Orthogonal linear group-subgroup pairs with the same invariants

The main theorem of Galois theory states that there are no finite group-subgroup pairs with the same invariants. On the other hand, if we consider complex linear reductive groups instead of finite groups, the analogous statement is no longer true: There exist counterexample group-subgroup pairs with the same invariants. However, it's possible to classify all these counterexamples for certain types of groups. In [16], we provided the classification for connected complex irreducible groups, and, in this paper, for connected complex orthogonal groups, i.e., groups that preserve some non-degenerate quadratic form.Comment: 27 pages, a part of PhD thesi

Power Laws are Logarithmic Boltzmann Laws

Multiplicative random processes in (not necessaryly equilibrium or steady state) stochastic systems with many degrees of freedom lead to Boltzmann distributions when the dynamics is expressed in terms of the logarithm of the normalized elementary variables. In terms of the original variables this gives a power-law distribution. This mechanism implies certain relations between the constraints of the system, the power of the distribution and the dispersion law of the fluctuations. These predictions are validated by Monte Carlo simulations and experimental data. We speculate that stochastic multiplicative dynamics might be the natural origin for the emergence of criticality and scale hierarchies without fine-tuning.Comment: latex, 9 pages with 3 figure

Spontaneous Scaling Emergence in Generic Stochastic Systems

We extend a generic class of systems which have previously been shown to spontaneously develop scaling (power law) distributions of their elementary degrees of freedom. While the previous systems were linear and exploded exponentially for certain parameter ranges, the new systems fulfill nonlinear time evolution equations similar to the ones encountered in Spontaneous Symmetry Breaking (SSB) dynamics and evolve spontaneously towards "fixed trajectories" indexed by the average value of their degrees of freedom (which corresponds to the SSB order parameter). The "fixed trajectories" dynamics evolves on the edge between explosion and collapse/extinction. The systems present power laws with exponents which in a wide range ($\alpha < -2.$) are universally determined by the ratio between the minimal and the average values of the degrees of freedom. The time fluctuations are governed by Levy distributions of corresponding power. For exponents $\alpha > -2$ there is no "thermodynamic limit" and the fluctuations are dominated by a few, largest degrees of freedom which leads to macroscopic fluctuations, chaos and bursts/intermitency.Comment: latex, 11 page

Variations in the total electron content of the ionosphere at mid-latitudes during quiet sun conditions

Faraday rotation effect on satellite signals used to determine ionospheric electron content at midlatitudes during quiet sun condition

Some aspects of core formation in Mercury

An evaluation of existing theories on the existence of the planet's metallic core is presented. Topics considered are: (1) magnetic fields; (2) surface geology; (3) cosmochemical models