Affine transformations Any set of so-called affine transformations that take the vector for each point, multiply it by a fixed matrix and then add a fixed vector, will yield nested patterns similar to those shown in the main text.

whenever there is sufficient uniformity to give a stable structure to space one can still think of something like parallel slices at different angles as representing motion at different fixed speeds. And the crucial point is that whenever the underlying system is causal invariant the exact same underlying rules will account for what one sees in slices at different angles. … And the remarkable point is then that this is also what seems to happen in physics.

But the discovery which launched what has become known as chaos theory is that at least in principle there can be systems whose sensitivity to their initial conditions is so great that no machine with fixed tolerances can ever be expected to yield repeatable results. … The result of this is that any change in the initial position of a point will be amplified by a factor of two at each step. … But what does this actually mean for the motion of a point in the material?

Iterated maps For maps of the form x  a x (1 - x) discussed on page 920 the attractor for small a is a fixed point, then a period 2 limit cycle, then period 4, 8, 16, etc.

And indeed for the first curve shown below, this procedure works just fine, and quickly leads to the lowest point. … And as a result, such blocks remain fixed and cannot disappear. … In the first case, the most obvious mechanical or mathematical procedure of continually going downhill will successfully lead one to the lowest point.

And in a system like a cellular automaton a halting problem can be set up by asking whether a cell at a particular position ever turns a particular color, or whether, more globally, the complete state of the system ever reaches a fixed point and no longer changes. … Mathematica works by taking its input and repeatedly applying transformation rules—a process which normally reaches a fixed point that is returned as the answer, but with definitions like x = x + 1 ( x having no value) formally does not.

= {}, AllNet[k], q = ISets[b = Map[Table[ Position[d, NetStep[net, #, a]] 〚 1, 1 〛 , {a, 0, k - 1}]&, d]]; DeleteCases[MapIndexed[#2 〚 2 〛 - 1  #1 &, Rest[ Map[Position[q, #] 〚 1, 1 〛 &, Transpose[Map[Part[#, Map[ First, q]]&, Transpose[b]]], {2}]] - 1, {2}], _  0, {2}]]] DSets[net_, k_:2] := FixedPoint[Union[Flatten[Map[Table[NetStep[net, #, a], {a, 0, k - 1}]&, #], 1]]&, {Range[Length[net]]}] ISets[list_] := FixedPoint[Function[g, Flatten[Map[ Map[Last, Split[Sort[Part[Transpose[{Map[Position[g, #] 〚 1, 1 〛 &, list, {2}], Range[Length[list]]}], #]], First[#1]  First[#2]&], {2}]&, g], 1]], {{1}, Range[2, Length[list]]}] If net has q nodes, then in general MinNet[net] can have as many as 2 q -1 nodes. … To obtain such trimmed networks one can apply the function TrimNet[net_] := With[{m = Apply[Intersection, Map[FixedPoint[ Union[#, Flatten[Map[Last, net 〚 # 〛 , {2}]]]&, #]&, Map[List, Range[Length[net]]]]]}, net 〚 m 〛 /.

In a rectangular region, the position is given by Mod[a t, {w, h}] and every point will be visited if the parameters have irrational ratios. In a region that contains fixed circular obstructions, the motion can become sensitively dependent on initial conditions. … For a system of balls in a region with cyclic boundaries, a complicated proof due to Yakov Sinai from the 1960s purports to show that every ball eventually visits every point in the region, and that certain simple statistical properties of trajectories are consistent with randomness.

Yet if one uses this measure of distance then the distance between two particles will always tend to remain fixed as the number of connections in the thread. But the point is that this measure of distance is in reality just a simple idealization of what is relevant in practice. … And while they can get from one point to another through the large number of connections that define the background space, they cannot in a sense fit through a small number of connections in a thread.

Huffman coding From a list p of probabilities for blocks, the list of codewords can be generated using Map[Drop[Last[#], -1] &, Sort[ Flatten[MapIndexed[Rule, FixedPoint[Replace[Sort[#], {{p0_, i0_}, {p1_, i1_}, pi___}  {{p0 + p1, {i0, i1}}, pi}] & , MapIndexed[List, p]] 〚 1, 2 〛 , {-1}]]]] -1 Given the list of codewords c , the sequence of blocks that occur in encoded data d can be uniquely reconstructed using First[{{}, d} //.