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The focus of this work has been to develop two families of wavelet solvers for the inner displacement boundary-value problem of elastostatics. Our methods are particularly suitable for the deformation analysis corresponding to geoscientifically relevant (regular) boundaries like sphere, ellipsoid or the actual Earth's surface. The first method, a spatial approach to wavelets on a regular (boundary) surface, is established for the classical (inner) displacement problem. Starting from the limit and jump relations of elastostatics we formulate scaling functions and wavelets within the framework of the Cauchy-Navier equation. Based on numerical integration rules a tree algorithm is constructed for fast wavelet computation. This method can be viewed as a first attempt to "short-wavelength modelling", i.e. high resolution of the fine structure of displacement fields. The second technique aims at a suitable wavelet approximation associated to Green's integral representation for the displacement boundary-value problem of elastostatics. The starting points are tensor product kernels defined on Cauchy-Navier vector fields. We come to scaling functions and a spectral approach to wavelets for the boundary-value problems of elastostatics associated to spherical boundaries. Again a tree algorithm which uses a numerical integration rule on bandlimited functions is established to reduce the computational effort. For numerical realization for both methods, multiscale deformation analysis is investigated for the geoscientifically relevant case of a spherical boundary using test examples. Finally, the applicability of our wavelet concepts is shown by considering the deformation analysis of a particular region of the Earth, viz. Nevada, using surface displacements provided by satellite observations. This represents the first step towards practical applications.

This paper deals with the handling of deformable linear objects (DLOs), such as hoses, wires, or leaf springs. It investigates usable features for the vision-based detection of a changing contact situation between a DLO and a rigid polyhedral obstacle and a classification of such contact state transitions. The result is a complete classification of contact state transitions and of the most significant features for each class. This knowledge enables reliable detection of changes in the DLO contact situation, facilitating implementation of sensor-based manipulation skills for all possible contact changes.

The Earth's surface is an almost perfect sphere. Deviations from its spherical shape are less than 0,4% of its radius and essentially arise from its rotation. All equipotential surfaces are nearly spherical, too. In consequence, multiscale modelling of geoscientifically relevant data on the sphere involving rotational symmetry of the trial functions used for the approximation plays an important role. In this paper we deal with isotropic kernel functions showing local support and (one-dimensional) polynomial structure (briefly called isotropic finite elements) for reconstructing square--integrable functions on the sphere. Essential tool is the concept of multiresolution analysis by virtue of the spherical up function. The main result is a tree algorithm in terms of (low--order) isotropic finite elements.

A hub location problem consists of locating p hubs in a network in order to collect and consolidate flow between node pairs. This thesis deals with the uncapacitated single allocation p-hub center problem (USApHCP) as a special type of hub location problem with min max objective function. Using the so-called radius formulation of the problem, the dimension of the polyhedron of USApHCP is derived. The formulation constraints are investigated to find out which of these define facets. Then, three new classes of facet-defining inequalities are derived. Finally, efficient procedures to separate facets in a branch-and-cut algorithm are proposed. The polyhedral analysis of USApHCP is based on a tight relation to the uncapacitated facility location problem (UFL). Hence, many results stated in this thesis also hold for UFL.

The Discrete Ordered Median Problem (DOMP) generalizes classical discrete location problems, such as the N-median, N-center and Uncapacitated Facility Location problems. It was introduced by Nickel [16], who formulated it as both a nonlinear and a linear integer program. We propose an alternative integer linear programming formulation for the DOMP, discuss relationships between both integer linear programming formulations, and show how properties of optimal solutions can be used to strengthen these formulations. Moreover, we present a specific branch and bound procedure to solve the DOMP more efficiently. We test the integer linear programming formulations and this branch and bound method computationally on randomly generated test problems.

Radiation therapy planning is always a tight rope walk between dangerous insufficient dose in the target volume and life threatening overdosing of organs at risk. Finding ideal balances between these inherently contradictory goals challenges dosimetrists and physicians in their daily practice. Today’s planning systems are typically based on a single evaluation function that measures the quality of a radiation treatment plan. Unfortunately, such a one dimensional approach cannot satisfactorily map the different backgrounds of physicians and the patient dependent necessities. So, too often a time consuming iteration process between evaluation of dose distribution and redefinition of the evaluation function is needed. In this paper we propose a generic multi-criteria approach based on Pareto’s solution concept. For each entity of interest - target volume or organ at risk a structure dependent evaluation function is defined measuring deviations from ideal doses that are calculated from statistical functions. A reasonable bunch of clinically meaningful Pareto optimal solutions are stored in a data base, which can be interactively searched by physicians. The system guarantees dynamical planning as well as the discussion of tradeoffs between different entities. Mathematically, we model the upcoming inverse problem as a multi-criteria linear programming problem. Because of the large scale nature of the problem it is not possible to solve the problem in a 3D-setting without adaptive reduction by appropriate approximation schemes. Our approach is twofold: First, the discretization of the continuous problem is based on an adaptive hierarchical clustering process which is used for a local refinement of constraints during the optimization procedure. Second, the set of Pareto optimal solutions is approximated by an adaptive grid of representatives that are found by a hybrid process of calculating extreme compromises and interpolation methods.

As the sustained trend towards integrating more and more functionality into systems on a chip can be observed in all fields, their economic realization is a challenge for the chip making industry. This is, however, barely possible today, as the ability to design and verify such complex systems could not keep up with the rapid technological development. Owing to this productivity gap, a design methodology, mainly using pre designed and pre verifying blocks, is mandatory. The availability of such blocks, meeting the highest possible quality standards, is decisive for its success. Cost-effective, this can only be achieved by formal verification on the block-level, namely by checking properties, ranging over finite intervals of time. As this verification approach is based on constructing and solving Boolean equivalence problems, it allows for using backtrack search procedures, such as SAT. Recent improvements of the latter are responsible for its high capacity. Still, the verification of some classes of hardware designs, enjoying regular substructures or complex arithmetic data paths, is difficult and often intractable. For regular designs, this is mainly due to individual treatment of symmetrical parts of the search space by backtrack search procedures used. One approach to tackle these deficiencies, is to exploit the regular structure for problem reduction on the register transfer level (RTL). This work describes a new approach for property checking on the RTL, preserving the problem inherent structure for subsequent reduction. The reduction is based on eliminating symmetrical parts from bitvector functions, and hence, from the search space. Several approaches for symmetry reduction in search problems, based on invariance of a function under permutation of variables, have been previously proposed. Unfortunately, our investigations did not reveal this kind of symmetry in relevant cases. Instead, we propose a reduction based on symmetrical values, as we encounter them much more frequently in our industrial examples. Let \(f\) be a Boolean function. The values \(0\) and \(1\) are symmetrical values for a variable \(x\) in \(f\) iff there is a variable permutation \(\pi\) of the variables of \(f\), fixing \(x\), such that \(f|_{x=0} = \pi(f|_{x=1})\). Then the question whether \(f=1\) holds is independent from this variable, and it can be removed. By iterative application of this approach to all variables of \(f\), they are either all removed, leaving \(f=1\) or \(f=0\) trivially, or there is a variable \(x'\) with no such \(\pi\). The latter leads to the conclusion that \(f=1\) does not hold, as we found a counter-example either with \(x'=0\), or \(x'=1\). Extending this basic idea to vectors of variables, allows to elevate it to the RTL. There, self similarities in the function representation, resulting from the regular structure preserved, can be exploited, and as a consequence, symmetrical bitvector values can be found syntactically. In particular, bitvector term-rewriting techniques, isomorphism procedures for specially manipulated term graphs, and combinations thereof, are proposed. This approach dramatically reduces the computational effort needed for functional verification on the block-level and, in particular, for the important problem class of regular designs. It allows the verification of industrial designs previously intractable. The main contributions of this work are in providing a framework for dealing with bitvector functions algebraically, a concise description of bounded model checking on the register transfer level, as well as new reduction techniques and new approaches for finding and exploiting symmetrical values in bitvector functions.

This publication tries to develop mathematical subjects for school from realistic problems. The center of this report are business planning and decision problems which occur in almost all companies. The main topics are: Calculation of raw material demand for given orders, consumption of existing stock and the lot sizing.

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.