## Fachbereich Mathematik

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- 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.

- On the classification of vector bundles on projective curves (1999)
- We consider the "representation type" of the classification problem of vector bundles on a projective curve. We prove that this problem is always either finite, or tame, or wild and we completely describe those curves which are of finite, resp. tame, vector bundle type. We also give a complete list of indecomposable vector bundles for the finite and tame cases.

- Geometry of families of nodal curves on the blown-up projective plane (1999)
- Let P2r be the projective plane blown up at r generic points. Denote by E0; E1; : : : ; Er the strict transform of a generic straight line on P2 and the exceptional divisors of the blown-up points on P2r respectively. We consider the variety Virr of all irreducible curves C with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility on families of reducible curves. For r ^ 9 we give the complete answer concerning the existence of nodal curves in Virr.

- Four characterizations of scalar-type operators with spectrum in a half-line (1997)
- \(C^0\)-scalar-type spectrality criterions for operators \(A\), whose resolvent set contains the negative reals, are provided. The criterions are given in terms of growth conditions on the resolvent of \(A\) and the semi-group generated by \(A\).These criterions characterize scalar-type operators on the Banach space \(X\), if and only if \(X\) has no subspace isomorphic to the space of complex null-sequences.