As an example, one can consider a generalization of the arithmetic systems discussed on page 122 —in which one has a whole number n , and at each step one finds the remainder after dividing by a constant, and based on the value of this remainder one then applies some specified arithmetic operation to n .

At a practical level, the one-dimensional character of data from radio signals makes it difficult for us to apply our visual systems—which remain our most powerful general-purpose analysis tools.

The sequential limit [in generalized substitution systems] Even when the order of applying rules does not matter, using the scheme of a sequential substitution system will often give different results.

Writing a Boolean function in DNF is the rough analog of applying Expand to a polynomial. Conjunctive normal form (CNF) Or[…] ∧ Or[…] ∧ … is the rough analog of applying Factor .

But for multiway systems where each rule p  q is accompanied by its reverse q  p , and such pairs are represented say by "AAB" ↔ "BBAA" , an equivalent operator system can immediately be obtained either from Apply[Equal, Map[Fold[#2[#1] &, x, Characters[#]] &, rules, {2}], {1}] or from (compare page 1172 ) Append[Apply[Equal, Map[(Fold[f, First[#], Rest[#]] &)[Characters[#]] &, rules, {2}], {1}], f[f[a, b], c]  f[a, f[b, c]]] where now objects like "A" and "B" are treated as constants—essentially functions with zero arguments.

Note that rules of the kind discussed on page 508 which involve replacing clusters of nodes can only apply when cycles in the cluster match those in the network.

If the coefficients inside all the sine functions are rational, then going from t = 0 to t = 2 π Apply[LCM, Map[Denominator, list]] yields a closed curve.

Given p = Array[Prime, Length[list], PrimePi[Max[list]] + 1] or any list of integers that are all relatively prime and above Max[list] (the integers in list are assumed positive) CRT[list_, p_] := With[{m = Apply[Times, p]}, Mod[Apply[Plus, MapThread[#1 (m/#2)^EulerPhi[#2] &, {list, p}]], m]] yields a number x such that Mod[x, p]  list .

The number of states with spatial period m is given by s[m_, k_]:= k m - Apply[Plus, Map[s[#, k] &, Drop[Divisors[m], -1]]] or equivalently s[m_, k_]:=Apply[Plus, (MoebiusMu[m/#] k # &)[Divisors[m]]] In a cellular automaton with a total of n cells, the maximum possible repetition period is thus s[n, k] .

At each step there is a number x between 0 and 1 that is updated by applying a fixed mapping.