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doi: 10.1002/sapm1974533217pmid: N/A
This paper deals with the derivation of “first approximations” to the solutions of the Orr‐Sommerfeld equation which are uniformly valid in a full neighborhood of the critical point. To this order the theory is remarkably simple. The essential elements in the theory are all well known from the older heuristic theories but its general structure is substantially different. The uniform approximations are also vastly simpler than the composite approximations obtained recently by the method of matched asymptotic expansions.
Halász, Sylvia; Kleitman, D. J.
doi: 10.1002/sapm1974533225pmid: N/A
In this paper we prove that the point in a triangle T maximizing the probability that it lies within a randomly chosen triangle within T is the center of T. A simple general closed expression for the indicated probability at any point P of T is obtained, though the proof proceeds by a symmetry argument on contributions to derivatives of the probability. It is also shown that the indicated probability is linearly related to the average over points in T of the product of the areas within T on either side of the line determined by P and . This gives the proof for n = 3 of the following conjecture of A. Prékopa: Fix a point P in an n − 1 dimensional simplex S. Take n independent points, uniformly distributed in S, and consider the probability ρ(P) of P being inside the random simplex determined by these n points. Conjecture: ρ(P) is maximal, when P is the center of gravity of S.
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