## Preprints (rote Reihe) des Fachbereich Mathematik

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#### Keywords

- average density (3)
- tangent measure distributions (3)
- Brownian motion (2)
- Palm distributions (2)
- average densities (2)
- density distribution (2)
- lacunarity distribution (2)
- occupation measure (2)
- order-two densities (2)
- Algebraic Geometry (1)
- Cantor sets (1)
- Complexity (1)
- Complexity and performance of numerical algorithms (1)
- Dirichlet series (1)
- Function of bounded variation (1)
- Hochschild homology (1)
- Hochschild-Homologie (1)
- Homologietheorie (1)
- Ill-Posed Problems (1)
- Improperly posed problems (1)
- Integral transform (1)
- Kallianpur-Robbins law (1)
- Linear Integral Equations (1)
- Local completeness (1)
- Moduli Spaces (1)
- Palm distribution (1)
- Quasi-identities (1)
- Rectifiability (1)
- Riemann-Siegel formula (1)
- Sheaves (1)
- Stratifaltigkeiten (1)
- Translation planes (1)
- Verschlüsselung (1)
- Vigenere (1)
- Zyklische Homologie (1)
- algebraic geometry (1)
- cusp forms (1)
- cyclic homology (1)
- fractals (1)
- geometric measure theory (1)
- geometry of measures (1)
- higher order (1)
- hyper-quasi-identities (1)
- hyperquasivarieties (1)
- intersection local time (1)
- invariant theory (1)
- limit models (1)
- locally maximal clone (1)
- log averaging methods (1)
- logarithmic average (1)
- logarithmic averages (1)
- moduli spaces (1)
- non-commutative geometry (1)
- order-three density (1)
- order-two density (1)
- ovoids (1)
- planar Brownian motion (1)
- preservation of relations (1)
- quadratic forms (1)
- quasivarieties (1)
- ratio ergodic theorem (1)
- singular spaces (1)
- singuläre Räume (1)
- strong theorems (1)

312

Vigenere-Verschlüsselung
(1999)

322

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.

296

We show that the occupation measure on the path of a planar Brownian motion run for an arbitrary finite time intervalhas an average density of order three with respect to thegauge function t^2 log(1/t). This is a surprising resultas it seems to be the first instance where gauge functions other than t^s and average densities of order higher than two appear naturally. We also show that the average densityof order two fails to exist and prove that the density distributions, or lacunarity distributions, of order threeof the occupation measure of a planar Brownian motion are gamma distributions with parameter 2.

293

Tangent measure distributions were introduced by Bandt and Graf as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by contractive mappings, which are not similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models of Bedford and Fisher.

295

Tangent measure distributions are a natural tool to describe the local geometry of arbitrary measures of any dimension. We show that for every measure on a Euclidean space and every s, at almost every point, all s-dimensional tangent measure distributions define statistically self-similar random measures. Consequently, the local geometry of general measures is not different from the local geometry of self-similar sets. We illustrate the strength of this result by showing how it can be used to improve recently proved relations between ordinary and average densities.

292

Symmetry properties of average densities and tangent measure distributions of measures on the line
(1995)

Answering a question by Bedford and Fisher we show that for every Radon measure on the line with positive and finite lower and upper densities the one-sided average densities always agree with one half of the circular average densities at almost every point. We infer this result from a more general formula, which involves the notion of a tangent measure distribution introduced by Bandt and Graf. This formula shows that the tangent measure distributions are Palm distributions and define self-similar random measures in the sense of U. Zähle.

338

320

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.

306

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 \(\mu \geq n/4\)

319

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.

313

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.

299

We propose a new discretization scheme for solving ill-posed integral equations of the third kind. Combining this scheme with Morozov's discrepancy principle for Landweber iteration we show that for some classes of equations in such method a number of arithmetic operations of smaller order than in collocation method is required to appoximately solve an equation with the same accuracy.

316

328

In this short note we prove some general results on semi-stable sheaves on P_2 and P_3 with arbitrary linear Hilbert polynomial. Using Beilinson's spectral sequence, we compute free resolutions for this class of semi-stable sheaves and deduce that the smooth moduli spaces M_{r m + s}(P_2) and M_{r m + r - s}(P_2) are birationally equivalent if r and s are coprime.