Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Curved Hexagonal Packings of Equal Disks in a Circle   For each k >= 1 and corresponding hexagonal number h(k) = 3k(k+1)+1, we
introduce m(k) = max[(k-1)!/ 2, 1] packings of h(k) equal disks inside a circle
which we call "the curved hexagonal packings". The curved hexagonal packing of
7 disks (k = 1, m(1)=1) is well known and the one of 19 disks (k = 2, m(2)=1)
has been previously conjectured to be optimal. New curved hexagonal packings of
37, 61, and 91 disks (k = 3, 4, and 5, m(3)=1, m(4)=3, and m(5)=12) were the
densest we obtained on a computer using a so-called "billiards" simulation
algorithm. A curved hexagonal packing pattern is invariant under a 60 degree
rotation. For k tending to infinity, the density (covering fraction) of curved
hexagonal packings tends to pi*pi/12. The limit is smaller than the density of
the known optimum disk packing in the infinite plane. We found disk
configurations that are denser than curved hexagonal packings for 127, 169, and
217 disks (k = 6, 7, and 8). In addition to new packings for h(k) disks, we
present new packings we found for h(k)+1 and h(k)-1 disks for k up to 5, i.e.,
for 36, 38, 60, 62, 90, and 92 disks. The additional packings show the
``tightness'' of the curved hexagonal pattern for k =< 5: deleting a disk does
not change the optimum packing and its quality significantly, but adding a disk
causes a substantial rearrangement in the optimum packing and substantially
decreases the quality.

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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
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The categories of this paper are: physics
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The categories of this paper are: q-alg
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: cs