Solve[Nest[f, x, p]  x, x], Im[#]  0 &] For x  a x (1 - x) the results usually cannot be expressed in terms of explicit radicals beyond period 2.

With two possible forms of behavior h[i] = 0 or 1 for initial condition i , an example of such a number is Sum[2 -i h[i], {i, ∞ }] . Closely related is the total probability for each form of behavior, given for example by Sum[2^-(Ceiling[Log[2, i]]) h[i], {i, ∞ }] .

And indeed in case (g), for example, one sees exactly r 1 linear growth, reflecting dimension 2. Similarly, in case (d) one sees r 0 growth, reflecting dimension 1, while in case (h) one sees r 2 growth, reflecting dimension 3.

For a cylinder, there are difficulties with boundary conditions at infinity, but the drag coefficient was nevertheless calculated by William Oseen in 1915 to be 8 π /(R (1/2 + Log[8/R] - EulerGamma)) . … For a vortex street no analytical solution has ever been found, and indeed it is only recently that the general paths of fluid elements have even been accurately deduced. … Over the range 50 ≲ R ≲ 150 vortices are found to be generated at a cylinder with almost perfect periodicity at a dimensionless frequency (Strouhal number) that increases smoothly from about 0.12 to 0.19.

In the first case shown, starting for example at position 4 the dot then visits positions 5, 0, 1, 2 and so on, at each step going from one node in the network to the next. … There are cycles which contain states that are visited repeatedly, and there can also be trees that represent transient states that can each only ever occur at most once in the evolution of the system. … The pictures below give corresponding results for a class 2 cellular automaton (rule 132).

Einstein equations In the absence of matter, the standard statement of the Einstein equations is that all components of the Ricci tensor—and thus also the Ricci scalar—must be zero (or formally that R ij = 0 ). … Note that the equation can also be written R μ ν = 8 π G (T μ ν - 1/2 T μ μ g μ ν /c 4 ) .) … (A fairly stringent example is 0 ≤ p ≤ ρ /3 —and whether this is actually true for non-trivial interacting quantum fields remains unclear.)

Most arose first as solutions to specific differential equations, typically in physics and astronomy; some arose as products, sums of series or inverses of other functions. In the mid-1800s it became clear that despite their different origins most of these functions could be viewed as special cases of Hypergeometric2F1[a, b, c, z] , and that the functions covered the solutions to all linear differential equations of a certain type. ( Zeta and PolyLog are parametric derivatives of Hypergeometric2F1 ; elliptic modular functions are inverses.)

And no doubt the processes which produce these patterns during the development of the organism can be idealized by simple 2D cellular automata. … Since the late 1970s, it has been common to assume that the response of a cell can be modelled by derivatives of Gaussians such as those shown below, or perhaps by Gabor functions given by products of trigonometric functions and Gaussians.

If one considers all 2 n possible sequences (say of 0's and 1's) of length n then it is straightforward to see that most of them must be more or less algorithmically random. For in order to have enough programs to generate all 2 n sequences most of the programs one uses must themselves be close to length n . (In practice there are subtleties associated with the encoding of programs that make this hold only for sufficiently large n .)

Implementation [of continuous cellular automata] The state of a continuous cellular automaton at a particular step can be represented by a list of numbers, each lying between 0 and 1. This list can then be updated using CCAEvolveStep[f_, list_List] := Map[f, (RotateLeft[list] + list + RotateRight[list])/3] CCAEvolveList[f_, init_List, t_Integer] := NestList[CCAEvolveStep[f, #] &, init, t] where for the rule on page 157 f is FractionalPart[3#/2] & while for the rule on page 158 it is FractionalPart[# + 1/4] & .