[Boolean] formula sizes There are a total of 2 2 n possible Boolean functions of n variables. The maximum number of terms needed to represent any of these functions in DNF is 2 n - 1 . The actual numbers of functions which require 0, 1, 2, ... terms is for n = 2 : {1, 9, 6} ; for n = 3 : {1, 27, 130, 88, 10} , and for n = 4 : {1, 81, 1804, 13472, 28904, 17032, 3704, 512, 26} .

But the number of systems that have traditionally been studied in mathematics and that are known to require only one 2-argument operator are fairly limited. … Pages 773 and 774 indicate that most axiom systems in mathematics involve operators with at most 2 arguments (there are exceptions in geometry). (Constants such as 1 or ∅ can be viewed as 0-argument operators.)

For s = 2 , k = 2 the largest results for all inputs of sizes 0 to 4 are {7, 17, 31, 49, 71} , all obtained with machine 1447. For n > 4 the largest results are 2 n + 2 - 3 , achieved for x = 2 n - 1 with machines 378 and 1351. For s = 3 , k = 2 the largest results for successive sizes are {25, 53, 159, 179, 1021, 5419} (often achieved by machine 600720; see below ) and for s = 2 , k = 3 {35, 83, 843, 8335} (often achieved by machine 840971).

The left edge of the pattern moves 1 cell every 2 steps; the boundary between repetition and randomness moves on average 0.17 cells per step.

The representation is not unique; a[n] = 2 n , n (n + 1) and (n + 1)!… 2 : BesselI[0, 2] - 1 ; n 2 n : Log[2] ; n 2 : π 2 /6 ; (3n - 1)(3n - 2): π √ 3 /9 ; 3 - 16n + 16n 2 : π /8 ; n n!

Implementation of general cellular automata With k colors and r neighbors on each side, a single step in the evolution of a general cellular automaton is given by CAStep[CARule[rule_List, k_, r_], a_List] := rule 〚 -1 - ListConvolve[k^Range[0, 2r], a, r + 1] 〛 where rule is obtained from a rule number num by IntegerDigits[num, k, k 2r + 1 ] .

The fact that √ 2 is not a rational number was discovered by the Pythagoreans. … It is known that Exp[n] and Log[n] for whole numbers n (except 0 and 1 respectively) are transcendental. It is also known for example that Gamma[1/3] and BesselJ[0, n] are transcendental.

(For large n this is approximately λ n with λ = 3/4 ; if 1- and 2-edged regions are allowed then λ = (3 + √ 3 )/6 ≃ 0.79 .) There may be some easy way to derive such results, but so far it has only been done using fairly sophisticated techniques from quantum field theory developed in the late 1970s. … Note that the networks obtained always have dimension 2 according to my definitions.

The total spin is always a fixed multiple of the basic unit ℏ : 1/2 for quarks and leptons, 1 for photons and other ordinary gauge bosons, 2 for gravitons, and in theory 0 for Higgs particles. … For small transformations, Spin( d ) is just the ordinary rotation group SO( d ), but globally it is its universal cover, or SU(2) in 3D. … Such objects have the property that they are not left invariant by 360° rotations, but only by 720° ones—a feature potentially fairly easy to reproduce with networks, perhaps even without definite integer dimensions.

3n+1 problem as cellular automaton If one writes the digits of n in base 6, then the rule for updating the digit sequence is a cellular automaton with 7 possible colors (color 6 works as an end marker that appears to the left and right of the actual digit sequence): {a_, b_, c_}  If[b  6, If[EvenQ[a], 6, 4], 3 Mod[a, 2] + Quotient[b, 2] /. 0  6 /; a  6] The 3n+1 problem can then be viewed as a question about the existence of persistent structure in this cellular automaton.