## Fachbereich Mathematik

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- On Risk Rates and Large Deviations in Finite Markov Chain Experiments (1997)
- The observation of an ergodic Markov chain asymptotically allows perfect identification of the transition matrix. In this paper we determine the rate of the information contained in the first n observations, provided the unknown transition matrix belongs to a known finite set. As an essential tool we prove new refinements of the large deviation theory of the empirical pair measure of finite Markov chains. Keywords: Markov Chain, Entropy, Bayes risk, Large Deviations.

- Shannon-Optimal Priors on iid Statistical Experiments Converge Weakly to Jeffreys Prior (1999)
- In 1979, J.M. Bernardo argued heuristically that in the case of regular product experiments his information theoretic reference prior is equal to Jeffreys' prior. In this context, B.S. Clarke and A.R. Barron showed in 1994, that in the same class of experiments Jeffreys' prior is asymptotically optimal in the sense of Shannon, or, in Bayesian terms, Jeffreys' prior is asymptotically least favorable under Kullback Leibler risk. In the present paper, we prove, based on Clarke and Barron's results, that every sequence of Shannon optimal priors on a sequence of regular iid product experiments converges weakly to Jeffreys' prior. This means that for increasing sample size Kullback Leibler least favorable priors tend to Jeffreys' prior.

- Differentiable families of measures O.G. Smolyanov Faculty of Mechanics and Mathematics (1998)
- The paper studies differential and related properties of functions of a real variable with values in the space of signed measures. In particular the connections between different definitions of differentiability are described corresponding to different topologies on the measures. Some conditions are given for the equivalence of the measures in the range of such a function. These conditions are in terms of socalled logarithmic derivatives and yield a generalization of the Cameron-Martin-Maruyama-Girsanov formula. Questions of this kind appear both in the theory of differentiable measures on infinite-dimensional spaces and in the theory of statistical experiments.

- Some formulae with logarithmic derivatives related to a quantization of some infinite-dimensional Hamiltonian (1996)
- Some formulae, containing logarithmic derivatives of (smooth) measures on infinitedimensional spaces, arise in quite different situations. In particular, logarithmic derivatives of a measure are inserted in the Schr"odinger equastion in the space consisting of functions that are square integrable with respect to this measure, what allows us to describe very simply a procedure of (canonical) quantization of infinite-dimensional Hamiltonian systems with the linear phase space. Further, the problem of reconstructing of a measure by its logarithmic derivative (that was posed in [1] independently of any applications) can be equivalent either to the problem of finding the "ground state" (considered as some measure) for infinite-dimensional Schr"odinger equation, or to the problem of finding an invariant measure for a stochastic differential equation (that is a central question of so-called stochastic quantization), or to the problem of recenstruc ting "Gibbsian measure by its specification" (i.e. by a collection of finite-dimensional conditional distributions). Logarithmic derivatives of some measure appear in Cameron-Martin-Girsanov-Maruyama formulae and in its generalizations related to arbitrary smooth measures; they allow also to connect these formulae and the Feynman-Kac formulae. This note discusses all these topics. Of course due to its shortness the presentation is formal in main, and precise analitical assumptions are usually absent. Actually only a list of formulae with small comments is given. Let us mention also that we do not consider at all so-called Dirichlet forms to which a great deal of literature is devoted (cf. [3] and references therein to the works of S. Alberion and others).

- Smooth probability measures and associated differential operators (1999)
- We compare different notions of differentiability of a measure along a vector field on a locally convex space. We consider in the L2-space of a differ entiable measure the analoga of the classical concepts of gradient, divergence and Laplacian (which coincides with the OrnsteinUhlenbeck operator in the Gaussian case). We use these operators for the extension of the basic results of Malliavin and Stroock on the smoothness of finite dimensional image measures under certain nonsmooth mappings to the case of non-Gaussian measures. The proof of this extension is quite direct and does not use any Chaos-decomposition. Finally, the role of this Laplacian in the procedure of quantization of anharmonic oscillators is discussed.

- Some reflections on and experiences with SPLIFS (1999)
- Starting from the uniqueness question for mixtures of distributions this review centers around the question under which formally weaker assumptions one can prove the existence of SPLIFs, in other words perfect statistics and tests. We mention a couple of positive and negative results which complement the basic contribution of David Blackwell in 1980. Typically the answers depend on the choice of the set theoretic axioms and on the particular concepts of measurability.

- Sudakov's typical marginals, random linear functionals and a conditional central limit theorem (1997)
- V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a highdimensional second order measure are close to each other in most directions. Extending this and a related result in the context of projection pursuit of P. Diaconis and D. Freedman [Dia84], we give for a probability measure P and a random (a.s.) linear functional F on a Hilbert space simple sufficient conditions under which most of the one-dimensional images of P under F are close to their canonical mixture which turns out to be almost a mixed normal distribution. Using the concept of approximate conditioning we deduce a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions.

- Differenzierbare Maße und Stochastische Analysis (1998)
- Bekanntlich gibt es keinen befriedigenden unendlich dimensionalen Ersatz für das Lebesgue-Mass. Andererseits lassen sich viele Techniken klassischer Analysis auch auf unendlich dimensionale Situationen übertragen. Eine Möglichkeit hierzu gibt die Theorie differenzierbarer Masse. Man definiert Richtungsableitungen für Masse ähnlich wie für Funktionen. Eines der zentralen Beispiele ist das Wiener-Mass. Stochastische Integration bezüglich der Brownschen Bewegung, insbesondere das Skorokhod-Integral ergeben sich in natürlicher Weise durch diesen Ansatz und auch die Grundideen des MalliavinKalküls lassen sich in diesem Rahmen einfach erläutern. Die Vorträge geben die meisten Beweise.

- Identifying a finite graph by its random walk (1999)
- In the following we illustrate by two examples and simplify the statement of the main result of a joint paper with Peter Scheffel

- The normalisation: a new algorithm, implementation and comparisons (1997)
- Primary decomposition of an ideal in a polynomial ring over a field belongs to the indispensable theoretical tools in commutative algebra and algebraic geometry. Geometrically it corresponds to the decomposition of an affine variety into irreducible components and is, therefore, also an important geometric concept.The decomposition of a variety into irreducible components is, however, slightly weaker than the full primary decomposition, since the irreducible components correspond only to the minimal primes of the ideal of the variety, which is a radical ideal. The embedded components, although invisible in the decomposition of the variety itself, are, however, responsible for many geometric properties, in particular, if we deform the variety slightly. Therefore, they cannot be neglected and the knowledge of the full primary decomposition is important also in a geometric context.In contrast to the theoretical importance, one can find in mathematical papers only very few concrete examples of non-trivial primary decompositions because carrying out such a decomposition by hand is almost impossible. This experience corresponds to the fact that providing efficient algorithms for primary decomposition of an ideal I ae K[x1; : : : ; xn], K a field, is also a difficult task and still one of the big challenges for computational algebra and computational algebraic geometry.All known algorithms require Gr"obner bases respectively characteristic sets and multivariate polynomial factorization over some (algebraic or transcendental) extension of the given field K. The first practical algorithm for computing the minimal associated primes is based on characteristic sets and the Ritt-Wu process ([R1], [R2], [Wu], [W]), the first practical and general primary decomposition algorithm was given by Gianni, Trager and Zacharias [GTZ]. New ideas from homological algebra were introduced by Eisenbud, Huneke and Vasconcelos in [EHV]. Recently, Shimoyama and Yokoyama [SY] provided a new algorithm, using Gr"obner bases, to obtain the primary decompositon from the given minimal associated primes.In the present paper we present all four approaches together with some improvements and with detailed comparisons, based upon an analysis of 34 examples using the computer algebra system SINGULAR [GPS]. Since primary decomposition is a fairly complicated task, it is, therefore, best explained by dividing it into several subtasks, in particular, while sometimes only one of these subtasks is needed in practice. The paper is organized in such a way that we consider the subtasks separately and present the different approaches of the above-mentioned authors, with several tricks and improvements incorporated. Some of these improvements and the combination of certain steps from the different algorithms are essential for improving the practical performance.