## Fachbereich Mathematik

### Refine

#### Year of publication

- 1996 (30) (remove)

#### Document Type

- Preprint (27)
- Article (1)
- Doctoral Thesis (1)
- Master's Thesis (1)

#### Keywords

- Boltzmann Equation (1)
- Cantor sets (1)
- Collision Operator (1)
- Evolutionary Integral Equations (1)
- Gröbner bases (1)
- Hamiltonian (1)
- K-cardinality trees (1)
- Kinetic Theory of Gases (1)
- Lagrange (1)
- Lavrentiev regularization (1)

- Enskog-like kinetic models for vehicular traffic (1996)
- In the present paper a general criticism of kinetic equations for vehicular traffic is given. The necessity of introducing an Enskog-type correction into these equations is shown. An Enskog-line kinetic traffic flow equation is presented and fluid dynamic equations are derived. This derivation yields new coefficients for the standard fluid dynamic equations of vehicular traffic. Numerical simulations for inhomogeneous traffic flow situations are shown together with a comparison between kinetic and fluid dynamic models.

- Orthogonal and non-orthogonal multiresolution analysis, scale discrete and exact fully discrete wavelet transform on the sphere (1996)
- Based on a new definition of delation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in case of band-limited wavelets.

- Multidimensionale Systemtheorie (1996)
- Die Theorie der mehrdimensionalen Systeme ist ein relativ junges Forschungsgebiet innerhalb der Systemtheorie, erste Arbeiten stammen aus den 70er Jahren. Hauptmotiv für das Studium multidimensionaler Systeme war die Notwendigkeit einer Erweiterung der Theorie der digitalen Filter, die in der klassischen, eindimensionalen Signalverarbeitung (zeitabhängige Signale) Anwendung finden, auf den Bereich der Bildverarbeitung, also auf zweidimensionale Signale.; Die Vorlesung beschäftigt sich daher in ihrem ersten Teil mit skalaren zweidimensionalen Systemen und beschränkt sich im wesentlichen auf den linearen Fall. Untersucht werden zweidimensionale Filter, ihre wichtigsten Eigenschaften, Kausalität und Stabilität, sowie ihre Zustandsraum- realisierungen, etwa die Modelle von Roesser und Fornasini-Marchesini. Parallelen und Unterschiede zur eindimensionalen Systemtheorie werden betont.; Im zweiten Teil der Vorlesung werden allgemeine höherdimensionale und multivariable Systeme behandelt. Für diese Systeme erweist sich der von Jan Willems begründete Zugang zur Systemtheorie, der sogenannte behavioral approach, als zweckmäßig. Grundlegende Ideen dieses Ansatzes sowie eine der wichtigsten Methoden zum Rechnen mit Polynomen in mehreren Variablen, die Theorie der Gröbnerbasen, werden vorgestellt.

- Some Estimates on the Boltzmann Collision Operator (1996)
- The paper presents some new estimates on the gain term of the Boltzmann collision operator. For Maxwellian molecules, it is shown that the L -norm of the gain term can be bounded in terms of the L1 and L -norm of the density function f. In the case of more general collision kernels, like the hard-sphere interaction potential, the gain term is estimated pointwise by the L -norm of the density function and the loss term of the Boltzmann collision operator.

- Asymptotic Behaviour of Self-Organizing Maps with Non-Uniform Stimuli Distribution (1996)
- Here the almost sure convergence of one dimensional Kohonen" s algorithm in its general form, namely, 2k point neightbour setting with a non-uniform stimuli distribution is proved. We show that the asymptotic behaviour of the algorithm is governed by a cooperative system of differential equations which in general is irreducible. The system of differential equation has an asymptotically stable fixed point which a compact subset of its domain of attraction will be visited by the state variable Xn infinitely often.

- Adaptive Load Balance Techniques in Parallel Rarefied Gas Simulations (1996)
- The paper presents some adaptive load balance techniques for the simulation of rarefied gas flows on parallel computers. It is shown that a static load balance is insufficient to obtain a scalable parallel efficiency. Hence, two adaptive techniques are investigated which are based on simple algorithms. Numerical results show that using heuristic techniques one can achieve a sufficiently high efficiency over a wide range of different hardware platforms.

- Singular Optimal Control - The State of the Art (1996)
- The purpose of this paper is to present the state of the art in singular optimal control. If the Hamiltonian in an interval \([t_1,t_2]\) is independent of the control we call the control in this interval singular. Singular optimal controls appear in many applications so that research has been motivated since the 1950s. Often optimal controls consist of nonsingular and singular parts where the junctions between these parts are mostly very difficult to find. One section of this work shows the actual knowledge about the location of the junctions and the behaviour of the control at the junctions. The definition and the properties of the orders (problem order and arc order), which are important in this context, are given, too. Another chapter considers multidimensional controls and how they can be treated. An alternate definition of the orders in the multidimensional case is proposed and a counterexample, which confirms a remark given in the 1960s, is given. A voluminous list of optimality conditions, which can be found in several publications, is added. A strategy for solving optimal control problems numerically is given, and the existing algorithms are compared with each other. Finally conclusions and an outlook on the future research is given.

- Some formulae with logarithmic derivatives related to a quantization of some infinite-dimensional Hamiltonian (1996)
- Some formulae, containing logarithmic derivatives of (smooth) measures on infinitedimensional spaces, arise in quite different situations. In particular, logarithmic derivatives of a measure are inserted in the Schr"odinger equastion in the space consisting of functions that are square integrable with respect to this measure, what allows us to describe very simply a procedure of (canonical) quantization of infinite-dimensional Hamiltonian systems with the linear phase space. Further, the problem of reconstructing of a measure by its logarithmic derivative (that was posed in [1] independently of any applications) can be equivalent either to the problem of finding the "ground state" (considered as some measure) for infinite-dimensional Schr"odinger equation, or to the problem of finding an invariant measure for a stochastic differential equation (that is a central question of so-called stochastic quantization), or to the problem of recenstruc ting "Gibbsian measure by its specification" (i.e. by a collection of finite-dimensional conditional distributions). Logarithmic derivatives of some measure appear in Cameron-Martin-Girsanov-Maruyama formulae and in its generalizations related to arbitrary smooth measures; they allow also to connect these formulae and the Feynman-Kac formulae. This note discusses all these topics. Of course due to its shortness the presentation is formal in main, and precise analitical assumptions are usually absent. Actually only a list of formulae with small comments is given. Let us mention also that we do not consider at all so-called Dirichlet forms to which a great deal of literature is devoted (cf. [3] and references therein to the works of S. Alberion and others).