A survey of Sylvester's problem and its generalizations

Summary LetP be a finite set of three or more noncollinear points in the plane. A line which contains two or more points ofP is called aconnecting line (determined byP), and we call a connecting lineordinary if it contains precisely two points ofP. Almost a century ago, Sylvester posed the disarmingly simple question:Must every set P determine at least one ordinary line? No solution was offered at that time and the problem seemed to have been forgotten. Forty years later it was independently rediscovered by Erdös, and solved by Gallai. In 1943 Erdös proposed the problem in the American Mathematical Monthly, still unaware that it had been asked fifty years earlier, and the following year Gallai's solution appeared in print. Since then there has appeared a substantial literature on the problem and its generalizations.In this survey we review, in the first two sections, Sylvester's problem and its generalization to higher dimension. Then we gather results about the connecting lines, that is, the lines containing two or more of the points. Following this we look at the generalization to finite collections of sets of points. Finally, the points will be colored and the search will be for monochromatic connecting lines.     

Polarization-selective grating mirrors used in the generation of radial polarization

Two novel methods to control the polarization of laser radiation are presented. The discrimination between different polarization distributions isperformed with a corrugation grating in the top high-index layer of a multilayer mirror, which couples the undesired polarization into a lossy waveguidemode of the multilayer. The generation of radially polarized radiation in a laser resonator is presented as a practical verification of the principle.This revised version was published online in May 2005. The Article Category was removed.This revised version was published online in August 2005 with a corrected cover date.     

Dynamical systems — Past and present

Plenary lecture at the International Congress of Mathematicians (Berlin 1998, August 18–27). Reprinted from Doc. Math. J., DMV, Extra Volume ICM I, 1998, pp. 381–402. ©J.Moser, 1998.     

Molecular thermodynamics on the basis of an equation of state for argon. II. Binary mixtures of weakly nonspherical molecules

Using a molecular perturbation theory based on an equation of state for pure argon, excess properties and vapor-liquid equilibria are predicted for various binary mixtures composed of weakly nonspherical molecules. The results are rather satisfactory and generally much better than obtained using typical empirical methods. It is further demonstrated that a binary parameter in the dispersion energy results in only modest improvement     

Strangelet search and particle production studies in Pb+Pb collisions at 158 AGeV/c at CERN SPS

We searched for long-lived strange quark matter particles, so-calledstrangelets, and studied particle and antiparticle production in Pb+Pb collisions at 158 GeV/c per nucleon at zero degree production angle. We give upper limits for the production of strangelets covering a mass to charge ratio up to 120 GeV/c ² and lifetimest _lab>1.2 μs and plot invariant differential production cross sections as a function of rapidity for a variety of particles.     

Indolalkaloide aus dem blätterpilz cortinarius infractus (agaricales)

From fruit bodies of the gilled agaric $\text{Cortinarius infractus}$ three new β-carboline derivatives infractin (1), 6-hydroxyinfractin (2) and infractopicrin (4) have been isolated. 4 is responsible for the bitter taste of the toadstool.