Download American Mathematical Monthly, volume 117, May 2010 by Daniel J. Velleman PDF

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By Daniel J. Velleman

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Additional resources for American Mathematical Monthly, volume 117, May 2010

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Below, several exercises are presented, which yield additional information around the topic discussed in this article (those that are relatively difficult are marked by (*) and need the method of transfinite induction). Exercise 1. Show that there exists no 1-homogeneous covering of R2 (respectively, of S2 ) with circles. Infer this fact from the more general result stating that there exists no 1-homogeneous covering of R2 (respectively, of S2 ) with Jordan curves. ) On the other hand, show that there exists a partition of A2 into countably many circles.

10 , X ) is an inverse gamma distribution with parame10 ters γ + 10α and i=1 λi + δ. Once again, this is an available distribution. Gelfand and Smith applied Gibbs sampling to the posterior distribution in the pumps model and obtained marginal posterior distributions for all the λi ’s and for β. The results were impressive: in relatively few iterations, the posterior samples recreated results obtained from other more involved integration methods. Some of the results of using Gibbs sampling with the pumps model are shown in Figure 5.

After suitable relabeling and permuting of rows and of columns, we may place 5+ in cell (4, 10) as shown in Figure 9. That entry does not appear in at least one of the three partial 428 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 Figure 8. 1 1 2 6− 5− 4− 4− 4− 4− 2 3 3 4 4 5 5 6 6+ 6 7 1 2 3 4 5 6 7+ 1 6− 5− 4− 4− 4− 4− 7 8 9 10 11 12 13 5− 4− 3− 3− 3− Figure 9. 1 1 2 6− 2 3 3 4 4 5 5+ 5 6 6+ 6 7 1 2 3 4 5 6 7+ 1 6− 5− 4− 3− 3− 3− 7 8 9 10 11 12 13 transversals found in rows 5 and 6.

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