Differential geometry Standard descriptions of properties like curvature—as used for example in general relativity—are normally based on differential geometry. … But in an arbitrary space things can be more complicated, and in general such a path will be a geodesic (see note below ) which can have a more complicated form. … (In Mathematica, the explicit form of such a tensor can be represented as a nested list for which TensorRank[list]  4 .)

Sometimes nonzero size is taken into account by inserting additional interaction parameters—as done in the 1950s with magnetic moments and form factors of protons and neutrons. … And indeed—apart from few rare suggestions to the contrary—the same is now assumed throughout mainstream practical particle physics for all of the basic particles that appear in the Standard Model.

But in physics it was still assumed that space itself must have a standard fixed Euclidean form—and that everything in the universe must just exist in this space.

If m is of the form 2 j , this implies a maximum period for any a of m/4 , achieved when MemberQ[{3, 5}, Mod[a, 8]] . … Each point in the 2D plots in the main text has coordinates of the form {n[i], n[i + 1]} where n[i + 1] = Mod[a n[i], m] . … Starting in the 1980s, the most common example has been the Data Encryption Standard (DES) introduced by the U.S. government (see page 1085 ).

Instead, what has normally been done is to take the array of spins to be in thermal equilibrium with a heat bath, so that, following standard statistical mechanics, each possible spin configuration occurs with probability Exp[- β e[s]] , where β is inverse temperature. … And what one sees at least roughly is that right around the phase transition there are patches of black and white of all sizes, forming an approximately nested random pattern.