Poisson point processes,excursions and stable processes in two-dimensional structures

Itô’s contributions lie at the root of stochastic calculus and of the theory of excursions. These ideas are also very useful in the study of conformally invariant two-dimensional structures, via conformal loop ensembles, excursions of Schramm–Loewner evolutions and Poisson point processes of Brownian loops.     

Zur projektiven Differentialgeometrie der Flächen,die von einer einparametrigen Schar von Kegelschnitten erzeugt werden

Ohne Zusammenfassung     

The true self-repelling motion

We construct and study a continuous real-valued random process, which is of a new type: It is self-interacting (self-repelling) but only in a local sense: it only feels the self-repellance due to its occupation-time measure density in the `immediate neighbourhood' of the point it is just visiting. We focus on the most natural process with these properties that we call `true self-repelling motion'. This is the continuous counterpart to the integer-valued `true' self-avoiding walk, which had been studied among others by the first author. One of the striking properties of true self-repelling motion is that, although the couple (X _t , occupation-time measure of X at time t) is a continuous Markov process, X is not driven by a stochastic differential equation and is not a semi-martingale. It turns out, for instance, that it has a finite variation of order 3/2, which contrasts with the finite quadratic variation of semi-martingales. One of the key-tools in the construction of X is a continuous system of coalescing Brownian motions similar to those that have been constructed by Arratia [A1, A2]. We derive various properties of X (existence and properties of the occupation time densities L _t (x), local variation, etc.) and an identity that shows that the dynamics of X can be very loosely speaking described as follows: −dX _t is equal to the gradient (in space) of L _t (x), in a generalized sense, even though x↦L _t (x) is not differentiable. Received: 15 April 1997 / Revised version: 30 January 1998     

Die zweifachen Blutelschen Kegelschnittflächen

A surface in projective space generated by a one parameter family of conics is called a conic surface of Blutel if the tangent planes of taken along a generating conic, envelop a quadratic cone. If the conjugate curves (with respect to the generating conics) are conics, too, we call a two-fold Blutel's conic surface. In an earlier paper [4] it was shown that the planes of both conic families, the generating and the conjugate one, belong to a pencil, each. The present paper completes these investigations by integrating the derivative equations (3), (8), (9), (10). As a final result, a complete classification of all these surfaces is given. They are all algebraic of at most fourth order and furthermore—besides the quadrics and certain degenerate cases—they are complex projectively equivalent to the cyclides of Dupin.     

Über polycyclische 2-Alken-1-on—Guanidin-Kondensate

Guanidine reacts with 1,3-diphenyl-2-propen-1-one to yield not only the dihydropyrimidine2 a ² and the aromatic compound3 a ², but also a bicyclic base, 2,4,6,8-tetraphenyl-2,8-dihydro-1H-pyrimido[1,2–a]pyrimidine (4). Action of 4-phenyl-3-buten-2-one on guanidine generates a 22-condensate, 7-methyl-4,5-diphenyl-4,4a,5,6,7,8,10,10a-octahydro-7,10a-methano-pyrimido[4,5–d]diazocine-2,9(1H, 3H)-diimine12 R besides the methylphenylpyrimidinamine3 b ². The structural formulae4 and12 are proved by NMR-spectra. The mass spectra and the mechanisms of the formation of4 and12 are also discussed.

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Herrn Prof. Dr.O. Hromatka zum 70. Geburtstag gewidmet.     

Some mathematical aspects of the scaling limit of critical two-dimensional systems

It has been observed long ago that many systems from statistical physics behave randomly on macroscopic level at their critical temperature. In two dimensions, these phenomena have been classified by theoretical physicists thanks to conformal field theory, that led to the derivation of the exact value of various critical exponents that describe their behavior near the critical temperature. In the last couple of years, combining ideas of complex analysis and probability theory, mathematicians have constructed and studied a family of random fractals (called ‘Schramm-Loewner evolutions’ or SLE) that describe the only possible conformally invariant limits of the interfaces for these models. This gives a concrete construction of these random systems, puts various predictions on a rigorous footing, and leads to further understanding of their behavior. The goal of this paper is to survey some of these recent mathematical developments, and to describe a couple of basic underlying ideas. We will also briefly describe some very recent and ongoing developments relating SLE, Brownian loop soups and conformal field theory.     

Zur projektiven Differentialgeometrie der Flächen,die von einer einparametrigen Schar von Kegelschnitten erzeugt werden

Ohne ZusammenfassungDie vorliegende Arbeit ist zusammen mit den noch folgenden Teilen II und III aus einer von der Naturwissenschaftlich-mathematischen Fakultät der Albert-Ludwigs-Universität in Freiburg i. Br. angenommenen Habilitationsschrift hervorgegangen.     

Über einander ein- und umbeschriebene Laplace-Ketten in mehrdimensionalen projektiven Räumen

Ohne ZusammenfassungMit 3 Textabbildungen     

Über sich schließende Figuren asymptotisch transformierter Kurven undW-transformierter Flächen

Ohne Zusammenfassung