A formal proof of the Kepler conjecture
Thomas C. Hales,Mark Adams,Gertrud Bauer,Dat Tat Dang,John Harrison,Truong Le Hoang,Cezary Kaliszyk,Victor Magron,Sean McLaughlin,Thang Tat Nguyen,Truong Quang Nguyen,Tobias Nipkow,Steven Obua,Joseph Pleso,Jason Rute,Alexey Solovyev,An Hoai Thi Ta,Trung Nam Tran,Diep Thi Trieu,Josef Urban,Ky Khac Vu,Roland Zumkeller +21 more
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This paper constitutes the official published account of the now completed Flyspeck project and describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants.Abstract:
This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the offi cial published account of the now completed Flyspeck project.read more
Citations
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The sphere packing problem in dimension 24
TL;DR: In this article, it was shown that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing.
Journal ArticleDOI
Monte Carlo convolution for learning on non-uniformly sampled point clouds
TL;DR: MCCNN as mentioned in this paper represents the convolution kernel itself as a multilayer perceptron, phrasing convolution as a Monte Carlo integration problem, using this notion to combine information from multiple samplings at different levels, and using Poisson disk sampling as a scalable means of hierarchical point cloud learning.
Journal ArticleDOI
Geometric constraints during epithelial jamming.
Lior Atia,Dapeng Bi,Yasha Sharma,Jennifer A. Mitchel,Bomi Gweon,Bomi Gweon,Stephan A. Koehler,Stephen J. DeCamp,Bo Lan,Jae Hun Kim,Rebecca Hirsch,Adrian F. Pegoraro,Kyu Ha Lee,Jacqueline R. Starr,David A. Weitz,Adam C. Martin,Jin-Ah Park,James P. Butler,James P. Butler,Jeffrey J. Fredberg +19 more
TL;DR: In this article, the authors show that cell shape and shape variability are mutually constrained through a relationship that is purely geometrical that is shown to govern processes as diverse as maturation of the pseudostratified bronchial epithelial layer cultured from non-asthmatic or asthmatic donors.
Journal ArticleDOI
The Role of the Mizar Mathematical Library for Interactive Proof Development in Mizar
Grzegorz Bancerek,Czesław Byliński,Adam Grabowski,Artur Korniłowicz,Roman Matuszewski,Adam Naumowicz,Karol Pąk +6 more
TL;DR: This paper shows the process of forming this library, the evolution of its design principles, and also presents some data showing its current use with the modern version of the Mizar proof checker, but also as a rich corpus of semantically linked mathematical data.
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Analytic Description of Primordial Black Hole Formation from Scalar Field Fragmentation
Eric Cotner,Alexander Kusenko,Alexander Kusenko,Misao Sasaki,Misao Sasaki,Misao Sasaki,Volodymyr Takhistov +6 more
TL;DR: In this article, the dynamics of a scalar field in the early universe can produce primordial black hole formation under mild assumptions regarding the scalar potential, which is a more generic phenomenon than was once thought.
References
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Book
Isabelle/HOL: A Proof Assistant for Higher-Order Logic
TL;DR: This presentation discusses Functional Programming in HOL, which aims to provide students with an understanding of the programming language through the lens of Haskell.
Book
Introduction to Interval Analysis
TL;DR: This unique book provides an introduction to a subject whose use has steadily increased over the past 40 years, and provides broad coverage of the subject as well as the historical perspective of one of the originators of modern interval analysis.
Proceedings ArticleDOI
seL4: formal verification of an OS kernel
Gerwin Klein,Kevin Elphinstone,Gernot Heiser,June Andronick,David Cock,Philip Derrin,Dhammika Elkaduwe,Kai Engelhardt,Rafal Kolanski,Michael Norrish,Thomas Sewell,Harvey Tuch,Simon Winwood +12 more
TL;DR: To the knowledge, this is the first formal proof of functional correctness of a complete, general-purpose operating-system kernel.
Book
Introduction to HOL: a theorem proving environment for higher order logic
Michael J. C. Gordon,Tom Melham +1 more
TL;DR: A tutorial on goal-directed proof: tactics and tacticals and theorem-Proving With HOL, a simple proof tool for goal-oriented proof of the binomial theorem.