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In the pictures below the liquid is divided into cells, with each cell having a temperature from 0 to 1, corresponding exactly to a continuous cellular automaton of the kind discussed on page 155 .
The smallest such t is given by MultiplicativeOrder[k, n] , which always divides EulerPhi[n] (see page 1093 ), and has a value between Log[k, n] and n - 1 , with the upper limit being attained only if n is prime.
The lengths of the longest cycles are given on page 951 ; all other cycles must have lengths which divide these.
In addition, patterns are often divided into three or four large bands, presumably in correspondence with features of the anatomy of the mantle.
Another definition (equivalent for any connected causal network) is that spacelike slices are what consistently divide a causal network into a past and a future.
For all one ever need do is to work out the remainder from dividing the position of a particular square by the size of the basic repeating block, and this then immediately tells one how to look up the color one wants.
And in this case any slice that consistently divides the causal network into a past and a future must correspond to a possible state of the underlying system—and any non-overlapping sequence of such slices must represent a possible evolution history for the system.
And as Claude Shannon argued in the 1940s, the length of message needed to be reasonably certain that only one key will satisfy this criterion is equal to the length of the key divided by the redundancy of the language in which the message is written—equal to about 0.5 for English (see below ).
One approach to factoring a number n is just to try dividing it by each of the numbers up to √ n .
Arithmetic coding
Consider dividing the interval from 0 to 1 into a succession of bins, with each bin having a width equal to the probability for some sequence of blocks to occur.