Abstract. We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to solve the sphere packing problem in dimensions 8 and 24.

Although most wireless terrestrial networks are based on twodimensional (2D) design, in reality, such networks operate in threedimensions (3D). Since most often the size (i.e., the length and the width) of such terrestrial networks is significantly larger than the differences in the third dimension (i.e., the height) of the nodes, the 2D assumption is somewhat justified and usually it does not lead to major inaccuracies. However, in some environments, this is not the case; the underwater, atmospheric, or space communications being such apparent examples. In fact, recent interest in underwater acoustic ad hoc and sensor networks hints at the need to understand how to design networks in 3D. Unfortunately, the design of 3D networks is surprisingly more difficult than the design of 2D networks. For example, proofs of Kelvin's conjecture and Kepler's conjecture required centuries of research to achieve breakthroughs,

: We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay star. We conjecture that the score of every Delaunay star is at most the score of the stars in the face-centered cubic and hexagonal close packings. This conjecture implies the Kepler conjecture. To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture. Contents: 1. Introduction, 2. The Program, 3. Quasi-regular Tetrahedra, 4. Quadrilaterals, 5. Restrictions, 6. Combinatorics, 7. The Method of Subdivision, 8. Explicit Formulas for Compression, Volume, and Angle, 9. Floating-Point Calculations. Appendix. D. J. Muder's Proof of Theorem 6.1. Sec...

We obtain computer assisted proofs of several spherical volume inequalities that appear in the analysis of semidefinite programming based approximation algorithms for Boolean constraint satisfaction problems. These inequalities imply, in particular, that the performance ratio achieved by the MAX 3-SAT approximation algorithm of Karloff and Zwick is indeed 7/8, as conjectured by them, and that the performance ratio of the MAX 3-CSP algorithm of the author is indeed ½. Other results are also implied. The computer assisted proofs are obtained using a system called REALSEARCH written by the author. This system uses interval arithmetic to produce rigorous proofs that certain collections of constraints in real variables have no real solution.

Abstract. We present a verified enumeration of tame graphs as defined in Hales ’ proof of the Kepler Conjecture and confirm the completeness of Hales ’ list of all tame graphs while reducing it from 5128 to 2771 graphs. 1

The kissing number k(3) is the maximal number of equal size nonoverlapping spheres in three dimensions that can touch another sphere of the same size. This number was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. The first proof that k(3) = 12 was given by Schütte and van der Waerden only in 1953. In this paper we present a new solution of the Newton-Gregory problem that uses our extension of the Delsarte method. This proof relies on basic calculus and simple spherical geometry. Keywords: Kissing numbers, thirteen spheres problem, Newton-Gregory problem, Legendre polynomials, Delsarte’s method

The series of papers in this volume gives a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than π / √ 18 ≈ 0.74048.... This is the oldest problem in discrete geometry and is an important part of Hilbert’s 18th problem. An example of a

Methods for deciding quantifier-free non-linear arithmetical conjectures over *** are crucial in the formal verification of many real-world systems and in formalised mathematics. While non-linear (rational function) arithmetic over *** is decidable, it is fundamentally infeasible: any general decision method for this problem is worst-case exponential in the dimension (number of variables) of the formula being analysed. This is unfortunate, as many practical applications of real algebraic decision methods require reasoning about high-dimensional conjectures. Despite their inherent infeasibility, a number of different decision methods have been developed, most of which have "sweet spots" --- e.g., types of problems for which they perform much better than they do in general. Such "sweet spots" can in many cases be heuristically combined to solve problems that are out of reach of the individual decision methods when used in isolation. RAHD ("Real Algebra in High Dimensions") is a theorem prover that works to combine a collection of real algebraic decision methods in ways that exploit their respective "sweet-spots." We discuss high-level mathematical and design aspects of RAHD and illustrate its use on a number of examples.

Abstract: An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition of R 3 into polyhedra. The polyhedra are divided into two classes. The first class of polyhedra, called quasi-regular tetrahedra, have density at most that of a regular tetrahedron. The polyhedra in the remaining class have density at most that of a regular octahedron (about 0.7209). Section

Machine interval arithmetic has become an important tool in computer assisted proofs in analysis. Usually, an interval arithmetic computation is just one of many ingredients in such a proof. The purpose of this contribution is to highlight and to summarize the role of interval arithmetic in some outstanding results obtained in computer assisted analysis. 'Outstanding' is defined through the observation that the importance of a mathematical result is at least to some extent indicated by the fact that it has been formulated as a 'conjecture' prior to its proof.