Continuous generalizations [of additive rules] Functions such as Binomial[t, n] and GegenbauerC[n, -t, -1/2] can immediately be evaluated for continuous t and n . The pictures on the right below show Sin[1/2 π a[t, n]] 2 for these functions (equivalent to Mod[a[t, n], 2] for integer a[t, n] ).

The formulas for local curvature as a function of arc length for each set of pictures are as follows: 1 (circle); s (Cornu spiral or clothoid); s 2 ; 1/Sqrt[s] (involute of circle); 1/s (logarithmic or equiangular spiral); 1/s 2 ; Exp[-s 2 ] ; Sin[s] ; s Sin[s] . The curvature functions f[s] can be thought of as specifying how much to turn a vehicle at every moment in order to keep it driving along the curve.

The whole procedure can be represented using a mathematical formula that involves either functions like Mod or more traditional functions like Sin .

Image averaging Walsh functions yield significantly better compression than simple successive averaging of 2×2 blocks of cells, as shown below.

Cellular automaton [Nand] formulas For 1 step, the elementary cellular automaton rules are exactly the 256 n = 3 Boolean functions. For 2 steps, they represent a small subset of the 2 32 n = 5 functions.

In fact, a fair fraction of all possible transformations based on algebraic functions will yield nested patterns. For typically the continuity of such functions implies that only a limited number of shapes not related by limited variations in local magnification can occur at any scale.

As discussed on page 1121 Schönfinkel introduced certain specific rules that he suggested could be used to build up functions defined in logic. … Most likely the reason is that building up functions on the basis of the structure of symbolic expressions has never seemed to have much obvious correspondence to the traditional mathematical view of functions as mappings. … Most likely the reason is that ever since the work of Bertrand Russell in the early 1900s it has generally been assumed that it is desirable to distinguish a hierarchy of different types of functions and objects—analogous to the different types of data supported in most programming languages.

The definition of randomness that we discussed in the previous section was based on the failure of the second of these two functions. … But in defining complexity we need to consider both functions of perception and analysis.

Higher-order logics In ordinary predicate—or so-called first-order—logic the objects x that ∀ x and ∃ x range over are variables of the kind used as arguments to functions (or predicates) such as f[x] . To set up second-order logic, however, one imagines also being able to use ∀ f and ∃ f where f is a function (say the head of f[x] ). … (Note however that this is possible in Henkin versions of higher-order logic that allow only limited function domains.)

[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 ).