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A125846
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Numerator of volume of best symplectic packing of n balls in 4-dimensional ball.
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5
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1, 1, 3, 1, 4, 24, 63, 288, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
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OFFSET
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1,3
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COMMENTS
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Explanation, figure, table, references in Traynor. McDuff and Polterovich's existence proof of these packings in nonexplicit; they result from the symplectic blow-up operation. Explicit constructions for n = 8 and n = 9 are still unknown. Biran showed that A125846(n) = A125847(n) = 1 for all n >= 9.
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REFERENCES
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P. Biran, Symplectic packing in dimension 4, Geom. Funct. Anal. 7 (1997), pp. 420-437, MR1466333.
D. McDuff and L. Polterovich, Symplectic packings and algebraic geometry, Invent. Math. 115 (1994), pp. 403-434, MR1262938.
Lisa Traynor, Book review (of Embedding problems in symplectic geometry, by Felix Schlenk, deGruyter Expositions in Mathematics, vol. 40, Berlin, 2005), Bull. Amer. Math. Soc. 43 (2006), pp. 593-597.
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LINKS
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FORMULA
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A125846(n)/A125847(n) is maximal symplectic packing density with n balls, as calculated by McDuff and Polterovich.
G.f.: x*(1 + 2*x^2 - 2*x^3 + 3*x^4 + 20*x^5 + 39*x^6 + 225*x^7 - 287*x^8)/(1 - x). - Elmo R. Oliveira, Aug 04 2024
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EXAMPLE
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For n = 1..9, the densities are 1, 1/2, 3/4, 1, 4/5, 24/25, 63/64, 288/289, 1.
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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