The evidence is based on applying various standard statistical tests of randomness, and remains somewhat haphazard. … Despite empirical evidence, no number expressed just in terms of standard mathematical functions has ever been rigorously proved to be normal. … One based on gradual extension of work by Richard Stoneham from 1971 is that numbers of the form Sum[1/(p n b p n ), {n, ∞ }] for prime p > 2 are normal in base b (for GCD[b, p]  1 ), and are transcendental.

Repetition in continuous systems A standard approach to partial differential equations (PDEs) used for more than a century is so-called linear stability analysis, in which one assumes that small fluctuations around some kind of basic solution can be treated as a superposition of waves of the form Exp[  k x] Exp[  ω t] . And at least in a linear approximation any given PDE then typically implies that ω is connected to the wavenumber k by a so-called dispersion relation, which often has a simple algebraic form.

The default form of evaluation for recursive functions implemented by all standard computer languages (including Mathematica) is the so-called leftmost innermost scheme, which attempts to find explicit values for each f[k] that occurs first, and will therefore never notice if f[k] in fact occurs only in the combination f[k] - f[k] .

always to have the same overall rounded form, essentially independent of their orientation with respect to the underlying grid. … And indeed the standard theory for gravity introduced in 1915 is precisely based on the notion that gravity can be viewed merely as a feature of space.

For even in the planar networks discussed on page 527 a great many different arrangements of connections can be viewed as being formed from different configurations of nearby pairs of non-planar persistent structures. … But the standard formalism of quantum theory says that this is not correct, and that in fact one has to look at so-called probability amplitudes, not ordinary probabilities.

I have tried in the index to give all names in the form they might be used on standardized documents in the modern U.S. I have done standard transliterations from non-Latin character sets. … I normally give formal versions of forenames—though for individuals I have personally known I give in the text the form of forenames I would normally use in addressing them.

Standard mitotic cell division normally produces identical copies of DNA—with random errors potentially leading for example to cancers. … But when meiosis forms egg and sperm cells they get only one version of each. … In the immune system blocks of DNA—and joins between them—are selected at random by microscopic chemical processes when antibodies are formed.

At first most such systems will probably tend either to be based on standard engineering, or to be quite direct emulations of human components that we see. … But at a microscopic scale something like the surface of a solid has in at least some form remarkably detailed information about its history.

The idea that even a vacuum without particles will have a complicated and in some ways random form also exists in standard quantum field theory in traditional physics.

The picture below shows a standard method of encrypting messages represented by sequences of black and white squares. … The idea is just to form an encrypting sequence by repeatedly cycling through the elements in the key.