## Fachbereich Mathematik

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#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (892)
- Fraunhofer (ITWM) (2)

Fast reconstruction formulae in x-ray computerized tomography demand the directions, in which the measurements are taken, to be equally distributed over the whole circle. In many applications data can only be provided in a restricted range. Here the intrinsic difficulties are studied by giving a singular value decomposition of the Radon transform in a restricted range. Practical limitations are deduced.

The Trippstadt Problem
(1984)

Close to Kaiserslautern is the town of Trippstadt, which, together with five other small towns forms a local administration unit (Verbandsgemeinde) called Kaiserslautern-Süd. Trippstadt has its own beautiful public swimming pool, which causes problems though; the cost for the upkeep of the pool is higher than the income and thus has to be divided among the towns belonging to the Verbandsgemeinde. Because of this problem the administration wanted to find out which fraction of the total number of pool visitors came from the different towns. They planned to ask each pool guest where he came from. They did this for only three days though because the waiting lines at the cashiers became unbearably long and they could see that because of this the total number of guests would decrease. Then they wondered how to find a better method to get the same data and that was when I was asked to help with the solution of the problem.

In these notes we will discuss some aspects of a problem arising in carindustry. For the sake of clarity we will set the problem into an extremely simplified scheme. Suppose that we have a body which is emitting sound, and that the sound is measured at a finite number of points around the body. We wish to determine the intensity of the sound at an observation point which is moving.

We want to study solid objects in real three dimensional space aiming at two issues:; (i1) modelling solids subject to boolean set algebra, including wire models,; (i2) determining the behaviour of moving solids, e.g. when they collide and the resulting points of contact.; ; This research has been initiated by the FORD Motor Company, Cologne. It is motivated by the intention to provide for a model of an automatical car gear, which gives a high precision basis to the optimization of moving tolerances.

Estimation of P(R kl/gleich S) is considered for the simple stress-strength model of failure. Using the Pareto and Power distributions together with their combined form a useful parametric solution is obtained and is illustrated numerically. It is shown that these models are also applicable when only the tails of distributions for R and S are considered. An application to the failure study concerning the fractures is also included.

Stability and Robustness Properties of Universal Adaptive Controllers for First Order Linear Systems
(1987)

The question: What is an adaptive controller? is as old as the word adaptive control itself. In this paper we will adopt a pragmatic viewpoint which identifies adaptive controllers with nonlinear feedback controllers, designed for classes (families) of linear systems. In contrast to classical linear feedback controllers which are designed for individual systems, these non-linear controllers are required to achieve a specific design objective (such as e.g. stability, tracking or decoupling) for a whole prescribed family of linear systems.