Inst. Mathematik

294 Items

Recent Submissions
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...

Exact and soft recovery of structured signals from atomic and total variation norm regularization

Flinth, Axel (2018)

This thesis treats regularization techniques of inverse problems using TV- and atomic norms. In the first part of the thesis, the two concepts are connected, making a unified treatment in the rest of the thesis possible. Concretely, it is proven that for reasonable dictionaries, atomic norms can be calculated with the help of TV -minimization, and that the TV -norm can be considered as an atomi...

Homotopy colimits – comparison lemmas for combinatorial applications

Welker, Volkmar ; Ziegler, Günter M. ; Živaljević, Rade T. (1999)

We provide a “toolkit” of basic lemmas for the comparison of homotopy types of homotopy colimits of diagrams of spaces over small categories. We show how this toolkit can be used in quite different fields of applications. We demonstrate this with respect to 1. Björner's “Generalized Homotopy Complementation Formula” [5], 2. the topology of toric varieties, 3. the study of homotopy types of arra...

Noise induced synchronization and related topics

Vorkastner, Isabell (2018)

This thesis focuses on noise induced synchronization. Noise induced synchronization describes the stabilizing effect of noise on the long-time dynamics of a random dynamical system. While the attractor in the absence of noise is not a single point, the random attractor collapses to a single random point under the addition of noise. In the first part, we consider a system that is known to synch...

Flows over time and submodular function minimization

Schlöter, Miriam (2018)

Many parts of our daily routine that we expect to "just work" rely on certain optimization processes that most of the time take place in a network. Ranging from streets, railroads or power networks to the telecommunications network used by the Internet and the networks induced by social networks: the list of structures that can be modeled as a network goes on and on and optimization processes o...

Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces

Emmrich, Etienne ; Grigorieff, Rolf Dieter (2006)

In this paper, we study the convergence of the finite difference discretization of a second order elliptic equation with variable coefficients subject to general boundary conditions. We prove that the scheme exhibits the phenomenon of supraconvergence on nonuniform grids, i.e., although the truncation error is in general of the first order alone, one has second order convergence. All error esti...

Radii minimal projections of polytopes and constrained optimization of symmetric polynomials

Brandenberg, René ; Theobald, Thorsten (2006)

We provide a characterization of the radii minimal projections of polytopes onto j-dimensional subspaces in Euclidean space . Applied to simplices this characterization allows to reduce the computation of an outer radius to a computation in the circumscribing case or to the computation of an outer radius of a lower-dimensional simplex. In the second part of the paper, we use this characterizati...

Supraconvergence and Supercloseness of a Discretisation for Elliptic Third-kind Boundary-value Problems on Polygonal Domains

Emmrich, Etienne (2007)

The third-kind boundary-value problem for a second-order elliptic equation on a polygonal domain with variable coefficients, mixed derivatives, and first-order terms is approximated by a linear finite element method with first-order accurate quadrature. The corresponding bilinear form does not need to be strongly positive. The discretisation is equivalent to a finite difference scheme. Although...

The framework of α-molecules

Schäfer, Martin (2018)

The theory of wavelets constitutes an integral part of modern harmonic analysis with many theoretical and practical applications. In engineering for example, wavelets are nowadays a popular tool for the efficient representation and approximation of functions. Much of their success thereby relies on the fact that they are more suited to represent smooth signals with singularities than traditiona...

A special evolution equation used in the analysis of the stochastic Navier-stokes equation

Lisei, Hannelore (2001)

We investigate a stochastic evolution equation of a special type. We use it to develop a linear approximation method for the solution of an equation of Navier-Stokes type.

Periodic discrete conformal maps

Hetrich-Jeromin, Udo ; McIntosh, Ian ; Norman, P. ; Pedit, Franz (2001)

Recently there has been much interest in the theory of discrete surfaces in 3-space and its connection with the discretization of soliton equations (see e.g. [3], [11] and references therein). In this article we study a discrete geometry which is the simplest example for both theories.

Vertex-facet incidences of unbounded polyhedra

Joswig, Michael ; Kaibel, Volker ; Pfetsch, Marc E. ; Ziegler, Günter M. (2001)

How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is bounded. In the case of a polyhedron that is simple and ``simplicial,'' i.e., a d-dimensional polyhedron that has d facets through each vertex and d vertices o...

Discrete constant mean curvature surfaces and their index

Polthier, Konrad ; Rossman, Wayne (2002)

We define triangulated piecewise linear constant mean curvature surfaces using a variational characterization. These surfaces are critical for area amongst continuous piecewise linear variations which preserve the boundary conditions, the simplicial structures, and (in the nonminimal case) the volume to one side of the surfaces. We then find explicit formulas for complete examples, such as disc...

Performance Of H-Lu Preconditioning For Sparse Matrices

Grasedyck, Lars ; Hackbusch, Wolfgang ; Kriemann, Ronald (2008)

In this paper we review the technique of hierarchical matrices and put it into the context of black-box solvers for large linear systems. Numerical examples for several classes of problems from medium- to large-scale illustrate the applicability and efficiency of this technique. We compare the results with those of several direct solvers (which typically scale quadratically in the matrix size) ...

On solving norm equations in global function fields

Gaál, István ; Pohst, Michael E. (2009)

The potential of solving norm equations is crucial for a variety of applications of algebraic number theory, especially in cryptography. In this article we develop general effective methods for that task in global function fields for the first time.

Constructing simplicial branched covers

Witte, Nikolaus (2009)

Izmestiev and Joswig described how to obtain a simplicial covering space (the partial unfolding) of a given simplicial complex, thus obtaining a simplicial branched cover [Adv. Geom. 3:191–255, 2003]. We present a large class of branched covers which can be constructed via the partial unfolding. In particular, for d ≤ 4 every closed oriented PL d-manifold is the partial unfolding of some polyto...

φ(Fn ) = Fm

Luca, Florian ; Nicolae, Florin (2009)

We show that 1, 2 and 3 are the only Fibonacci numbers whose Euler functions are also Fibonacci numbers.

Uncertainty exploration

Meißner, Julie (2018)

Combinatorial optimization captures a wide range of applications such as infrastructure design and scheduling of tasks. For a fixed number of possible new infrastructure connections or tasks to schedule, there are exponentially many infrastructure networks and schedules that can be created from them. Additionally, a major challenge is data uncertainty. If, for example, the construction cost of ...

Tensor methods for the numerical solution of high-dimensional parametric partial differential equations

Pfeffer, Max (2018)

This thesis deals with tensor methods for the numerical solution of parametric partial differential equations (PDEs). Because of their parametric dependence, these differential equations exhibit a high dimensionality. Numerical methods like the Galerkin method quickly reach their limits, since the resulting linear system grows exponentially with the number of parameters. This is known as the cu...

Wait-and-see strategies in polling models

Aurzada, Frank ; Beck, Sergej ; Scheutzow, Michael (2012)

We consider a general polling model with N stations. The stations are served exhaustively and in cyclic order. Once a station queue falls empty, the server does not immediately switch to the next station. Rather, it waits at the station for the possible arrival of new work (“wait-and-see”) and, in the case of this happening, it restarts service in an exhaustive fashion. The total time the serve...