Astrophysical Fractals: Interstellar Medium and Galaxies

The interstellar medium is structured as a hierachy of gas clouds, that looks self-similar over 6 orders of magnitude in scales and 9 in masses. This is one of the more extended fractal in the Universe. At even larger scales, the ensemble of galaxies looks also self-similar over a certain ranges of scales, but more limited, may be over 3-4 orders of magnitude in scales. These two fractals appear to be characterized by similar Hausdorff dimensions, between 1.6 and 2. The various interpretations of these structures are discussed, in particular formation theories based on turbulence and self-gravity. In the latter, the fractal ensembles are considered in a critical state, as in second order phase transitions, when large density fluctuations are observed, that also obey scaling laws, and look self-similar over an extended range.Comment: 30 pages, 6 figures, Proceedings of "The Chaotic Universe", Roma colloquium, 1-5 Feb 99, World Scientific Advanced Series in Astrophysics and Cosmology, ed. V. Gurzadyan, Li-Zhi Fang and Remo Ruffin

An extension of McDiarmid's inequality

We derive an extension of McDiarmid's inequality for functions $f$ with bounded differences on a high probability set ${\cal Y}$ (instead of almost surely). The behavior of $f$ outside ${\cal Y}$ may be arbitrary. The proof is short and elementary, and relies on an extension argument similar to Kirszbraun's theorem.Comment: Note (4 pages