## Preprints (rote Reihe) des Fachbereich Mathematik

- 316
- On the distribution of the maximum of sums of dependent random variables (1999)
- The study of queuing theory brings us to the problems of finding to find the limit distribution of the maximal sum of a sequence of random variables and of estimating how close this distribution is to the distribution of the sum.

- 313
- Optimal Order Results for a Class of Regularizazion Methodes Using Unbounded Operators (1999)
- A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an priori parameter choice strategy, it is shown that the method yields optimal order. Error estimates have also been obtained under stronger assumptions on the the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper includes, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also recent results of Tautenhahn (1996) in the setting Hilbert scales.

- 310
- Locally Maximal Clones II (1999)

- 308
- Error estimates for band-limited spherical regularization wavelets in some inverse problems of satellite geodesy (1999)
- In this paper we discuss a special class of regularization methods for solving the satellite gravity gradiometry problem in a spherical framework based on band-limited spherical regularization wavelets. Considering such wavelets as a reesult of a combination of some regularization methods with Galerkin discretization based on the spherical harmonic system we obtain the error estimates of regularized solutions as well as the estimates for regularization parameters and parameters of band-limitation.

- 305
- Applications of Number Theory to Ovoids and Translation Planes (1999)
- In this paper we show that for each prime p=7 there exists a translation plane of order p^2 of Mason-Ostrom type. These planes occur as 6-dimensional ovoids being projections of the 8-dimensional binary ovoids of Conway, Kleidman and Wilson. In order to verify the existence of such projections we prove certain properties of two particular quadratic forms using classical methods form number theory.

- 304
- Clones preserving a quasi-order (1999)
- It is proved that if a finite non-trivial quasi-order is nota linear order then there exist continuum many clones, whichconsist of functions preserving the quasi-order and containall unary functions with this property. It is shown that, fora linear order on a three-element set, there are only 7 suchclones

- 303
- Density theorems for the intersection local times of planar Brownian motion (1999)
- We show that the intersection local times \(\mu_p\) on the intersection of \(p\) independent planar Brownian paths have an average density of order three with respect to the gauge function \(r^2\pi\cdot (log(1/r)/\pi)^p\), more precisely, almost surely, \[ \lim\limits_{\varepsilon\downarrow 0} \frac{1}{log |log\ \varepsilon|} \int_\varepsilon^{1/e} \frac{\mu_p(B(x,r))}{r^2\pi\cdot (log(1/r)/\pi)^p} \frac{dr}{r\ log (1/r)} = 2^p \mbox{ at $\mu_p$-almost every $x$.} \] We also show that the lacunarity distributions of \(\mu_p\), at \(\mu_p\)-almost every point, is given as the distribution of the product of \(p\) independent gamma(2)-distributed random variables. The main tools of the proof are a Palm distribution associated with the intersection local time and an approximation theorem of Le Gall.

- 302
- Brakhage's implicit iteration method and Information Complexity of equations with operators having closed range (1999)
- An a posteriori stopping rule connected with monitoringthe norm of second residual is introduced forBrakhage's implicit nonstationary iteration method, applied to ill-posed problems involving linear operatorswith closed range. It is also shown that for someclasses of equations with such operators the algorithmconsisting in combination of Brakhage's method withsome new discretization scheme is order optimal in the sense of Information Complexity.