In the mid-1800s the idea emerged of a vector potential whose curl gives the magnetic field, and it was soon recognized—notably by James Clerk Maxwell —that any function whose curl vanishes (and that can therefore normally be written as a gradient) could be added to the vector potential without affecting the magnetic field. … 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.

At a mathematical level, following work by Joseph Fourier around 1810 it became clear by the mid-1800s how any sufficiently smooth function could be decomposed into sums of sine waves with frequencies corresponding to successive integers.

If one allows trigonometric functions, any equation for integers can be converted to one for real numbers; for example x 2 + y 2  z 2 for integers is equivalent to Sin[ π x] 2 + Sin[ π y] 2 + Sin[ π z] 2 + (x 2 + y 2 - z 2 ) 2  0 for real numbers.

If the evolution of rule 30 can be set up as on page 704 to emulate any Boolean function then the problem considered here is immediately equivalent to satisfiability.

Category theory can be viewed as a formalization of operations on abstract data types in computer languages—though unlike in Mathematica it normally requires that functions take a single bundle of data as an argument.

(Iterating the related function BitXor[i, 2i] yields numbers whose digit sequences correspond to the rule 60 cellular automaton).

This function appeared on page 870 in the discussion of binomial coefficients modulo 2, and will appear again in several other places in this book.

Singular behavior [in PDEs] An example of an equation that yields inconsistent behavior is the diffusion equation with a negative diffusion constant: ∂ t u[t, x]  - ∂ xx u[t, x] This equation makes any variation in u as a function of x eventually become infinitely rapid.

The specific form of the continuous generalization of the modulo 2 function used is λ [x_] := Exp[-10 (x - 1) 2 ] + Exp[-10 (x - 3) 2 ] Each cell in the system is then updated according to λ [a + c] for rule 90, and λ [a + b + c + b c] for rule 30.

If a system is additive it means that one can work out how the system will behave from any initial condition just by combining the patterns ("Green's functions") obtained from certain basic initial conditions—say ones containing a single black cell. … But in general one can apply to each cell value any function σ that obeys the so-called Cauchy functional equation σ [x+y]  σ [x] + σ [y] . … But one can also imagine setting up systems whose states are continuous functions of position. ϕ then defines a mapping from one such function to another.