The sphere packing problem in dimension 24
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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.Abstract:
Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.read more
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Kissing numbers of closed hyperbolic manifolds
TL;DR: In this paper, an upper bound for the number of shortest closed geodesics in a closed hyperbolic manifold of any dimension in terms of its volume and systole was established.
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Moment methods in energy minimization: New bounds for Riesz minimal energy problems
TL;DR: In this article, a converging hierarchy of optimization problems is constructed to lower the ground state energy of interacting particle systems. But it is not known whether the second step of this hierarchy may be sharp throughout a phase transition and may be universally sharp for 5-particles on the unit sphere.
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Towards a proof of the 24-cell conjecture
TL;DR: In this paper, the uniqueness of maximum kissing arrangements in 4-dimensional Euclidean space and the 24-cell conjecture were studied. And the lattice packing D4 was shown to be the densest sphere packing in 4 dimensions.
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Wiener’s problem for positive definite functions
D V Gorbachev,S. Yu. Tikhonov +1 more
TL;DR: In this paper, the sharp constant for positive definite functions in Wiener's inequality was studied and a lower bound of 2.401 ℓ + o(1))n was obtained.
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Compact Packings of Space with Two Sizes of Spheres
TL;DR: It is shown that the compact packings of Euclidean three-dimensional space with two sizes of spheres are exactly those obtained by filling with spheres of size 2-1, and the octahedral holes of a close-packing of sphere of size 1.
References
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TL;DR: The Handbook of Mathematical Functions with Formulas (HOFF-formulas) as mentioned in this paper is the most widely used handbook for mathematical functions with formulas, which includes the following:
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TL;DR: A program to prove the Kepler conjecture on sphere packings is described and it is shown that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.
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TL;DR: In this paper, the authors provided an overview of spherical codes and designs, and derived bounds for the cardinality of spherical A-codes in terms of the Gegenbauer coefficients of polynomials compatible with A.
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The Selberg trace formula for PSL (2, IR)
TL;DR: The selberg trace formula (version A) as mentioned in this paper is a trace formula for the Poincare series and the spectral decomposition of L2(? \H,?).