## Fachbereich Mathematik

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#### Erscheinungsjahr

- 1995 (39) (entfernen)

#### Dokumenttyp

- Preprint (23)
- Wissenschaftlicher Artikel (10)
- Bericht (4)
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- Vorlesung (1)

#### Volltext vorhanden

- ja (39) (entfernen)

#### Schlagworte

- Boltzmann Equation (3)
- Numerical Simulation (3)
- Hysteresis (2)
- Boundary Value Problems (1)
- CAQ (1)
- Evolution Equations (1)
- Fatigue (1)
- Hybrid Codes (1)
- Non-linear wavelet thresholding (1)
- Palm distributions (1)

- On Matroids with Multiple Objectives (1995)
- In this paper we investigate two optimization problems for matroids with multiple objective functions, namely finding the pareto set and the max-ordering problem which conists in finding a basis such that the largest objective value is minimal. We prove that the decision versions of both problems are NP-complete. A solution procedure for the max-ordering problem is presented and a result on the relation of the solution sets of the two problems is given. The main results are a characterization of pareto bases by a basis exchange property and finally a connectivity result for proper pareto solutions.

- Lexicographic Max-Ordering - A Solution Concept for Multicriteria Combinatorial Optimization (1995)
- In this paper we will introduce the concept of lexicographic max-ordering solutions for multicriteria combinatorial optimization problems. Section 1 provides the basic notions of multicriteria combinatorial optimization and the definition of lexicographic max-ordering solutions. In Section 2 we will show that lexicographic max-ordering solutions are pareto optimal as well as max-ordering optimal solutions. Furthermore lexicographic max-ordering solutions can be used to characterize the set of pareto solutions. Further properties of lexicographic max-ordering solutions are given. Section 3 will be devoted to algorithms. We give a polynomial time algorithm for the two criteria case where one criterion is a sum and one is a bottleneck objective function, provided that the one criterion sum problem is solvable in polynomial time. For bottleneck functions an algorithm for the general case of Q criteria is presented.

- Connectedness of Efficient Solutions in Multiple Criteria Combinatorial Optimization (1995)
- In multiple criteria optimization an important research topic is the topological structure of the set \( X_e \) of efficient solutions. Of major interest is the connectedness of \( X_e \), since it would allow the determination of \( X_e \) without considering non-efficient solutions in the process. We review general results on the subject,including the connectedness result for efficient solutions in multiple criteria linear programming. This result can be used to derive a definition of connectedness for discrete optimization problems. We present a counterexample to a previously stated result in this area, namely that the set of efficient solutions of the shortest path problem is connected. We will also show that connectedness does not hold for another important problem in discrete multiple criteria optimization: the spanning tree problem.

- An Overview of the Method of Smoothed Particle Hydrodynamics (1995)
- This report is intended to provide an introduction to the method of SmoothedParticle Hydrodynamics or SPH. SPH is a very versatile, fully Lagrangian, particle based code for solving fluid dynamical problems. Many technical aspects of the method are explained which can then be employed to extend the application of SPH to new problems.

- A Model for the Cloudiness of Fabrics (1995)
- Cloudy inhomogenities in artificial fabrics are graded by a fast method which is based on a Laplacian pyramid decomposition of the fabric image. This band-pass representation takes into account the scale character of the cloudiness. A quality measure of the entire cloudiness is obtained as a weighted mean over the variances of all scales.

- Locally Supported Kernels for Spherical Spline Interpolation (1995)
- By the use of locally supported basis functions for spherical spline interpolation the applicability of this approximation method is spread out since the resulting interpolation matrix is sparse and thus efficient solvers can be used. In this paper we study locally supported kernels in detail. Investigations on the Legendre coefficients allow a characterization of the underlying Hilbert space structure. We show now spherical spline interpolation with polynomial precision can be managed with locally supported kernels, thus giving the possibility to combine approximation techniques based on spherical harmonic expansions with those based on locally supported kernels.

- On a New Condition for Strictly Positive Definite Functions on Spheres (1995)
- Recently, Xu and Cheney (1992) have proved that if all the Legendre coefficients of a zonal function defined on a sphere are positive then the function is strictly positive definite. It will be shown in this paper, that even if finitely many of the Legendre coefficients are zero, the strict positive definiteness can be assured. The results are based on approximation properties of singular integrals, and provide also a completely different proof of the results ofXu and Cheney.

- Learning and Replication of Periodic Signals in Neural-Like Networks (1995)
- The paper describes the concepts and background theory for the analysis of a neural-like network for learning and replication of periodic signals containing a finite number of distinct frequency components. The approach is based on the combination of ideas from dynamic neural networks and systems and control theory where concepts of dynamics, adaptive control and tracking of specified time signals are fundamental. The proposed procedure is a two stage process consisting of a learning phase when the network is driven by the required signal followed by a replication phase where the network operates in an autonomous feedback mode whilst continuing to generate the required signal to a desired acccuracy for a specified time. The analysis draws on currently available control theory and, in particular, on concepts from model reference adaptive control.