Particle Candidates of Ultrahigh Energy Cosmic Rays

We discuss candidates for trans-GZK cosmic rays observed in a variety of detectors. Three types of primaries are represented among the abstracts submitted to this meeting: neutrin os causing a Z-burst, protons arising from the decay of ultra-heavy metastable particles and neutrinos within the framework of low scale string-like models of unification. We attempt to evaluate the relative merits of these schemes. No definite conclusion can be reached at this time. However, we point out that some schemes are more credible/predictive than others. Data to be gathered by the Pierre Auger observatories as well as orbiting detectors (OWL, Airwatch...) should be able to decide between the various schemes.Comment: 15 pages, LaTex. Substantially revised to take into account the discussion at HEP2001, Budapest July 200

Multidimensional hyperbolic billiards

The theory of planar hyperbolic billiards is already quite well developed by having also achieved spectacular successes. In addition there also exists an excellent monograph by Chernov and Markarian on the topic. In contrast, apart from a series of works culminating in Sim\'anyi's remarkable result on the ergodicity of hard ball systems and other sporadic successes, the theory of hyperbolic billiards in dimension 3 or more is much less understood. The goal of this work is to survey the key results of their theory and highlight some central problems which deserve particular attention and efforts

Characteristic free description of semi-invariants of 2×22\times 2 matrices

A minimal homogeneous generating system of the algebra of semi-invariants of tuples of two-by-two matrices over an infinite field of characteristic two or over the ring of integers is given. In an alternative interpretation this yields a minimal system of homogeneous generators for the vector invariants of the special orthogonal group of degree four over a field of characteristic two or over the ring of integers. An irredundant separating system of semi-invariants of tuples of two-by-two matrices is also determined, it turns out to be independent of the characteristic.Comment: A crucial reference to a paper of A. Lopatin was adde