A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface

Given a graph $G$ cellularly embedded on a surface $\Sigma$ of genus $g$, a cut graph is a subgraph of $G$ such that cutting $\Sigma$ along $G$ yields a topological disk. We provide a fixed parameter tractable approximation scheme for the problem of computing the shortest cut graph, that is, for any $\varepsilon >0$, we show how to compute a $(1+ \varepsilon)$ approximation of the shortest cut graph in time $f(\varepsilon, g)n^3$. Our techniques first rely on the computation of a spanner for the problem using the technique of brick decompositions, to reduce the problem to the case of bounded tree-width. Then, to solve the bounded tree-width case, we introduce a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which may be of independent interest

Quantifying Homology Classes

We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements' size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in $O(\beta^4 n^3 \log^2 n)$ time, where $n$ is the size of the simplicial complex and $\beta$ is the Betti number of the homology group. Third, we discuss different ways of localizing homology classes and prove some hardness results

Comparing Heegaard and JSJ structures of orientable 3-manifolds

The Heegaard genus g of an irreducible closed orientable 3-manifold puts a limit on the number and complexity of the pieces that arise in the Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. For example, if p of the complementary components are not Seifert fibered, then p < g. This result generalizes work of Kobayashi. The Heegaard genus g also puts explicit bounds on the complexity of the Seifert pieces. For example, if the union of the base spaces of the Seifert pieces has Euler characteristic X and there are a total of f exceptional fibers in the Seifert pieces, then f - X is no greater than 3g - 3 - p.Comment: 30 pages, 10 figure

The Magnitude-Size Relation of Galaxies out to z ~ 1

As part of the Deep Extragalactic Evolutionary Probe (DEEP) survey, a sample of 190 field galaxies (I_{814} <= 23.5) in the ``Groth Survey Strip'' has been used to analyze the magnitude-size relation over the range 0.1 < z < 1.1. The survey is statistically complete to this magnitude limit. All galaxies have photometric structural parameters, including bulge fractions (B/T), from Hubble Space Telescope images, and spectroscopic redshifts from the Keck Telescope. The analysis includes a determination of the survey selection function in the magnitude-size plane as a function of redshift, which mainly drops faint galaxies at large distances. Our results suggest that selection effects play a very important role. A first analysis treats disk-dominated galaxies with B/T < 0.5. If selection effects are ignored, the mean disk surface brightness (averaged over all galaxies) increases by ~1.3 mag from z = 0.1 to 0.9. However, most of this change is plausibly due to comparing low luminosity galaxies in nearby redshift bins to high luminosity galaxies in distant bins. If this effect is allowed for, no discernible evolution remains in the disk surface brightness of bright (M_B < -19) disk-dominated galaxies. A second analysis treats all galaxies by substituting half-light radius for disk scale length, with similar conclusions. Indeed, at all redshifts, the bulk of galaxies is consistent with the magnitude-size envelope of local galaxies, i.e., with little or no evolution in surface brightness. In the two highest redshift bins (z > 0.7), a handful of luminous, high surface brightness galaxies appears that occupies a region of the magnitude-size plane rarely populated by local galaxies. Their wide range of colors and bulge fractions points to a variety of possible origins.Comment: 19 pages, 12 figures. Accepted for publication in the Astrophysical Journa

Constructions and Noise Threshold of Hyperbolic Surface Codes

We show how to obtain concrete constructions of homological quantum codes based on tilings of 2D surfaces with constant negative curvature (hyperbolic surfaces). This construction results in two-dimensional quantum codes whose tradeoff of encoding rate versus protection is more favorable than for the surface code. These surface codes would require variable length connections between qubits, as determined by the hyperbolic geometry. We provide numerical estimates of the value of the noise threshold and logical error probability of these codes against independent X or Z noise, assuming noise-free error correction