The rise of communications technology in the early 1900s led to work on quantitative theories of communication, and for example in 1928 Ralph Hartley suggested that an objective measure of the information content of a message with n possible forms is Log[n] . … In 1948 Claude Shannon suggested using a measure of information based on p Log[p] , and there quickly developed the notion that this could be used to find the fundamental redundancy of any sequence of data, independent of its possible meaning (compare page 1071 ).

[History of] exact solutions Some notable cases where closed-form analytical results have been found in terms of standard mathematical functions include: quadratic equations (~2000 BC) ( Sqrt ); cubic, quartic equations (1530s) ( x 1/n ); 2-body problem (1687) ( Cos ); catenary (1690) ( Cosh ); brachistochrone (1696) ( Sin ); spinning top (1849; 1888; 1888) ( JacobiSN ; WeierstrassP ; hyperelliptic functions); quintic equations (1858) ( EllipticTheta ); half-plane diffraction (1896) ( FresnelC ); Mie scattering (1908) ( BesselJ , BesselY , LegendreP ); Einstein equations (Schwarzschild (1916), Reissner–Nordström (1916), Kerr (1963) solutions) (rational and trigonometric functions); quantum hydrogen atom and harmonic oscillator (1927) ( LaguerreL , HermiteH ); 2D Ising model (1944) ( Sinh , EllipticK ); various Feynman diagrams (1960s-1980s) ( PolyLog ); KdV equation (1967) ( Sech etc.); Toda lattice (1967) ( Sech ); six-vertex spin model (1967) ( Sinh integrals); Calogero–Moser model (1971) ( Hypergeometric1F1 ); Yang–Mills instantons (1975) (rational functions); hard-hexagon spin model (1979) ( EllipticTheta ); additive cellular automata (1984) ( MultiplicativeOrder ); Seiberg–Witten supersymmetric theory (1994) ( Hypergeometric2F1 ).

And given t elements operating in parallel one can consider the class NC studied by Nicholas Pippenger in 1978 of computations that can be done in a number of steps that is at most some power of Log[t] .

It is often convenient to fit s n for large n to the form 2 h n , where h is the so-called spatial (topological) entropy (see page 1084 ), given by Log[2, κ ] .

Large geometrical patterns of logging were for example briefly visible after snow in 1961 near Cochrane, Canada—as captured by an early weather satellite.

The total lengths of these chains (corresponding to the depth of the evaluation tree) seem to increase roughly like Log[n] for all the rules on this page.

Starting in the late 1940s the development of information theory began to suggest connections between randomness and inability to compress data, but emphasis on p Log[p] measures of information content (see page 1071 ) reinforced the idea that block frequencies are the only real criterion for randomness.

The smallest examples that show other behavior are: • r[z, r[s, s]] , which is 1/2#(# + 1)& , with quadratic growth • r[z, r[s, c[s, s]]] , which is 2 # + 1 - # - 2 & , with exponential growth • r[z, r[s, p[2]]] , which is 2^Ceiling[Log[2, # + 2]] - # - 2 & , which shows very simple nesting • r[z, r[c[s, z], z]] , which is Mod[#, 2]& , with repetitive behavior • r[z, r[s, r[s, s]]] which is Fold[1/2#1(# + 1) + #2 &, 0, Range[#]]& , growing like 2 2 x . r[z, r[s, r[s, r[s, p[2]]]]] is the first function to show significantly more complex behavior, and indeed as the picture below indicates, it already shows remarkable randomness. From its definition, the function can be written as Fold[Fold[2^Ceiling[Log[2, Ceiling[(#1 + 2)/(#2 + 2)]]] (#2 + 2) - 2 - #1 &, #2, Range[#1]] &, 0, Range[#]]& Its first zeros are at {4, 126, 813, 966, 1166, 1177, 1666, 1897} .

For m > 1 , the value of n for which m  Fibonacci[n] is Round[Log[GoldenRatio, √ 5 m]] .

The resulting curve has a nested form, with envelope n^Log[3, 2] .