Experimental results for the Poincar\'e center problem (including an Appendix with Martin Cremer)

We apply a heuristic method based on counting points over finite fields to the Poincar\'e center problem. We show that this method gives the correct results for homogeneous non linearities of degree 2 and 3. Also we obtain new evidence for Zoladek's conjecture about general degree 3 non linearitiesComment: 16 pages, 2 figures, source code of programs at http://www-ifm.math.uni-hannover.de/~bothmer/strudel/. Added references, the result of Example 6.2 is not new. Added two new sections on rationally reversible systems. The 4th codim 7 component we saw only experimentally can now also be identified geometrical

Significance of log-periodic signatures in cumulative noise

Using methods introduced by Scargle in 1978 we derive a cumulative version of the Lomb periodogram that exhibits frequency independent statistics when applied to cumulative noise. We show how this cumulative Lomb periodogram allows us to estimate the significance of log-periodic signatures in the S&P 500 anti-bubble that started in August 2000.Comment: 14 pages, 7 figures; AMS-Latex; introduction rewritten, some points of the exposition clarified. Author-supplied PDF file with high resolution graphics is available at http://btm8x5.mat.uni-bayreuth.de/~bothmer

Geometric Syzygies of Canonical Curves of even Genus lying on a K3-Surface

Based on a recent result of Voisin [2001] we describe the last nonzero syzygy space in the linear strand of a canonical curve C of even genus g=2k lying on a K3 surface, as the ambient space of a k-2-uple embedded P^{k+1}. Furthermore the geometric syzygies constructed by Green and Lazarsfeld [1984] from g^1_{k+1}'s form a non degenerate configuration of finitely many rational normal curves on this P^{k+1}. This proves a natural generalization of Green's conjecture [1984], namely that the geometric syzygies should span the space of all syzygies, in this case.Comment: 29 pages; 5 figure

Reply to [arXiv:1201.5347] "Comment on 'Vortex-assisted photon counts and their magnetic field dependence in single-photon superconducting detectors'"

We argue that cutoff in the London model cannot be settled without use of the microscopic theory

The transcendental lattice of the sextic Fermat surface

We prove that the integral polarized Hodge structure on the transcendental lattice of a sextic Fermat surface is decomposable. This disproves a conjecture of Kulikov related to a Hodge theoretic approach to proving the irrationality of the very general cubic fourfold.Comment: 15 pages; v2: minor changes, streamlined the argument in Section