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We prove lower bounds for energy in the Gaussian core model, in which point particles interact via a Gaussian potential.
1941
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1942
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1945
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1975
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F. H. Stillinger, Phase transitions in the Gaussian core system , J. Chem. Phys. 65
1976
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G. A. Kabatyanskii and V. I. Levenshtein, Bounds for packings on a sphere and in space (Russian), Problemy Peredači Informacii 14
1978
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V. I. Levenšteĭn, On bounds for packings in n n -dimensional Euclidean space (Russian), Dokl. Akad. Nauk SSSR 245
1979
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1981
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1985
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1994
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G. N. Watson, A Treatise on the Theory of Bessel Functions , reprint of the second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. MR1349110
1995
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J. J. Holt and J. D. Vaaler, The Beurling-Selberg extremal functions for a ball in Euclidean space , Duke Math. J. 83
1996
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1997
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J. Ramanathan, Methods of Applied Fourier Analysis , Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 1998. MR1632358 doi: 10.1007/978-1-4612-1756-5
1998
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G. E. Andrews, R. Askey, and R. Roy, Special Functions , Encyclopedia of Mathematics and its Applications 71
1999
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D. V. Gorbachev, An extremal problem for entire functions of exponential spherical type, which is connected with the Levenshteĭn bound for the density of a packing of ℝ n \mathbb{R}^{n} by balls (Russian), Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inform. 6
2000
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H. Cohn, New upper bounds on sphere packings II , Geom. Topol. 6
2002
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H. Cohn and Y. Zhao, Sphere packing bounds via spherical codes , Duke Math. J. 163
2010
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F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions , National Institute of Standards and Technology, U.S. Department of Commerce, Washington, DC and Cambridge University Press, Cambridge, 2010. MR2723248 http://dlmf.nist.gov/
2010
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S. Vance, Improved sphere packing lower bounds from Hurwitz lattices , Adv. Math. 227
2011
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H. Cohn and J. Woo, Three-point bounds for energy minimization , J. Amer. Math. Soc. 25
2012
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2002
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H. Cohn and N. Elkies, New upper bounds on sphere packings I , Ann. of Math. (2) 157
2003
Cited alongside, same era.
L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis , reprint of the second edition, Classics in Mathematics, Springer-Verlag, Berlin, 2003. MR1996773 doi: 10.1007/978-3-642-61497-2
2003
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P. Sarnak and A. Strömbergsson, Minima of Epstein’s zeta function and heights of flat tori , Invent. Math. 165
2006
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S. Torquato and F. H. Stillinger, New conjectural lower bounds on the optimal density of sphere packings , Experiment. Math. 15
2006
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H. Cohn and A. Kumar, Universally optimal distribution of points on spheres , J. Amer. Math. Soc. 20
2007
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2008
Cited alongside, same era.
2009
Cited alongside, same era.
2013
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R. A. Horn and C. R. Johnson, Matrix Analysis , second edition, Cambridge University Press, 2013. MR2978290
2013
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A. Venkatesh, A note on sphere packings in high dimension , Int. Math. Res. Not. IMRN 2013
2013
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A. Selberg, Collected Papers II , reprint of the 1991 edition, Springer Collected Works in Mathematics, Springer, Heidelberg, 2014. MR3308963
2014
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B. Simon, Real Analysis , A Comprehensive Course in Analysis, Part 1, American Mathematical Society, Providence, RI, 2015. MR3408971
2015
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P. G. Boyvalenkov, P. D. Dragnev, D. P. Hardin, E. B. Saff, and M. M. Stoyanova, Universal lower bounds for potential energy of spherical codes , Constr. Approx. 44
2016
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2017
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