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#### Erscheinungsjahr

- 1998 (35) (entfernen)

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- Preprint (30)
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#### Schlagworte

- coset enumeration (2)
- particle methods (2)
- subgroup problem (2)
- Analysis (1)
- Boltzmann Equation (1)
- CFD (1)
- Complexity (1)
- Dirichlet series (1)
- Funktionalanalysis (1)
- Gröbner base (1)

#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (35) (entfernen)

In this paper the kinetic model for vehicular traffic developed in [3,4] is considered and theoretical results for the space homogeneous kinetic equation are presented. Existence and uniqueness results for the time dependent equation are stated. An investigation of the stationary equation leads to a boundary value problem for an ordinary differential equation. Existence of the solution and some properties are proved. A numerical investigation of the stationary equation is included.

Groups can be studied using methods from different fields such as combinatorial group theory or string rewriting. Recently techniques from Gröbner basis theory for free monoid rings (non-commutative polynomial rings) respectively free group rings have been added to the set of methods due to the fact that monoid and group presentations (in terms of string rewriting systems) can be linked to special polynomials called binomials. In the same mood, the aim of this paper is to discuss the relation between Nielsen reduced sets of generators and the Todd-Coxeter coset enumeration procedure on the one side and the Gröbner basis theory for free group rings on the other. While it is well-known that there is a strong relationship between Buchberger's algorithm and the Knuth-Bendix completion procedure, and there are interpretations of the Todd-Coxeter coset enumeration procedure using the Knuth-Bendix procedure for special cases, our aim is to show how a verbatim interpretation of the Todd-Coxeter procedure can be obtained by linking recent Gröbner techniques like prefix Gröbner bases and the FGLM algorithm as a tool to study the duality of ideals. As a side product our procedure computes Nielsen reduced generating sets for subgroups in finitely generated free groups.

We present a particle method for the numerical simulation of boundary value problems for the steady-state Boltzmann equation. Referring to some recent results concerning steady-state schemes, the current approach may be used for multi-dimensional problems, where the collision scattering kernel is not restricted to Maxwellian molecules. The efficiency of the new approach is demonstrated by some numerical results obtained from simulations for the (two-dimensional) BEnard's instability in a rarefied gas flow.

In this paper we present a domain decomposition approach for the coupling of Boltzmann and Euler equations. Particle methods are used for both equations. This leads to a simple implementation of the coupling procedure and to natural interface conditions between the two domains. Adaptive time and space discretizations and a direct coupling procedure leads to considerable gains in CPU time compared to a solution of the full Boltzmann equation. Several test cases involving a large range of Knudsen numbers are numerically investigated.

For the determination of the earth" s gravity field many types of observations are available nowadays, e.g., terrestrial gravimetry, airborne gravimetry, satellite-to-satellite tracking, satellite gradiometry etc. The mathematical connection between these observables on the one hand and gravity field and shape of the earth on the other hand, is called the integrated concept of physical geodesy. In this paper harmonic wavelets are introduced by which the gravitational part of the gravity field can be approximated progressively better and better, reflecting an increasing flow of observations. An integrated concept of physical geodesy in terms of harmonic wavelets is presented. Essential tools for approximation are integration formulas relating an integral over an internal sphere to suitable linear combinations of observation functionals, i.e., linear functionals representing the geodetic observables. A scale discrete version of multiresolution is described for approximating the gravitational potential outside and on the earth" s surface. Furthermore, an exact fully discrete wavelet approximation is developed for the case of band-limited wavelets. A method for combined global outer harmonic and local harmonic wavelet modelling is proposed corresponding to realistic earth" s models. As examples, the role of wavelets is discussed for the classical Stokes problem, the oblique derivative problem, satellite-to-satellite tracking, satellite gravity gradiometry, and combined satellite-to-satellite tracking and gradiometry.

Application of Moment Realizability Criteria for Coupling of the Boltzmann and Euler Equations
(1998)

The moment realizability criteria have been used to test the domains of validity of the Boltzmann and Euler Equations. With the help of this criteria teh coupling of the Boltzmann and Euler equations have been performed in two dimensional spatial space. The time evolution of domain decompositions for such equations have been presented in different time steps. The numerical resulta obtained from the coupling code have been compared with those from the pure Boltzmann one.

In order to improve the distribution system for the Nordic countries the BASF AG considered 13 alternative scenarios to the existing system. These involved the construction of warehouses at various locations. For every scenario the transportation, storage, and handling cost incurred was to be as low as possible, where restrictions on the delivery time were given. The scenarios were evaluated according to (minimal) total cost and weighted average delivery time. The results led to a restriction to only three cases, involving only one new warehouse each. For these a more accurate model for the cost was developped and evaluated, yielding results similar to a simple linear model. Since there were no clear preferences between cost and delivery time, the final decision was chosen to represent a compromise between the two criteria.

Knowledge about the distribution of a statistical estimator is important for various purposes like, for example, the construction of confidence intervals for model parameters or the determiation of critical values of tests. A widely used method to estimate this distribution is the so-called bootstrap which is based on an imitation of the probabilistic structure of the data generating process on the basis of the information provided by a given set of random observations. In this paper we investigate this classical method in the context of artificial neural networks used for estimating a mapping from input to output space. We establish consistency results for bootstrap estimates of the distribution of parameter estimates.

Convex Analysis
(1998)

Preface Convex analysis is one of the mathematical tools which is used both explicitly and indirectly in many mathematical disciplines. However, there are not so many courses which have convex analysis as the main topic. More often, parts of convex analysis are taught in courses like linear or nonlinear optimization, probability theory, geometry, location theory, etc.. This manuscript gives a systematic introduction to the concepts of convex analysis. A focus is set to the geometrical interpretation of convex analysis. This focus was one of the reasons why I have decided to restrict myself to the finite dimensional case. Another reason for this restriction is that in the infinite dimensional case many proofs become more difficult and more technical. Therefore, it would not have been possible (for me) to cover all the topics I wanted to discuss in this introductory text in the infinite dimensional case, too. Anyway, I am convinced that even for someone who is interested in the infinite dimensional case this manuscript will be a good starting point. When I offered a course in convex analysis in the Wintersemester 1997/1998 (upon which this manuscript is based) a lot of students asked me how this course fits in their own studies. Because this manuscript will (hopefully) be used by some students in the future, I will give here some of the possible statements to answer this very question. - Convex analysis can be seen as an extension of classical analysis, in which still we get many of the results, like a mean-value theorem, with less assumptions on the smoothness of the function. - Convex analysis can be seen as a foundation of linear and nonlinear optimization which provides many tools to handle concepts in optimization much easier (for example the Lemma of Farkas). - Finally, convex analysis can be seen as a link between abstract geometry and very algorithmic oriented computational geometry. As already explained before, this manuscript is based on a one semester course and therefore cannot cover all topics and discuss all aspects of convex analysis in detail. To guide the interested reader I have included a list of nice books about this subject at the end of the manuscript. It should be noted that the philosophy of this course follows [3], [4] and THE BOOK of modern convex analysis [6]. The geometrical emphasis however, is also related to intentions of [1].^L