With a rule given in this form, a single step in the evolution of the Turing machine can be implemented with the function TMStep[rule_List, {s_, a_List, n_}] /; (1 ≤ n ≤ Length[a]) := Apply[{#1, ReplacePart[a, #2, n], n + #3}&, Replace[{s, a 〚 n 〛 }, rule]] The evolution for many steps can then be obtained using TMEvolveList[rule_, init_List, t_Integer] := NestList[TMStep[rule, #]&, init, t] An alternative approach is to represent the complete state of the Turing machine by MapAt[{s, #}&, list, n] , and then to use TMStep[rule_, c_] := Replace[c, {a___, x_, h_List, y_, b___}  Apply[{{a, x, #2, {#1, y}, b}, {a, {#1, x}, #2, y, b}} 〚 #3 〛 &, h /. rule]] The result of t steps of evolution from a blank tape can also be obtained from (see also page 1143 ) s = 1; a[_] = 0; n = 0; Do[{s, a[n], d} = {s, a[n]} /. rule; n += d, {t}]

(Related objects have also arisen in connection with Hölder exponents for discontinuous functions.)

But in the course of the 1700s and 1800s formulas were successfully found for solutions to a great many problems in mathematical physics (see note below )—at least when suitable special functions (see page 1091 ) were introduced. … In the 1820s it was shown that quintic equations cannot in general be solved in terms of radicals (see page 1137 ), and by the 1890s it was known that degree 7 equations cannot in general be solved even if elliptic functions are allowed. Around 1890 it was then shown that the three-body problem could not be solved in general in terms of ordinary algebraic functions and integrals (see page 972 ).

Color vision The three types of color-sensitive cone cells on the human retina each have definite response curves as a function of wavelength.

In the 1600s and 1700s, however, the development of calculus and notions of continuous functions made use of more deductive methods.

And the reason for this is that with s = {1, 2} the function Union[Complement[s, a], b] corresponding to a  b only ever gets to the 3 elements {{1}, {2}, {1, 2}} .

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

Despite empirical evidence, no number expressed just in terms of standard mathematical functions has ever been rigorously proved to be normal.

In 1926 Erwin Schrödinger then suggested a partial differential equation for the wave functions of particles like electrons. … But by the end of the 1960s theoretical progress seemed blocked by basic questions about functions of several complex variables, and predictions that were made did not seem to work well. … In condensed matter physics there are nevertheless cases like the Kondo model where exact solutions have been found, and where the effective energy function for electrons happens to be roughly the same as in a relativistic theory.)

The transitions between these states have probabilities given by m[Map[Length, list]] where m[s_] := With[{q = FoldList[Plus, 0, s]}, ReplacePart[ RotateRight[IdentityMatrix[Last[q]], {0, 1}], 1/Length[s], Flatten[Outer[List, Rest[q], Drop[q, -1] + 1], 1]]] The average spectrum of sequences generated according to these probabilities can be obtained by computing the correlation function for elements a distance r apart ξ [list_, r_] := With[{w = (# - Apply[Plus, #]/Length[#] &)[ Flatten[list]]}, w .