But a low-level language such as Fortran , C or Java always stores a fixed number of digits, typically around 53, in its standard double-precision floating-point representation of numbers. … But if the computer gives a fixed number of digits at each step, then additional digits must be filled in on the right. … The point is that the presence of randomness makes the system behave on different steps as if it were evolving from slightly different initial conditions.

An alternative formulation is to ask whether for all n FixedPoint[(3#/2^IntegerExponent[#, 2] + 1)/2 &, n]  2 With the rule n  If[EvenQ[n], 5n/2, (n + 1)/2] used in the main text, the sequence produced repeats if n ever reaches 2, 4 or 40 (and possibly higher numbers).

With rules set up in this way, each step in the evolution of a network system is given by NetEvolveStep[{depth_Integer, rule_List}, list_List] := Block[ {new = {}}, Join[Table[Map[NetEvolveStep1[#, list, i] &, Replace[NeighborNumbers[list, i, depth], rule]], {i, Length[list]}], new]] NetEvolveStep1[s : {___Integer}, list_, i_] := Follow[list, i, s] NetEvolveStep1[{s1 : {___Integer}, s2 : {___Integer}}, list_, i_] := Length[list] + Length[ AppendTo[new, {Follow[list, i, s1], Follow[list, i, s2]}]] The set of nodes that can be reached from node i is given by ConnectedNodes[list_, i_] := FixedPoint[Union[Flatten[{#, list 〚 # 〛 }]] &, {i}] and disconnected nodes can be removed using RenumberNodes[list_, seq_] := Map[Position[seq, #] 〚 1, 1 〛 &, list 〚 seq 〛 , {2}] The sequence of networks obtained on successive steps by applying the rules and then removing all nodes not connected to node number 1 is given by NetEvolveList[rule_, init_, t_Integer] := NestList[(RenumberNodes[#, ConnectedNodes[#, 1]] &)[ NetEvolveStep[rule, #]] &, init, t] Note that the nodes in each network are not necessarily numbered in the order that they appear on successive lines in the pictures in the main text.

Given an original DNF list s , this can be done using PI[s, n] : PI[s_, n_] := Union[Flatten[ FixedPointList[f[Last[#], n] &, {{}, s}] 〚 All, 1 〛 , 1]] g[a_, b_] := With[{i = Position[Transpose[{a, b}], {0,1}]}, If[Length[i]  1 && Delete[a, i] === Delete[b, i], {ReplacePart[a, _, i]}, {}]] f[s_, n_] := With[ {w = Flatten[Apply[Outer[g, #1, #2, 1] &, Partition[Table[ Select[s, Count[#, 1]  i &], {i, 0, n}], 2, 1], {1}], 3]}, {Complement[s, w, SameTest  MatchQ], w}] The minimal DNF then consists of a collection of these prime implicants.

But even though successive vortices are formed at fixed intervals, irregularities can develop as the array of vortices goes downstream, and such irregularities seem to occur at lower Reynolds numbers for flows past plates than cylinders. … The drag coefficient remains almost constant at a value around 1 until R ≃ 3 × 10 5 , at which point it drops precipitously for a while.

Basic quantum mechanics is set up to describe how fixed numbers of particles behave—say in externally applied electromagnetic or other fields. … As mentioned on page 1044 , ordinary quantum field theory in effect deals only with point particles.