Such uniformity in frequency is implied by standard quantum theory to exist in the idealized zero-point fluctuations of a free quantum field—with direct consequences for such semiclassical phenomena as the Casimir effect and Hawking radiation.

The basic idea of an operator system is to work with expressions such as (p ∘ q) ∘ ((q ∘ r) ∘ p) built up using some operator ∘ , and then to consider for example what equivalences may exist between such expressions. … And in principle one can use this method to establish any equivalence that exists between any expressions with an operator of any specific form.

Localized structures [in Turing machines] Even when the overall behavior of a Turing machine is complicated, it is possible for simple localized structures to exist, much as in cellular automata such as rule 110.

This suggests that to determine whether a repetitive pattern with repeating blocks of size n exists may in general take a number of steps which grows more rapidly than any polynomial in n .

For n < 9 there are always some sequences in which no runs of length 3 exist. … If one has a sufficiently long sequence, therefore, just knowing that a run of equally spaced identical elements exists in it does not narrow down at all what the sequence actually is, and can so cannot ultimately be considered a useful regularity.

In the context of electromagnetism alone, there would be no particular reason to expect this (unless magnetic monopoles exist).

By setting up the precise initial conditions that exist at the beginning of the whole picture it would certainly in principle be possible to get such behavior.

I suspect that universal examples with blocks of the same size exist with just 3 colors.

Systematic methods now exist to find conserved quantities that are given by integrals of polynomials of any given degree in the dependent variables and their derivatives.

In the pictures above we have seen various examples of individual networks that might exist at a particular step in the evolution of a network system.