In each case they start with the number 1, then successively multiply by the specified multiplier, keeping only the rightmost 31 digits in the base 2 representation of the number obtained at each step. … The last two pictures in each row above give the distribution of points whose coordinates in two and three dimensions are obtained by taking successive numbers from the linear congruential generator. If the output from the generator was perfectly random, then in each case these points would be uniformly distributed.

Indeed, as we saw on page 141 , taking square roots can for example generate seemingly random digit sequences. Many mathematicians may object that digit sequences are just too fragile an entity to be worth studying. … Indeed, we will see later in this book that large classes of digit sequences can be considered equivalent with respect to computational operations, but these classes are quite different ones from those that are considered equivalent with respect to mathematical operations.

One might have thought that in going say from f[2000] to f[2001] there would only ever be a small change. … In case (d), the fluctuation in each f[n] turns out to be essentially just the number of 1's that occur in the base 2 digit sequence for n . And in case (c), the fluctuations are determined by the total number of 1's that occur in the digit sequences of all numbers less than n .

But now what the cellular automata do is to take specifications of positions of cells, and then in effect compute directly from these the colors of cells. The way things are set up the initial conditions for these cellular automata consist of digit sequences of numbers that give positions. The color of a particular cell is then found by evolving for a number of steps equal to the length of these input digit sequences.

Then—somewhat in analogy to retrieving closest memories—one can take a sequence of length n that one receives and find the codeword that differs from it in the fewest elements. … Defining PM[s_] := IntegerDigits[Range[2 s - 1], 2, s] blocks of data of length m can be encoded with Join[data, Mod[data . Select[PM[s], Count[#, 1] > 1 &], 2]] while blocks of length n (and at most one error) can be decoded with Drop[(If[#  0, data, MapAt[1 - # &, data, #]] &)[ FromDigits[Mod[data .

Structures [in rule 110] The persistent structures shown can be obtained from the following {n, w} by inserting the sequences IntegerDigits[n, 2, w] between repetitions of the background block b : {{152, 8}, {183, 8}, {18472955, 25}, {732, 10}, {129643, 18}, {0, 5}, {152, 13}, {39672, 21}, {619, 15}, {44, 7}, {334900605644, 39}, {8440, 15}, {248, 9}, {760, 11}, {38, 6}} The repetition periods and distances moved in each period for the structures are respectively {{4, -2}, {12, -6}, {12, -6}, {42, -14}, {42, -14}, {15, -4}, {15, -4}, {15, -4}, {15, -4}, {30, -8}, {92, -18}, {36, -4}, {7, 0}, {10, 2}, {3, 2}} Note that the periodicity of the background forces all rule 110 structures to have periods and distances given by {4, -2} r + {3, 2} s where r and s are non-negative integers. … Extended versions of (b) and (c) can be obtained from Flatten[{IntegerDigits[1468, 2], Table[ IntegerDigits[102524348, 2], {n}], IntegerDigits[v, 2]}] where n is a non-negative integer and v is one of {1784, 801016, 410097400, 13304, 6406392, 3280778648} Note that in most cases multiple copies of the same structure can travel next to each other, as seen on page 290 .

Connection [of 2D substitution systems] with digit sequences Just as in the 1D case discussed on page 891 , the color of a cell at position {i, j} in a 2D substitution system can be determined using a finite automaton from the digit sequences of the numbers i and j . At step n , the complete array of cells is Table[If[FreeQ[Transpose[IntegerDigits[{i, j}, k, n]], form], 1, 0], {i, 0, k n - 1}, {j, 0, k n - 1}] where for the pattern on page 187 , k = 2 and form = {0, 1} . … Note that the excluded pairs of digits are in exact correspondence with the positions of which squares are 0 in the underlying rules for the substitution systems.

And from this distribution one can tell with Statistics of block frequencies for various sequences. … Sequence (f) is formed by concatenating base 2 digits of successive integers. … Sequence (h) is the base 2 digits of π .

In both cases it then turns out that h can be obtained from (see note above ) h[a_, b_] := FromDigits[g[ListConvolve[ IntegerDigits[a, k], IntegerDigits[b, k], {1, -1}, 0]], k] where for multiplication rules g = Identity and for additive cellular automata g = Mod[#, k] & . For multiplication rules, there are normally carries (handled by FromDigits ), but for power cellular automata, these have only limited range, so that g = Mod[#, k α ] & can be used. … If h = Times , then as discussed in the note above, the most obvious procedure for evaluating h[a, b] would involve about m n operations, where m and n are the numbers of digits in a and b .

And if this were so, then it would mean that the initial conditions for systems like the shift map would naturally have digit sequences that are almost always random. … For what it says is that the randomness we see somehow comes from randomness that is already present—but it does not explain where that randomness comes from. … Yet the fact that systems like (a) and (b) can intrinsically generate randomness even from simple initial conditions does not mean that they