Powers of three in base 2 The n th row in the pattern shown can be obtained simply as IntegerDigits[3 n , 2] . … The fraction seems to tend to 1/2. If one looks only at the rightmost s columns of the pattern, one sees repetition—but the period of the repetition grows like 2 s .

With more than two piles it was discovered in 1901 that one player can in general force the other to lose by arranging that after each of their moves Apply[BitXor, h]  0 , where h is the list of heights. … If one allows only specific numbers of objects to be taken at each step a nested pattern is again obtained.

Note that the pictures shown indicate only which states yield which states—not for example in how many ways the rules can be applied to a given state to yield a given new state. … The rule for the system can then be stated in terms of a difference vector—which for {"AB"  "BBB", "ABB"  "AAAB"} is {{-1, 2}, {2, -1}} .

But with the setup I use, there is only a partial ordering of events, defined by causal connections. … For successive numbers of elements the total numbers of these are 1, 2, 5, 16, 63, 318, 2045, 16999, ...

So long as one only ever looks at the original input and final output it turns out that one can construct a system that exhibits undecidability but is not universal. … Note however that this result is extremely specific to looking only at what is considered output from the system, and that inside the system there are all sorts of components that are definitely universal.

I give in full those forenames that I believe are or were most commonly used by a particular individual; for other forenames (including for example Russian patronymics) I give only initials.

Only in the 1990s, however, did exact minimization of complex DNF expressions become common. Minimization of Boolean expressions with depth larger than 2 has been considered off and on since the late 1950s, and became popular in the 1990s in connection with the BDDs discussed above.

Biological [forms of] perception Animals can process data not only from visual or auditory sources (as discussed on pages 577 and 585 ), but also from mechanical, thermal, chemical and other sources.

The fact that there are a million nerve fibers going from the eye to the brain, but only about 30,000 going from the ear to the brain means that while it takes several million bits per second to transmit video of acceptable quality, a few tens of thousands of bits are adequate for audio (NTSC television is 5 MHz; audio CDs 22 kHz; telephone 8 kHz).

This lack of dependence makes it somewhat inevitable that the only rules that end up being emulated in this way are ones with very simple behavior. In any 1D cellular automaton the color of a particular cell can always be determined from the colors t steps back of a block of 2 r t + 1 cells (compare pages 605 and 960 ).