On the Sylvester-Gallai and the orchard problem for pseudoline arrangements

We study a non-trivial extreme case of the orchard problem for $12$ pseudolines and we provide a complete classification of pseudoline arrangements having $19$ triple points and $9$ double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of B\"or\"oczky. Since Melchior's inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Gr\"unbaum's problems. We formulate some open problems which involve our new examples of line arrangements.Comment: 5 figures, 11 pages, to appear in Periodica Mathematica Hungaric

Complex Matrix Models and Statistics of Branched Coverings of 2D Surfaces

We present a complex matrix gauge model defined on an arbitrary two-dimensional orientable lattice. We rewrite the model's partition function in terms of a sum over representations of the group U(N). The model solves the general combinatorial problem of counting branched covers of orientable Riemann surfaces with any given, fixed branch point structure. We then define an appropriate continuum limit allowing the branch points to freely float over the surface. The simplest such limit reproduces two-dimensional chiral U(N) Yang-Mills theory and its string description due to Gross and Taylor.Comment: 21 pages, 2 figures, TeX, harvmac.tex, epsf.tex, TeX "big

On a new (21_4) polycyclic configuration

When searching for small 4-configurations of points and lines, polycyclic configurations, in which every symmetry class of points and lines contains the same number of elements, have proved to be quite useful. In this paper we construct and prove the existence of a previously unknown (21_4) conguration, which provides a counterexample to a conjecture of Branko Gr\"unbaum. In addition, we study some of its most important properties; in particular, we make a comparison with the well-known Gr\"unbaum-Rigby configuration. We show that there are exactly two (21_4) geometric polycyclic configurations and seventeen (21_4) combinatorial polycyclic configurations. We also discuss some possible generalizations.Comment: 22 pages, 12 figure

Automorphisms of order three on numerical Godeaux surfaces

We prove that a numerical Godeaux surface cannot have an automorphism of order three

Deletion constructions of symmetric 4-configurations. Part I.

By deletion constructions we mean several methods of generation of new geometric configurations by the judicious deletion of certain points and lines, and introduction of other lines or points. A number of such procedures have recently been developed in a systematic way. We present here one family of such constructions, and will describe other families in the following parts

Measured Radiation and Background Levels During Transmission of Megawatt Electron Beams Through Millimeter Apertures

We report measurements of photon and neutron radiation levels observed while transmitting a 0.43 MW electron beam through millimeter-sized apertures and during beam-off, but accelerating gradient RF-on, operation. These measurements were conducted at the Free-Electron Laser (FEL) facility of the Jefferson National Accelerator Laboratory (JLab) using a 100 MeV electron beam from an energy-recovery linear accelerator. The beam was directed successively through 6 mm, 4 mm, and 2 mm diameter apertures of length 127 mm in aluminum at a maximum current of 4.3 mA (430 kW beam power). This study was conducted to characterize radiation levels for experiments that need to operate in this environment, such as the proposed DarkLight Experiment. We find that sustained transmission of a 430 kW continuous-wave (CW) beam through a 2 mm aperture is feasible with manageable beam-related backgrounds. We also find that during beam-off, RF-on operation, multipactoring inside the niobium cavities of the accelerator cryomodules is the primary source of ambient radiation when the machine is tuned for 130 MeV operation.Comment: 9 pages, 11 figures, submitted to Nuclear Instruments and Methods in Physics Research Section