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A proof of the Kepler conjecture
 Math. Intelligencer
, 1994
"... This section describes the structure of the proof of ..."
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Cited by 110 (11 self)
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This section describes the structure of the proof of
The Flyspeck Project
"... Abstract. This article gives an introduction to a longterm project called Flyspeck, whose purpose is to give a formal verification of the ..."
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Cited by 16 (2 self)
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Abstract. This article gives an introduction to a longterm project called Flyspeck, whose purpose is to give a formal verification of the
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 9 (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
Can padic integrals be computed
"... This article gives an introduction to arithmetic motivic integration in the context of padic integrals that arise in representation theory. A special case of the fundamental lemma is interpreted as an identity of Chow motives. 1 ..."
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Cited by 7 (2 self)
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This article gives an introduction to arithmetic motivic integration in the context of padic integrals that arise in representation theory. A special case of the fundamental lemma is interpreted as an identity of Chow motives. 1
A proof of the dodecahedral conjecture
, 1998
"... This article gives a proof of Fejes Tóth’s Dodecahedral conjecture: the volume of a Voronoi polyhedron in a threedimensional packing of balls of unit radius is at least the volume of a regular dodecahedron of unit inradius. 1 ..."
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Cited by 5 (2 self)
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This article gives a proof of Fejes Tóth’s Dodecahedral conjecture: the volume of a Voronoi polyhedron in a threedimensional packing of balls of unit radius is at least the volume of a regular dodecahedron of unit inradius. 1
exact linear arithmetic decision procedure
, 2009
"... using floatingpoint computations to help an ..."
unknown title
, 2002
"... Can padic integrals be computed? This article gives an introduction to arithmetic motivic integration in the context of padic integrals that arise in representation theory. A special case of the fundamental lemma is interpreted as an identity of Chow motives. 1 ..."
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Can padic integrals be computed? This article gives an introduction to arithmetic motivic integration in the context of padic integrals that arise in representation theory. A special case of the fundamental lemma is interpreted as an identity of Chow motives. 1
© 2006 Springer Science+Business Media, Inc. Historical Overview of the Kepler Conjecture
"... Abstract. This paper is the first in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the facecentered cubic packing. After some preliminary comments about the facecentered cubic and ..."
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Abstract. This paper is the first in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the facecentered cubic packing. After some preliminary comments about the facecentered cubic and hexagonal close packings, the history of the Kepler problem is described, including a discussion of various published bounds on the density of sphere packings. There is also a general historical discussion of various proof strategies that have been tried with this problem. 1.
Annals of the Japan Association for Philosophy of Science Vol.21 (2013) 21～35 21 Mathematical Knowledge: Motley and Complexity of Proof
"... Modern mathematics is, to my mind, a complex edifice based on conceptual constructions. The subject has undergone something like a biological evolution, an opportunistic one, to the point that the current subject matter, methods, and procedures would be patently unrecognizable a century, certainly ..."
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Modern mathematics is, to my mind, a complex edifice based on conceptual constructions. The subject has undergone something like a biological evolution, an opportunistic one, to the point that the current subject matter, methods, and procedures would be patently unrecognizable a century, certainly two centuries, ago. What has been called “classical mathematics ” has indeed seen its day. With its richness, variety, and complexity any discussion of the nature of modern mathematics cannot but accede to the primacy of its history and practice. As I see it, the applicability of mathematics may be a driving motivation, but in the end mathematics is autonomous. Mathematics is in a broad sense selfgenerating and selfauthenticating, and alone competent to address issues of its correctness and authority. What brings us mathematical knowledge? The carriers of mathematical knowledge are proofs, more generally arguments and constructions, as embedded in larger contexts.1 Mathematicians and teachers of higher mathematics know this, but it should be said. Issues about competence and intuition can be raised as well as factors of knowledge involving the general dissemination of analogical or inductive reasoning