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By Daniel J. Velleman

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J. E. Hearnshaw and M. S. Paterson, Problems drive, Eureka 27 (1964) 6–8 and 39–40. html. 11. C. P. Jargodzki and F. Potter, Mad About Physics: Braintwisters, Paradoxes, and Curiosities, John Wiley, New York, 2001. 12. P. B. Johnson, Leaning tower of lire, Amer. J. Phys. 23 (1955) 240. 13. G. M. , Clarendon, Oxford, 1907. 14. M. Paterson and U. Zwick, Overhang, in Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’06), Society for Industrial and Applied Mathematics, Philadelphia, 2006, 231–240.

For stacks with a small number of blocks, we enumerated all possible combinatorial stack structures and numerically optimized each of them. For larger numbers of blocks this approach is clearly not feasible and we had to use various heuristics to cut down the number of combinatorial structures considered. The stacks of Figures 3, 4, 5, and 6 were found using extensive numerical experimentation. The stacks of Figures 3, 4, and 5 are optimal, while the stacks of Figure 6 are either optimal or very close to being so.

Let f ∈ Cc,0 finite variances, their zero bias distributions exist, so in particular, σn2 E f (Yn∗ ) = E Yn F(Yn ) for all n. By (21), since y F(y) is in Cb , we obtain σ 2 lim E f (Yn∗ ) = lim σn2 E f (Yn∗ ) = lim E[Yn F(Yn )] = E[Y F(Y )] = σ 2 E f (Y ∗ ). 1. We now provide the proof of the partial converse to the Lindeberg-Feller theorem. 3. 2 implies Wn∗ →d Z ∗ . But Z is a fixed point of the zero bias transformation, hence Wn∗ →d Z. 1 yields that X In ,n → p 0, and Slutsky’s lemma (30) now gives that Wn + X ∗In ,n = Wn∗ + X In ,n →d Z.

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American Mathematical Monthly, volume 116, number 1, january 2009 by Daniel J. Velleman


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