## Fachbereich Mathematik

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#### Keywords

- Mixture Models (2)
- 2-d kernel regression (1)
- Algorithmics (1)
- Earth's disturbing potential (1)
- Gauge Distances (1)
- Geometric Ergodicity (1)
- Identifiability (1)
- Inverses Problem (1)
- Legendre Wavelets (1)
- Lineare Integralgleichung (1)

- On Geometric Ergodicity of CHARME Models (2007)
- In this paper we consider a CHARME Model, a class of generalized mixture of nonlinear nonparametric AR-ARCH time series. We apply the theory of Markov models to derive asymptotic stability of this model. Indeed, the goal is to provide some sets of conditions under which our model is geometric ergodic and therefore satisfies some mixing conditions. This result can be considered as the basis toward an asymptotic theory for our model.

- A note on the identifiability of the conditional expectation for the mixtures of neural networks (2007)
- We consider a generalized mixture of nonlinear AR models, a hidden Markov model for which the autoregressive functions are single layer feedforward neural networks. The non trivial problem of identifiability, which is usually postulated for hidden Markov models, is addressed here.

- Quantile Sieve Estimates for Time Series (2007)
- We consider the problem of estimating the conditional quantile of a time series at time \(t\) given observations of the same and perhaps other time series available at time \(t-1\). We discuss sieve estimates which are a nonparametric versions of the Koenker-Bassett regression quantiles and do not require the specification of the innovation law. We prove consistency of those estimates and illustrate their good performance for light- and heavy-tailed distributions of the innovations with a small simulation study. As an economic application, we use the estimates for calculating the value at risk of some stock price series.

- Scheduling and Location (ScheLoc): Makespan Problem with Variable Release Dates (2007)
- While in classical scheduling theory the locations of machines are assumed to be fixed we will show how to tackle location and scheduling problems simultaneously. Obviously, this integrated approach enhances the modeling power of scheduling for various real-life problems. In this paper, we present in an exemplary way theory and a solution algorithm for a specific type of a scheduling and a rather general, planar location problem, respectively. More general results and a report on numerical tests will be presented in a subsequent paper.

- Splines on the 3-dimensional Ball and their Application to Seismic Body Wave Tomography (2007)
- In this paper we construct spline functions based on a reproducing kernel Hilbert space to interpolate/approximate the velocity field of earthquake waves inside the Earth based on traveltime data for an inhomogeneous grid of sources (hypocenters) and receivers (seismic stations). Theoretical aspects including error estimates and convergence results as well as numerical results are demonstrated.

- Minimum Cut Bases in Undirected Networks (2007)
- Given an undirected, connected network G = (V,E) with weights on the edges, the cut basis problem is asking for a maximal number of linear independent cuts such that the sum of the cut weights is minimized. Surprisingly, this problem has not attained as much attention as its graph theoretic counterpart, the cycle basis problem. We consider two versions of the problem, the unconstrained and the fundamental cut basis problem. For the unconstrained case, where the cuts in the basis can be of an arbitrary kind, the problem can be written as a multiterminal network flow problem and is thus solvable in strongly polynomial time. The complexity of this algorithm improves the complexity of the best algorithms for the cycle basis problem, such that it is preferable for cycle basis problems in planar graphs. In contrast, the fundamental cut basis problem, where all cuts in the basis are obtained by deleting an edge, each, from a spanning tree T is shown to be NP-hard. We present heuristics, integer programming formulations and summarize first experiences with numerical tests.

- Polyhedral Analysis of Uncapacitated Single Allocation p-Hub Center Problems (2007)
- In contrast to p-hub problems with a summation objective (p-hub median), minmax hub problems (p-hub center) have not attained much attention in the literature. In this paper, we give a polyhedral analysis of the uncapacitated single allocation p-hub center problem (USApHCP). The analysis will be based on a radius formulation which currently yields the most efficient solution procedures. We show which of the valid inequalities in this formulation are facet-defining and present non-elementary classes of facets, for which we propose separation problems. A major part in our argumentation will be the close connection between polytopes of the USApHCP and the uncapacitated p-facility location (pUFL). Hence, the new classes of facets can also be used to improve pUFL formulations.

- Convergent Finite Element Discretizations of the Density Gradient Equation for Quantum Semiconductors (2007)
- We study nonlinear finite element discretizations for the density gradient equation in the quantum drift diffusion model. Especially, we give a finite element description of the so--called nonlinear scheme introduced by {it Ancona}. We prove the existence of discrete solutions and provide a consistency and convergence analysis, which yields the optimal order of convergence for both discretizations. The performance of both schemes is compared numerically, especially with respect to the influence of approximate vacuum boundary conditions.

- Local Smoothing Methods in Image Processing (2007)
- In this article a new data-adaptive method for smoothing of bivariate functions is developed. The smoothing is done by kernel regression with rotational invariant bivariate kernels. Two or three local bandwidth parameters are chosen automatically by a two-step plug-in approach. The algorithm starts with small global bandwidth parameters, which adapt during a few iterations to the noisy image. In the next step local bandwidths are estimated. Some general asymptotic results about Gasser-Müller-estimators and optimal bandwidth selection are given. The derived local bandwidth estimators converge and are asymptotically normal.

- New heuristics for the minimum fundamental cut basis problem (2007)
- Given an undirected connected network and a weight function finding a basis of the cut space with minimum sum of the cut weights is termed Minimum Cut Basis Problem. This problem can be solved, e.g., by the algorithm of Gomory and Hu [GH61]. If, however, fundamentality is required, i.e., the basis is induced by a spanning tree T in G, the problem becomes NP-hard. Theoretical and numerical results on that topic can be found in Bunke et al. [BHMM07] and in Bunke [Bun06]. In the following we present heuristics with complexity O(m log n) and O(mn), where n and m are the numbers of vertices and edges respectively, which obtain upper bounds on the aforementioned problem and in several cases outperform the heuristics of Schwahn [Sch05].