Restoration of isotropy on fractals

We report a new type of restoration of macroscopic isotropy (homogenization) in fractals with microscopic anisotropy. The phenomenon is observed in various physical setups, including diffusions, random walks, resistor networks, and Gaussian field theories. The mechanism is unique in that it is absent in spaces with translational invariance, while universal in that it is observed in a wide class of fractals.Comment: 11 pages, REVTEX, 3 postscript figures. (Compressed and encoded figures archived by "figure" command). To appear in Physical Review Letter

3D Photoionisation Modelling of NGC 6302

We present a three-dimensional photoionisation and dust radiative transfer model of NGC 6302, an extreme, high-excitation planetary nebula. We use the 3D photoionisation code Mocassin} to model the emission from the gas and dust. We have produced a good fit to the optical emission-line spectrum, from which we derived a density distribution for the nebula. A fit to the infrared coronal lines places strong constraints on the properties of the unseen ionising source. We find the best fit comes from using a 220,000 K hydrogen-deficient central star model atmosphere, indicating that the central star of this PN may have undergone a late thermal pulse. We have also fitted the overall shape of the ISO spectrum of NGC 6302 using a dust model with a shallow power-law size distribution and grains up to 1.0 micron in size. To obtain a good fit to the infrared SED the dust must be sufficiently recessed within the circumstellar disk to prevent large amounts of hot dust at short wavelengths, a region where the ISO spectrum is particularly lacking. These and other discoveries are helping to unveil many properties of this extreme object and trace it's evolutionary history.Comment: 8 pages, 4 figures; for the proceedings of "Asymmetric Planetary Nebuale IV," R. L. M. Corradi, A. Manchado, N. Soker ed

Upper estimate of martingale dimension for self-similar fractals

We study upper estimates of the martingale dimension $d_m$ of diffusion processes associated with strong local Dirichlet forms. By applying a general strategy to self-similar Dirichlet forms on self-similar fractals, we prove that $d_m=1$ for natural diffusions on post-critically finite self-similar sets and that $d_m$ is dominated by the spectral dimension for the Brownian motion on Sierpinski carpets.Comment: 49 pages, 7 figures; minor revision with adding a referenc

Transition density of diffusion on Sierpinski gasket and extension of Flory's formula

Some problems related to the transition density u(t,x) of the diffusion on the Sierpinski gasket are considerd, based on recent rigorous results and detailed numerical calculations. The main contents are an extension of Flory's formula for the end-to-end distance exponent of self-avoiding walks on the fractal spaces, and an evidence of the oscillatory behavior of u(t,x) on the Sierpinski gasket.Comment: 11 pages, REVTEX, 2 postscript figure