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An element m in a message is encoded as c = PowerMod[m, d, n] . It can then be decoded as PowerMod[c, e, n] , where e = PowerMod[d, -1, EulerPhi[n]] .
Power spectra [of random processes] Many random processes in nature show power spectra Abs[Fourier[data]] 2 with fairly simple forms. … Other pure power laws 1/f α are also sometimes seen; the pictures below show some examples. … (There was confusion in the late 1980s when theoretical studies of self-organized criticality failed correctly to take squares in computing power spectra.)
But the crucial point is that because of the way the system was constructed there is nevertheless a simple formula for the color of each cell: it is given just by a particular digit in the number obtained by raising the multiplier to a power equal to the number of steps. … For if one was to work out the value of a power m t by explicitly performing t multiplications, this would be very similar to explicitly following t steps of cellular automaton evolution.
A typical result is that correlations between colors of different cells fall off like a power of distance—with the specific power depending only on general features of the nested patterns formed, and not on most details of the system.
natural selection is often touted as a force of almost arbitrary power, I have increasingly come to believe that in fact its power is remarkably limited.
Indeed, even after say 1,048,576 steps—or any number of steps that is a power of two—the array of cells produced always turns out to correspond just to a simple superposition of two or three shifted copies of the initial conditions.
way to get such an answer in a number of steps that increases not exponentially but only like a power. … So what this implies is that to answer questions about the t -step behavior of a multiway system can take any ordinary Turing machine a number of steps that increases faster than any power of t .
Generating functions [for regular languages] The sequences in a regular language can be thought of as corresponding to products of non-commuting variables that appear as coefficients in a formal power series expansion of a generating function.
Also of relevance are intensity or power spectra, obtained as the square of these spectra.
Non-power bases One can consider representing numbers by Sum[a[n] f[n], {n, 0, ∞ }] where the f[n] need not be k n .
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