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A proof of the Kepler conjecture
 Math. Intelligencer
, 1994
"... This section describes the structure of the proof of ..."
Abstract

Cited by 108 (11 self)
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This section describes the structure of the proof of
Computational Method for CryoprobeLayout Optimization via Finite Sphere Packing Daigo Tanaka
, 2007
"... Cryosurgery is the destruction of undesired tissues by freezing, for example, in prostate cryosurgery. Minimallyinvasive cryosurgery is currently performed by means of an array of cryoprobes, each in the shape of a long hypodermic needle. The optimal arrangement of the cryoprobes, which is known to ..."
Abstract
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Cryosurgery is the destruction of undesired tissues by freezing, for example, in prostate cryosurgery. Minimallyinvasive cryosurgery is currently performed by means of an array of cryoprobes, each in the shape of a long hypodermic needle. The optimal arrangement of the cryoprobes, which is known to have a dramatic effect on the quality of the cryoprocedure, is based on the cryosurgeon’s experience. This thesis focuses on an automated computerized technique for cryosurgery planning, in an effort to improve the quality of cryosurgery. A twophase optimization method is proposed for this purpose, based on two previous and independent developments. Phase I is based on a bubblepacking method, a sphere packing method in the finite domain previously used as an efficient method for finite elements meshing. Phase II is based on a forcefield analogy method, which has proven to be robust at the expense of a typically long runtime. Both optimization phases are used to seek an optimal layout and an optimal insertion depth of cryoprobes. Twodimensional and threedimensional models of the prostate and urethral warmer have been reconstructed from patient’s ultrasound images. The quality of planning for each method was evaluated based on bioheat transfer
ON A STRONG VERSION OF THE KEPLER CONJECTURE
, 2013
"... We raise and investigate the following problem which one can regard as a very close relative of the densest sphere packing problem. If the Euclidean 3space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface ar ..."
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We raise and investigate the following problem which one can regard as a very close relative of the densest sphere packing problem. If the Euclidean 3space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells? In particular, we prove that the average surface area in question is always at least √24 = 13.8564.... 3