Convex majorization with an application to the length of critical paths
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1. (Y) for all non-negative, non-decreasing convex functions φ (X is convexly smaller than Y) if and only if, for all .2.Let H be the Hardy–Littlewood maximal function HY(x) = E(Y – X | Y > x). Then HY(Y) is the smallest random variable exceeding stochastically all random variables convexly smaller than Y.3.Let X1X2 · ·· Xn be random variables with given marginal distributions, let I1,I2, ···, Ik be arbitrary non-empty subsets of {1,2, ···, n} and let M = max (M is the completion time of a PERT network with paths Ij, and delay times Xi.) The paper introduces a computation of the convex supremum of M in the class of all joint distributions of the Xi's with specified marginals, and of the ‘bottleneck probability' of each path.
1979 ◽
Vol 16
(03)
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pp. 671-677
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1961 ◽
Vol 2
(2)
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pp. 229-242
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1996 ◽
Vol 10
(2)
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pp. 229-241
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1979 ◽
Vol 16
(04)
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pp. 794-802
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1994 ◽
Vol 144
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pp. 431-434
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1979 ◽
Vol 44
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pp. 357-372
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1977 ◽
Vol 35
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pp. 210-211
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