The pictures on the facing page show what happens if one successively multiplies a number by various constant factors, and then looks at the digit sequences of the numbers that result. … And one of the reasons for the popularity of linear congruential generators is that with fairly straightforward mathematical analysis it is possible to tell exactly what multiplication factors will maximize this repetition period.

In 1918 Hermann Weyl tried to reproduce electromagnetism by adding the notion of an arbitrary scale or gauge to the metric of general relativity (see page 1028 )—and noted the "gauge invariance" of his theory under simultaneous transformation of electromagnetic potentials and multiplication of the metric by a position-dependent factor. Following the introduction of the Schrödinger equation in quantum mechanics in 1926 it was almost immediately noticed that the equations for a charged particle in an electromagnetic field were invariant under gauge transformations in which the wave function was multiplied by a position-dependent phase factor. … And after a few earlier attempts, Yang–Mills theories were introduced in 1954 by extending the notion of a phase factor to an element of an arbitrary non-Abelian group.

In a first approximation, the slowdown factor is the refractive index. … The effective mass for massive particles increases by a factor 1/Sqrt[1 - v 2 /c 2 ] at speed v , making it take progressively more energy to increase v .

The "quality" of the image is determined by how many weights are kept; a typical default quality factor, used say by Export in Mathematica, is 75.

Two sine functions Sin[a x] + Sin[b x] can be rewritten as 2 Sin[1/2(a + b) x] Cos[1/2(a - b) x] (using TrigFactor ), implying that the function has two families of equally spaced zeros: 2 π n/(a + b) and 2 π (n + 1/2)/(b - a) .

But the exact number of steps in each case depends on the prime factors of the numbers that define the system.

And with this assumption, n should increase by a factor of 5/2 half the time, and decrease by a factor close to 1/2 the rest of the time—so that after t steps it should be multiplied by an overall factor of about ( √ 5 /2) t . … If one applies the same kind of argument to the standard 3n+1 problem, then one concludes that n should on average decrease by a factor of √ 3 /2 at each step, making it unsurprising that at least in most cases n eventually reaches the value 1.

In the type of system shown on the facing page , it turns out that the repetition period is maximal whenever the number of positions moved at each step shares no common factor with the total number of possible positions—and this is achieved for example whenever either of these quantities is a prime number.

But what we now see is that in fact all the different forms that are observed are in effect just consequences of the The effects of varying five simple features of the rule for the growth of a mollusc shell: (a) the overall factor by which the size increases in the course of each revolution; (b) the relative amount by which the opening is displaced downward at each revolution; (c) the size of the opening relative to the overall size of the shell; (d) the elongation of the opening; (e) the orientation of elongation in the opening.

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.