Ring Star Formation Rates in Barred and Nonbarred Galaxies

Nonbarred ringed galaxies are relatively normal galaxies showing bright rings of star formation in spite of lacking a strong bar. This morphology is interesting because it is generally accepted that a typical ring forms when material collects near a resonance, set up by the pattern speed of a bar or bar-like perturbation. Our goal in this paper is to examine whether the ring star formation properties are related to the non-axisymmetric gravity potential in general. For this purpose, we obtained H{\alpha} emission line images and calculated the line fluxes and star formation rates (SFRs) for 16 nonbarred SA galaxies and four weakly barred SAB galaxies with rings. For comparison, we combine our observations with a re-analysis of previously published data on five SA, seven SAB, and 15 SB galaxies with rings, three of which are duplicates from our sample. With these data, we examine what role a bar may play in the star formation process in rings. Compared to barred ringed galaxies, we find that the inner ring SFRs and H{\alpha}+[N ii] equivalent widths in nonbarred ringed galaxies show a similar range and trend with absolute blue magnitude, revised Hubble type, and other parameters. On the whole, the star formation properties of inner rings, excluding the distribution of H ii regions, are independent of the ring shapes and the bar strength in our small samples. We confirm that the deprojected axis ratios of inner rings correlate with maximum relative gravitational force Q_g; however, if we consider all rings, a better correlation is found when local bar forcing at the radius of the ring, Q_r, is used. Individual cases are described and other correlations are discussed. By studying the physical properties of these galaxies, we hope to gain a better understanding of their placement in the scheme of the Hubble sequence and how they formed rings without the driving force of a bar.Comment: 55 pages; 21 figures and 9 tables. Article has been accepted for publication in the Astronomical Journa

The 4-Body Problem in a (1+1)-Dimensional Self-Gravitating System

We report on the results of a study of the motion of a four particle non-relativistic one-dimensional self-gravitating system. We show that the system can be visualized in terms of a single particle moving within a potential whose equipotential surfaces are shaped like a box of pyramid-shaped sides. As such this is the largest $N$-body system that can be visualized in this way. We describe how to classify possible states of motion in terms of Braid Group operators, generalizing this to $N$ bodies. We find that the structure of the phase\textcolor{black}{{} space of each of these systems yields a large variety of interesting dynamics, containing regions of quasiperiodicity and chaos. Lyapunov exponents are calculated for many trajectories to measure stochasticity and previously unseen phenomena in the Lyapunov graphs are observed.Comment: 40 pages, 23 figures, to appear in Journal of Mathematical Physic

Dirichlet fundamental domains and complex-projective varieties

We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.Comment: 1 figur

Deconstructing Approximate Offsets

We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using CGAL, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter \delta; its running time additionally depends on \delta. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011, submitted to DC

Scaling Limits for Minimal and Random Spanning Trees in Two Dimensions

A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in $\R^d$. Tightness of the distribution, as $\delta \to 0$, is established for the following two-dimensional examples: the uniformly random spanning tree on $\delta \Z^2$, the minimal spanning tree on $\delta \Z^2$ (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in $\R^2$ with density $\delta^{-2}$. In each case, sample trees are proven to have the following properties, with probability one with respect to any of the limiting measures: i) there is a single route to infinity (as was known for $\delta > 0$), ii) the tree branches are given by curves which are regular in the sense of H\"older continuity, iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds one, iv) there is a random dense subset of $\R^2$, of dimension strictly between one and two, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints, v) branching occurs at countably many points in $\R^2$, and vi) the branching numbers are uniformly bounded. The results include tightness for the loop erased random walk (LERW) in two dimensions. The proofs proceed through the derivation of scale-invariant power bounds on the probabilities of repeated crossings of annuli.Comment: Revised; 54 pages, 6 figures (LaTex