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#### Erscheinungsjahr

- 1998 (35) (entfernen)

#### Dokumenttyp

- Preprint (30)
- Vorlesung (3)
- Wissenschaftlicher Artikel (1)
- Diplomarbeit (1)

#### 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)

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.

Monomial representations and operations for Gröbner bases computations are investigated from an implementation point of view. The technique ofvectorized monomial operations is introduced and it is shown how it expedites computations of Gröbner bases. Furthermore, a rank-based monomialrepresentation and comparison technique is examined and it is concluded that this technique does not yield an additional speedup over vectorizedcomparisons. Extensive benchmark tests with the Computer Algebra System SINGULAR are used to evaluate these concepts.

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.

In this paper we study a particular class of \(n\)-node recurrent neural networks (RNNs).In the \(3\)-node case we use monotone dynamical systems theory to show,for a well-defined set of parameters, that,generically, every orbit of the RNN is asymptotic to a periodic orbit.Then, within the usual 'learning' context of NeuralNetworks, we investigate whether RNNs of this class can adapt their internal parameters soas to 'learn' and then replicate autonomously certain external periodic signals.Our learning algorithm is similar to identification algorithms in adaptivecontrol theory. The main feature of the adaptation algorithm is that global exponential convergenceof parameters is guaranteed. We also obtain partial convergence results in the \(n\)-node case.

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.

In the present paper we investigate the Rayleigh-Benard convection in rarefied gases and demonstrate by numerical experiments the transition from purely thermal conduction to a natural convective flow for a large range of Knudsen numbers from 0.02 downto 0.001. We address to the problem how the critical value for the Rayleigh number defined for incompressible vsicous flows may be translated to rarefied gas flows. Moreover, the simulations obtained for a Knudsen number Kn=0.001 and Froude number Fr=1 show a further transition from regular Rayleigh-Benard cells to a pure unsteady behavious with moving vortices.

In the following an introduction to the level set method will be givenso that one becomes aware of the arising problems, which lead to the needof reinitialization. The problems concerning reinitialization itself will be analysed more detailed and a solution for area loss will be proposed. This solution consists in a combination of the commonly used PDE for reinitialization and extrapolation around the zero level set. Numericalexperiments show rather satisfactory results as far as area loss and computation of curvature are concerned.

We prove that there exists a positive \(\alpha\) such thatfor any integer \(\mbox{$d\ge 3$}\) and any topological types \(\mbox{$S_1,\dots,S_n$}\) of plane curve singularities, satisfying \(\mbox{$\mu(S_1)+\dots+\mu(S_n)\le\alpha d^2$}\), there exists a reduced irreducible plane curve of degree \(d\) with exactly \(n\) singular points of types \(\mbox{$S_1,\dots,S_n$}\), respectively. This estimate is optimal with respect to theexponent of \(d\). In particular, we prove that for any topological type \(S\) there exists an irreducible polynomial of degree \(\mbox{$d\le 14\sqrt{\mu(S)}$}\) having a singular point of type \(S\).

On a family F of probability measures on a measure space we consider the Hellinger and Kullback-Leibler distances. We show that under suitable regulari ty conditions Jeffreys' prior is proportional to the k-dimensional Hausdorff measure w.r.t. Hellinger dis tance respectively to the k2 -dimensional Hausdorff measure w.r.t. Kullback-Leibler distance. The proof i s based on an area-formula for the Hausdorff measure w.r.t. to generalized distances.