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Quasi- (that is, Sub-) Random Sequences in Software Writer PDF417 in Software Quasi- (that is, Sub-) Random Sequences




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7.8 Quasi- (that is, Sub-) Random Sequences use software pdf417 integrated toinsert pdf417 in software SQL Server 2000/2005/2008/2012 . . .

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1 0 .2 .4 .6 .8 points 1 to 1024 1 Figure 7.8.1.

First 10 pdf417 2d barcode for None 24 points of a two-dimensional Sobol sequence. The sequence is generated numbertheoretically, rather than randomly, so successive points at any stage know how to ll in the gaps in the previously generated distribution..

Vk goes from present t o absent (or vice versa) only every 2k 1 steps. Antonov and Saleev s contribution was to show that instead of using the bits of the integer j to select direction numbers, one could just as well use the bits of the Gray code of j , G.j /.

(For a quick review of Gray codes, look at 22.3.) Now G.

j / and G.j C 1/ differ in exactly one bit position, namely in the position of the rightmost zero bit in the binary representation of j (adding a leading zero to j if necessary). A consequence is that the j C 1st Sobol -Antonov-Saleev number can be obtained from the j th by XORing it with a single Vi , namely with i the position of the rightmost zero bit in j .

This makes the calculation of the sequence very ef cient, as we shall see. Figure 7.8.

1 plots the rst 1024 points generated by a two-dimensional Sobol sequence. One sees that successive points do know about the gaps left previously, and keep lling them in, hierarchically. We have deferred to this point a discussion of how the direction numbers Vi are generated.

Some nontrivial mathematics is involved in that, so we will content ourselves with a cookbook summary only: Each different Sobol sequence (or component of an n-dimensional. Degree 1 2 3 4 5 6 7 8 barcode pdf417 for None 9 0 (i.e., x C 1).

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