Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

CosmoHammer: Cosmological parameter estimation with the MCMC Hammer

We study the benefits and limits of parallelised Markov chain Monte Carlo (MCMC) sampling in cosmology. MCMC methods are widely used for the estimation of cosmological parameters from a given set of observations and are typically based on the Metropolis-Hastings algorithm. Some of the required calculations can however be computationally intensive, meaning that a single long chain can take several hours or days to calculate. In practice, this can be limiting, since the MCMC process needs to be performed many times to test the impact of possible systematics and to understand the robustness of the measurements being made. To achieve greater speed through parallelisation, MCMC algorithms need to have short auto-correlation times and minimal overheads caused by tuning and burn-in. The resulting scalability is hence influenced by two factors, the MCMC overheads and the parallelisation costs. In order to efficiently distribute the MCMC sampling over thousands of cores on modern cloud computing infrastructure, we developed a Python framework called CosmoHammer which embeds emcee, an implementation by Foreman-Mackey et al. (2012) of the affine invariant ensemble sampler by Goodman and Weare (2010). We test the performance of CosmoHammer for cosmological parameter estimation from cosmic microwave background data. While Metropolis-Hastings is dominated by overheads, CosmoHammer is able to accelerate the sampling process from a wall time of 30 hours on a dual core notebook to 16 minutes by scaling out to 2048 cores. Such short wall times for complex data sets opens possibilities for extensive model testing and control of systematics.Comment: Published version. 17 pages, 6 figures. The code is available at http://www.astro.ethz.ch/refregier/research/Software/cosmohamme

Private Prisons and the New Marketplace for Crime

A saner and safer prison policy in the United States begins by ending the scourge of the private prison corporation and returning crime and punishment to public function. We continue by radically reimagining our sentencing policies and reducing them significantly for non-violent crimes. We end the War on Drugs, once and for all, and completely reconfigure our drug and prison policy by legalizing and regulating marijuana use and providing health services to addicts of harder drugs and using prison for only violent drug kingpins and cartel bosses. We stop the current criminalization of immigration in its tracks and block the private prison lobby from influencing legislation in our current immigration policy debates. We provide prisoners a fair wage for work done in prison, allowing them a re-entry account upon release filled with the money they earned while working in prison. We provide humane and habitable prison cells populated by one inmate, as saner and safer crime and punishment policies will imprison far fewer American citizens. At their core, private prisons reflect a continuation of policies that have tainted the criminal justice system with perceptions of arbitrariness, unfairness, and injustice. As this article has shown, the continued proliferation of private prisons does not save taxpayers money, increase prison safety, or elevate the conditions of the prison environment. Conversely, they do the opposite. Inmates are being physically abused, denied medical care, and forced to endure inhumane living conditions, as corporations like CCA and GEO Group realize higher profits from a marketplace in which prisoners are in high demand. Indeed, CCA is a textbook example of the grave injustices that can occur when profit maximization clashes with human dignity. The time has arrived for private prisons to be eliminated and for legislators and courts to realize that this experiment is one that has failed. Until that time comes, Congress should implement purpose-driven reforms to ensure that private prisons can no longer be institutions where inmates have rights but no remedies