Web hosting ecommerce - 92 RANDOM NUMBERS 3.3.4 in each subcube of
92 RANDOM NUMBERS 3.3.4 in each subcube of the unit cube, when the unit cube has been divided into 64 subcubes of size $ X 4 X a; this same generator might yield completely empty subsquares of the unit square, when the unit square has been divided into 64 subsquares of size & x isl. Since we increase our expectations in lower dimensions, a separate test for each dimension is required. It is not always true that vt 5 milt, although this upper bound is valid when the points form a rectangular grid. For example, it turns out that uz = m > m in Fig. 8, because a nearly hexagonal structure brings the m points closer together than would be possible in a strictly rectangular arrrangement. In order to develop an algorithm that computes vt efficiently, we must look more deeply at the associated mathematical theory. Therefore a reader who is not mathematically inclined is advised to skip to part D of this section, where the spectral test is presented as a plug-in method accompanied by several examples. On the other hand, we shall see that the mathematics behind the spectral test requires only some elementary manipulations of vectors. Some authors have suggested using the minimum number Nt of parallel covering lines or hyperplanes as the criterion, instead of the maximum distance l/z,+ between them. However, this number does not appear to be as important as the concept of accuracy defined above, because it is biased by how nearly the slope of the lines or hyperplanes matches the coordinate axes of the cube. For example, the 20 nearly vertical lines that cover all the points of Fig. 8 are actually l/J328 units apart, and this might falsely imply an accuracy of one part in &%, or perhaps even of one part in 20. The true accuracy of only one part in $% is realized only for the larger family of 21 lines with a slope of 7/15; another family of 24 lines, with a slope of -11/13, also has a greater inter-line distance than the 20-line family, since l/m > l/m. The precise way in which families of lines act at the boundaries of the unit hypercube does not seem to be an especially clean or significant criterion; however, for those people who prefer to count hyperplanes, it is possible to compute Nt using a method quite similar to the way in which we shall calculate vt (see exercise 16). *B. Theory behind the test. In order to analyze the basic set (2), we start with the observation that ( ujx + (1 + a + *. . + aj- )c mod 1 $j(x) = m > We can get rid of the mod 1 operation by extending the set periodically, making infinitely many copies of the original t-dimensional hypercube, proceed- ing in all directions. This gives us the set L= ;+k$g+kz,…,q+kt integer x, kl, lcz, . . . , Ict {( >I I = vi+ ;+kl,;+ks,…, 5 + kt> integer :c, ICI, k2, . . . , kt , r (
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