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.
In 1969, Fejes Toth conjectured that in Euclidean 3-space any packing of equal balls such that each ball is touched by twelve others consists of hexagonal layers. This article verifies this conjecture.
The first phase of the ATLAS (Australia Telescope Large Area Survey) project surveyed a total 7 square degrees down to 30 micro Jy rms at 1.4 GHz and is the largest sensitive radio survey ever attempted. We report on the scientific achievements of ATLAS to date and plans to extend the project as a path finder for the proposed EMU (Evolutionary map of the Universe) project which has been designed to use ASKAP (Australian Square Kilometre Array Pathfinder).
