The so-called elementary functions (logarithms, exponentials, trigonometric and hyperbolic functions, and their inverses) were mostly introduced before about 1700. … Most arose first as solutions to specific differential equations, typically in physics and astronomy; some arose as products, sums of series or inverses of other functions. In the mid-1800s it became clear that despite their different origins most of these functions could be viewed as special cases of Hypergeometric2F1[a, b, c, z] , and that the functions covered the solutions to all linear differential equations of a certain type. ( Zeta and PolyLog are parametric derivatives of Hypergeometric2F1 ; elliptic modular functions are inverses.)

Encodings [of cellular automaton rules] Generalizing the setup in the main text one can say that a cellular automaton i can emulate j if there is some encoding function ϕ that encodes the initial conditions a j for j as initial conditions for i , and which has an inverse that decodes the outcome for i to give the outcome for j . With evolution functions f i and f j the requirement for the emulation to work is f j [a j ]  InverseFunction[ ϕ ][f i [ ϕ [a j ]]] In the main text the encoding function is taken to have the form Flatten[a /. rules] —where rules are say {1  {1, 1}, 0  {0, 0}} —with the result that the decoding function for emulations that work is Partition[ ã , b] /. … Various questions about encoding functions ϕ have been studied over the past several decades in coding theory.

And if one assumes that this is a general feature then one can formally derive for any a the result 1/2 (1 - g[a t InverseFunction[g] [1 - 2x]]) where g is a function that satisfies the functional equation g[a x]  1 + (a/2) (g[x] 2 - 1) When a = 4 , g[x] is Cosh[Sqrt[2 x]] . … But in general for arbitrary a there is no standard mathematical function that seems to satisfy the functional equation. … In addition, from a known replication formula for an elliptic or other function one can often construct an iterated map whose behavior can be expressed in terms of that function.

(The function s[d] has a maximum around d = 5.26 , then decreases rapidly with d .) … To next order the result is s[d] r d (1 - RicciScalar r 2 /(6(d + 2)) + (5 RicciScalar 2 - 3 RiemannNorm + 8 RicciNorm - 18 Laplacian[RicciScalar])r 4 /(360 (d + 2)(d + 4)) + …) where the new quantities involved are RicciNorm = Norm[RicciTensor, {g, g}] RiemannNorm = Norm[Riemann, {g, g, g, Inverse[g]}] Norm[t_, gl_] := Tr[Flatten[t Dual[t, gl]]] Dual[t_, gl_]:= Fold[Transpose[#1 . Inverse[#2], RotateLeft[ Range[TensorRank[t]]]] &, t, Reverse[gl]] Laplacian[f_] := Inner[D, Sqrt[Det[g]] (Inverse[g] .

But in general one can apply to each cell value any function σ that obeys the so-called Cauchy functional equation σ [x+y]  σ [x] + σ [y] . … If ⊕ has an inverse, so that it defines a group, then the only continuous (Lie group) examples turn out to be combinations of ordinary addition and modular addition (the group U(1)). … But one can also imagine setting up systems whose states are continuous functions of position. ϕ then defines a mapping from one such function to another.

Within say a surface whose points {x 1 , x 2 , … } are obtained by evaluating an expression e as a function of parameters p (so that for example e = {x, y, f[x, y]} , p = {x, y} for a Plot3D surface) the metric turns out to be given by (Transpose[#] . # &) [Outer[D, e, p]] In ordinary Euclidean space a defining feature of geometry is that the shortest path between two points is a straight line. … Γ 〚 i 〛 , {i, d}, {j, d}, {k, d}] where the so-called Christoffel symbol Γ ij k is Γ = With[{gi = Inverse[g]}, Table[Sum[ gi 〚 l, k 〛 ( ∂ p 〚 j 〛 g 〚 i, l 〛 + ∂ p 〚 i 〛 g 〚 j, l 〛 - ∂ p 〚 l 〛 g 〚 j, l 〛 ), {l, d}], {i, d}, {j, d}, {k, d}]]/2 There are d 4 elements in the nested lists for Riemann , but symmetries and the so-called Bianchi identity reduce the number of independent components to 1/12 d 2 (d 2 - 1) —or 20 for d = 4 . … Inverse[g]]

Following the work of Lars Onsager around 1944, it turns out that an exact solution in terms of traditional mathematical functions can be found in this case. … (Another analytical result is that for e~e 0 correlations between pairs of spins can be expressed in terms of Painlevé functions.) … 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.

(This is analogous to what happens for example in classical diffraction theory, where there is an analog of the path integral—with ℏ replaced by inverse frequency—whose stationary points correspond through the so-called eikonal approximation to rays in geometrical optics.) … Discretizing yields lattice gauge theories with energy functions involving for example Cos[ θ i - θ j ] for color directions at adjacent sites.