A formal proof of the Kepler conjecture

Hales, Thomas C.; Adams, M. R.; Bauer, Gertrud; Dang, Dat Tat; Harrison, John; Hoang, Truong Le; Kaliszyk, Cezary; Magron, Victor; McLaughlin, S.; Nguyen, Thang; Nguyen, Truong Q.; Nipkow, Tobias; Obua, Steven; Pleso, Joseph; Rute, Jason; Solovyev, Alexey; Ta, An Hoai Thi; Tran, Trung; Trieu, Diep Thi; Urban, Josef; Vu, Ky Khac; Zumkeller, Roland

doi:10.48550/arxiv.1501.02155

Cited by 14 publications

(19 citation statements)

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“…The term random packing implies the lack of crystalline order in the packing, even at short ranges. Without this restriction, the largest volume fraction obtained by packing of nonoverlapping spheres is 𝜋/3 √ 2 ≈ 0.74 [10].…”

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confidence: 99%

The random packing density of nearly spherical particles

Kallus¹

2016

Soft Matter

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Obtaining general relations between macroscopic properties of random assemblies, such as density, and the microscopic properties of their constituent particles, such as shape, is a foundational challenge in the study of amorphous materials. By leveraging existing understanding of the random packing of spherical particles, we estimate the random packing density for all sufficiently spherical shapes. Our method uses the ensemble of random packing configurations of spheres as a reference point for a perturbative calculation, which we carry to linear order in the deformation. A fully analytic calculation shows that all sufficiently spherical shapes pack more densely than spheres. Additionally, we use simulation data for spheres to calculate numerical estimates for nonspherical particles and compare these estimates to simulations.

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confidence: 99%

The random packing density of nearly spherical particles

Kallus¹

2016

Soft Matter

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“…Giving a rigorous proof requires a genuine idea, but there exist short, elementary proofs [8]. The three-dimensional problem was solved by Hales [9] via a lengthy and complex computer-assisted proof, which was extraordinarily difficult to check but has since been completely verified using formal logic [10].…”

Section: Sphere Packingmentioning

confidence: 99%

“…Instead, each dimension has its own idiosyncracies and charm. Understanding the densest sphere packing in R 8 tells us only a little about R 7 or R 9 , and hardly anything about R 10 .…”

Section: Sphere Packingmentioning

confidence: 99%

A Conceptual Breakthrough in Sphere Packing

Cohn¹

2017

Notices Amer. Math. Soc.

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“…The classical packing problem inquires as to the maximum number of 2D disks of radius R that can be positioned entirely within a 2D unit square, but there are many variants of this problem and packing is very much an active area of research. For example, a complete formal proof of Kepler's conjecture, that heavily relies on computers, was only published recently [23]. Many conjectures remain open and several problems can benefit if we can use a computer to produce or validate different large packing configurations fast.…”

Section: A Combinatorial Optimizationmentioning

confidence: 99%

Testing Fine-Grained Parallelism for the ADMM on a Factor-Graph

Hao

Oghbaee

Rostami

et al. 2016

2016 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW)

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Abstract-There is an ongoing effort to develop tools that apply distributed computational resources to tackle large problems or reduce the time to solve them. In this context, the Alternating Direction Method of Multipliers (ADMM) arises as a method that can exploit distributed resources like the dual ascent method and has the robustness and improved convergence of the augmented Lagrangian method. Traditional approaches to accelerate the ADMM using multiple cores are problem-specific and often require multi-core programming. By contrast, we propose a problem-independent scheme of accelerating the ADMM that does not require the user to write any parallel code. We show that this scheme, an interpretation of the ADMM as a message-passing algorithm on a factor-graph, can automatically exploit fine-grained parallelism both in GPUs and shared-memory multi-core computers and achieves significant speedup in such diverse application domains as combinatorial optimization, machine learning, and optimal control. Specifically, we obtain 10-18x speedup using a GPU, and 5-9x using multiple CPU cores, over a serial, optimized C-version of the ADMM, which is similar to the typical speedup reported for existing GPU-accelerated libraries, including cuFFT (19x), cuBLAS (17x), and cuRAND (8x).

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A formal proof of the Kepler conjecture

Cited by 14 publications

References 0 publications

The random packing density of nearly spherical particles

The random packing density of nearly spherical particles

A Conceptual Breakthrough in Sphere Packing

Testing Fine-Grained Parallelism for the ADMM on a Factor-Graph

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