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Two-player Knock 'em Down
We analyze the two-player game of Knock 'em Down, asymptotically as the
number of tokens to be knocked down becomes large. Optimal play requires mixed
strategies with deviations of order sqrt(n) from the naive law-of-large numbers
allocation. Upon rescaling by sqrt(n) and sending n to infinity, we show that
optimal play's random deviations always have bounded support and have marginal
distributions that are absolutely continuous with respect to Lebesgue measure.Comment: 15 pages, 1 figure. v2 has minor revision
Isomorphism in expanding families of indistinguishable groups
For every odd prime and every integer there is a Heisenberg
group of order that has pairwise
nonisomorphic quotients of order . Yet, these quotients are virtually
indistinguishable. They have isomorphic character tables, every conjugacy class
of a non-central element has the same size, and every element has order at most
. They are also directly and centrally indecomposable and of the same
indecomposability type. The recognized portions of their automorphism groups
are isomorphic, represented isomorphically on their abelianizations, and of
small index in their full automorphism groups. Nevertheless, there is a
polynomial-time algorithm to test for isomorphisms between these groups.Comment: 28 page
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Number of Pages: 2Integrative BiologyGeological Science
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