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

By Daniel J. Velleman

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

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Rev. Lett. 58 (1987) 86–88. 86 61. M. Tanner, Tools for Statistical Inference: Methods for Exploration of Posterior Distributions and Likelihood Functions, Springer-Verlag, New York, 1993. 62. M. Tanner and W. Wong, The calculation of posterior distributions by data augmentation, J. Amer. Statist. Assoc. 82 (1987) 528–540. 2307/2289457 63. L. Tierney, Markov chains for exploring posterior distributions, Ann. Statist. 22 (1994) 1701–1728. 1214/aos/1176325750 64. S. M. Ulam, Adventures of a Mathematician, Scribner, New York, 1976.

An argument similar to the previous one shows that the family of circles {C(x) : x ∈ X } forms a k-homogeneous covering of R2 . We thus conclude that (1) is valid. The effective equivalence (1) ⇔ (2) directly indicates that there is a close relationship (of duality type) between the k-homogeneous coverings of R2 with circles of diameter 1 and the (k, 1)-Sierpi´nski sets. By virtue of Example 2, this circumstance allows us to give an effective construction of a (2k, 1)-Sierpi´nski set. In connection with the statement above, let us remark that it is unknown whether there exists an effective example of a (k, 1)-Sierpi´nski set (or, equivalently, of a khomogeneous covering of R2 by circles of diameter 1) for an odd natural number k ≥ 3.

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. Exercise 2. Give an effective example of a 1-homogeneous covering of the space R3 with circles. Let us stress that the circles from Exercise 2 are not required to be of one and the same radius.

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