History of universality In Greek times it was noted as a philosophical matter that any single human language can be used to describe the same basic range of facts and processes. … The idea of having an abstract procedure that can be fed a range of different inputs had precursors in antiquity in the use of letters to denote objects in geometrical constructions, and in the 1500s in the introduction of symbolic formulas and algebraic variables.

But one might imagine that across the much broader range of computational systems that I have considered in this book—and that presumably occur in nature—there could nevertheless still be great differences in the rates at which given computations can be done.

But particularly when the original PDE is nonlinear one often finds that Im[ ω ] < 0 for some range of k —implying an instability which causes modes with certain spatial wavelengths to grow.

As emphasized by Benoit Mandelbrot in the 1970s and 1980s, topography and contour lines (notably coastlines) seem to show apparently random structure on a wide range of scales—with definite power laws being measured in quite a few cases.

All such methods yield signals that remain roughly in the range of frequencies { ω - δ , ω + δ } where δ is the data rate in s[t] . … And when CDMA methods are used there can be spreading over a significant range of frequencies—with regularities being recognizable only if one knows or can cryptanalyze LFSR sequences.

Defining PM[s_] := IntegerDigits[Range[2 s - 1], 2, s] blocks of data of length m can be encoded with Join[data, Mod[data .

Starting with an ordinary base 2 digit sequence, one prepends a unary specification of its length, then a specification of that length specification, and so on: (Flatten[{Sign[-Range[1 - Length[#], 0]], #}] &)[ Map[Rest, IntegerDigits[Rest[Reverse[NestWhileList[ Floor[Log[2, #] &, n + 1, # > 1 &]]],2]]] (d) Binary-coded base 3.

The examples shown in detail in the main text all have the feature that the block size b and number of steps t are matched, so that r t = b (where the range r = 1 for elementary rules).

In coal and petroleum a continuous range of alkanes occur.

In this book such offset lists are always taken to be in the order given by Sort , so that for range r rules in d dimensions the order is the same as Flatten[Array[List, Table[2r + 1, {d}], -r], d - 1] .