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92
New upper bounds on sphere packings
, 2001
"... Abstract. We develop an analogue for sphere packing of the linear programming bounds for errorcorrecting 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 s ..."
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Cited by 40 (5 self)
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Abstract. We develop an analogue for sphere packing of the linear programming bounds for errorcorrecting 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.
Coverage and Connectivity in ThreeDimensional Networks
 In MobiCom
, 2006
"... 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 ..."
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Cited by 24 (0 self)
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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,
Sphere Packings I
 Discrete Comput. Geom
, 1996
"... : 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 relate ..."
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Cited by 22 (6 self)
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: 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 facecentered 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. Quasiregular Tetrahedra, 4. Quadrilaterals, 5. Restrictions, 6. Combinatorics, 7. The Method of Subdivision, 8. Explicit Formulas for Compression, Volume, and Angle, 9. FloatingPoint Calculations. Appendix. D. J. Muder's Proof of Theorem 6.1. Sec...
Lowconnectivity and Fullcoverage Three Dimensional Wireless Sensor Networks
 In Proc. of ACM MobiHoc
, 2009
"... Lowconnectivity and fullcoverage three dimensional Wireless Sensor Networks (WSNs) have many realworld applications. By low connectivity, we mean there are at least k disjoint paths between any two sensor nodes in a WSN, where k ≤ 4. In this paper, we design a set of patterns for these networks. ..."
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Cited by 14 (4 self)
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Lowconnectivity and fullcoverage three dimensional Wireless Sensor Networks (WSNs) have many realworld applications. By low connectivity, we mean there are at least k disjoint paths between any two sensor nodes in a WSN, where k ≤ 4. In this paper, we design a set of patterns for these networks. In particular, we design and prove the optimality of 1 and2connectivity patterns under any value of the ratio of communication range rc over sensing range rs, amongregular lattice deployment patterns. We further propose a set of patterns to achieve 3 and4connectivity patterns and investigate the evolutions among all the proposed lowconnectivity patterns. Finally, we study the proposed patterns under several practical settings.
Computer Assisted Proof of Optimal Approximability Results
, 2002
"... 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 3S ..."
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Cited by 13 (4 self)
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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 3SAT approximation algorithm of Karloff and Zwick is indeed 7/8, as conjectured by them, and that the performance ratio of the MAX 3CSP 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.
Flyspeck i: Tame graphs
 International Joint Conference on Automated Reasoning, volume 4130 of LNCS
, 2006
"... 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 ..."
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Cited by 12 (2 self)
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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
Combined Decision Techniques for the Existential Theory of the Reals
 CALCULEMUS
, 2009
"... Methods for deciding quantifierfree nonlinear arithmetical conjectures over *** are crucial in the formal verification of many realworld systems and in formalised mathematics. While nonlinear (rational function) arithmetic over *** is decidable, it is fundamentally infeasible: any general decisi ..."
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Cited by 10 (5 self)
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Methods for deciding quantifierfree nonlinear arithmetical conjectures over *** are crucial in the formal verification of many realworld systems and in formalised mathematics. While nonlinear (rational function) arithmetic over *** is decidable, it is fundamentally infeasible: any general decision method for this problem is worstcase 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 highdimensional 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 "sweetspots." We discuss highlevel mathematical and design aspects of RAHD and illustrate its use on a number of examples.
Multiplicity of Generation, Selection, and Classification Procedures for Jammed HardParticle Packings
 J. Phys. Chem. B
, 2001
"... this paper, we will focus our attention on the question of what is really meant by a "jammed" hardparticle system. The answer to this question is quite subtle, and a failure to appreciate the nuances involved has resulted in considerable ambiguity in the literature on this question. Yet a precise d ..."
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Cited by 10 (5 self)
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this paper, we will focus our attention on the question of what is really meant by a "jammed" hardparticle system. The answer to this question is quite subtle, and a failure to appreciate the nuances involved has resulted in considerable ambiguity in the literature on this question. Yet a precise definition for the term "jammed" is a necessary first step before one can undertake a search for jammed structures in a meaningful way. We will show that there is a multiplicity of definitions for jammed structures. For simplicity and definiteness, we will restrict ourselves to equisized ddimensional hard spheres in ddimen sional Euclidean space. Of particular concern will be the cases of equisized hard circular disks (d ) 2) and equisized hard spheres (d ) 3)
The kissing problem in three dimensions
 Discrete Comput. Geom
"... 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 Schüt ..."
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Cited by 10 (5 self)
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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 NewtonGregory 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, NewtonGregory problem, Legendre polynomials, Delsarte’s method
New conjectural lower bounds on the optimal density of sphere packings
 MATH
, 2006
"... Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a 100yearold lower bound on the maximal packing density due to Minkowski in ddimensional Euclidea ..."
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Cited by 9 (5 self)
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Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a 100yearold lower bound on the maximal packing density due to Minkowski in ddimensional Euclidean space Rd. The asymptotic behavior of this bound is controlled by 2−d in high dimensions. Using an optimization procedure that we introduced earlier [TS02] and a conjecture concerning the existence of disordered sphere packings in Rd, we obtain a conjectural lower bound on the density whose asymptotic behavior is controlled by 2−0.77865...d, thus providing the putative exponential improvement of Minkowski’s bound. The conjecture states that a hardcore nonnegative tempered distribution is a pair correlation function of a translationally invariant disordered sphere packing in Rd for asymptotically large d if and only if the Fourier transform of the autocovariance function is nonnegative. The conjecture is supported by two explicit analytically characterized disordered packings, numerical packing constructions in low dimensions, known necessary conditions that only have relevance in very low dimensions, and the fact that we can recover the forms of known rigorous lower bounds. A byproduct of our approach is an asymptotic conjectural lower bound on the average kissing number whose behavior is controlled by 20.22134...d, which is to be compared to the best known asymptotic lower bound on the individual kissing number of 20.2075...d. Interestingly, our optimization procedure is precisely the dual of a primal linear program devised by Cohn and Elkies [CE03] to obtain upper bounds on the density, and hence has implications for linear programming bounds. This connection proves that our density estimate can never exceed the CohnElkies upper bound, regardless of the validity of our conjecture.