Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.Comment: 15 page

High impact activities in parks: best management practice and future research

Off-road driving, horseriding, rock climbing and similar activities can be lucrative for tour operators and important for local recreational groups, but contentious for management of national parks and protected areas, both because of safety and liability and because of potentially high environmental impacts. These include spreading weeds and pathogens, starting fires and crushing bird nests on beaches, amongst many others. In Australia as elsewhere, off-road vehicles and horses are allowed only in some places in some parks, and often only under permit. We have very little reliable scientific information to-date on just how serious these impacts may be, and on how well they might be managed through minimal-impact practices such as vehicle washdowns, stockfeed processing and seasonal closures. Such information can only be obtained through site specific ecological studies of the plants, animals and watercourses likely to be affected, differentiating tourism impacts from natural causes and fluctuations. This report examines management strategies for these activities worldwide and in Australia. Suggestions for best management practice and future research agendas are set

A Dense Packing of Regular Tetrahedra

We construct a dense packing of regular tetrahedra, with packing density $D > >.7786157$.Comment: full color versio

In search of dying radio sources in the local universe

Up till now very few dying sources were known, presumably because the dying phase is short at centimeter wavelengths. We therefore have tried to improve the statistics on sources that have ceased to be active, or are intermittently active. The latter sources would partly consist of a fossil radio plasma left over from an earlier phase of activity, plus a recently restarted core and radio jets. Improving the statistics of dying sources will give us a better handle on the evolution of radio sources, in particular the frequency and time scales of radio activity. We have used the WENSS and NVSS surveys, in order to find sources with steep spectral indices, associated with nearby elliptical galaxies. In the cross correlation we presently used only unresolved sources, with flux densities at 1.4 GHz larger than 10 mJy. The eleven candidates thus obtained were observed with the VLA in various configurations, in order to confirm the steepness of the spectra, and to check whether active structures like flat-spectrum cores and jets are present, perhaps at low levels. We estimated the duration of the active and relic phases by modelling the integrated radio spectra using the standard models of spectral evolution. We have found six dying sources and three restarted sources, while the remaining two candidates remain unresolved also with the new VLA data and may be Compact Steep Spectrum sources, with an unusually steep spectrum. The typical age of the active phase, as derived by spectral fits, is in the range 10^7 - 10^8 years. For our sample of dying sources, the age of the relic phase is on average shorter by an order of magnitude than the active phase.Comment: 21 pages, 17 figures, accepted by A&A. For a version with high quality figures, see http://erg.ca.astro.it/preprints/dying2007