Imprints of log-periodic self-similarity in the stock market

Detailed analysis of the log-periodic structures as precursors of the financial crashes is presented. The study is mainly based on the German Stock Index (DAX) variation over the 1998 period which includes both, a spectacular boom and a large decline, in magnitude only comparable to the so-called Black Monday of October 1987. The present example provides further arguments in favour of a discrete scale-invariance governing the dynamics of the stock market. A related clear log-periodic structure prior to the crash and consistent with its onset extends over the period of a few months. Furthermore, on smaller time-scales the data seems to indicate the appearance of analogous log-periodic oscillations as precursors of the smaller, intermediate decreases. Even the frequencies of such oscillations are similar on various levels of resolution. The related value $\lambda \approx 2$ of preferred scaling ratios is amazingly consistent with those found for a wide variety of other complex systems. Similar analysis of the major American indices between September 1998 and February 1999 also provides some evidence supporting this concept but, at the same time, illustrates a possible splitting of the dynamics that a large market may experience.Comment: 13 pages, LaTeX-REVTeX, 4 PS figures. Significantly extended version to appear in The European Physical Journal

Complementarity in quantum walks

We study discrete-time quantum walks on $d$-cycles with a position and coin-dependent phase-shift. Such a model simulates a dynamics of a quantum particle moving on a ring with an artificial gauge field. In our case the amplitude of the phase-shift is governed by a single discrete parameter $q$. We solve the model analytically and observe that for prime $d$ there exists a strong complementarity property between the eigenvectors of two quantum walk evolution operators that act in the $2d$-dimensional Hilbert space. Namely, if $d$ is prime the corresponding eigenvectors of the evolution operators obey $|\langle v_q|v'_{q'} \rangle| \leq 1/\sqrt{d}$ for $q\neq q'$ and for all $|v_q\rangle$ and $|v'_{q'}\rangle$. We also discuss dynamical consequences of this complementarity. Finally, we show that the complementarity is still present in the continuous version of this model, which corresponds to a one-dimensional Dirac particle.Comment: 5+7 pages, 2 figures, comments welcom

"Magnetoscan": A Modified Hall Probe Scanning Technique for the Detection of Inhomogeneities in Bulk High Temperature Superconductors

We present a novel technique for the investigation of local variations of the critical current density in large bulk superconductors. In contrast to the usual Hall probe scanning technique, the sample is not magnetized as a whole before the scan, but locally by a small permanent magnet, which is fixed near the Hall probe, during the scanning process. The resulting signal can be interpreted as a qualitative measure of the local shielding currents flowing at the surface.Comment: 8 pages, 4 figure

Fermi-Edge Superfluorescence from a Quantum-Degenerate Electron-Hole Gas

We report on the observation of spontaneous bursts of coherent radiation from a quantum-degenerate gas of nonequilibrium electron-hole pairs in semiconductor quantum wells. Unlike typical spontaneous emission from semiconductors, which occurs at the band edge, the observed emission occurs at the quasi-Fermi edge of the carrier distribution. As the carriers are consumed by recombination, the quasi-Fermi energy goes down toward the band edge, and we observe a continuously red-shifting streak. We interpret this emission as cooperative spontaneous recombination of electron-hole pairs, or superfluorescence, which is enhanced by Coulomb interactions near the Fermi edge. This novel many-body enhancement allows the magnitude of the spontaneously developed macroscopic polarization to exceed the maximum value for ordinary superfluorescence, making electron-hole superfluorescence even more "super" than atomic superfluorescence.Comment: 10 pages, 5 figure