Where to stand when playing darts?

Franzen, Björn G.; Steif, Jeffrey E.; Wästlund, Johan

doi:10.48550/arxiv.2010.00566

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Addendum: Fallibility1

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2022

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“…9. Acknowledgements I am grateful to Ljuben Mutafchiev [13], Lennart Bondesson [43,44] and Jeffrey Steif [45] for helpful discussions. Vaclav Kotesovec provided relevant asymptotic expansions in [23].…”

Section: Addendum: Fallibilitymentioning

confidence: 99%

Components and Cycles of Random Mappings

Finch¹

2022

Preprint

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Each connected component of a mapping {1, 2, . . . , n} → {1, 2, . . . , n} contains a unique cycle. The largest such component can be studied probabilistically via either a delay differential equation or an inverse Laplace transform. The longest such cycle likewise admits two approaches: we find an (apparently new) density formula for its length. Implications of a constraint -that exactly one component exists -are also examined. For instance, the mean length of the longest cycle is (0.7824...)√ n in general, but for the special case, it is (0.7978...)√ n, a difference of less than 2%.

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Section: Addendum: Fallibilitymentioning

confidence: 99%