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205
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 23 (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.
An overview of the Kepler conjecture
"... 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 pr ..."
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Cited by 18 (2 self)
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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
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
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 17 (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.