دورية
On the Height of a Random Set of Points in a d-Dimensional Unit Cube
| العنوان: | On the Height of a Random Set of Points in a d-Dimensional Unit Cube |
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| المؤلفون: | Breimer, Eric, Goldberg, Mark, Kolstad, Brian, Magdon-Ismail, Malik |
| المصدر: | Experimental Mathematics; January 2001, Vol. 10 Issue: 4 p583-597, 15p |
| مستخلص: | We investigate, through numerical experiments, the asymptotic behavior of the length Hd(n) of a maximal chain (longest totally ordered subset) of a set of n points drawn from a uniform distribution on the d-dimensional unit cube VD= [0, 1]d. For d ≥ 2, it is known that cd(n) = Hd(n)/n1/dconverges in probability to a constant Cd< e, with Iimd→∞Cd= e. For d = 2, the problem has been extensively studied, and it is known that C2= 2; Cdis not currently known for any d ≥ 3. Straightforward Monte Carlo simulations to obtain Cdhave already been proposed, and shown to be beyond the scope of current computational resources. In this paper, we present a computational approach which yields feasibleexperiments that lead to estimates for Cd. We prove that Hd(n) can be estimated by considering only those chains close to the diagonal of the cube. A new conjecture regarding the asymptotic behavior of cd(n) leads to even more efficient experiments. We present experimental support for our conjecture, and the new estimates of Cdobtained from our experiments, for d ∈ {3,4,S,6}. |
| قاعدة البيانات: | Supplemental Index |
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Full Text Finder | تدمد: | 10586458 1944950x |
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| DOI: | 10.1080/10586458.2001.10504678 |