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87
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 37 (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.
What Are All the Best Sphere Packings in Low Dimensions?
 DISCRETE & COMPUTATIONAL GEOMETRY 9
, 1995
"... We describe what may be all the best packings of nonoverlapping equal spheres in dimensions n < 10, where "best" means both having the highest density and not permitting any local improvement. For example, the best fivedimensional sphere packings are parametrized by the 4colorings of the onedimen ..."
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Cited by 12 (4 self)
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We describe what may be all the best packings of nonoverlapping equal spheres in dimensions n < 10, where "best" means both having the highest density and not permitting any local improvement. For example, the best fivedimensional sphere packings are parametrized by the 4colorings of the onedimensional integer lattice. We also find what we believe to be the exact numbers of "uniform" packings among these, that is, those in which the automorphism group acts transitively. These assertions depend on certain plausible but as yet unproved postulates. Our work may be regarded as a continuation of Lfiszl6 Fejes T6th's work on solid packings.
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 12 (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 11 (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
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)
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 (6 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.
Proving bounds for real linear programs in isabelle/HOL (Extended Abstract)
 THEOREM PROVING IN HIGHER ORDER LOGICS (TPHOLS 2005), VOLUME 3603 OF LECT. NOTES IN COMP. SCI
, 2005
"... The Flyspeck project [3] has as its goal the complete formalization of Hales’ proof [2] of the Kepler conjecture. The formalization has to be carried out within a mechanical theorem prover. For our work described in this paper, we have chosen the generic proof assistant Isabelle, tailored to Higher ..."
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Cited by 8 (0 self)
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The Flyspeck project [3] has as its goal the complete formalization of Hales’ proof [2] of the Kepler conjecture. The formalization has to be carried out within a mechanical theorem prover. For our work described in this paper, we have chosen the generic proof assistant Isabelle, tailored to HigherOrder Logic (HOL) [4]. In the following, we will refer to this environment as Isabelle/HOL. An important step in Hales ’ proof is the maximization of about 10 5 real linear programs. The size of these linear programs (LPs) varies, the largest among them consist of about 2000 inequalities in about 200 variables. The considered LPs have the important property that there exist a priori bounds on the range of the variables. The situation is further simplified by our attitude towards the linear programs: we only want to know wether the objective function of a given LP is bounded from above by a given constant K. Under these assumptions, Hales describes [1] a method for obtaining an arbitrarily precise upper bound for the maximum value of the objective function of an LP. This method still works nicely in the context of mechanical theorem