Inst. Mathematik

289 Items

Recent Submissions
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...

Isotropic and coisotropic subvarieties of Grassmannians

Kohn, Kathlén (2018)

In this thesis, we study subvarieties of Grassmannians which are characterized by certain rank one conditions on their tangent or conormal spaces. Although these concepts seem to be very abstract at first sight, we will see that many of these subvarieties are naturally associated to underlying projective varieties. Typically such varieties consist of linear spaces which meet a projective variet...

Stable marriage and roommates problems with restricted edges: complexity and approximability

Cseh, Ágnes ; Manlove, David F. (2016-04-16)

In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the comp...

Tropicalization of del Pezzo surfaces

Ren, Qingchun ; Shaw, Kristin ; Sturmfels, Bernd (2016-03-30)

We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schläfli graphs, respectively. There are 27 trees on each tropical ...

Presolving techniques and linear relaxations for cumulative scheduling

Heinz, Stefan (2018)

In the mathematical optimization community the term scheduling usually describes the computation of a sequential plan for a set of jobs w.r.t. a set of side conditions such as precedence constraints and resource restrictions. Thereby, a certain objective should be fulfilled which can be for example to minimize the latest completion time of all jobs. The sequential plan can be an ordering of the...

On the dimension of posets with cover graphs of treewidth 2

Joret, Gwenael ; Micek, Piotr ; Trotter, William T. ; Wang, Ruidong ; Wiechert, Veit (2016-06-01)

In 1977, Trotter and Moore proved that a poset has dimension at most 3 whenever its cover graph is a forest, or equivalently, has treewidth at most 1. On the other hand, a well-known construction of Kelly shows that there are posets of arbitrarily large dimension whose cover graphs have treewidth 3. In this paper we focus on the boundary case of treewidth 2. It was recently shown that the dimen...