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206
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 67 (7 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
 PROCEEDINGS OF THE 12TH ANNUAL INTERNATIONAL CONFERENCE ON MOBILE COMPUTING AND NETWORKING
, 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 dimensio ..."
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Cited by 46 (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, whereas their
Formal Proof
, 2008
"... There remains but one course for the recovery of a sound and healthy condition—namely, that the entire work of the understanding be commenced afresh, and the mind itself be from the very outset not left to take its own course, but guided at every step; and the business be done as if by machinery. ..."
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Cited by 27 (1 self)
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There remains but one course for the recovery of a sound and healthy condition—namely, that the entire work of the understanding be commenced afresh, and the mind itself be from the very outset not left to take its own course, but guided at every step; and the business be done as if by machinery.
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 24 (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 22 (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 ..."
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Cited by 18 (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.
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 18 (7 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.
Jammed HardParticle Packings: From Kepler to Bernal and Beyond
, 2010
"... Understanding the characteristics of jammed particle packings provides basic insights into the structure and bulk properties of crystals, glasses, and granular media, and into selected aspects of biological systems. This review describes the diversity of jammed configurations attainable by frictionl ..."
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Cited by 18 (3 self)
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Understanding the characteristics of jammed particle packings provides basic insights into the structure and bulk properties of crystals, glasses, and granular media, and into selected aspects of biological systems. This review describes the diversity of jammed configurations attainable by frictionless convex nonoverlapping (hard) particles in Euclidean spaces and for that purpose it stresses individualpacking geometric analysis. A fundamental feature of that diversity is the necessity to classify individual jammed configurations according to whether they are locally, collectively, or strictly jammed. Each of these categories contains a multitude of jammed configurations spanning a wide and (in the large system limit) continuous range of intensive properties, including packing fraction φ, mean contact number Z, and several scalar order metrics. Application of these analytical tools to spheres in three dimensions (an analog to the venerable Ising model) covers a myriad of jammed states, including maximally dense packings (as Kepler conjectured), lowdensity strictlyjammed tunneled crystals, and a substantial family of amorphous packings. With respect to the last of these, the current approach displaces the
Ten methods to bound multiple roots of polynomials
 J. Comput. Appl. Math. (JCAM
"... Abstract. Given a univariate polynomial P with a kfold multiple root or a kfold root cluster near some z̃, we discuss various different methods to compute a disc near z ̃ which either contains exactly or contains at least k roots of P. Many of the presented methods are known, some are new. We are ..."
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Abstract. Given a univariate polynomial P with a kfold multiple root or a kfold root cluster near some z̃, we discuss various different methods to compute a disc near z ̃ which either contains exactly or contains at least k roots of P. Many of the presented methods are known, some are new. We are especially interested in rigorous methods, that is taking into account all possible effects of rounding errors. In other words every computed bound for a root cluster shall be mathematically correct. We display extensive test sets comparing the methods under different circumstances. Based on the results we present a hybrid method combining five of the previous methods which, for given z̃, i) detects the number k of roots near z ̃ and ii) computes an including disc with in most cases a radius of the order of the numerical sensitivity of the root cluster. Therefore, the resulting discs are numerically nearly optimal. 1. Introduction and notation. Throughout the paper denote by P = n∑ ν=0 pνz ν ∈ C[z] a (real or