Computing A Glimpse of Randomness

A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly non-computable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 64 bits of a Chaitin Omega: 0000001000000100000110001000011010001111110010111011101000010000. Full description of programs and proofs will be given elsewhere.Comment: 16 pages; Experimental Mathematics (accepted

The Differences of Star Formation History Between Merging Galaxies and Field Galaxies in the EDR of the SDSS

Based on the catalog of merging galaxies in the Early Data Release (EDR) of the Sloan Digital Sky Survey (SDSS), the differences of star formation history between merging galaxies and field galaxies are studied statistically by means of three spectroscopic indicators the 4000-\r{A} break strength, the Balmer absorption-line index, and the specific star formation rate. It is found that for early-type merging galaxies the interactions will not induce significant enhancement of the star-formation activity because of its stability and lack of cool gas. On the other hand, late-type merging galaxies always in general display more active star formation than field galaxies on different timescales within about 1Gyr. We also conclude that the mean stellar ages of late-type merging galaxies are younger than those of late-type field galaxies.Comment: 9 pages, 4 figures, accepted for publication in PAS

Stability of Magnetized Disks and Implications for Planet Formation

This paper considers gravitational perturbations in geometrically thin disks with rotation curves dominated by a central object, but with substantial contributions from magnetic pressure and tension. The treatment is general, but the application is to the circumstellar disks that arise during the gravitational collapse phase of star formation. We find the dispersion relation for spiral density waves in these generalized disks and derive the stability criterion for axisymmetric $(m=0)$ disturbances (the analog of the Toomre parameter $Q_T$) for any radial distribution of the mass-to-flux ratio $\lambda$. The magnetic effects work in two opposing directions: on one hand, magnetic tension and pressure stabilize the disk against gravitational collapse and fragmentation; on the other hand, they also lower the rotation rate making the disk more unstable. For disks around young stars the first effect generally dominates, so that magnetic fields allow disks to be stable for higher surface densities and larger total masses. These results indicate that magnetic fields act to suppress the formation of giant planets through gravitational instability. Finally, even if gravitational instability can form a secondary body, it must lose an enormous amount of magnetic flux in order to become a planet; this latter requirement represents an additional constraint for planet formation via gravitational instability and places a lower limit on the electrical resistivity.Comment: accepted in Ap