But in fact, having just specified a block of length x + 2 r t in the initial conditions, the cellular automaton rule then uniquely determines the color of every cell in the patch, allowing a total of at most s[t, x] = k x + 2 r t configurations.

Intrinsically defined curves With curvature given by a function f[s] of the arc length s , explicit coordinates {x[s], y[s]} of points are obtained from (compare page 1048 ) NDSolve[{x'[s]  Cos[ θ [s]], y'[s]  Sin[ θ [s]], θ '[s]  f[s], x[0]  y[0]  θ [0]  0}, {x, y, θ }, {s, 0, s max }] For various choices of f[s] , formulas for {x[s], y[s]} can be found using DSolve : f[s] = 1: {Sin[ θ ], Cos[ θ ]} f[s] = s: {FresnelS[ θ ], FresnelC[ θ ]} f[s] = 1/ √ s : √ θ {Sin[ √ θ ], Cos[ √ θ ]} f[s] = 1/s: θ {Cos[Log[ θ ]], Sin[Log[ θ ]]} f[s] = 1/s 2 : θ {Sin[1/ θ ], Cos[1/ θ ]} f[s] = s n : result involves Gamma[1/n, ±  θ n/n ] f[s] = Sin[s] : result involves Integrate[Sin[Sin[ θ ]], θ ] , expressible in terms of generalized Kampé de Fériet hypergeometric functions of two variables.

(It is much easier to implement in Mathematica—as discussed above—since there functions like BitXor can operate on integers of any length.)

A number is said to be "normal" in a particular base if every digit and every block of digits of any length occur with equal frequency.

One might think that a more efficient approach would be to start with the trivial length t digit sequence for c t in base c , then to find a particular base k digit just by converting to base k .

If m = k s -1 then it turns out that interchanging a pair of adjacent length s blocks in list never affects the result.

An example with 8 registers and 41 instructions is: or {d[4, 40], i[5], d[3, 9], i[3], d[7, 4], d[5, 14], i[6], d[3, 3], i[7], d[6, 2], i[6], d[5, 11], d[6, 3], d[4, 35], d[6, 15], i[4], d[8, 16], d[5, 21], i[1], d[3, 1], d[5, 25], i[2], d[3, 1], i[6], d[5, 32], d[1, 28], d[3, 1], d[4, 28], i[4], d[6, 29], d[3, 1], d[5, 24], d[2, 28], d[3, 1], i[8], i[6], d[5, 36], i[6], d[3, 3], d[6, 40], d[4, 3]} Given any register machine, one first applies the function RMToRM2 from page 1114 , then takes the resulting program and initial condition and finds an initial condition for the URM using R2ToURM[prog_, init_] := Join[init, With[ {n = Length[prog]}, {1 + LE[Reverse[prog] /.

(Note that if an axiom system does manage to reproduce logic in full then as indicated on page 814 its consequences can always be derived by proofs of limited length, if nothing else by using truth tables.)

For whether one does calculations by hand, by mechanical calculator or by electronic computer, one always needs an explicit representation for numbers, typically in terms of a sequence of digits of a certain length.

Fractal dimensions [of additive cellular automata] The total number of nonzero cells in the first t rows of the pattern generated by the evolution of an additive cellular automaton with k colors and weights w (see page 952 ) from a single initial 1 can be found using g[w_, k_, t_] := Apply[Plus, Sign[NestList[Mod[ ListCorrelate[w, #, {-1, 1}, 0], k] &, {1}, t - 1]], {0, 1}] The fractal dimension of this pattern is then given by the large m limit of Log[k,g[w, k,k m + 1 ]/g[w, k, k m ]] When k is prime it turns out that this can be computed as d[w_, k_:2] := Log[k,Max[Abs[Eigenvalues[With[ {s = Length[w] - 1}, Map[Function[u, Map[Count[u, #] &, #1]], Map[Flatten[Map[Partition[Take[#, k + s - 1], s, 1] &, NestList[Mod[ListConvolve[w, #], k] &, #, k - 1]], 1] &, Map[Flatten[Map[{Table[0, {k - 1}], #} &, Append[#, 0]]] &, #]]] &[Array[IntegerDigits[#, k, s] &, k s - 1]]]]]]] For rule 90 one gets d[{1, 0, 1}] = Log[2, 3] ≃ 1.58 .