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Time series
Sequences of continuous numerical data are often known as time series, and starting in the 1960s standard models for them have consisted of linear recurrence relations or linear differential equations with random noise continually being added. … As discussed on page 919 it was already realized in the 1970s that even without external random noise nonlinear models could produce time series with seemingly random features.

Generating functions [for regular languages]
The sequences in a regular language can be thought of as corresponding to products of non-commuting variables that appear as coefficients in a formal power series expansion of a generating function.

Computer printouts
The printouts show a series of elementary cellular automata started from random initial conditions (see page 232 ).

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Forms of a binary operator satisfying the constraints of a series of different axiom systems.

The argument is based on showing that an algebraic function always exists for which the coefficients in its power series correspond to any given nested sequence when reduced modulo some p . … But then there is a general result that if a particular sequence of power series coefficients can be obtained from an algebraic (but not rational) function modulo a particular p , then it can only be obtained from transcendental functions modulo any other p —or over the integers.

[Cognitive] child development
As children get older their thinking becomes progressively more sophisticated, advancing through a series of fairly definite stages that appear to be associated with an increasing ability to handle generalization and abstraction.

Mathematical properties [of branching model]
If an element c of the list b is real, so that there is a stem that goes straight up, then the limiting height of the center of the pattern is obtained by summing a geometric series, and is given by 1/(1 - c) .

Many sine functions
Adding many sine functions yields a so-called Fourier series (see page 1074 ). … Other so-called Fourier series in which the coefficient of Sin[m x] is a smooth function of m for all integer m yield similarly simple results.

These numbers can also be obtained as the coefficients of x n in the series expansion of x ∂ x Log[ ζ [m, x]] , with the so-called zeta function, which is always a rational function of x , given by
ζ [m_, x_] := 1/Det[IdentityMatrix[Length[m]] - m x]
and corresponds to the product over all cycles of 1/(1 - x n ) .

If we assume that the density varies slowly with position and time, then we can make series expansions such as
f[x + dx, t] f[x , t] + ∂ x f[x, t] dx + 1/2 ∂ xx f[x, t] dx 2 + …
where the coordinates are scaled so that adjacent cells are at positions x - dx , x , x + dx , etc. … And from this it follows that
f[x, t + dt] c (f[x - dx, t] + f[x + dx, t]) + (1 - 2c)f[x, t]
Performing a series expansion then yields
f[x, t] + dt ∂ t f[x, t] f[x, t] + c dx 2 ∂ xx f[x, t]
which in turn gives exactly the usual 1D diffusion equation ∂ t f[x, t] ξ ∂ xx f[x, t] , where ξ is the diffusion coefficient for the system.