Nonlinear feedback shift registers Linear feedback shift registers of the kind discussed on page 974 can be generalized to allow any function f (note the slight analogy with cyclic tag systems): NLFSRStep[f_, taps_, list_] := Append[Rest[list], f[list 〚 taps 〛 ]] With the choice f=IntegerDigits[s, 2, 8] 〚 8 - # . {4, 2, 1} 〛 & and taps = {1, 2, 3} this is essentially a rule s elementary cellular automaton. … One set of computations concerned functions f[{w_, x_, y_, z_}] := Mod[w + y + z + x y + x z + y z, 2] (apparently chosen to have balance between 0's and 1's that would minimize correlations). … And as noted by Nicolaas de Bruijn in 1946 there are 2 2 n - 1 -n such paths with length 2 n , and thus this number of functions f out of the 2 2 n possible must yield sequences of maximal length.

The evaluation of functions with attribute Flat in Mathematica provides an example of confluence. … Showing only the arguments to f , the pictures below illustrate how the flat functions Xor and And are confluent, while the non-flat function Implies is not.

Gradient descent [in constraint satisfaction] A standard method for finding a minimum in a smooth function f[x] is to use FixedPoint[# - a f'[#] &, x 0 ] If there are local minima, then which one is reached will depend on the starting point x 0 .

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, ∞ }] . (Z transform or generating function methods can be applied directly only for substitution systems with rules such as {1  list, 0  1 - list} .) … It is related to the product of sawtooth functions given by Product[Abs[Mod[2 s ω , 2, -1]], {s, t}] .

One can scan a quadrant of an infinite grid using the σ function on page 1127 , or one can scan a whole grid by for example going in a square spiral that at step t reaches position (1/2(-1) # ({1, -1}(Abs[# 2 - t] - #) + # 2 - t- Mod[#, 2]) &)[ Round[ √ t ]]

The bottom plot gives the repetition period for this system as a function of its size; for odd n this period is equal to MultiplicativeOrder[2, n] .

Repetition periods for various cellular automata as a function of size.

But on a larger scale the brain seems to be organized into areas with very definite functions. … Certain higher mental functions are known to be localized in definite areas of the brain, though within these areas there is often variability between individuals.

One might imagine that it should be possible to set up a function f[i, n] which if given successive integers i would give the n th base 2 digit in every possible real number. … Analogously, one might imagine that it should be possible to have a function f[i, n] which enumerates all possible programs that always halt, and specifies a digit in their output when given input n .

But with suitable weights one can reproduce many functions. … But out of the 2 2 n possible Boolean functions of n inputs, only 14 (out of 16) can be obtained for n = 2 , 104 (out of 256) for n = 3 , 1882 for n = 4 , and 94304 for n = 5 . … By introducing enough hidden units it is then possible—just as in the formulas discussed on page 616 —to reproduce essentially any function.