Exact beta function from the holographic loop equation of large-N QCD_4

We construct and study a previously defined quantum holographic effective action whose critical equation implies the holographic loop equation of large-N QCD_4 for planar self-avoiding loops in a certain regularization scheme. We extract from the effective action the exact beta function in the given scheme. For the Wilsonean coupling constant the beta function is exacly one loop and the first coefficient agrees with its value in perturbation theory. For the canonical coupling constant the exact beta function has a NSVZ form and the first two coefficients agree with their value in perturbation theory.Comment: 42 pages, latex. The exponent of the Vandermonde determinant in the quantum effective action has been changed, because it has been employed a holomorphic rather than a hermitean resolution of identity in the functional integral. Beta function unchanged. New explanations and references added, typos correcte

Mass Distributions of HST Galaxy Clusters from Gravitational Arcs

Although N-body simulations of cosmic structure formation suggest that dark matter halos have density profiles shallower than isothermal at small radii and steeper at large radii, whether observed galaxy clusters follow this profile is still ambiguous. We use one such density profile, the asymmetric NFW profile, to model the mass distributions of 11 galaxy clusters with gravitational arcs observed by HST. We characterize the galaxy lenses in each cluster as NFW ellipsoids, each defined by an unknown scale convergence, scale radius, ellipticity, and position angle. For a given set of values of these parameters, we compute the arcs that would be produced by such a lens system. To define the goodness of fit to the observed arc system, we define a chi^2 function encompassing the overlap between the observed and reproduced arcs as well as the agreement between the predicted arc sources and the observational constraints on the source system. We minimize this chi^2 to find the values of the lens parameters that best reproduce the observed arc system in a given cluster. Here we report our best-fit lens parameters and corresponding mass estimates for each of the 11 lensing clusters. We find that cluster mass models based on lensing galaxies defined as NFW ellipsoids can accurately reproduce the observed arcs, and that the best-fit parameters to such a model fall within the reasonable ranges defined by simulations. These results assert NFW profiles as an effective model for the mass distributions of observed clusters.Comment: Submitted to ApJ, 14 figures include

An Efficient Algorithm for Computing High-Quality Paths amid Polygonal Obstacles

We study a path-planning problem amid a set $\mathcal{O}$ of obstacles in $\mathbb{R}^2$, in which we wish to compute a short path between two points while also maintaining a high clearance from $\mathcal{O}$; the clearance of a point is its distance from a nearest obstacle in $\mathcal{O}$. Specifically, the problem asks for a path minimizing the reciprocal of the clearance integrated over the length of the path. We present the first polynomial-time approximation scheme for this problem. Let $n$ be the total number of obstacle vertices and let $\varepsilon \in (0,1]$. Our algorithm computes in time $O(\frac{n^2}{\varepsilon ^2} \log \frac{n}{\varepsilon})$ a path of total cost at most $(1+\varepsilon)$ times the cost of the optimal path.Comment: A preliminary version of this work appear in the Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithm