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

By Daniel J. Velleman

ISBN-10: 0883851857

ISBN-13: 9780883851852

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

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For a positive integer n, an nth-order Latin square consists of n 2 cells arranged in a square with each cell containing one symbol from a given set of n symbols. Each row and each column contains all n symbols. To put it another way, no symbol is duplicated in any row or in any column. It is customary to use the integers 1, 2, . . , n as the set of symbols. Two Latin squares of order n are said to be orthogonal if, when one is placed on top of the other, all n 2 possible ordered pairs of symbols appear.

S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon, Optimization by simulated annealing: An experimental evaluation; Part I, Graph partitioning, Oper. Res. 37 (1989) 865–892. 865 42. , Optimization by simulated annealing: An experimental evaluation; Part II, Graph coloring and number partitioning, Oper. Res. 39 (1991) 378–406. 378 43. L. Kelvin, Nineteenth century clouds over the dynamical theory of heat and light, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 2 (1901) 1–40.

In this case, let y be a point of A2 such that y = yn (n = 0, 1, . . , m − 1), ||y − z|| = 1/2 (z ∈ Z \ {xm }), ||y − xm || = 1/2. The existence of y follows from the facts that the circle in A2 centered at xm with radius 1/2 is countably infinite and the first two requirements concerning y rule out only finitely many points on this circle, so there are infinitely many y satisfying all three requirements. Again, we define Cm as the circle with center y and radius 1/2. , as having the least index in the sequence {xn : n = 0, 1, .

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


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