But in practice the most accurate measurements show phenomena such as 1/f noise, presumably as a result of features of the detector and perhaps of electromagnetic fields associated with decay products.

The inequivalent commutative monoids with up to k = 4 colors are (in total there are 1, 2, 5, 19, 78, 421, 2637, … such objects): For k = 2 , r = 1 the number of rules additive with respect to these is respectively: {8, 9} ; for k = 2 , r = 2 : {32, 33} ; for k = 3 , r = 1 : {28, 27, 35, 244, 28} ; for k = 4 , r = 1 : {1001, 65, 540, 577, 126, 4225, 540, 9065, 757, 408, 65, 133, 862, 224, 72, 72, 91, 4096, 64} It turns out to be possible to show that any rules ϕ additive with respect to some addition operation ⊕ must work by applying that operation to values associated with cells in their neighborhood. … 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)).

So far in my life the primary computer hardware systems I have used have been: Elliott 903 (1973-6); IBM 370 (1976-8); CDC 7600 (1978-9); VAX 11/780 (1980-2); Sun-1, 2, Ridge 32 (1982-4); CM-1 (1985); Sun-3 (1985-8); SPARC (1988-91); NeXT (1991-4); HP 700 (1995-6); PC (1996- ).

Huffman coding with a large number of codewords will approach this if all the p[i] are powers of 1/2. … For p[i] that are not powers of 1/2, non-integer length codewords would be required.

The longest of these are respectively {57, 94, 42, 57, 55, 53, 179, 157} and occur for theorems {(((a ⊼ a) ⊼ b) ⊼ b)  (((a ⊼ b) ⊼ a) ⊼ a), (a ⊼ (a ⊼ (a ⊼ a)))  (a ⊼ ((a ⊼ b) ⊼ b)), (((a ⊼ a) ⊼ a) ⊼ a)  (((a ⊼ a) ⊼ b) ⊼ a), (((a ⊼ a) ⊼ b) ⊼ b)  (((a ⊼ b) ⊼ a) ⊼ a), (a ⊼ ((b ⊼ b) ⊼ a))  (b ⊼ ((a ⊼ a) ⊼ b)), ((a ⊼ a) ⊼ a)  ((b ⊼ b) ⊼ b), ((a ⊼ a) ⊼ a)  ((b ⊼ b) ⊼ b), ((a ⊼ a) ⊼ a)  ((b ⊼ b) ⊼ b)} Note that for systems that do not already have it as an axiom, most theorems use the lemma (a ⊼ b)  (b ⊼ a) which takes respectively {6, 1, 8, 49, 8, 1, 119, 118} steps to prove.

Note that for x of the form p π /q , the k = ∞ sum is just ( π /q/(2q)) 2 Sum[Sin[n 2 p π /q]/Sin[n π /(2q)] 2 , {n, q - 1}] The pictures below show Sum[Cos[2 n x], {n, k}] (as studied by Karl Weierstrass in 1872). … They can be thought of as having dimensions 2 - a and smoothed power spectra ω -(1 + 2a) .

Network constraint systems Cases (a), (f) and (p) allow all networks that do not contain respectively cycles of length 1 (self-loops), cycles of length 3 or less, and cycles of length 5 or less. … For templates involving nodes out to distance one, there are 13 minimal sets in the sense of page 941 , of which only 6 contain just one template, 6 contain two and 1 contains three.

2D representations [of substitution systems] Individual sequences from 1D substitution systems can be displayed in 2D by breaking them into a succession of rows.

(A way to do this for pairs of non-negative integers is to use σ [{x_, y_}] := 1/2(x + y)(x + y + 1) + x .) … Any real number x can be represented as a set of integers using for example Rest[FoldList[Plus, 1, ContinuedFraction[x]]] but except when x is rational this list is not finite.

General associative [cellular automaton] rules With a cellular automaton rule in which the new color of a cell is given by f[a 1 , a 2 ] (compare page 886 ) it turns out that the pattern generated by evolution from a single non-white cell is always nested if the function f has the property of being associative or Flat . … The result can also be generalized to cellular automata with basic rules involving more than two elements—since if f is Flat , f[a 1 , a 2 , a 3 ] is always just f[f[a 1 , a 2 ], a 3 ] .