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

A Dense Packing of Regular Tetrahedra

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

Certification of Bounds of Non-linear Functions: the Templates Method

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table

Combined oil spill modelling and shoreline sensitivity analysis for contingency planning in the Irish Sea

Offshore oil spills often result in severe environmental and socio-economic consequences. This work focuses on a busy, yet poorly studied part of NW Europe, the Irish Sea, to assess the impact of future oil spills on the nearby coast. By integrating numerical models and shoreline sensitivity analyses for two confined areas, Liverpool Bay and Milford Haven, this work acknowledges wind direction and speed as principal controls on the movement of oil under winter/storm conditions and in shallow waters. Ocean currents play a secondary role, but are significant in deeper waters and in low-wind summer conditions. The temporal elements used in the modelling thus stress that when the spill occurs is just as important as where. As a corollary, the fate of spilled oil is determined in this work for distinct scenarios and types. Response strategies are recommended to minimise the impact of future spills on coastal populations

Contact numbers for congruent sphere packings in Euclidean 3-space

Continuing the investigations of Harborth (1974) and the author (2002) we study the following two rather basic problems on sphere packings. Recall that the contact graph of an arbitrary finite packing of unit balls (i.e., of an arbitrary finite family of non-overlapping unit balls) in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have in Euclidean 3-space. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. Finally, we are interested also in the following more special version of the above problem. Namely, let us imagine that we are given a lattice unit sphere packing with the center points forming the lattice L in Euclidean 3-space (and with certain pairs of unit balls touching each other) and then let us generate packings of n unit balls such that each and every center of the n unit balls is chosen from L. Just as in the general case we are interested in finding good estimates for the largest contact number of the packings of n unit balls obtained in this way.Comment: 18 page

On a computer-aided approach to the computation of Abelian integrals

An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four

Mathematical practice, crowdsourcing, and social machines

The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question answering system {\it mathoverflow} contains around 40,000 mathematical conversations, and {\it polymath} collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of "soft" aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a "social machine", a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent Computer Mathematics, CICM 2013, July 2013 Bath, U

From crystal to amorphopus: a novel route towards unjamming in soft disk packings

It is presented a numerical study on the unjamming packing fraction of bi- and polydisperse disk packings, which are generated through compression of a monodisperse crystal. In bidisperse systems, a fraction f_+ = 40% up to 80% of the total number of particles have their radii increased by \Delta R, while the rest has their radii decreased by the same amount. Polydisperse packings are prepared by changing all particle radii according to a uniform distribution in the range [-\Delta R,\Delta R]. The results indicate that the critical packing fraction is never larger than the value for the initial monodisperse crystal, \phi = \pi/12, and that the lowest value achieved is approximately the one for random close packing. These results are seen as a consequence of the interplay between the increase in small-small particle contacts and the local crystalline order provided by the large-large particle contacts.Comment: two columns, 14 pages, 12 figures, accepted for publication in Eur. Phys. J.

Three-dimensional random Voronoi tessellations: From cubic crystal lattices to Poisson point processes

We perturb the SC, BCC, and FCC crystal structures with a spatial Gaussian noise whose adimensional strength is controlled by the parameter a, and analyze the topological and metrical properties of the resulting Voronoi Tessellations (VT). The topological properties of the VT of the SC and FCC crystals are unstable with respect to the introduction of noise, because the corresponding polyhedra are geometrically degenerate, whereas the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. For weak noise, the mean area of the perturbed BCC and FCC crystals VT increases quadratically with a. In the case of perturbed SCC crystals, there is an optimal amount of noise that minimizes the mean area of the cells. Already for a moderate noise (a>0.5), the properties of the three perturbed VT are indistinguishable, and for intense noise (a>2), results converge to the Poisson-VT limit. Notably, 2-parameter gamma distributions are an excellent model for the empirical of of all considered properties. The VT of the perturbed BCC and FCC structures are local maxima for the isoperimetric quotient, which measures the degre of sphericity of the cells, among space filling VT. In the BCC case, this suggests a weaker form of the recentluy disproved Kelvin conjecture. Due to the fluctuations of the shape of the cells, anomalous scalings with exponents >3/2 is observed between the area and the volumes of the cells, and, except for the FCC case, also for a->0. In the Poisson-VT limit, the exponent is about 1.67. As the number of faces is positively correlated with the sphericity of the cells, the anomalous scaling is heavily reduced when we perform powerlaw fits separately on cells with a specific number of faces

Cell shape analysis of random tessellations based on Minkowski tensors

To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic models. In the context of image analysis of synthetic and biological materials, this question is central to the problem of inferring information about formation processes from spatial measurements of resulting random structures. We address this question by a theory-based simulation study of shape indices derived from Minkowski tensors for a variety of tessellation models. We focus on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, and Gibbs hard-core and random sequential absorption processes as well as for Laguerre tessellations of polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data are complemented by mechanically stable crystalline sphere and disordered ellipsoid packings and area-minimising foam models. We find that shape indices of individual cells are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of the shape indices between many of these tessellation models. Given a realization of a tessellation, these shape indices can narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by density-resolved volume-shape correlations.Comment: Chapter of the forthcoming book "Tensor Valuations and their Applications in Stochastic Geometry and Imaging" in Lecture Notes in Mathematics edited by Markus Kiderlen and Eva B. Vedel Jense