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But my guess is that its most important function is quite mundane: just as muscles build up lactic acid waste products, so also I suspect synapses in the brain build up waste products, and these can only safely be cleared out when the brain is not in normal use.

In studying specific instances of objects like groups one often represents elements as products of constants or generators, and then for example specifies the group by giving relations between these products.

After t steps a continuous approximation to the spectrum is Product[1 - Exp[2 s ω ], {s, t}] , which is an example of a type of product studied by Frigyes Riesz in 1918 in connection with questions about the convergence of trigonometric series. It is related to the product of sawtooth functions given by Product[Abs[Mod[2 s ω , 2, -1]], {s, t}] . … After t steps a continuous approximation to the spectrum is Product[1 + Exp[3 s 2 ω ], {s, t}] .

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.

The system generates a dark gray stripe on the left at all positions that correspond to any product of numbers other than 1.

Since numbers can be factored uniquely into products of powers of primes, a number can be specified by a list in which 1's appear at the positions of the appropriate Prime[m] n (which can be sorted by size) and 0's appear elsewhere, as shown below.

But in practice the most accurate measurements show phenomena such as 1/f noise, presumably as a result of features of the detector and perhaps of electromagnetic fields associated with decay products.

Arithmetic systems [emulating register machines]
Given the program for a register machine with nr registers in the form on page 896 , an arithmetic system which emulates it can be obtained from
RMToAS[prog_, nr_] := With[{p = Length[prog], g = Product[Prime[j], {j, nr}]}, {p g, Sort[Flatten[MapIndexed[ With[{n = First[#2] - 1}, #1 /. … The evolution of the arithmetic system is given by
ASEvolveList[{n_, rules_}, init_, t_] := NestList[(Mod[#, n] /. rules)[#] &, init, t]
Given a value m obtained in the evolution of the arithmetic system, the state of the register machine to which it corresponds is
{Mod[m, p] + 1, Map[Last, FactorInteger[ Product[Prime[i], {i, nr}] Quotient[m, p]]] - 1}
Note that it is possible to have each successive step involve only multiplication, with no addition, at the cost of using considerably larger numbers overall.

States versus colors [in Turing machines]
The total number of possible Turing machines depends on the product s k .

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 ) .