Ten Conferences WORDS: Open Problems and Conjectures

In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding informations concerning advances in problems solving

Radial Mixing in Galactic Disks: The Effects of Disk Structure and Satellite Bombardment

We use a suite of numerical simulations to investigate the mechanisms and effects of radial migration of stars in disk galaxies like the Milky Way (MW). An isolated, collisionless stellar disk with a MW-like scale-height shows only the radial "blurring" expected from epicyclic orbits. Reducing the disk thickness or adding gas to the disk substantially increases the level of radial migration, induced by interaction with transient spiral arms and/or a central bar. We also examine collisionless disks subjected to gravitational perturbations from a cosmologically motivated satellite accretion history. In the perturbed disk that best reproduces the observed properties of the MW, 20% of stars that end up in the solar annulus 7 kpc < R < 9 kpc started at R < 6 kpc, and 7% started at R > 10 kpc. This level of migration would add considerable dispersion to the age-metallicity relation of solar neighborhood stars. In the isolated disk models, the probability of migration traces the disk's radial mass profile, but in perturbed disks migration occurs preferentially at large radii, where the disk is more weakly bound. The orbital dynamics of migrating particles are also different in isolated and perturbed disks: satellite perturbations drive particles to lower angular momentum for a given change in radius. Thus, satellite perturbations appear to be a distinct mechanism for inducing radial migration, which can operate in concert with migration induced by bars and spiral structure. We investigate correlations between changes in radius and changes in orbital circularity or vertical energy, identifying signatures that might be used to test models and distinguish radial migration mechanisms in chemo-dynamical surveys of the MW disk.Comment: Full resolution paper available at http://www.astronomy.ohio-state.edu/~bird/bkw11_fullres.pdf . 13 pages, 12 figures; emulate MNRAS format. Accepted for publication in MNRA

Graph Isomorphism for unit square graphs

In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time. Since the recognition problem for this class of graphs is NP-hard we can not rely on standard techniques for geometric graphs based on constructing a canonical realization. Instead, we develop new techniques which combine structural insights into the class of unit square graphs with understanding of the automorphism group of such graphs. For the latter we introduce a generalization of bounded degree graphs which is used to capture the main structure of unit square graphs. Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.Comment: 31 pages, 6 figure