Similar phenomena at different scales: Black Holes, the Sun, Gamma-ray Bursts, Supernovae, Galaxies and Galaxy Clusters

Many similar phenomena occur in astrophysical systems with spatial and mass scales different by many orders of magnitudes. For examples, collimated outflows are produced from the Sun, proto-stellar systems, gamma-ray bursts, neutron star and black hole X-ray binaries, and supermassive black holes; various kinds of flares occur from the Sun, stellar coronae, X-ray binaries and active galactic nuclei; shocks and particle acceleration exist in supernova remnants, gamma-ray bursts, clusters of galaxies, etc. In this report I summarize briefly these phenomena and possible physical mechanisms responsible for them. I emphasize the importance of using the Sun as an astrophysical laboratory in studying these physical processes, especially the roles magnetic fields play in them; it is quite likely that magnetic activities dominate the fundamental physical processes in all of these systems. As a case study, I show that X-ray lightcurves from solar flares, black hole binaries and gamma-ray bursts exhibit a common scaling law of non-linear dynamical properties, over a dynamical range of several orders of magnitudes in intensities, implying that many basic X-ray emission nodes or elements are inter-connected over multi-scales. A future high timing and imaging resolution solar X-ray instrument, aimed at isolating and resolving the fundamental elements of solar X-ray lightcurves, may shed new lights onto the fundamental physical mechanisms, which are common in astrophysical systems with vastly different mass and spatial scales. Using the Sun as an astrophysical laboratory, "Applied Solar Astrophysics" will deepen our understanding of many important astrophysical problems.Comment: 22 pages, 13 figures, invited discourse for the 26th IAU GA, Prague, Czech Republic, Aug. 2006, to be published in Vol. 14 IAU Highlights of Astronomy, Ed. K.A. van der Hucht. Revised slightly to match the final submitted version, after incorporating comments and suggestions from several colleagues. A full-resolution version is available on request from the author at [email protected]

Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense

Recent works have shown that random triangulations decorated by critical ($p=1/2$) Bernoulli site percolation converge in the scaling limit to a $\sqrt{8/3}$-Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by SLE$_6$ in two different ways: 1. The triangulation, viewed as a curve-decorated metric measure space equipped with its graph distance, the counting measure on vertices, and a single percolation interface converges with respect to a version of the Gromov-Hausdorff topology. 2. There is a bijective encoding of the site-percolated triangulation by means of a two-dimensional random walk, and this walk converges to the correlated two-dimensional Brownian motion which encodes SLE$_6$-decorated $\sqrt{8/3}$-LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield (2014); this is sometimes called $\textit{peanosphere convergence}$. We prove that one in fact has $\textit{joint}$ convergence in both of these two senses simultaneously. We also improve the metric convergence result by showing that the map decorated by the full collection of percolation interfaces (rather than just a single interface) converges to $\sqrt{8/3}$-LQG decorated by CLE$_6$ in the metric space sense. This is the first work to prove simultaneous convergence of any random planar map model in the metric and peanosphere senses. Moreover, this work is an important step in an ongoing program to prove that random triangulations embedded into $\mathbb C$ via the so-called $\textit{Cardy embedding}$ converge to $\sqrt{8/3}$-LQG.Comment: 55 pages; 13 Figures. Minor revision according to a referee report. Accepted for publication at EJ

A two-species competition model on Z^d

We consider a two-type stochastic competition model on the integer lattice Z^d. The model describes the space evolution of two ``species'' competing for territory along their boundaries. Each site of the space may contain only one representative (also referred to as a particle) of either type. The spread mechanism for both species is the same: each particle produces offspring independently of other particles and can place them only at the neighboring sites that are either unoccupied, or occupied by particles of the opposite type. In the second case, the old particle is killed by the newborn. The rate of birth for each particle is equal to the number of neighboring sites available for expansion. The main problem we address concerns the possibility of the long-term coexistence of the two species. We have shown that if we start the process with finitely many representatives of each type, then, under the assumption that the limit set in the corresponding first passage percolation model is uniformly curved, there is positive probability of coexistence.Comment: 16 pages, 2 figure

Shape anisotropy of polymers in disordered environment

We study the influence of structural obstacles in a disordered environment on the size and shape characteristics of long flexible polymer macromolecules. We use the model of self-avoiding random walks on diluted regular lattices at the percolation threshold in space dimensions d=2, 3. Applying the Pruned-Enriched Rosenbluth Method (PERM), we numerically estimate rotationally invariant universal quantities such as the averaged asphericity A_d and prolateness S of polymer chain configurations. Our results quantitatively reveal the extent of anisotropy of macromolecules due to the presence of structural defects.Comment: 8 page

Spatial networks with wireless applications

Many networks have nodes located in physical space, with links more common between closely spaced pairs of nodes. For example, the nodes could be wireless devices and links communication channels in a wireless mesh network. We describe recent work involving such networks, considering effects due to the geometry (convex,non-convex, and fractal), node distribution, distance-dependent link probability, mobility, directivity and interference.Comment: Review article- an amended version with a new title from the origina