## Preprints (rote Reihe) des Fachbereich Mathematik

- 322
- The finite-section approximation for ill-posed integral equations on the half-line (2001)
- Integral equations on the half of line are commonly approximated by the finite-section approximation, in which the infinite upper limit is replaced by apositie number called finite-section parameter. In this paper we consider the finite-section approximation for first kind intgral equations which are typically ill-posed and call for regularization. For some classes of such equations corresponding to inverse problems from optics and astronomy we indicate the finite-section parameters that allows to apply standard regularization techniques. Two discretization schemes for the finite-section equations ar also proposed and their efficiency is studied.

- 321
- Homological Mirror Symmetry in Dimension One (2000)
- In this paper we complete the proof began by A. Polishchuk and E. Zaslow of a weak version of Kontsevich's symmetry conjecture for elliptic curves.

- 320
- Presentation of power-ordered sets (2000)
- Power-ordered sets are not always lattices. In the case of distributive lattices we give a description by disjoint of chains. Finite power-ordered sets have a polarity. We introduct the leveled lattices and show examples with trivial tolerance. Finally we give a list of Hasse diagrams of power-ordered sets.

- 319
- Pathwise Kallianpur-Robbins laws for Brownian motion in the plane (1998)
- The Kallianpur-Robbins law describes the long term asymptotic behaviour of the distribution of the occupation measure of a Brownian motion in the plane. In this paper we show that this behaviour can be seen at every typical Brownian path by choosing either a random time or a random scale according to the logarithmic laws of order three. We also prove a ratio ergodic theorem for small scales outside an exceptional set of vanishing logarithmic density of order three.

- 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.