable hierarchies. This is a continuous counterpart of the pluri-Lagrangian (or Lagrangian multiform) theory of integrable lattice systems. We derive the multi-time Euler Lagrange equations in their full generality for hierarchies of two-dimensional systems, and construct a pluri-Lagrangian formulation of the potential Korteweg-de V...

Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization

Bobenko, Alexander I. ; Sechelmann, Stefan ; Springborn, Boris (2016)

We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrila...

Constructing solutions to the Björling problem for isothermic surfaces by structure preserving discretization

Bücking, Ulrike ; Matthes, Daniel (2016)

In this article, we study an analog of the Björling problem for isothermic surfaces (that are a generalization of minimal surfaces): given a regular curve γ in R3 and a unit normal vector field n along γ, find an isothermic surface that contains γ, is normal to n there, and is such that the tangent vector γ′ bisects the principal directions of curvature. First, we prove that this problem is uni...

On the variational interpretation of the discrete KP equation

Boll, Raphael ; Petrera, Matteo ; Suris, Yuri B. (2016)

We study the variational structure of the discrete Kadomtsev-Petviashvili (dKP) equation by means of its pluri-Lagrangian formulation. We consider the dKP equation and its variational formulation on the cubic lattice ZNas well as on the root latticeQ(AN). We prove that, on a lattice of dimension at least four, the corresponding Euler-Lagrange equations are equivalent to the dKP equation.

Complex line bundles over simplicial complexes and their applications

Knöppel, Felix ; Pinkall, Ulrich (2016)

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of André Weil on the classificati...

DGD gallery: storage, sharing, and publication of digital research data

Joswig, Michael ; Mehner, Milan ; Sechelmann, Stefan ; Techter, Jan ; Bobenko, Alexander I. (2016)

We describe a project, called the DGD Gallery, whose goal is to store geometric data and to make it publicly available. The DGD Gallery offers an online web service for the storage, sharing, and publication of digital research data.

Discrete complex analysis on planar Quad-graphs

Bobenko, Alexander I. ; Günther, Felix (2016)

We develop further a linear theory of discrete complex analysis on general quad-graphs, extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph leads to generalizations as well as to new proofs of previously known discrete analogs of classical theorems. New results include in particular di...

S-conical CMC surfaces. Towards a unified zheory of discrete surfaces with constant mean curvature

Bobenko, Alexander I. ; Hoffmann, Tim (2016)

We introduce a novel class of s-conical nets and, in particular, study s-conical nets with constant mean curvature. Moreover we give a unified description of nets of various types: circular, conical and s-isothermic. The later turn out to be interpolating between the circular net discretization and the s-conical one.

Exponential decay ofcovariances for the supercritical membrane model

Bolthausen, Erwin ; Cipriani, Alessandra ; Kurt, Noemi (2017)

We consider the membrane model, that is the centered Gaussian field on Zdwhose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose aδ-pinning condition, giving a reward of strengthεfor the field to be 0 at any site of the lattice. In this paper we prove that in dimensionsd≥5covariances of the pinned field decay at least exponentially, as opposed to the field without...

Shadow price of information in discrete time stochastic optimization

Pennanen, Teemu ; Perkkiö, Ari-Pekka (2017)

The shadow price of information has played a central role in stochastic optimization ever since its introduction by Rockafellar and Wets in the mid-seventies. This article studies the concept in an extended formulation of the problem and gives relaxed sufficient conditions for its existence. We allow for general adapted decision strategies, which enables one to establish the existence of soluti...

How unsplittable-flow-covering helps scheduling with job-dependent cost functions

Höhn, Wiebke ; Mestre, Julián ; Wiese, Andreas (2017)

Generalizing many well-known and natural scheduling problems, scheduling with job-specific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such a...

An On-line competitive algorithm for coloring bipartite graphs without long induced paths

Micek, Piotr ; Wiechert, Veit (2016)

The existence of an on-line competitive algorithm for coloring bipartite graphs is a tantalizing open problem. So far there are only partial positive results for bipartite graphs with certain small forbidden graphs as induced subgraphs. We propose an on-line competitive coloring algorithm for P9-free bipartite graphs.