All such methods yield signals that remain roughly in the range of frequencies { ω - δ , ω + δ } where δ is the data rate in s[t] . … In general to send many signals together one just needs to associate each with a function f[i, t] orthogonal to all other functions f[j, t] (see page 1072 ). … (Some such systems are however being replaced by lower-power phased array systems.)

One general result, however, is that all so-called Sturmian sequences Round[(n + 1) a + b] - Round[n a + b] with a an irrational number must yield discrete spectra. … The generating function for the sequence (with 0 replaced by -1) satisfies f[z]  (1 - z) f[z 2 ] , so that f[z]  Product[1 - z 2 n , {n, 0, ∞ }] .

Mathematica is available from Wolfram Research for all standard computer systems; much more information about it can be found on the web, especially from www.wolfram.com . … They should run without any change in all subsequent versions of Mathematica, and the majority will also run in prior versions, all the way back to Mathematica 1 (released 1988) or Mathematica 2 (released 1990). Most of the programs require only the language component of Mathematica—and not its mathematical knowledge base—and so should run in all software systems powered by Mathematica, in which language capabilities are enabled.

Random number generators A fairly small number of different types of random number generators have been used in practice, so it is possible to describe all the major ones here. … The original suggestion made by Derrick Lehmer in 1948 was to take a number n and at each step to replace it by Mod[a n, m] . … Page 962 showed diagrams that represent the evolution for all possible starting values of n .

And the way this is normally captured is by saying that when an observation is made the vector of amplitudes is somehow replaced by its projection s into a subspace corresponding to the outcome seen—with the probability of getting the outcome being taken to be determined by s . … For even if one takes these processes to be pure quantum ones, what I believe is that in almost all cases appropriate idealized limits of them will reproduce what are in effect the usual rules for observations in quantum theory. … But at least for a single particle, the Schrödinger equation is in all ways linear, and so it cannot support any kind of real sensitivity to initial conditions, or even to parameters.

(One can have lists instead of strings, replacing StringJoin by Flatten .) … (In most PCP systems, including all the examples shown except (a) and (g), one string is always systematically longer than the other.)