Morphing of Triangular Meshes in Shape Space

We present a novel approach to morph between two isometric poses of the same non-rigid object given as triangular meshes. We model the morphs as linear interpolations in a suitable shape space $\mathcal{S}$. For triangulated 3D polygons, we prove that interpolating linearly in this shape space corresponds to the most isometric morph in $\mathbb{R}^3$. We then extend this shape space to arbitrary triangulations in 3D using a heuristic approach and show the practical use of the approach using experiments. Furthermore, we discuss a modified shape space that is useful for isometric skeleton morphing. All of the newly presented approaches solve the morphing problem without the need to solve a minimization problem.Comment: Improved experimental result

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

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