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- Fachbereich Mathematik (35) (remove)

In this paper we study the space-time asymptotic behavior of the solutions and derivatives to th incompressible Navier-Stokes equations. Using moment estimates we obtain that strong solutions to the Navier-Stokes equations which decay in \(L^2\) at the rate of \(||u(t)||_2 \leq C(t+1)^{-\mu}\) will have the following pointwise space-time decay \[|D^{\alpha}u(x,t)| \leq C_{k,m} \frac{1}{(t+1)^{ \rho_o}(1+|x|^2)^{k/2}} \]
where \( \rho_o = (1-2k/n)( m/2 + \mu) + 3/4(1-2k/n)\), and \(|a |= m\). The dimension n is \(2 \leq n \leq 5\) and \(0\leq k\leq n\) and \(\mu \geq n/4\)

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.

Wavelet transform originated in 1980's for the analysis of seismic signals has seen an explosion of applications in geophysics. However, almost all of the material is based on wavelets over Euclidean spaces. This paper deals with the generalization of the theory and algorithmic aspects of wavelets to a spherical earth's model and geophysically relevant vector fields such as the gravitational, magnetic, elastic field of the earth.A scale discrete wavelet approach is considered on the sphere thereby avoiding any type of tensor-valued 'basis (kernel) function'. The generators of the vector wavelets used for the fast evaluation are assumed to have compact supports. Thus the scale and detail spaces are finite-dimensional. As an important consequence, detail information of the vector field under consideration can be obtained only by a finite number of wavelet coefficients for each scale. Using integration formulas that are exact up to a prescribed polynomial degree, wavelet decomposition and reconstruction are investigated for bandlimited vector fields. A pyramid scheme for the recursive computation of the wavelet coefficients from level to level is described in detail. Finally, data compression is discussed for the EGM96 model of the earth's gravitational field.

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.

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.

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.

In this paper we derive nonparametric stochastic volatility models in discrete time. These models generalize parametric autoregressive random variance models, which have been applied quite successfully to nancial time series. For the proposed models we investigate nonparametric kernel smoothers. It is seen that so-called nonparametric deconvolution estimators could be applied in this situation and that consistency results known for nonparametric errors- in-variables models carry over to the situation considered herein.

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.

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.