Properties [of example symbolic system] All initial conditions eventually evolve to expressions of the form Nest[ ℯ , ℯ , m] , which then remain fixed. … During the evolution the rule can apply only to the inner part FixedPoint[Replace[#, ℯ [x_]  x] &, expr] of an expression. … For all initial conditions this depth seems at first to increase linearly, then to decrease in a nested way according to FoldList[Plus, 0, Flatten[Table[ {1, 1, Table[-1, {IntegerExponent[i, 2] + 1}]}, {i, m}]]] This quantity alternates between value 1 at position 2 j and value j at position 2 j - j + 1 .

The last axiom is a schema (see page 1156 ) that states the principle of mathematical induction: that if a statement is valid for a = 0 , and its validity for a = b implies its validity for a = b + 1 , then it follows that the statement must be valid for all a . … Axioms 3, 4 and 6 can then be replaced by a × b  b × a , a × (b × c)  (a × b) × c and ( Δ a) × ( Δ a × b)  Δ a × ( Δ b × ( Δ a)) . … It was shown by Raphael Robinson in 1950 that universality is also achieved by the Robinson axioms for reduced arithmetic (usually called Q) in which induction—which cannot be reduced to a finite set of ordinary axioms (see page 1156 )—is replaced by a single weaker axiom.

Smooth iterated maps In the main text, all the functions used as mappings consist of linear pieces, usually joined together discontinuously. … (An important result discovered by Mitchell Feigenbaum in 1975 is that this basic setup is universal to all smooth maps whose functions have a single hump.) … In the special case a = 4 , it turns out that replacing x by Sin[ π u] 2 makes the mapping become just u  FractionalPart[2 u] , revealing simple shift map dependence on the initial digit sequence.

In the lattice version in physics one typically considers what happens to averages over all possible configurations of a system if one does a so-called blocking transformation that replaces blocks of elements by individual elements.

One gets extremely similar results with a type A Eden model in which one just randomly selects a cell from all the ones adjacent to the cluster. With a grid of cells set up in advance, each step in this type of Eden model can be achieved with AStep[a_] := ReplacePart[a, 1, (# 〚 Random[ Integer, {1, Length[#]}] 〛 &)[Position[(1 - a)Sign[ ListConvolve[{{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}, a, {2, 2}]], 1]]] This implementation can readily be extended to generalized aggregation models (see below ).

Comments on Mathematica functions CenterList works by first creating a list of n 0's, then replacing the middle 0 by a 1. … CAStep uses the fact that Mathematica can manipulate all the elements in a list at once.

In basic logic any statement that is true for all possible assignments of truth values to variables can always be proved from the axioms of basic logic. In 1930 Kurt Gödel showed a similar result for pure predicate logic: that any statement that is true for all possible explicit values of variables and all possible forms of predicates can always be proved from the axioms of predicate logic. … A typical issue is that in, say, ∀ x ( ∃ y ( ¬ x  y)) , x and y are dummy variables whose specific names are not supposed to be significant; yet the names become significant if, say, x is replaced by y .

Note that in this representation, effects can propagate all the way across the system in a single step. … For each complete update of a rule 90 sequential cellular automaton, the pictures below show results with (a) left-to-right scan, (b) random ordering of all cells, the same for each pass through the whole system, (c) random ordering of all cells, different for different passes, (d) completely random ordering, in which a particular cell can be updated twice before other cells have even been updated once. … The following will update triples of cells in the specified order by using the function f : OrderedUpdate[f_, a_, order_]:= Fold[ReplacePart[ #1, f[Take[#1, {#2 - 1, #2 + 1}]], #2] &, a, order] A random ordering of n cells corresponds to a random permutation of the form Fold[Insert[#1, #2, Random[Integer, Length[#1]] + 1] &, {}, Range[n]]

In most places in the space of all possible field configurations, the value of s will vary quite quickly between nearby configurations. … In cases like QED and QCD the most obvious solutions to the classical equations are ones in which all fields are zero. … In studying quantum field theories it has been common to consider effectively replacing time coordinates t by  t to go from ordinary Minkowski space to Euclidean space (see page 1043 ).

This means that same rules should apply if one not only reverses the direction of time (T), but also simultaneously inverts all spatial coordinates (P) and conjugates all charges (C), replacing particles by antiparticles. … Originally it was assumed that C, P and T would all separately be invariances, as they are in classical mechanics.