## Fachbereich Mathematik

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#### Keywords

- Up Functions (2)
- Abel integral equations (1)
- Approximation (1)
- Bessel functions (1)
- Decomposition and Reconstruction Schemes (1)
- Dynamic cut (1)
- Earliest arrival augmenting path (1)
- Euler's equation of motion (1)
- Glättung (1)
- Glättungsparameterwahl (1)
- Gravitationsfeld (1)
- Harmonische Spline-Funktion (1)
- Hochschild homology (1)
- Hochschild-Homologie (1)
- Homologietheorie (1)
- Inverse problems in Banach spaces (1)
- Inverses Problem (1)
- L-curve Methode (1)
- Label correcting algorithm (1)
- Label setting algorithm (1)
- Lavrentiev regularization for equations with monotone operators (1)
- Locally Supported Radial Basis Functions (1)
- Mehrkriterielle Optimierung (1)
- Mehrskalenanalyse (1)
- Multiple criteria analysis (1)
- Multiresolution Analysis (1)
- Multisresolution Analysis (1)
- Network flows (1)
- Quasi-identities (1)
- Reflection (1)
- Spherical (1)
- Spherical Wavelets (1)
- Stratifaltigkeiten (1)
- Wavelet (1)
- Zyklische Homologie (1)
- associated Legendre functions (1)
- conditional quantiles (1)
- consecutive ones property (1)
- cyclic homology (1)
- elasticity problem (1)
- hyper-quasi-identities (1)
- hypergeometric functions (1)
- hyperquasivarieties (1)
- incident wave (1)
- kernel estimate (1)
- non-commutative geometry (1)
- parameter choice (1)
- quantile autoregression (1)
- quasi-P (1)
- quasi-SH (1)
- quasi-SV (1)
- quasivarieties (1)
- refraction (1)
- set covering (1)
- singular spaces (1)
- singuläre Räume (1)
- stop location (1)
- time series (1)
- triclinic medium (1)
- uniform consistency (1)
- value-at-risk (1)

In this paper, the reflection and refraction of a plane wave at an interface between .two half-spaces composed of triclinic crystalline material is considered. It is shown that due to incidence of a plane wave three types of waves namely quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH) will be generated governed by the propagation condition involving the acoustic tensor. A simple procedure has been presented for the calculation of all the three phase velocities of the quasi waves. It has been considered that the direction of particle motion is neither parallel nor perpendicular to the direction of propagation. Relations are established between directions of motion and propagation, respectively. The expressions for reflection and refraction coefficients of qP, qSV and qSH waves are obtained. Numerical results of reflection and refraction coefficients are presented for different types of anisotropic media and for different types of incident waves. Graphical representation have been made for incident qP waves and for incident qSV and qSH waves numerical data are presented in two tables.

SST (satellite-to-satellite tracking) and SGG (satellite gravity gradiometry) provide data that allows the determination of the first and second order radial derivative of the earth's gravitational potential on the satellite orbit, respectively. The modeling of the gravitational potential from such data is an exponentially ill-posed problem that demands regularization. In this paper, we present the numerical studies of an approach, investigated in [24] and [25], that reconstructs the potential with spline smoothing. In this case, spline smoothing is not just an approximation procedure but it solves the underlying compact operator equation of the SST-problem and the SGG-problem. The numerical studies in this paper are performed for a simplified geometrical scenario with simulated data, but the approach is designed to handle first or second order radial derivative data on a real satellite orbit.

In this paper we consider set covering problems with a coefficient matrix almost having the consecutive ones property, i.e., in many rows of the coefficient matrix, the ones appear consecutively. If this property holds for all rows it is well known that the set covering problem can be solved efficiently. For our case of almost consecutive ones we present a reformulation exploiting the consecutive ones structure to develop bounds and a branching scheme. Our approach has been tested on real-world data as well as on theoretical problem instances.

We study a possiblity to use the structure of the regularization error for a posteriori choice of the regularization parameter. As a result, a rather general form of a selection criterion is proposed, and its relation to the heuristical quasi-optimality principle of Tikhonov and Glasko (1964), and to an adaptation scheme proposed in a statistical context by Lepskii (1990), is discussed. The advantages of the proposed criterion are illustrated by using such examples as self-regularization of the trapezoidal rule for noisy Abel-type integral equations, Lavrentiev regularization for non-linear ill-posed problems and an inverse problem of the two-dimensional profile reconstruction.

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 an estimate which we get by inverting a kernel estimate of the conditional distribution function, and prove its asymptotic normality and uniform strong consistency. We illustrate the good performance of the estimate for light and heavy-tailed distributions of the innovations with a small simulation study.

A new class of locally supported radial basis functions on the (unit) sphere is introduced by forming an infinite number of convolutions of ''isotropic finite elements''. The resulting up functions show useful properties: They are locally supported and are infinitely often differentiable. The main properties of these kernels are studied in detail. In particular, the development of a multiresolution analysis within the reference space of square--integrable functions over the sphere is given. Altogether, the paper presents a mathematically significant and numerically efficient introduction to multiscale approximation by locally supported radial basis functions on the sphere.

Hyperquasivarieties
(2003)

We define a class of topological spaces (LCNT-spaces) which come together with a nuclear Frechet algebra. Like the algebra of smooth functions on a manifold, this algebra carries the differential structure of the object. We compute the Hochschild homology of this object and show that it is isomorphic to the space of differential forms. This is a generalization of a result obtained by Alain Connes in the framework of smooth manifolds.